ESTIMATING DEMAND FOR DIFFERENTIATED PRODUCTS
WITH CONTINUOUS CHOICE AND VARIETY-SEEKING:
AN APPLICATION TO THE PUZZLE OF UNIFORM PRICING
a dissertation
submitted to the department of economics
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
Robert Stanton McMillan
March 2005
c Copyright by Robert Stanton McMillan 2005
All Rights Reserved
ii
I certify that I have read this dissertation and that, in my
opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Peter Reiss
(Principal Adviser)
I certify that I have read this dissertation and that, in my
opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Frank Wolak
I certify that I have read this dissertation and that, in my
opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Liran Einav
Approved for the University Committee on Graduate Studies.
iii
Abstract
Retailers typically sell many different products from the same manufacturer at the same
price. I consider retailer-based explanations for this uniform pricing puzzle, estimating the
counterfactual profits that would be lost by a retailer switching from a non-uniform to a
uniform pricing regime in the carbonated soft drink category. In order to calculate this
profit difference, I develop a new structural model of demand that improves on existing
work by more closely matching several key features of the data. These key features include
the fact that households may be variety-seeking, that they make continuous choices, and
that they choose from a large number of products. Using household-level panel data on
purchases of carbonated soft drinks, I estimate that the retail store I observe earned an
additional $36.56 (1992 dollars) in average weekly profits by charging non-uniform prices.
This corresponds to roughly a 3% difference in profits, and suggests that when a retail store
faces even a relatively small cost to determine the optimal set of non-uniform prices, it may
be optimal to charge the same price for many products.
iv
Acknowledgements
I thank my advisers Peter Reiss and Frank Wolak for their guidance and advice. I also
thank Pat Bajari, Michaela Draganska, Liran Einav, Cristobal Huneeus, Navin Kartik, Tom
MaCurdy, Mikko Packalen, James Pearce, and many others at the Stanford Economics
department for helpful discussions. Finally, I thank my parents, my brother, and Helen
Chabot. Without their support and encouragement, this would never have come to fruition.
v
Contents
Abstract
iv
Acknowledgements
v
1 Introduction
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Demand-side Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
An Overview of Menu Costs . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.4
Estimating Menu Costs with Soft Drink Data . . . . . . . . . . . . . . . . .
13
1.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2 A Model of Continuous Demand and Variety-Seeking
25
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2
Previous Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3
The Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.4
Behavior of the Model When There Are Two Inside Goods . . . . . . . . .
34
2.4.1
Demand Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.4.2
Engel Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.4.3
The A Matrix and Some Additional Remarks . . . . . . . . . . . . .
56
2.5
Choice Behavior Generated by the Model . . . . . . . . . . . . . . . . . . .
58
2.6
The Costs of Misspecification . . . . . . . . . . . . . . . . . . . . . . . . . .
68
2.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
2.8
Appendix A: Restrictions Yielding Concavity . . . . . . . . . . . . . . . . .
80
2.8.1
Two Goods, Two Characteristics . . . . . . . . . . . . . . . . . . . .
80
2.8.2
Two Goods, Three Characteristics . . . . . . . . . . . . . . . . . . .
81
vi
2.8.3
2.9
Three Goods, Three Characteristics . . . . . . . . . . . . . . . . . .
82
Appendix B: Analytic Solutions to the Two-Inside Good Case: Details . . .
83
2.10 Appendix C: Additional Figures
. . . . . . . . . . . . . . . . . . . . . . . .
3 Investigating the Costs of Uniform Pricing
87
94
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
3.1.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
3.1.2
Data: Soft Drinks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
3.1.3
Estimating Demand . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
3.1.4
Counter-Factuals and Preview of Main Result . . . . . . . . . . . . .
98
3.1.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
Empirical Demand Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
3.2.1
Product-Level Demand Model . . . . . . . . . . . . . . . . . . . . . .
101
3.2.2
Store Choice Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
3.2.3
Residual Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
108
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
3.3.1
IRI Basket Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
3.3.2
Dominick’s Finer Foods Data . . . . . . . . . . . . . . . . . . . . . .
117
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
3.4.1
Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
3.4.2
Store Choice Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
3.4.3
Counter-Factuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
Interpreting “Lost” Profits . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135
3.5.1
Menu Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135
3.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
3.7
Appendix 3.A: Numerically Solving The Utility Function . . . . . . . . . . .
139
3.8
Appendix 3.B: Modeling Heterogeneity of Preferences . . . . . . . . . . . .
140
3.9
Appendix 3.C: Analysis of the Panel Composition . . . . . . . . . . . . . . .
141
3.1
3.2
3.3
3.4
3.5
References
146
vii
List of Tables
1.1
Distribution of Purchase Occasions by Number of Items and Number of UPCs
Purchased (all Carbonated Soft Drinks) . . . . . . . . . . . . . . . . . . . .
1.2
22
Distribution of Purchase Occasions by Number of Items and Number of UPCs
Purchased (Among Top 25 Carbonated Soft Drinks) . . . . . . . . . . . . .
23
2.1
Pairs of (ε1 , ε2 ) corresponding to the demand curves in Figures 2.1-2.7 . . .
36
2.2
Pairs of (ε1 , ε2 ) corresponding to the demand curves in Figures 2.1-2.30 . .
49
2.3
Marginal Cost Configurations . . . . . . . . . . . . . . . . . . . . . . . . . .
69
2.4
Example Product Universe Number One . . . . . . . . . . . . . . . . . . . .
70
2.5
Example Product Universe Number Two . . . . . . . . . . . . . . . . . . . .
75
2.6
Summary of Dominance Conditions . . . . . . . . . . . . . . . . . . . . . . .
87
3.1
Distribution of Purchase Occasion Expenditure by Store, All Purchase Occasions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
111
Distribution of Purchase Occasion Expenditure by Store, Purchases Made by
Panelists Who Visited Store A at least Once. . . . . . . . . . . . . . . . . .
112
3.3
Variety and Size Distribution of in the Dataset, grouped by Manufacturer .
113
3.4
Summary Statistics for Prices and Quantities Sold at Store A, grouped by
Manufacturer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
3.5
Characteristics of Products in the Dataset, grouped by Manufacturer . . . .
116
3.6
Parameter Estimates from Structural Model of Product Choice . . . . . . .
119
3.7
Selected Own and Cross Price Elasticities from Homogenous Logit Model of
Product Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8
119
Matrix of Estimated Average Own and Cross-Price Elasticities from Structural Model of Product Choice . . . . . . . . . . . . . . . . . . . . . . . . .
viii
120
3.9
Coefficients on Demographic Variables from Specifications III and VI of the
Conditional Logit Model of Store Choice . . . . . . . . . . . . . . . . . . . .
125
3.10 Coefficients on Price Index Variables for Specifications I-VI of Conditional
Logit Model of Store Choice. Standard errors are in brackets. . . . . . . . .
126
3.11 Coefficients on Price Indices for Specifications VII and VIII of Conditional
Logit Model of Store Choice. Standard errors are in brackets. . . . . . . . .
127
3.12 Summary Statistics for Marginal Costs (in Dollars per 12oz Serving) Implied
by the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
3.13 Summary Statistics on the Differences Between Observed Non-Uniform Prices
and “Optimal” Uniform Prices (in Dollars per 12oz Serving) . . . . . . . . .
132
3.14 Uniform and Non-Uniform Prices, Quantities and Estimated Profits for the
Week of July 7, 1991. Prices and marginal costs reported are in cents per
12oz Serving. Quantity is measured in 12oz servings. Profits are measured
in dollars. All prices and profits are in nominal terms. The total difference
in profits for the week from the two pricing strategies is $61.52. The 3L size
of Pepsi was not offered in this week. . . . . . . . . . . . . . . . . . . . . . .
134
3.15 Size and Composition of Households in Panel . . . . . . . . . . . . . . . . .
142
3.16 Age Distribution of Primary Male and Female in Households in Panel . . .
143
3.17 Income Distribution of Panel Households . . . . . . . . . . . . . . . . . . . .
143
3.18 Summary Statistics for Population Living Near Stores in the Panel
144
ix
. . . .
List of Figures
1.1
Graph of Prices for Coke and Pepsi at Store B, 6/91-6/93 . . . . . . . . . .
16
1.2
Graph of Prices for Coke and Pepsi at Store A, 6/91-6/93 . . . . . . . . . .
17
1.3
Graph of Prices Ratios of Pepsi Varieties to Regular Pepsi at Store B, 6/91-6/93 18
1.4
Graph of Prices Ratios of Pepsi Varieties to Regular Pepsi at Store A, 6/91-6/93 19
2.1
Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
"
1
0.5
0.5
1
#
, β1 = 5.5, β2 = 5, ρ1 = ρ2 = 0.5, p2 =
$0.50, w = 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Demand for good one and two as a function of p1 , for 40th and 60th percentiles of ε1 and ε2 . A =
"
1 0
#
, β1 = 5.5, β2 = 5, ρ1 = ρ2 = 0.5, p2 =
0 1
$0.50, w = 60. Compare to Figures 2.3 and 2.4. . . . . . . . . . . . . . . . .
2.3
"
#
1 0
0 1
, β1 = 6.875, β2 = 5, ρ1 = 0.4, ρ2 =
0.5, p2 = $0.50, w = 60. Compare to Figures 2.2 and 2.4. . . . . . . . . . .
"
1 0
#
0 1
, β1 = β2 = 5, ρ1 = ρ2 = 0.5, p2 =
$0.50, w = 60. Compare to Figures 2.2 and 2.3 . . . . . . . . . . . . . . . .
42
Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
"
1
0.5
0.5
1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.5, p2 =
$0.50, w = 60. Compare to Figure 2.1. . . . . . . . . . . . . . . . . . . . . .
2.6
41
Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
2.5
40
Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
2.4
37
44
Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
"
1
0.5
0.5
1
#
, β1 = 6.875, β2 = 5, ρ1 = 0.4, ρ2 =
0.5, p2 = $0.50, w = 60. Compare to Figures 2.1 and 2.5. . . . . . . . . . .
x
45
2.7
Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
"
1
0.999
0.999
1
#
, β1 = 5.5, β2 = 5, ρ1 = ρ2 = 0.5, p2 =
$0.50, w = 60. Compare to Figure 2.1. . . . . . . . . . . . . . . . . . . . . .
2.8
47
Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th percentiles of ε1 and ε2 . Each
"
symbol represents an increase of $0.20 in w. A =
1
0.5
0.5
1
#
, β1 = 5.5, β2 =
5, ρ1 = ρ2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9
50
Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th percentiles of ε1 and ε2 . Each
symbol represents an increase of $0.20 in w. A =
"
1
0.5
0.5
1
#
, β1 = β2 = 5, ρ1 =
ρ2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.10 Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th percentiles of ε1 and ε2 . Each
"
symbol represents an increase of $0.10 in w. A =
1 0
#
0 1
, β1 = 5.5, β2 =
5, ρ1 = ρ2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.11 Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th percentiles of ε1 and ε2 . Each
"
symbol represents an increase of $0.10 in w. A =
1 0
#
0 1
, β1 = β2 = 5, ρ1 =
ρ2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
2.12 Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th percentiles of ε1 and ε2 . Each
symbol represents an increase of $0.10 in w. A =
"
1 0
0 1
#
, β1 = 6.875, β2 =
5, ρ1 = 0.4, ρ2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.13 Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th percentiles of ε1 and ε2 . Each
symbol represents an increase of $0.40 in w. A =
"
1
0.999
0.999
1
#
, β1 = 5.5, β2 =
5, ρ1 = ρ2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2.14 Graph of logit choices as a function of error terms . . . . . . . . . . . . . .
59
2.15 Graph of purchases (broken into groups) as a function of error terms. A =
"
1 0
0 1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = $0.30, w = $30. The probabili-
ties of the four regions are:
"
0.21 0.09
#
0.49 0.21
. . . . . . . . . . . . . . . . . . . . . .
61
2.16 Negative Lognormal Probability Distribution of εj . Mean=e0.5 , Variance=e2 −
e, Mode=e−1 , Median=1. ln(−εj ) ∼ N (0, 1). . . . . . . . . . . . . . . . . .
xi
62
2.17 Graph of purchases (broken into groups) as a function of error terms. A =
"
1 0
0 1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 1. The probabilities
of the seven regions are:

0.09 0.13 0.09





0.14 0.13 
0.27
0.09

. . . . . . . . . . . . . . . . . . . . . . .
2.18 Graph of purchases as a function of error terms. A =
"
1 0
#
0 1
63
β1 = 7, β2 =
5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 30. The probabilities of the four regions
are:
"
0.24 0.05
0.57 0.13
#
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
2.19 Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
"
#
10
0
0
10
, β1 = β2 = 0.5, ρ1 = ρ2 = 0.4, p1 = p2 =
0.3, w = 30. The probabilities of the four regions are:
"
0.21 0.09
#
0.49 0.21
. . . . . . .
65
2.20 Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
"
1
0.1
0.1
1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 =
0.3, w = 30. The probabilities of the four regions are:
"
0.28 0.07
#
0.38 0.27
. . . . . . .
66
2.21 Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
"
1 0.5
0
1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 =
0.3, w = 30. The probabilities of the four regions are:
"
0.57 0.05
#
0.24 0.15
. . . . . . .
66
2.22 Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
"
1 1
0 1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w =
30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
2.23 Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were
grouped as: {1,3},{2,4}. The marginal
costs
corresponding
to the
horizontal


 


1 0

 0 1


 0 0

0 0
axis are shown in Table 2.3. A =
0 0

0 0 


1 0 

0 1
,β=
3
 
 5 
 
 
 7 
 
9
,ρ=
0.05


 0.08 




 0.07 


0.09
. . . . . . .
72
2.24 Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were
grouped as: {1,3},{2,4}. The marginal
costs
corresponding
to the
horizontal


 


1 0 0 0



1 0 0 



 0 0 1 0 
0 0 0 1
axis are shown in Table 2.3. A = 0
,β=
xii
9
 
 5 
 
 
 7 
 
9
,ρ=
0.05


 0.08 




 0.07 


0.09
. . . . . . .
73
2.25 Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were
grouped as: {1,3},{2,4}. The marginal
costs
corresponding
to the
horizontal

 



1 0

 0 1


 0 0

0 0
0 0

0 0 


1 0 

0 1
axis are shown in Table 2.3. A =
,β=
1
 
 5 
 
 
 7 
 
9
,ρ=
0.15


 0.08 




 0.07 


0.09
. . . . . . .
74
2.26 Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were
grouped as: {1,3},{2,4}. This corresponds to uniform pricing by “diet” characteristic. the The marginal
costs
corresponding
to the
horizontal axis are

 



1 1 0 1



1 0 0 



 1 0 1 1 
1 0 1 0
shown in Table 2.3. A = 1
,β=
5
 
 5 
 
 
 5 
 
5
,ρ=
0.05


 0.07 




 0.07 


0.09
. . . . . . . . . . . .
76
2.27 Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were
grouped as: {1,2},{3,4}. This corresponds to uniform pricing by “Coke” or
“Pepsi” characteristics. The marginal
costs
corresponding
to the
horizontal

 



1 1

 1 1


 1 0

1 0
0 1

0 0 


1 1 

1 0
,β=
axis are shown in Table 2.3. A =
5
 
 5 
 
 
 5 
 
5
,ρ=
0.05


 0.07 




 0.07 


0.09
. . . . . . .
77
2.28 Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were
grouped as: {1,2},{3,4}. This corresponds to uniform pricing by “Coke” or
“Pepsi” characteristics. The marginal
costs
corresponding
to the
horizontal






1 1

 1 1


 1 0

1 0
0 1

0 0 


1 1 

1 0
axis are shown in Table 2.3. A =
,β=
3.5


 5.5 




 3.5 


3.5
,ρ=
0.05


 0.07 




 0.07 


0.09
. . . . . .
78
2.29 Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
"
1
0.999
0.999
1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.5, p2 =
$0.50, w = 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
2.30 Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
"
1
0.999
0.999
1
#
, β1 = 6.875, β2 = 5, ρ1 = 0.4, ρ2 =
0.5, p2 = $0.50, w = 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
2.31 Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
"
1 0
0 1
#
, β1 = β2 = 5, ρ1 = 0.5, ρ2 = 0.4, p1 = p2 =
0.3, w = 30. The probabilities of the four regions are:
xiii
"
0.23 0.06
0.55 0.15
#
. . . . . . .
90
2.32 Graph of purchases as a function of error terms. The groupings are the same
"
as in Figure 2.15. A =
1
0.02
0.02
1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 =
0.3, w = 30. The probabilities of the four regions are:
"
0.22 0.08
#
0.47 0.23
. . . . . . .
90
2.33 Graph of purchases as a function of error terms. The groupings are the same
"
as in Figure 2.15. A =
1
0.25
0.25
1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 =
0.3, w = 30. The probabilities of the four regions are:
"
0.36 0.05
#
0.22 0.37
. . . . . . .
91
2.34 Graph of purchases as a function of error terms. The groupings are the same
"
as in Figure 2.15. A =
1
0.1
0.1
1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 =
0.3, w = 1. The probabilities of the seven regions are:

0.16 0.11 0.07




0.10 0.11 

0.23
0.16

. . . . . . .
91
2.35 Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
"
#
10
1
1
10
, β1 = β2 = 0.5, ρ1 = 0.5, ρ2 = 0.4, p1 = p2 =
0.3, w = 30. The probabilities of the four regions are:
"
0.28 0.07
#
0.38 0.28
. . . . . . .
92
2.36 Graph of purchases as a function of error terms. The groupings are the same
"
as in Figure 2.15. A =
1 0.08
0
1
#
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 =
0.3, w = 30. The probabilities of the four regions are:
"
0.27 0.08
#
0.45 0.20
. . . . . . .
92
2.37 Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
"
1 0.5
0
#
1
, β1 = 7, β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 =
0.3, w = 30. The probabilities of the four regions are:
"
0.65 0.03
#
0.22 0.10
. . . . . . .
93
2.38 Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
"
1 0.5
0
1
#
, β1 = β2 = 5, ρ1 = 0.5, ρ2 = 0.4, p1 = p2 =
0.3, w = 30. The probabilities of the four regions are:
3.1
"
0.66 0.04
0.19 0.11
#
. . . . . . .
Graph of the Price and Implied Marginal Cost (in cents per 12oz serving) for
a 2L Bottle of Regular Pepsi, 6/91-6/93 . . . . . . . . . . . . . . . . . . . .
3.2
130
Graph of the Maximum Difference Across Products (in cents per 12oz serving) Between a Product’s Uniform and Non-Uniform Prices, 6/91-6/93 . . .
3.4
130
Graph of the Average Markup (in cents per 12oz serving) Across Products,
6/91-6/93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
93
131
Graph of the Difference Between Profits from Uniform and Non-Uniform
Price Strategies, as a Percent of the Profits Earned at Non-Uniform Prices,
6/91-6/93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
136
3.5
Graph of the Counterfactual Dollars Lost from Charging Uniform Prices,
6/91-6/93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
136
Chapter 1
Introduction
1.1
Introduction
Retailers typically sell many different products from the same manufacturer at the same
price. For example, in the yogurt category, all flavors of six ounce Dannon Fruit-on-theBottom yogurt are sold at one price, while all flavors of six ounce Yoplait Original yogurt
are sold at a second (uniform) price. This practice is common in many product categories,
including frozen dinners, ice cream and salsa. In other product categories, however (e.g.,
frozen juice), items are typically sold at different prices within manufacturer brands. While
not true at all times, in all stores, and for all products, the extent of these uniform prices
across different retailers, product categories, and time is stunning. Why is it optimal for
the retailer to sell many different items at the same price? Consider the following anecdotal
examples:
Tea and Juice Although many teas and juices are sold at uniform prices, there are notable
exceptions. Frozen orange juice is almost always priced differently from other frozen
juice. Within the premium juice category (e.g., brands such as Odwalla and Naked
Juice), prices are frequently completely non-uniform. Similarly, although most teas
are sold at uniform prices, some varieties of tea are frequently sold at a higher price.
These non-uniform prices seem to correlate with marked differences in marginal costs.
Although most tea leaves cost roughly the same, some cost more to produce. Similarly,
differences in cost and juiciness across different fruits can lead to different marginal
costs for the same volume of liquid. Furthermore, both tea and juice are products
1
CHAPTER 1. INTRODUCTION
2
that are difficult for manufacturers to adjust the amount of input. If manufacturers
adjust the amount of real juice or tea leaves, consumers are likely to notice.
Wine Different varieties of wine from the same vineyard and vintage are typically sold at
the same price when they are sold for less than $15 a bottle. For example, the muchreviewed Charles Shaw wines are $1.99, regardless of variety (e.g., Merlot, Cabernet
Sauvignon, Shiraz, Gamay Beaujolais, Chardonnay, or Sauvignon Blanc). However,
for more expensive wine, different varieties are generally priced non-uniformly. As
prices increase, we see more and larger deviations from uniform prices.
Clothing Within a particular style, clothing is typically sold at the same price for different
colors and sizes. There are however, exceptions to this rule: while S,M,L and XL sizes
are typically the same price, many retailers charge more for XXXL and “tall” sizes.
These sizes generally cost the retailer more, either because of the amount of fabric
used, or because average costs are higher due to lower volumes. Also, although men’s
shirts are usually priced uniformly across colors, striped shirts are frequently priced
differently. Retailers frequently claim that this is because striped shirts are a different
style than solid colored shirts, with different demand. Finally, upscale clothing stores
are more likely to charge different prices for different colors.
Books Books, even harlequin romance novels, are not sold at uniform prices. Even different books by a single author generally have different prices. At first this seems
puzzling. But unlike many products, books are frequently sold with a suggested retail price stamped on their cover or dust jacket. Furthermore, this price is the same
everywhere. While many retailers offer lower prices (e.g., discounts for New York
Times Bestsellers), these are nearly always offered as a percentage difference from
this suggested price. If one is willing to take as given the constraint that most books
are sold at a single price (or at most 2-3 prices) nationally, it becomes clear that the
demand for different books is almost certainly quite different.
While many products exhibit uniform prices, there seem to be clear patterns characterizing products that are not priced uniformly. Products with ostensibly different marginal
costs, such as different flavors of tea, varieties of frozen juice, and odd sizes of clothing are
frequently sold at different prices. Other products, with demand that clearly differs across
varieties such as different colors of designer clothes, colors of cars, expensive varieties of
CHAPTER 1. INTRODUCTION
3
wine also tend to be priced differently.
A suggestive pattern emerges: unless there are clearly additional profits from nonuniform pricing, there is a strong tendency towards uniform prices. If marginal costs are
sufficiently different, we tend to see non-uniform prices. If demand for the products is
sufficiently different, we tend to see non-uniform prices. If prices are sufficiently high, we
tend to see non-uniform prices. This suggests that managerial menu costs on the part of
the retailer may be able to explain the observed uniform pricing behavior. In determining
what price to charge, the retailer incurs a cost, most obviously the opportunity cost of a
price-setting executive’s time. Given this cost, it may be optimal for the retailer to group
products with similar costs and demands and sell them at a single price.
This dissertation is devoted to the careful examination of this managerial menu cost
explanation. In order to roughly test this hypothesis, by measuring the menu costs that
would be required to rationalize this behavior, I carefully estimate a structural model of
the residual demand curve faced by the retailer. Using this demand model, I calculate the
counter-factual difference between the profit (defined as revenue minus cost of goods sold)
a retailer earns by charging uniform and non-uniform prices. Only a structural model will
allow me to predict demand – and hence calculate profits – at counter-factual prices. I
interpret this profit difference as the additional managerial menu cost (incurred when using
non-uniform prices instead of uniform ones) that would be sufficient to induce a retailer
to use uniform rather than non-uniform prices. I find that in a product category where
uniform prices are typically found, the necessary managerial menu costs average roughly
3% of profits, or $35 per week, per store, per category (in 1992 dollars).1 According to
Levy, Bergen, Dutta & Venable (1997), any price change would, at a minimum involve a
category pricing manager. In this time period, they report that such a manager typically
earned roughly $100,000 annually. Assuming a 40 hour work week, this corresponds to a
wage of roughly $50 per hour. Hence, my finding can be seen as saying the following: “For
the category I consider, charging non-uniform prices nets the retailer an additional average
$35 per week, per store. If charging non-uniform prices instead of uniform prices takes up
more than about an hour of a pricing executive’s time per store per category, the retailer
will find it more profitable to simply charge uniform prices.”
1
For reference, one 2004 dollar is equivalent to $0.72 1991 dollars, $0.74 1992 dollars, or $0.76 1993
dollars, the years covered by the data in this dissertation.
CHAPTER 1. INTRODUCTION
4
The remainder of the chapter explores alternative explanations proposed by the literature, and describes my approach in more detail. Given the pervasiveness of uniform pricing
in retail environments, the dearth of papers on the subject is quite surprising. To my knowledge, only Orbach & Einav (2001) confront the uniform pricing puzzle directly (and then
only in the movie industry). They observe that tickets for movies that are a priori expected
to be blockbusters are sold at the same price as movies that are a priori expected to be
box office bombs. Unfortunately, they are hamstrung by the fact that they never observe
non-uniform pricing for movies, and are unable to find any convincing explanation.
The answer to the uniform pricing puzzle must revolve around the key question: How do
retailers set prices? Menu costs are not the only potential explanation. Indeed, in addition
to the menu cost explanation, there are a variety of explanations, which fall into two groups:
demand-side (consumer-based) and supply-side (retailer-based) explanations.
Potential demand-side explanations involve explicit consumer preferences for uniform
prices. Based on discussions with price-setters, Kashyap (1995) and Canetti, Blinder &
Lebow (1998) find that many firms believe they face a kinked demand curve, containing socalled “price points” where marginal revenue is discontinuous. Two consumer preferences
that might yield a “uniform price” price point are that more prices makes it harder to figure
out what to buy, and that more prices make the consumer feel that the retailer is trying to
take advantage of her. Shugan (1980) and Hauser & Wernerfelt (1990) develop theoretical
models of costly optimization and Draganska & Jain (2001) find some evidence of this in
yogurt. Kahneman, Knetsch & Thaler (1986) look at consumers’ perceptions of fairness,
and show that consumers may perceive unfairness in retailers pricing policies. Evidence
that these perceptions of fairness can affect demand can be found in recent popular press
surrounding actions by Coke (Hays 1997) and Amazon (Heun 2001).
In addition to the “null” hypothesis that the observed prices are actually optimal in a
traditional supply and demand framework, the two principal supply-side explanations are
that the observed uniform pricing behavior is either driven by menu costs or stems from an
attempt by retailers to soften price competition.
This is the explanation favored by Ball & Mankiw (2004), who use it to explain “sticky”
prices. The puzzle of uniform pricing across differentiated products is closely related to
the long-standing macro-economic issue of sticky prices. Sticky prices can be thought of as
uniform prices for products that are differentiated by time – an example of inter-temporal
price uniformity. Addressing this issue, Ball & Mankiw (2004) suggest that much of the
CHAPTER 1. INTRODUCTION
5
observed inter-temporal price uniformity can be explained by the menu costs associated
with “the time and attention required of managers to gather the relevant information and
make... decisions.”(p.24-25).
Leslie (2004) also believes that menu costs play a role in pricing decisions. He finds
that Broadway theaters can earn higher profits by charging prices that differ across seats.
While he finds that charging different prices for different seating categories results in higher
profits, he cannot explain why theaters use only two or three different categories. Because
he does not observe seat-level price variation he cannot estimate the implied menu costs.
Although they do not describe it as such, Chintagunta, Dubé & Singh (2003) provide
a measurement of implied menu costs. They find that multi-store retail chains can earn
higher profits by charging different prices in different geographic areas. They predict that if
retailers charged a different price in each store (rather than using only three or four different
menus of prices for over 80 stores), the chain would have earned an additional $10,000 per
week in the orange juice category alone! Such a large result seems unreasonable in light
of the salary levels found in Levy et al. (1997), and may be due to the potential demandside explanations discussed above or the restrictive assumptions that they make about the
nature of competition between retailers in order to identify the demand.
Taking a different tack and inferring managerial menu costs from salary data, Levy et al.
(1997) guesstimate that the annual price-setting managerial costs are $2.3-$2.9 million at
the chain level, which translate to average annual per-store costs of roughly $7000.2 These
are average, not marginal costs, however.
In addition to menu costs, another possible supply-side explanation for uniform pricing is
that it leads to a softening of price competition. There is an extensive literature investigating
the effects of multi-market interaction on firms’ abilities to collude (for example Nevo (2001)
looks at the case of breakfast cereal, while Carlton (1989) suggests this as an explanation
for inter-temporal price uniformity). Most relevant to my analysis is Corts (1998) who
shows that firms engaged in multi-market competition may prefer to commit themselves to
charging the same price in both markets in order to soften price competition. Corts models
the interaction between two firms which compete in a Bertrand setting in two markets.
In his model, the two firms have identical costs, but these costs differ across markets. If
they are able to charge different prices in each market, the firms drive the price down to
marginal cost. However, if they are able to restrict themselves to charge the same price
2
These figures are in 1992 dollars. One 1992 dollar is equivalent to $1.35 2004 dollars.
CHAPTER 1. INTRODUCTION
6
in both markets, then they are able to earn positive profits in expectation. Viewing two
different flavors as different markets, his result suggests that retailers may benefit if they are
able to tacitly agree to charge fewer prices than the number of distinct products that they
sell. Alternatively, charging fewer prices may allow colluding firms to more easily detect
cheating. However, both of these theories of collusion require retaliation by the cartel in
the face of detected cheating. This does not coincide with what I see in my data: I see
repeated non-uniform pricing by one retailer that is not met by non-uniform pricing by any
other retailers in my sample. As we will see in more detail later (see Figures 1.1-1.4), I do
not see the stores in my data retaliating through the use of non-uniform prices in the face
of deviations.
Finally, there has also been a growing body of literature exploring the implications of
line length – the dual of the uniform pricing puzzle (See, for example, Draganska & Jain
(2001), Bayus & Putsis (1999), and Kadiyali, Vilcassim & Chintagunta (1999)). These
papers model the retailer’s decision to add products to a “line”. This literature has taken
for granted that all the products in a line have the same price. Indeed, many authors have
defined a product line as the set of products from a single manufacturer sold by the retailer
at a uniform price. Rather than examine the pricing decision, this literature has focused
exclusively on the decision of whether to introduce additional products. Clearly in addition
to facing the problem of maximizing product line length, manufacturers face the decision
of when to price products differently, that is, when to split a line. To my knowledge, this
question has not been directly addressed by the line length literature.
Although many of these explanations for uniform pricing are plausible (and may be the
cause for some cases of uniform pricing), this dissertation focuses primarily on the menu
cost explanation, which is essentially a story of bounded rationality on the part of the
retailer. There are several reasons for this focus. The first reason for this focus is that the
menu cost explanation uses standard assumptions concerning consumer choice behavior,
and at a minimum will provide a useful benchmark for comparison when considering other
explanations. Second, it is certainly plausible that for many goods, menu costs may be
able to explain uniform pricing. Third, while there is reason to believe that demand-side
explanations may play a role, I am currently limited by a lack of data. Exploring demandside explanations in more detail cannot be done with the data presently available.3 By
3
Doing so would, at a minimum, require exogenous switching between uniform and non-uniform pricing
strategies. In addition, Kahneman et al. (1986) suggest that consumer responses to alternative price behavior
are heavily dependent on framing (e.g., explanations for alternative pricing behavior).
CHAPTER 1. INTRODUCTION
7
contrast, if I assume that consumers do not have explicit preferences for uniform prices, it
is possible to examine the implications of uniform pricing for retailers.
This dissertation estimates the economic profits that retailers appear to lose by following
optimal uniform pricing strategies. I compare the expected profit earned under under the
actual pricing regime, as well as the expected profits earned under several alternative pricing
strategies, such as having one price for each size, one per unit price for each manufacturer,
and price for each manufacturer-brand-size. These lost profits place bounds on the retailer’s
costs of implementing these alternative pricing regimes. In the next section, I describe the
potential demand-side explanation in more detail, before continuing (in section 1.3) to
develop my framework for estimating the menu costs to the retailer from following nonuniform pricing strategies.
1.2
Demand-side Explanations
In this section, I briefly discuss “fairness” and costly consumer optimization and their
implications for uniform pricing. As mentioned in the introduction, the assumption that
uniform pricing is entirely driven by supply-side factors is quite strong. One might reasonably suppose that the source of uniform pricing lies with the consumer. Unfortunately, when
examining these demand-side explanations, my maintained assumption that consumer preferences are stable across different stores is far less tenable. To see this, consider two markets:
A and B. Suppose that market A has uniform prices while market B has non-uniform prices.
One cannot simultaneously assume that the demand systems in these two markets are the
same, while assuming that the cause of the uniform pricing lies with the demand system.
Estimation of the demand system in the case of explicit demand-side preferences for uniform prices would require either truly exogenous changes between uniform and non-uniform
prices (as in a controlled experiment), or much stronger assumptions. However, even if we
observed apparently exogenous price variation, with the same households making purchases
under both uniform and non-uniform pricing regimes, evidence suggests that many of these
demand-side explanations are subject to framing issues.
In thinking about fairness and its effects on pricing and consumer demand, we need to
consider two questions: (1) Do consumers believe the products are priced fairly? and (2)
If consumers believe that products are priced unfairly, what do they do? In an attempt to
better understand consumer ideas about fairness, Kahneman et al. (1986) surveyed roughly
CHAPTER 1. INTRODUCTION
8
100 Canadian households. Their results showed that people’s perceptions regarding whether
a firm’s actions were fair or unfair depended on the way the actions were framed. For
example, 38% of respondents thought that it was fair for a firm to lower wages 7% during a
recession with no inflation, while 78% of respondents thought it was fair for a firm to raise
wages only 5% during a recession with 12% inflation. In several other questions, they found
that people typically believe that a firm is acting unfairly when price differences stem from
changes in market power, but that they believe firms act fairly when price differences are
the result of shifting costs.
Since, in the absence of market power, prices are driven to marginal cost, the implication
of this finding is that when prices are non-uniform, consumers may believe (rightly or
wrongly) that the only fair reason for non-uniform prices is that the marginal costs (or
perhaps average costs) for the products are different.
Presumably if consumers perceive that they are being treated unfairly, they may react
negatively. Although I am unaware of any studies that systematically look at consumer behavior when consumers perceive that firms are acting unfairly, it is reasonable to conjecture
that the consumer response would be to decrease purchases of either the products priced
unfairly, the category these products are in, or the store that sells them. Hence, if a retailer
charges non-uniform prices for products that consumers perceive to have equal costs, the
firm may face decreased demand.
Two rather extreme real-world examples bear this out. First, in late 1999 it was revealed
that Coca-Cola was considering reprogramming some of its vending machines to charge
higher prices during warmer weather (Hays 1997). This met with quick disapproval by consumers, who threatened a boycott of all Coca-Cola products if the temperature-contingent
pricing was implemented. Hasty back-pedalling by Coca-Cola ensued. Later, in the fall
of 2000, consumers discovered that Amazon.com was charging different prices to different
consumers, both randomly and based on purchase histories. After threats by consumers to
boycott the company, the company agreed to discontinue the practice (Heun 2001). These
are two extreme examples, but they illustrate that consumers may respond to non-uniform
prices by decreasing overall demand.
The principal implication of the fairness theory is that if products are priced nonuniformly, consumers may ask themselves whether there is a cost-based justification. Unfortunately this leaves a great deal unexplained. Can retailers that wish to sell at non-uniform
prices explain their actions in a framework that is palatable to consumers? Two pieces of
CHAPTER 1. INTRODUCTION
9
evidence suggest that they can. The first is that consumers do not protest the existence of
sales. If all products are sold at marginal cost, consumers should interpret sales as evidence
that goods are sold above marginal cost. The second, more convincing piece of evidence
that consumer wrath can be mitigated is the recent change in pricing regimes undertaken
by many major league baseball teams. Formerly sold at the same price for a particular
seat, professional baseball tickets have recently switched to non-uniform prices. The price
of a ticket now depends on the quality of the opponent (Fatsis 2002). This has been accomplished through aggressive framing by teams. Games against poorly performing teams have
been “discounted”, while tickets against “premium” opponents are “a few dollars more”.
Another rationalization for a non-standard demand system centers on the way that
consumers optimize. Grocery stores sell thousands of products; an average store in a large
grocery chain might carry more than 14,000 unique items. Even within a single category,
they frequently stock several hundred different items. As discussed in Shugan (1980) and
Hauser & Wernerfelt (1990), having to choose from such a large number of products may
make it difficult for the household to figure out which bundle of goods offers it the greatest
utility.4 Uniform prices may make it easier for consumers to decide between products: If
two products are the same price, then consumers need only consider the relative utility of
the goods, rather than deciding between the more expensive good and the bundle consisting
of the less expensive good and the other goods that could be purchased with the difference.
However, it is also possible to imagine that if products are too close to each other in
characteristics space, uniform prices confound the choice process. In this case, consumers
may find it easier to decide between two items when their prices are different. To my
knowledge, the behavioral choice literature has not yet addressed the issue of how uniform
prices affect choice, instead focusing on the effects of varying numbers and varieties of goods.
The literature on computational costs for consumers is still relatively undeveloped, but
in specifying the consumer’s utility function, some authors have tried including product
variety in the consumer’s utility function. Draganska & Jain (2001) specify consumer i’s
indirect utility from choosing good j with characteristics Xjt and price pjt as:
0
uijt = Xjt
β − αpjt + f (ljt ) + ijt
(1.1)
4
The related “choice overload” problem is currently an active research area in behavioral marketing.
Looking at jams and jellies, Iyengar & Lepper (2000) find evidence that consumers feel overwhelmed when
faced with choosing between too many alternatives, decreasing their purchase probability.
CHAPTER 1. INTRODUCTION
10
where f is a parametric quadratic function and ljt is the number of products in the same line
as good j. Consistent with Iyengar’s results, they find that consumer utility first increases
with line length, and then decreases. Similarly, Ackerberg & Rysman (2004) assume that
household utility varies with the total number of products offered. They estimate both
additive and multiplicative effects, and find that both effects significantly alter the estimated
price elasticities.
Unfortunately, experimental results have yet to reveal why consumers should care only
about the line length for the line that the product is in, rather than all products in the
category, or all products within some visual radius. Further examination of consumers’
perceptions of fairness and the factors that influence the costs of consumers’ utility maximization are clearly fertile ground for future research. In the interim, however, we can
investigate the plausibility of the menu costs explanation.
1.3
An Overview of Menu Costs
Assuming that uniform pricing is driven entirely by supply-side factors, then clearly the
first-order question is: “How much profit is forgone?” To answer this question, I begin by
assuming that retailers are able to choose among a variety of different pricing strategies of
differing sophistication. These strategies are functions that map each period’s state space
to a vector of prices. Consider the array of potential pricing strategies that a retailer of
carbonated soft drinks might choose from:
• Charge a constant percentage markup of 30% over wholesale price.
• Charge a markup of a constant amount of $0.25 over wholesale price.
• Charge a constant percentage markup of x% over wholesale price, where x is chosen
optimally.
• Charge a single per unit price (e.g., $0.25 per 12 ounces) for all soft drinks, regardless
of size or flavor.
• Charge a single per unit price for all soft drinks from the same manufacturer, regardless
of size, but charge different prices across manufacturers.
• Charge a single per unit price for all soft drinks of the same size, regardless of manufacturer, but charge different prices across sizes.
CHAPTER 1. INTRODUCTION
11
• Charge a different price for each product (e.g., one price for a 2-Liter bottle of Diet
Coke, a second price for a 2-Liter bottle of Coke Classic, a third price for a 2-Liter
bottle of Diet Pepsi, etc.).
• Charge two different per-unit prices for all soft drinks, but determine the groupings
of products and these two price levels optimally.
In this context, uniform pricing by manufacturer-brand-size and completely non-uniform
prices are just two of many potential pricing strategies. The retailer’s implementation costs
for these pricing strategies clearly differ. For example, charging a constant markup of 30%
on all products requires no knowledge on the part of the retailer about the residual demand
curve that it faces. In fact, many books on applied pricing for small retailers (e.g., Burstiner
(1997)) suggest that they simply charge a 100% markup on their entire inventory, a practice
known as “keystone pricing”. By contrast, charging a different (and profit maximizing price)
for each product requires intimate knowledge on the part of the retailer of the residual
demand curve that it faces. It must be cognizant not only of consumers’ preferences, but
also of the current state space (competitors’ current prices, current advertising activity,
current wholesale prices, holiday periods, etc.). Furthermore, mapping these state variables
to optimal prices for each product involves solving a high dimensional optimization problem
every period.
The following framework is useful for analyzing the retailer’s decision process. Assume
that each period, the retailer maximizes expected profits, less menu costs:
Expected
Profit
Expected
=
this Period
Revenue
Expected
−
from Sales
Cost
of Goods
−
Menu Costs
this Period
(1.2)
The menu costs incurred by the retailer each period can be thought of as having two
components:
Menu Costs
this Period
Maintenance
=
Costs
(Recurring)
Upgrade
+
Costs
(1.3)
(if any, Non-Recurring)
The first component of the current period’s menu cost is the recurring cost of maintaining
the current pricing strategy. Obviously, this cost would vary depending on the pricing
CHAPTER 1. INTRODUCTION
12
strategy chosen. A simple pricing strategy, such as “charge the same prices as last period”
would incur zero maintenance costs. But in the case of a complex strategy, this cost could
be quite high. Each period, the retailer would have to learn the current state space (entering
wholesale prices into a computer pricing program, learning competitors’ prices, etc.) and
apply the pricing rule to determine the prices for that period. Even within the class of
complicated pricing rules, costs might vary. Due to the difficulties inherent in numerical
optimization, uni-dimensional pricing strategies (e.g., charging a single optimal markup) are
much easier to implement (and hence require less managerial time) than high-dimensional
pricing strategies.
The second component of the current period’s menu cost is the cost of upgrading to a
better pricing strategy. This fixed cost is non-recurring (or at least infrequently recurring).
If the retailer decides to use the same pricing strategy as in the previous period, no upgrade
costs would be incurred. But if, for example, a retailer decided to switch from a “charge
the same prices as last period” strategy to a “charge the profit-maximizing price for each
good” strategy, they would potentially have to incur several costs.
First, in order to learn their demand curve, the retailer may want to introduce exogenous
price variation.5 In addition to the opportunity cost of the time it takes for a manager to
determine these prices, this experimentation involves forgone profits, because it explicitly
requires charging prices that are believed to be non-optimal. Fortunately, this experimentation is required only infrequently – when it is believed that the structural parameters of
the demand system have changed.
Second, the retailer must analyze the data to recover the structural parameters of the
demand system. Here the retailer faces a choice regarding the level of sophistication used in
estimating the demand curve. For example, the retailer could employ a homogenous logit
model, a heterogenous logit model, or another model (such as that found in this paper) in
estimating demand. In choosing among alternative models, the retailer must trade off the
opportunity cost of a manager’s time, or the cost of hiring consulting services against the
expected cost of mis-specification. Like the costs associated with experimentation, this cost
must be incurred only when it is believed that the structural parameters of the demand
system have changed. Third, the retailer may potentially have to purchase software (such
as optimization software) to allow it to convert the estimated demand system to a set of
5
As discussed in more detail later, it is not possible to estimate the cross-price elasticities between two
products if their relative prices are constant.
CHAPTER 1. INTRODUCTION
13
optimal prices.
Unlike the maintenance costs, which are incurred each period, these upgrade costs would
only be incurred by the retailer infrequently, when the retailer perceives that the upgrade
costs are less than the present discounted value of the additional profits gained from following the new pricing strategy.6 This means that the upgrade costs must be incurred
whenever there is reason to believe that the structural parameters of the demand system
have changed. If the demand system changes substantially over time, or across distance,
this will lead to additional upgrade costs. For example, if Coke is more popular in some
areas, while Diet Coke is more popular in other areas, then demand must be estimated
separately in each of these areas. This explains why even large chains might charge uniform
prices - because demand may differ structurally across geographic areas, and hence it may
need to be re-estimated for each area, eliminating returns to scale in upgrade costs. Similarly (though less likely), the retailer will have to re-estimate demand more frequently in
areas where the distribution of consumers’ preferences change frequently. We are less likely
to see costly-to-implement pricing strategies when the retailer cannot expect to recoup the
upgrade costs.
For tractability, this dissertation assumes a static model, and estimates the additional
per-period maintenance costs for non-uniform pricing relative to uniform pricing. Estimating a dynamic structural model would have to include a model of the retailer’s expectations
about the demand curve it would find if it experimented, as well as the retailer’s expectations how marginal costs and demand would change over time, and their competitor’s
actions.
1.4
Estimating Menu Costs with Soft Drink Data
In order to actually calculate the implied menu costs, I must learn both the marginal costs
and the demand curves faced by the retailer. In practice, I only observe data on weekly
prices and quantities purchased by households. Economic theory suggests I can recover
the marginal costs if I know the residual demand function, but I must observe price and
6
I choose to remain agnostic about the process that allows the retailer to form expectations about the
additional profit to be gained from upgrading to a new pricing system without actually implementing the
system.
CHAPTER 1. INTRODUCTION
14
quantity data that includes variation in prices - I must observe a retailer charging nonuniform prices.7 I solve this problem by considering a product that is frequently (but not
always) priced uniformly: different flavors of carbonated soft drinks. As I discuss in more
detail below (and in chapter 2), although several demand models currently exist, all either
ignore potential variety-seeking behavior by consumers, or cannot be feasibly estimated
for more than a handful of products. Given that the proposed counter-factual pricing
experiments involve adjusting the relative prices between different varieties, there is reason
to expect that allowing for this variety-seeking behavior – in the form of negative crossprice elasticities – may be important. Chapter 2 investigates the neccesity of accounting
for this variety-seeking behavior in more detail, comparing the results from counter-factual
exercises using my new model and the traditional logit model, and shows that menu cost
estimates may be substantially incorrect if the wrong model is used.
Within the soft drink category, there is often a great deal of price variation, both for the
same product over time and between products from different manufacturers. Indeed, Coke
and Pepsi frequently alternate promotion weeks, with Coke on sale one week and Pepsi on
sale the next. This behavior can be seen in Figures 1.1 and 1.2, which plot the weekly price
(normalized to 12-ounce servings) of 2-Liter containers of Coke and Pepsi over a two year
period at two different stores. In contrast to the price variation seen in these figures, soft
drinks are typically sold at uniform prices by manufacturer-brand-size. Within a size, all
flavors of Pepsi are typically sold at one price, and all flavors of Coke at another uniform
price. Figure 1.3 illustrates this by plotting the ratios of the price of Diet Pepsi and Diet
Caffeine Free Pepsi to Regular Pepsi. From the graph, it is easy to see that at this store
7
Strictly speaking this is not quite true. For example, it is possible to empirically estimate a homogenous
logit model and hence derive cross-price elasticities between two goods whose price ratio is constant as long
as the their price levels are changing (i.e., get identification off sales). To see this, consider the case with two
products and an outside good, with product dummies for characteristics. In this case, a homogenous logit
model of household choice implies a system of 2 linear equations, where the dependent variable in equation j
is the log of the ratio of the market share of good j to the market share of the outside good: ln(sjt ) − ln(s0t ).
The independent variables are the price of good j and indicator variables for each good. Because the logit
assumes that the coefficients on these prices and dummy variables are equal in each equation, the system
collapses to a single equation with three variables: price and indicator variables for the two goods. This
equation is usually estimated by OLS. If the two goods are sold at the same price, say $1 in every period,
clearly the price variable is co-linear with the product dummies, and the model cannot be estimated. If,
however, the two goods are sold at different prices in each period, but at the same price relative to each
other, then the model can be estimated. Unfortunately, the identification in this case is coming entirely from
the functional form. The logit model assumes that households choose the alternative yielding the highest
indirect utility. Because indirect utility functions are homogenous of degree zero – meaning that a change
in the level of all prices is equivalent to a change in the level of income – the effect of this price variation is
equivalent to variation in household income.
CHAPTER 1. INTRODUCTION
15
(the same store as in Figure 1.1), the three Pepsi UPCs8 were always sold at the same
price. However, not all stores charged uniform prices during this time. Figure 1.4 shows
the same price ratios as in Figure 1.3 but at another store (the same store as in Figure
1.2). The graph for this store clearly shows a great deal more variation in the prices of
different flavors of Pepsi. This variation allows us to estimate demand separately for each
Pepsi variety. Similar price variation among varieties of other manufacturer brands at this
store allows us to estimate demand for many different items.
8
The Universal Product Code (UPC), also sometimes known as a Store Keeping Unit (SKU) is a number
that uniquely identifies each product/size. For example, a 2-Liter plastic bottle of Caffeine-Free Diet CocaCola has a different UPC than a 2-Liter plastic bottle of Diet Coca-Cola, or a 12oz can of Caffeine-Free Diet
Coca-Cola.
35
Cents per 12oz Serving
20
25
30
08jun1991
15
05dec1991
Coke
Pepsi
02jun1992
Week
29nov1992
Prices of 2L (67.6oz) sizes at Store B
Figure 1.1: Graph of Prices for Coke and Pepsi at Store B, 6/91-6/93
28may1993
CHAPTER 1. INTRODUCTION
16
35
Cents per 12oz Serving
15
20
25
30
08jun1991
10
05dec1991
Coke
Pepsi
02jun1992
Week
29nov1992
Prices of 2L (67.6oz) sizes at Store A
Figure 1.2: Graph of Prices for Coke and Pepsi at Store A, 6/91-6/93
28may1993
CHAPTER 1. INTRODUCTION
17
1.4
Price Ratio
1
1.2
.8
05dec1991
Diet Pepsi
Diet Caffeine Free Pepsi
02jun1992
Week
29nov1992
28may1993
Price Ratios of Pepsi Varieties to Regular Pepsi, 2L (67.6oz) size at Store B
08jun1991
.6
Figure 1.3: Graph of Prices Ratios of Pepsi Varieties to Regular Pepsi at Store B, 6/91-6/93
CHAPTER 1. INTRODUCTION
18
1.4
Price Ratio
1
1.2
.8
05dec1991
Diet Pepsi
Diet Caffeine Free Pepsi
02jun1992
Week
29nov1992
28may1993
Price Ratios of Pepsi Varieties to Regular Pepsi, 2L (67.6oz) size at Store A
08jun1991
.6
Figure 1.4: Graph of Prices Ratios of Pepsi Varieties to Regular Pepsi at Store A, 6/91-6/93
CHAPTER 1. INTRODUCTION
19
CHAPTER 1. INTRODUCTION
20
As an aside, note that Figures 1.3 and 1.4 (and the puzzle addressed in this dissertation)
highlight two features of the data: (1) the prices of the three goods are moving in lock-step
(i.e., Figure 1.3 is essentially two straight lines) and (2) they are all the same price (i.e.,
these lines are at 1). Although this dissertation considers (2), observing only (1) but not
(2) would also be explained by menu costs. For example, if a retailer knew that Diet Coke
was more popular than Coke (but not how this varied with their absolute prices or the
prices of other products) then one unsophisticated pricing strategy that would result from
this would be to always charge $0.10 more for Diet Coke than Coke.
Given this dataset, I need a model of consumer demand that will allow me to estimate
the residual demand curve in period t for each product j: Qjt (·). It is important to note
that I need to estimate the residual demand function faced by a single store. Unless the
retailer is a monopolist, this is the not the same as the market demand function faced
by all stores. The residual demand function reflects the presence of other stores in the
market. The difference is that the residual demand function accounts for the fact that the
prices charged at other stores affect demand at store A. This means that if store B has a
clearance sale, I should expect demand at store A to decline. Several previous empirical
demand studies have ignored this aspect for the very good reason that in most datasets,
this information is simply not available – prices for other stores are not observed. In my
case, however, I observe the prices charged at four other competing stores. Furthermore,
as explained later, the additional stores in the dataset were chosen precisely because they
were the stores that shoppers were most likely to visit.9
To estimate the retailer’s residual demand curve, I begin by decomposing the household’s
demand for soft drinks into two parts. I assume that conditional on going shopping (a
process that, following the existing literature, I take to be exogenous) the household first
chooses which store to shop at. Then, conditional on the choice of store, the household
chooses the bundle of goods from that store that maximizes their utility. This means that
the residual demand faced by a particular store in a given week is equal to the sum over all
households (that went shopping in that week) of the probability that the household chose
that, multiplied by their expected purchases, conditional on choosing that store. In other
9
Throughout this dissertation I assume that the retailer uses a best-response to other retailers, and does
not account for the fact that deviations may lead to changes in rivals’ pricing strategy. This is the same as
the criterion for a Nash equilibrium.
CHAPTER 1. INTRODUCTION
21
words, this equation:
Et [Q(p)] =
X
E [i’s purchases|i goes to A] · P
i
!
i’s characteristics
i goes to A and prices at all stores
(1.4)
describes the residual demand curve faced by store A in week t. Although I estimate
a structural model of product choice, conditional on store choice (my approach to this
problem is the main subject of the next chapter), for tractability, I estimate a reduced form
model of store choice. Ideally, I would like the household’s entire choice problem to satisfy
utility maximization. One way to do this would be to model the household’s choice of store
as a multinomial logit choice, where the mean indirect utilities are derived from the optimal
bundles that the household could have selected at each store. Unfortunately, this approach
is computationally very burdensome. As an approximation to this, I model the household’s
choice of store as a multinomial logit choice, where the indirect utility from each store varies
with household and purchase-occasion characteristics, as well as price indices for various
categories from each store.
Many authors have used the logit model to estimate household demand for differentiated
products. The principal advantage of the logit is that it is very easy to estimate. A
significant drawback of the logit is that in its traditional form, it does not allow for varietyseeking behavior by households – all cross-price elasticities are constrained to be positive.
In the applications for which the logit was originally developed, where each agent makes
a single choice among several competing alternatives, this is not a problem. It becomes
problematic, however, in applications where agents’ choices are not exclusive. Such nonexclusive choices occur in many settings, but are particularly frequent in grocery stores.
Empirical evidence from households’ purchase behavior of soft drinks suggests that
these patterns are present in the data. As shown in Table 1.1, households typically buy
several units of the same product, and/or several different products, within a single purchase
occasion. This is also evident, though to a lesser degree, when I consider only purchases
among the top 25 UPCs by sales in Table 1.2.
Several alternatives exist to the traditional logit model to account for potential complementarities that arises when households buy bundles. One approach that does account for
complementarities is the AIDS model of (Deaton & Muellbauer 1980), but this approach
is can only be applied to aggregate data. Because using household-level data offers the
CHAPTER 1. INTRODUCTION
22
Table 1.1: Distribution of Purchase Occasions by Number of Items and Number of UPCs
Purchased (all Carbonated Soft Drinks)
Total Number
of Items Purchased
1
2
3
4
5
6
7
8
9
10+
Total
of
1
1,332
967
177
370
25
122
7
83
15
169
3,267
Total Number
UPCs Purchased
2
3
4 5+
0
0
0
0
377
0
0
0
195
86
0
0
111
45 16
0
50
22
9
3
46
22
5
2
23
13
4
2
16
17
1
1
11
3
4
0
39
11
5
4
868 219 44 12
Total
1,332
1,344
458
542
109
197
49
118
33
228
4,410
This table shows the distribution of household purchase occasions across multiple units and multiple
products, replicating a similar table found in (Dubé 2001).
potential for increasing the number of observations – it gives us significantly more information over a shorter time period – it seems wasteful to aggregate potentially useful variation.
Furthermore, because the AIDS approach is non-hedonic and does not use product characteristics, it can only be estimated when all products are offered in all weeks. If even one
product is missing in a given week, the AIDS approach cannot use that week. Recent work
by Israilevich (2004) has attempted to ameliorate this problem with mixed success.
As I will discuss in more detail in chapter 2, several other approaches to this problem have been suggested, including modifications to the logit by Gentzkow (2004), Hendel
(1999), Dubé (2001), and Chan (2002) or by taking a different approach altogether, as in
Kim, Allenby & Rossi (2002). All of these attempts however, have had significant shortcomings. Models based closely on the logit require that the econometrician specify the number
of potential bundles that the household can choose, while other models suffer computational difficulties that severely restrict the number of products that can be included in the
analysis. This dissertation improves on these existing models by developing and estimating
a model that contains the flexibility to capture variety-seeking choice behavior as well as
the scalability to handle continuous choice and larger product spaces. Using the parameter
estimates from this new structural model, I am able to calculate the implied profits that
CHAPTER 1. INTRODUCTION
23
Table 1.2: Distribution of Purchase Occasions by Number of Items and Number of UPCs
Purchased (Among Top 25 Carbonated Soft Drinks)
Total Number
of Items Purchased
1
2
3
4
5
6
7
8
9
10+
Total
Total Number
of UPCs Purchased
1
2
3 4
1,163
0
0 0
274 108
0 0
86
38
6 0
49
23
3 1
17
23
5 2
25
15
6 0
5
6
2 0
8
3
3 0
3
2
2 0
60
5
2 0
1,690 223 29 3
5
0
0
0
0
0
0
0
0
0
1
1
Total
1,163
382
130
76
47
46
13
14
7
68
1,946
This table shows the distribution of household purchase occasions across multiple units and multiple
products, replicating a similar table found in (Dubé 2001).
the retailer earned following its non-uniform pricing strategy as well as the profits that it
would have earned had it followed a strategy of uniform prices.
1.5
Conclusion
This chapter has described the puzzle of uniform pricing for differentiated products, and has
covered many anecdotal facts. These anecdotes suggest that managerial menu costs on the
part of the retailer may be able to explain the observed uniform pricing behavior. In order to
test this hypothesis, and measure the menu costs that would be required to rationalize this
behavior, I need to carefully estimate a structural model of the residual demand curve faced
by the retailer. Only a structural model will allow me to predict demand at counter-factual
prices.
The remainder of the dissertation proceeds as follows: chapter 2 lays out the problems
with the existing demand estimation literature as it pertains to my data. Building on this
literature, I develop a new model and explore its characteristics in detail.
Chapter 3 uses this model to estimate household-level demand, using grocery data.
These estimates are then used to perform the counter-factual experiments described in this
CHAPTER 1. INTRODUCTION
24
chapter. By comparing the expected profit earned by a single retailer at the weekly prices
actually charged to the expected profit that same retailer would have earned, had it charged
uniform prices that were optimal in each week (subject only to the restriction that they
be uniform by manufacurer-brand-size), I am able to infer that the retailer would have
experienced a profit loss of roughly $36.56 (in 1992 dollars) per week if it had charged
uniform, rather than non-uniform prices. This leads me to conclude that relatively small
managerial menu costs associated with weekly price optimization are sufficient to lead to
the observed behavior by many other retailers – that of uniform prices.
Chapter 2
A Model of Continuous Demand
and Variety-Seeking
2.1
Introduction
As discussed in chapter 1, existing demand models are unable to simultaneously address
several key features that are frequently encountered in consumer choice data (and that
are found in my dataset). These three key features are: that households may be varietyseeking, that they make continuous choices, and that they choose from a large number of
products. This chapter begins by exploring in more detail the shortcomings of existing
models in dealing with these features. I consider both deviations from the the classic
AIDS model, as well as deviations from the traditional multinomial logit model.1 After
identifying these shortcomings in the existing literature, I develop a new demand model that
allows for variety-seeking households that make continuous choices from a large number of
products. This new model is based on the hedonic framework of many existing models;
each good is represented by a vector of characteristics. However, it involves solving a
direct utility function, subject to a budget constraint in order to compute a household’s
demand for a vector of products. By allowing households’ preferences to be nonlinear in
the characteristics, I am able to model variety-seeking behavior. In addition, the use of a
direct utility function and budget constraint enables us to capture households’ continuous
1
With the exception of Kim et al. (2002), the current literature has tended to use the logit framework
(hedonic utility and an i.i.d. extreme value idiosyncratic shock) to incorporate variety-seeking behavior,
assuming that idiosyncratic shocks are i.i.d. across choices.
25
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
26
choice in a way that reflects economic theory. Finally, I am able to estimate the model
despite the large number of products by using the Method of Simulated Moments, which is
significantly less computationally costly than Simulated Maximum Likelihood proposed by
other authors. I discuss this last point in chapter 3.
After developing the new model, I demonstrate the additional features that my model
offers. I demonstrate the advantages of the model in several ways. First, I look at the
demand and Engel curves generated by the model, showing how the parameters affect
their slope, curvature, and reservation prices. Second, I look at how the model distributes
probabilities determining propensities to purchase different bundles of goods. This shows
how the parameters and the characteristics matrix affect the extensive margin. Finally, I
generate data based on the new model, and estimate the profit difference between uniform
and non-uniform prices using both the new model and the logit model. I find that the logit
estimates of the profit differences are not robust to the misspecification.
2.2
Previous Literature
Existing demand models are unable to address simultaneously: complementarities between
goods, continuous choice, and a large number of products. Given that households typically
purchase several different soft drink products within the same purchase occasion, there is
reason to believe that complementarities exist in my data. This behavior is summarized in
table 1.1, which replicates a similar table in Dubé (2001). It shows the frequency with which
households buy either multiple units of the same product, or several different products,
where a product is defined in this case to be a UPC.
One approach that does account for these complementarities is the Almost Ideal Demand
System model of Deaton & Muellbauer (1980). The AIDS model is appealing because it
does not require the econometrician to locate the products in characteristics space, and
allows for a relatively unconstrained system of cross-price elasticities. However, using the
AIDS model here is problematic for two reasons. First, the approach can only be applied
to aggregated data. In my case, aggregating the data would ignore the potentially useful
variation contained in the household-level data. Second, because the traditional AIDS
approach is not hedonic and does not use product characteristics, it can only be estimated
when all products are offered in all weeks. If even one product is missing in a given week,
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
27
the AIDS approach cannot use that week.2 This problem is not readily apparent when
dealing with data that is aggregated across several sizes to the brand level (see, for example,
Hausman, Leonard & Zona (1994)). But when dealing with UPC-level data, the problem
can frequently become severe. In my data, nearly every week has some product that is
not offered. Recent work by Israilevich (2004) has attempted to ameliorate this problem
by parameterizing the prices of the “missing” goods in weeks when they are not offered.
However, he reports difficulty in getting the estimates to converge when there are more
than a handful of missing good-weeks.
Although the traditional multinomial logit does not allow for purchases of more than
one product, there are two obvious ways to remedy this. One way is to view the purchase
of several products at once as a set of independent choices. This is the approach implicitly
taken by papers using aggregate data, such as Berry, Levinsohn & Pakes (1995) and Nevo
(2001). Recall that the traditional logit model with two inside goods, with mean utilities
V1 and V2 , and an outside good with mean utility equal to zero gives the following choice
probabilities:
P (Choose outside good) =
and
P (Choose good j) =
1
1+
eV1
+ eV2
eVj
1 + eV1 + eV2
Adapting this as suggested above, we have:
P (q1 , q2 , z) = P
=
Choose
!q1
Good One
eV1
1 + eV1 + eV2
q 1 ·
·P
Choose
!q2
Good Two
eV2
1 + eV1 + eV2
q 2 ·
Choose
·P
!z
Outside Good
1
z
1 + eV1 + eV2
where q1 , q2 , and z are the quantities of the different goods chosen.
However, two problems exist. First, the econometrician must specify the number of
choices that the household makes. In the traditional logit model, each household is assumed
to make a single choice: Cmax = q1 + q2 + z = 1. But when the household could potentially
make more than one choice, the econometrician must specify the number. This is the
2
The AIDS model leads to an econometric model in which each week is an observation, with one equation
for each product (predicting the market share of that product), and the prices of each good entering as
explanatory variables in all equations. Hence, if I do not observe a single product’s price and demand in a
particular week, I cannot use that week as an observation.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
28
problem that Berry et al. (1995) and Nevo (2001) refer to as needing to specify the size of the
market, or the share of the outside good. We will explore this problem in more detail below.
The second problem with this model is that it is not able to capture complementarities
between products – all cross-price elasticities are constrained to be positive. This can be
seen by noting that in this case E[q1 ] = Cmax ∗ P (Choose good 1), which is a strictly
decreasing function of V2 .
A second potential way to tweak the traditional logit model, that does allow for complementarities, is to assume that the household chooses from among a large number of possible
bundles. This amounts to assuming that the household aggregates the characteristics of
each product in the bundle to create a kind of meta-product. In this case:
exp (V1 q1 + V2 q2 )
(q1i ,q2j ,z)∈C exp (V1 q1i + V2 q2j )
P (q1 , q2 , z) = P
where the set of possible choices is defined as3
C = {(q1i , q2j , z)|q1i ≥ 0, q2j ≥ 0, z ≥ 0, q1i + q2j + z = Cmax }
In order to make the sum in the denominator finite, it is necessary to restrict the number
of potential bundles. This may be done by discretizing q1 and q2 (and, by implication, z).
Now we have:
E(q1 ) =
1 + eV1
eV1 + 2e2V1 + eV1 +V2
+ eV2 + eV1 +V2 + e2V1 + e2V2
when Cmax = 2. Or:
E(q1 ) =
1 + eV1
eV1 + 2e2V1 + 3e2V1 + eV1 +V2 + 2e2V1 +V2 + eV1 +2V2
+ eV2 + eV1 +V2 + e2V1 + e2V2 + e2V1 +V2 + e2V2 + V1 + e3V1 + e3V2
when Cmax = 3. I critique this approach in more detail by following Gentzkow (2004).
Gentzkow generalizes the above model by parameterizing the marginal utility from each
potential bundle of goods. Essentially, he estimates the mean utility for each good (relative
to the outside good), as well as the mean utility from each possible bundle. This approach
is best understood in the context of the following example. Suppose that there are two
3
For example, when the number of choices Cmax = 2, there are six potential choices, with indirect utilities
given by: u(1, 0, 1) = V1 + ε1,0,1 , u(1, 1, 0) = V1 + V2 + ε1,1,0 , u(2, 0, 0) = 2V1 + ε2,0,0 , u(0, 2, 0) = 2V2 + ε0,2,0 ,
and u(0, 0, 2) = ε0,0,2 , where the ε’s are i.i.d. extreme value (Gumbel) distributed.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
29
goods: coffee and cream. Then at time t, household i will receive indirect utility
uit (coffee) = Γcoffee + εi,t,coffee
from choosing coffee,
uit (cream) = Γcream + εi,t,cream
from choosing cream, or
uit (coffee & cream) = Γcoffee & cream + εi,t,coffee & cream
from choosing coffee and cream (where, in general, Γcoffee & cream 6= Γcoffee + Γcream ), or
uit (outside good) = εi,t,0
from choosing the outside good, where the Γ’s are all estimated parameters, and the ε’s
are all i.i.d. extreme value (Gumbel) distributed. This approach has several significant
drawbacks in this setting.
The first problem with applying Gentzkow’s model in this setting is that it cannot handle large numbers of products, because the number of parameters in the model increases
exponentially with the number of products. In the above example, there are three parameters: Γcoffee ,Γcream , and Γcoffee & cream . If we add another alternative, say sugar, then we add
four(!) additional parameters: Γsugar , Γcoffee & sugar ,Γcream & sugar , and Γcoffee & cream & sugar .
This problem becomes even more severe when, as in this case, consumers make non-binary
choices. Suppose I could buy two coffees. Or twenty. The parameter space explodes.
The second problem with applying Gentzkow’s model is that it requires that the econometrician specify exactly how many choices each household makes. This is fine in his
application, where the choice is the weekly decision to subscribe to a newspaper or not, but
it is difficult to say how many times you have made the choice between having a coffee (with
or without cream) in any reasonable time frame. Once? Twice? Zero? One might think
that it would be possible to make the number of choices depend upon the household’s actual
expenditure and the price index via a budget constraint. But this means that the number
of choices depends on the price level. This serves to expose an underlying problem with the
logit model of demand. It predicts that all goods will be purchased with positive probability
regardless of their price. Hendel (1999)’s solution to this problem of is to assume that the
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
30
number of choices made follow a Poisson distribution whose mean varies with observable
characteristics about the household (or firm in his case).
The third problem with Gentzkow’s model in this setting is that even if we deal with
non-binary choice sets by placing restrictions on the parameters to reduce their number, it
is not possible to place restrictions on the distribution of the error terms of the bundles.
Returning to the 2-good example, εi,t,coffee and εi,t,cream are distributed independently of
εi,t,coffee & cream (as well as of each other). Allowing for the choice of two coffees ({coffee &
coffee}) would give us:
uit (coffee & coffee) = Γcoffee & coffee + εi,t,coffee & coffee
Where εi,t,coffee is again distributed independently of εi,t,coffee & coffee . These independence
assumptions are justified only by the computational convenience that they deliver – closed
form solutions for the choice probabilities.
Although Hendel does not address the complementarity issue, he allows for continuous
choice by essentially assuming that the utility from the non-price product characteristics
is concave in quantity. Hence, while the traditional logit assumes that the mean utility
from alternative j with characteristics vector Xj and price pj is uj = βXj , Hendel proposes
uj = (βXj qj )γ − αpj , where qj is the quantity of good j consumed, and γ is a curvature
parameter. Chan (2002) also proposes a continuous choice model based on the logit, but
it predicts positive consumption of all characteristics and infinite consumption of some
products.
Kim et al. (2002) adopt a different approach. They write down the household’s direct
utility function – which they assume is additively separable in products – and solve the
Kuhn-Tucker conditions. Unfortunately, estimating their model requires using (simulated)
maximum likelihood, which becomes very computationally intensive when considering more
than a handful of products as it involves integrating a normal distribution that has number
of dimensions equal to the number of products in the choice space.
My model is a hybrid between Kim et al. (2002)’s and Chan’s. I improve on Chan
by explicitly solving for the household’s budget constraint, by extending it to UPC-level
demand, and modelling the panel aspect of the data. The non-linearity also allows me to
use a more flexible matrix of characteristics, with more characteristics than products. I
also use physical characteristics, rather than brand-level (as in Chan) or product-level (as
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
31
in Kim et al. (2002)) dummy variables.
2.3
The Model
To formulate a model that accounts for variety-seeking, continuous choice, and a large choice
set, I begin by writing down a household-level direct utility function over bundles of goods.
This section describes the model in detail.
In order to reduce the dimensionality of the parameter space, I use a hedonic approach
and assume that, as in the logit model, households derive utility from product characteristics. This means that each product j ∈ J is completely described by a vector of
characteristics of length C.4 The menu faced by the household can be represented by a
J × C matrix A where the rows of A are the products, and the columns are characteristics.
Hence, the A matrix can be thought of as a stacked matrix consisting of the Xj ’s from the
logit model – although, unlike the logit model, the A matrix can have more characteristics
than products (C > J).
More concretely, consider an example with two characteristics (Diet and Cola) and two
available products (Diet Coke and Diet 7Up). Because it is a diet cola, Diet Coke has
characteristic vector [1 1].5 As a diet non-cola, Diet 7Up has characteristic vector [1 0].
Stacking these two products’ characteristic vectors yields the following A matrix:
"
A=
1 1
#
1 0
Again following the logit, I assume that a household’s utility function is additively separable
in these characteristics. Unlike the logit, however, I allow households the ability to consume
multiple units of a single product, as well as consuming several different products. In
particular, I assume that household i myopically maximizes the utility function:
Uit (qit , zit ) =
X
βc (A0c qit + 1)ρc + ε0it qit + zit
(2.1)
c∈ C
with respect to qit and zit . Ac is the cth column of A, qit is a column vector of length J
4
Abusing notation, I use J to mean both the set of products and the number of elements in that set.
Note that although in this case the product characteristics are indicator variables, in general they need
only be non-negative. For example, in the estimated model one of the characteristics is the number of
milligrams of caffeine per 12-ounce serving.
5
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
32
comprising household i’s purchases at time t of the J goods described by A, zit is the amount
of outside good consumed, and βc and ρc are the characteristic-specific scalar components
of β and ρ. The J dimensional vector εit represents the household/shopping-occasion marginal utility shocks, which are observed by the utility-maximizing household, but not by
the econometrician. The household maximizes this utility function subject to the budget
constraint:
X
pjt qjt + zt ≤ wit
(2.2)
j∈ J
where wit is the household’s total grocery expenditure in week t. The imposition of the
budget constraint is a significant difference between my model and logit-based models.
Since the logit assumes that the household is constrained by making a fixed number of
choices, rather than by a budget constraint, the logit model places positive probability that
households will purchase any particular product, regardless of its price. This conflicts with
standard economic theory. By contrast, my model assigns zero probability to the household
purchasing products once their prices pass a certain threshold.
I assume that εit is i.i.d. across products, time, and households, and negatively lognormally distributed on the interval (−∞, 0). These error terms represent purchase-occasion
specific idiosyncratic taste shocks for particular items that are observed by the household,
but not by the econometrician, such as a sudden aversion to consuming Diet Coke. In
principle, these error-terms may be correlated across products within a purchase occasion.
This could stem from characteristic-specific shocks, such as a sudden preference for low
calorie or diet beverages. Additionally, the shocks may be heteroscedastic, either across
time or, more likely, across products. However, for simplicity (and to reduce the number of
parameters required to estimate the model), I restrict these shocks to be i.i.d, and assume
they follow a negative standard log-normal distribution.
The reason that these taste shocks are bounded from above relates to the fact that I
assume that the household does not satiate on the idiosyncratic taste characteristic. The
taste shocks can be thought of as a log-normally distributed product and purchase-occasion
specific characteristic with parameter values β = −1 and ρ = 1. I make these parametric
restrictions because my data do not contain sufficient variation to identify them. As shown
in the next section, the result of these restrictions is that, pj = pj − εj is the effective
price to the household of the product. If pj < 0 then the household inelastically spends
its entire budget on good j. This behavior is not reasonable. Hence, it is necessary to
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
33
bound εit from above in order to prevent unreasonable choice behavior. If, for example, the
realization of εitj is greater than the price of good j, a household may never consume the
outside good on that purchase occasion, regardless of the level of wt . Therefore, to avoid
these negative effective prices I must, at a minimum have εj < pj . This means that I must
have −∞ < εj < pj . However, because it makes little theoretical sense to have the support
of the distribution of the error terms dependent upon prices, I impose the restriction εj < 0.
Within the class of distributions having one-sided support, the log-normal distribution is
desirable because it offers flexible variation with two parameters (although I do not do so
here).
The ρ parameters introduce the nonlinearity that is key to the model’s ability to capture
households’ continuous choice behavior and potential preference for variety. Unfortunately,
because of this nonlinearity, it is not possible to assign clear-cut interpretations to the
parameters as one can do with a linear model. With this caveat, I provide some economic intuition for these parameters. βc ρc is the maximum marginal utility the household
can receive from characteristic c, and hence is a measure the household’s preference for a
particular characteristic, (similar to the familiar coefficients on characteristics in the logit
model).6 At the most basic level, the sign of βc determines whether the household considers
characteristic c to be a good or a bad. Positive βc ’s imply that the household receives positive marginal utility from that characteristic, while negative βc ’s imply negative marginal
utility from that characteristic. At a secondary level, the magnitude of βc determines the
strength of a household’s preferences for a particular characteristic.7 However, as seen in
the next section, the impact of βc is affected in a nonlinear way by the satiation rate, ρc .
The parameter ρc is the satiation rate for that characteristic - it determines the rate at
which the marginal utility for characteristic c changes. Values of ρc closer to one mean that
the household’s marginal utility from additional units of that characteristic changes more
slowly. Also, in addition to increasing βc ρc , changes in ρc affect the behavior of households’
demand curves as prices approach zero. For this reason, throughout much of the analysis
in this section when I vary ρc , I hold βc ρc constant.
Returning to the example, if we substitute the A matrix into the utility function, we
6
When βc > 0 (and hence 0 < ρc < 1), marginal utility is positive, but decreasing. When βc < 0 (and
hence 1 < ρc ), marginal utility is both negative and decreasing.
7
In order to capture household heterogeneity, in chapter 3 I allow the taste parameters β to vary across
households. This allows us to capture persistent household-specific preferences, such as a taste for Coke or
Pepsi.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
34
can see that the household maximizes:
Uit (qit , zit ) = βDiet (qi,t,DietCoke + qi,t,Diet7U p + 1)ρDiet + βCola (qi,t,DietCoke + 1)ρCola +
εi,t,DietCoke qi,t,DietCoke + εi,t,Diet7U p qi,t,Diet7U p + zit
with respect to qit and zit , subject to:
pt,DietCoke qi,t,DietCoke + pt,Diet7U p qi,t,Diet7U p + zit ≤ wit
Unlike the logit, which offers closed form solutions for the households’ expected purchases, this model generally does not have such simple solutions, even for particular values
of ε. Because this model differs so substantially from familiar approaches, the remainder of
the chapter will both show the choice behavior generated by the model, and demonstrate
that this behavior includes both complementarities and continuous choice.
2.4
Behavior of the Model When There Are Two Inside
Goods
This demand model differs significantly from many of the existing demand models discussed
above. For this reason, we would like to know what the demand curves and Engel curves
associated with the model look like. The primary purpose of this section is to show a
variety of typical demand and Engel curves from the model and to relate the features we
see in these curves to the mathematics of the model. The secondary purpose is to show
how these features vary as we change the parameters of the model and the characteristics
of the products in the choice set – providing us with economic interpretations of the taste
parameters. In order to allow greater analytical clarity, this section deals exclusively with
the case when there are two inside goods, each with two differentiating characteristics, and
an outside good. When possible, I solve for the analytical solutions. However, in many cases
closed-form solutions are not obtainable, in these cases I determine the effect of changes in
the parameters or the A matrix on the solution.
The general form of the characteristic matrix with two goods and two characteristics is:
"
A=
a11 a12
a21 a22
#
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
35
With preference parameters β and ρ, the household’s utility function is thus:
U (q1 , q2 , z) = β1 (a11 q1 + q2 a21 + 1)ρ1 + β2 (a12 q1 + a22 q2 + 1)ρ2 + ε1 q1 + ε2 q2 + z
The household’s objective is to choose q1 , q2 and z, to maximize this utility function subject
to a budget constraint:
p1 q 1 + p 2 q 2 + z ≤ w
Where w is the household’s grocery expenditure, qj is the number of units of good j consumed, pj is the price per unit of that good, and z is the number of units of outside good
consumed (and whose price I normalize to one). I also impose non-negativity constraints for
q1 , q2 and z. Assuming that the concavity conditions derived in the Appendix are satisfied,
we can solve this optimization problem by forming the Lagrangian:
L(q1 , q2 , z, λ0 , λ1 , λ2 , λ3 ) = β1 (a11 q1 + q2 a21 + 1)ρ1 + β2 (a12 q1 + a22 q2 + 1)ρ2 +
ε1 q 1 + ε2 q 2 + z
+λ0 (w − p1 q1 − p2 q2 − z) + λ1 q1 + λ2 q2 + λ3 z
The mechanics of the solutions to this Lagrangian discussed in this section are derived in
the Appendix.
2.4.1
Demand Curves
As mentioned earlier, because of the high degree of nonlinearity present in the household’s
demand and indirect utility functions, it is not possible to assign clear-cut interpretations
to the parameters in my model as one can do with the logit model. The purpose of this
subsection is to use the demand curves generated by the model to gain insight into both the
characteristics of the model and the way in which the parameters affect the choice behavior
it generates.
This subsection begins by showing a series of typical demand curves generated by the
model. These curves have three main features. First, although difficult to fully appreciate
in the two-good case, the model generates a large number of corner solutions. This should
not be surprising, since this behavior is one of the key features of the data that the model
was intended to match. Second, the demand curves are usually at least somewhat kinked at
the point where the household shifts from substituting between goods of differing marginal
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
36
utilities. This is a consequence of my method of introducing corner solutions. Third, the
household-level own and cross price elasticities exhibit a large amount of variation with
respect to prices. In particular, many households have own-price and cross-price elasticities
of zero over substantial regions of the price space.
After covering these typical demand curves, I show the effects of changes in the parameters β and ρ, as well as the effects of changes in A, the product characteristics matrix,
as it moves between a diagonal and a row-dependent matrix. In brief, βρ has the largest
effect upon the demand curve, while ρ has the greatest impact on demand when prices are
very low.
"
Figure 2.1 shows some typical demand curves for the two good model when A =
1
0.5
0.5
1
#
.
The left half of the figure contains four different own-price demand curves for good one, each
from a different combination of ε1 and ε2 – the idiosyncratic product-specific shocks. The
right half of the figure graphs the corresponding demand curves for good two as a function
of the price of good one. These four curves correspond to the potential combinations of the
40th and 60th percentile values of the distributions of ε1 and ε2 – which are approximately
-1.49 and -1.82.
Table 2.1 lists the (ε1 , ε2 ) pairs corresponding to the curves in Figures 2.1-2.7. I use
these same values in all the demand curve graphs in this subsection, as well as the Engel
curves shown in the following subsection.
Table 2.1: Pairs of (ε1 , ε2 ) corresponding to the demand curves in Figures 2.1-2.7
Curve
ε1
ε2
①
−1.49 −1.49
②
−1.49 −1.82
③
−1.82 −1.49
④
−1.82 −1.82
With this in mind, the interior solutions (when all three goods are consumed) are:
a22
h
q̂1 =
i
1
ρ1 −1
− a21
h
(p2 −ε2 )a11 −(p1 −ε1 )a21
β2 ρ2 (a11 a22 −a21 a12 )
i
1
ρ2 −1
− a22 + a21
(2.3)
a11 a22 − a21 a12
a11
q̂2 =
(p1 −ε1 )a22 −(p2 −ε2 )a12
β1 ρ1 (a11 a22 −a21 a12 )
h
(p2 −ε2 )a11 −(p1 −ε1 )a21
β2 ρ2 (a11 a22 −a21 a12 )
i
1
ρ2 −1
− a12
h
(p1 −ε1 )a22 −(p2 −ε2 )a12
β1 ρ1 (a11 a22 −a21 a12 )
a11 a22 − a21 a12
i
1
ρ1 −1
− a11 + a12
(2.4)
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
37
Figure 2.1: Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
1 0.5
0.5 1
, β1 = 5.5, β2 = 5, ρ1 = ρ2 = 0.5, p2 = $0.50, w = 60
q (p | p =$0.50)
1
q (p | p =$0.50)
2
2
1
0.9
0.9
0.8
0.8
0.7
0.7
2
0.6
Price of Good 1
Price of Good 1
1
1
0.5
0.4
1
1
2
2
0.6
4
1
0.5
0.4
0.3
0.3
0.2
0.2
3
4
3
0.1
0
0
0.1
2
4
q1
6
8
0
0
1
2
3
4
5
q2
ẑ = w − p1 q1 − p2 q2
(2.5)
The ˆ’s are to remind the reader that these can be considered candidate solutions. In many
cases we will not have an interior solution. This will occur if either of the terms in brackets
are negative, or if q̂1 < 0, q̂2 < 0, or ẑ < 0. I discuss each of these possibilities below.
In Figure 2.1, households are at interior solutions on those portions of the curves where
both q1 > 0 and q2 > 0. For example, the portion of curve ①, covering roughly 0.2 < p1 <
0.54 is at an interior solution. Plugging in values into the above equations tells us that the
relevant demand equations for this segment of the demand curve are (holding p2 constant
at 0.5):
q1 ≈ −9.38 ∗ (2.49 − p1)−2 + 5.67 ∗ (p1 + 0.5)−2 − .667
(2.6)
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
38
q2 ≈ 18.75 ∗ (2.49 − p1)−2 − 2.84 ∗ (p1 + 0.5)−2 − .667
(2.7)
While the household is at an interior solution, the own-price elasticity of good one is:
∂q1 p1
∂p1 q1
=
−p1
q1 det A2
a222
β1 ρ1 (1−ρ1 )
p1 a22 −p2 a12
β1 ρ1 det A
ρ1
1−ρ1
p1 a22 −p2 a12
β1 ρ1 det A
ρ1
1−ρ1
p2 a11 −p1 a21
β2 ρ2 det A
ρ2
1−ρ2
+
a221
β2 ρ2 (1−ρ2 )
p2 a11 −p1 a21
β2 ρ2 det A
ρ2
1−ρ2
+
a21 a11
β2 ρ2 (1−ρ2 )
(2.8)
and the cross-price elasticity is:
∂q1 p2
∂p2 q1
=
p2
q1 det A2
a22 a12
β1 ρ1 (1−ρ1 )
(2.9)
Note several things about these demand equations and the elasticities they imply. First,
at interior solutions, the household’s expenditure level w does not enter the demand function
for the inside goods – making the compensated and uncompensated demand elasticities
equal. This is because at the interior, all three goods are purchased, and any additional
wealth will be spent on the outside good. Second, the cross-price elasticity in this case (two
goods and two characteristics) is always positive. This is specific to the two-inside-good
case, and is due to the adding-up constraint. Third, when the A matrix is diagonal, the
cross-price elasticities are all zero (at the interior solutions).
Since I only show demand curves here for the inside goods, a brief word is in order about
demand for the outside good. Some households with very low expenditure levels (small w),
or particularly strong preferences for inside goods (very high β or ρ, or very small ε1 or ε2 )
may find it optimal not to purchase any of the outside good. These households spend all of
their grocery expenditure on the inside good. While I will consider these cases later in this
section when I discuss the Engel curves generated by the model, all of the demand curves
in this section have a sufficient expenditure level ($60) that some outside good is always
purchased.
Returning to the example of curve ① in Figure 2.1, as p1 increases above about 0.54,
the household with demand curve ① reaches a corner solution for good one. At this point,
q̂1 < 0, ẑ > 0, and the household’s demand function for good two becomes:
q1 = 0, z = w − p2 q2 and q2 = max 0, min
q2∗ ,
w
p2
(2.10)
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
39
Where q2∗ is the solution to the equation:8
a21 β1 ρ1 (a21 q2∗ + 1)ρ1 −1 + a22 β2 ρ2 (a22 q2∗ + 1)ρ2 −1 + ε2
=1
p2
(2.11)
Note that p1 does not enter Equation 2.10. This means that once a household’s demand for
a good goes to zero, the cross-price elasticity between that good and all other goods is zero.
This is the reason that the demand curve shown on the right, which plots q2 as a function
of p1 becomes vertical as p1 > 0.54.
A similar effect occurs on the portion of the beginning of curve ①, when q̂2 moves away
from zero and the household switches from consuming only good one to also consuming
good two. In this range, covering roughly 0 < p1 < 0.2, we have ẑ > 0 and q̂2 < 0, which
yields the demand curve
q2 = 0, z = w − p1 q1 and q1 = max 0, min
q1∗ ,
w
p1
(2.12)
Where q1∗ is the solution to the equation:
a11 β1 ρ1 (a11 q1∗ + 1)ρ1 −1 + a12 β2 ρ2 (a12 q1∗ + 1)ρ2 −1 + ε1
=1
p1
(2.13)
Again, note that once q̂2 < 0, the cross-price elasticity between goods one and two is zero.
Before moving on, the last feature to recognize about the model is that (although the
behavior is not shown in these figures) it is possible that the household will never want to
consume any good one (or, symmetrically, good two), even when its price is zero. This is
because the household may receive a small negative realization of ε2 , and a large negative
realization of ε1 . This makes them unwilling to purchase good one, even at a price of zero.
Having covered the basic characteristics of the demand curves, we’d like to know how
changes in the parameters lead to changes in the demand curves. These changes are clearest
when the product characteristic matrix is the identity matrix, so we’ll consider that case
first, before returning to the non-diagonal case.
When the product characteristics matrix is diagonal, the characteristics are productspecific. A real-world example of this would be if the available goods were: food, shelter,
and a composite outside good. One implication of this is that the household’s utility function
8
Note that the concavity conditions guarantee that this solution, if positive, is unique.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
40
Figure 2.2: Demand for good one and two as a function of p1 , for 40th and 60th
percentiles of ε1 and ε2 . A = 10 01 , β1 = 5.5, β2 = 5, ρ1 = ρ2 = 0.5, p2 = $0.50, w = 60.
Compare to Figures 2.3 and 2.4.
q (p | p =$0.50)
1
q (p | p =$0.50)
2
2
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Price of Good 1
Price of Good 1
1
1
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0
0.5
1
1.5
2
0
0.2
2.5
2
0.5
0.3
0
1
0.4
q1
0.6
q2
0.8
1
is additively separable in the two goods, which means that changes in the price of good two
can only affect demand for good one (and vice versa) through the budget constraint. Figure
2.2 shows some typical demand curves when the A matrix is diagonal. Even though these
graphs each contain four different demand curves – just like Figure 2.1 – in both the left
and right panes, each curve has another curve stacked on top of it. This is because in this
case, Equations 2.3 and 2.4 reduce to:
q̂1 =
p1 − ε 1
β1 ρ1
1
ρ1 −1
−1
(2.14)
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
q̂2 =
p2 − ε 2
β2 ρ2
1
ρ2 −1
41
−1
(2.15)
Hence, as long as the budget constraint is not binding on the interior goods (i.e., p1 q̂1 +
p2 q̂2 ≤ w), the cross-price elasticities in this case are zero, for all price combinations of p1
and p2 .
Figure 2.3: Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
1 0
0 1
, β1 = 6.875, β2 = 5, ρ1 = 0.4, ρ2 = 0.5, p2 = $0.50, w = 60. Compare to Figures 2.2 and
2.4.
q (p | p =$0.50)
1
q (p | p =$0.50)
2
2
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Price of Good 1
Price of Good 1
1
1
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0
0.5
1
q1
1.5
2
2
0.5
0.3
0
1
0
0.2
0.4
0.6
q2
0.8
1
Figure 2.3 is identical to Figure 2.2, except that ρ1 has decreased from 0.50 to 0.40, and
β1 has increased from 5.5 to 6.875 in order to hold their product, β1 ρ1 , constant. The reason
I hold β1 ρ1 constant is to isolate the second-derivative effect from the first-derivative effect
(that is, isolate the rate of change in the marginal utility from the maximal marginal utility).
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
42
The most noticeable difference between the two graphs is that the quantity demanded of
good one at p1 = 0 decreases substantially, a change from roughly 2.4 units to 1.75 units
for the 40th percentile value of ε1 , with a similar changes for the 40th percentile. At the
same time, there is very little change in the reservation price, the price at which demand
for good one goes to zero. This will make it relatively difficult to empirically estimate ρ
precisely as distinct from βρ without observing demand when prices are close to zero.
Figure 2.4: Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A = 10 01 , β1 = β2 = 5, ρ1 = ρ2 = 0.5, p2 = $0.50, w = 60.
Compare to Figures 2.2 and 2.3
q (p | p =$0.50)
1
q (p | p =$0.50)
2
2
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Price of Good 1
Price of Good 1
1
1
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0
0.5
1
q1
1.5
2
2
0.5
0.3
0
1
0
0.2
0.4
0.6
q2
0.8
1
Contrast this with what occurs when we decrease β1 as in Figure 2.4, from 5.5 to 5.0.
In this case, there is also some movement in the behavior of the demand curve as p1 goes
to zero. But more significant is the effect on the reservation price for good one, which
decreases by nearly a third for the 60th percentile group of 1 , with a similar decrease for
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
43
the 40th percentile as well.
To see why changes to β1 have a much larger effect than changes to ρ1 in determining
the reservation price, consider the following equations governing the reservation prices of
the two goods. Since good one is the sole source of characteristic one, there are only two
constraints. First the cost of marginal utility from consuming good one must less than
the cost of marginal utility from consuming the outside good. Second, good one must not
be “crowded out” by good two, if the household’s budget is not great enough to buy any
outside good. Mathematically, this means that good one will be consumed as long as:
(
)
β2 ρ2 a22 (a22 pw2 + 1)ρ2 −1
β1 ρ1 a11
> max 1,
p1
p2
(2.16)
In this case, increasing β1 , ρ1 , or a11 linearly increases the reservation price. An increase
in β1 of 0.25 causes an increase in the reservation price by $0.25. If the budget constraint
is binding for the interior goods, then the second term in braces is greater than one, and
then decreasing β2 , ρ2 , or a22 all decrease the reservation price of good one. An increase in
β2 by 10% would increase the reservation price for good one by at least 9%.
By contrast, changes in ρ1 have a much larger effect on q1 than changes in β1 when p1
is close to zero. The reason for this can be seen by evaluating Equation 2.14 at p1 = 0, and
comparing the derivatives with respect to β1 and ρ1 (holding β1 ρ1 constant):
∂q1 (p1 = 0)
=
∂β1
∂q1 (p1 = 0)
=
∂ρ1
β1 ρ1
−ε1
β1 ρ1
−ε1
1
1−ρ1
· log
1
1−ρ1
· β1−1
1
1 − ρ1
· (1 − ρ1 )−2
(2.17)
(2.18)
When evaluated at the parameter values β1 = 5, ρ1 = 0.5, the effect of a change in ρ
(holding βρ constant) has an effect more than seven times larger. If βρ is not held constant,
the effect of a change in ρ is even greater.
"
Returning to the original case, with A =
1
0.5
0.5
1
#
, the parameters β and ρ have similar
effects as above, as can be seen in Figures 2.5 and 2.6, which respectively show the effects
of decreasing β1 and of decreasing ρ1 while holding β1 ρ1 constant.
Unlike the case with a diagonal A matrix, however, here the products overlap significantly. This means that much less simplification is possible, and as a result, interpretation
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
44
Figure 2.5: Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
1 0.5
0.5 1
, β1 = β2 = 5, ρ1 = ρ2 = 0.5, p2 = $0.50, w = 60.
Compare to Figure 2.1.
q (p | p =$0.50)
1
q (p | p =$0.50)
2
2
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Price of Good 1
Price of Good 1
1
1
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0
2
4
q1
6
8
2
0.5
0.3
0
1
0
0
1
2
3
4
5
q2
becomes more difficult. In general, demand for good one is positive as long as:
a22
h
(p1 −ε1 )a22 −(p2 −ε2 )a12
β1 ρ1 (a11 a22 −a21 a12 )
i
1
ρ1 −1
− a21
h
(p2 −ε2 )a11 −(p1 −ε1 )a21
β2 ρ2 (a11 a22 −a21 a12 )
i
1
ρ2 −1
− a22 + a21 > 0
(2.19)
n
o
However, if p2 < min β1 ρ1 a21 + β2 ρ2 a22 , p1 aa22
, we also require the second condition that:
12
p 1 < p2
a11 β1 ρ1 (a21 pw2 + 1)1−ρ1 + a12 β2 ρ2 (a22 pw2 + 1)1−ρ2
a21 β1 ρ1 (a21 pw2 + 1)1−ρ1 + a22 β2 ρ2 (a22 pw2 + 1)1−ρ2
(2.20)
The first condition just says that q̂1 > 0. The second is a condition that assures that
it is “cheaper” (loosely defined) to buy characteristic one from good one than it is to buy
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
45
Figure 2.6: Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
1 0.5
,
β
=
6.875,
β
=
5,
ρ
1
2
1 = 0.4, ρ2 = 0.5, p2 = $0.50, w = 60. Compare to Figures
0.5 1
2.1 and 2.5.
q (p | p =$0.50)
1
q (p | p =$0.50)
2
2
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Price of Good 1
Price of Good 1
1
1
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0
2
4
q1
6
2
0.5
0.3
0
1
0
0
1
2
3
4
5
q2
characteristic one from good two. This brings us to a discussion of what I call “dominance”.
In addition to interior solutions, this model also allows for corner solutions. Indeed this
is one of the key features of the model. In many cases the household will choose not to
purchase one or more of the goods. The household will choose to purchase no units of an
interior good, say good one, if either: (1) given the amount of good two that the household
has purchased, the cost of marginal utility from good one is greater than the cost of marginal
utility from the outside good or, (2) all of good one’s characteristics can be obtained more
cheaply by buying good two.
The first possibility for non-interior solutions is that which has already been discussed
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
46
above: if any of ẑ, q̂1 , or q̂2 is less than zero, demand for that good will be zero, and if two
of the three predicted consumption values ẑ, q̂1 , and q̂2 are negative, then the household
spends its entire budget on the third (positive) good. Hence, q̂1 < 0 and ẑ < 0, implies that
the the solution is (0, pw2 , 0), while q̂2 < 0 and ẑ < 0 implies that the solution is ( pw1 , 0, 0).
Finally, q̂1 < 0 and q̂2 < 0 implies that the household spends all of its money on the outside
good.
The second possibility occurs when one of the terms in the brackets is negative. When
this occurs, I say that one good “dominates” the other. While this is nearly always a
redundant condition when there are only two inside goods, it occurs with greater frequency
when more goods are considered. From a household’s perspective, one good is dominated by
another if all of it’s desirable characteristics can be obtained more cheaply from some other
good (or combination of goods). Note that because of the idiosyncratic shocks (the ε’s) this
will vary from household to household. There are two ways for good one to be dominated
by good two. If good one’s principal9 characteristic is bad (has a negative β), then unless
good one is a cheaper source of the other characteristic than good two it will have zero
demand. Similarly, if good one’s non-principal characteristic is good (has a positive β) then
unless good one is a cheaper source for it’s principal characteristic than good two it will
have zero demand. Mathematically, if one good dominates the other, we will have either
or10
(p1 − ε1 )a22 − (p2 − ε2 )a12
<0
β1 ρ1 (a11 a22 − a21 a12 )
(2.21)
(p2 − ε2 )a11 − (p1 − ε1 )a21
<0
β2 ρ2 (a11 a22 − a21 a12 )
(2.22)
The signs of the determinant of A, and the β’s determine which inequality implies dominance
of which good (this is discussed in more detail in the Appendix).
This condition generally remains in the background, but when the products move closer
together in the characteristics space, this becomes more of a possibility. For example,
Figure 2.7 illustrates what happens when the two products are nearly perfect substitutes:
A=
"
1
0.999
0.999
1
#
. In this case, good one will dominate good two if: 0.999(p1 − ε1 ) < p2 − ε2
and good two will dominate good one if 0.999(p2 − ε2 ) < p1 − ε1 . Obviously, one good or
9
In the two-good/two-characteristic case, if det A > 0, good one’s principal characteristic is the first
characteristic. If det A < 0, good one’s principal characteristic is the second characteristic. In the three
good case, the intuition is less clear, but the mathematics are similar.
10
It is not possible for both inequalities to hold simultaneously.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
47
the other is dominant except in a very small range of prices.
Figure 2.7: Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
1
0.999
,
β
=
5.5,
β
=
5,
ρ1 = ρ2 = 0.5, p2 = $0.50, w = 60. Compare to Figure 2.1.
1
2
0.999
1
q (p | p =$0.50)
1
q (p | p =$0.50)
2
2
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Price of Good 1
Price of Good 1
1
1
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0
5
10
q1
15
2
0.5
0.3
0
1
0
0
2
4
q2
6
8
The feature of the model that makes it possible for one product to dominate the other
is the same feature that reduces the dimensionality of the parameter space: assuming that
every product is completely represented by a set of many observable characteristics and one
unobservable characteristic. These features are also apparent in Figures 2.29 and 2.30. In
sharp contrast to Figure 2.4, increasing β1 has essentially no effect on the reservation price
of good one. This is because, as the price of good one increases, households are substituting
not to the outside good, as they were when the A matrix was diagonal, but rather they
are substituting to good two. And since an increase in β1 affects both good one and good
two nearly equally, we see no effect on the reservation price of good one. As noted earlier,
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
48
where we do see a large effect, in both Figures 2.29 and 2.30 is in the quantity demanded,
particularly when prices are low.
2.4.2
Engel Curves
In addition to allowing corners in the inside goods, the model also allows corner solutions
in the outside good - in some cases the household may not desire to consume any outside
good. In this case, we will have ẑ ≤ 0 in equation 2.5 (and therefore z = 0, due to
the non-negativity constraint). Because the household’s utility from the inside goods is
strictly concave, this situation is dependent on the household’s expenditure level. Once the
household’s budget reaches the point where it purchases positive amounts of the outside
good, further increases in the household’s budget will not result in any increase in their
purchases of the inside goods. This subsection explores this case in more detail by examining
the Engel curves generated by the model.11
The four main characteristics of these Engel curves are as follows. First, they contain
many corners: in many cases, the marginal utility households receive from one good is
strictly greater than that from another over some range of expenditure levels. In the twogood case, this means that many of the Engel curves begin (and may even end) by following
one of the axes. Second, relating to these corners, the Engel curves tend to be piecewise
linear (or nearly linear). The third main characteristic of these Engel curves is that they
terminate. Unlike many other models, the households in my model always reach a satiation
point with respect to each of the interior goods. Beyond this point, any additional income
is spent on the outside good. Finally, the Engel curves in this model may be negatively
sloped for some range of expenditure levels. This behavior only occurs when the A matrix is
non-diagonal, and reflects the fact that as households’ expenditure levels get high enough to
allow them to purchase both goods, the second good may provide some of the characteristics
formerly obtained through the first good – leading the household to consume less of the
first good. This subsection highlights these characteristics, as well as illustrating the effects
on the Engel curves of changes in the parameters and the product characteristics matrix.
In contrast to the discussion of demand curves in the previous subsection, in which I
was able to provide closed-form solutions for many cases, when ẑ < 0 there is no general
11
One feature of these graphs that may require explanation is that at the end of the Engel curves, the
symbols are sometimes closer together. This is due to the fact that the household reaches it’s satiation point
for the inside goods somewhere during the last incremental increase in w.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
49
closed-form solution, making interpretation of the Engel curves more difficult. When ẑ < 0,
q̂1 > 0 and q̂2 > 0 the solutions are
z = 0, q2 =
w − p1 q 1
and q1 = q1∗
p2
(2.23)
where q1∗ is the solution to
ρ1 −1
w − p1 q 1
(p2 a11 β1 ρ1 − p1 a21 β1 ρ1 ) a11 q1 + a21
+1
+ p2 ε 1 =
p2
ρ2 −1
w − p1 q 1
+1
(p1 a22 β2 ρ2 − p2 a12 β2 ρ2 ) a12 q1 + a22
+ p1 ε 2
p2
(2.24)
(2.25)
Figure 2.8 plots the Engel curves that these equations generate for an identical set of
parameters to those in Figure 2.1. Here we have returned to the case where A =
"
1
0.5
0.5
1
#
.
Each pane of the graph represents a price pair, (p1 , p2 ), and each curve within the panes
represents a (ε1 , ε2 ) pair. The parameters are the same in this Figure as in Figure 2.1. I
chose to display these particular percentiles to provide substantial variation in the choices
while not allowing the variation from one percentile to completely dominate the other in
the figures.
Table 2.2: Pairs of (ε1 , ε2 ) corresponding to the demand curves in Figures 2.1-2.30
Curve
ε1
ε2
#
−1.49 −1.82
M
−1.49 −1.49
−1.82 −1.82
O
−1.82 −1.49
Table 2.2 lists the symbols that correspond with (ε1 , ε2 ) pairs in Figures 2.8-2.13. These
symbols serve both to mark the curves, and to denote the expenditure levels associated
with those points on the Engel curves. Although they vary slightly from one Figure to
the other, each symbol in Figure 2.8 reflects an increase in the expenditure level of $0.20.
Hence, in Figure 2.8, the Engel curves only continue from an expenditure level of zero to
(at most) an expenditure level of $2.20. After $2.20 (in Northeast pane), the household
becomes satiated with respect to the interior goods, and does not increase its expenditure
on the inside goods, regardless of the total expenditure level.
Looking at Figure 2.8, both the Northeast and Southwest panes have similar patterns.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
50
Figure 2.8: Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th
percentiles
of ε1 and ε2 . Each symbol
1 0.5
represents an increase of $0.20 in w. A = 0.5
,
β
=
5.5,
β2 = 5, ρ1 = ρ2 = 0.5.
1
1
P1=0.25, P2=0.5
P1=0.5, P2=0.5
4
4
3
3
q2
5
q2
5
2
2
1
1
0
0
1
2
3
4
0
5
0
1
2
3
q1
q1
P1=0.25, P2=0.25
P1=0.5, P2=0.25
4
4
3
3
5
4
5
q2
5
q2
5
4
2
2
1
1
0
0
1
2
3
q1
4
5
0
0
1
2
3
q1
The O pair of ε’s leads that household to consume only good two, while the # pair leads
that household to consume only good one. Because p1 = p2 in both of these panes, the
curves for pairs M and , which both have ε1 = ε2 , share the same path. The curve for M
goes further however, because the ε’s are closer to zero in this case, and therefore lead the
household to consume more of both goods before switching to the outside good.
In the Northwest pane, only the O curve results in purchases of good two. It may seem
surprising that the Engel curve in this pane has a negative slope for part of its length. This is
due to the fact that initially characteristic one offers the highest marginal utility, which leads
the household to consume only good one, which is the better source of characteristic one.
Eventually, however, characteristic two offers higher marginal utility than one. However,
since good two contains characteristic one, as well as characteristic two, the household is
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
51
able to cut back on its consumption of good one, leading to the observed negative slope.
Also, the Northwest and Southeast panes are similar, although reflected about the 45◦
line. The reason that the non-corner Engel curve terminates in a larger value of q1 in the
Southeast than it does of q2 in the Northwest is that β1 is slightly greater than β2 .
Figure 2.9: Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th
of ε1 and ε2 . Each symbol
percentiles
1 0.5
represents an increase of $0.20 in w. A = 0.5 1 , β1 = β2 = 5, ρ1 = ρ2 = 0.5.
P1=0.25, P2=0.5
P1=0.5, P2=0.5
3
3
q2
4
q2
4
2
2
1
1
0
0
1
2
3
0
4
0
1
2
3
q1
q1
P1=0.25, P2=0.25
P1=0.5, P2=0.25
3
3
q2
4
q2
4
4
2
2
1
1
0
0
1
2
3
4
0
0
1
2
q1
3
4
q1
Figure 2.9 shows that this asymmetry disappears when both β1 = β2 and ρ1 = ρ2 .
The Figure considers the effect on the Engel curves from a slight decrease in β1 from 5.5
to 5.0, holding all else constant. Now the Northwest and Southeast panes are completely
symmetric. Additionally, there is a slight decrease in the satiation points for both goods,
although good one is affected more, since it contains more of the first characteristic.
The Engel curves change significantly when we make the A matrix more diagonal, as
seen in Figure 2.10, which plots the Engel curves when A =
"
1 0
0 1
#
. Notice that unlike the
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
52
Figure 2.10: Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th
percentiles
of ε1 and ε2 . Each symbol
represents an increase of $0.10 in w. A = 10 01 , β1 = 5.5, β2 = 5, ρ1 = ρ2 = 0.5.
1
0.8
0.8
0.6
0.6
q2
1
0.4
0.4
0.2
0.2
0
q2
P1=0.5, P2=0.5
0
0.5
0
1
1
P1=0.25, P2=0.25
P1=0.5, P2=0.25
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.5
q1
1
0
0
q1
q2
q2
P1=0.25, P2=0.5
0.5
1
0
0
0.5
q1
1
q1
three previous sets of curves, none of the cures in this graph show corner solutions in either
inside good at the satiation point. The result of this is fewer overlapping curves. This
change is due to the fact that neither product can substitute for the other in characteristic
space. Good one is now the only source of characteristic one, and good two is the only
source of characteristic two. These changes aside, there are similarities. The Northeast and
Southwest panes are still very similar to each other, with the lower prices in the Southwest
pane leading to higher levels of consumption before satiation.
Notice as well that in all four panes, the prices and ε’s of good two have no effect on
the satiation point of good one, and vice-versa. This can be seen in the fact that the O and
the have the same value of q1 at the satiation point, and that the M and the # also have
the same value of q1 at the satiation point. This independence is due to the fact that the
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
53
products have no overlapping characteristics, and is true for all the Engel curves with this
A matrix.
Figure 2.11: Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th percentiles
of ε1 and ε2 . Each symbol
represents an increase of $0.10 in w. A = 10 01 , β1 = β2 = 5, ρ1 = ρ2 = 0.5.
1
0.8
0.8
0.6
0.6
q2
1
0.4
0.4
0.2
0.2
0
q2
P1=0.5, P2=0.5
0
0.2
0.4
0.6
0.8
0
1
0.6
P1=0.5, P2=0.25
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.2
0.4
P1=0.25, P2=0.25
1
0
0.2
q1
1
0
0
q1
q2
q2
P1=0.25, P2=0.5
0.4
0.6
0.8
1
q1
0
0
0.2
0.4
0.6
0.8
1
0.8
1
q1
Figures 2.11 and 2.12 are very similar to Figure 2.10. The difference is that in Figure
2.11, β1 has been decreased from 5.5 to 5.0, and in Figure 2.12 ρ1 has been decreased from
0.05 to 0.04, while holding β1 ρ1 = 2.5 constant.
The main effect of decreasing β in Figure 2.11, is to decrease the satiation point for
good one in all four panes. Because good two does not contain any of characteristic one,
however, the satiation points for good two remain unchanged. A second effect of the change
in β1 is that now we have β1 = β2 and ρ1 = ρ2 . This means that when ε1 = ε2 , as in the
Northeast and Southwest panes for the M and pairs, the Engel curve moves along the 45◦
line.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
54
Figure 2.12: Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th percentiles of ε1 and ε2 . Each symbol
represents an increase of $0.10 in w. A = 10 01 , β1 = 6.875, β2 = 5, ρ1 = 0.4, ρ2 = 0.5.
P1=0.5, P2=0.5
1
1
0.8
0.8
0.6
0.6
q2
q2
P1=0.25, P2=0.5
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
q1
0.8
0
1
0
0.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
q1
0.8
0.6
q1
0.8
1
P1=0.5, P2=0.25
q2
q2
P1=0.25, P2=0.25
0.4
1
0
0
0.2
0.4
0.6
q1
0.8
1
Finally, in Figure 2.13, I show the effects of a dramatically different A matrix. In this
case, A =
"
1
0.999
0.999
1
#
, making the products nearly perfect substitutes. The Northwest and
Southeast panes are most representative of actual choice behavior in this situation. The
Northeast and Southwest panes show knife-edge behavior for the M and pairs. Because
the products have nearly identical characteristics, the β’s and ρ’s make little difference in
distinguishing between goods one and two. Here, ε1 and ε2 have the largest effect on which
good is purchased. Except in the rare cases (as shown here) when p1 − ε1 = p2 − ε2 , only
one of the two goods will be purchased.
To recap, this subsection has illustrated the behavior of the Engel curves generated by
the model. These curves have several features that differentiate them from many other
Engel curves. First, as shown throughout this subsection, all of the Engel curves terminate,
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
55
Figure 2.13: Demand for good one and two as a function of expenditure level, w, for four
different values of p1 and p2 , for 40th and 60th percentiles
of ε1 and ε2 . Each symbol
1
0.999
represents an increase of $0.40 in w. A = 0.999
,
β
=
5.5, β2 = 5, ρ1 = ρ2 = 0.5.
1
1
6
5
5
4
4
q2
6
3
3
2
2
1
1
0
q2
P1=0.5, P2=0.5
7
0
2
4
0
6
4
P1=0.25, P2=0.25
P1=0.5, P2=0.25
7
6
6
5
5
4
4
3
3
2
2
1
1
0
2
q1
7
0
0
q1
q2
q2
P1=0.25, P2=0.5
7
2
4
q1
6
0
0
2
4
6
6
q1
signifying satiation in the inside goods. Second, the curves tend to be primarily piece-wise
linear, often sharply changing direction once a certain expenditure level is reached. This
behavior is closely related to the many corners found in the model, and is generated by
the fact that I have assumed that households’ utility is derived primarily from the product
characteristics, rather than the products themselves. Finally, although changes in β and ρ
do have significant effects on the Engel curves, much of the variation we see is generated
by changes in the product characteristics matrix. The next subsection discusses this fact in
more detail.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
2.4.3
56
The A Matrix and Some Additional Remarks
As the previous two subsections have illustrated, the structure of the A matrix can have a
large impact on the choice behavior that the model predicts. In this subsection, therefore, I
summarize the restrictions and recommendations for A matrices in empirical applications.
First, all of the elements of A must be non-negative. Essentially, the elements of A
should be thought of as a ranking. For example, due to the fact that all errors are negative,
a good with all zeros will never be purchased.
Second, while it is possible for A to have more rows than columns, if this is the case
then A will have greater row rank than column rank. This means that one good can
be synthesized by one or more other goods. Therefore only one of these two (possibly
composite) goods will be chosen (whichever has the higher value for Aj1 /(pj − εj )). The
number of un-dominated products that it can contain (i.e., the maximum number of different
products that will be purchased on a single purchase occasion) is thus limited to the number
of columns.
Third, in estimation, it is a good idea to allow for more characteristics than products.
Consider the two-good case. In the two-good case, it’s not possible to have negative crossprice elasticities. The closest my model can come to complements is having cross-price
elasticities of zero. In the two-good two-characteristic
case, the cross-price elasticities are
#
"
1 0
. Similarly, the closest the model can come
zero (at interior solutions) when A =
0 1
to a world with perfect substitutes is when the two products have" identical
# characteristics.
1 1
. In this case, we
Again, in the two-good, two characteristic case, this requires A =
1 1
will never observe households purchasing both goods simultaneously. In order to allow the
parameters of the model to determine the probabilities that the household buys both goods
together it is necessary for the econometrician to choose an
" A matrix
# that accommodates
1 1 0
both extremes. One such A matrix in this case is: A =
. In this case, as β1
1 0 1
approaches zero, the cross-price elasticities will approach zero. Conversely when β2 and β3
approach zero, each household has a discontinuity point where they are perfectly elastic
between good one and good two (this occurs when p1 − ε1 = p2 − ε2 ).
Fourth, as we have seen throughout this section, the specification of the A matrix
greatly influences the possible choice behavior of the model. Unlike many other models, as
currently formulated, even the scale chosen for measuring each characteristic is important
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
57
in determining some of the potential quantity behavior. For example, if we scale all of the
the elements of a particular characteristic of A by a positive scalar λ, we have:
β1 (a11 q1 + a21 q2 + 1)ρ1 =
β1
β1
(λa11 q1 + λa21 q2 + λ)ρ1 6= ρ1 (λa11 q1 + λa21 q2 + 1)ρ1
λ ρ1
λ
Note that the λ is not simply absorbed by the β. The size of λ affects the closeness of
the corner. The effect is that as the elements of A increase in magnitude for a particular
characteristic, we are less likely to see corner solutions for that characteristic. In principle,
this problem could be eliminated by parameterizing the distance from the corner, and substituting a characteristic-specific parameter for each “1”. I do not do so here because, like
Kim et al. (2002), I lack sufficient variation in my data. In practice, it may be partially
parameterized by comparing results from several alternative scalings. If the predicted purchases for a particular product are frequently very close to, but slightly greater than zero,
this indicates that the additive term (the “1”) for the characteristic most closely identified
with that product (the characteristic for which that product is the cheapest source) is too
small relative to the the scale of that characteristic. Hence, the scale of that characteristic
should be decreased.
Finally, I have assumed that households’ decisions are static, and that they do not
condition current purchases on expectations of future prices (or past purchases). Within this
context, although the A matrix above is treated as time invariant throughout this chapter,
in the next chapter I include feature and display as time-varying characteristics, with the
matrix A varying from week to week. Implicit in using these time-varying characteristics is
the assumption that households do not condition current purchases on their expectations
of future changes in the characteristics matrix.
Before proceeding further, I emphasize several significant assumptions, normalizations,
and operational differences from existing models that are worth highlighting and explaining
in more detail. These are:
• I normalize ρ > 0. This is because it is very difficult to distinguish the choice behavior
produced by negative ρ’s from positive ones. It is also theoretically unappealing to
have the derivatives of the utility function alternate signs.
• As derived in the Appendix, in the case with two inside goods and two product
characteristics the utility function is concave (assuming det A > 0 and ρ > 0) if and
only if βc (ρc − 1) ≤ 0 ∀c ∈ C. When there are more than two goods or characteristics,
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
58
this is a sufficient, but not necessary condition.
• Although the model extends the existing literature by allowing households to have
preferences for variety, the model does not currently allow these preferences for variety
to be time-varying. For example, this could be violated if households occasionally had
a party. This could be modeled by introducing correlation across the εj ’s within a
purchase occasion. The reason I assume that the εj ’s are i.i.d. both across goods
and across time is for computational tractability (doing so would further increase the
number of parameters to estimate). In future work, this distribution could be easily
parameterized.
In addition to these, there are several other key differences between this model and most
other demand models. First, as noted previously, unlike much of the existing literature,
my model uses an explicit budget constraint. This is a necessary feature in modeling
bundles. There must be some way in which the household’s choice set is restricted from
considering an infinite number of potential bundles. Because the logit model does not
include a budget constraint, small departures from it on this front would place positive
probability on purchasing items at near-infinite prices.
2.5
Choice Behavior Generated by the Model
Section 2.4 showed that the model can generate a wide variety of choice behavior. The
purpose of this section is to show that the model has a large amount of freedom in allocating probability to these different outcomes. The characteristics matrix does place a
large amount of structure on the “shape” of the error space decomposition, but within this
structure the parameters are able to “stretch” the divisions in several directions. This is
in striking contrast to the logit model, which can only reposition the the way in which the
error space is divided.
The logit model requires the econometrician to specify the functional form for the mean
indirect utilities from each choice, V0 , V1 , and V2 . Frequently, the specification Vi = βXi −
αpi is used, where Xi is a vector of characteristics of product i, pi is the price of good i,
and β and α are parameters. The price and characteristics of the outside good are usually
normalized to zero, and hence V0 = 0.
The logit model roughly analogous to the example with two inside goods12 is one with
12
The model is only roughly analogous because, in addition to the reasons mentioned above, this particular
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
59
two inside goods and one outside good. The household receives three marginal utility shocks
(one for each good) that are unobserved by the econometrician, and that are assumed to be
distributed i.i.d. according to a Gumbel distribution. Hence, if V 1, V 2, and V 0 (normalized
to zero,) are the mean indirect utilities, the household chooses among the three goods to
maximize:
max {V 0i = ε0i , V 1i = V 1 + ε1i , V 2i = V 2 + ε2i }
(2.26)
Figure 2.14: Graph of logit choices as a function of error terms
Buy Outside Good
1
ε0
−V1
Buy Good 2
Buy Good 1
0
−V2
ε1
ε2
−V2+1
−V1+1
Figure 2.14 graphs these logit choices as a function of the logit error terms. To understand the graph, it is helpful to think of the shape depicted as a paper airplane descending
at a 45◦ angle into the page. The arrows along the axes indicate the directions of increase
for ε0 , ε1 , and ε2 . Although this orientation is non-standard, I believe it shows the graph
more clearly. V 1 and V 2 are the mean indirect utilities that the household receives from
choosing good one or good two respectively. V 0, the mean utility from the outside good is
normalized to zero, and is not shown. The paper airplane shape divides the graph into three
regions. The top region represents the error space that results in the household choosing
the outside good (i.e, good 0). The lower right region is the region of the error space that
specification justifies the inclusion of the αpi term by assuming that there is a second outside good, with
price normalized to one, on which the household spends any remaining money.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
60
results in the household choosing good 2, while the lower left represents a choice of good
1. The fuselage of the “plane”, where all three planes intersect, goes through the point
(−V 1, −V 2, 0). Recall that in the logit model, the ε’s are i.i.d. and follow an extreme value
(Gumbel) distribution. This distribution is very similar to the standard normal distribution, but has thicker tails. The important point is that there are only two parameters, V 1
and V 2 that completely determine how the error space affects the choice outcomes. The
probability of the occurrence of any particular point in this error space is given by
−ε0 −e−ε1 −e−ε2
P (ε0 , ε1 , ε2 ) = e−ε0 −ε1 −ε2 e−e
To illustrate the richer error structure provided by my model, the remainder of this
section shows how the different parts of the model affect the patterns of choice behavior
that it generates. I look at several different configurations of the A matrix (specified by
the econometrician and the data), including the extreme examples of diagonal, square, and
triangular, as well as several intermediate cases. For each of these configurations, I look
at the effects of changing the parameters, scaling the A matrix, and a binding budget
constraint.
Figure 2.15 shows the distribution of choices resulting from the error structure in my
model when the A matrix is diagonal.13 The parameters used to generate the figure are
β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 30, and A =
"
1 0
0 1
#
. Because my model is
continuous, rather than discrete, I have discretized the choices into the four regions shown in
Figure 2.15 in order to allow for better comparison. The regions in this graph (and most of
the rest of the graphs in this section) are color-coded as follows: The dark rectangular region
in the Northeast corner is the region of the error space in which neither good one nor good
two is purchased. The light rectangle in the Southwest corner represents the region where
both goods are purchased. The two rectangles in the Northwest and Southeast represent
regions where only one of the two goods is purchased.
For reference, Figure 2.16 graphs the probability distribution of εj . Recall that the εj ’s
are assumed to be i.i.d. across products, purchase-occasions, and households.
The probabilities of each region are given in the Figure’s description.14 These probabilities were obtained by simulating ε draws. Although the regions in Figure 2.15 are
rectangular, and allow for a relatively simple analytical transformation, in general this will
13
14
Due to rounding, probabilities may not sum to one.
In each case, the probabilities have been rounded, and hence, may not sum to one.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
61
Figure 2.15: Graph of purchases (broken into groups) as a function of error terms. A =
1 0
0 1
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = $0.30, w = $30. The probabilities of the
four regions are:
0.21 0.09
0.49 0.21














Don’t Buy
Good Two 
















Buy Some of
Good Two 


|
{z
}
|
Buy Some of
Good One
{z
}
Don’t Buy
Good One
not be the case. As we will see in subsequent graphs, the regions are often very irregular.
For this reason, simulation methods are necessary.
Figure 2.17 illustrates the effect of the budget constraint. The majority of the graph is
identical to Figure 2.15. The only differences between the figures occur in the areas where
either ε1 or ε2 are small in magnitude. In these areas, the regions that are slightly lighter
represent the areas of the error space where the budget constraint is binding. That is,
regions of the graph in which z = 0. Except where noted, the probability of these regions
in the other graphs in this section are near-zero.
The only effect of the budget constraint on the propensity to purchase the two goods
can be seen by noting that there are now 45◦ lines between the point of intersection of
the purchase boundaries and the axes. The slope of the lines comes from the ratio of the
prices of the two goods. This is because when the budget constraint is binding on the
interior goods, no outside good is purchased (z = 0). Therefore, assuming that neither
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
62
Figure 2.16: Negative Lognormal Probability Distribution of εj . Mean=e0.5 ,
Variance=e2 − e, Mode=e−1 , Median=1. ln(−εj ) ∼ N (0, 1).
0.7
0.6
Probability Density
0.5
0.4
0.3
0.2
0.1
0
0
−0.5
−1
−1.5
−2
−2.5
ε
−3
−3.5
−4
−4.5
−5
good dominates the other, Equation 2.24 gives us:
ρ1 −1
w − p1 q 1
p2 ε1 − p1 ε2 = − (p2 a11 β1 ρ1 − p1 a21 β1 ρ1 ) a11 q1 + a21
+1
p2
ρ2 −1
w − p1 q 1
(p1 a22 β2 ρ2 − p2 a12 β2 ρ2 ) a12 q1 + a22
+1
p2
(2.27)
(2.28)
Hence, the relative amounts of q1 and q2 purchased are determined by the weighted sum
p2 ε1 − p1 ε2 of the error terms. This leads to the linear transitions seen in the Southwest
corner of the Figure.
The effects of increasing the parameters β1 and ρ1 are shown in Figures 2.18 and 2.31.
Both β and ρ have similar effects on the household’s propensity to purchase. Higher values
of either β1 or ρ1 mean that good one is purchased even at more negative realizations of ε1 .
However, as discussed in section 2.4.1, ρ has a greater impact than β on the amount that
the household purchases, conditional on purchase.
The effects of increasing the scale of the A matrix can be seen in Figure 2.19, which
shows that the propensity to purchase can be held roughly constant by adjusting β to
compensate for changes in the scale of the A matrix. Here I multiply all of the elements of
A by 10, while dividing β by 10. The propensity to purchase shown in this figure is nearly
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
63
Figure 2.17: Graph of purchases (broken into groups) as a function of error terms. A =
1 0
0 1
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 1. The probabilities of the seven
regions are:



0.09 0.13 0.09
0.14 0.13 
0.27
0.09
identical to that shown in Figure 2.15. As noted in subsection 2.4.3, while the propensity
to purchase remains the same, the amounts purchased are affected by the transformation.
Thus far in this section, we have considered only the case when A is a diagonal matrix
– each good has its own characteristic. Looking back, one can see that in all of the graphs
thus far, ε1 does not affect the probability of purchasing good two, and vice versa. The next
few figures illustrate the changes in choice behavior as the A matrix becomes non-diagonal.
As we move to a progressively more non-diagonal matrices, such as in Figures 2.20 and 2.32
(in the Appendix), we begin to see interaction effects. For values of ε1 or ε2 below roughly
-2.5, Figures 2.32 and 2.15 are nearly identical. The only difference in these regions is
that the boundary lines, where the household is indifferent between buying and not buying,
have shifted outwards very slightly. For values of ε closer to zero however, it is clear that
even a relatively small amount of the off diagonal characteristics can significantly alter the
outcome space. Realizations of ε1 that are close to zero lead to large purchases of good one.
Because good one now contains some of the second characteristic, this leads to a decrease
in the demand for good two. The degree of curvature in the Southwest portion of the graph
is determined by setting qb1 = 0 in Equation 2.3, or qb2 = 0 in Equation 2.4. This effect is
even more marked in Figure 2.33, which moves A even further from a diagonal form.
Figure 2.34 is analogous to Figure 2.17, but with a different A matrix. It shows the
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
64
Figure 2.18: Graph of purchases as a function of error terms. A = 10 01
β1 = 7, β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3,
w = 30.
The probabilities of the four regions
0.05
are: 0.24
0.57 0.13
effects of a binding budget constraint on choice behavior. As discussed earlier (see Equation
2.27), there are 45◦ lines between the points of intersection of the purchase boundary and
the axes, representing the boundary between purchasing only one inside good and both
inside goods.
As was the case in Figures 2.18 and 2.31, increasing β1 or ρ1 have the effect of stretching
the graph in the horizontal direction. That is, ε1 can be more negative, and we will still
observe households buying good one.
Figure 2.35 shows the effect of scaling the A matrix. As was the case with Figure 2.19,
we can achieve very similar cutoffs in the boundaries between buying and not buying by
decreasing the β’s to compensate for increasing the scale of the A matrix.
Having considered the cases of diagonal and square A matrices, we turn our attention to
the transition from diagonal to progressively more triangular characteristics matrix. Figures
2.22 and 2.36 show how this progression affects choice behavior. Notice that unlike in Figure
2.20, where both purchase boundaries in the Southwest corner of the graph were curved,
here we only have curvature in the transition from buying to not buying good two. This
comes directly from the fact that when a21 = 0, the boundary equation from Equation 2.3
is linear in ε1 and ε2 , while the boundary equation from setting qb2 = 0 in Equation 2.4 is
nonlinear in ε1 and ε2 .
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
65
Figure 2.19: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
10 0
0 10
, β1 = β2 = 0.5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 30. The
probabilities of the four regions are:
0.21 0.09
0.49 0.21
As the figures throughout this section have shown, my model is able to flexibly assign
probabilities to different outcomes. Although the structure of the A matrix has a significant
effect on the choice probabilities and their variation, in general the parameters β and ρ are
accommodate observed purchase probabilities such as those shown in Table 1.1.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
66
Figure 2.20: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
1 0.1
0.1 1
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 30. The
probabilities of the four regions are:
0.28 0.07
0.38 0.27
Figure 2.21: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
1 0.5
0 1
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 30. The
probabilities of the four regions are:
0.57 0.05
0.24 0.15
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
67
Figure 2.22: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
1 1
0 1
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 30.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
2.6
68
The Costs of Misspecification
Although the previous sections have shown the additional flexibility of the choice model
developed in this chapter, given the added complexity in estimating the model, it seems
worth investigating whether the effort to achieve additional flexibility is worthwhile. For
example, if the true data generating process is that described in this chapter, how far off
would we be if we estimated demand using a traditional logit model? In particular, how
poorly does the logit perform in making the calculations described in chapter one (and those
I plan to make in chapter three) – the profit difference between uniform and non-uniform
prices. As the Monte Carlo results in this section show, depending on the parameters and
the A matrix, the answers given by the logit may be significantly off.
Throughout this section, I consider a model with four inside goods and one outside
good. Although my model can handle more product characteristics than products (an A
matrix that has more columns than rows), here I use a square A matrix to avoid tilting the
results in favor of my model.
For the Monte Carlo results in this section, I have made the following assumptions:
1. I assume that 5000 shoppers visited the store each week, and that their budgets were
distributed normally, with a mean of $30, and a standard deviation of $20. This
reflects the actual distribution of budgets in my data. The assumption about the
number of shoppers is essentially a normalization, and affects only the amount of
simulation error introduced.
2. I consider several different parameter combinations and also show the effects of changes
in β and ρ.
3. I consider two potential product characteristic spaces, one in which each characteristic
is product-specific, and another more representative of the soft drink category, which
is the focus of the following chapter.
I consider several potential sets of parameters and product characteristics. In each case, I
proceed as follows. First, I draw one set of ε’s for each household. Then for each set of ε’s, at
a particular set of prices (more on this in a moment), I solve for the optimal consumption
bundle for each household. After solving for each household’s demand, I aggregate the
realized demand, and optimize over the price space, first allowing for non-uniform prices,
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
69
and then restricting the prices to be uniform. This gives me the true optimal uniform and
non-uniform prices.
In order to illustrate how the relationship between the logit predictions and the true
model depends on the marginal costs of the products, I calculate prices and resulting profit
differences under 34 = 81 different combinations of marginal costs. Doing so also shows
how changes in the products marginal costs translate to changes in the profit difference
between uniform and non-uniform prices. Each figure uses the same sequence of marginal
costs, numbered 1 through 81. The cost configurations used for each week are shown in
Table 2.3.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
c1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
c2
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
Table 2.3: Marginal Cost Configurations
c3
c4
c1
c2
c3
c4
c1
0.1 0.1 28 0.2 0.1 0.1 0.1 55 0.3
0.1 0.2 29 0.2 0.1 0.1 0.2 56 0.3
0.1 0.3 30 0.2 0.1 0.1 0.3 57 0.3
0.2 0.1 31 0.2 0.1 0.2 0.1 58 0.3
0.2 0.2 32 0.2 0.1 0.2 0.2 59 0.3
0.2 0.3 33 0.2 0.1 0.2 0.3 60 0.3
0.3 0.1 34 0.2 0.1 0.3 0.1 61 0.3
0.3 0.2 35 0.2 0.1 0.3 0.2 62 0.3
0.3 0.3 36 0.2 0.1 0.3 0.3 63 0.3
0.1 0.1 37 0.2 0.2 0.1 0.1 64 0.3
0.1 0.2 38 0.2 0.2 0.1 0.2 65 0.3
0.1 0.3 39 0.2 0.2 0.1 0.3 66 0.3
0.2 0.1 40 0.2 0.2 0.2 0.1 67 0.3
0.2 0.2 41 0.2 0.2 0.2 0.2 68 0.3
0.2 0.3 42 0.2 0.2 0.2 0.3 69 0.3
0.3 0.1 43 0.2 0.2 0.3 0.1 70 0.3
0.3 0.2 44 0.2 0.2 0.3 0.2 71 0.3
0.3 0.3 45 0.2 0.2 0.3 0.3 72 0.3
0.1 0.1 46 0.2 0.3 0.1 0.1 73 0.3
0.1 0.2 47 0.2 0.3 0.1 0.2 74 0.3
0.1 0.3 48 0.2 0.3 0.1 0.3 75 0.3
0.2 0.1 49 0.2 0.3 0.2 0.1 76 0.3
0.2 0.2 50 0.2 0.3 0.2 0.2 77 0.3
0.2 0.3 51 0.2 0.3 0.2 0.3 78 0.3
0.3 0.1 52 0.2 0.3 0.3 0.1 79 0.3
0.3 0.2 53 0.2 0.3 0.3 0.2 80 0.3
0.3 0.3 54 0.2 0.3 0.3 0.3 81 0.3
c2
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
c3
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
c4
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
70
Using the 81 “weeks” of data generated by the different marginal cost configurations, I
estimated a homogenous logit model of demand. The homogenous logit model is particularly
convenient in this case, because it yields the estimating equation:
ln(sjt ) − ln(s0t ) = βXj − αpjt + ηjt
(2.29)
where sjt is the share of good j in “week” t, and s0t is the share of the outside good in that
week. As is well known, the econometrician is required to specify the size of the market - and
hence the share of the outside good in each week. After experimenting with different values,
I found that in general, the logit profit difference predictions did not change significantly
for a wide range of market size definitions.
I consider two product characteristic possibilities. First, I consider the case when each
product is identified by a product-specific characteristic. The example given in Table 2.4
is one where each of the four products: Coke, Diet Coke, Pepsi, and Diet Pepsi have no
demand-relevant characteristics in common. Specifying the A matrix requires two steps.
The first step is to assume its structure, as I have done. The second step is to assume
the scale of each of its columns. Here, I set each of the entries either “1” or “0”. It
is important to stress that this choice is not a normalization. Choosing the scale of each
product characteristic is equivalent to choosing the scale of the additive constant (the “+1”)
in each of the households’ sub-utility functions. This choice affects the probability that
households are at a corner for that characteristic. In future work, I hope to estimate these
scales (or equivalently, parameterize the “+1”), but for now I assume the scale.
Products:
Coke
Diet Coke
Pepsi
Diet Pepsi
Table 2.4: Example Product Universe Number One
Characteristics:

Coke Diet Coke Pepsi Diet Pepsi
1 0 0 0
 0 1 0 0
X
0
0
0
A=
 0 0 1 0
0
X
0
0
0 0 0 1
0
0
X
0
0
0
0
X
→




This product characteristic structure ensures that the products are poor substitutes for
one another. Each product is the sole source of its characteristic – there is no overlapping.
Hence, this case reflects a world in which households have a greater preference for variety.
Figure 2.23 plots the percentage by which the expected profits from optimal non-uniform
prices are lower than the expected profits from optimal uniform prices (when I restrict
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
71
p1 = p3 and p2 = p4 ). The vertical axis in the figure measures:
ΠN on−U nif orm − ΠU nif orm
ΠN on−U nif orm
where
ΠN on−U nif orm = max{p1 ,p2 } Q1 (p)(p1 − c1 ) + Q2 (p)(p2 − c2 ) + Q3 (p)(p3 − c3 ) + Q4 (p)(p4 − c4 )
and
ΠU nif orm = max
{p1 ,p2 }
Q1 (p)(p1 − c1 ) + Q2 (p)(p2 − c2 ) + Q3 (p)(p1 − c3 ) + Q4 (p)(p2 − c4 )
For each marginal cost configuration (“week”) I plot both the “true” loss, which assumes
uses my model is the true model and uses it to generate the Q’s in the equations above, as
well as the loss predicted by the logit model, which uses the parameters estimated from the
fake data to generate the Q’s.
In general, the “actual” profit differences, (i.e. assuming that my model is the data
generating process,) are typically 5% or less, with no week higher than 20%. The lost
profits are greater when the marginal costs of the products grouped are farther apart. The
logit estimates correlate with this, but predict much larger differences across the board.
Weeks 3, 6, 9, 30, 33, 36, 57, 60, and 63 show the largest difference in expected profits
when pricing uniformly. These weeks correspond to the weeks with the greatest differences
between the costs of goods two and four.
As seen in the figure, the logit model’s predictions deviate significantly from the truth,
particularly in weeks 9, 36, and 63. As mentioned above, these are weeks when the actual
marginal costs of the products within the uniform clusters are significantly different, and
the true profit difference is greater. However, the logit does not consistently predict the
magnitude of the profit differences. For example, weeks 3, 30, and 57 all have large actual
profit differences, but while the logit estimates are too high in other cases, in these they are
correct, or are too low.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
72
Figure 2.23: Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were grouped as:
{1,3},{2,4}. The marginal costs corresponding
to the horizontal
axis
are shown in Table



2.3. A
1 0
 0 1
=
 0 0
0 0
0
0
1
0
0
0 
,
0 
1
β=
3
 5 
 ,
 7 
9
ρ=
0.05
 0.08 


 0.07 
0.09
Percentage Profit Decrease from Uniform Price Restriction
70%
60%
% Profit Decrease
50%
40%
Our Model
Logit
30%
20%
10%
0%
0
.
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Marginal Cost Configuration
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
73
Figure 2.24: Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were grouped as:
{1,3},{2,4}. The marginal costs corresponding
to the horizontal
axis
are shown in Table



2.3. A
1 0
 0 1
=
 0 0
0 0
0
0
1
0
0
0 
,
0 
1
β=
9
 5 
 ,
 7 
9
ρ=
0.05
 0.08 


 0.07 
0.09
Percentage Profit Decrease from Uniform Price Restriction
100%
90%
80%
% Profit Decrease
70%
60%
Our Model
Logit
50%
40%
30%
20%
10%
0%
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Marginal Cost Configuration
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
74
Figure 2.24 shows the predicted profit differences when I increase β1 from 3 to 9. This
change can be thought of as directly increasing demand for good one. The effect of this
change is to increase the profits lost by pricing uniformly in many cases. What we observe,
is that now the weeks with the greatest profit difference from uniform pricing are: 9, 18, 27,
57, 66, and 75. Now, the cost differences between goods one and three appear to be driving
the profit differences. When the cost of good one is close to the cost of good three the actual
lost profits are relatively small. The logit is again an imprecise predictor, exaggerating weeks
with large profit differences, and under-estimating weeks with moderate profit losses.
Figure 2.25: Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were grouped as:
{1,3},{2,4}. The marginal costs corresponding
to the horizontal
axis
are shown in Table



2.3. A
1 0
 0 1

= 0 0
0 0
0
0
1
0
0
0 
,
0 
1
β=
1
 5 
 ,
 7 
9
ρ=
0.15
 0.08 


 0.07 
0.09
Percentage Profit Decrease from Uniform Price Restriction
70%
60%
% Profit Decrease
50%
40%
Our Model
Logit
30%
20%
10%
0%
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Marginal Cost Configuration
Figure 2.24 shows the predicted profit differences when we hold β1 ρ1 = 0.15, and increase
ρ1 from 0.05 to 0.15. This change can be thought of as decreasing the rate of satiation for
good one. The effect of this change is to increase the profits lost by pricing uniformly in
some cases, but decrease them significantly when the marginal costs of goods one and three
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
75
are closer. This figure can be seen as an amalgam of Figures 2.23 and 2.24. Differences
in marginal costs between goods two and four drive much of the variation, but the middle
third of the weeks (weeks 28 through 54) show comparatively smaller differences, coming
from a lack of large cost differences between goods one and three.
Products:
Coke
Diet Coke
Pepsi
Diet Pepsi
Table 2.5: Example Product Universe Number Two
Characteristics:


Soda Coke Pepsi Diet
1 1 0 0
 1 1 0 1 
X
X
0
X

A=
 1 0 1 0 
X
X
0
0
1 0 1 1
X
0
X
X
X
0
X
0
→
These have shown the effects when the A matrix is diagonal, a world in which demand
for one good does not significantly affect demand the the others. Next, I consider the case
where there is significantly more overlap. n particular, I assume the A matrix is as shown
in Table 2.5. In this case, the four demand-relevant characteristics are: Soda, Coke, Pepsi,
and Diet. Again, I choose to set each positive characteristic to “1”. Because the logit
demand model is linear, it’s not able to match this product characteristic matrix exactly
- the constant term is co-linear with the pair of brand dummy variables. Therefore, in
this case, I omit one of the brand dummy variables in estimating the logit for this second
characteristics.
In Figures 2.26 and 2.28, I vary the groupings of the products. First grouping them
by
 
whether or not they are diet sodas, and then grouping them by brand. I chose β =

0.05


 0.07 




 0.07 


0.09
5
 
 5 
 
 
 5 
 
5
and

ρ=
in order to simulate an environment where households cared most about whether
a product was a diet drink, and then about the brand.
Figure 2.26 shows both larger median profit losses for the true model, and less variation
with respect to changes in marginal costs. The logit model, however, predicts much greater
profit differences.
Figure 2.27 uses the same parameters as Figure 2.26, but now groups the products by
brand, putting Coke and Diet Coke in one group, and Pepsi and Diet Pepsi in another.
Here the logit actually does a remarkably good job of predicting the differences in expected
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
76
Figure 2.26: Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were grouped as:
{1,3},{2,4}. This corresponds to uniform pricing by “diet” characteristic. the The
marginal costs corresponding
to the
in Table 2.3.

 horizontal
  axis are shown

A
1 1
 1 1

= 1 0
1 0
0
0
1
1
1
0 
,
1 
0
β=
5
 5 
 ,
 5 
5
ρ=
0.05
 0.07 


 0.07 
0.09
Percentage Profit Decrease from Uniform Price Restriction
30%
% Profit Decrease
25%
20%
Our Model
Logit
15%
10%
5%
0%
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Marginal Cost Configuration
profits. The picture changes dramatically, however, if we change the parameters slightly.
Figure Figure 2.28 decreases all of the β’s by 30% from 5.0 to 3.5. Although the true
percentage profit differences remain more-or-less unchanged, the logit predictions change
dramatically. At low levels they are too low, and at high levels they are too high.
This section has shown that, at least for this application, measuring the difference in
expected profits between uniform and non-uniform prices, the logit model does not perform
well when it does not accurately describe the data generating process. I conclude that in
this case, the additional complexity of my model is worthwhile. Unfortunately, I have been
unable to determine factors that affect the degree to which the logit model miscalculates
the result. Learning to recognize cases in advance, when it is more important to estimate
demand using my model (and incur the associated additional costs) than to use the logit
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
77
Figure 2.27: Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were grouped as:
{1,2},{3,4}. This corresponds to uniform pricing by “Coke” or “Pepsi” characteristics.
The marginal costs corresponding
to
axis
are shown in Table 2.3.

 the horizontal
 

A
1 1
 1 1

= 1 0
1 0
0
0
1
1
1
0 
,
1 
0
β=
5
 5 
 ,
 5 
5
ρ=
0.05
 0.07 


 0.07 
0.09
Percentage Profit Decrease from Uniform Price Restriction
30%
% Profit Decrease
25%
20%
Our Model
Logit
15%
10%
5%
0%
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Marginal Cost Configuration
as an approximation is an area for future research.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
78
Figure 2.28: Graph of percentage decrease in expected profits from optimal uniform prices
versus optimal non-uniform Prices. For the uniform case, products were grouped as:
{1,2},{3,4}. This corresponds to uniform pricing by “Coke” or “Pepsi” characteristics.
The marginal costs corresponding
to the horizontal
axis




 areshown in Table 2.3.
A
1 1
 1 1

= 1 0
1 0
0
0
1
1
1
0 
,
1 
0
β=
3.5
 5.5 


 3.5 ,
3.5
ρ=
0.05
 0.07 


 0.07 
0.09
Percentage Profit Decrease from Uniform Price Restriction
70%
60%
% Profit Decrease
50%
40%
Our Model
Logit
30%
20%
10%
0%
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Marginal Cost Configuration
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
2.7
79
Conclusion
The logit demand model has been a workhorse of demand estimation for over thirty years.
Even at its most basic form, it is a powerful and easily-estimated model. And yet, even
with recent extensions, it has significant shortcomings. These shortcomings are most apparent when the model is applied in a setting with continuous choice, with many products,
and when households may be variety-seeking. This chapter has described the state of the
current demand estimation literature, and its continued shortcomings with respect to these
challenges. In response to these challenges, I developed a new model, able to accommodate
these features.
Although retaining the hedonic framework of the logit model, I introduce significant
nonlinearity to the household’s utility function. Allowing for continuous choice forces us to
use a budget constraint to limit the household’s purchase behavior. As we will see in more
detail in the following chapter, these changes come at significant computational cost.
To justify this cost, and because this model represents a departure from much of the
existing demand literature, a large portion of this chapter has been spent exploring and
dissecting the workings of the new model. By showing the demand and Engel curves that
it generates, as well as the possible probability distributions over various choices, I showed
that the model was indeed able to deliver realistic choice patterns.
Additionally, this chapter showed that for my own application, looking at the difference in expected profits from uniform and non-uniform pricing, the logit model performs
quite poorly when compared to the new model. This finding helps to justify the added
computational complexity.
As computers become faster, and more household-level data becomes available, I expect
the techniques employed in this dissertation should continue to gain momentum. The strategy of writing down a direct utility function, and numerically maximizing it with respect
to a budget constraint yields much more freedom than previous approaches. Unlike the
work of Kim et al. (2002), which uses Simulated Maximum Likelihood, my approach using
Method of Simulated Moments does not require high dimensional numerical integration,
and thus allows for a much larger product space. The particular utility function assumed
in this dissertation is both flexible, concave, and parsimonious with parameters.
Finally, this demand model has a variety of potential uses beyond the uniform pricing
application. For example, although I do not do so in this paper, modeling households’
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
80
purchases of bundles in this new way allows for a variety of additional counterfactuals that
were previously not possible. It is now possible to calculate the effects of promotions of the
kind: “buy good A, get good B free”, as well as more complex models of cross-category
interactions.
2.8
Appendix A: Restrictions Yielding Concavity
In order to ensure that the necessary first-order conditions of the Lagrangian are also
sufficient, as well as to reflect the predictions of economic theory, we would like to restrict
the parameters of the model to make utility a concave function of consumption. The utility
function is concave if and only if its hessian matrix is negative semi-definite, which in turn
is true if and only if all of the hessian’s eigenvalues are all real and non-positive.
2.8.1
Two Goods, Two Characteristics
In the case of only two interior goods, the utility function is concave (and the hessian’s
eigenvalues are all real and non-positive) if and only if the following three conditions are
satisfied: (1) h11 ≤ 0, (2) h22 ≤ 0, and (3) h11 h22 −h12 h21 ≥ 0, where hij =
∂2U
∂qi ∂qj
represents
the i, jth element of the Hessian.
With two characteristics and two goods,
h11 = a211 β1 ρ1 (ρ1 − 1)(a11 q1 + a21 q2 + 1)ρ1 −2 + a212 β2 ρ2 (ρ2 − 1)(a12 q1 + a22 q2 + 1)ρ2 −2
h22 = a221 β1 ρ1 (ρ1 − 1)(a11 q1 + a21 q2 + 1)ρ1 −2 + a222 β2 ρ2 (ρ2 − 1)(a12 q1 + a22 q2 + 1)ρ2 −2
and
h12 = h21 = a11 a21 β1 ρ1 (ρ1 − 1)(a11 q1 + a21 q2 + 1)ρ1 −2 + a12 a22 β2 ρ2 (ρ2 − 1)(a12 q1 + a22 q2 + 1)ρ2 −2
If we define:
T1 = β1 ρ1 (ρ1 − 1)(a11 q1 + a21 q2 + 1)ρ1 −2
T2 = β2 ρ2 (ρ2 − 1)(a12 q1 + a22 q2 + 1)ρ2 −2
T 1 = (a11 q1 + a21 q2 + 1)ρ1 −2
T 2 = (a12 q1 + a22 q2 + 1)ρ2 −2
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
81
then we can see that
h11 = a211 T1 + a212 T2
h22 = a221 T1 + a222 T2
h12 = a11 a21 T1 + a12 a22 T2
Also, note that T j ≥ 0.
h22 h11 − h221 = (a211 T1 + a212 T2 )(a221 T1 + a222 T2 ) − (a11 a21 T1 + a12 a22 T2 )2
(2.30)
= a211 a222 T1 T2 + a212 a221 T1 T2 − 2a11 a12 a21 a22 T1 T2
(2.31)
= T1 T2 (a211 a222 + a212 a221 − 2a11 a12 a21 a22 )
(2.32)
= T1 T2 (det A)2
(2.33)
Hence,
h22 h11 − h221 ≥ 0 ⇔
(2.34)
β1 ρ1 (ρ1 − 1)β2 ρ2 (ρ2 − 2)(det A)2 ≥ 0
(2.35)
Combining these three conditions, for the two good, two characteristic case, a necessary
and sufficient condition for concavity (assuming det A 6= 0) is:
βj ρj (ρj − 1) ≤ 0, j = 1, 2
2.8.2
Two Goods, Three Characteristics
As in the previous case, we require (1) h11 ≤ 0, (2) h22 ≤ 0, and (3) h11 h22 − h12 h21 ≥ 0.
In this case,
h11 = a211 β1 ρ1 (ρ1 − 1)(a11 q1 + a21 q2 + 1)ρ1 −2 + a212 β2 ρ2 (ρ2 − 1)(a12 q1 + a22 q2 + 1)ρ2 −2 +
a213 β3 ρ3 (ρ3 − 1)(a13 q1 + a23 q2 + 1)ρ3 −2
h22 = a221 β1 ρ1 (ρ1 − 1)(a11 q1 + a21 q2 + 1)ρ1 −2 + a222 β2 ρ2 (ρ2 − 1)(a12 q1 + a22 q2 + 1)ρ2 −2 +
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
82
a223 β3 ρ3 (ρ3 − 1)(a13 q1 + a23 q2 + 1)ρ3 −2
and
h12 = h21 = a11 a21 β1 ρ1 (ρ1 − 1)(a11 q1 + a21 q2 + 1)ρ1 −2 + a12 a22 β2 ρ2 (ρ2 − 1)(a12 q1 + a22 q2 + 1)ρ2 −2 +
a13 a23 β3 ρ3 (ρ3 − 1)(a13 q1 + a23 q2 + 1)ρ3 −2
Following the previous subsection, if we define:
T3 = β3 ρ3 (ρ3 − 1)(a13 q1 + a23 q2 + 1)ρ3 −2
T 3 = (a13 q1 + a23 q2 + 1)ρ3 −2
then we can see that
h11 = a211 T1 + a212 T2 + a213 T2
h22 = a221 T1 + a222 T2 + a223 T2
h12 = a11 a21 T1 + a12 a22 T2 + a13 a23 T3
As before, we find that
βj ρj (ρj − 1) ≤ 0, j = 1, 2
is a sufficient condition, however it is no longer necessary.
2.8.3
Three Goods, Three Characteristics
The case of three goods and three characteristics does not offer any additional intuition,
but it is included for completeness, to show that the result extends to higher dimensions.
In this case the hessian is negative semi-definite if and only if: (1) h11 ≤ 0, (2) h22 ≤ 0, (3)
h33 ≤ 0, (4)h11 h22 − h12 h21 ≥ 0, (5)h11 h33 − h13 h31 ≥ 0 (6)h22 h33 − h23 h32 ≥ 0, and (7)
h11 h22 h33 + 2h12 h23 h31 − h11 h223 − h22 h213 − h33 h212 + h11 h22 h33 ≤ 0. Once again,
βj ρj (ρj − 1) ≤ 0, j = 1, 2, 3
is a sufficient, but not necessary condition.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
2.9
83
Appendix B: Analytic Solutions to the Two-Inside Good
Case: Details
If the concavity conditions derived in the previous Appendix are satisfied, we can solve the
household’s optimization problem by forming the Lagrangian:
L(q1 , q2 , z, λ0 , λ1 , λ2 , λ3 ) = β1 (a11 q1 + q2 a21 + 1)ρ1 + β2 (a12 q1 + a22 q2 + 1)ρ2 +
ε1 q 1 + ε2 q 2 + z
+λ0 (w − p1 q1 − p2 q2 − z) + λ1 q1 + λ2 q2 + λ3 z
The Lagrangian yields the following equations for a local maximum:
∂L
∂q1
= a11 β1 ρ1 (a11 q1 + a21 q2 + 1)ρ1 −1 + a12 β2 ρ2 (a12 q1 + a22 q2 + 1)ρ2 −1 + ε1 − λ0 p1 + λ1 = 0
(2.36)
∂L
∂q2
ρ1 −1
= a21 β1 ρ1 (a11 q1 + a21 q2 + 1)
ρ2 −1
+ a22 β2 ρ2 (a12 q1 + a22 q2 + 1)
+ ε 2 − λ 0 p2 + λ 2 = 0
(2.37)
∂L
= 1 − λ0 + λ 3 = 0
∂z
(2.38)
λ0 (w − p1 q1 − p2 q2 − z) = 0
λ1 q1 = 0
λ2 q2 = 0
λ3 z = 0
λ0 , λ 1 , λ 2 , λ 3 ≥ 0
First, note that we can substitute λ0 = 1 + λ3 . Next, if we define:
α1 = (a11 q1 + a21 q2 + 1)ρ1 −1
α2 = (a12 q1 + a22 q2 + 1)ρ2 −1
Then Equations 2.36 and 2.37 represent a linear system with two equations and two unknowns α1 and α2 . Assuming that the determinant of A is non-zero15 , and that β1 β2 ρ1 ρ2 6=
15
If det A = 0, then the characteristics of one good are equal to a multiple of the characteristics of the other.
Therefore only one of the two inside goods will be chosen (whichever has the higher value for Aj1 /(pj − εj )).
This “dominance” concept will be discussed later.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
84
0 then the solution to this equation is:
α1 =
(p1 + λ3 p1 − ε1 − λ1 )a22 − (p2 + λ3 p2 − ε2 − λ2 )a12
β1 ρ1 (a11 a22 − a21 a12 )
α2 =
(p2 + λ3 p2 − ε2 − λ2 )a11 − (p1 + λ3 p1 − ε1 − λ1 )a21
β2 ρ2 (a11 a22 − a21 a12 )
Therefore:
(p1 + λ3 p1 − ε1 − λ1 )a22 − (p2 + λ3 p2 − ε2 − λ2 )a12
β1 ρ1 (a11 a22 − a21 a12 )
1
ρ1 −1
(p2 + λ3 p2 − ε2 − λ2 )a11 − (p1 + λ3 p1 − ε1 − λ1 )a21
β2 ρ2 (a11 a22 − a21 a12 )
1
ρ2 −1
a11 q1 + a21 q2 + 1 =
a12 q1 + a22 q2 + 1 =
Continuing, this gives us:
1
ρ1 −1
a22 δ1
q1 =
1
ρ2 −1
− 1 − a21 δ2
−1
a11 a22 − a21 a12
1
1
ρ −1
ρ −1
a11 δ2 2 − 1 − a12 δ1 1 − 1
q2 =
a11 a22 − a21 a12
Where for convenience, we define
δ1 =
(p1 + λ3 p1 − ε1 − λ1 )a22 − (p2 + λ3 p2 − ε2 − λ2 )a12
β1 ρ1 (a11 a22 − a21 a12 )
δ2 =
(p2 + λ3 p2 − ε2 − λ2 )a11 − (p1 + λ3 p1 − ε1 − λ1 )a21
β2 ρ2 (a11 a22 − a21 a12 )
We begin by examining the interior solution, when all three goods are consumed. In
this case λ1 = λ2 = λ3 = 0 and λ0 = 1, and the solutions are:
a22
q̂1 =
1
ρ1 −1
− a21
(p2 −ε2 )a11 −(p1 −ε1 )a21
β2 ρ2 (a11 a22 −a21 a12 )
1
ρ2 −1
− a22 + a21
(2.39)
a11 a22 − a21 a12
a11
q̂2 =
(p1 −ε1 )a22 −(p2 −ε2 )a12
β1 ρ1 (a11 a22 −a21 a12 )
(p2 −ε2 )a11 −(p1 −ε1 )a21
β2 ρ2 (a11 a22 −a21 a12 )
1
ρ2 −1
− a12
(p1 −ε1 )a22 −(p2 −ε2 )a12
β1 ρ1 (a11 a22 −a21 a12 )
a11 a22 − a21 a12
1
ρ1 −1
− a11 + a12
(2.40)
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
ẑ = w − p1 q1 − p2 q2
85
(2.41)
The ˆ’s are to remind the reader that these can be considered candidate solutions – there
are several ways in which this interior solution may break down, leading to a non-interior
solution. The above solutions may yield q̂1 < 0, q̂2 < 0, or ẑ < 0. Additionally, if either
of the δ’s is negative, then the above solution will predict complex consumption. I explore
each of these cases in the next few paragraphs.
Assume for the moment that both δ’s are positive. Then if q̂1 < 0 and ẑ < 0, then the
solution is (0, pw2 , 0), while if q̂2 < 0 and ẑ < 0, then the solution is ( pw1 , 0, 0).
If ẑ > 0 and q̂1 < 0, then
∗ w
q1 = 0, z = w − p2 q2 and q2 = max 0, min q2 ,
p2
(2.42)
Where q2∗ is the solution to the equation16 :
a21 β1 ρ1 (a21 q2∗ + 1)ρ1 −1 + a22 β2 ρ2 (a22 q2∗ + 1)ρ2 −1 + ε2
=1
p2
Similarly, if ẑ > 0 and q̂2 < 0, then
∗ w
q2 = 0, z = w − p1 q1 and q1 = max 0, min q1 ,
p1
(2.43)
Where q1∗ is the solution to the equation:
a11 β1 ρ1 (a11 q1∗ + 1)ρ1 −1 + a12 β2 ρ2 (a12 q1∗ + 1)ρ2 −1 + ε1
=1
p1
Finally, if ẑ < 0, q̂1 > 0 and q̂2 > 0 then we have λ1 = λ2 = 0 and λ0 = 1 + λ3 . Rather
than trying to solve for λ3 , in this case it’s easier to return to the first order conditions,
which in this case give us:
16
λ0 p1 = a11 β1 ρ1 (a11 q1 + a21 q2 + 1)ρ1 −1 + a12 β2 ρ2 (a12 q1 + a22 q2 + 1)ρ2 −1 + ε1
(2.44)
λ0 p2 = a21 β1 ρ1 (a11 q1 + a21 q2 + 1)ρ1 −1 + a22 β2 ρ2 (a12 q1 + a22 q2 + 1)ρ2 −1 + ε2
(2.45)
Note that the concavity conditions guarantee that this solution, if positive, is unique.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
86
Dividing the first equation by the second yields:
a11 β1 ρ1 (a11 q1 + a21 q2 + 1)ρ1 −1 + a12 β2 ρ2 (a12 q1 + a22 q2 + 1)ρ2 −1 + ε1
p1
=
p2
a21 β1 ρ1 (a11 q1 + a21 q2 + 1)ρ1 −1 + a22 β2 ρ2 (a12 q1 + a22 q2 + 1)ρ2 −1 + ε2
Substituting in q2 =
w−p1 q1
p2
and simplifying:
w − p1 q 1
(p2 a11 β1 ρ1 − p1 a21 β1 ρ1 ) a11 q1 + a21
+1
p2
ρ1 −1
w − p1 q 1
+1
(p1 a22 β2 ρ2 − p2 a12 β2 ρ2 ) a12 q1 + a22
p2
Hence the solution is
z = 0, q2 =
+ p2 ε 1 =
ρ2 −1
+ p1 ε 2
w − p1 q 1
and q1 = q1∗
p2
(2.46)
where q1∗ is the solution to
ρ1 −1
w − p1 q 1
+1
+ p2 ε 1 =
(p2 a11 β1 ρ1 − p1 a21 β1 ρ1 ) a11 q1 + a21
p2
ρ2 −1
w − p1 q 1
+1
+ p1 ε 2
(p1 a22 β2 ρ2 − p2 a12 β2 ρ2 ) a12 q1 + a22
p2
(2.47)
(2.48)
Now, consider the case we would have either δ1 < 0 or δ2 < 0 at an interior solution.
I.e., when
or
(p1 − ε1 )a22 − (p2 − ε2 )a12
<0
β1 ρ1 (a11 a22 − a21 a12 )
(2.49)
(p2 − ε2 )a11 − (p1 − ε1 )a21
<0
β2 ρ2 (a11 a22 − a21 a12 )
(2.50)
Closer inspection reveals two insights: (1) these are mutually exclusive cases and (2) they
correspond to “dominance” of one inside good by the other. Basically, one good is dominated
by another if you can get all of it’s desirable characteristics more cheaply from some other
good (or combination of goods). There are two ways for good one to be dominated by good
two. If good one’s principal17 characteristic is bad (has a negative β), then unless good
one is a cheaper source of the other characteristic than good two it will have zero demand.
17
In the two-good/two-characteristic case, if det A > 0, good one’s principal characteristic is the first
characteristic. If det A > 0, good one’s principal characteristic is the second characteristic. In the three
good case, the intuition is less clear, but the mathematics are similar.
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
87
Similarly, if good one’s non-principal characteristic is good (has a positive β) then unless
good one is a cheaper source for it’s principal characteristic than good two it will have zero
demand. In what follows, I define pj = pj − εj . pj can be thought of as the effective price
for the household of one unit of good j.
Table 2.6: Summary of Dominance Conditions
a12
a22
a11
a21
det A β1
β2
Dominant
p1 − p2
p1 − p2
Good
>0 >0
>0
1
>0 <0
<0
2
>0
>0
<0
2
>0
<0
>0
1
<0 >0
<0
2
<0 <0
>0
1
<0
>0
>0
1
<0
<0
<0
2
The inequalities in equations 2.49 and 2.50 imply that good one is dominated by good
two (and thus has zero demand) if any of A1-A4 are met:
Condition A1: det A > 0, β1 < 0, and
Condition A2: det A > 0, β2 > 0, and
Condition A3: det A < 0, β1 > 0, and
Condition A4: det A < 0, β2 < 0, and
a12
p1
a11
p1
a12
p1
a11
p1
<
<
<
<
a22
p2 .
a21
p2 .
a22
p2 .
a21
p2 .
Note that for a particular set of parameters (A, β, ρ), at most one of these conditions is
relevant. Similarly, good two is dominated by good one if any of B1-B4 hold:
Condition B1: det A > 0, β1 > 0, and
Condition B2: det A > 0, β2 < 0, and
Condition B3: det A < 0, β1 < 0, and
Condition B4: det A < 0, β2 > 0, and
a22
p2
a21
p2
a22
p2
a21
p2
<
<
<
<
a12
p1 .
a11
p1 .
a12
p1 .
a11
p1 .
If good two is dominated by good one, then the solution is given by Equation 2.43,
while if good one is dominated by good two the solution is given by Equation 2.42. These
conditions are summarized in Table 2.6.
2.10
Appendix C: Additional Figures
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
88
Figure 2.29: Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
1
0.999
0.999
1
, β1 = β2 = 5, ρ1 = ρ2 = 0.5, p2 = $0.50, w = 60
q (p | p =$0.50)
1
q (p | p =$0.50)
2
2
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Price of Good 1
Price of Good 1
1
1
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0
5
10
q1
15
2
0.5
0.3
0
1
0
0
2
4
q2
6
8
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
89
Figure 2.30: Demand for good one and two as a function of p1 , for the 40th and 60th
percentiles of ε1 and ε2 . A =
1
0.999
,
β
=
6.875,
β2 = 5, ρ1 = 0.4, ρ2 = 0.5, p2 = $0.50, w = 60
1
0.999
1
q (p | p =$0.50)
1
q (p | p =$0.50)
2
2
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Price of Good 1
Price of Good 1
1
1
0.5
0.4
0.4
0.3
0.2
0.2
0.1
0.1
0
5
10
q1
15
2
0.5
0.3
0
1
0
0
2
4
6
q2
8
10
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
90
Figure 2.31: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
1 0
0 1
, β1 = β2 = 5, ρ1 = 0.5, ρ2 = 0.4, p1 = p2 = 0.3, w = 30. The
probabilities of the four regions are:
0.23 0.06
0.55 0.15
Figure 2.32: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
1
0.02
0.02
1
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 30. The
probabilities of the four regions are:
0.22 0.08
0.47 0.23
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
91
Figure 2.33: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
1
0.25
0.25
1
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 30. The
probabilities of the four regions are:
0.36 0.05
0.22 0.37
Figure 2.34: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
1 0.1
0.1 1
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 1. The
probabilities of the seven regions are:



0.16 0.11 0.07
0.10 0.11 
0.23
0.16
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
92
Figure 2.35: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
10 1
1 10
, β1 = β2 = 0.5, ρ1 = 0.5, ρ2 = 0.4, p1 = p2 = 0.3, w = 30.
The probabilities of the four regions are:
0.28 0.07
0.38 0.28
Figure 2.36: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
1 0.08
0
1
, β1 = β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 30. The
probabilities of the four regions are:
0.27 0.08
0.45 0.20
CHAPTER 2. CONTINUOUS DEMAND AND VARIETY-SEEKING
93
Figure 2.37: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
1 0.5
0 1
, β1 = 7, β2 = 5, ρ1 = ρ2 = 0.4, p1 = p2 = 0.3, w = 30. The
probabilities of the four regions are:
0.65 0.03
0.22 0.10
Figure 2.38: Graph of purchases as a function of error terms. The groupings are the same
as in Figure 2.15. A =
1 0.5
0 1
, β1 = β2 = 5, ρ1 = 0.5, ρ2 = 0.4, p1 = p2 = 0.3, w = 30.
The probabilities of the four regions are:
0.66 0.04
0.19 0.11
Chapter 3
Investigating the Costs of Uniform
Pricing
3.1
3.1.1
Introduction
Overview
Retailers typically sell many different products from the same manufacturer at the same
price. As mentioned in chapter one, there are a large number of potential explanations for
this uniform pricing behavior. One potential explanation is that retailers face managerial
menu costs, and hence find it costly to charge the optimal price for each product in each
period. It is this explanation that I investigate in more depth in this chapter. In order to
assess the plausibility of the menu costs explanation, I would like to know how large these
menu costs would have to be to lead retailers to charge uniform prices.
The following framework is useful for analyzing the retailer’s decision process. Assume
that each period, the retailer maximizes expected profits, less menu costs:
Expected
Profit
this Period
Expected
=
Revenue
Expected
−
from Sales
Cost
of Goods
−
Menu Costs
this Period
(3.1)
My approach to measuring the menu costs that would rationalize the observed uniform
pricing behavior is to look at the counterfactual expected profits earned by the retailer
94
CHAPTER 3. COSTS OF UNIFORM PRICING
95
under uniform and non-uniform pricing strategies. That is, for each week, I compute:

Minimum Menu Cost
this Period
that Rationalizes
Uniform Prices


= 







Expected
Expected
Revenue from Sales
Cost of Goods
this Period
−
this Period
with Non-Uniform
with Non-Uniform
Prices
Prices
Expected
Expected
Revenue from Sales
Cost of Goods
this Period
−
this Period
with Uniform
with Uniform
Prices
Prices


−

(3.2)





(3.3)
The left-hand side of this equation corresponds to the amount of profit that the retailer
actually earned by charging non-uniform prices instead of uniform prices. Therefore, the
model predicts that if the retailer had faced menu costs higher than this amount, I would
have observed that retailer charging uniform prices if the choice was made on a week-byweek basis. More likely, as discussed in chapter one, the uniform versus non-uniform pricing
strategy decision is made infrequently. This would mean that the relevant profit difference
to consider would be the discounted value of the sum of the profit differences over several
weeks. The difference between the profits earned under uniform and non-uniform prices
depends only on the demand function faced by the store, and it’s marginal costs.
In order to make this comparison empirically, it is necessary to learn (a) the demand
system faced by the retailer and (b) the cost structure faced by the retailer. Learning
these two pieces in order to perform the counterfactual experiment posed above, requires
developing a structural model of demand and supply.
3.1.2
Data: Soft Drinks
The main problem in calculating this potential profit difference between uniform and nonuniform pricing strategies is in estimating demand. Specifically, I must estimate a demand
system for different varieties. As discussed in chapter one, if these different varieties always
sell at the same price, it is not possible to identify demand for the different varieties. After
much searching, I found a dataset that contained the necessary variation. In this dataset,
one retailer charged non-uniform prices for different flavors of carbonated soft drinks.
CHAPTER 3. COSTS OF UNIFORM PRICING
96
Like many products, carbonated soft drinks are typically priced uniformly by manufacturerbrand-size. For example, at a typical retailer, 2-Liter bottles of Pepsi, Diet-Pepsi, and DietCaffeine-Free Pepsi all typically sell at the same price, relative to each other. This behavior
can be seen in Figure 1.3. By contrast, a typical retailer will sell Coke and Pepsi at different
prices, as seen in Figure 1.1. In my dataset, I observe a retailer charging non-uniform prices
for soft drinks. An example of this variation can be seen in Figure 1.4. It is this variation
(and similar variation in other soft drink varieties at this retailer) that I use to identify
demand for individual varieties of soft drinks.
There are a number of features that make the carbonated soft drink category amenable
to this investigation. First, the soft drink category has a large number of products, and
a large number of different varieties of similar products. Second, contrary to many other
product categories, these products could plausibly be grouped in a number of alternative
ways. This stems from the fact that the product packaging for soft drinks is largely identical
across brands and manufacturers. The label on a 2-Liter bottle of Coke may differ from its
Pepsi counterpart, but the physical shape of the bottle is identical. This physical similarity
is much different than other product categories, like yogurt, where consumers might be less
likely to accept line pricing by flavors. Third, soft drinks are the most frequently purchased
item in scanner data. According to scanner data, in most product categories, the median
household makes a purchase only a couple of times per year. By contrast, the median
household in the sample purchases soft drinks on eleven occasions over the sample period
of two years. This gives us the hope of obtaining reasonably good estimates.
There are, however, several potential drawbacks to looking at carbonated soft drinks.
First, anecdotal evidence suggests that consumers may stockpile soft drinks. This would be
problematic for me, since the demand model I developed in chapter two does not account
for dynamics. However, contrary to expectations, in their descriptive paper, Hendel & Nevo
(2002) find no evidence of stockpiling of soft drinks. This is not definitive however, since
they also find little evidence of stockpiling in detergents, while their structural paper on
detergents (Hendel & Nevo 2002) does find such effects.
A second potential problem with using soft drinks is that retailers may view the soft
drink category as a “loss-leader” category, pricing it low in order to drive store traffic.
Although it features in some theoretical work (see for example, Hess & Gerstner (1987))
there is little empirical evidence of loss leader behavior. In a dataset covering the same
geographic area and time period as my own in which I observe actual retail and wholesale
CHAPTER 3. COSTS OF UNIFORM PRICING
97
prices, I do not observe negative margins except in a handful of cases. Nevertheless, it is
possible for loss-leader behavior to subtly depress soft drink prices without actually pushing
them below marginal costs. To the extent that cross-category loss-leader behavior by the
retailer does occur, it would only affect the assumptions I use to recover marginal costs. My
method would interpret low prices as evidence of low marginal costs, when in fact prices
would be low in order to drive store traffic. My approach could be modified to account
for cross-category loss-leader behavior, but I lack strong candidates for an alternative to
assuming the retailer maximizes profits on a category-by-category basis, and simultaneously
estimating demand for several product categories is beyond the scope of this paper.
A third potential problem with the soft drink industry is the argument that the soft
drink category is different because the two main manufacturers, Coke and Pepsi, have
extraordinarily strong brands and, as a result, may exert pressure on retailers to price their
products in a particular way. If Coke and Pepsi exerted influence over pricing, it does not
present a problem for my demand estimation, which relies solely on the fact that I observe
price variation. It could potentially affect my profit calculations. However, what I see in
the data does not appear to be consistent with the Coke and Pepsi influencing pricing (at
least not at the store that charged non-uniform prices). What I observe in the data is one
store charging non-uniform prices, and all other stores charging uniform prices. It seems
highly implausible that Coke and Pepsi would (or could) dictate to every other store in the
area to charge uniform prices, but allow the retailer I observe to charge non-uniform prices.
3.1.3
Estimating Demand
Having found a dataset containing sufficient price variation to identify demand for individual
varieties, it became clear that demand for carbonated soft drinks present a fourth potential
problem: it has several key features that make existing demand models unsuitable. These
features are that: choice is continuous, there are a large number of products, and households
may be variety-seeking. As noted in chapter two, existing demand models are unable to
simultaneously handle these features. This led me to develop the new model of demand
described in chapter two.
In calculating the retailer’s counterfactual expected profit, the relevant demand function
to consider is the residual demand function, which reflects the presence of other stores in the
market. The difference between residual demand and market demand is that the residual
demand function accounts for the fact that the prices charged at other stores affect demand
CHAPTER 3. COSTS OF UNIFORM PRICING
98
at any particular store. One can think of the residual demand curve as having several
components, i.e., that in purchasing carbonated soft drinks, households make a series of
decisions. They must decide whether to shop, where to shop, what to buy, and how much
to buy.
With respect to the first stage of this process, although some authors, such as Kahn &
Schmittlein (1989) have investigated the household’s decision of when to shop, little progress
has been made in identifying the factors that affect this process. Therefore, following Bell
& Lattin (1998), Rhee & Bell (2002), and others, I take the household’s decision to go
shopping, as well as their shopping budget on that occasion to be exogenously determined
and uncorrelated with all unobservables.
Moving to the next level of the decision tree, rather than develop a structural model
that incorporates the household’s simultaneous decision of where to shop, what to buy,
and in what quantity1 , I follow Bell & Lattin (1998) in decomposing the decision into two
conditionally independent parts. Thus the demand at store A in week t is:
Et [Qt (pt )] =
X
i
E [i’s purchases|i goes to A] · P
!
i’s characteristics
i goes to A and prices at all stores
(3.4)
In order to estimate this residual demand function Qt (pt ) for carbonated soft drinks
faced by the retailer, I follow a two-step approach described in more detail in section 3.2.
First in section 3.2.1, I briefly review the demand model from the previous chapter, which
I apply in to estimating a structural model of product choice, conditional on store choice.
Then, in section 3.2.2, I describe the model of store choice.
3.1.4
Counter-Factuals and Preview of Main Result
The second step required to calculate the profit difference suggested in Equation 3.3 is to
recover the cost structure faced by the retailer. In particular, to calculate the implied managerial menu costs, I need to make assumptions about how the retailer set its prices during
the time period of my data. This step is essential to recovering the retailer’s marginal costs.
If I had marginal cost data for the retailer, I would not need to make these assumptions
1
A fully structural model would involve calculating the household’s expected utility, net of travel costs,
from visiting each of the stores in its choice set. Such a model would also involve consumers forming
expectations of the menu of prices they would face at each store. Given my estimation procedure, this
approach is far too computationally burdensome. Instead, I approximate this choice, by assuming that
households’ choices among the stores in my sample follow a logit choice model. This model represents an
approximation of the true model and does not directly correspond to any model of utility maximization. I
use it because it is computationally cheap, and because I believe the approximation is reasonable.
CHAPTER 3. COSTS OF UNIFORM PRICING
99
about the competitive environment. The key assumption I choose to make is the the retailer
faced Bertrand competition with the other local grocery stores in my sample, as well as other
retailers outside my sample. I further assume that the retailer maximized profits for the
soft drink category separately from its other categories (i.e., I assume that the retailer did
not use loss leaders), faced constant marginal costs (which may have varied from product
to product and week to week) function, and that they charged the profit-maximizing price
for each product (UPC). Although I could choose from a multitude of alternative assumptions about the retailer’s price-setting strategy, among them those listed in section 1.3, it
is difficult to know how to select among them without further information. Together, these
assumptions imply that in each week t, the retailer chooses prices for each soft drink j that
maximize the single-period expected profit2 function for carbonated soft drinks:
Et [Πt ] =
X
Et [Qjt (pt )] (pjt − cjt )
(3.5)
j∈J
where pt is the vector of prices pjt , cjt is the marginal cost of good j in week t, and the
retailer’s expectation is taken over the idiosyncratic household-level preference shocks.
These assumptions about the retailer’s price-setting strategy allow me to recover the
implied marginal cost for each product from the retailer’s first-order conditions for profit
maximization. The assumption of profit maximization implies that:
X
∂Et [Qkt (pt )]
∂Et [Πt ]
= Et [Qjt (pt )] +
(pkt − ckt )
=0
∂pjt
∂pjt
(3.6)
k∈J
in each week t, for each of the products j sold by the retailer. This gives me a series of J
equations and J unknowns – the cjt ’s – for each week.
I then use these marginal costs to calculate the retailer’s expected profit for each week
under uniform and non-uniform pricing strategies. To calculate the retailer’s expected
profits from non-uniform pricing, I calculate:
Et [Πt ] =
X
Et [Qjt (pt )] (pjt − cjt )
j∈J
2
The retailer’s expectations are over the unknown (to the retailer and the econometrician) realizations
of the households’ idiosyncratic preference shocks.
CHAPTER 3. COSTS OF UNIFORM PRICING
100
at the actual prices. Because I do not observe the retailer charging uniform prices, calculating the retailer’s expected profits from uniform pricing involves one final step: I must
numerically solve for the hypothetically optimal uniform prices – the set of prices that maximize expected profits (based on the demand system and marginal costs that I estimate)
subject to the constraint that they be uniform by manufacturer-brand-size.
3.1.5
Conclusion
Comparing the difference in profits on a week-by-week basis between the uniform and nonuniform pricing strategies, I find that although it varies from week to week depending
primarily on changes in the marginal costs, the average weekly difference in profits is $36.56
(in nominal dollars). After adjusting for inflation, my estimate of the total difference in
profits between the two pricing strategies over the entire two year sample is $5,135. This
suggests that single-store grocery retailers will not find it profitable to learn what optimal
non-uniform prices they should charge if this learning costs them more than roughly $2,500
per year. However, it also suggests that, absent demand-side factors not measured in this
paper, many large grocery chains may be leaving money on the table with respect to pricing
in the soft drink category.
3.2
Empirical Demand Model
As mentioned in the introduction, the distinction between market demand and residual
demand is an important one. Several previous empirical demand studies have ignored
this aspect for the very good reason that in most datasets, this information is simply not
available – prices for other stores are not observed. In my case, however, I observe the prices
charged at four other competing stores. Furthermore, the additional stores in the dataset
were chosen precisely because they were the stores that shoppers were most likely to switch
to.
I assume that the households’ choice process is as follows. First, I assume that an
exogenous process governs consumers’ decision of whether to shop in a given week, as well
as their total grocery expenditure in that week. Second, conditional on deciding to shop,
I model households’ store-choice decision using a conditional logit. Then, conditional on
a household’s choice of store, I assume that they optimally allocate expenditure between
soft drinks and all other groceries. Hence, the residual demand faced by store A is equal to
CHAPTER 3. COSTS OF UNIFORM PRICING
101
the sum over all households i (that went shopping in that week) of the probability that the
household chose store A, multiplied by their expected purchases qit , conditional on choosing
store A. The resulting expected residual demand vector faced by store A in week t is:
Et [Q(p)] =
X
E [i’s purchases|i goes to A] · P
i
!
i’s characteristics
i goes to A and prices at all stores
(3.7)
or, more formally:
E [Q(pt |Ωobs,t , Ωunobs,t )] =
X
E [qit |parameters, Ωobs,t ] · P (A|parameters, Ωobs,t )
(3.8)
i
where Ωobs,t and Ωunobs,t represent observed and unobserved variables specifying the state
of the world in week t. For tractability, I assume that the store choice decision is made
independently of the households’ soft drink purchase decision. Economically, this rules out,
for example, going to store A because it is the only store that carries product j. More
importantly, it also rules out going to a particular store based on a purchase-occasionspecific shock. This means that I am assuming that households’ idiosyncratic preference
shocks for varieties of soft drinks are independent of their idiosyncratic shock for their
store-choice decision. Unfortunately, this means that the complete demand model is not
consistent with utility maximization.3 The next two subsections describe the specification
and estimation procedure used for each of these two components of residual demand on
more detail.
3.2.1
Product-Level Demand Model
As discussed earlier, it is a fundamental feature of the grocery market for carbonated soft
drinks that households make continuous choices among a large number of products, and
may be variety-seeking. Recall that in Table 1.1, nearly 70% of the households’ purchase
occasions involve the purchase of more than one unit, and roughly 25% of purchase occasions
involve the purchase of more than one different UPC. As discussed in chapter two, existing
demand models were unable to accommodate these features. Therefore, in modeling demand
for soft drinks I use the household-level demand model developed in the previous chapter,
which I briefly review here.
3
For example, the household’s choice of store is assumed to be conditionally independent of the household’s preferences for particular items. This is reflected in my use of a generic average price, rather than an
average of the bundle of products the household would purchase.
CHAPTER 3. COSTS OF UNIFORM PRICING
102
In order to reduce the dimensionality of the parameter space, I assume that, as in the
logit models considered above, households derive utility from product characteristics. Each
product j (shown in Table 3.3) can then be expressed as a vector of C different characteristics
(described in section 3.3.1), and the menu faced by the household can be represented by a
J × C matrix A where the rows of A are the products, and the columns are characteristics.
Hence, A is just a stacked matrix consisting of the Xjt ’s from the logit model.
To illustrate how the dataset fits into the model from chapter two, consider an example
with just two characteristics (Diet and Cola) and three available products (Coke, Diet Coke,
and Diet 7Up). Because it is a non-diet cola, Coke has characteristic vector [0 1].4 As a
diet cola, Diet Coke has characteristic vector [1 1]. Finally, as a diet non-cola, Diet 7Up
has characteristic vector [1 0]. Stacking these three products’ characteristic vectors yields
the following A matrix5 :

0 1




A=
1
1


1 0
Again following the logit, I assume that a household’s utility function is additively separable in these characteristics. Unlike the logit, however, my model allows households the
ability to consume multiple units of a single product, as well as consuming several different
products. In particular, I assume that in week t, household i myopically maximizes the
utility function6 :
Uit (qt , zt ) =
X
βc (A0ct qt )ρc + ε0it qt + zt
(3.9)
c∈ C
with respect to the vector qt and the scalar zt . Act is the cth column of A in week t, qt is
a column vector of length J comprising the household’s purchases of the J goods described
by A, zt is the amount of outside good consumed, and βc and ρc are the characteristicspecific scalar parameters for that household. The J dimensional vector εit represents
4
Note that although in this case the product characteristics are indicator variables, in general they need
only be non-negative. For example, in the estimated model one of the characteristics is the number of
milligrams of caffeine per 12-ounce serving.
5
Although the A matrix shown here is time invariant, the empirical A matrix will typically vary from
week to week, because I include feature and display as time-varying characteristics.
6
For simplicity, the model assumes households are homogenous in their preferences (β and ρ). Extending
the model to account for heterogeneity (observed or unobserved) is straightforward, though computationally
burdensome. The relevant details, difficulties and tradeoffs are discussed in the Appendix.
CHAPTER 3. COSTS OF UNIFORM PRICING
103
the household/shopping-occasion marginal utility shock, which is observed by the utilitymaximizing household, but not by the econometrician.7 The household maximizes this
utility function subject to the budget constraint:
X
pjt qjt + zt ≤ wit
(3.10)
j∈ J
where wit is the household’s total grocery expenditure in the store in week t. Returning to
the three-good example above and substituting the A matrix into the utility function, we
can see that the household maximizes 8 :
Uit (qt , zt ) = βDiet (qDietCoke,t + qDiet7U p,t )ρDiet + βCola (qCoke,t + qDietCoke,t )ρCola +
εi,Coke,t qCoke + εi,DietCoke,t qDietCoke,t + εi,Diet7U p,t qDiet7U p,t + zt
with respect to q and z, subject to:
pCoke,t qCoke,t + pDietCoke,t qDietCoke,t + pDiet7U p,t qDiet7U p,t + zt ≤ wit
The model is estimated using the Method of Simulated Moments (MSM), developed
independently by McFadden (1989) and Pakes & Pollard (1989)9 . This estimation method
uses the fact that the expectation of the difference between the expected purchases and the
actual, observed purchases, is zero at the true parameter values. More formally, I use the
(|J| + 2) ∗ (|J| + 1) moment conditions:
I
T
1 XX
(qit − E[qit |β, ρ, pt , wit ]) xit
HI,T,R (β, ρ) =
IT
(3.11)
i=1 t=1
7
I currently assume that εit is i.i.d. across products, time, and households, and negatively log-normally
distributed on the interval (−∞, 0). It is necessary to bound εit from above in order to prevent unreasonable
choice behavior. If, for example, the realization of εitj is greater than the price of good j, a household may
never consume the outside good on that purchase occasion, regardless of the level of wt .
8
In reality, households are forced to choose between the discrete sizes offered by the store. I make no
attempt to model this feature of the data, as it introduces an extraordinary amount of computational cost
with little clear return. In the estimation, I do not restrict the predicted purchases to the discrete purchases
that the household could have actually made, instead allowing purchase quantities to vary continuously.
Although Dubé (2001) suggests selecting the purchasable grid point adjacent to the unconstrained maximum,
this is not numerically feasible in my case as it would require the examination of 225 ' 3.3 × 107 points for
each household maximization.
9
Gouriéroux & Monfort (1996) contains the best summary and discussion of various simulation estimators
and their properties that I have found.
CHAPTER 3. COSTS OF UNIFORM PRICING
104
where qit is the household’s vector of actual purchases. Each moment is an average across
all purchase occasions of the difference between expected purchases and the actual, observed
purchases, interacted with the instruments. The vector of instruments, xit consists of all
exogenous variables in the model (more on this in the following paragraphs), namely: the
prices of each good, the household’s budget, and a constant.
Because exact computation is infeasible, I simulate E[qit |β, ρ, pt , wit ] by drawing R = 30
sets of εit ’s (which I hold constant as I search the parameter space). Hence, I use the fact
that:
R
E[qit |β, ρ, wit , pit ] ≈
1 X
q(β, ρ, wit , pt , εrit )
R
(3.12)
r=1
Using these moments, I define my estimates as minimizing the distance function:
b = argmin (HI,T,R (ρ, β))0 W (HI,T,R (ρ, β))
[b
ρ, β]
(3.13)
Ideally, I would implement this as a two-stage procedure. The first stage of this procedure would involve choosing the weighting matrix, and finding consistent estimates of the
parameters. The second stage takes these estimates and uses them to calculate the optimal
weighting matrix, and then re-estimates the parameters. In practice, however, estimation
currently takes several weeks. Therefore, I report only the (inefficient) first-stage estimates.
Each of these two stages of estimation consists of iterating over several steps, which I
review now:
1. Choose starting values for the parameters: β and ρ.
2. Take the actual characteristics of the households in the sample that went shopping in
that week. In this case, a household is completely characterized by it’s budget (wit ).
This amounts to assuming that the retailer knows the distribution of the households
that would go shopping (not necessarily at its store) in each week.
3. Draw R sets of εit ’s for each observed purchase occasion. I use R = 30. This means
drawing R∗(Number of Purchase Occasions)∗(Number of Products)= 30∗16008∗25 =
12, 006, 000 i.i.d. negative lognormal random numbers. These random numbers are
held constant across iterations.
4. Using the expenditures from the actual purchase occasion as the budget constraint,
and the actual menu of prices in the week of the purchase occasion, take the current
CHAPTER 3. COSTS OF UNIFORM PRICING
105
parameters and solve explicitly (numerically) the household’s utility maximization
problem. This step is non-trivial and accounts for the bulk of the computational power
involved in this estimation procedure. This means solving R∗(Number of Purchase
Occasions)=30 ∗ 16008 = 480, 240 utility maximization problems for each iteration
of the parameter values. I discuss this process, and suggest numerical algorithms at
greater length in the Appendix.
5. For each purchase occasion, average over the R different purchase vectors to get the
expected purchases for that purchase occasion at the current set of parameter values.
6. Using the difference between the actual vector of purchases on that purchase occasion and the expected purchases calculated in Step 4, calculate the current moment
equations.
7. Interact these moment equations with the weighting matrix W to calculate the current
distance function. The weighting matrix is of dimension (|J| + 2) ∗ (|J| + 1) by
(|J| + 2) ∗ (|J| + 1).
8. Using a numerical minimization algorithm10 , choose a new set of parameter values (β
and ρ) and repeat steps until a minimum is found.
In addition to the computational cost, this simulation method forces me to make assumptions about the distribution of the unobservables (ε). The assumption that I choose to
make is that these unobservables are distributed independently of all observable variables.
In particular, I assume that they are distributed independently of prices. The distributions
of ε could be made dependent upon prices (or other observables). I do not do so here for
two reasons. First, it seems at least plausible that brand, holiday, feature, and display
variables account for much of the potential for unobserved correlation between price and
these unobservables, but to the extent that the retailer observes time-varying changes in
the household error terms, the unconditional distribution of the error term will differ significantly from its distribution conditional on prices. In this case my estimates may be
both biased and inconsistent. Short of simulation, I cannot think of a way to “bound”
the effects of violations of this assumption. The second, and principal justification for this
assumption, is that it is crucial in making the estimation tractable. Even implementing a
10
I have had the most success using the simplex-based E04CCF routine available from the Numerical
Algorithms Group (NAG).
CHAPTER 3. COSTS OF UNIFORM PRICING
106
recursive routine to match predicted with observed market shares (as in Berry et al. (1995))
would be prohibitively computationally expensive. Economically, with respect to prices, I
am assuming that the retailer does not observe any (or at least does not adjust prices in
response to) time-varying changes in the distribution of the idiosyncratic demand shocks.
Given this assumption that the idiosyncratic shocks are distributed independently of prices,
it is internally consistent to use prices as instruments (since they will be orthogonal to the
difference between the actual and the expected demands).
3.2.2
Store Choice Model
As noted earlier, I need to estimate the store’s residual demand, not market demand. The
true process by which households choose where to shop is almost certainly related to their
decisions about exactly what to purchase once they get there. However, I am unaware of any
paper that simultaneously models the household’s store choice and product-level purchasing
decisions. A fully structural model would involve calculating the household’s expected
utility, net of travel costs, from visiting each of the stores in its choice set. Such a model
would also involve consumers forming expectations of the menu of prices they would face at
each store. Furthermore, the effects of these prices on store choice would almost certainly
depend on the household’s expected shopping basket on that purchase occasion. Given
my estimation procedure, this approach is far too computationally burdensome. Instead,
I approximate this choice, by assuming that households’ choices among the stores in my
sample follow a logit choice model. This model represents an approximation, of the true
model and does not directly correspond to any model of utility maximization. I use it
because it is computationally cheap, and because I believe the approximation is reasonable.
The model I estimate assumes that in week t, conditional on going shopping, household
i derives indirect utility:
0
uist = Dist
δ 0 − p0st δ 1 + ηist
(3.14)
from choosing store s at time t, where Dist is a vector of household characterstics interacted
with store indicator variables, p is a vector of price indices for several product categories at
store s, including soft drinks, and δ 0 and δ 1 are vectors of parameters.11 I defer discussion
of the exact specifications and discussion of the estimated coefficients to section 3.4.2.
This model implicitly assumes that households form expectations about current prices
11
In principle, the elements of δ 0 and δ 1 could be allowed to vary across households, though I do not do
so here.
CHAPTER 3. COSTS OF UNIFORM PRICING
107
(see Ho, Tang & Bell (1998)). I estimate several specifications using current prices, implicitly
assuming that the household is able to perfectly forecast (or learn) these prices. I also
estimate several other specifications using prices from the previous two weeks as predictors of
store choice12 , though in general I do not find that prices substantially influence households’
choice of store. These findings are consistent with those of Hoch, Dreze & Purk (1994) who
also find that consumers are largely inelastic to short term price changes in their choice of
store.
Although I am not aware of any papers that simultaneously address the household’s
decision of what to buy and where to shop, there is an extensive literature on store choice,
which I will not attempt to summarize in detail here. Instead I focus on the portions of
that literature that I have included in the specification of this model.
I follow Bell, Ho & Tang (1998) and Leszczyc, Sinha & Timmermans (2000) in incorporating household-level demographics and find that these are both statistically and economically significant in predicting store choice. While Rhee & Bell (2002), find that once
unobserved heterogeneity is accounted for, shoppers’ demographic characteristics are not
statistically significant in predicting the probability of switching, they do not allow the effects of these characteristics to vary across stores. Although I do not control for unobserved
heterogeneity, I find that the effect of household characteristics vary on a store-by-store
basis.
After demographic variables, I find that one of the most significant predictors of store
choice is whether the household visited the store in the previous two weeks. This is consistent
with the finding by Rhee & Bell (2002), who find that households are highly path-dependent
in their choice of store. However, this may simply be controlling for unobserved time-varying
heterogeneity among consumers.
I also account for the possibility that, as suggested by Bell & Lattin (1998), households
with higher expenditure levels tend to prefer stores with certain pricing formats. Specifically,
they found that households with large average expenditure levels tend to prefer so-called
Every Day Low Price (EDLP) stores to High-Low stores whose prices fluctuate more wildly
from week-to-week. To account for this effect, I interacted the household’s expenditure level
for the purchase occasion with store indicator variables. This allows shoppers who expect
to have high (or low) expenditure levels to seek out specific stores.
12
I also experimented with using longer lags, but found that they did not improve the predictive power of
the model.
CHAPTER 3. COSTS OF UNIFORM PRICING
108
Finally, as mentioned earlier, I take the household’s decision to go shopping to be governed by an exogenous process. This is consistent with evidence in Chiang, Chung & Cremers (2001) who find that consumers decision to shop is largely unaffected by marketing
mix variables.
3.2.3
Residual Demand
I complete the construction of the residual demand system by bringing together the product
choice and store choice models. The residual demand faced by store A is equal to the sum
over all households (that went shopping in that week) of the probability that the household
chose store A, multiplied by their expected purchases, conditional on choosing store A.
Hence, the expected demand system faced by store A in week t is:
X
(3.15)
"
#
R
0 δ0 − p δ1
X 1 X
exp
D
At
iAt
≈
q(ρ, β, pt , wit , εrit ) · P
0 δ0 − p δ1)
R
exp
(D
st
s∈{A,B,C,D,E}
ist
(3.16)
Et [Q(p)] =
E [qit |ρ, β, pt , wit ] · P A|Dist , pAt , p−At
i
i
r=1
In calculating this expected demand, I follow similar steps to those used in the estimation:
1. Take the estimated values of the parameters: β and ρ.
2. Draw R sets of εit for each observed purchase occasion. I use R = 30. These random
numbers need not be the same as those used to estimate the parameters above. Note
that in this case the number of purchase occasions is the total number of store trips
(to any store) made in that week, not just the purchase occasions from store A.
3. Using the expenditures from the actual purchase occasions as the budget constraints,
and the actual menu of prices in the week of the purchase occasion, take the current
parameters and solve explicitly solve the household’s utility maximization problem
(using numerical methods discussed in the Appendix).
4. For each purchase occasion, average over the R different purchase vectors to get the
expected purchases for that purchase occasion at the current set of parameter values.
5. For each purchase occasion, multiply these expected purchases by the probability of
choosing store A in that week.
CHAPTER 3. COSTS OF UNIFORM PRICING
3.3
109
Data
Although a trip to nearly any store offers many cases of uniform pricing, I focus exclusively
on grocery stores. There are a number of reasons for restricting my attention to grocery
stores. First, they offer literally thousands of examples of uniform pricing. Second, given
that grocery stores carry a large number of products,13 one might expect them to use
relatively sophisticated pricing techniques. Third, grocery stores are a significant portion
of the economy. In 1997, U.S. grocery stores had sales in excess of $368 billion, with
roughly 100,000 establishments (U.S. Economic Census, 1997). Fourth, with few exceptions,
groceries are not characterized by consumer uncertainty. Consumers are presumably quite
familiar with products’ characteristics as well as their preferences over these characteristics.
For example, there is not much uncertainty about what will be inside when you pop open a
can of Diet Coke. Finally, grocery stores conveniently offer the availability of scanner panel
data.
This paper utilizes two Chicago-area grocery datasets, both of which have already been
extensively studied. The principal dataset used has store-level price and quantity data for a
geographic cluster of five stores, from several different chains. It also contains a householdlevel component that allows us to observe the purchase patterns of individual households.
The second dataset used has store-level price, quantity, and cost data for all the grocery
stores of a single chain - Dominick’s Finer Foods. This section describes each of these
datasets in more detail, and briefly takes a rough look at the representativeness of the
panel.
3.3.1
IRI Basket Data
For a two-year period from 1991 to 1993, Information Resources Incorporated (IRI) collected
a panel dataset in urban Chicago. This dataset has both aggregate and micro components.
The aggregate component consists of weekly price and quantity14 data at the store/UPC
level for several different product categories at five geographically close stores. Throughout
the paper, these stores are referred to as stores A through E. As mentioned earlier, one
of these stores, which I will call store A, charged non-uniform prices for carbonated soft
drinks during this period. The micro-level component of this dataset contains carbonated
13
14
A typical grocery store carries over 14,000 different products.
Quantity sold includes sales to all customers, not just those in the panel.
CHAPTER 3. COSTS OF UNIFORM PRICING
110
soft drink purchase histories for 548 households at these five grocery stores over the twoyear period. The dataset also contains the households’ total grocery expenditure on each
purchase occasion. IRI paid these households to use a special electronic card that recorded
their purchases when they shopped at these stores. For the majority of the analysis, I use
only a subset of these households consisting of 262 households that visited store A (the
store at which I estimate demand) at least once during the two year sample period.
According to the documentation provided with the data, these five stores and 548 panelists were selected by IRI using two criteria: First, although very little information is
available on the actual sampling procedures used, IRI tried to create a stratified random
sample of households, reflective of the population in the area. Second, in order to avoid
the effects of unobserved market fluctuations, it was IRI’s goal to, as much as possible,
achieve a closed system. That is, IRI tried to include the stores that the households in
the panel would be most likely to shop at, in order to observe as large a fraction of their
grocery expenditure as possible. That IRI achieved this goal is supported by the fact for the
vast majority of the households, grocery expenditure at stores within the sample universe
appears to be fairly constant over time.
Tables 3.1 and 3.2 show the distribution of households’ expenditure at different stores
for all households, as well as for those who shopped at store A at least once. The mean
weekly expenditure by a household shopping at Store A was $22, while the median was
$15.15 This is less than stores B and C, but similar to stores D and E. Even households
that shopped at store A at least once tended to spend more at these other stores, although
the majority ($350,000) of their total expenditure of $610,000 over the period was at store
A.
15
For clarity, all dollar references in this section are nominal. Hence, prices in 1991 use 1991 dollars,
etc. I use this approach because retailers and wholesale prices over the period do not appear to move with
inflation. For reference, one 1991 dollar is equivalent to $1.39 2004 dollars, a 1992 dollar is equivalent to
$1.35 2004 dollars, and a 1993 dollar is equivalent to $1.31 2004 dollars.
Store
Store
Store
Store
Store
Total
Store
A
B
C
D
E
Table 3.1: Distribution of Purchase Occasion Expenditure by Store, All Purchase Occasions
Number of
Expenditure in Dollars
Purchase Mean Standard Minimum
25th
Median
75th
Maximum
Total
Occasions
Deviation
Percentile
Percentile
($000’s)
16,008
22
21
0.25
8
15
27
228
350
10,063
38
36
0.16
14
27
50
281
390
13,733
44
38
0.14
17
33
60
378
600
7,835
22
30
0.34
6
12
23
325
170
5,637
26
28
0.34
9
16
33
254
150
53,516
31
32
0.14
10
20
40
378
1,700
CHAPTER 3. COSTS OF UNIFORM PRICING
111
Table 3.2: Distribution of Purchase Occasion Expenditure by Store, Purchases Made by Panelists Who Visited Store A at
least Once.
Store
Number of
Expenditure in Dollars
Purchase Mean Standard Minimum
25th
Median
75th
Maximum
Total
Occasions
Deviation
Percentile
Percentile
($000’s)
Store A
16,008
22
21
0.25
8
15
27
228
350
Store B
615
32
27
1.24
14
23
41
177
20
Store C
2,234
34
38
0.14
13
24
40
378
8
Store D
6,435
15
29
0.34
5
10
17
188
99
Store E
3,479
19
19
0.34
7
13
26
201
68
Total
28,782
21
23
0.14
8
14
26
378
610
CHAPTER 3. COSTS OF UNIFORM PRICING
112
CHAPTER 3. COSTS OF UNIFORM PRICING
113
Unlike many previous papers which have estimated brand-level demand, this paper
estimates UPC-level demand. Over a two year period, a typical grocery store sells over 200
different items in the carbonated soft drink category. The vast majority of these products
are offered only rarely, or quickly enter and exit. Because this paper uses panel purchases to
estimate demand, and many of these products are only rarely (or never) purchased by the
panel, it not practical to estimate the households’ demand for them. Instead, I estimate the
households’ demand for the 25 products with the largest market share by volume. These
products represent 71% of the Store A’s carbonated soft drink sales by volume, and 69% of
their total soft drink sales by dollar value.
The products included in the analysis are shown in Table 3.3. Of these, three varieties
(8 items) were distributed by the Coca Cola Corp., two varieties (8 items) were distributed
by Pepsi Co., two varieties (4 items) were distributed by Dr. Pepper/7Up, two varieties (4
items) were distributed by the Royal Crown Corp., and one variety (1 items) was distributed
by an independent producer under a private label.
Table 3.3: Variety and Size Distribution of in the Dataset, grouped by Manufacturer
Manufacturer
Variety
Coca Cola
Coke
Diet Coke
Diet CF Coke
Pepsi
Diet Pepsi
RC
Diet Rite
7Up
Diet 7Up
Private Label
Pepsico
RC Corp.
DP/7Up
PL
Total Number of Items
Number of 12oz servings (Liters in parentheses)
1
5.63
6
8.45
12
24
(0.36) (2.0) (2.13) (3.0) (4.26)
(8.52)
!
1
!
!
!
!
!
!
!
!
8
!
!
2
!
!
2
!
!
!
!
!
!
!
!
!
!
!
!
5
7
Number of
Sizes Avail.
3
3
2
4
4
1
3
2
2
1
25
Some descriptive statistics on the price and sales volume for these products is shown
in Table 3.4. The price of a 12-ounce serving of carbonated soft drink varied from a high
of $0.49 as part of a 12-pack of 12-ounce cans of Diet Coke, to a low of $0.12 for a single
can of the Private Label cola. Most products appear to have had either an end-of-aisle
display, or a mention in the store’s circular in between one-third to one-half of the weeks.
The notable exceptions to this were the 3L bottle of Pepsi, the 2L bottle of Diet 7up, and
the store brand which received significantly less advertising (as measured by circular and
display activity).
2L Bottle
12-pack 12oz cans
24-pack 12oz cans
2L Bottle
12-pack 12oz cans
24-pack 12oz cans
12-pack 12oz cans
24-pack 12oz cans
2L Bottle
6-pack 12oz cans
12-pack 12oz cans
24-pack 12oz cans
2L Bottle
3L Bottle
12-pack 12oz cans
24-pack 12oz cans
2L Bottle
2L Bottle
3L Bottle
24-pack 12oz cans
2L Bottle
6-pack 12oz cans
2L Bottle
24-pack 12oz cans
12oz Can
Mean
Price
($)
0.22
0.31
0.27
0.23
0.31
0.27
0.33
0.27
0.22
0.39
0.31
0.26
0.23
0.20
0.31
0.26
0.21
0.37
0.21
0.26
0.17
0.21
0.18
0.26
0.19
Maximum
Price
($)
0.32
0.47
0.34
0.32
0.49
0.34
0.47
0.34
0.32
0.48
0.47
0.34
0.32
0.20
0.47
0.34
0.32
0.48
0.32
0.34
0.32
0.32
0.20
0.32
0.22
S.D.
of
Price
0.06
0.09
0.06
0.06
0.10
0.06
0.10
0.06
0.06
0.10
0.10
0.06
0.06
0.00
0.10
0.06
0.06
0.11
0.06
0.06
0.02
0.05
0.03
0.05
0.03
Mean
Price
at DFF
0.26
0.34
0.24
0.26
0.34
0.25
0.36
0.21
0.25
0.22
0.33
0.20
0.25
0.04
0.34
0.24
0.25
0.39
0.24
0.19
0.19
0.24
0.19
0.21
NA
Mean DFF
Wholesale
Price
0.23
0.29
0.18
0.23
0.31
0.19
0.30
0.15
0.22
0.17
0.28
0.15
0.22
0.03
0.30
0.18
0.22
0.30
0.21
0.14
0.15
0.19
0.15
0.18
NA
S.D. of DFF
Wholesale
Price
0.04
0.06
0.07
0.05
0.04
0.08
0.04
0.06
0.04
0.09
0.08
0.06
0.05
0.05
0.05
0.07
0.04
0.10
0.05
0.05
0.05
0.04
0.00
0.02
NA
Mean Number
of Servings Sold
Per Week
30
29
17
21
28
28
15
14
32
13
25
50
14
7
21
23
20
15
12
14
30
13
10
19
20
All prices are in nominal Dollars per 12-ounce serving. Source: IRI and DFF Data.
Private Label
Diet 7up
7up
RC
Diet Rite
Diet Pepsi
Pepsi
Diet Caffeine Free Coke
Diet Coke
Coke
Minimum
Price
($)
0.12
0.17
0.17
0.12
0.17
0.15
0.17
0.17
0.12
0.24
0.16
0.17
0.12
0.20
0.10
0.17
0.12
0.17
0.12
0.17
0.12
0.12
0.11
0.17
0.12
S.D. of Number
of Servings Sold
Per Week
26
30
27
27
31
41
20
22
23
15
26
84
16
7
24
35
16
22
16
24
17
14
20
34
13
% of Weeks on
End of Aisle
Display
47
44
38
47
33
38
22
33
46
17
38
42
40
2
46
42
57
28
55
43
31
37
11
37
2
% of Weeks
Featured in
Circular
61
51
46
56
49
45
40
45
58
33
50
52
56
1
50
47
54
32
53
44
35
24
9
31
25
Table 3.4: Summary Statistics for Prices and Quantities Sold at Store A, grouped by Manufacturer
% Market
Share
by Volume
3.93
3.22
2.67
2.65
3.19
3.95
2.06
2.09
5.03
1.60
2.61
5.23
2.02
1.41
2.06
3.19
2.74
1.67
1.49
2.03
5.88
1.68
1.43
2.67
4.13
% Market
Share
by Sales
3.63
3.77
2.61
2.26
3.78
4.16
2.37
2.26
4.68
2.33
2.91
4.96
1.86
1.25
2.30
2.99
2.37
2.23
1.17
1.92
4.51
1.41
1.00
2.56
3.39
CHAPTER 3. COSTS OF UNIFORM PRICING
114
CHAPTER 3. COSTS OF UNIFORM PRICING
115
Both the traditional logit model and the new model I propose reduce the dimensionality
of the demand system parameter space by assuming that households’ preferences over products are driven by product characteristics. The store’s residual demand Qjt (·) for product j
is denominated in twelve ounce servings of carbonated soft drink. The characteristics used
in the analysis are: calories (per 12 ounce serving), milligrams of sodium (per 12 ounce serving), milligrams of caffeine (per 12 ounce serving), grams of sugar (per 12 ounce serving),
as well as indicator variables for the presence of citric acid, phosphoric acid, and whether it
is a diet drink. These physical characteristics were obtained by contacting the manufacturers of the products, and, to the best of my knowledge, represent the characteristics of the
products during the relevant time period. I also include indicator variables for size, brand,
and whether it was featured in store A’s weekly circular, or an in-store display (in store A),
as well as a constant common to all products. These characteristics were chosen based on
earlier work by Dubé (2001). These characteristics are the elements of the A matrix, and
are shown in Table 3.5. This table also shows the number of weeks that each product was
available. For example, the 12-pack of 12-ounce cans of Diet Pepsi was unavailable for 16
of the 104 weeks, while the 24-pack of 12-ounce cans of Diet Caffeine Free Coke, and the
24-pack of 12-ounce cans of Diet Caffeine Free Coke were not available for 15 weeks.
Private Label
2L Bottle
12-pack 12oz cans
24-pack 12oz cans
2L Bottle
12-pack 12oz cans
24-pack 12oz cans
12-pack 12oz cans
24-pack 12oz cans
2L Bottle
6-pack 12oz cans
12-pack 12oz cans
24-pack 12oz cans
2L Bottle
3L Bottle
12-pack 12oz cans
24-pack 12oz cans
2L Bottle
2L Bottle
3L Bottle
24-pack 12oz cans
2L Bottle
6-pack 12oz cans
2L Bottle
24-pack 12oz cans
12oz Can
104
99
99
104
99
99
100
89
104
104
90
100
104
94
88
101
104
104
104
98
104
104
104
89
104
Weeks
Sold
140
140
140
0
0
0
0
0
150
150
150
150
0
0
0
0
160
0
0
0
140
140
0
0
140
Calories
50
50
50
40
40
40
40
40
37.5
37.5
37.5
37.5
37.5
37.5
37.5
37.5
50
45
45
45
75
75
45
45
50
Sodium
(mg)
39
39
39
0
0
0
0
0
40.5
40.5
40.5
40.5
0
0
0
0
42
0
0
0
39
39
0
0
39
Sugar
(g)
34
34
34
45
45
45
0
0
38
38
38
38
36
36
36
36
43
48
48
48
0
0
0
0
34
Caffeine
(mg)
Contains
Phosphoric
Acid
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
Contains
Citric
Acid
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
0
0
0
0
1
1
1
1
0
1
1
1
0
0
1
1
0
Diet
% of Weeks
Featured
in Circular
47
32
42
46
32
41
32
44
44
20
36
42
44
1
33
42
32
20
5
23
45
22
44
39
20
% of Weeks on
End of Aisle
Display
35
27
32
36
23
32
19
31
28
11
27
33
27
2
30
37
26
29
6
34
42
21
42
38
3
Characteristics are per 12oz serving. Source: Coca Cola Corp., Pepsico, Royal Crown Corp., and Cadbury Beverages.
Private Label
7up
DP/7up
Diet 7up
RC
Diet Rite
Diet Pepsi
Pepsi
Diet Caffeine Free Coke
Royal Crown
Pepsico
Coke
Coca Cola
Size
Table 3.5: Characteristics of Products in the Dataset, grouped by Manufacturer
Diet Coke
Variety
Manufacturer
CHAPTER 3. COSTS OF UNIFORM PRICING
116
CHAPTER 3. COSTS OF UNIFORM PRICING
3.3.2
117
Dominick’s Finer Foods Data
I supplement the IRI data with demographic and wholesale price data from Dominick’s
Finer Foods, a grocery chain located in the greater Chicago metropolitan area. From
1989 to 1997, through an arrangement with the University of Chicago Graduate School of
Business, Dominick’s kept track of store-level, weekly unit sales and price for every UPC
symbol for a number of product categories, including carbonated soft drinks. This dataset
has weekly store sales totals in these product categories as well as store area demographics
pulled from census data. In addition, the dataset contains the actual wholesale prices that
Dominick’s paid for each good. For a more thorough description of the dataset, see Hoch
et al. (1994).16
Finally, because this dataset covers the same time and geographic area as the IRI dataset,
one can match many of the products across the two datasets, giving us a measure of the
wholesale prices for these products. To the best of my knowledge this matching has not
been done in previous work.
3.4
3.4.1
Results
Structural Model
The parameter estimates and standard errors17 from the structural model are presented in
Table 3.6. As discussed in chapter 2, the nonlinearity of the model makes these parameter
estimates difficult to interpret directly, however we can make some inferences, particularly
relative to each other.
The characteristic that appears to have the largest affect on households’ soft drink
purchasing decisions is the indicator variable for whether the product is sold on a holiday.
This characteristic has both the largest maximal marginal effect18 (βholiday ρholiday ) and a β
16
This dataset is publicly available from the Kilts Center at the University of Chicago Graduate School
of Business web page:
http://gsbwww.uchicago.edu/kilts/research/db/dominicks/
√ b
17
The standard errors shown in Table 3.6 are calculated using the
N (0, V ) as the
fact that:
n(θn −θ0 0 ) →d −1
−1
0
0
1
number of observations n goes to infinity. V = [G0 W G0 ] G0 W Ω0 (1 + R ) W G0 [G0 W G0 ] , where W is
P I PT
∂E[qit |θ,pt ,wit ]
b IT = 1
the weighting matrix. I obtain a consistent estimator of V by using G
i=1
t=1 xit
IT
PI PT ∂θ
1
b
b
b
and θ = [β, ρb]. I use the estimated parameters β and ρ to compute ΩIT = IT i=1 t=1 ((qit −
b ρ
b ρ
E[qit |β,
b, pt , wit ])xit )((qit − E[qit |β,
b, pt , wit ])xit )0 which is a consistent estimator of Ω0 . For further discussion, see Gouriéroux & Monfort (1996). I use a diagonal weighting matrix (W ), with the elements scaled
by the sum of squares of each of the instruments.
18
The term “maximal marginal effect” is somewhat misleading. It is actually the most positive marginal
CHAPTER 3. COSTS OF UNIFORM PRICING
118
that is statistically significantly greater than zero. The effect of the Holiday characteristic is
offset somewhat by the fact that ρholiday is quite close to zero. This means that although the
marginal utility from soda on holidays starts off higher, its second derivative is more negative
than for other characteristics. Taken together, these two facts imply that households are
more likely to purchase soft drinks on holidays, but not likely to substantially increase the
quantity that they purchase.
After the Holiday characteristic, the Coke and Pepsi characteristics have the largest
effects, based on their high maximal marginal utility values. Interestingly, while βP epsi is
statistically significantly different from zero, βCoke is not. Given that both βP epsi and ρP epsi
are relatively large, consumers differentially prefer Pepsi (and Coke, to a lesser degree) to
other soft drinks – they are both more likely to purchase these brands, and more likely to
purchase more of them.
At the other end of the spectrum, due to the fact that both their β’s and their ρ’s
are relatively small, Sodium and Caffeine do not appear to significantly affect consumers
purchasing behavior (although their coefficients are imprecisely estimated).
More readily interpretable than the parameter estimates are the own and cross-price
elasticities that they imply. In the logit model, the cross-price elasticities are infamously
dependent upon market shares. These restrictions are evident in Table 3.7, which shows a
matrix of own and cross price elasticities for several sizes and varieties of Coke and Pepsi.
The logit model predicts, for example, that the demand for 24-packs (288oz) of Diet Coke
increases by 0.46% from either a 1% increase in the price of a 2-liter bottle of Diet Coke or
a 1% increase in the price of a 2-liter bottle of Pepsi. This contrasts sharply with the price
elasticities implied by my structural model, presented in Table 3.8. As clearly seen, my
model allows for a rich pattern of cross-price elasticities. In addition to allowing cross-price
elasticities to vary across products, my model also allows for negative off-diagonal price
elasticities, suggesting that complementarities exist. For example, if households tend to
purchase both 12-packs of Coke and 6-packs of 7Up, then an increase in the price of Coke
may cause more people to buy Pepsi, but it will also lead people both to buy less Coke and
to buy less of the Coke/7Up bundle. This flexibility is not possible using the traditional
logit model.
effect. When βc is negative, ρc is necessarily greater than one, and hence, the marginal effect is increasing
in magnitude as quantity increases.
CHAPTER 3. COSTS OF UNIFORM PRICING
119
Table 3.6: Parameter Estimates from Structural Model of Product Choice
β
Characteristic
Constant
Calories
Sugar
Sodium
Caffeine
Phosphoric Acid
Citric Acid
Cola
Flavored
Single Serving
288oz
Diet
Coke
Diet Coke
Pepsi
7up
RC
Jewel
Holiday
Feature
Display
Units
per 12oz
g per 12oz
mg per 12 oz
mg per 12oz
Indicator Variable
”
”
”
”
”
”
”
”
”
”
”
”
”
”
”
coeff.
0.0001
-0.0001
0.0054
0.0000
0.0000
0.5431
0.1674
0.2588
0.3671
0.3174
-0.0003
-0.0076
0.5836
-0.0115
0.4477
0.2333
0.2380
0.4449
5.3081
0.2232
0.4830
ρ
s.e.
(0.5435)
(0.6231)
(1.2754)
(0.8227)
(0.3318)
(0.1842)
(0.0708)
(0.1187)
(0.1610)
(0.0311)
(0.0046)
(0.0059)
(0.3054)
(0.0062)
(0.0216)
(0.1140)
(0.0906)
(0.3412)
(1.8195)
(0.1304)
(0.4114)
coeff.
0.7471
1.2581
0.8134
0.1608
0.0957
0.3742
0.6585
0.4188
0.5139
0.5690
1.6013
1.3875
0.4090
1.6398
0.4764
0.5892
0.4824
0.3507
0.0586
0.3683
0.2712
s.e.
(0.6382)
(0.5957)
(0.5749)
(0.2259)
(0.0737)
(0.2111)
(0.2603)
(0.3062)
(0.3254)
(0.0640)
(0.4515)
(0.4267)
(0.2517)
(0.0793)
(0.0778)
(0.2415)
(0.1868)
(0.2108)
(0.0305)
(0.2217)
(0.2434)
Maximal
Marginal
Effect
(βρ)
0.0001
-0.0001
0.0044
0.0000
0.0000
0.2032
0.1102
0.1084
0.1886
0.1806
-0.0005
-0.0105
0.2387
-0.0189
0.2133
0.1374
0.1148
0.1560
0.3111
0.0822
0.1310
This table shows parameter estimates from the structural model under two different specifications. The right-most column is the product of β and ρ.
Table 3.7: Selected Own and Cross Price Elasticities from Homogenous Logit Model of
Product Choice
Coke
Diet Coke
Pepsi
Diet Pepsi
288oz
2L
288oz
2L
288oz
2L
288oz
2L
Coke - 288oz
-2.398 0.015 0.046 0.014 0.073 0.019 0.051 0.013
Coke - 2L
0.052 -1.992 0.046 0.014 0.073 0.019 0.051 0.013
Diet Coke - 288oz 0.052 0.015 -2.404 0.014 0.073 0.019 0.051 0.013
Diet Coke - 2L
0.052 0.015 0.046 -2.001 0.073 0.019 0.051 0.013
Pepsi - 288oz
0.052 0.015 0.046 0.014 -2.203 0.019 0.051 0.013
Pepsi - 2L
0.052 0.015 0.046 0.014 0.073 -2.002 0.051 0.013
Diet Pepsi - 288oz 0.052 0.015 0.046 0.014 0.073 0.019 -2.392 0.013
Diet Pepsi - 2L
0.052 0.015 0.046 0.014 0.073 0.019 0.051 -2.000
Diet Rite, 2L Bottle
Diet Rite, 24-pack 12oz cans
RC, 2L Bottle
Diet Pepsi, 2L Bottle
Diet Pepsi, 24-pack 12oz cans
Diet Pepsi, 12-pack 12oz cans
Diet Pepsi, 3L Bottle
Pepsi, 6-pack 12oz cans
Pepsi, 2L Bottle
Pepsi, 24-pack 12oz cans
Pepsi, 12-pack 12oz cans
Private Label, 12oz Can
Diet Caffeine Free Coke, 24-pack 12oz cans
Diet Caffeine Free Coke, 12-pack 12oz cans
Diet Coke, 2L Bottle
Diet Coke, 24-pack 12oz cans
Diet Coke, 12-pack 12oz cans
Coke, 2L Bottle
Coke, 24-pack 12oz cans
Coke, 12-pack 12oz cans
Diet 7up, 2L Bottle
Diet 7up, 24-pack 12oz cans
7up, 6-pack 12oz cans
7up, 2L Bottle
7up, 2L Bottle
-4.158 0.054 0.056 0.131 0.023 0.014 0.036 -0.015 -0.001 0.008 -0.005 0.000 0.013 0.030 0.020 0.035 0.044 0.001 -0.002 0.000 0.007 0.018 0.006 0.003 0.006
7up, 6-pack 12oz cans
0.081 -7.921 0.049 0.121 0.040 0.034 0.034 0.011 0.025 -0.003 0.024 0.013 0.026 0.056 0.024 0.028 0.061 -0.008 0.033 0.016 -0.001 0.012 0.000 0.026 -0.003
Diet 7up, 24-pack 12oz cans
0.082 -0.003 -3.409 0.109 0.034 0.019 -0.049 0.064 0.039 0.011 0.061 0.033 0.015 0.035 0.026 0.000 0.025 0.007 0.028 0.034 0.016 -0.020 0.023 0.057 0.023
Diet 7up, 2L Bottle
0.063 0.072 0.036 -6.704 -0.019 -0.010 -0.009 0.010 0.008 0.014 0.006 0.008 -0.010 0.005 0.002 0.009 0.001 0.010 0.012 0.009 0.016 -0.004 0.019 0.011 0.026
Coke, 12-pack 12oz cans
0.052 -0.131 0.020 -0.056 -4.584 0.071 0.088 0.084 0.068 0.090 0.100 0.064 0.063 0.109 0.049 0.065 0.078 0.028 0.083 0.045 0.056 0.042 0.060 0.054 0.076
Coke, 24-pack 12oz cans
0.048 0.055 0.028 -0.032 0.119 -2.971 0.134 0.093 0.108 0.119 0.109 0.082 0.070 0.106 0.059 0.070 0.127 0.033 0.083 0.056 0.073 0.059 0.071 0.076 0.065
Coke, 2L Bottle
0.027 0.028 -0.017 -0.022 0.094 0.053 -4.048 0.074 0.043 0.065 0.067 0.037 0.032 0.039 0.022 0.050 0.074 0.026 0.010 0.012 0.045 0.034 0.033 0.022 0.068
Diet Coke, 12-pack 12oz cans
0.020 -0.113 0.041 0.070 0.088 0.070 0.105 -4.757 0.075 0.134 0.188 0.067 0.044 0.081 0.045 0.052 0.054 0.054 0.065 0.053 0.066 0.021 0.090 0.072 0.093
Diet Coke, 24-pack 12oz cans
-0.003 0.155 0.057 0.068 0.111 0.121 0.143 0.113 -2.891 0.138 0.140 0.115 0.058 0.120 0.057 0.049 0.094 0.070 0.137 0.079 0.098 0.028 0.111 0.094 0.145
Diet Coke, 2L Bottle
0.012 0.029 0.006 0.027 0.081 0.044 0.072 0.103 0.055 -3.981 0.093 0.043 0.017 0.037 0.019 0.041 0.054 0.041 0.038 0.028 0.055 0.017 0.069 0.036 0.082
Diet Caffeine Free Coke, 12-pack 12oz cans 0.000 -0.057 0.040 0.030 0.099 0.069 0.110 0.180 0.083 0.117 -4.979 0.071 0.042 0.100 0.044 0.047 0.081 0.055 0.093 0.063 0.085 0.018 0.093 0.071 0.104
Diet Caffeine Free Coke, 24-pack 12oz cans 0.003 -0.013 0.048 0.044 0.123 0.089 0.124 0.099 0.115 0.133 0.113 -2.516 0.060 0.057 0.055 0.056 0.084 0.066 0.075 0.071 0.074 0.025 0.131 0.080 0.102
Private Label, 12oz Can
0.041 0.036 0.016 -0.039 0.088 0.057 0.083 0.066 0.044 0.049 0.059 0.044 -2.191 0.068 0.059 0.065 0.069 0.032 0.054 0.047 0.053 0.041 0.050 0.062 0.059
Pepsi, 12-pack 12oz cans
0.060 0.053 0.033 0.033 0.124 0.067 0.097 0.110 0.056 0.069 0.116 0.045 0.060 -4.085 0.060 0.075 0.074 0.055 0.101 0.051 0.042 0.046 0.068 0.066 0.092
Pepsi, 24-pack 12oz cans
0.105 -0.131 0.058 0.055 0.151 0.109 0.110 0.121 0.097 0.108 0.156 0.088 0.125 0.212 -2.591 0.133 0.121 0.150 0.199 0.121 0.137 0.113 0.138 0.124 0.146
Pepsi, 2L Bottle
0.080 -0.052 0.003 0.098 0.071 0.050 0.094 0.061 0.034 0.073 0.053 0.036 0.052 0.069 0.057 -3.286 0.096 0.075 0.089 0.059 0.096 0.063 0.105 0.049 0.087
Pepsi, 6-pack 12oz cans
0.076 0.037 0.033 0.012 0.061 0.064 0.124 0.087 0.050 0.067 0.095 0.045 0.051 0.127 0.047 0.088 -6.029 0.076 0.123 0.059 0.121 0.066 0.109 0.070 0.092
Diet Pepsi, 3L Bottle
-0.002 0.000 0.006 0.013 0.029 0.017 0.044 0.060 0.031 0.057 0.050 0.028 0.019 0.044 0.039 0.050 0.078 -2.972 0.040 0.038 0.096 0.039 0.075 0.042 0.058
Diet Pepsi, 12-pack 12oz cans
0.006 -0.008 0.041 0.040 0.072 0.054 0.039 0.068 0.068 0.077 0.100 0.053 0.051 0.114 0.063 0.094 0.090 0.088 -4.340 0.064 0.117 0.057 0.088 0.091 0.130
Diet Pepsi, 24-pack 12oz cans
0.006 -0.095 0.064 0.090 0.112 0.084 0.065 0.109 0.110 0.114 0.168 0.090 0.084 0.132 0.097 0.118 0.120 0.113 0.147 -3.122 0.126 0.085 0.134 0.112 0.126
Diet Pepsi, 2L Bottle
0.017 0.011 0.011 0.075 0.027 0.022 0.052 0.049 0.032 0.060 0.067 0.029 0.020 0.014 0.036 0.054 0.086 0.069 0.049 0.047 -4.250 0.041 0.069 0.054 0.063
RC, 2L Bottle
0.052 0.074 -0.027 -0.005 0.077 0.061 0.097 0.034 0.024 0.042 0.039 0.021 0.050 0.065 0.066 0.091 0.113 0.087 0.057 0.060 0.095 -1.825 0.094 0.060 0.089
RC, 3L Bottle
0.010 0.021 0.009 0.029 0.024 0.014 0.023 0.030 0.023 0.045 0.035 0.024 0.012 0.022 0.015 0.029 0.055 0.030 0.027 0.020 0.033 0.018 -4.157 0.026 0.068
Diet Rite, 24-pack 12oz cans
0.006 0.074 0.052 0.053 0.096 0.058 0.056 0.135 0.066 0.074 0.092 0.050 0.056 0.069 0.054 0.054 0.097 0.063 0.075 0.057 0.083 0.047 0.103 -3.693 0.128
Diet Rite, 2L Bottle
0.008 0.002 0.009 0.042 0.039 0.016 0.044 0.041 0.027 0.045 0.036 0.019 0.016 0.030 0.019 0.029 0.037 0.035 0.031 0.028 0.037 0.022 0.064 0.041 -4.980
RC, 3L Bottle
Table 3.8: Matrix of Estimated Average Own and Cross-Price Elasticities from Structural Model of Product Choice
CHAPTER 3. COSTS OF UNIFORM PRICING
120
CHAPTER 3. COSTS OF UNIFORM PRICING
3.4.2
121
Store Choice Model
As discussed in section 3.2.2, I estimate a conditional logit model of store choice, where
the choice is conditional on shopping at one of the stores that we observe.19 The results
from eight different specifications of the store choice model are presented in Tables 3.9, 3.9,
and 3.11. Specifications I-III have identical demographic variables, but involve progressively
longer lagged price indices. Specifications IV-VI are nearly identical to I-III. They differ only
in the fact that they include an additional variable that measures whether the household
made a purchase at the store in the previous two weeks. Specifications VII and VIII
simultaneously incorporate price indices from several periods for a subset of products, both
with and without the lagged store choice variable.20
The two main results of my analysis of store choice are: (1) that observable demographics
significantly affect households’ choice of store, even after incorporating a measure of path
dependence, and (2) for the product categories for which we have data, households are (at
least in the short term) relatively inelastic with respect to store choice.
Table 3.9 presents the coefficients on the demographic variables in Specifications III and
VI. These coefficients remain essentially unchanged with respect to different combinations
of price index variables. The demographic variables include an indicator variable equal to
one if the household made a purchase (of any kind) at the store in the previous two weeks.
This lagged store choice variable accounts for two things: First, it acts like a household-level
fixed effect, and second, it accounts for fact that it is easier to shop at a store when you
know that store’s layout.21
With respect to observable demographics, I find that people with lower incomes, were
less likely to shop at stores A and B, and more likely to shop at stores C, D and E.
Having an unemployed female in the household at the beginning of the sample period was
a significant factor in store choice (although having an unemployed male was not), with
unemployed female households much more likely to shop at store C. Non-white households
were more likely to shop at store A, and households that subscribed to a newspaper were
19
For two different sets of stores, there are a non-trivial number of households that shop at both stores.
This is true for stores A&D and B&C. I treat going to both stores as a separate alternative. In generating
the price variables in this case, I use the lower of the two price indices of the stores in the bundle.
20
Note: The coefficients are all measured with respect to Store C. If a household made multiple purchases
at the same store in the same week, I collapsed these into a single purchase occasion.
21
In the case of the “bundled” stores, the variable is created slightly differently. For example, if you went
to stores A and D last two weeks, then this week the variable would be one for store A, store D, and the
bundle of stores A and D. If you only went to Store A in the last two weeks, then this week the variable
would be one for store A, zero for store D, and 0.5 for the bundle of stores A and D.
CHAPTER 3. COSTS OF UNIFORM PRICING
122
substantially more likely to shop at stores A, D, and to a lesser extent E. They are less likely
to shop at store B. Furthermore, the coefficients on these demographic variables are largely
invariant to the other aspects of the specification (depending primarily on whether lagged
store choice is included). For this reason, I only present these coefficients for Specifications
III and VI in Table 3.9.
Although the effects of expenditure levels on store choice are statistically significant,
they are not economically so.22
As mentioned in section 3.2.2, I explored a variety of specifications for the store choice
model, including lags of price indices, alternative measures of price, additional demographic
variable, and additional index variables measuring the fraction of the category that was
featured in the store circular or an end-of-aisle display.
The evidence from the effects of price on store choice were less encouraging, though,
as noted earlier, they are in substantial agreement with the literature. The coefficients
for price of Cookies and Detergent have the expected sign, and are statistically significant.
Unfortunately, although some of the price index variables are significant, many are only
significant at the five percent level. Given the number of coefficients, it is not surprising
that a subset are statistically significant. Furthermore, many of the price index variables
do not have the expected sign. Cat food, bar soap, and yogurt, for example, both have
statistically significantly positive coefficients in several specifications. This suggests that
these price indices may be capturing effects other than (i.e., that they are correlated with
an omitted variable). This gives me less confidence in interpreting the coefficient on soft
drinks, which (although the point estimates do not move too wildly) is only significant when
I do not account for path dependence.
While I do not report their results here, I also estimated models using alternative price
indices, including the Stone price index and a variety of indices measuring the extent of
discounts offered. These alternative measures of price did not appear to have any effect on
store choice.
22
I explored using logged expenditure, as well as nonlinear effects from expenditure levels, but the effects
were not substantially different.
CHAPTER 3. COSTS OF UNIFORM PRICING
3.4.3
123
Counter-Factuals
Marginal Costs
In order to use the estimated demand system to recover estimates of the expected profits
lost from uniform pricing, I return to the steps discussed in section 1.3. As discussed there,
in order to recover the marginal costs for each product in each week, I need to make an
assumption about the actual price-setting behavior of the retailer during the sample period.
The assumption I choose to make is that the retailer maximizes total weekly profit for the
soft drink category, and charges the profit-maximizing price for each product in each week.
This assumption implies the following J first-order conditions (one for each good j) for each
week t:
X
∂Πt,N on−U nif orm
∂Et [Qkt (pt )]
= Et [Qjt (pt )] +
(pkt − ckt )
=0
∂pjt
∂pjt
(3.17)
k∈J
By taking these first-order conditions numerically23 , I am able to solve the system of J
equations and J unknowns for each week – the cjt ’s – and recover the implied weekly
marginal costs for each product. Note that in recovering these marginal costs, the level of
Qjt drops out. That is, the implied marginal cost is independent of the total number of
households shopping in that week.
In recovering the marginal costs from the first-order conditions of the retailer, I am
implicitly assuming that the demand system that I have estimated is the true demand
system (and by association, that it is the demand system that store A used in setting its
prices), and that in each week, the retailer knows the distribution of the budgets of the
households. Solving this system of equations, gives me the implied marginal cost cjt for
each good, at store A, during week t.
Table 3.12 contains summary statistics for these implied marginal costs. In general, the
estimated marginal costs are substantially lower than the wholesale prices reported in the
Dominick’s dataset (taken from a geographically proximate competing grocery retailer) and
shown in Table 3.4. This discrepancy may be explained by the fact that store A is part of
a large chain, and therefore may have received preferential wholesale prices. Additionally,
these implied marginal costs may be capturing the effects of slotting allowances or nonlinearities in wholesale prices (such as block discounts) not accounted for in the Dominick’s
23
I do this by (1) choosing a fixed number of households, (2) simulating demand from these households
at the observed prices at store A in week t, (3) numerically taking the derivatives of demand for each good
with respect to all other goods.
CHAPTER 3. COSTS OF UNIFORM PRICING
124
data (see Israilevich (2004)).
If I am underestimating the true marginal costs, the likely source would be that suggest
that either retailers are pricing non-optimally with respect to the soft drink category, that
the estimated demand model is incorrect, or that the assumed supply model is incorrect
(e.g., retailers may be engaging in cross-category subsidization).
CHAPTER 3. COSTS OF UNIFORM PRICING
125
Table 3.9: Coefficients on Demographic Variables from Specifications III and VI of the
Conditional Logit Model of Store Choice
Specification III
Constant
Expenditure ($)
Log(Income)
Unemployed
Female
Unemployed
Male
Non-white
Subscribes to
Newspaper
Household Has
No Kids
A
4.633
[0.190]
-0.012
[0.000]
-0.443
[0.019]
-0.768
[0.038]
-0.053
[0.069]
0.921
[0.035]
0.428
[0.034]
-0.920
[0.209]
A and D
4.595
[0.235]
-0.010
[0.001]
-0.507
[0.024]
-0.479
[0.049]
-0.542
[0.097]
0.243
[0.045]
0.089
[0.045]
-16.726
[742.304]
B
2.38
[0.203]
-0.001
[0.000]
-0.256
[0.020]
-0.070
[0.036]
-0.427
[0.078]
0.271
[0.037]
-0.463
[0.038]
-1.071
[0.248]
Store
B and C
D
1.18
-0.544
[0.279]
[0.276]
0.002
-0.014
[0.000]
[0.001]
-0.246
0.030
[0.028]
[0.027]
0.209
-0.640
[0.049]
[0.055]
-0.176
-2.183
[0.105]
[0.256]
-0.710
-0.907
[0.063]
[0.067]
-0.311
0.361
[0.052]
[0.044]
1.752
-16.315
[0.170]
[713.396]
E
-4.904
[0.296]
-0.012
[0.001]
0.452
[0.028]
-1.110
[0.060]
0.127
[0.106]
-0.819
[0.065]
0.150
[0.043]
-0.807
[0.299]
Specification VI
(Adds an Indicator Variable for Whether the Household Visited that Store in the Last Two Weeks)
Store
A
A and D
B
B and C
D
E
Constant
2.586
3.887
2.462
1.849
-0.679
-1.668
[0.309] [0.321]
[0.294] [0.306]
[0.387]
[0.432]
Expenditure ($)
-0.011 -0.007
0.000
0.002
-0.013
-0.014
[0.001] [0.001]
[0.001] [0.001]
[0.001]
[0.001]
Log(Income)
-0.263 -0.410
-0.261 -0.260
0.027
0.178
[0.030] [0.032]
[0.029] [0.030]
[0.037]
[0.041]
Unemployed Female -0.552 -0.306
-0.026 0.288
-0.586
-0.631
[0.062] [0.065]
[0.052] [0.054]
[0.080]
[0.086]
Unemployed Male
0.331
-0.027
-0.278 -0.013
-1.42
0.091
[0.120] [0.131]
[0.112] [0.113]
[0.280]
[0.165]
Non-white
0.551
0.093
0.090
-0.747
-0.642
-0.538
[0.056] [0.060]
[0.054] [0.066]
[0.082]
[0.086]
Subscribes to
0.450
0.151
-0.400 -0.254
0.475
0.373
Newspaper
[0.055] [0.059]
[0.053] [0.056]
[0.065]
[0.067]
Does not Have Kids 1.088
-14.635
-0.368 1.773
-13.314
1.455
[0.310] [631.998] [0.279] [0.197]
[676.928] [0.372]
CHAPTER 3. COSTS OF UNIFORM PRICING
126
Table 3.10: Coefficients on Price Index Variables for Specifications I-VI of Conditional
Logit Model of Store Choice. Standard errors are in brackets.
Product
Category
Bacon
BBQ Sauce
Butter
Cat Food
Cereal
Cleansers
Coffee
Cookies
Crackers
Detergents
Eggs
Fabric Softener
Frozen Pizza
Hot Dogs
Ice Cream
Peanut Butter
Snacks
Bar Soap
Soft Drinks
Sugarless Gum
Toilet Tissue
Yogurt
Shopped at Store
in Past 2 Weeks
Number of
Observations
Pseudo R2
Log Likelihood
Effect on Market Share from a
increase in Soft Drink Prices:
Before:
After:
274,337
Specification
III
IV
Prices
Current
Lagged 2 Weeks Prices
-0.036
0.019
[0.023]
[0.032]
0.139
-0.101
[0.095]
[0.128]
-0.023
-0.002
[0.068]
[0.092]
1.070
0.286
[0.283]
[0.412]
0.166
-0.139
[0.052]
[0.074]
-0.041
0.039
[0.062]
[0.088]
0.010
0.123
[0.036]
[0.049]
-0.526
-0.437
[0.071]
[0.101]
0.038
-0.090
[0.043]
[0.062]
-0.042
-0.095
[0.020]
[0.028]
0.011
-0.101
[0.057]
[0.084]
-0.091
-0.131
[0.054]
[0.076]
-0.032
-0.031
[0.032]
[0.042]
0.105
-0.028
[0.032]
[0.044]
-0.002
-0.015
[0.036]
[0.049]
0.003
-0.194
[0.044]
[0.064]
-0.270
-0.160
[0.061]
[0.085]
0.288
0.465
[0.064]
[0.089]
-0.082
-0.026
[0.032]
[0.044]
0.058
0.026
[0.069]
[0.094]
-0.002
-0.042
[0.046]
[0.063]
0.681
0.149
[0.105]
[0.148]
4.372
[0.029]
273,489
277,011
V
Prices
Lagged 1 Week
-0.042
[0.032]
0.087
[0.130]
-0.069
[0.094]
0.181
[0.422]
0.159
[0.074]
-0.055
[0.091]
-0.007
[0.050]
-0.291
[0.103]
-0.001
[0.062]
-0.003
[0.028]
0.051
[0.085]
-0.157
[0.078]
-0.026
[0.046]
0.043
[0.045]
-0.035
[0.051]
-0.072
[0.065]
-0.198
[0.085]
0.324
[0.089]
-0.037
[0.045]
0.058
[0.096]
0.055
[0.063]
-0.053
[0.150]
4.376
[0.029]
274,337
VI
Prices
Lagged 2 Weeks
-0.063
[0.032]
0.180
[0.132]
-0.056
[0.094]
0.718
[0.417]
0.240
[0.075]
-0.217
[0.091]
-0.001
[0.049]
-0.410
[0.102]
0.065
[0.062]
0.026
[0.028]
-0.303
[0.083]
-0.065
[0.078]
-0.042
[0.045]
0.122
[0.045]
0.029
[0.050]
0.092
[0.061]
-0.237
[0.085]
0.470
[0.088]
-0.019
[0.045]
0.209
[0.096]
-0.008
[0.064]
0.869
[0.151]
4.386
[0.029]
273,489
0.124
-67,458
0.1241
-66,800
0.1246
-66,586
0.5243
-36,632
0.5282
-35,982
0.5291
-35,821
.3405
.3395
.3407
.3384
.3414
.3395
.3405
.3402
.3407
.3404
.3412
.3412
I
Current
Prices
0.022
[0.023]
-0.121
[0.093]
-0.065
[0.067]
1.217
[0.284]
-0.123
[0.051]
0.129
[0.061]
0.094
[0.035]
-0.468
[0.071]
-0.023
[0.043]
-0.064
[0.020]
0.073
[0.057]
-0.086
[0.054]
-0.010
[0.031]
0.015
[0.032]
-0.051
[0.036]
-0.091
[0.045]
-0.173
[0.061]
0.304
[0.064]
-0.041
[0.031]
0.007
[0.069]
-0.066
[0.045]
0.432
[0.105]
II
Prices
Lagged 1 Week
-0.005
[0.023]
-0.019
[0.094]
-0.050
[0.068]
0.993
[0.288]
0.006
[0.052]
0.101
[0.062]
0.060
[0.036]
-0.479
[0.071]
-0.01
[0.043]
-0.076
[0.020]
0.149
[0.058]
-0.086
[0.055]
-0.028
[0.033]
0.059
[0.032]
-0.048
[0.037]
-0.093
[0.046]
-0.247
[0.061]
0.252
[0.065]
-0.101
[0.032]
-0.012
[0.069]
-0.003
[0.045]
0.195
[0.105]
277,011
CHAPTER 3. COSTS OF UNIFORM PRICING
127
Table 3.11: Coefficients on Price Indices for Specifications VII and VIII of Conditional
Logit Model of Store Choice. Standard errors are in brackets.
Product
Category
Cat Food
Cereal
Coffee
Cookies
Detergents
Eggs
Hot Dogs
Peanut Butter
Salty Snacks
Bar Soap
Fabric Softener
Soft Drinks
Yogurt
Shopped at Store
in Past 2 Weeks
Number of
Observations
Pseudo R2
Log Likelihood
Effect on Market Share from a
increase in Soft Drink Prices:
Before:
After:
Current
Prices
0.657
[0.317]
-0.129
[0.054]
0.091
[0.036]
-0.372
[0.078]
-0.049
[0.021]
0.023
[0.060]
-0.036
[0.032]
-0.034
[0.048]
-0.069
[0.069]
0.241
[0.077]
-0.064
[0.059]
-0.008
[0.033]
0.327
[0.109]
Specification
VII
Prices Lagged Prices Lagged
1 week
2 weeks
0.469
0.715
[0.335]
[0.316]
-0.019
0.126
[0.055]
[0.055]
0.009
0.002
[0.036]
[0.036]
-0.189
-0.365
[0.081]
[0.077]
-0.036
-0.014
[0.021]
[0.021]
0.148
-0.043
[0.061]
[0.059]
0.013
0.074
[0.032]
[0.033]
-0.064
0.025
[0.049]
[0.046]
-0.220
-0.134
[0.072]
[0.069]
0.035
0.198
[0.083]
[0.075]
-0.081
-0.025
[0.062]
[0.058]
-0.101
-0.081
[0.033]
[0.033]
0.037
0.586
[0.108]
[0.107]
273,489
Current
Prices
-0.192
[0.461]
-0.153
[0.077]
0.162
[0.050]
-0.391
[0.112]
-0.076
[0.029]
-0.183
[0.088]
-0.060
[0.045]
-0.131
[0.066]
-0.152
[0.095]
0.362
[0.106]
-0.126
[0.083]
0.010
[0.046]
0.075
[0.156]
Specification
VIII
Prices Lagged Prices Lagged
1 week
2 weeks
0.283
0.681
[0.485]
[0.460]
0.121
0.247
[0.077]
[0.077]
-0.066
-0.003
[0.049]
[0.050]
-0.021
-0.248
[0.117]
[0.112]
0.042
0.035
[0.029]
[0.029]
0.112
-0.337
[0.089]
[0.086]
0.005
0.105
[0.045]
[0.047]
-0.072
0.090
[0.068]
[0.063]
-0.122
-0.135
[0.096]
[0.094]
-0.075
0.384
[0.113]
[0.103]
-0.097
0.008
[0.087]
[0.083]
-0.070
-0.018
[0.045]
[0.046]
-0.182
0.812
[0.154]
[0.153]
4.389
[0.029]
273,489
0.1255
-66,524
0.5296
-35,786
0.3414
0.3370
0.3414
0.3406
7up, 2L Bottle
7up, 6-pack 12oz cans
Diet 7up, 24-pack 12oz cans
Diet 7up, 2L Bottle
Coke, 12-pack 12oz cans
Coke, 24-pack 12oz cans
Coke, 2L Bottle
Diet Coke, 12-pack 12oz cans
Diet Coke, 24-pack 12oz cans
Diet Coke, 2L Bottle
Diet Caffeine Free Coke, 12-pack 12oz cans
Diet Caffeine Free Coke, 24-pack 12oz cans
Private Label, 12oz Can
Pepsi, 12-pack 12oz cans
Pepsi, 24-pack 12oz cans
Pepsi, 2L Bottle
Pepsi, 6-pack 12oz cans
Diet Pepsi, 3L Bottle
Diet Pepsi, 12-pack 12oz cans
Diet Pepsi, 24-pack 12oz cans
Diet Pepsi, 2L Bottle
RC, 2L Bottle
RC, 3L Bottle
Diet Rite, 24-pack 12oz cans
Diet Rite, 2L Bottle
Product
Mean
Marginal
Cost
0.123
0.310
0.148
0.148
0.160
0.093
0.099
0.165
0.085
0.094
0.165
0.080
0.036
0.149
0.082
0.087
0.249
0.077
0.170
0.109
0.111
0.014
0.084
0.116
0.105
Std. Dev. of
Marginal
Cost
0.090
0.204
0.088
0.096
0.120
0.079
0.075
0.133
0.084
0.080
0.127
0.080
0.034
0.130
0.077
0.068
0.136
0.014
0.137
0.074
0.067
0.035
0.033
0.062
0.068
Number of
Weeks w/
Negative Costs
6
2
3
3
2
12
10
3
19
16
3
18
19
3
14
14
1
0
4
9
3
29
6
1
3
Mean
Markup
($)
0.108
0.081
0.119
0.083
0.179
0.181
0.150
0.179
0.189
0.154
0.180
0.190
0.154
0.190
0.191
0.161
0.164
0.123
0.171
0.167
0.134
0.165
0.108
0.147
0.113
Std. Dev.
of Markup
0.035
0.135
0.032
0.046
0.048
0.026
0.027
0.054
0.031
0.032
0.048
0.029
0.018
0.078
0.024
0.023
0.080
0.014
0.088
0.023
0.023
0.020
0.017
0.023
0.025
Mean
Margin (%)
53.2
27.7
49.1
41.7
58.5
71.1
65.3
58.8
74.8
67.6
58.4
75.5
82.9
61.6
75.0
69.3
43.7
61.6
55.6
64.7
58.5
93.8
58.1
58.7
56.0
Table 3.12: Summary Statistics for Marginal Costs (in Dollars per 12oz Serving) Implied by the Model
CHAPTER 3. COSTS OF UNIFORM PRICING
128
CHAPTER 3. COSTS OF UNIFORM PRICING
129
As is frequently the case (see Villas-Boas (2002)) my estimates imply that in some weeks,
for some products, marginal costs are negative. Although this seems economically bizarre,
in theory these could be explained by slotting allowances. In practice, (see Israilevich
(2004)) the arrangements between retailers and manufacturers regularly involves nonlinear
contracting schemes such as block discounts (which imply negative marginal costs over some
regions). To the extent that my assumption of constant marginal costs is violated, I may
be picking up some of these nolinearities.
Figure 3.1 shows a typical path of prices and implied marginal costs over time, The key
feature to notice here is that intertemporal marginal cost (e.g., wholesale price) variation is
responsible for nearly all of the inter-temporal price variation. This feature is mirrored in
the Dominick’s data. Figure 3.2 plots the average markup in cents per 12oz serving over the
sample period implied by the derived marginal costs. The average average markup across
products appears to be roughly fourteen cetns per 12oz serving, with occasional spikes
upwards (and one large spike downwards). Together with the low standard deviations on
margins shown in Table 3.12, this also agrees with what is observed in the Dominick’s data.
“Optimal” Uniform Prices
In order to calculate the profits the firm would have earned by following a uniform pricing
strategy, I must first solve for the “optimal” uniform prices. I do this by restricting the prices
each week to be uniform by manufacturer-brand-size24 . Then for each week, I numerically
solve for the set of prices that maximizes expected profits, subject to this restriction.
Table 3.13 presents summary statistics on the differences between these “optimal” uniform prices and the non-uniform prices actually charged by the retailer. Contrary to (my)
expectations, the majority of the differences were not uniformly positive or negative. That
is, in some weeks the non-uniform price was higher than the optimal uniform price, while
in other weeks it was lower. In hindsight, this is actually suggested by the variation in
the price ordering of the varieties in Figure 1.4. Furthermore, for all but two products (2L
containers of RC and Diet Rite), the average difference between the non-uniform and the
optimal uniform prices was less than one cent per 12oz serving.
24
For example, I restrict 12-packs of 12oz cans of Coke, Diet Coke, and Diet Caffeine Free Coke to all sell
at the same price each week, although I allow this price to vary across weeks.
CHAPTER 3. COSTS OF UNIFORM PRICING
130
30
20
10
0
Cents per 12 Ounce Serving
40
Figure 3.1: Graph of the Price and Implied Marginal Cost (in cents per 12oz serving) for a
2L Bottle of Regular Pepsi, 6/91-6/93
08jun1991
05dec1991
02jun1992
29nov1992
28may1993
Week
Price of 2L Pepsi (288oz size)
Cost of 2L Pepsi (288oz size)
Figure 3.2: Graph of the Average Markup (in cents per 12oz serving) Across Products,
6/91-6/93
18
16
14
12
10
Cents per 12 Ounce Serving
20
Average Markup Over Time
08jun1991
05dec1991
02jun1992
Week
29nov1992
28may1993
CHAPTER 3. COSTS OF UNIFORM PRICING
131
Figure 3.3: Graph of the Maximum Difference Across Products (in cents per 12oz serving)
Between a Product’s Uniform and Non-Uniform Prices, 6/91-6/93
0
Cents per 12oz Serving
5
10
15
Maximum Price Difference Per Serving
Between Uniform and Non-Uniform Prices
08jun1991
05dec1991
02jun1992
Week
29nov1992
28may1993
Table 3.13: Summary Statistics on the Differences Between Observed Non-Uniform Prices and “Optimal” Uniform Prices
(in Dollars per 12oz Serving)
Greatest
Greatest
Product
Mean Std. Dev. of
Increase
Decrease
Difference
Difference from Uniform from Uniform
7up, 2L Bottle
-0.001
0.015
0.145
0.028
Diet 7up, 2L Bottle
-0.001
0.016
0.152
0.023
Coke, 12-pack 12oz cans
0.004
0.024
0.053
0.161
Coke, 24-pack 12oz cans
0.000
0.018
0.111
0.135
Coke, 2L Bottle
0.001
0.005
0.021
0.023
Diet Coke, 12-pack 12oz cans
-0.002
0.012
0.054
0.035
Diet Coke, 24-pack 12oz cans
0.001
0.011
0.046
0.084
Diet Coke, 2L Bottle
0.001
0.006
0.021
0.018
Diet Caffeine Free Coke, 12-pack 12oz cans
-0.001
0.010
0.051
0.030
Diet Caffeine Free Coke, 24-pack 12oz cans
-0.001
0.007
0.046
0.016
Pepsi, 12-pack 12oz cans
-0.005
0.031
0.232
0.031
Pepsi, 24-pack 12oz cans
0.000
0.004
0.017
0.019
Pepsi, 2L Bottle
-0.001
0.005
0.031
0.011
Diet Pepsi, 12-pack 12oz cans
-0.001
0.022
0.184
0.046
Diet Pepsi, 24-pack 12oz cans
-0.002
0.007
0.039
0.013
Diet Pepsi, 2L Bottle
0.002
0.009
0.019
0.065
RC, 2L Bottle
0.011
0.015
0.014
0.056
Diet Rite, 2L Bottle
-0.028
0.037
0.125
0.023
CHAPTER 3. COSTS OF UNIFORM PRICING
132
CHAPTER 3. COSTS OF UNIFORM PRICING
133
Profit Differences
Again using my marginal cost estimates and the actual quantities sold, I can estimate
the profits that the store A actually earned in each week. In addition, by simulating
expected demand, I can calculate the profits that the firm expected to earn each week. The
difference between these two numbers is that the former is scaled by the number of shoppers
who actually went shopping in that week. These give me a measure of the profits earned
by the firm under the non-uniform pricing regime. Comparing the these expected profit
figures yields a weekly estimate of the percentage profit decrease that store A would have
experienced if it had charged uniform prices.
Table 3.14 shows the detailed results of these calculations for a typical week of the
sample: the week beginning July 7, 1991. Several features are apparent. The first is that
demand is strongly skewed towards the lowest priced products. The ten products priced
at $0.21 cents per 12oz serving or lower sell by far the lasrgest share of the quantity. The
second feature is that many of the prices are the same or nearly the same under both
uniform and non-uniform pricing policies. In this week, store A actually charged the same
price for 24-packs of 12oz cans of both Coke and Diet Coke. Because I assume (in order to
identify the marginal costs) that the retailer charged the optimal prices in each week, the
results are skewed towards finding a smaller estimate of the profit difference. Third, much
of the increase in profits comes from a significant decrease in the price of a single product:
2L bottles of Royal Crown (RC) cola. Finally, I note in passing that demand for some goods
increased, in spite of an increase in the price going from the uniform to the non-uniform.
This can be attributed to the effects of cross-price elasticities – the prices of many other
goods also moved.
7up, 2L Bottle
7up, 6-pack 12oz cans
Diet 7up, 24-pack 12oz cans
Diet 7up, 2L Bottle
Coke, 12-pack 12oz cans
Coke, 24-pack 12oz cans
Coke, 2L Bottle
Diet Coke, 12-pack 12oz cans
Diet Coke, 24-pack 12oz cans
Diet Coke, 2L Bottle
Diet Caffeine Free Coke, 12-pack 12oz cans
Diet Caffeine Free Coke, 24-pack 12oz cans
Private Label, 12oz Can
Pepsi, 12-pack 12oz cans
Pepsi, 24-pack 12oz cans
Pepsi, 2L Bottle
Pepsi, 6-pack 12oz cans
Diet Pepsi, 12-pack 12oz cans
Diet Pepsi, 24-pack 12oz cans
Diet Pepsi, 2L Bottle
RC, 2L Bottle
RC, 3L Bottle
Diet Rite, 24-pack 12oz cans
Diet Rite, 2L Bottle
Product
31.9
50.9
20.8
31.9
47.1
20.8
32.0
47.1
20.8
32.0
47.1
20.8
17.0
48.8
20.8
31.8
50.4
48.8
20.8
31.8
19.7
20.1
32.5
19.7
pU nif orm
58
12
5381
8
57
5609
83
81
5788
83
70
5944
3050
68
9099
284
108
51
8337
94
2703
298
285
367
QU nif orm
31.8
48.0
20.8
31.8
47.0
20.8
31.8
47.0
21.4
31.9
47.0
20.8
17.0
47.0
20.8
31.8
48.0
47.0
20.7
31.8
14.1
20.0
31.2
31.7
pN on−U nif orm
57
24
5406
10
58
5538
85
81
5463
86
71
5896
3016
87
8964
274
151
69
8283
90
4495
296
348
16
QN on−U nif orm
Marginal
Cost
24.9
58.5
6.6
29.1
33.5
0.7
18.8
32.8
-0.7
20.4
32.9
-0.7
1.1
31.0
0.0
19.0
40.8
31.6
2.9
20.5
-1.7
9.7
18.9
24.5
ΠU nif orm
($)
4.05
-0.90
761.30
0.23
7.80
1126.52
10.88
11.58
1242.81
9.58
10.01
1272.49
484.78
12.04
1887.23
36.41
10.27
8.79
1492.67
10.75
578.59
30.78
38.73
-17.43
ΠN on−U nif orm
($)
3.94
-2.55
764.38
0.25
7.76
1115.18
10.98
11.48
1209.34
9.82
9.99
1265.51
479.22
13.88
1862.89
35.04
10.80
10.69
1479.67
10.20
708.66
30.44
42.75
1.16
Table 3.14: Uniform and Non-Uniform Prices, Quantities and Estimated Profits for the Week of July 7, 1991. Prices and
marginal costs reported are in cents per 12oz Serving. Quantity is measured in 12oz servings. Profits are measured in
dollars. All prices and profits are in nominal terms. The total difference in profits for the week from the two pricing
strategies is $61.52. The 3L size of Pepsi was not offered in this week.
CHAPTER 3. COSTS OF UNIFORM PRICING
134
CHAPTER 3. COSTS OF UNIFORM PRICING
135
The weekly expected differences in profits are presented in Figures 3.4 and 3.5. I estimate a distinct mass point at zero lost profits. As noted above, this is largely due to my
assumption that, in each week, the prices charged by store A were optimal. This assumption
implies that for weeks in which store A actually charged uniform prices, it could not have
expected to lose any profits doing so.
My estimates imply that by charging uniform, rather than non-uniform prices, the
retailer would have lost $36.56 per week in profit, or a total of $3,803 over the two year
period of my sample. The prospect of earning an additional $3,803 in profits (roughly $5,135
2004 dollars) over a two-year period for the soft drink category may seem small, but this is
only a single store in a much larger chain of more than 100 stores. If the chain were able
to realize similar profit increases at other stores in the chain, a rough estimate of the profit
increase would be over $250,000 dollars per year in 2004 dollars. This would presumably be
more than enough to hire an empirical economist to determine the optimal prices for each
product in each store in each week. Furthermore, this estimate is solely for the soft drink
category. While it is not clear what the results would be for other categories, similar profit
increases may be possible.
3.5
3.5.1
Interpreting “Lost” Profits
Menu Costs
As mentioned in the introduction, if we set aside demand-side explanations, the two reasons
for retailers to charge uniform prices are: to reduce menu costs and to soften price competition with other retailers. In the event that the difference between uniform and non-uniform
prices is close to zero, this would suggest that retailers do not expect to lose much (if any)
profit by charging uniform prices. On the other hand, if the predicted profit differences are
positive, we must try to differentiate between these (and potentially other) explanations for
the hypothetical “lost profits”. When talking to store managers, the most frequently offered
explanation for the observed price uniformity is some form of menu costs. When pressed,
Safeway store managers respond that the reason for uniform pricing is that it is “too much
trouble” to price every good separately. In understanding what is meant by “too much
trouble” it is important to distinguish between two different kinds of menu costs: physical
menu costs and managerial menu costs.
One type of menu cost comes from the costs associated with physically changing prices.
CHAPTER 3. COSTS OF UNIFORM PRICING
136
10
5
0
Percentage Difference Between
Uniform and Non-Uniform Prices
15
Figure 3.4: Graph of the Difference Between Profits from Uniform and Non-Uniform Price
Strategies, as a Percent of the Profits Earned at Non-Uniform Prices, 6/91-6/93
08jun1991
05dec1991
02jun1992
29nov1992
28may1993
Week
Figure 3.5: Graph of the Counterfactual Dollars Lost from Charging Uniform Prices,
6/91-6/93
0
100
Dollars
200
300
400
500
Profit Difference in Dollars
Between Uniform and Non-Uniform Prices
08jun1991
05dec1991
02jun1992
Week
29nov1992
28may1993
CHAPTER 3. COSTS OF UNIFORM PRICING
137
According to Tony Mather, Director Business Systems, Safeway (U.K.): “Pricing at the
moment is very labor-intensive. Shelf-edge labels are batch printed, manually sorted and
changed by hand while customers are out of the store.”25 Levy et al. (1997) estimate the
average menu costs to a large chain-owned grocery store for physically changing a single
price tag to be $0.52. To put this in perspective, a typical large grocery store usually
changes the price tags on about 4,000 items each week, changing as many as 14,000 tags in
some weeks. Although their study was conducted on behalf of a company selling electronic
price display tags, their stated aim was to put a lower bound on menu costs and they report
that grocery store executives generally agreed with their findings.
One might think physical menu costs promote uniform pricing – that stores reduce their
physical menu costs by charging uniform prices. However, two pieces of evidence suggest
that physical menu costs do not explain uniform prices. First, grocery stores typically
post prices for every UPC even when they are uniformly priced. Hence, the physical menu
costs are the same, regardless of whether the prices are priced uniformly or non-uniformly.
Second, in cases where the physical menu cost is presumably small or insignificant, we still
observe uniform prices. Even grocery stores that have implemented electronic display tags
and that can change prices throughout the store at the touch of a button from the store’s
central computer continue to charge uniform prices. Furthermore, online grocery stores –
who presumably have nearly zero physical menu costs – also sell at uniform prices.
A second kind of menu cost, and one that has not typically been discussed in the literature is the managerial cost associated with figuring out what price to charge for that
product. While academic papers generally assume that retailers learn optimal prices costlessly, this is an abstraction from reality. In order to learn its demand function, a retailer
must experiment by charging a variety of prices – introducing exogenous price variation –
and this experimentation can be costly. In addition, the retailer may have to hire personnel
or consulting services to determine “optimal” prices. These costs may not be insubstantial.
A recent article in Business Week (Keenan 2003) suggests that implementing the advanced
techniques offered by pricing consultants typically requires a “12-month average installation” time and a price that “start[s] at around $3 million.” If the additional expected profit
to be gained from charging different prices for two products is less than the cost of figuring
out what those prices should be, then we will see uniform prices.
This suggests that the store’s choice of whether to follow a uniform or non-uniform
25
http://www.symbol.com/uk/Solutions/case study safeway.html
CHAPTER 3. COSTS OF UNIFORM PRICING
138
pricing strategy is more likely a long-term decision rather than a week-by-week decision.
In this case, the relevant cost to consider is the present discounted value of the sum of the
lost expected profits across weeks and represents the one-time or infrequent cost either of
experimentation or consulting services.
Managerial menu costs also suggest a reason that pricing strategies may vary across
stores – leading some stores to charge uniform prices while others charge non-uniform prices.
Pricing decisions for most large grocery chains are made at the chain level. Store managers
at these chains typically receive the week’s prices electronically from company headquarters,
and are only responsible for making sure that price labels are printed and placed on shelves.
This centralization allows large chains to spread out these managerial costs across many
stores. However, evidence suggests that even large chains may be influenced by managerial
menu costs. Chintagunta et al. (2003) document the fact that Dominick’s Finer Foods
grouped its stores into three different categories based on the levels of competition the
stores faced, with each of roughly one hundred stores charging one of three menus of prices.
Such pricing heuristics presumably lower managerial costs by reducing the dimensionality
of the optimal pricing problem, but at the cost of non-optimal prices.
Other pricing heuristics seem to be in widespread use. Both small retailers and large
grocery stores26 frequently use constant-markup pricing heuristics, such as pricing all goods
at wholesale cost plus a fixed percentage or amount. The apparent widespread use of these
pricing heuristics may explain why soft drink prices tend to vary dramatically over time, but
not cross-sectionally – while wholesale prices move a good deal over time, wholesale prices
are typically uniform within manufacturer-brand. Unfortunately, this raises the question of
why manufacturers would choose to price their products uniformly.
26
Data suggests that Dominick’s Finer Foods (described in section 5) frequently followed a constantmarkup pricing strategy.
CHAPTER 3. COSTS OF UNIFORM PRICING
3.6
139
Conclusion
In retail environments, many differentiated products are sold at uniform prices. Explanations for this behavior can be grouped into demand-side and supply-side explanations.
Lacking the necessary data to investigate demand-side explanations, I look at supply-side
explanations. Using grocery store scanner panel data and household grocery purchase histories, I examine the market for carbonated soft drinks – a product that is frequently, but
not always, sold at uniform prices – and evaluate the validity of several supply-side explanations. To do this, I develop a new structural model of household demand for carbonated
soft drinks. Using the estimated demand system, I conduct the counter-factual experiment
of forcing the prices of a particular store to be uniform, and comparing the resulting profits
to the non-uniform case. The results from the new structural model suggest that uniform
pricing leads to a total profit loss for the retailer over the two year sample period, of roughly
$5,135 in 2004 dollars.
This result suggests that there are additional profits to be earned from non-uniform
pricing, under the assumption that the retailer charged optimal prices. Clearly, however, it
may not be profitable for single-store retailers to take advantage of this opportunity. Without the benefits of multiple stores over which to spread the managerial costs of determining
optimal prices, single store retailers may find it optimal to charge uniform prices. Unfortunately, this “scale” explanation cannot be the whole story. Anecdotal evidence suggests that
Walmart charges uniform prices for many products, even though that company has almost
certainly realized most returns to scale with respect to managerial menu costs. Although
additional research is necessary regarding demand-side reactions to non-uniform pricing,
these results suggest that pricing managers, particularly those at large retail chains, should
be aware of potential additional profits available from non-uniform pricing. Moreover, they
suggest that for single-store retailers, relatively small managerial menu costs are able to
generate the observed behavior.
3.7
Appendix 3.A: Numerically Solving The Utility Function
The key ingredient to this estimation procedure is the ability to quickly and reliably solve
the household’s constrained utility maximization problem. In addition to theoretical reasons
for imposing concavity in the household’s utility function, without this restriction, solving
for the household’s optimal bundle would be difficult if not impossible. When the utility
CHAPTER 3. COSTS OF UNIFORM PRICING
140
function is concave, this is much easier, and gradient-based numerical methods give good
results.
It is imperative that the numerical solutions to the household’s optimization problem
be correct. If the solutions are not the true optimal bundles, the parameter estimates will
not be consistent or unbiased. To perform these optimizations, I have employed several
different numerical methods, with varying degrees of success. Because there is no way to
analytically verify the solution when using large numbers of goods and/or characteristics,
I use the best solution from a large number of randomly drawn starting values with the
Subplex and Nelder-Mead optimization routines as the “true” solution. The following three
optimization algorithms have proven useful:
• NAG E04UGF - This algorithm is based on the SNOPT/NPSOL packages and currently gives the best results. Unlike the other optimization packages used, this uses a
user-supplied analytic gradient. With a single randomized starting value, the “true”
solution is found at least 99% of the time. With two randomized starting values, this
increases to 100%.
• NAG E04CCF - This is the Numerical Algorithm Group’s FORTRAN implementation
of the Nelder-Mead simplex method. It also gives good results, but, because it does
not use gradients, takes much longer.
• Subplex - Subplex is a subspace-searching simplex method for the unconstrained optimization of general multivariate functions. Like the Nelder-Mead simplex method
it generalizes, the subplex method is well suited for optimizing noisy objective functions. Subplex was developed by Tom Rowan at Oak Ridge National Laboratory and
is described in: T. Rowan, ”Functional Stability Analysis of Numerical Algorithms”,
Ph.D. thesis, Department of Computer Sciences, University of Texas at Austin, 1990.
Subplex tends to be less consistent at finding the correct solution, but occasionally
significantly improves on the solutions in the above two methods.
3.8
Appendix 3.B: Modeling Heterogeneity of Preferences
Although the product-level demand model I estimate does not incorporate household-level
heterogeneity, this would be a relatively straightforward extension for future work. The
greatest obstacle to estimating such a model is that it increases the number of parameters
CHAPTER 3. COSTS OF UNIFORM PRICING
141
to estimate. I explored specifications including discrete types driven by observable demographics including household size and median expenditure level, but the MSM distance
function in this case was poorly behaved. Similarly, one could estimate a model of unobserved heterogeneity, but the greatest obstacle in this case would be a dramatic increase
in the required computing power. As mentioned earlier, in order to get a sufficiently good
estimate of the expected purchases, it is necessary to use at least R = 30 simulations.
Unfortunately, with current processing power, this means that estimation takes several
weeks. Incorporating additional parameters increases the difficulty of the MSM distance
function optimization. Incorporating unobserved heterogeneity requires at least an order
of magnitude increase in the number of simulations, making such a venture prohibitively
computationally expensive for this application at the present time, although it might be
possible by using a smaller set of products (which would allow the utility function to be
solved more quickly).
3.9
Appendix 3.C: Analysis of the Panel Composition
This paper makes extensive use of the purchase histories of the households in the IRI
dataset. Hence, one would like to know whether these households are indeed representative
of the households that typically shop at these stores. Fortunately, in addition to purchase
histories, the IRI dataset contains demographic information for each of the households. As
seen in Table 3.15, the mean household size for the panel is 1.9, with a standard deviation
of 1.23. Additionally, nearly all of the households have children, and more than 1/3 have
a retired female. Table 3.16 shows the age distribution for the primary man and woman
in the household. For both men and women in the panel, the median age appears to be in
the range of 55-64, although these numbers may be skewed by the large number of possible
non-responses. Finally, Table 3.17 shows the distribution of income among households in
the panel, with a median in the range of $20,000-25,000.
CHAPTER 3. COSTS OF UNIFORM PRICING
142
Table 3.15: Size and Composition of Households in Panel
Total Number of Households
262
Median Number of Members in Household
1
Mean Number of Members in Household (s.d.)
1.75 (1.06)
Fraction of Households with No Kids
.004
Kids Aged 0-5
.015
Kids Aged 6-11
.069
Kids Aged 12-18
.061
Kids Aged 18+
.889
with a Retired Male
.168
Retired Female
.347
Sample consists of all households that shopped at store A at least once during the two year
period. Source: IRI Data.
65+
115
43.9
59
22.5
10k
52
19.9
10-12k
14
5.3
12-15k
26
9.9
15-20k
16
6.1
Household Income
20-25k 25-35k 35-45k
14
28
23
5.3
10.7
8.8
45-55k
20
7.6
55-65k
19
7.3
65-75k
12
4.6
Sample consists of all households that shopped at store A at least once during the two year period. Source: IRI Data.
Num. of Hhds in Cat.
% of Hhds
N.R.
5
1.9
Table 3.17: Income Distribution of Panel Households
Sample consists of all households that shopped at store A at least once during the two year period. Source: IRI Data.
Table 3.16: Age Distribution of Primary Male and Female in Households in Panel
Age
None Present/
No Response 18-29 30-34 35-44 45-44 55-64
Num. of Hhds w/ Primary Female Aged
35
1
8
26
37
40
% of Hhds
13.4
0.4
3.1
9.9
14.1
15.3
Num. of Hhds w/ Primary Male Aged
145
2
2
11
19
24
% of Hhds
55.3
0.8
0.8
4.2
7.3
9.2
75k+
33
12.6
CHAPTER 3. COSTS OF UNIFORM PRICING
143
Demographic data for population in US Census tracts surrounding four Dominick’s Finer Foods stores located in close
geographical proximity to the stores in the panel. No information is available on the population size of these areas. Source:
Market Metrics, based on 1990 US Census Data.
Table 3.18: Summary Statistics for Population Living Near Stores in the Panel
W
X
Y
Z
Mean Number of Members in Household
1.55
2.74
2.53
2.15
Fraction of Households with 1 Member
.614
.270
.324
.426
with 2 Members
.280
.277
.288
.302
with 3 or 4 Members
.092
.309
.269
.193
with 5+ Members
.014
.144
.119
.079
Fraction of Women that have No Kids
.881
.708
.689
.738
that have Kids Aged 0-5
.060
.144
.152
.142
that have Kids Aged 6-17
.059
.147
.159
.121
Fraction of Population Retired
.094
.172
.169
.124
Median (s.d.) Household Income ($000’s)
31.1 (25.9) 25.4 (20.4) 24.1 (21.3) 26.5 (23.9)
Fraction of Households with Income<$15k
.088
.133
.153
.152
CHAPTER 3. COSTS OF UNIFORM PRICING
144
CHAPTER 3. COSTS OF UNIFORM PRICING
145
In the end, it is difficult to know how to assess these numbers. Ideally, I would like
to know how they compare to the population of shoppers at store A. A rough proxy for
this population is shown in Table 3.18. This table displays demographic information on the
population surrounding four stores owned by Dominick’s Finer Foods that are located in
close geographic proximity to store A. Several stark differences are apparent. Households in
the panel are much more likely to have children than those in the surrounding population.
They also tend to have lower incomes, with a substantially larger fraction of them earning
below $15,000 per year. These differences are suggestive that the households in the panel
are quite different from those in the background population, but no more than that. If
typical shoppers at store A also differ from the background population, my panel may
still be representative. Regardless, it is important to note that the ability to draw broad
inferences from household-level data hinges critically on the representativeness of the panel.
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