Toward a Formal Model of Steiner’s Dual Stage Theory of
Manufacturing and Retailing
By Michael P. Lynch
Author’s Note: This paper provides the complete mathematical model of Steiner’s dual
stage theory that underlies the calculations and simulations given in my paper “Why
economists are wrong to neglect retailing and how Steiner’s theory provides an
explanation of important regularities” published in The Antitrust Bulletin, Vol.
XLIX, No. 4, Winter, 2004].
Last Revised: 7/05/2004
© Michael P. Lynch, 2004
Purpose
The purpose of this paper is to build a formal model of one aspect of Robert L. Steiner’s
Dual Stage theory of manufacturing and retailing 1 complex enough to capture its major
elements, yet simple enough to solve analytically, or at least to simulate using a
spreadsheet. I focus on the competition between and among manufacturers who have
succeeded in establishing their brand name on a national level (LBs), private label
manufacturers who provide retailers with store brands (SBs) tailored to compete directly
with a given LB, and the retailers who stock them both. Steiner’s theory is much richer
than the model developed here. For example, Faris and Albion describe four stages of
Steiner’s theory regarding manufacturers, starting with an unadvertised brand, then
initially advertising it, a growth phase, and finally a “maturity” phase where the LB is
carried by all retailers in the category. 2 The model presented here deals only with the
final stage, when a brand has achieved 100% penetration. The reason for limiting my
model to a single phase has everything to do with complexity, and nothing to do with a
lack of interest in the complete informal theory Steiner has spelled out, with detailed
historical examples, in more than 25 papers published over the last three decades.
The structure of this paper is as follows. First, I review the empirical regularities or
“stylized facts” that Steiner’s theory seeks to explain, and list certain other facts taken as
given that the theory doesn’t seek to explain. Then I summarize Steiner’s explanation of
these stylized facts and explain how I have tried to model them mathematically. Next, I
present a model involving retailers who each stock a leading nationally advertised brand
(LB) and a store brand (SB) and the LB manufacturer. Profit-maximizing behavior in this
model produces many of the regularities described by Steiner and others. Finally, I
briefly sketch a generalized model where retailers carry an indefinite number of different
brands in different categories.
The Facts to Be Explained
Steiner’s theory offers an explanation of two striking empirical regularities. First, despite
the fact that LBs almost invariably have higher retail prices than corresponding SBs,
retailers usually price “leading national brands” [LBs] so that they yield lower dollar and
percentage gross margins 3 [$gms and %gms] than on competing fringe brands or on
“store brands” [SBs]. This is the first inverse association: the more prominent the brand,
the lower its retail margin. Second, manufacturers of LBs tend to have high
manufacturing margins on these products compared to manufacturing margins on fringe
1
See Steiner, Basic Relationships in Consumer Goods Industries, Research in Marketing, vol. 7, pp.165208 (1984) and The Inverse Association between the Margins of Manufacturers and Retailers, Review of
Industrial Organization, vol.8 #6, 1993 for perhaps the most complete statements of his theory.
2
Faris, Paul and Mark Albion, “The Impact of Advertising on the Price of Consumer Products,” Journal of
Marketing, Vol. 44, Summer 1980, 27 – 30.
3
The retail dollar gross margin ($gm) is the retail price less the manufacturer price. The %gm is the $gm
divided by the retail price. In this paper I ignore wholesalers and the distinction between the retail gross
margin which excludes the wholesaling function and the gross distribution margin which includes it.
2
brands and SBs. This is the second inverse association: the more prominent the brand,
the lower its retail margin and the higher its manufacturing margin. The data in Table 1
illustrates both inverse associations for soft drinks as sold in a Chicago area supermarket
called Dominick’s Finer Foods over the period from 1987 to 1997. The table also shows
a third regularity, one so obvious, that it is often overlooked. The price of the LB is
virtually always higher than that of the SB. In table 1, the median SB sells at a 26%
discount to the LB. I use this regularity as one of the building blocks of the model. The
demand for the SB is highly dependent on the price of the LB and little or no units of the
SB will be sold unless its price is below that of the LB. This reflects the LBs “reputation
premium”. 4
Column 1 shows the ratio of the invoice cost of the leading brand to the invoice cost of
the store brand. I assume that the invoice cost is equal to the manufacturer price in each
case. I have chosen a scale for all prices shown in Tables 1 and 2, by taking the
manufacturers’ price for the store brands in each row to be $1. Comparing column 6 to
column 8 row by row, illustrates the first inverse association. For example, the
percentage gross margin (%gm) on Coke Classic is 13% versus 49% on the comparable
store brand. The median %gm for the leading national brands is 18% versus 48% for the
store brands. The %gms on SB soft drinks is generally more than twice as high as that on
LB soft drinks. Moreover, the lowest %gms (12% to 15%) are on the most popular
brands and most heavily advertised brands, Pepsi and Coke. The highest %gms (20%39%) are on lesser known brands such Sunkist Orange, RC Cola and various Schweppes
products. Similarly, comparing columns 5 and 7 row by row shows that dollar gross
margins ($gms) are higher on SBs than on LBs. For example, the $gm on Coke Classic
is 35 cents versus 96 cents on the comparable SB. The median $gm on SBs is 94 cents
versus 46 cents on the LBs. Note that the $gms on SBs are considerably higher, despite
the fact their retail prices are lower than the LBs they are designed to compete with
(compare columns 2 &3). Column 4 shows the percentage discount at which the SBs are
offered relative to the comparable LBs. The median discount from the LB retail price is
26%, with a high for Canada Dry Ginger Ale of more than 32% to a low of only 9% for
Sunkist Orange. So both dollar and percentage retail gross margins are higher on soft
drink store brands than on the leading brands they are designed to compete with.
If we assume that the manufacturers’ prices on SBs are a decent approximation to the
marginal cost of the comparable LB, then Table 1 also provides evidence for the second
inverse association. 5 Coke Classic, for example, would have a manufacturing margin of
4
A term coined by Dorothea Braithwaite in “The Economic Effects of Advertising,” Economic journal,
1928, XXXVIII, 16-37. For questionnaire derived estimates of reputation premia for various LBs see Raj
Sethuraman, “What Makes Consumers Pay More for National Brands Than for Store Brands: Image or
Quality?” MSI WORKING PAPER # 00-110 2000
5
This entails two separate and certainly controvertible propositions. First, that the quality of the LB and SB
are essentially equivalent and second, that the manufacturer price of the SB, which presumably at least
covers fixed costs, is a decent approximation of the marginal cost of the LB. On these questions see Barsky
et al., supra, at pp. 12 – 24. They conclude “In summary, we believe there is enough evidence to suggest
that using private label product prices to infer national brand costs is a reasonable assumption in this
industry. There is reason to believe, therefore, that this measure of markup can be appropriate for at
least some categories and products in this industry. Further, since the private label will
have some markup, and the nationally branded products have advantages on size and scale
3
over 57% ($1.35/$2.35). Although we don’t have any direct evidence on the
manufacturing margin earned by the SB supplier, its marginal cost would have to be
much less than half of Coke’s to have as high a margin. Given that the SB supplier likely
operates at a smaller scale than Coke, it seems very unlikely that its marginal costs are
substantially lower than Coke’s. Note also that the best known brands have the highest
estimated % manufacturing margins (%mms). All the Pepsi and Coke products have
%mms in excess of 55%, whereas RC Cola is around 47%, A&W Root Beer 42 – 44%
and Sunkist Orange is again the lowest at 32%.
in production, packaging and negotiation on input prices, we believe that private label
product prices provide a conservative measure of these costs.”
4
Table 1
Leading Brand
Ratio
LB
Mfr
Price
to SB
Mfr
Price
Leading
Brand
Retail
Price*
Store
Brand
Retail
Price*
% Discount
from
LB
Price
1
2
3
4
$2.70
$2.50
$2.63
$2.57
$2.52
$3.12
$3.14
$2.48
$2.03
$2.39
$3.07
$3.47
$2.05
$2.57
$1.96
$1.92
$1.91
$1.82
$1.81
$2.30
$2.32
$2.09
$1.75
$2.18
$2.34
$2.34
$1.73
$1.81
27.5%
23.1%
27.5%
29.1%
28.1%
26.5%
25.9%
16.0%
13.8%
9.1%
23.7%
32.4%
15.3%
29.6%
Dollar
Gross
% Dollar % Dollar
% LB
Mar- Gross Gross Gross LB
Mfr
gin on Mar- Mar- Mar- Mfr
MarLead- gin on gin on gin on Margin
LB
SB
SB
gin
ing
Brand
5
$0.35
$0.33
$0.40
$0.32
$0.52
$0.84
$0.85
$0.60
$0.25
$0.93
$1.01
$1.11
$0.33
$0.32
6
13.0%
13.0%
15.3%
12.3%
20.6%
27.0%
27.0%
24.2%
12.3%
39.0%
32.9%
32.0%
16.0%
12.3%
7
$0.96
$0.92
$0.91
$0.82
$0.81
$1.30
$1.32
$1.09
$0.75
$1.18
$1.34
$1.34
$0.73
$0.81
8
48.9%
47.9%
47.6%
45.0%
44.8%
56.5%
57.0%
52.0%
42.8%
54.1%
57.3%
57.3%
42.3%
44.6%
9
$1.35
$1.17
$1.23
$1.25
$1.00
$1.28
$1.29
$0.88
$0.78
$0.46
$1.06
$1.36
$0.72
$1.25
10
Coke Classic
Schwepps Ginger Ale
Pepsi Cola N/R
Pepsi Cola Diet N/R
Barq's Root Beer
Schweppes Tonic N/R
Schweppes Diet Tonic N/R
R.C. Cola
A&W Rootbeer Reg
Sunkist Orange
Schwepps Ginger Ale
Canada Dry Ginger Ale
A&W Rootbeer SF
Diet Coke
2.35
2.17
2.23
2.25
2.00
2.28
2.29
1.88
1.78
1.46
2.06
2.36
1.72
2.25
57.4%
53.9%
55.2%
55.6%
50.0%
56.1%
56.3%
46.8%
43.8%
31.5%
51.5%
57.6%
41.9%
55.6%
Column Medians
Column Minimum
Column Maximum
2.20 $2.57 $1.94 26.2% $0.46 18.3% $0.94 48.4% $1.20 54.5%
1.46 $2.03 $1.73 9.1% $0.25 12.3% $0.73 42.3% $0.46 31.5%
2.36 $3.47 $2.34 32.4% $1.11 39.0% $1.34 57.3% $1.36 57.6%
* Assumes all SB mfr prices equal to $1.
Source: Barsky, Robert, Mark Bergen, Shantanu Dutta, Daniel Levy, “What Can the Price Gap between Branded and Private Label
Products Tell Us about Markups?” Presented at the NBER Conference on Research in Income and Wealth:Scanner Data and Price
Indexes,.September 15-16, 2000 [Revised: September 13, 2001], Table A-2, Only those brands with 300 or more observations
were used. Available at: : papers.nber.org/papers/W8426
5
Another empirical regularity is the long term stability of percentage retail margins by
type of retailer. 6 Despite the great volatility of margins on individual products from
week to week, the overall margin for particular types of retailer shows very little
variability. According to the Annual Census of Retail Trade, the %gm for all retail
stores averaged 32% over the period 1986 – 1998. It varied little: a low of 31%, a high
32.4% with a coefficient of variation of only 1.2%. Different types of retailers have
characteristic %gms that vary little over time. For example, from 1986 to 1998, the %gm
of apparel and accessory stores varied from a low of 40.7% to a high of 44.7% with a
coefficient of variation 2.5%. Grocery stores varied from 22.9% to 25.9% with a
coefficient of variation of 4.4%, although they exhibited a clear upward trend. Gasoline
stations had the lowest margins, averaging 20.6% and the highest variation with a
coefficient of variation of 7.3%. The stability of retail margins, raises questions
concerning the usual practice of economists in dealing with pass-throughs of general
upstream cost increases. By and large, economists typically assume that upstream cost
increases will be passed through dollar-for-dollar or 100% pass-through, or that retailers
will not mark-up an upstream cost increase. Steiner has argued that this is not the way
retailers behave 7 , that they in fact attempt to preserve their %gms in the category. The
model below implies that retailers will nearly preserve their pre-upstream cost increase
%gm, i.e., that the pass-though rate will be 100% plus a mark-up closely related to the
original %gm, rather than a 100% as typically assumed by economists.
What the Model Assumes But Does Not Explain
The model assumes that some manufacturers are successful in establishing their products
as leading national brands, but that this same avenue is not open to the manufacturers of
private labels. The theory doesn’t attempt to explain why some brands achieve high
visibility and others don’t. It just assumes there are some of both. In the real world, it
may be hard to understand why some brands become so successful and luck seems to be
sometimes an important factor. Levi’s jeans, for example, probably owe a considerable
part of their success to the fact that they were worn by James Dean and Marlon Brando in
two extremely influential movies. 8 In any case, LBs and SBs are assumed not explained.
Steiner’s Explanation for the Inverse Associations
Because of advertising, consumers recognize that the LB at one store is identical to the
LB at another store. In contrast, consumers do not know if a store brand at one store is
identical, except in name, to a store brand at a different store. Consumers are more likely
to choose a store to shop on the basis of the availability and price of the LB, than on the
basis of the availability and price of a given store’s SB. 9 Thus, the elasticity of demand,
6
See Lynch, and Steiner AAI papers.
Steiner, The Third Relevant Market, Antitrust Bulletin, vol. 65, #3 (2000).
8
Steiner, Jeans: Vertical Restraints and Efficiency, in Larry Duetsch, Editor, Industry Studies, Prentice
Hall (1993).
9
In my 1986 paper I added that consumers might choose which store to shop at by comparing different
stores’ LB prices, even if they intended to buy few, if any, LBs. See Lynch, “The Steiner Effect: A
7
6
as seen by a retailer, will be higher on an LB than on fringe and store brands. To
maximize profits, retailers will set retail prices using the so-called “Lerner Rule” so that
the %gm on each product will equal the inverse of the elasticity of demand for that
product as perceived by the retailer. Thus, the first inverse association is explained. The
second follows from the assumption that successful national advertising leads to a lower
elasticity of demand as seen by an LB manufacturer relative to an SB manufacturer.10
Now the Lerner Rule implies that each manufacturer will set his factory price so that the
percentage manufacturer margin (%mm) will be equal to the inverse of the relevant
elasticity of demand. Thus, the second inverse association is explained. Further, Steiner
asserts that the relative power of manufacturers/depends on whether consumers are more
“…disposed to switch brands within stores rather than stores within brand.” 11 In the
former case, retailers will dominate manufacturers, in the latter, manufacturers will be in
the driver’s seat.
Modeling Choices
At the core of Steiner’s theory is the relative strength of two different consumer choices:
which store to shop at and, at a given store, which brand to buy. Therefore, a model of
his theory must, at a minimum have at least two retail outlets and each retailer must carry
at least two different brands. Here I begin with the case where each retailer is assumed to
carry both an LB and an SB that competes directly with the LB. Thus the retailer deals in
only one category, but carries two goods in the category. The number of retailers (R), on
the other hand, is allowed to vary and could be large. In fact, I assume that retailers can
either enter the business if it appears to offer higher than normal profits or leave if it does
not, i.e., that R will be just that number of retailers that will eliminate any excess profits.
I make this choice for two reasons. First, I believe that entry is unusually easy in
retailing. If some product or product category is expected to produce higher dollar gross
margins per foot of shelf space than the products currently occupying that space, then an
existing retailer can simply reallocate some of the existing shelf space occupied by lower
margin items to new higher margin generating items. Of course, entirely new stores can
also enter. So I believe that zero profits is a “realistic” assumption. Second, this
assumption leads to refutable predictions and so makes the theory less “empirically
empty” than it otherwise might be. I also assume that all firms vary their prices to
compete, and take the prices of competing firms as given, i.e., they behave as CournotNash players using Bertrand pricing. The model is a special case of Chamberlain’s large
number monopolistic competition model.
I also assume retailers must incur sunk costs to contract with employees and to occupy
space of fR per unit time. This expenditure on overhead costs will support any level and
Prediction from a Monopolistically Competitive Model Inconsistent With Any Combination of Pure
Monopoly or Competition,” Working Paper No. 141, Bureau of Economics, Federal Trade Commission,
August 1986, p.4.
10
I omit two important elements of Steiner’s explanation of an LB manufacturer’s demand curve, the retail
penetration of the brand and the retail dealer support of the brand. The omission is because I have not
succeeded in modeling them, not because I think they’re unimportant.
11
“Basic Relationships…” supra note 1 at p. 202.
7
mixture of LB and a SB product transactions, that is, no maximum capacity and neither
economies or diseconomies of scope.
Missing Elements in This Model
Perhaps the most important element of Steiner’s theory not reflected in this model is
the situation of manufacturers of brands that are perhaps nationally advertised, but are
not stocked by a significant number of retailers. For this type of manufacturer, the
brand’s retail “penetration” – the market share of the category held by retailers who
stock the brand, is crucial. Changes in the factory price will generally lead to changes
in penetration and so the demand a lesser national brand manufacturer faces will be
more elastic than the LB manufacturer. Incorporating lesser national brands into the
analysis would add another layer of complexity to an already complex model.
Moreover, the explicit analytic solution presented here depends on assuming
symmetry, i.e., that in equilibrium, all retailers choose the same prices, sell the same
quantities etc. Retail penetration, however, is only significant when stores behave
differently; some stores carry the lesser national brands, others do not. Without
symmetry, I doubt that an analytic solution is possible.
Other important omissions are: (1) individual consumer demand curves for the category
are totally inelastic, (2) once a given amount of fixed costs are incurred, it is assumed that
any quantity of retail transactions or products can be produced, (3) retailers only offer
two products in one category, whereas real retailers carry tens of thousands of products in
hundreds of categories and (4) there is only one leading brand whereas most categories
have multiple brands that are nationally advertised. I will later sketch a way of
expanding the model to an arbitrary number of categories without adding substantial
complexity. The effects of these omissions may significantly change certain conclusions.
Hopefully, we may be able to investigate these, in the future, through simulation studies.
How Consumers Choose Where to Shop
Consumers must first choose where to shop. Assume there are N potential customers
in a market area served by R stores. The key assumption is that consumers choose a
store on the basis of LB prices only, though other non-price factors, e.g. an individual
consumer’s distance from the store, also matter. The rationale is that consumers know
that LBs are identical at different stores, whereas SBs may vary in quality. Consumers
may also believe that comparative prices on LBs are a good guide to comparative
prices of all the items carried by different stores. Consumers know that SB prices are
always lower than LB prices, so they may rationally choose stores on the basis of LB
prices, even though they may buy an SB product. In any case, it is assumed that
consumers know LB prices before choosing which store to visit. As shown
previously, 12 if the consumer’s choice between any two stores depends only on the
ratio of the LB prices at those two stores (and a couple of other non-restrictive
12
See Lynch, supra, footnote 2.
8
assumptions), then the share or fraction of the N consumers attracted by the ith store is
given by a relatively simple function,
(1) si =
mi pi− μ
R
∑m
k
pk− μ
1
for i =1, ..R where R is the number of retail stores in the market and μ is a parameter that
measures the “visibility” of the LB product among stores, and mi is a parameter
measuring non-price factors affecting the attractiveness of store i to consumers. If all
stores had the same prices, for example, then store i’s share of the retail market would mi
divided by the sum of all the ms. In general, the parameter mi can reflect special
conditions for retailer i (such as being within a more densely populated area than some
other stores). The visibility parameter, μ, measures the sensitivity of consumers to the
ratio of LB prices at different stores. If R = 5, and all mi are equal, then if store 1’s price
was 10% below all others, then it would garner 25% of all customers if μ = 3, 30% if μ =
5 etc. The elasticity of the number of customers as seen by an individual retailer is,
(2)
ε Rsp = −
LB
pLB ∂si
= (1 − si ) μ
si ∂pLB
For example, if μ = 5 and the store’s share is 20%, then the share elasticity is 4. So the
higher the visibility and the lower the market share, the higher the elasticity of demand
seen by the retailer. I will later assume that the visibility parameter can be increased
through advertising. An increase in visibility implies a larger consumer response to a
given difference in prices among stores. 13 Thus the “visibility” of the LB brand is
closely related to the willingness of customers to switch retail stores in order to get the
LB. I will later assume that N, the number of consumers in the market who wish items in
the product category can be increased through manufacturer advertising.
Choice between LB and SB within a given store
Consumers will buy at most one unit of either the LB or the SB product, but not both.
Consumers vary with respect to the “reputation” premium, r, they are willing to pay for
the LB relative to the SB. Reputation premia are uniformly distributed between 0 and
rmax. If the jth consumer places a premium of rj on LB, for example, he will select LB
over SB at store i if pLBi is less than or equal to pSBi + rj. 14 If, on the other hand, pLBi is
13
Steiner used the example of bridges connecting previously separated retail markets as an analogy for the
effects of advertising on retail competition. See his “Intrabrand Competition – Stepchild of Antitrust”,
Antitrust Bulletin, vol.36, #1, (1991) 170-172.
14
We also assume that consumer j has a maximum reservation price of rp + rj where rp is the
(common) reservation price that all consumers share for SB products.
9
greater than pSbi + rj, then consumer j will buy one unit of the SB so long as its price is
lower than the consumer reservation price, rp, for the SB. The quantities sold at the at
the ith store will be,
⎡ r − ( pLBi − pSBi ) ⎤
(3) qLBi = si N ⎢ max
⎥ , if pSB < pLB <= pSB + rmax
rmax
⎣
⎦
and,
⎡ p − pSBi ⎤
(4) qSBi = si N ⎢ LBi
⎥ , if pSB < pLB <rp
⎣ rmax
⎦
Note that the expressions in the brackets on the right hand side of equations (3) and
(4) are the shares of the customers visiting store i that choose the LB and SB
respectively and that the quantities sum to si N.
Retailers: Pricing the Store Brand
Retailers now choose prices to maximize category profits. Retail profit for store i is
given by,
(5) π i = ( pLBi − mLB ) qLBi + ( pSBi − mSB ) qSBi − f Ri
Or,
⎡
⎛ r − ( pLBi − pSBi ) ⎞
⎛ ( pLBi − pSBi ) ⎞ ⎤
(6) π i = si N ⎢$ gmLBi ⎜ max
⎟ + $ gmSBi ⎜
⎟ ⎥ − f Ri = si N ( $ gmpci )
r
r
⎢⎣
max
max
⎝
⎠
⎝
⎠ ⎥⎦
Where $gmpci stands for dollar gross margin per customer, that is, the share weighted
average of the two gross margins shown in the bracket on the left hand side of equation
(6).
For convenience, I rewrite equation (6) as,
⎡
⎛ ( pLBi − pSBi ) ⎞ ⎤
⎟ ⎥ − f Ri
rmax
⎝
⎠ ⎦⎥
(7) π i = si N ⎢$ gmLBi + ( $ gmSBi − $ gmLBi ) ⎜
⎣⎢
Now set the partial derivative of (7) above with respect to pSBi equal to zero. Note that si
is independent of the price of the SB.
(8)
⎡
⎛ 1
∂π i
= si N ⎢( $ gmSB − $ gmLB ) ⎜ −
∂pSBi
⎢⎣
⎝ rmax
⎞ ⎛ pLB − pSB
⎟−⎜
⎠ ⎝ rmax
⎞⎤
⎟⎥ = 0
⎠ ⎥⎦
10
Multiply both sides of 8 by rmax/Nsi (assumed >0) to get,
(9) pSBi − mSBi − pLBi + mLBi − pLBi + pSBi = 0
Now solve (9) for the profit maximizing SB retail price,
⎛ m − mSB
*
(10) pSBi
= pLBi − ⎜ LB
2
⎝
Δm
⎞
⎟ = pLBi − 2
⎠
The remarkably simple equation (10) says that the profit maximizing retail price of the
store brand will equal the retail price of the leading brand less the average difference in
the two products’ manufacturer selling prices. We see below that the store brand prices
predicted by equation (10) and the actual store brand prices for soft drinks chosen by
Dominick’s are surprisingly good. The store brand will always sell at a discount relative
to the leading brand and that discount will equal the average difference between the two
factory prices.
Note also that equation (10) implies,
(11) $ gmSB − $ gmLB =
Δm
2
So if the factory price of the leading brand is higher than the factory price of the store
brand, then the profit maximizing dollar gross margin on the store brand will be higher
than the margin on the leading brand. Moreover, since the retail price of SB must be
lower than the retail price of LB, the percentage retail gross margin on the SB will be
higher than on the LB item. Thus, equations (10) and (11) imply the first inverse
association holds for both dollar and percentage margins.
The price differential in equation (10) also determines the store level market shares of the
two items (see equations (3) and (4) above).
I now use equation (10) to eliminate pSBi from the remaining equations. Equation (7) can
now be written as,
2
⎡
⎛ Δm ⎞ ⎛ 1
(12) π i = si N ⎢$ gmLBi + ⎜
⎟ ⎜
⎝ 2 ⎠ ⎝ rmax
⎣⎢
⎞⎤
⎟ ⎥ − f Ri = si N $ gmpci − f Ri
⎠ ⎦⎥
Note that $gmpci, defined above as the share weighted average margin per customer, is
now equal to the expression in the bracket on the left hand side.
11
Retailers: Pricing the Leading Brand
Setting the partial derivative of equation (12) with respect to pLBi to zero leads to,
(13)
∂π i
∂s
= si N [1] + [$ gmpci ] i N = 0
∂pLB
∂pLB
Equation (13) can be rearranged to,
(14)
pLB
p ∂si
= − LB
= μ (1 − si )
si ∂pLB
[$ gmpci ]
Solve equation (14) for si ,
(15) si =
μ $ gmpci − pLB
μ $ gmpci
Finally, entry and exit assures that retailers only recover fixed costs, or that,
(16) si N ( $ gmpc ) = f Ri or,
(17) si =
f Ri
N ( $ gmpci )
By equating (15) and (17), multiplying both sides of the resulting equation by μ$gmpci,
we get,
(18) μ ( $ gmpci ) − pLBi = μ
f Ri
N
Finally, we can solve (18) for the profit-maximizing retail price for the LB,
(19) p
*
LBi
2
⎛ μ ⎞⎡
⎛ Δm ⎞ ⎛ 1
=⎜
⎟ ⎢ mLB − ⎜
⎟ ⎜
⎝ 2 ⎠ ⎝ rmax
⎝ μ − 1 ⎠ ⎢⎣
⎞ f Ri ⎤
⎥
⎟+
⎠ N ⎥⎦
Note the similarity and the differences from the “Lerner monopoly mark-up” equation.
The first term in parenthesis on the right hand side is akin to the usual mark-up with the
visibility parameter, μ, playing the role of the elasticity as seen by the retailer. While μ is
closely related to the elasticity of the number of customers attracted by changes in the
price of the LB, it is not equal to the elasticity of demand faced by the retailer for the
12
leading brand. The bracket on the right contains three terms. The first term is simply the
manufacturer’s price for the LB and corresponds precisely to the input price in the Lerner
theorem. The second term is an offset due to competition from the store brand. The
larger the gap between the LB and SB factory prices, the larger the offset. The offset is
proportional to the square of the gap. The third term reflects the requirement that each
store just cover its fixed costs and no more. Note that no individual retailer prices
explicitly to recover fixed costs. Rather retailers choose prices to maximize their gross
margin dollars from the category. However, entry and exit enforce recovery of individual
retailer fixed costs and that is why the equilibrium price contains a term reflecting them.
Symmetric Case: Retailers
To further simplify, I now assume that all retailers have identical parameters, that is,
(20)
mi = m
f Ri = f R
for i = 1,…R
The subscript “i” can now be dropped from equations (10) and (18) giving the retail
prices of each brand at each store. The equilibrium number of retailers, R*, is determined
by
Ngmpc*
fR
The larger the number of customers, and the larger the average dollar gross margin per
customer, the larger the number of retailers will be. On the other hand, the higher the
fixed costs of each retailer, the fewer there will be.
(21) R* =
Manufacturers
Private Label Manufacturer
I assume that manufacturing store brands is highly competitive and that there are a large
number of firms that will enter or exit if mSB differs from the cost of producing the
private label, including a normal rate of return on investment. I treat mSB as a parameter.
Leading Brand Manufacturer: the Optimal Factory Price
I assume that the LB manufacturer has constant marginal cost, mcLB, and fixed costs of
fLB. He chooses a price, m*LB , and an advertising expenditure, A*, to maximize profits.
(22) π LB = ( mLB − mcLB ) QLB − f LB − A
Where from equations (3) and (10),
13
⎛ r − ( pLB − pSB ) ⎞
⎛
Δm ⎞
(23) QLB = N ⎜ max
⎟ = N ⎜1 −
⎟
rmax
⎝ 2rmax ⎠
⎝
⎠
The partial derivative of equation (22) is,
(24)
⎛
∂π LB
∂Q
Δm
= QLB + ( mLB − mcLB ) LB = N ⎜1 −
∂mLB
∂mLB
⎝ 2rmax
⎞
( mLB − mcLB )
⎟− N
2rmax
⎠
Finally, setting the expression in (24) equal to zero and solving for the LB factory price
yields,
⎛ m + mcLB ⎞
*
(25) mLB
= rmax + ⎜ SB
⎟
2
⎝
⎠
Equation (25) says that the profit-maximizing factory price for the LB is equal to the
maximum reservation premium that any consumer would pay for the brand plus the
average of the factory price of the store brand and the LB firm’s marginal cost.
Leading Brand Manufacturer: the Optimal Advertising Expenditure
Finally, I assume that the LB manufacturer can, through advertising, affect the total
number N of potential customers who might buy a brand in the category, its maximum
reputation premium rmax any consumer is willing to pay for the LB, and the visibility, μ,
of the brand. It is at this point that the complexity of the model overwhelmed my ability
to find an analytical solution. I simply assume that,
N = α 0 Aα1
(26) rmax = α 2 Aα 3
μ = α 4 + α 5 rmax
Where all the alpha constants are greater than zero and the exponents α1 and α3 are less
than 1. So, in all three equations, advertising exhibits decreasing returns to scale.
Spreadsheet Model
For given values of N, rmax and μ, the equations derived above allow one to calculate the
retail prices of the LB and SB products, their market shares and the factory price and
quantity sold of the LB. The factory price of the SB is taken as given, as is the marginal
and fixed cost of the LB manufacturer. I have constructed a spreadsheet incorporating
these equations that can be used to find the profit-maximizing level of advertising
expenditure and so simulate the entire model. Begin by choosing an arbitrary
14
expenditure on advertising. Then use the three equations listed above in (26) to compute
N, rmax and μ. Now use equation (25) to calculate the optimal LB factory price. Given
the LB and SB factory prices, rmax and μ, you can now use equation (19) to calculate the
optimal retail price for the LB. Given p*LB, use equation (10) to calculate the optimal
retail price of the SB. Given the retail prices, we can now calculate the quantities and
market shares of both products, and so on. We now have the optimal LB factory price
and the quantity sold, and can compute the LB manufacturer’s profit for any given
advertising expenditure. I now use the “solver” function in Excel to find that level of
advertising expenditure that maximizes the LB manufacturer’s profit. The spreadsheet
automatically calculates all the prices and quantities corresponding to that level of
advertising expenditure.
Implication1: Retailers Will Set Higher Dollar and Percentage Gross Margins on SBs
Than on LBs
In the model, it is very clear why profit-maximizing retailers will always set higher
$rgms on SBs. Take the Coke Classic and its competing store brand as shown in the first
row of Table 1 as an example. For every unit of Classic Coke sold, the retailer keeps 35
cents as a contribution toward covering fixed cost including profits. Suppose, contrary to
fact, Dominick’s had priced the SB at $1.35, instead of $1.96 as observed. Then the
$gms would be same on both brands. The retailer would presumably sell more units of
the store brand, but he would not increase his category gross margin dollars by one
penny. Why? Because every additional unit of the SB sold means one fewer unit of the
LB sold. If the retailer priced the SB at $1.34, he would actually reduce category profits
despite higher SB unit sales. If he set the SB price at $1.36, however, he would increase
category profits so long as SB sales didn’t fall to zero. At the higher price, some SB
customers will switch to Classic Coke, but those who continue to choose the SB will
yield a net increase of one penny per unit sold to the gross margin dollars earned in the
category. It is the difference between the $gms that counts. There is a limit to how much
the retailer can raise the price of the SB. In the model, it is assumed that at an SB price
of $1.96, the SB share of unit sales would fall to zero. So the model says that if the SB is
carried at all, a profit maximizing retailer will always put a higher dollar margin on the
SB than on the LB. Since the retail price of the SB must always be lower than the retail
price of the LB, it follows that the %gm on the SB will always be higher than on the LB.
Thus the first inverse association is explained.
The model, however, does more than just confirm the first inverse association; it
determines the exact amount by which the SB must be discounted from the LB. Because
of the reputation premium Coke Classic commands, the SB must always sell at a discount
to it. In the model, how much of a discount depends on the maximum reputation
premium any customer would pay. The actual discount chosen by Dominick’s is shown
in Table 1 to be 84 cents per unit. Under the assumptions given above, the model
provides a surprisingly simple formula for calculating the category profit maximizing
price for the SB, given the retail price for the LB. The model implies that the optimal SB
15
price is equal to the LB retail price less the average difference between the manufacturer
prices. For given manufacturer prices (here $2.35 and $1) and a given Classic Coke retail
price of $2.70, the profit maximizing retail price for the SB will be $2.03. The actual
price chosen by Dominick’s was $1.96. Table 2 shows the predicted versus the actual
prices for all the soft drink brands shown in Table 1. In the model, it will always be true
that retailers will choose higher margins on the SB than on the LB.
16
Table 2
Predicted versus Actual Store Brand Soft Drink Prices,
Dominick's Data, 1987 – 1997
Leading Brand
Predicted Actual
Pred.
less %Error
Actual
Coke Classic
Schwepps Ginger Ale
Pepsi Cola N/R
Pepsi Cola Diet N/R
Barq's Root Beer
Schweppes Tonic N/R
Schweppes Diet Tonic N/R
R.C. Cola
A&W Rootbeer Reg
Sunkist Orange
Schwepps Ginger Ale
Canada Dry Ginger Ale
A&W Rootbeer SF
Diet Coke
$2.03
$1.91
$2.02
$1.94
$2.02
$2.48
$2.49
$2.04
$1.64
$2.16
$2.54
$2.79
$1.69
$1.94
$1.96
$1.92
$1.91
$1.82
$1.81
$2.30
$2.32
$2.09
$1.75
$2.18
$2.34
$2.34
$1.73
$1.81
$0.07
-$0.01
$0.11
$0.12
$0.21
$0.19
$0.17
-$0.04
-$0.11
-$0.01
$0.20
$0.45
-$0.05
$0.13
3.41%
-0.48%
5.43%
6.23%
10.25%
7.52%
6.76%
-2.14%
-6.72%
-0.57%
7.73%
15.96%
-2.83%
6.89%
Column Average
Column Median
Column Minimum
Column Maximum
$2.12
$2.02
$1.64
$2.79
$2.02
$1.94
$1.73
$2.34
$0.10 4.10%
$0.12 5.83%
-$0.11 -6.72%
$0.45 15.96%
Source: Barsky, Robert, Mark Bergen, Shantanu Dutta, Daniel
Levy, “What Can the Price Gap between Branded and Private
Label Products Tell Us about Markups?” Presented at the NBER
Conference on Research in Income and Wealth:Scanner Data
and Price Indexes,.September 15-16, 2000 [Revised: September
13, 2001], Available at: : papers.nber.org/papers/W8426.
17
Implication2: the Strong Steiner Effect
Figure 1 illustrates the strong Steiner effect, i.e., the retail price falls despite the rise in
the factory price. Initially the LB factory price is $1.04 and its retail price is $1.74. Then
there is a change in the effectiveness of advertising. A given dollar spent on advertising
now reaches a larger number of people. The figure is generated by changing the
parameter α1 in equation (26)), the power to which A is raised, from .4 to .41. The new
equilibrium entails higher advertising expenditure, a larger number of potential
customers, a higher maximum reputation premium and increased visibility. The LB
manufacturer raises his price by two cents to $1.06, but the retail price actually falls by
one cent to $1.73. The succeeding points in the figure show the equilibrium prices as α1
increases by .01, until it is equal to .5. At that point, the LB factory price has been raised
by 48 cents, from $1.04 to $1.52, but the retail price has fallen by 11 cents to $1.63.
In the model, it is not always true that the strong Steiner Effect obtains. The effects of
increased advertising on the visibility parameter, μ, and on the number of potential
customers must out weigh its affect on increasing the reputation premium and hence the
LB factory price.
18
LB Factory and Retail Prices as a Function of
Increasing Advertising Effectiveness on the Size of
the Market
Prices ($ per unit)
$1.80
$1.60
$1.40
LB Factory Price
LB Retail Price
$1.20
$1.00
$0.80
$0.60
0
100
200
300
400
500
Number of Potential Customers (000s)
Figure 1
19
Implication 3: The First (Relative) Inverse Association
Figure 2 reflects the same simulation as Figure 2, but this time plots the %gms as the
advertising parameter α1 increases from .4 to .5 by steps of .01. The LB %gm falls from
40% to under 7% because of increased visibility and the larger number of potential
customers. However, the SB retail margin also falls, from about 42% to 27%. The SB
must compete with the now lower priced LB at retail, so its dollar and percentage
margins must fall. This result is related to, but not quite the same as, the first inverse
association discussed by Steiner. Although the SB %gm increases relative to the LB
margin, both margins fall with increasing advertising effectiveness and visibility. Figure
3 shows that the dollar gross margin of the LB as a percentage of the share weighted
category margin (gmpci) falls from slightly under 100% to less than 55%, while the SB
margin increases from slightly more than 100% to about 188%.
20
%Retail Margin
LB and SB %Retail Margins As A Function of
Increasing Advertising Effectiveness on the Size
of the Market
45.0%
40.0%
35.0%
30.0%
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
%GM-LB
%GM-SB
0
100
200
300
400
500
Number of Potential Customers (000s)
Figure 2
$GM as % of Category
$GM
Inverse Relative Association Between LB and SB
Dollar Retail Margins Relative to the Average
Category as a Function of Increased Advertising
Effectiveness
200.0%
150.0%
$GMLB/$GMPC
$GMSB/$GMPC
100.0%
50.0%
0.0%
0
200
400
600
Number of Potential Customers
(000s)
Figure 3
21
Inverse Association of Percentage LB Margins at
Retail and Manufacturing Levels: The Steiner
Effect
80.0%
% Margin
70.0%
60.0%
50.0%
40.0%
%GMLB
%MMLB
30.0%
20.0%
10.0%
0.0%
0
100
200
300
400
500
Number of Potential Customers (000s)
Figure 4
Implication 4: The Second Inverse Association
Figure 4 again uses the same simulation to illustrate the second inverse association. As
the brands popularity and visibility increases, the LB factory %margins go up, but retail
margins decline.
22
LB Retail Price ($ per
unit)
LB Retail Pass-Through of Factory Price Increases More
Than 100%
$4.00
$3.50
$3.00
$2.50
$2.00
$1.50
$1.00
$1.00
LB Retail Price
LB Retail 100% PT
$1.50
$2.00
$2.50
LB Factory Price ($ per unit)
Figure 5
Implication 5: Pass-throughs of General Upstream Cost Increases Will Always be at a
Rate of More Than 100%
Figure 5 illustrates a general result of the model, general upstream manufacturer cost
increases will be passed through to the consumer in a manner that preserves the original
retail %gm. We begin with the same initial situation presented in figures 1 – 4. In this
example, none of the initial advertising parameters change. Instead, both the factory SB
price and the LB manufacturers marginal cost are assumed to increase by 10 cents at each
step. Equation (25) implies that the LB factory price will by rise by 10 cents. Given that
the difference in the factory prices is unchanged, equation (10) implies that the difference
between the two retail prices will be unchanged. Equation (19) now implies that the LB
factory price will be marked-up by a factor of μ/(μ-1). The SB retail price will rise by
exactly the same dollar amount as the LB retail price, since the differential between the
two remains unchanged. Thus the pass-through rate for a uniform upstream cost increase
will exceed 100% by one divided by the visibility parameter, μ.
23
Broader View of Steiner’s Theory
The model sketched above applies to one category, whereas real retailers carry hundreds
of categories. Although there are some new complications to be faced when extending
the model to many categories, it does suggest the following broad view of how retail and
manufacturing margins are determined. Each category contains at least one LB.
Equation (1) can be generalized to cover any number of categories. The higher the
visibility of the LB, the lower the LB’s retail margin. LBs in different categories have
different visibilities and this accounts, in part, for differences among category margins.
Whether a low margin on an LB leads to low category margins depends on ratio of the
LB and SB factory prices and the reputation discount that the SB must offer to compete
with the relevant LB. The ratio of LB to SB factory prices will be lower, the greater the
extent of manufacturing economies of scale available to the LB manufacturer.
The major difficulty in extending the model to many categories, however, is that the
quantity that a consumer buys in one category is not independent of the quantity she buys
in another. Due to transportation economies of scale, it will often be rational to buy items
in many different categories during a particular store visit. Thus, inter-category
purchases are heavily interdependent, and this interdependence needs to be explicitly
modeled. This requires a theory of how consumers select a store from which to buy a
bundle of goods, when transportation is subject to economies of scale. Robert W.
Bacon’s book, Consumer Spatial Behaviour: A Model of Purchasing Decisions over
Space and Time, Clarendon Press Oxford; 1984, constitutes a brilliant start, but also
shows how difficult such a theory will be.
There is little one can say of a general nature concerning the overall efficiency of the
equilibria generated by this model. I am afraid that one of the prices to be paid in moving
to more realistic model recognizing economies of scale is the absence of any sweeping
statements saying X is clearly the most efficient arrangement. The best we can do, and it
is difficult enough, is to analyze specific situations in hopes of finding more or less
efficient modes of organization.
24