Price Setting with Customer Retention Concerns
Luigi Paciello
,
Andrea Pozzi
,
Nicholas Trachter
EIEF
Università di Roma “Tor Vergata”
May 2013
Paciello, Pozzi, Trachter (EIEF)
1 / 29
Intro
Customer retention concerns
I
I
Departure from purely static demand
Distinguish between two margins of demand
I
I
I
Intensive: P ↑⇒ Q ↓
Extensive: “chunks” of demand (i.e. customers) are gained or lost
If there are frictions it is hard to regain lost customers
Paciello, Pozzi, Trachter (EIEF)
2 / 29
Intro
What we do (and aim to do)
1) Novel evidence on shopping behavior from supermarket scanner data
I
freq. of shopping, freq. of customers exiting, dist. of basket price changes
I
measure elasticity of customer exit to own basket price
2) GE model of price setting with customer retention concerns
I
features
I
customers incur random search cost to leave producer of homogenous good
I
persistent idiosyncratic productivity shocks make producers heterogenous
3) Bring together data and model to estimate some key parameter
Paciello, Pozzi, Trachter (EIEF)
3 / 29
Intro
Why do we care?
Framework allows to make three main contributions
I optimal price setting policy
I
quantify cost pass-through/real price rigidity
I
quantify relevance for economy response to aggregate TFP shocks
Paciello, Pozzi, Trachter (EIEF)
4 / 29
data
Data description
1) Scanner data on grocery purchases at a large supermarket chain
I two years of data (2004-2006)
I information on timing, quantities, and prices of purchased goods (UPC’s)
2) Store data
I weekly revenue and quantities for each UPC
I
>200 stores (representative of all price areas) in 10 states
3) Census data to match demographic characteristics at block group level.
Paciello, Pozzi, Trachter (EIEF)
5 / 29
data
Descriptive statistics: shopping behavior
Longitudinal variation
Expenditure per trip (USD)
Days elapsed between trips
Mean
Std.Dev.
25th pctile
75th pctile
56
4.2
66
7.5
12
1
78
5
Cross-sectional variation
Expenditure per trip (USD)
Days elapsed between trips
Number of trips (Avg across hhid’s)
Paciello, Pozzi, Trachter (EIEF)
Mean
Std.Dev.
25th pctile
75th pctile
69
7.7
150
40
11.6
128
40
3.4
66
87
8.6
200
6 / 29
data
Construction of key variables
1) Price of a customer’s basket
I
match household to own complete value price series
I
I
can be done for a subset of households
weighted average of prices of goods in the agent’s basket
I
I
basket is set of UPCs purchased by agent over the two years
the weight of each UPC is its share of total expenditure for this agent
2) Exit from customer base
I
an agent is no longer in the customer base when it does not visit the
chain for n=4 consecutive weeks
I
we attribute the exit to the week of the last visit
Paciello, Pozzi, Trachter (EIEF)
7 / 29
data
Identifying times of customer leaving
Exiting = Missing at least 4 consecutive weeks
Date of exit: last date of shopping
Mean Std.Dev.
Longitudinal variation
Duration as customer (weeks)
39
38
Cross-sectional variation
Duration as customer (weeks)
61
38
Paciello, Pozzi, Trachter (EIEF)
25th pctile
75th pctile
7
78
21
99
8 / 29
data
0
Fraction of households
.2
.4
.6
Probability of exiting the customer base
0
Paciello, Pozzi, Trachter (EIEF)
.05
.1
.15
Probability of leaving
.2
.25
9 / 29
data
Descriptive statistics: prices
Table: Descriptive statistics on basket price changes
Shoppers Basket
# of UPC in the basket
∆p
|∆p|
%|∆p| > 1%
%|∆p| > 5%
%|∆p| > 10%
Paciello, Pozzi, Trachter (EIEF)
Stores Basket
Mean
290
-0.01%
2.9%
Std.dev.
172
4.3%
3.1%
Mean
1,430
-0.04%
2.4%
Std.dev.
1,051
3.3%
2.4%
73.5
16.8
4.1
-
68.6%
11.5%
2.6%
-
10 / 29
data
The effect of prices on probability of leaving
Dep. variable: Indicator for leaving the chain in that week
(mean=0.042; st.dev.=0.20)
Exitht = b0 + b1 dpht + Xh0 b2 + Xt0 b3 + εht
dph
(1)
(2)
0.031**
0.023**
(0.015)
(0.012)
dpstore
-0.013
(0.022)
Observations
Dem. controls
Time f.e.
Paciello, Pozzi, Trachter (EIEF)
103,507
No
Yes
91,555
Yes
Yes
11 / 29
model
Summing up
I
Persistence in the customer-retailer relationship
and
I
Sensitivity to prices of break in the customer-retailer relationship
Suggest frictions in customer’s search for a new retailer.
Next: What are the implications for price setting and pass-through?
Paciello, Pozzi, Trachter (EIEF)
12 / 29
model
Setup
3 types of agents:
I measure one of firms producing good 1 (price p) - object of interest
I
I
competitive rep. firm producing good 2 (price q)
rep. household: a worker and measure Γ > 1 of shoppers
Timing:
1. At beginning of t, each shopper matched to a producer j of good 1.
2. idiosyncratic productivity shocks realize
3. each producer set price
4. each shopper draws own search cost ψ i.i.d. ∼ G(·), ψ ∈ [0, +∞)
I
I
if incur cost ψ, randomly assigned to a new producer
otherwise, stays with current producer
5. consumption occurrs
6. the match is exogenously disrupted w.p. τ
Paciello, Pozzi, Trachter (EIEF)
13 / 29
model
Production technology
Production of good 1: linear in labor, y1,j = zj l1,j .
Productivity z: K values z1 < z2 < · · · < zk < · · · < zK ,
Pr[ zj0 = s zj = k ] = µks .
Production of good 2: linear in labor, y2 = l2 .
Paciello, Pozzi, Trachter (EIEF)
14 / 29
model
The household’s problem
Preferences:
"
E0
∞
X
Z
β
t
t=0
L(t)1+φ
u(ci (t))di −
−
1+φ
#
Z
χi (t)ψi (t) di
where
u(ci (t)) =
ci (t)1−γ
1−γ
ci (t) = ω 1−θ c1,i (t)θ + (1 − ω)1−θ c2,i (t)θ
1/θ
, θ < 1 , ω ∈ (0, 1)
Assumptions:
I no inventories/full perishability
I
no within-household insurance/non-verifiability of shopping
Paciello, Pozzi, Trachter (EIEF)
15 / 29
model
Household’s intra-temporal decisions
Consumption decision:
u0 (pi , I) = max
θ
θ
ω 1−θ c1,i
+ (1 − ω)1−θ c2,i
(1−γ)/θ
1−γ
c1,i ,c2,i
subject to
pi c1,i + q c2,i = I ,
Labor decision:
"
max Et
L(t)
∞
X
β
τ −t
τ =t
Z
L(t)1+φ
u0 (pi (t), I(t))di −
1+φ
subject to
Z
#
1
Γ I(t) = W (t) L(t) +
πj (t)dj
0
Paciello, Pozzi, Trachter (EIEF)
16 / 29
model
Shopper’s search decision
Conjecture equilibrium Markovian structure for prices:
P ≡ {p1 , p2 , . . . , pk , . . . , pK }
Value of staying (V̄ (p, k , Pj )) is
"
#
Z
K
n
o
X
j
0
j
u0 (p) + β τ V̂ + (1 − τ )
µkk 0 max V̄ (pk 0 , k , P ), V̂ − ψ g(ψ) dψ ,
k 0 =1
Outside option: V̂
Exogenous probability of leaving: τ
Paciello, Pozzi, Trachter (EIEF)
17 / 29
model
Optimal search policy
p̄k : price below which no-one leaves, i.e.
V̄ (p̄k , k ; P) = V̂ .
ψ̄(p, p̄k ): threshold rule s.t. shopper leaves if ψ ≤ ψ̄(p, p̄k ), i.e.
0
if p ≤ p̄k ,
ψ̄(p, p̄k ) =
u0 (p̄k ) − u0 (p) otherwise ,
ψ̄(p, p̄k ) is strictly increasing in p and strictly decreasing in p̄k , for all p ≥ p̄k .
Paciello, Pozzi, Trachter (EIEF)
18 / 29
model
Firm’s problem: customer base
mj ≡ customer base of producer j at the beginning of the period.
mj+ ≡ customer base of producer j after stay\leave decisions are made.
δ ≡ arrival rate of new shoppers, proportional to mj .
Producer j’s shoppers dynamics:
h
i
mj+ = mj (1 + δ) − mj G ψ̄(p, p̄kj ) = mj δ + 1 − G ψ̄(p, p̄kj )
, (1)
h
i
mj0 = mj+ (1 − τ ) = mj δ + 1 − G ψ̄(p, p̄kj ) (1 − τ ) .
(2)
Producer j’s period profits:
W
π(p, zk ) = c1 (p) p −
zk
p̂k ≡ argmax p π(p, zk )
Paciello, Pozzi, Trachter (EIEF)
19 / 29
model
Firm’s problem: Construction of value function
Value function of firm j with productivity k
F̃ (zk , mj )
= max mj+ π(p, zk ) + β̂
s.t.
p
X
µks F̃ (zs , mj0 )
for each k ,
s
eq. for customer base dynamics and static profits
Contraction Mapping theorem implies that F̃ (zk , mj ) = mj F̃ (zk , 1) = mj F (zk )
where
!
K
X
0
µk k 0 F (pk 0 , k )
π(p, zk ) + (1 − τ )β̂
F (p, k ) = δ + 1 − G ψ̄(p, p̄k )
k 0 =1
pk∗
≡ argmax p F (p, k )
Paciello, Pozzi, Trachter (EIEF)
20 / 29
model
Firm’s problem: caveats
1. F (zk ) may not be a contraction ⇒ Include probability of firm death
2. The problem of the firm is not globally concave in p ⇒ FOC can be used
only because we prove the following
Proposition: The value function for a firm with productivity zk , i.e. F (zk ), is
single-peaked, attaining its maximum at p = pk∗ solving the first order
condition of the problem.
Moreover, let p̂k denote the price that maximizes static profits for a firm with
current productivity zk , i.e. ∂π(p̂k , zk )/∂p = 0. If p̄k ≥ p̂k , then pk∗ = p̂k . If
p̄k < p̂k , then pk∗ ∈ (p̄k , p̂k ).
Paciello, Pozzi, Trachter (EIEF)
21 / 29
model
Case 2: p̂k > p̄k
Per-Cu stomer F irm Valu e
Per-Cu stomer F irm Valu e
Case 1: p̂k ≤ p̄k
p ∗ = p̂
p̄
p
I
I
p̂
p
p̂k > p̄k ⇒ pk∗ ∈ (p̄k , p̂k ) OR p̂k ≤ p̄k ⇒ pk∗ = p̂k .
if p̂k > p̄k , ”k ” has two effects on pk∗ :
I
I
I
p∗
p̄
standard cost channel: k ↑ ⇒ zk ↑ ⇒ pk∗ ↓
novel search channel: k ↑ ⇒ p̄k ↑↓ ⇒ pk∗ ↑↓
persistent productivity + search costs give non-monotonicity of pk∗ .
Paciello, Pozzi, Trachter (EIEF)
22 / 29
model
Equilibrium
All producers choose same pricing polic:y P∗ = {pk∗ }k ≤K
f (m, k ): invariant distribution of (m, k ) solves
f (m, y ) =
K
X
µky f
k =1
Γ=
δ=
K Z
X
k =1 0
PK
k =1
Z
∞
M(k ) ≡
m
,k
δ + 1 − G ψ̄(pk∗ , p̄k ) (1 − τ )
!
,
∞
m f (m, k ) dm
M(k ) G ψ̄(pk∗ , p̄k )
τ
+
Γ
1−τ
m f (m, k ) dm
0
Shopper outside option:
V̂ =
K
X
µ∗k V̄ (p∗k , k , P∗ )
k =1
Paciello, Pozzi, Trachter (EIEF)
23 / 29
model
Statistics of interest
(Observable) Avg. elasticity of probability of exit to price changes:
b≡
K
X
M(k )
k=1
Γ
G(ψ̄(pk∗ e , p̄k )) − G(ψ̄(pk∗ , p̄k ))
,
→0
lim
(Observable?) Avg. change in probability of exit conditional on |∆p| > 0:
κ≡
K
X
M(k )
k =1
Γ
K
X
k 0 |pk∗0 >pk∗
µk 0 |p∗0 >pk∗ G(ψ̄(pk∗0 , p̄k 0 )) − G(ψ̄(pk∗ , p̄k )) ,
k
(Model Output) Pass-through:
ξ≡
K X
K
X
k
Paciello, Pozzi, Trachter (EIEF)
k 0 6=k
µ∗k
µkk 0 log(pk∗0 ) − log(pk∗ )
.
1 − µkk log(zk 0 ) − log(zk )
24 / 29
model
Parameter choices
Parameter/function
Value
Method
Model
Data
0.999
Standard
Calibrated parameters
Discount factor, β
Inter. elast. of subst., γ
1
Standard
Disutility of labor, φ
1
Frisch elasticity = 1
5/6
Avg. markup in 10%-20% range
Elast. of subst. good 1 and 2, θ
Estimated parameters
Productivity process, ln zt = ρ ln zt−1 + σt
OLS using cost data on a panel of stores
Persistence parameter, ρ
0.84
Standard deviation parameter, σ
0.035
Distribution of search cost, g(ψ)
Simulated method of moments
Shape parameter, λ1
??
Target b
Scale parameter, λ2
??
Target κ
0.55
Target grocery share of disp. income (0.38)
Expenditure share, ω
Paciello, Pozzi, Trachter (EIEF)
25 / 29
model
Results
Avg. elasticity of customer base (b)
Avg. pass-through (ξ)
0.9
0.8
5
0.7
Pass-Th rou gh : ξ
Sensitivity of Breaks to Prices: b, in %
6
4
3
2
0.6
0.5
0.4
0.3
0.2
1
0.1
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Scale of Search Costs: λ 2
Paciello, Pozzi, Trachter (EIEF)
0.08
0.09
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Scale of Search Costs: λ 2
26 / 29
model
Non-monotonicity in prices
Optimal Price vs Productivity
Probability of leaving vs Productivity
0.1
0.3
λ2 =
λ2 =
λ2 =
λ2 =
Log-Op timal Price: p ∗k
0.26
0.24
0.22
0.2
0.18
λ2 =
λ2 =
λ2 =
λ2 =
0.09
Prob ab ilty of B reak: G( ψ̄ k)
0.28
0.01
0.10
0.20
0.30
0.01
0.10
0.20
0.30
0.08
0.07
0.06
0.05
0.04
0.03
0.16
0.02
0.14
0.12
−0.2
0.01
−0.15
−0.1
−0.05
0
0.05
Log-Productivity: log(z k )
Paciello, Pozzi, Trachter (EIEF)
0.1
0.15
0
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Log-Productivity: log(z k )
27 / 29
model
Extensions
1. Returning customers
2. Loyal customers
Paciello, Pozzi, Trachter (EIEF)
28 / 29
model
What’s next
Response to aggregate productivity shock
Comparison to linear model.
Paciello, Pozzi, Trachter (EIEF)
29 / 29
appendix
Construction of key variables
1) Price of a customer’s basket
I
match household to own complete value price series
I
I
can be done for a subset of households
weighted average of prices of goods in the agent’s basket
I
I
basket is set of UPCs purchased by agent over the two years
the weight of each UPC is its share of total expenditure for this agent
2) Exit from customer base
I
an agent is no longer in the customer base when it does not visit the
chain for n=4 consecutive weeks
I
we attribute the exit to the week of the last visit