When is Price Discrimination
Profitable?
Eric T. Anderson
Kellogg School of Management
James Dana
Kellogg School of Management
Motivation
• Price Discrimination by a Monopolist
– Offer multiple products of differing qualities
– Distort quality sold to low value consumers
(Mussa and Rosen, 1978)
• But, price discrimination is not always optimal,
and certainly not always used
– Stokey (1979)
– Salant (1989)
Research Agenda
• Develop prescriptive tools to evaluate when
price discrimination is profitable.
• Applications
– Advance Purchase Discounts
• Screening using reduced flexibility
– Intertemporal Price Discrimination
• Screening using consumption delays
– “Damaged” Goods
• Screening using reduced features
– Versioning Information Goods
– Coupons
Key Assumption:
Quality is Constrained
• Commonly Made Assumption
– Explicit
• Salant (1989)
– Usually implicit and underemphasized
•
•
•
•
Coupons (Anderson and Song, 2004)
Intertemporal Price Discrimination (Stokey, 1978)
Damaged Goods (Deneckere and McAfee, 1996)
Versioning (Bhargava and Choudhary)
Case 1: Two Types
• Assumptions
– Two consumer types, i  {H,L}, with mass ni
– Utility: Vi(q)
– Cost: c(q)
• Unconstrained Quality
• Constrained Quality
– Upper Bound is q=1
Three Options
• Sell just one product to just the high value consumers
– Set the price at high type’s willingness to pay
• Sell just one product, but price it to sell to both the high
and the low value consumers
– Set the price at low type’s willingness to pay
• Sell one product designed for the high types and second
product designed for the low types.
– Price the low type’s product at their willingness to pay
– Price the high type’s product at their willingness to pay or where they
are just indifferent between their product and the low type’s product,
whichever is higher.
– Lower the quality of the low type’s product to “screen” the high value
consumers
Unconstrained Quality
c’(q)
V’H(q)
V’L(q)
qL
q*L
q*H
Constrained Quality
BnH > AnL
D
c’(q)
B
V’H(q)
CnL > DnH
C
A
V’L(q)
q 1
q*L
q*H
Result
• Conditions for Price Discrimination
BnH  AnL
CnL  DnH
• Rewrite these as
nH
C
A


C  D nL  nH A  B
• A necessary condition is
A B A

CD C
Constrained Quality
A B
A
CD
C
  
necessary
condition:
V q,  c q is
necessary
condition:
 log supermodular
D
c’(q)
B
V’H(q)
C
A
V’L(q)
q 1
q*L
q*H
Log Supermodularity
A twice differentiable function F(q,) is
everywhere log supermodular if and only if
F(q1 , )
F(q2 , )
is increasing in  for all q1 > q 2
or equivalently
F(q1 , )  F(q2 , )
F(q2 , )
is increasing in  for all q1 > q 2
Case 1:
Two Types, Two Products
Consumers
Firm
Two Types:  , 
Cost: c(q)
Mass: n, n
Offers: (q, t ), (q, t)
Utility: V (q,  )  t
Results
Claim 1
There exists a distribution of consumers, i.e. n and n,
for which the seller offers multiple qualities if and only if
  
ö for some qö  1.
V (q, )  c(q) is log supermodular on  ,  q,1
For all other distributions of consumer types, the firm offers q  1.
Figure
Both Qualities Offered
Pareto
Improvement


 V (q, )  c(q) / q
 V (q, )  c(q) / q
V (1, )  c(1)
V (1, )  c(1)
max
q
V (q, )  c(q)
V (q, )  c(q)
n
nn
Case 2:
Continuum of Types and Qualities
Consumers
Firm
Type:    , 
Distribution: f ( )
Cost: c(q)
Utility: V (q, )  p(q)
Offer: p(q)
Results
Proposition:
a) If V(q,) – c(q) is log submodular then
the firm sells a single quality
b) If V(q,) – c(q) is log supermodular then
the firm sells multiple qualities
Results
• Corollary:
If V(q,) = h()g(q) and c(q) > 0 then the firm
sells multiple products if
c q 
g q 

c q  q g q  q
for all q, and the firm sells a single product if
c q 
g q 

c q  q g q  q
Applications
• Intertemporal Price Discrimination
• Damaged Goods
• Coupons
• Versioning Information Goods
• Advance Purchase Discounts
Intertemporal Price Discrimination
• Stokey (1979), Salant (1989)
– U(t,) = d t
– Product Cost: k(t) = cd t
• Transformation
– q= d t
– This gives us: V(q,) – c(q) = q – cq
• Results
– This is not log supermodular
Intertemporal Price Discrimination
• More general utility function – Stokey (1979)
– U(t,) = g(t)
Price discrimination is feasible if g (t) < 0
But
 ln q 
V  q,    cq   g 
  cq
 ln d 
is log submodular, if g (t) ≤ 0 and c ≥ 0, so price
discrimination never optimal.
Intertemporal Price Discrimination
• More general cost function: c(q)
– The surplus function
 q  c q 
is log supermodular if and only if

c q 

c q
q
or marginal cost > average cost
Damaged Goods
• Model from Deneckere and McAfee (1996)
– Continuum of types with unit demands
– Two exogenous quality levels: qL and qH
– V(qH,) = ,
V(qL,) = l()
• V(q,) - c(q) is log supermodular if
l   
1

  cH l    cL
• With some additional transformations, we recover the necessary and
sufficient condition of Deneckere and McAfee.
Coupons
• Model from Anderson and Song (2004)
–
–
–
–
Consumers uniformly distributed on  , 
No Coupon Used: V(,N) = a + b
Coupon Used: V(,C) = a + b – H()
Product Cost: c
Coupon Cost: l
• V(q,) – c(q), q{C,N} is log supermodular if
  H   


    c     H    c  l
Versioning Information Goods
• Information Goods  No Marginal Cost
• Literature
– Shapiro and Varian (1998)
– Varian (1995, 2001)
– Bhargava and Choudhary (2001, 2004)
• Versioning profitable only if
V  ,qH 
V  ,qL 

V  ,qL  
V  ,qH  
When are Advance Purchase
Discounts Profitable?
James Dana
Kellogg School of Management