When is Price Discrimination Profitable?

Eric T. AndersonKellogg School of Management

James DanaKellogg 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

c’(q)

V’H(q)

V’L(q)

q*L q*

HqL

Unconstrained Quality

Constrained Quality

c’(q)

V’H(q)

V’L(q)

q*L q*

H1q

A

B

C

DBnH > AnL

CnL > DnH

Result

• Conditions for Price Discrimination

• Rewrite these as

• A necessary condition is

H

L H

nC A

C D n n A B

H L

L H

Bn An

Cn Dn

A B A

C D C

Constrained Quality

c’(q)

V’H(q)

V’L(q)

q*L q*

H1q

A

B

C

D

necessary condition:

A B

C D

A

C necessary condition: V q, c q is log supermodular

Log Supermodularity

A twice differentiable function F(q,) is everywhere log supermodular if and only if

or equivalently

F(q1,)

F(q2,)

is increasing in for all q1 > q

2

F(q1, ) F(q

2, )

F(q2, )

is increasing in for all q1 > q

2

Case 1:Two Types, Two Products

Two Types: ,

Mass: ,

Utility: ( , )

Consumers

n n

V q t

Cost:

Offers: (q, t ), (q, t)

Firm

c(q)

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

V (q,) c(q) is log supermodular on , öq,1 for some öq 1.

For all other distributions of consumer types, the firm offers q 1.

Both Qualities Offered

Pareto Improvement

V (q, ) c(q) / q

V (q, ) c(q) / q

max

q

V (q, ) c(q)

V (q, ) c(q)

V (1, ) c(1)

V (1, ) c(1)

n

n n

Figure

Case 2:Continuum of Types and Qualities

Consumers

Type: ,

Distribution: f ( )

Utility: V (q, ) p(q)

Firm

Cost: c(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

for all q, and the firm sells a single product if

c q c q q

g q

g q q

c q c q q

g 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,) = t

– Product Cost: k(t) = ct

• Transformation– q= t

– This gives us: V(q,) – c(q) = q – cq

• Results– This is not log supermodular

• More general utility function – Stokey (1979)– U(t,) = g(t)

Price discrimination is feasible if g (t) < 0

But

is log submodular, if g (t) ≤ 0 and c ≥ 0, so price discrimination never optimal.

Intertemporal Price Discrimination

ln,

ln

qV q cq g cq

• More general cost function: c(q)– The surplus function

is log supermodular if and only if

or marginal cost > average cost

Intertemporal Price Discrimination

c q c q

q

q c q

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,) = ()

• V(q,) - c(q) is log supermodular if

• With some additional transformations, we recover the necessary and sufficient condition of Deneckere and McAfee.

1

cH

cL

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:

• V(q,) – c(q), q{C,N} is log supermodular if

,

c

H

H c

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 DanaKellogg School of Management