1. When is Price Discrimination Profitable?
Eric T. Anderson Kellogg School of Management
James Dana Kellogg School of Management

2. 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)
Much of the literature deals with existence results.
Show that there exist conditions under which price discrimination is profitable. Also show that there exist conditions under which price discrimination is not profitable.

3. 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

4. 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)
Constraint has force
First best is constrained quality
Jones: working paper on information goods
Johnson: fighting brands, exogenous qualities
DM: Damaged goods

5. Case 1: Two Types Assumptions Unconstrained Quality
Two consumer types, i  {H,L}, with mass ni
Utility: Vi(q)
Cost: c(q)
Unconstrained Quality
Constrained Quality
Upper Bound is q=1
Details on other assumptions are in the paper.
Q=1. Constraint has force. Binding for social planner.

6. 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

7. Unconstrained Quality
c’(q)
V’H(q)
V’L(q) qL
q*L
q*H

8. Constrained Quality c’(q) V’H(q) V’L(q) q*L q*H A B C D BnH > AnL
CnL > DnH
For simplicity, we have assumed two exogenous quality levels: q=1, and q lower.
Assume that qupper is first best quality – constraint has force
The question now is whether to price discriminate and offer two quality levels.
Start with selling q=1 to all types. Profits is (A+C) (nH + nL)
Price Discrimination: Gain B*nH and give up A*nL so B*nH > A*nL
II) Start with selling q=1 to type H. Profit (A+B+C+D)nH
Price Discrimination: Gain C*nL and give up D*nH so C*nL > A*nH
If the firm sells one quality to all types, the markup over cost is A+C. Low types get zero surplus, high types get B+D surplus.
If the firm sells to two qualities, profit on low types is C (give up A*n_low). But, can charge an extra B*n_high.
If the firm sells one quality to high types, the markup over cost is A+C + B+D.

9. Result Conditions for Price Discrimination Rewrite these as
A necessary condition is

10. Constrained Quality c’(q) V’H(q) V’L(q) q*L q*H A B C D
For High types, D+C is the TOTAL surplus available when offering low quality. A+B is the marginal increase in surplus when going from low to high quality.
For Low types, C is the TOTAL surplus available when offering low quality. A is the marginal increase in surplus when going from low to high quality.
Our main result is that the ratio of:
marginal increase in net surplus to
total net surplus
must be increasing in consumer type

11. Log Supermodularity
A twice differentiable function F(q,q) is everywhere log supermodular if and only if
or equivalently

12. Case 1: Two Types, Two Products

13. Results
Claim 1

14. Figure
Intuition:
At the far right, we have many High types. The optimal strategy is q=1 and a high price. As the number of high types decreases, price disc becomes optimal. Two effects occur: Low Types are now served AND high types are offered a lower price.
At the far left, there are so few high types that the firm sells q=1 to everyone. As we add more high types, price discrimination becomes feasible. But, now the high types pay a higher price for the same quality q=1. So, while price discrimination occurs we don’t get a Pareto Improvement.

15. Case 2: Continuum of Types and Qualities

16. Results
Proposition:
If V(q,q) – c(q) is log submodular then the firm sells a single quality
If V(q,q) – c(q) is log supermodular then the firm sells multiple qualities
More buyers
PI if price of high type falls
Claim 5: two exogenous quality levels
Claim 6: relax assumption on quality constraint a bit (binding for some, but not all consumers)

17. Results
Corollary:
If V(q,q) = h(q)g(q) and c(q) > 0 then the firm sells multiple products if
for all q, and the firm sells a single product if

18. Applications Intertemporal Price Discrimination Damaged Goods Coupons
Versioning Information Goods
Advance Purchase Discounts

19. Intertemporal Price Discrimination
Stokey (1979), Salant (1989)
U(t,q) = qd t
Product Cost: k(t) = cd t
Transformation
q= d t
This gives us: V(q,q) – c(q) = qq – cq
Results
This is not log supermodular
We can show that Salant is weakly a subset of Claim 2.

20. Intertemporal Price Discrimination
More general utility function – Stokey (1979)
U(t,q) = qg(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.

21. Intertemporal Price Discrimination
More general cost function: c(q)
The surplus function
is log supermodular if and only if
or marginal cost > average cost

22. Damaged Goods Model from Deneckere and McAfee (1996)
Continuum of types with unit demands
Two exogenous quality levels: qL and qH
V(qH,q) = q, V(qL,q) = l(q)
V(q,q) - c(q) is log supermodular if
With some additional transformations, we recover the necessary and sufficient condition of Deneckere and McAfee.

23. Coupons Model from Anderson and Song (2004)
Consumers uniformly distributed on
No Coupon Used: V(q,N) = a + qb
Coupon Used: V(q,C) = a + qb – H(q)
Product Cost: c Coupon Cost: l
V(q,q) – c(q), q{C,N} is log supermodular if

24. 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

25. When are Advance Purchase Discounts Profitable?
James Dana Kellogg School of Management