When is Price Discrimination Profitable?

1. When is Price Discrimination Profitable? Eric T. AndersonKellogg School of Management James DanaKellogg 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)

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)

5. 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

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. D c’(q) B V’H(q) A C V’L(q) q*L q*H Constrained Quality BnH > AnL CnL > DnH

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

10. D c’(q) B V’H(q) A C V’L(q) q*L q*H Constrained Quality

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

15. Case 2:Continuum of Types and Qualities

16. Results Proposition: • IfV(q,q) – c(q)is log submodular then the firm sells a single quality • IfV(q,q) – c(q)is log supermodular then the firm sells multiple qualities

17. Results • Corollary: IfV(q,q) =h(q)g(q) and c(q) > 0 then the firm sells multiple products if for allq, 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

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 DanaKellogg School of Management