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**When is Price Discrimination Profitable?**

Eric T. Anderson Kellogg School of Management James Dana Kellogg School of Management

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**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.

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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

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**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

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**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.

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**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

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**Unconstrained Quality**

c’(q) V’H(q) V’L(q) qL q*L q*H

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**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.

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**Result Conditions for Price Discrimination Rewrite these as**

A necessary condition is

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**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

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

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**Case 1: Two Types, Two Products**

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Results Claim 1

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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.

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**Case 2: Continuum of Types and Qualities**

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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)

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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

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**Applications Intertemporal Price Discrimination Damaged Goods Coupons**

Versioning Information Goods Advance Purchase Discounts

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**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.

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**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.

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**Intertemporal Price Discrimination**

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

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**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.

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**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

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**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

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**When are Advance Purchase Discounts Profitable?**

James Dana Kellogg School of Management

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