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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)

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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)

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Case 1: Two Types Assumptions –Two consumer types, i {H,L}, with mass n i –Utility: V i (q) –Cost: c(q) Unconstrained Quality Constrained Quality –Upper Bound is q=1

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Three Options Sell just one product to just the high value consumers –Set the price at high types 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 types willingness to pay Sell one product designed for the high types and second product designed for the low types. –Price the low types product at their willingness to pay –Price the high types product at their willingness to pay or where they are just indifferent between their product and the low types product, whichever is higher. –Lower the quality of the low types product to screen the high value consumers

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c(q) V H (q) V L (q) q*Lq*L q*Hq*H qLqL Unconstrained Quality

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Constrained Quality c(q) V H (q) V L (q) q*Lq*L q*Hq*H A B C D Bn H > An L Cn L > Dn H

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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*Lq*L q*Hq*H A B C D

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Log Supermodularity A twice differentiable function F(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

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Case 2: Continuum of Types and Qualities

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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

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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

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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, ) = t –Product Cost: k(t) = c t Transformation – q= t –This gives us: V(q, ) – c(q) = q – cq Results –This is not log supermodular

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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

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

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Damaged Goods Model from Deneckere and McAfee (1996) –Continuum of types with unit demands –Two exogenous quality levels: q L and q H –V(q H, ) =, V(q L, ) = ( ) V(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(,N) = a + b –Coupon Used: V(,C) = a + b – H( ) –Product Cost: cCoupon Cost: V(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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