When is Price Discrimination Profitable?

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When is Price Discrimination Profitable?

• Eric T. AndersonKellogg School of Management
• James DanaKellogg 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 ni
• Utility Vi(q)
• Cost c(q)
• Unconstrained Quality
• Constrained Quality
• Upper Bound is q1

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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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Unconstrained Quality
c(q)
VH(q)
VL(q)
qL
qH
qL
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Constrained Quality
BnH gt AnL
CnL gt DnH
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Result

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

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Constrained Quality
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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 1Two Types, Two Products
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Results

• Claim 1

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Figure
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Case 2Continuum 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

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Results

• Corollary
• If V(q,q) h(q)g(q) and c(q) gt 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

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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) lt 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 gt 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 DanaKellogg School of Management