Intertemporal price discrimination with two products

• Author: Jean‐Charles Rochet,John Thanassoulis
• Published date: 01 December 2019
• Date: 01 December 2019
• DOI: http://doi.org/10.1111/1756-2171.12301

RAND Journal of Economics

Vol.50, No. 4, Winter 2019

pp. 951–973

Intertemporal price discrimination

with two products

Jean-Charles Rochet∗

and

John Thanassoulis∗∗

We study the two-product monopoly profit maximization problem for a seller who can commit

to a dynamic pricing strategy. We show that if consumers’ valuations are not strongly ordered,

then optimality for the seller can require intertemporal price discrimination: the seller offers

a choice between supplying a complete bundle now, or delaying the supply of a component of

that bundle until a later date. For general valuations, we establish a sufficient condition for such

dynamic pricing to be more profitable than mixed bundling. So we show that the established

no-discrimination-across-time result does not extend to two-product sellers under standard taste

distributions.

1. Introduction

In celebrated work, Stokey (1979) studied how a monopoly seller of a single good could

change prices over time so as to maximize profits. It would seem natural that by lowering prices

through time, those who value the good most can be induced to buy the good earlier than those

who value the good least, and so generate higher profits. Stokey (1979) demonstrated that for a

seller who could commit to a path for prices and who had an inventory which could be restocked,

such time-varying prices were suboptimal. The seller woulddo best by setting a price and sticking

to it.

Yet marketing scholars have prominently argued that lowering prices over time to subsets

of consumers, chosen based on purchase history, is profitable. Rossi, McCulloch, and Allenby

(1996) calculated that, 20 years ago, using purchase history to target price promotions could raise

promotional revenues by a factor of two and a half times. More recently it has been noted that

∗SFI@UNIGE, University of Z¨

urich, and TSE; Jean-Charles.Rochet@unige.ch.

∗∗University of Warwick and CEPR; john.thanassoulis@wbs.ac.uk.

Wewould like to thank the Editor, David Myatt, and three anonymous referees for invaluableguidance which has greatly

improvedthis article. We would also like to thank audiences at the CEPR IO 2015 workshopin Z ¨

urich, the Mid-westtheor y

conference 2016 in Rochester, ESEM 2016 in Geneva, the Columbia-Duke-MIT-Northwestern IO Theory Conference

2016, the Paris meeting of CREST 2016, INFMA 2016 in Nanjing China, and the Berlin 2017 IO Day. We are very

grateful for the helpful comments from ¨

Ozlem Bedre-Defolie, Chris Dance, Sergiu Hart, Justin Johnson, Paul Klemperer,

Jiangtao Li, Yossi Spiegel, and Roland Strausz. Any errors remain our own.

C2019, The RAND Corporation. 951

952 / THE RAND JOURNAL OF ECONOMICS

this is an under-estimate, and the profit potential of personalizing pricing based on prior purchase

history appears to be greatest for online stores (Zhang and Wedel, 2009; Taylor, 2004). However,

most marketing work on this issue has considered the question of who to target using purchase

history data, and with what offers, separately (Prinzie and Van den Poel, 2005; Reutterer et al.,

2006).

The purpose of this article is to derive useful and interpretable sufficient conditions under

which a seller’s profit is increased by using prices which change over time, when the seller is not

bound by capacity constraints and can commit to a path for prices.

We first study a two-good seller serving a consumer distribution with two types. We show

that the optimal selling strategy hinges on whether or not the consumers’ valuations are strongly

ordered. We define the valuations to not be strongly ordered if the two consumer types do not

satisfy the standard Spence-Mirrlees sorting conditions: the consumer type valuing the bundle

most does not also value each component most. We show that dynamic pricing is optimal in this

case if and only if the consumers valuing the bundle most are numerous enough in the population.

The most profit is achieved by using a strategy wedescribe as dynamic pricing on the cross-sell1:

the seller offers a choice between supplying the complete bundle now, or delaying the supply

of a component of that bundle until a later date. That is, the cross-sell to complete the bundle

is delayed.

This rationalizes a prominent sales approach in marketing. As an example, consider a

company providing pay-TV services. Let us suppose that type aconsumers care little for sports

on TV but value films and drama highly, whereas type bconsumers value sports more than type

ado, films and drama less than type a’s, and yet value the overall TV package the most. This

describes that consumer types aand bare not strongly ordered. Our work predicts that if the

volume of sports-loving type bconsumers is large enough, then the seller would optimally serve

type bconsumers with the bundle of sports and films and drama, whereas type aconsumers

would initially purchase just films and drama and receivetargeted price offers which decline over

time to add sports to their package; and they would purchase sports after delay.

Extending our analysis to continuous demand distributions is challenging. However, here

too we can offer a simple sufficient condition under which the strategyof dynamic pricing on the

cross-sell is a profitable addition to mixed bundling prices. The sufficient condition is to establish

whether or not the cross-partial derivative of the profit function with respect to the bundle price

and a component price, evaluated at current mixed bundling prices, is negative. If it is, then

profits can be increased by offering a price reduction after some time for the cross-sell from the

component to the whole bundle. We further demonstrate that the sufficiency condition applies

also when consumers have complementarities or substitutabilities in demand.

This partial derivative sufficiency condition is attractive in its simplicity and can be im-

plemented analytically and numerically. Analytically, we demonstrate that dynamic pricing on

the cross-sell is a profitable addition to the best mixed bundling prices when consumers have

valuations given by variations on the uniform distribution, exhausting most of the cases studied

in the literature. Computationally, we demonstrate that dynamic pricing on the cross-sell is more

profitable if valuations are normally distributed with sufficient negative correlation.

The negative cross-partial condition allows for an economic intuition. The cross-partial of

the profit function with respect to the bundle good and a component is a function of the density

of consumers who are indifferent between buying the component and the bundle. A leading way

in which this cross-partial becomes negative is if the density of consumers is downward sloping

across this group. An appropriate dynamic price for the cross-sell can be found, which attracts just

these consumers. The slope of the density function then ensures that more consumers upgrade to

the cross-sell, than downgrade from the bundle to the delayed cross-sell, raising profits.

1We borrow from the marketingliterature combining cross-selling and dynamic pricing. See Blattberg, Kim, and

Neslin (2008) and Ferrell and Hartline (2012) for textbook treatments.

C

The RAND Corporation 2019.