Prices and Deadweight Loss in Multiproduct Monopoly

• Author: JIM Y. JIN,GERALD PECH,MICHAEL TRÖGE,RABAH AMIR
• DOI: http://doi.org/10.1111/jpet.12173
• Published date: 01 June 2016
• Date: 01 June 2016

PRICES AND DEADWEIGHT LOSS IN MULTIPRODUCT MONOPOLY

RABAH AMIR

University of Iowa

JIM Y. JIN

University of St Andrews

GERALD PECH

KIMEP

MICHAEL TR ¨

OGE

ESCP-Europe

Abstract

The paper investigates prices and deadweight loss in multiproduct

monopoly with linear interrelated demand and constant marginal

costs. We show that, with commonly used models for linear demand

such as the Bowley demand and vertically or horizontally differenti-

ated demand, the price for each good is independent of demand cross-

effects and of the characteristics and number of other goods. This

contrasts with the oft-expressed view that prices critically depend on

demand cross-effects. We also show that for these linear models, the

deadweight loss due to monopoly amounts to half the total monopoly

profit. Finally, we show how a production subsidy might restore social

efficiency.

1. Introduction

Most firms do not only sell one, but many interrelated products. For example, super-

markets sell a multitude of substitutes, complements, or independent goods. Airlines

and railway companies sell tickets with different conditions for the same route and oil

corporations sell gasoline in petrol stations that differ by their locations.

This paper examines multiproduct monopoly (MPM) facing linear demand for

differentiated goods and constant unit costs and shows that optimal prices and welfare

loss can be expressed in a very simple way. As in the textbook case of a single product

monopoly, the monopoly price of each good is the average of its own inverse demand intercept

and its own marginal cost,and is thus independent of the characteristics of other products,

Rabah Amir, Tippie School of Business, University of Iowa, W210 John Pappajohn Business Building,

Iowa City, IA 52242-1000 (rabah-amir@uiowa.edu). Jim Y. Jin, School of Economics and Finance, Uni-

versity of St Andrews, Fife, UK (jyj@st-andrews.ac.uk). Gerald Pech, Department of Economics, College

of Social Sciences, KIMEP, 4 Abay Avenue, Almaty 050010, Kazakhstan (gpech@kimep.kz). Michael

Tr ¨

oge, ESCP-Europe, 79, Avenue de la R´

epublique, 75543 Paris cedex 11, France.

Wewish to thank Gautam Gowrisankaran, Stanley Reynolds, and David Ulph for insightful comments.

Received October 10, 2013; Accepted November 15, 2014.

C2016 Wiley Periodicals, Inc.

Journal of Public Economic Theory, 18 (3), 2016, pp. 346–362.

346

Multiproduct Monopoly 347

the interactions between products, and the number of products sold. In contrast, a common

view in economics as well as marketing is that monopoly prices critically depend

on cross-demand effects.1In particular, a somewhat prevalent misconception is that

MPM prices should be lower for complements and higher for substitutes, relative to

independent goods (see Section 2). Though intuitively plausible along a seemingly

natural line of argument, this conclusion is actually invalid!

We obtain these elementary results for MPM facing three commonly used linear de-

mand structures, corresponding to the three examples cited above: the standard Bowley

(1924) or Shubik (1959) models for demand with heterogeneous products, vertically

(quality) differentiated products, and horizontally (spatially) differentiated products.

As seen below, this multitude of demand models is motivated by the diversity of eco-

nomic settings where the issue of MPM pricing has been historically analyzed, often by

founding fathers of modern industrial economics.

Our result on MPM prices is interesting in its own right, but it can also be used

to address a number of further questions. In particular, we show that this result helps

to solve the complex but important problem of deadweight loss in MPM. In a single

product monopoly with linear demand, deadweight loss is half the monopoly profit. Ex-

ploiting the property that prices of existing goods do not change when a new product

is added to a product line, we can show that monopoly profit and the deadweight loss

always rise proportionally. Consequently, deadweight loss in MPM will also be half the

monopoly profit regardless of how many, or what types of, products are added. This

surprising result holds in all the models of linear demand we consider, despite funda-

mentally different welfare functions.

Based on an extensive literature search going back to the beginnings of neoclassic

economics, we believe that these simple properties of linear MPM have not been fully

uncovered.2While the complete solution of MPM pricing seems to have largely eluded

economists’ attention, some features of our results have emerged in the marketing lit-

erature. Shugan and Desiraju (2001) show that monopoly prices of two vertically differ-

entiated products do not depend on each other’s costs. Moorthy (2005) and Besanko,

Dub´

e, and Gupta (2005) show that with linear demand MPM prices do not respond to

cost changes of other products. Neither of these studies derives the general solution for

prices or explores the full scope of MPM pricing.3

The present results may be useful in various contexts. A direct practical implication

is that even in the presence of strong product interactions neglecting such relations

1For example, Reibstein and Gatignon (1984) argue in a seminal marketing paper that “The optimal

price is extremely sensitive to the inclusion or the exclusion of the cross-elasticities” (p. 266).

2The first formal analysis of the MPM goes back to Wicksell (1901, 1934). Edgeworth (1925) analyzes

railway fares of different classes but does not give explicit solutions. Hotelling (1932) provides a numer-

ical example with first- and second-class railway tickets. Robinson (1933) formally solves the problem

of a monopolist selling in different markets, but explicitly excludes price interdependence such as in

“the case of first- and third-class railway fares, analyzed by Edgeworth” (Robinson, 1933, p. 181). Coase

(1946) goes beyond Robinson’s analysis and examines monopoly prices for two interrelated products

using verbal and graphical arguments but again does not provide a mathematical solution. Holton

(1957) considers the MPM problem of a supermarket selling interrelated products arguing that, “su-

permarket operators do indeed establish prices with not only price elasticities but cross-elasticities in

mind.” Finally, Selten (1970) formally addresses the problem of a multiproduct monopolist facing lin-

ear demand but does not provide the simple properties of monopoly prices.

3Several authors have pointed out the similarity of the MPM problem to the Ramsey tax (Ramsey

1927), which maximizes social welfare for a certain level of tax revenue. Yet, the Ramsey problem is not

identical to unconstrained monopoly profit maximization. For example, TenRaa (2009) shows that the

structure of monopoly prices often differs from that induced by the Ramsey tax.