A critique on the theory of linear price monopoly

Nir Dagan

This Version January 23, 2000
First version distributed in January 15, 1998

The latest version of this article is permanently available at: <http://www.nirdagan.com/research/199803/>

  1. Abstract
  2. Introduction
  3. Background: existing informal claims
  4. The model and the theorem
  5. Conclusion
  6. References


We show that the absence of friction between consumers is not sufficient for linear pricing by a monopolist. On the contrary, it is shown that this situation allows the monopolist to achieve perfect discrimination.

Keywords: Monopoly, linear pricing, friction, price discrimination.

JEL: C72, D40.

1. Introduction

Under what conditions a monopolist will charge a linear price? Although this question is a fundamental one, the literature devoted hardly any attention to it. An informal argument asserts that the ability of consumers to trade with each other and the absence of friction among them forces the monopolist to charge a uniform linear price.

This paper shows that the informal arguments are false. We construct a formal model that captures the assumptions of the informal arguments. However, we show that without any additional assumptions uniform linear pricing does not occur. On the contrary, we show that when consumers can exhaust all market opportunities and trade without transaction costs, the monopolist will implement the outcome of perfect discrimination. This result holds even if the monopolist is forced to anonymous selling and knows only the aggregate demand function and his own cost function.

We model the monopoly situation as a two stage game. In the first stage the monopolist offers menus of trades to all consumers. In the second stage the consumers choose trades from the menus offered to them and trade with each other.

Like in the informal arguments, it is assumed that the consumers can sign binding contracts on re-trading commodities that they buy from the monopolist. That is, if the monopolist offers one consumer to buy at a low price and to another at a high price, these two consumers can agree upon that the low price consumer will buy from the monopolist a quantity that will be resold to the other consumer in a prearranged price.

The market mechanism of the second stage is not defined. Only general properties of the equilibrium correspondence are assumed.

The game is solved by backward induction, by assuming that the equilibrium correspondence of the market mechanism in the second stage is efficient and individually rational, and that the monopolist, who correctly anticipates the consumers' behavior, is maximizing his profit.

Giving the monopolist the first mover advantage is consistent with the assumption of the classical monopoly and oligopoly models, who also assume that the firms dictate the terms of trade. The assumption that the the second stage outcome is efficient captures the idea that the consumers can fully exhaust all market opportunities, and do not have any transaction cost or other friction. Clearly, it does not imply that the outcome of trade is Pareto efficient, since the monopolist is not playing at this second stage. Individual rationality means that each consumer weakly prefers the equilibrium outcome than trading directly with the monopolist or not trading at all.

2. Background: existing informal claims

Before going into our model, we provide two informal descriptions of situations in which allegedly linear pricing will prevail. These provide the motivation for the assumptions of our model.

Kreps (1990) in the opening statement of his chapter about monopoly (page 299) writes:

"The theory of monopoly is, on the face of things, very simple and straightforward. But behind this simple and straightforward theory lie some deep and interesting questions."

However, even the sceptical Kreps does not seem to doubt that certain conditions would lead to linear pricing. In page 306 he writes:

"The standard theory presumes that the monopoly must charge linear prices. That is, the monopoly sets a price per unit, and costumers may buy as many/few units as they wish at that price. The usual justification for this assumption is that the good is bought anonymously and can be resold. If the monopoly, say, give a quantity discount, then someone would buy a large amount (at a discount) and resell it in smaller lots to consumers who don't want quite as many. If the monopoly were to try to markup the price for larger lots, then the consumer desiring a large lot would buy a number of small lots (or have her friends buy small lots and resell them to her)."

The following is from Tirole (1988, page 134):

"...It is clear that if the transaction (arbitrage) costs between two consumers are low, any attempt to sell a given good to two consumers at different prices runs into the problem that the low-price consumer buys the good to resell it to the high-price one. For instance, the introduction of quantity discounts...implies that, in the absence of transaction costs between consumers, only one consumer buys the product and resells it to the other consumers. For example, if each consumer is buying according to the "two-part tariff" T(q)=A+pq (where A>0 is a fixed fee and p is the marginal price), only one consumer will pay the fixed fee. Hence, if there are many consumers, everything is almost as if the manufacturer sold at a linear (or uniform) price. If consumers can arbitrage perfectly, the producer is generally forced to charge a uniform or fully linear price: T(q)=pq.

Transactions costs offer a clue as when price discrimination is feasible..."

It should be noted that the two above quotations are parts of introductions to the topic of price discrimination (or non-linear pricing). It is argued later in the above texts that in the absence of the conditions leading to linear pricing, price discrimination will occur. That is, the above conditions are deemed necessary and sufficient for linear pricing.

3. The model and the theorem

3.1 The economic setting

The set of consumers is {1,...,n}. There are two commodities: the one produced by the monopolist and money. Every consumer i has a quasi-linear utility function with the corresponding decreasing demand function Di(p), where Di(0) is finite and for some price z, Di(z)=0

The aggregate demand function is D(p)=D1(p)+···+Dn(p), and CS(p) is the aggregate consumers' surplus at price p, which is equivalent to the area below the demand curve between the quantities 0 and D(p).

The monopolist has a cost function y=C(·) that assigns a monetary cost to each quantity produced. C(·) is assumed to be non-negative, non-decreasing, and continuous in the strictly positive region.

We assume that the revenue function pD(p) and the profit function pD(p)-C(D(p)) are concave.

3.2 The monopoly game

The game has two stages:

  1. The monopolist announces a two part tariff T(q)=A+pq, where A is a non-negative fixed fee and p is the marginal price.
  2. The market of the consumers trading with each other and with the monopolist resolves in an efficient and individually rational manner.

The assumptions on the second stage imply that for every two part tariff T(q)=A+pq, the consumers together will pay the fixed fee at most once and will buy:

We shall assume neutrality that says that in the case of A=CS(p)-pD(p) both 0 and D(p) are predicted by the second stage equilibrium correspondence.


Given the assumptions of efficiency, individual rationality, and neutrality of the second stage equilibrium correspondence, the game has a unique subgame perfect equilibrium outcome, in which the monopolist chooses a tariff T(q)=A*+p*q that satisfies:

and the consumers buy q=D(p*). All the consumers are indifferent between the equilibrium and the no-trade outcome.

Clearly, if C(·) is differentiable, p* would satisfy: MC(D(p*))=p*, where MC(·) is the marginal cost function.


This outcome can be supported as equilibrium by considering the selection of the equilibrium correspondence in the second stage to be the one which in case of indifference the consumers buy a positive amount.

As of uniqueness of outcome, note that the monopolist profit cannot be higher than that of the above mentioned equilibrium due to individual rationality. Moreover it cannot be lower since choosing p* and A*-e for small and positive e will result in the consumers buying q=D(p*).

4. Conclusion

Never trust textbooks.

5. References