# Student

ISyE 6230 Economic Decision Analysis

Matt Drake

Spring 2005

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Preliminaries

Before we begin our discussion of price discrimination and its economic motivations, we must ﬁrst (re)introduce a few topics in microeconomics that will prove useful in our subsequent analysis.

1.1

Demand

In order to determine how economic agents should determine prices and quantities of production, we must specify how consumers in a market will react to various prices. The most common method of characterizing consumer behavior is by specifying a demand function that determines the quantity demanded at each price oﬀered by the seller. We can also invert the demand function and characterize the price at which a certain quantity is demanded.

For the balance of this essay, we will consider deterministic demand functions which relate exactly what will be sold at a given price. Admittedly demand is generally stochastic, varying randomly from observation to observation, selling period to selling period. While we could introduce a stochastic demand element into the following discussions, deterministic demand allows us to concentrate on the economic insights from the models presented without the complications that accompany random demand. The following1 are several common demand functions utilized in price discrimination analysis. Although these functions are only dependent on the price of the good itself, they can be generalized to include the prices of other goods and other variables as well such as income. 1. Linear demand

Q(p) = a − bp

In the linear demand model, both of the parameters (a and b) are positive. The parameter a can be thought of as the market potential for the good, since this is the amount that would be sold if the price was zero; b is a measure of the consumers’ price sensitivity. 2. Cobb-Douglas (Constant Elasticity) demand

Q(p) = ap−

In the Cobb-Douglas demand model, parameters a and are positive. Again the parameter a is a market scaling parameter. We will see below that is equal to the price elasticity of demand. Remark 1 We will assume that our demand functions are downward sloping. This means that the quantity demanded decreases as the price of the good increases. A suﬃcient condition for demand to be downward sloping is that the consumers’ utility functions are quasi-linear. In order to get a sense of how the quantity demanded changes as the price changes, we look at the (own) price elasticity of demand.

Deﬁnition 1 Price elasticity of demand ( ) is the percentage change in quantity demanded divided by the percentage change in price.

1 The material in this section is common in any microeconomics textbook. The majority of the notation and concepts contained herein is from Tirole (1988).

1

Since we have assumed downward-sloping demand, the price elasticity of demand will always be negative (because quantity demanded and price are inversely proportional). In order to avoid confusion, we will consider the absolute ratio of the two percentage changes. In terms of the demand function, Q(p), we have =−

∂Q(p) p

.

∂p Q(p)

Consider a proﬁt-maximizing monopolist that faces demand of Q(p) and production cost C(q). This monopolist chooses the proﬁt-maximizing price, pm according to pm ≡ arg max{pQ(p) − C(Q(p))}.

(1)

Solving (1) yields the following important identity.

Theorem 1 (Inverse Elasticity Rule)

price pm .

pm −C (Q(pm ))

pm

=

1

where

is the price elasticity of demand at

The Inverse Elasticity Rule states that the inverse of the elasticity of demand at the monopoly price equals the gross proﬁt margin on the good. Substituting qm ≡ Q(pm ) into the Inverse Elasticity identity yields the familiar result that a monopolist produces the quantity where its marginal revenue equals its marginal cost. This results in a price that exceeds the marginal cost of the good, which we shall see below is the “socially-optimal” price (denoted by ps in Figure...

References: [1] K. Carroll and D. Coates (1999). Teaching price discrimination: Some clariﬁcation. Southern Economic

Journal, 66(2), 466-480.

[2] G. Debreu. Theory of Value. Wiley, New York, 1959.

[3] S. Happel and M. Jennings (1995). Herd them together and scalp them. The Wall Street Journal,

February 23, page A-14, column 4.

[4] L. Phlips. The Economics of Price Discrimination. Cambridge University Press, New York, 1983.

[5] A. Pigou. The Economics of Welfare. Macmillan, London, 1920.

[6] J. Tirole. The Theory of Industrial Organization. MIT Press, Cambridge, MA, 1988.

[7] H. Varian. Microeconomic Analysis. Third Edition. W.W. Norton & Company, New York, 1992.

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