The welfare effects of third-degree price discrimination with nonlinear demand functions.

The welfare effects of third-degree price discrimination are analyzed when demand in one market is an additively shifted version of demand in the other market and both markets are served with uniform pricing. Social welfare is lower with discrimination if the slope of demand is log concave or the convexity of demand is nondecreasing in the price. The demand functions commonly used in models of imperfect competition satisfy at least one of these sufficient conditions.

1. Introduction

* In 1999, Coca Cola admitted that it was developing a vending machine that would raise the price of a Coke when the external temperature increased above a certain level. The subsequent negative publicity, however, induced Coca Cola to abandon any plans it might have had to introduce this type of machine. This paper addresses the general question: What are the welfare effects of allowing a monopolist to practise third-degree price discrimination rather than requiring it to set a uniform price in all markets? A firm practices third-degree price discrimination when it classifies customers into separate markets using observed characteristics and sets different prices in these markets. When price discrimination is allowed, a monopolist earns higher profits and individual consumers gain or lose, depending on whether the discriminatory prices in their markets are below or above the uniform price. The impact of discrimination on total welfare, defined as aggregate consumer surplus plus profits, can go either way. In one well-known case, discrimination definitely lowers welfare. If all markets are served with uniform pricing, demand functions are linear and marginal cost is constant, then total welfare is lower with price discrimination than with uniform pricing. This is because total output is the same in the two cases. Pigou (1929) and Robinson (1969) gave early proofs of this result.

This paper analyzes the welfare effects of discrimination with a more general demand function. Suppose that demand is Q = a + bq(p), where a [greater than or equal to] 0 and b > 0, q(p) is the underlying demand function and p denotes the price. The additive and multiplicative terms, a and b, respectively, vary across markets. Linear demand is a special case. The price elasticity of demand falls as a/b increases, so the monopoly price is an increasing function of a/b. There are two interpretations of the demand function. Friedman (1987) presents a model of the demand for heating where the external temperature enters the demand function additively, so the shift factor, a, might represent the effect of the external temperature on demand. An alternative interpretation involves geographical discrimination. Suppose that a single firm, say a supermarket, restaurant or cinema chain, sells in different towns. In a given town, there are a committed consumers who always buy the product and b price-sensitive consumers, each with demand of q(p). The demographic composition of each town is defined by the ratio of committed to price-sensitive customers, a/b. Discrimination is not feasible within a town, maybe because of arbitrage possibilities, but it is feasible to discriminate across towns and a profit-maximizing firm will want to do so if a/b varies across towns. The UK's Competition Commission investigated supermarket pricing and found that "pricing might also respond to local demographics" in addition to local competitive pressures (Competition Commission, 2000). For simplicity, the analysis is presented with a alone varying, but all the results also hold when b also varies across markets.

The main result is that discrimination lowers welfare for all underlying demand functions typically used in theoretical and econometric models of imperfect competition, as long as all markets are served. With this demand structure, social welfare with discrimination is lower than that with uniform pricing if welfare with discrimination is a concave function of a lb. Two sufficient conditions for concavity of the welfare function are presented. An important role is played by the ratio of the curvature of the slope of the demand function to the curvature of the demand function itself. If this ratio is at most 1, which is equivalent to the slope of demand being log concave, then welfare falls with discrimination. Many demand functions have log-concave slopes. Some demand functions, though, such as those in the isoelastic class, do not satisfy this condition. Nevertheless, welfare is also lower with discrimination for a general class of such functions, which are characterized by a curvature that does not decrease with the price. Welfare can rise with discrimination when a large concentration of low-value consumers induces the firm to cut price substantially when discrimination is allowed and two examples are given. In general, though, welfare only rises with discrimination under very delicate assumptions in this model.

Price discrimination has the undesirable effect of ensuring that marginal utilities differ between consumers and thus output is distributed inefficiently, but this negative effect may be offset if total output is higher with discrimination. Varian (1985), building on the analysis of Schmalensee (1981), shows that a necessary condition for discrimination to raise welfare above the uniform-pricing level is that total output increases. (1) The output effects in the model can be found by applying the general formula given by Holmes (1989), who corrected the "adjusted-concavity" criterion of Robinson (1969) and also pioneered the analysis of price discrimination in oligopoly. The problem with the output test is that it does not always produce conclusive results. When output is known to increase, this does not imply that welfare rises because an output increase is necessary for welfare to rise, but not sufficient. For some demand functions, the effect of discrimination on output cannot be determined.

If price discrimination opens up new markets, then welfare is likely to increase and, indeed, weak Pareto improvements can be achieved if one market is served with uniform pricing and a new one is opened when discrimination is allowed (Hausman and Mackie-Mason, 1988). The focus in the present paper is on the case Where all markets are served with uniform pricing and assumptions are made to rule out the important possibility considered by Hausman and Mackie-Mason. The model considers only a pure monopolist. Price discrimination is also common in industries with competition, such as airlines, and Coca Cola itself is subject to significant competitive pressure. Stole (forthcoming) surveys models of competitive price discrimination (see also Armstrong and Vickers, 2001; Armstrong, 2006).

The structure of the paper is as follows. Section 2 contains the model of discriminatory and uniform pricing and the welfare framework. Section 3 provides general sufficient conditions for welfare to fall with discrimination. The effect of discrimination on output is considered in Section 4. The possibility of discrimination raising welfare is covered in Section 5. Conclusions are in Section 6.

2. The model

* The utility function is U(Q - a) = U(q), where Q is the quantity consumed, a [greater than or equal to] 0 is the shift factor and q [equivalent to] Q - a [greater than or equal to] 0. This function is increasing, strictly concave and differentiable four times. The price is p [greater than or equal to] 0. Utility maximization implies that U'(Q - a) = p, so demand is Q = a + q(p) and a acts as an additive shift factor. Call q(p) the underlying demand function, which satisfies q'(p) < 0 because of the strict concavity of utility. There is a choke price, if, [bar.p], which satisfies the following properties: q(p) = 0 and Q = 0 for p > [bar.p], Q = a + q([bar.p]) [greater than or equal to] 0 when p = [bar.p], and q(p) > 0 for p < [bar.p]. At any price above the choke price, demand is zero and underlying demand is strictly positive for prices below the choke price. The choke price is the maximum feasible price that the monopolist can set, and it may be thought of as the price of a substitute for the product. (2) Without a choke price, the profit function would be unbounded as profits are at least (p - c)a, where c is marginal cost, and with a positive a, it would pay to send the price to infinity. Consumer surplus is U(q(p)) - pq(p) - pa.

One interpretation of the demand function, following Friedman (1987), is that a is minus one times the external temperature, Q is the amount of energy purchased for heating and q = Q - a is the consumer's comfort level, which falls if the weather becomes colder but can be restored if the consumption of energy for heating rises to match. The extension to the case where the underlying demand function is also subject to multiplicative shifts, so Q = a + bq(p) for b > 0, is straightforward and is discussed in Appendix 1. What matters is the ratio a/b rather than a itself. There is no loss of generality in assuming that b is constant and equal in all markets. The second version has b price-sensitive customers, each with demand function q(p), and a committed customers who each buy one unit. The underlying demand function, q(p), can be thought of as the probability that a price-sensitive customer buys a single unit. The choke price, in this interpretation, is the reservation price of the price-sensitive customer with the highest valuation, which also equals the reservation price of the committed customers.