Adjusted concavity and the output effect under monopolistic price discrimination.

• Author: Cheung, Francis K.
• Position: Communications

1. Introduction

   The motive of engaging in third degree monopolistic price discrimination is rather simple, but its welfare implication is not. A necessary condition for price discrimination to improve welfare, as demonstrated by Schmalensee |5~ and Varian |8~, is that output increases under price discrimination. Because of this well-known result, a focal point of analysis of the social impact of price discrimination has been on the direction of change in output. The few general rules advanced to determine output change under price discrimination can be applied to three different situations: (i) all demand curves are linear, (ii) demand curves fall into two groups with opposing general curvatures, and (iii) all demand curves have similar general curvature.

   Definite answers have been obtained for the first two cases.(1) Regarding the third case, less complete results have been developed. Robinson |4~ introduces the "adjusted concavity" criterion that the direction of output change is determined by the relative size of the adjusted concavity of the two demand curves valued at the simple monopoly output levels. This criterion, as noted by Robinson herself and further elaborated by Edwards |1~, depends critically upon the local assumption that the demand elasticities in the two markets are not very different.(2) Greenhut and Ohta |3~ confirm that, without the local assumption, the adjusted concavity criterion could lead to a wrong prediction of output change by constructing an example with two constant elasticity demand functions. Formby, Layson, and Smith |2~ further show that the adjusted concavity criterion always leads to a wrong prediction for the class of constant elasticity demand functions. Schmalensee |5~ then claims that there is apparently no simple criteria to determine the output effect of price discrimination for the general multiple markets situation.(3) However, Shih, Mai, and Liu |6~ demonstrate that a general criterion can be obtained.

   The general criterion set forth by Proposition 2 in Shih, Mai, and Liu requires information about the slopes of the marginal revenue functions at some output levels which are between simple monopoly and discriminating output levels. A more applicable version--their Corollary 2.3--is obtained by imposing conditions on the rate of change of the slope of the marginal revenue functions, which are conditions on the third derivatives of the demand functions. Furthermore, their analysis leads them to conclude that "Robinson's adjusted concavity ... alone can not determine the output effect" |6, 157~.

   In this paper, it is demonstrated that Robinson's adjusted concavity can lead to definite conclusions on the output effect of price discrimination. Specifically, if the maximum (minimum) value of adjusted concavity over the range of output levels between the simple monopoly and discriminatory outputs in each of the weak markets is less (greater) than or equal to the minimum (maximum) value of adjusted concavity over the corresponding ranges of output in all of the strong markets, then total output under discrimination will be greater (less) than that under simple monopoly when all demand curves are strictly convex (concave). For the cases where the adjusted concavity condition is retained and the general curvature of the demand curves is reversed, examples are provided to show that definite answers can not be obtained.

2. Analysis

   Consider a monopolist producing a homogeneous good at a constant unit cost c and selling it in n separate markets with (inverse) demand function |p.sub i~(|q.sub.i~) for market i (i = 1,....,n). If third degree price discrimination is allowed, the firm chooses an output |q.sub.i~ to maximize its profit ||Pi~.sub.i~(|q.sub.i~) = ||p.sub.i~(|q.sub.i~) - c~|q.sub.i~ in market i. Assume that all profit functions are strictly concave (i.e., ||Pi~|double prime~.sub.i~(|q.sub.i~) |is less than~ 0). The first order conditions are

   |p|prime~.sub.i~(|q.sub.i~)|q.sub.i~ + |p.sub.i~(|q.sub.i~) - c = 0. (1)

   Dividing (1) by |p|prime~.sub.i~(|q.sub.i~) and summing over i give the total output

   |Mathematical Expression Omitted~,

   where hat ( ) of a variable denotes its optimal value under price discrimination.

   Under simple monopoly, the firm chooses a total output q to maximize its total profit |p(q) - c~q, where p(q) is the (inverse) aggregate demand function. Assume that all markets are served under simple monopoly. The first order condition gives the total...