A Note on Third-degree Price Discrimination

VARIAN [1985] AND SCHWARTZ [1990] PROVED THAT, very generally, monopolistic third-degree price discrimination decreases aggregate welfare if total output falls (a conjecture which dates back to the work of A. C. Pigou, [1920]). In particular, these authors adopted a representative consumer approach (by assuming quasi-linear preferences) and used revealed-preference arguments. En passant, their proofs (Varian, [1985: p.875] and Schwartz, [1990: p.1261]) also show that welfare cannot improve if total output is unchanged. However, in a recent paper in this Journal, Adachi [2002], by using linear interdependent demands (‘goods’ are formally q-complements: see e.g. Deaton and Muellbauer, [1980: p.57]), appears to suggest that welfare may increase even if total output does not change. In further contrast to the latter result, this note aims to prove that welfare must decrease under price discrimination if total output remains constant. In fact, our argument is based on a simple and well-known result of consumer theory. To see it, let us refer to the model in Schwartz [1990]. Consumers have quasi-linear preferences, and thus we can treat aggregate consumer behavior as if it were due to a “representative consumer” with indirect utility function v(p) + y0, where p is the vector of prices that the monopolist requires for his product (which is homogeneous, so that total cost depends only on total output) in n (possibly different and interdependent) markets, and y0 is the total endowment of the numéraire (the Marshallian composite commodity). By quasi-linearity, v(⋅) is convex and aggregate (social) welfare can be written as W(p) = π(p) + v(p) + y0, where π(p) is the monopolist’s profit function. We wish to compare the case of a uniform price vector p (p = pι, where ι is the relevant unit vector) with the case of a discriminatory price vector p ( for at least some , j d i p p ≠ j i ≠ n j i , 1 , = ), under the assumption that total output does