The Comparative Statics of Industry Long-Run Equilibrium

Abstract

The paper rigorously derives a result due to Samuelson under very weak regularity conditions: if there is free entry into an industry where producers competitively minimize cost, and if the minimum average cost of plant output is small relative to total industry output, then the industry total cost function will be approximated by a cost function which is linear in output. This industry cost function is dual to a well-behaved constant returns to scale production function, the large output or asymptotic production function. The paper also shows that the asymptotic production function generally exhibits more input substitutability than the plant production function. L'analyse de l'equilibre de longue periode de l'industrie par la methode de statique comparative. Ce memoire presente une derivation rigoureuse d'un theoreme de Samuelson a partir de postulats assumant des conditions de regularite assez peu contraignantes: s'il n'y a pas de barrieres a l'entree dans l'industrie, si les producteurs y minimisent les cofits concurrentiellement, et si le taux de production optimal au niveau de l'6tablissement est petit par rapport au taux de production pour l'industrie dans son entier, alors la fonction de cout total de l'industrie peut ktre approximee par une fonction lin6aire par rapport au taux de production. Cette fonction de coat de l'industrie est le duel d'une fonction de production aux rendements constants a l'echelle, la fonction de production asymptotique. Le m6moire montre que la fonction de production asymptotique implique une plus grande capacit6 de substitution entre facteurs de production que la fonction de production au niveau de l'6tablissement. THEOREM: In the absence of institutional elements of monopoly, production functions that are universally of strict first-degree homogeneity lead under laissez-faire to perfect (or pure) competition no matter how 'differentiated' are the products consumers demand. COROLLARY: In the absence of institutional elements of monopoly, imperfection of competition presupposes some deviations from constancy of returns. The terribly important welfare and market problem of differentiation of product arises only because of these deviations (which are inevitable in the real world, because at small enough output, first-degree homogeneity is an empirical absurdity). P.A. Samuelson (1967, 128) The financial assistance of the Canada Council is gratefully acknowledged. The author also acknowledges the helpful comments of two referees and Keizo Nagatani. Canadian Journal of Economics/Revue canadienne d'Economique, XIV, No. 1 February/fevrier 1981. Printed in Canada/Imprime au Canada. 0008-4085 / 81 / 0000-0078 $01.50 ? 1981 Canadian Economics Association This content downloaded from 207.46.13.168 on Sat, 09 Apr 2016 06:54:28 UTC All use subject to http://about.jstor.org/terms The comparative statics of industry long-run equilibrium / 79 In the above quotation, Samuelson indicates the importance to economics of the issue of constancy of returns to scale of a firm's production function. Samuelson, drawing on the earlier work of Joseph (1933), goes on to show how the possibility of replicating plants or firms in an industry with no barriers to entry leads to an industry production function which exhibits approximate constant returns to scale for large industry outputs. This is an extremely important result which deserves to find its way into textbooks on intermediate microeconomic theory. The fact that it has not is perhaps due to the fact that Samuelson's article was published in a book rather than a journal. A contributing factor may be that Samuelson's 'proofs' are not easy to follow. The purpose of this note is twofold: (i) we provide rigorous proofs of Samuelson's results (and some closely related work by Silberberg (1974a, 1974b) and Maurice (1972)) under very weak regularity conditions and we also establish some modest extensions to these authors' results (e.g., equation (22) which establishes that industry input substitutability is always equal to or greater than plant substitutability), and (ii) we provide geometric interpretations of our results and thus make the material accessible to teachers of microeconomic theory. The reader should also note the closely related work of Novshek and Sonnenschein (1978, 1980). Proofs of theorems are collected in an appendix.' We assume that the technology of a single plant in an industry can be described by means of an N input production function F where y = F(x), y is output and x (xI, x2, .. XN)T is a vector of inputs. Assumption 1: F is continuous from above over x : ON.2 Assumption 2: F is a non-negative function with F(ON) = 0 and F(x *) > 0 for at least one x > ON. It can be shown that assumption 1 implies that the cost function C(y, p) minx {pTx : F(x) ? y } (1) is well defined for all strictly positive input price vectors p > ON and all output levels y E Y, where Y can be taken to be the smallest convex set containing the range of F.3 Usually (i.e., in economics textbooks), Y = {y 0 y < + oo} so that any positive output level is producible by the plant, but we do not require this rather strong assumption. 1 If the reader is acquainted with duality theory at the level presented in Woodland (1977) or Darrough and Southey (1977), the proofs should be easy to follow. 2 This means that the set of input that can produce at least output level y, L(y) {x:F(x) ? y }, is a closed set. This very weak regularity condition cannot be contradicted by empirical data. 3 It can be shown (see Diewert, 1980) that assumption 1 implies that C will have the following properties: non-negativity, linear homogeneity, non-decreasingness, continuity and concavity in inputs prices p for fixed y, non-decreasingness and continuity from below in y for fixed p. This content downloaded from 207.46.13.168 on Sat, 09 Apr 2016 06:54:28 UTC All use subject to http://about.jstor.org/terms