On the Shape of the Production Possibility Frontier with More Commodities than Primary Factors

Abstract

The purpose of this paper is to investigate the shape and the properties of the production possibility frontier (p.p.f.) when the number of commodities (n) exceeds that of primary factors (m). It is a generalization of the author's recent paper (Inoue [1982]) which investigates the specific case of three commodities and two primary factors. Although it is widely known that the p.p.f. cannot be strictly concave to the origin when there are more commodities than primary factors (see Khang [1971], Chipman [1972] and Khang and Uekawa [1973]) the shapes and the properties the p.p.f. can assume in this case have not been classified. Perhaps the exceptional one is Melvin [1968] who argues that in general it is a nonnegative part of a cone, and in a special case, is a nonnegative part of a cylinder for three commodities and two primary factors case. By assuming the neoclassical technologies -i.e., production functions are homogeneous of degree one, strictly quasi-concave, twice continuously differentiable and well-behaved no joint production, no interindustry flows, perfect competition in both the commodities and factors markets, and perfectly inelastic supply of primary factors, first it is shown that if the prices of all commodities are linearly independent when the relative factor prices vary, then the p.p.f. is in general a ruled surface (i.e., not strictly concave to the origin) which is neither a nonnegative part of a cone nor a cylinder, while in a special case it is a nonnegative part of a cone (Theorem 1). Second, in the case of a cone though the amounts of commodities are indeterminate for given commodity prices and factor endowments, (i, the ratio of the amounts of commodity i on those of commodity n measured from the vertex of cone -i.e., i=(yi-ci)/(yn,-c,) where yi is the amounts of commodity i, and c=(cc,..., c) is the vertex of cone -, i=1,..., n is seen to be determined by relative commodity prices only; it does not depend on the amounts of factor endowments (Corollary 1). Third, if the relative prices of some commodities are constant when the relative factor prices vary, then the p.p.f. is in general a nonnegative part of a cylinder while in a special case is a nonnegative part of a plane (Theorem 2). Fourth, in the case of a cylinder, the composite commodities are defined naturally. Then for the composite commodities, the p.p.f. is classified in Theorem 1. Further, if the number of the composite