Production, Prices, and the Theory of Jointly-Derived Input Demand Functions

Abstract

The theory of derived input demand has been examined intensively for the situation in which there is only one variable input. Only special cases, however, of the theory of jointly-derived input demand functions have been investigated. In 1938 both Mosak and Allen analysed related factor demands under the assumption of perfect competition in the commodity and factor markets.2 Somewhat later Samuelson studied the same problem without the assumption of perfect competition in the product market.3 However, he imposed a different restrictive condition, namely that the firm under consideration be confined to movements along a given production isoquant in response to changes in relative factor prices. The purpose of this article is to generalize the theory of jointlyderived input demands by removing some of the restrictive assumptions contained in earlier studies. In particular, the following assumptions are made: (a) product demand is a function of and varies inversely with product' price, the prices of related commodities remaining unchanged throughout ;4 (b) the firm in question is a perfect competitor in the factor market; and (c) there is a given production function whose first partial derivatives are positive and monotonically decreasing over the relevant range of input quantities. In the case of two variable inputs the results are quite clear and readily explicable in terms of well-known economic magnitudes. Section II is devoted to an exhaustive analysis of this special case. The general case is presented in Section III. The results, in some instances, are not as precise as in the two-factor model. Nonetheless, the general properties of jointly-derived input demand functions may be stated straightforwardly.