A Note on the Elasticity of Derived Demand in the N-Factor Case

Abstract

Many years ago, Alfred Marshall gave four rules which purported to show how the elasticity of derived demand for a factor of production depended on various parameters [6, p. 385].2 These laws of derived demand were re-stated by Pigou [12, pp. 682-5], and in 1932 Hicks developed a formula which gave the price elasticity of derived demand for a factor of production in terms of the price elasticity of demand for the product, the price elasticity of supply of a competing factor, the first factor's share and the elasticity of substitution between the two factors [3, pp. 241-6], and Hicks used his formula to test the validity of Marshall's four laws. As articles are still being written on the Hicks formula (see Bronfenbrenner [2] and Muth [11]), it may be of some interest to see how the formula stands up when the number of factors is greater than two. Also, since Hicks's derivation of his ingenious formula is somewhat hard to follow, it may be of some interest to use a rather different technique in our derivation. We assume that the production of output y is subject to the constraints implied by an N factor, constant-returns-to-scale production function f(x1,..., XN); that producers take input prices p1,..., PN as fixed and competitively minimize the cost of producing a given output level y; that producers sell output at a price equal to minimum unit cost; and that industry output is determined by the condition that demand and supply of output be equal. Since each producer's technology is represented by the same constantreturns-to-scale production function, it is well known that even though industry output may be perfectly well determined, the output of each individual producer is not. In what follows, we aggregate the individual producers into a single industry producer and develop our formula for the elasticity of derived demand assuming a constant-returns-to-scale industry production function. Let us define the industry unit cost function by c(p1, .. ., PN). This function tells us what the minimum cost of producing one unit of output is when input prices are PI, * . * PN.3