On the relationship between production functions and input-output analysis with fixed value shares

Abstract

this paper we examine the necessary and sufficient conditions for constancy of the value input-output coefficients. In his well-known article Klein1 studied the same problem. Nevertheless, there is a striking difference between his approach and ours. Klein assumes that all inputand output prices are known, either as fixed or as given functions of input demand. He subsequently shows that every production function which is zero-homogeneous in all inputs and outputs yields constant value shares of inputs in the outputs, assuming optimal allocation of both. This is true whenever one specific vector of inputand output prices is considered. We, however, derive conditions of invariance of value shares for any exogenously determined variable vector of input prices and output prices. This additional requirement restricts the class of admissible production functions. First, we consider the case of a single homogeneous output per sector. We show that constancy of value input-output coefficients holds good if and only if the production function is of the linear-homogeneous CobbDouglas type. The sufficient condition is well-known, but contrarily to common belief the necessity of the condition has as yet not been established.