A Note on Houthakker's Aggregate Production Function in a Multifirm Industry

ONE OF THE common problems facing an economist dealing with production functions is the problem of aggregation of factors. In a rather neglected paper, Houthakker advances an ingenious approach for explaining the possibility of finding a neoclassical production function for an industry even when production within each of the firms (or cells)3 is done according to a fixed coefficients production function. These fixed proportions vary in a regular way from one cell to another so that the overall input-output relationship takes the form of a regular neoclassical production function. As Solow notices in a survey article on production functions4 this paper has been forgotten and not followed in any direction. In this note we try to reverse Houthakker's procedure and to show how each neoclassical production function implies some density function or distribution function over the cells. We here do it for CES production functions, but it will be obvious that the same method applies to any production function. Following Houthakker we normalize the cells so that each of them is capable of producing one unit of output. Each cell has a requirement, say t, of the variable factor and this requirement varies from one cell to another. If the wage rate in terms of output produced is p then all the cells with tp < 1 will produce a unit of output, all others will be idle. Assume that we are given a density function of the various cells by g(t). Output produced will then be Q = f Pg(t)dt and the input used A f f'IP tg(t)dt. By eliminating i/p one gets a relationship between Q and A. In this way Houthakker has shown that a Pareto distribution implies a CobbDouglas production function. Notice that the relationship between Q and A the cumulated product and factor used-is the familiar Lorenz curve. Assume that the overall relationship between output and the variable factor follows a CES production function with elasticity of substitution (a) smaller than 1;