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A Paradigm Shift from Production Function to Production Copula: Statistical Description of Production Activity of Firms

Heterogeneity of economic agents is emphasized in a new trend of macroeconomics. Accordingly the new emerging discipline requires one to replace the production function, one of key ideas in the conventional economics, by an alternative which can take an explicit account of distribution of firms' production activities. In this paper we propose a new idea referred to as production copula; a copula is an analytic means for modeling dependence among variables. Such a production copula predicts value added yielded by firms with given capital and labor in a probabilistic way. It is thereby in sharp contrast to the production function where the output of firms is completely deterministic. We demonstrate empirical construction of a production copula using financial data of listed firms in Japan. Analysis of the data shows that there are significant correlations among their capital, labor and value added and confirms that the values added are too widely scattered to be represented by a production function. We employ four models for the production copula, that is, trivariate versions of Frank, Gumbel and survival Clayton and non-exchangeable trivariate Gumbel; the last one works best.