A paradigm shift from production function to production copula: statistical description of production activity of firms

The heterogeneity of economic agents is emphasized in a new trend in macroeconomics. Accordingly, the new emerging discipline requires one to replace the production function, one of the key ideas in conventional economics, by an alternative that can take explicit account of the distribution of firms' production activities. In this paper we propose a new idea referred to as a production copula; a copula is an analytic means for modeling the dependence among variables. Such a production copula predicts the value added 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 the empirical construction of a production copula using financial data of listed Japanese firms. Analysis of the data shows that there are significant correlations among 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 latter was found to be the best.