  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.
