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

Abstract

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.

Keywords:

• Correlation
• Copulas
• Statistical methods
• Econophysics
• Heterogeneity of agents
• Corporate finance

JEL Classification::

• D24, D30, L11•E10, KUCP-2183

Acknowledgements

We would like to thank Em. Prof. Masanao Aoki for encouraging us to write this paper. Part of this research was financially supported by the Hitachi Research Institute, by the Program for Promoting Methodological Innovation in Humanities and Social Sciences by Cross-Disciplinary Fusing of the Japan Society for the Promotion of Science and by the Ministry of Education, Science, Sports, and Culture, Grants-in-Aid for Scientific Research (B), Grant No. 22300080 (2010–12). We also thank the Yukawa Institute for Theoretical Physics at Kyoto University. Discussions during the YITP Workshop YITP-W-07-16 on ‘Econophysics III: Physical Approach to Social and Economic Phenomena’ were useful in completing this paper.

Notes

†We have excluded one manufacturer with negative Y from our database.

‡To state this fact, we discard the data points for the top 1% of firms in each panel of figure 1, which are considerably depressed compared with the power-law behavior. Such a cut-off may be ascribed to finite-size effects in data collection.

†For θ < 0, should be replaced by the pseudo-inverse , which is equal to in 0 ≤ t ≤ η_C(0) but set to be 0 beyond t = η_C(0).

†Cumulant functions are usually defined in terms of the PDFs in place of the CDFs (Hansen and McDonald 2006).