Returns to scale with a Cobb-Douglas production function for four small Northern Italian firms

Osti, Davide (2022): Returns to scale with a Cobb-Douglas production function for four small Northern Italian firms.

Abstract

With this piece of evidence, I try to shed light upon the effects of fixed and variable costs on revenues for four firms operating in the sectors of lathing and milling, packaging machine construction, mechanical component production and shoe parts building, all four in the vicinity of Bologna, Italy, through the estimation of a linear bivariate simultaneous equation model where variable and fixed costs explain revenues; with a sample of eleven/twelve years of annual data for each firm, and find that a marginal increase in variable costs lead to more than proportional increases in revenues; similarly for fixed costs; I consider both contemporaneous regressions and distributed lags ones. I further estimate a Cobb-Douglas production function, in order to find out whether the returns to scale are increasing, constant or decreasing comparing various estimation methods: OLS, instrumental variable method, dynamic panel methods, as well as the Levinsohn and Petrin 2003 method, first separately for each single firm and then pooling the individual firms' samples in a panel; I find support for the hypothesis of slightly increasing returns to scale with the baseline Cobb-Douglas transformed in logarithms with capital, labour and materials as inputs.

• Item Type: MPRA Paper
• Language: English
• Keywords: production functions, returns to scale, cobb - douglas, stochastic frontier model, non linear least squares, production sets
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C33 - Panel Data Models ; Spatio-temporal Models; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C36 - Instrumental Variables (IV) Estimation; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation; D - Microeconomics > D2 - Production and Organizations > D22 - Firm Behavior: Empirical Analysis

References

[1] \textsc{Sydney N. Afriat}, ``Efficiency Estimation of Production Functions'', \emph{International Economic Review}, 1972, vol. 13, issue 3, 568-98;\\

[2] \textsc{Daniel A. Ackergberg, Kevin Caves, and Garth Frazer}, ``Identification properties of recent production function estimators'', \emph{Econometrica}, vol. 83, no. 6, November 2015, 2411-2451;\\

[3] \textsc{Dennis Aigner, Takeshi Amemiya}, and \textsc{Dale J. Poirier}, ``On the Estimation of Production Frontiers: Maximum Likelihood Estimation of the Parameters of a Discontinuous Density Function'', \emph{International Economic Review}, Vol. 17, No. 2 (Jun., 1976), pp. 377-396;\\

[4] \textsc{Dennis J. Aigner} and \textsc{S. F. Chu}, ``On Estimating the Industry Production Function'', \emph{The American Economic Review}, Vol. 58, No. 4 (Sep., 1968), pp. 826-839;\\

[5] \textsc{Dennis Aigner, C. A. Knox - Lovell}, and \textsc{Peter Schmidt}, ``Stochastic Frontier Production Function Models'', \emph{Journal of Econometrics}, Volume 6, Issue 1, July 1977, Pages 21-37;\\

[6] \textsc{Manuel Arellano} and \textsc{Stephen Bond}, ``Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations'', \emph{Review of Economic Studies}, 1991, 58, 277-297;\\

[7] \textsc{E. J. Borowski} and \textsc{J. M. Borwein}, \emph{Dizionario Collins della Matematica}, Gremese, 1989;\\

[8] \textsc{Charles W. Cobb} and \textsc{Paul H. Douglas}, ``A Theory of Production'', \emph{The American Economic Review}, Vol. 18, No. 1, \emph{Supplement, Papers and Proceedings of the Fortieth Annual Meeting of the American Economic Association} (Mar., 1928), pp. 139-165 ;\\

[9] \textsc{Phoebus J. Dhrymes}, ``On Devising Unbiased Estimators of the Parameters of the Cobb-Douglas Production Function'', \emph{Econometrica}, Vol. 30, No. 2 (Apr., 1962), pp. 297-304;\\

[10] \textsc{W. Erwin Diewert and Kevin J. Fox}, ``On the estimation of returns to scale, technical progress nad monopolistic markups'', \emph{Journal of Econometrics}, 145 (2008), 174-193;\\

[11] \textsc{Michael K. Evans} - \emph{Macroeconomic Activity. Theory, Forecasting and Control}, 1969, A Harper International Edition, Philadephia;\\

[12] \textsc{M. J. Farrell}, ``The Measurement of Productive Efficiency'', \emph{Journal of the Royal Statistical Society, Series A (General)}, Vol. 120, No. 3 (1957), pp. 253-290;\\

[13] \textsc{William Greene}, \emph{Econometric Analysis}, Prentice Hall, Upper Saddle River, New Jersey, 2012;\\

[14] \textsc{Zvi Griliches and Jacques Mairesse}, ``Production functions: the search for identification'', \emph{NBER working paper series}, March 1995, no. 5067;\\

[15] \textsc{Zvi Griliches} and \textsc{Vidar Ringstad}, ``Economies of Scale and the Form of the Production Function'', North Holland, 1971;\\

[16] \textsc{Zhezhi Houa, Shunan Zhaob}, and \textsc{Subal C. Kumbhakar}, ``The GMM estimation of semi-parametrics spatial stochastic frontier models'', \emph{European Journal of Operational Research}, Volume 305, Issue 3, 16 March 2023, Pages 1450-1464\\

[17] \textsc{Lawrence Klein}, ``The Specification Of Regional Econometric Models'', \emph{Papers in Regional Science}, Wiley Blackwell, January 1953, vol. 23(1), pages 105-116;\\

[18] \textsc{James Levinsohn and Amil Petrin}, ``Estimating production functions using inputs to control for unobservables'', \emph{Review of Economic Studies}, 2003, vol. 70, 317-341;\\

[19] \textsc{Jacques Mairesse}, ``Comparison of Production Function Estimates on the French and Norwegian Census of Manufacturing Industries'', in F. L. Altmann, O. Kyn and H. I. Wagener, eds., \emph{On the Measurement of Factor Productivities; Theoretical Problems and Empirical Results}, Papers and Proceedings of the $2^{nd}$ Reisenberg Symposium (Gottingen: Vandenbroeck and Ruprecht, 1976);\\

[20] \textsc{Jacob Marschak} and \textsc{William H. Andrews, Jr.}, ``Random Simultaneous Equations and the Theory of Production'', \emph{Econometrica}, Vol. 12, No. 3/4 (Jul. - Oct., 1944), pp. 143-205;\\

[21] \textsc{Andreu Mas Colell, Michael D. Whinston}, and \textsc{Jerry R. Green}, \emph{Microeconomic Theory}, \emph{Oxford University Press}, New York, Oxford, 1995, pp. 128-136;\\

[22] \textsc{Marjorie B. McElroy}, ``Additive general error models for production, cost, and derived demand or share systems'', \emph{Journal of Political Economy}, 1987, vol. 95, no. 4, 737-757;\\

[23] \textsc{Yair Mundalak}, and \textsc{Irving Hoch}, ``Consequence of Alternative Specifications in Estimation of Cobb - Douglas Production Functions'', \emph{Econometrica}, Vol. 33, No. 4 (Oct., 1965), pp. 814-828;\\

[24] \textsc{G. Steven Olley and Ariel Pakes}, ``The dynamics of productivity in the telecommunications equipment industry'', \emph{Econometrica}, Vol. 64, No. 6, November 1966, 1263-1297;\\

[25] \textsc{Amil Petrin, Brian Poi, and James Levinsohn}, ``Production function estimation in Stata using inputs to control for unobservables'', \emph{The Stata Journal}, 2004, 4, no. 2, 113-123;\\

[26] \textsc{James b. Ramsey} and \textsc{Paul Zarembka}, ``Specification Error Tests and Alternative Functional Forms of Aggregate Production Functions}, \emph{Journal of the American Statistical Association}, Sept. 1971, vol. 66, no. 335, pp. 471 - 477;\\

[27] \textsc{C. P. Timmers}, ``Using a Probabilistic Frontier Production Function to Measure Technical Efficiency'', \emph{Journal of Political Economy}, Jul. - Aug., 1971, Volume 79, Number 4, pp. ;\\

[28] \textsc{A. A. Walters}, ``Production and Cost Functions, an Econometric Survey'', \emph{Econometrica}, Vol. 31, No. 1/2 (Jan. - Apr., 1963), pp. 1-66;\\

[29] \textsc{Knut Wicksell}, ``Den 'kritiska punkten' i lagen for jordbrukets aftagande produktivitet'', \emph{Economisk Tidskrift}, 18, 1916;\\

[30] \textsc{De Min Wu}, ``Estimation of the Cobb-Douglas Production Function'', \emph{Econometrica}, Vol. 43, No. 4 (Jul., 1975), pp. 739-744;\\

[31] \textsc{Mahmut Yasar, Rafal Raciborski, and Brian Poi}, ``Production function estimation in Stata using the Olley and Pakes method'', \emph{The Stata Journal}, 2008. 8, no.2, 221-231;\\

[32] \textsc{A. Zellner, J. Kmenta}, and \textsc{J. Drèze}, ``Specification and Estimation of Cobb - Douglas Production Function Models'', \emph{Econometrica}, Vol. 34, No. 4 (Oct., 1966), pp. 784-795;