Constructing the Cobb–Douglas Production Function Based on Different Pareto-Distributed Capacities of Production Factors

Abstract. This paper considers a probabilistic approach to obtaining production functions: the capacities of production factors (the ratios of the factors to their per-unit values) are treated as independent random variables, and the characteristics of the Leontief function (the quantity of output) are found accordingly. The cumulative distribution function of the minimum from a set of independent random variables is used. The cases studied are the factor capacities with the Lomax, Pareto Type IV, and generalized Weibull distributions. Their discrete analogs are considered, in particular, the Zipf Type IV law. In all the cases, the median of the quantity of output is expressed through the medians of the factor capacities as a Cobb–Douglas production function under relatively small shape parameters of the distributions. The CES production function is constructed in the case of the generalized Weibull distribution with large shape parameters.

Keywords: production function, Cobb–Douglas production function, probabilistic approach, Leontief’s production function, Pareto distribution, Lomax distribution, generalized Weibull distribution, median.