Probabilistic Production Function of Machinery: Reliability Interpretation of the Leontief Technology

Abstract

The question of dividing factors of production into groups is brought up; the definition of the capacity of a group is suggested as a theoretical quantity of output given by only the group's inputs. A possible relation between groups' capacities and probabilistic production functions is described. It is proposed to treat the failure times of machinery as factors of production. Based on the Leontief technology, assuming that the machines' failure times are independently Weibull and Lomax distributed, the probabilistic CES and Cobb-Douglas production functions are respectively obtained for a production line, and the influence of the distributions' parameters on the hazard rate curves of the production line’s machines is examined. The hypothesis of equality of the Weibull distributions' shape parameters in the derivation of the probabilistic CES production function is justified by the reliability rationale. Based on the Leontief technology, assuming that the machines' failure times have independent generalized Weibull distributions, the probabilistic CES and Cobb-Douglas production functions are obtained in special cases, and the shape of the hazard rate curve is analyzed.