A Dynamic Probabilistic Version of the Aoki–Yoshikawa Sectoral Productivity Model

Enrico Scalas,Ubaldo Garibaldi

doi:10.5018/economics-ejournal.ja.2009-15

Abstract

Abstract In this paper, the authors explore a dynamical version of the Aoki and Yoshikawa model (AYM) for an economy driven by demand. They show that when an appropriate Markovian dynamics is taken into account, the AYM has different equilibrium distributions depending on the form of transition probabilities. In the version of the dynamic AYM presented here, transition probabilities depend on a parameter c tuning the choice of a new sector for workers leaving their sector. The solution of Aoki and Yoshikawa is recovered only in the case c = 0. All the other possible cases give different equilibrium probability distributions, including the Bose–Einstein distribution.

Highlights

Enrico Scalas was put in condition of performing this work by a local research grant of Università del Piemonte Orientale. In their recent book Reconstructing Macroeconomics, Masanao Aoki and Hiroshi Yoshikawa (2007) present a model used to derive the amount of production factor ni for the i-th economic sector, based on an exogenously given demand D and given di¤erent levels of productivity ai for each economic sector i
More speci...cally, let us suppose that an economy is made up of g sectors of size ni, where, as written before, ni is the amount of production factor used in sector i
Aoki and Yoshikawa are interested in ...nding the probability distribution of production factors across sectors, that is the distribution of the occupation vector n = (n1; n2; : : : ; ng) www.economics-ejournal.org when statistical equilibrium is reached

Summary

Introduction

In their recent book Reconstructing Macroeconomics, Masanao Aoki and Hiroshi Yoshikawa (2007) (see Yoshikawa 2003) present a model used to derive the amount of production factor ni for the i-th economic sector, based on an exogenously given demand D and given di¤erent levels of productivity ai for each economic sector i. Aoki and Yoshikawa are interested in ...nding the probability distribution of production factors across sectors, that is the distribution of the occupation vector n = (n1; n2; : : : ; ng). One can introduce con...gurations x = (x1; x2; : : : ; xn), with xi 2 f1; :::; gg, where xi = j means that the i-th worker is active in sector j; the number of distinct con...gurations belonging to a given occupation vector is W (xjn) : n! Boltzmann noticed that, when statistical equilibrium is reached, the probability (n) of an accessible occupation state is proportional to W (xjn); this means that n!. The Ehrenfest-Brillouin Model and its relationship with the AYM will be the subject of the section

Markovian Dynamics for the AYM

Discussion and Conclusions