Rational Planning in Exponentially Growing Input — Output Systems Using Labor Values

Abstract

In this paper the problem of an optimal choice of the technology is investigated within the framework of a multiperiod exponentially growing input-output substitution system. Let t = 0, 1,2,..., T denote the discrete period index, representing for example a day, a month or a year. The letter T indicates the length of the production process measured in periods. We consider a closed economic system decomposed into n sectors, each producing one commodity within a period using homogeneous labor and commodities produced in the previous period as inputs. Each sector j (1 ⩽j ⩽ n) has available a finite set K _j of methods of production. In order to produce one unit of commodity j (in sector j) using the method k ∈K _j in period t (1 ⩽ t ⩽ T) a direct labor input of a _0j (k)/(1 + α)^t-1 units of time say in hours as well as a _ij (k) units of commodities i (1 ⩽ i ⩽ n) are required. The constant α > ™ 1 represents the rate of (labor augmenting) technical progress. So as to simplify our mathematics we shall assume that α_0j (k) > 0 for all j and k.