INTERTEMPORAL OPTIMIZATION IN AN AGRARIAN MODEL WITH Z‐ACTIVITIES*

Abstract

AbstractThis paper examines the optimal characteristics of an agrarian growth model in which two production activities are distinguished, viz., one producing the usual agricultural (consumption) goods F, and the other consisting of labor‐using Z‐activities directed toward substitute consumption for industrial goods and augmentation of agricultural capital.Certain restrictive assumptions on the form of the utility function are made which reflect in part the nature of the three types of consumption goods being distinguished. Optimum conditions are derived using classical calculus of variations, yielding additional variables in the form of shadow prices. Non‐negativity of gross investment is shown to be the only relevant boundary constraint and is accordingly taken into account in the analysis. The phase diagram involving the capital‐labor ratio and the shadow price of Z‐goods indicates that the unique stationary point is a saddle‐point. Some segment of the two stable branches then represents the interior solution to the infinite horizon problem; if the initial capital‐labor ratio is ≪ sufficiently high ≫, the optimal path will entail zero gross investment in the initial phase. Where the period of optimization is finite, different patterns of optimum capital accumulation and valuation will emerge, depending on the initial and terminal values of the capital‐labor ratio. Two cases are illustrated, the optimal path in each being shown to exhibit the turnpike property.It follows that optimal growth may be achieved by directly controlling the rate of capital accumulation or by having the planning authority set the shadow price of Z‐goods. Other means of optimization are in fact available, e.g., controlling imports of industrial consumption goods. That one policy instrument is sufficient to attain the single objective is consistent with the well‐known proposition in the theory of economic policy.