Abstract
We consider a stationary, productive technology which is described mathematically by a closed convex cone (see (T. 1) and (T. 3)). Labour appears in an explicit form as a factor of production. Free disposal of labour and commodity inputs is assumed (see T. 2)). Storage activities may also exist (see (T. 4)). In defining feasible plans we assume, besides the usual technological restrictions (see (F. 2), (F. 3), and (F. 4)) that population and labour force grow at a geometrical rate which is independent of time and of the level of economic activity (see (L)). Our criterion for optimality of consumption is obtained by discounting a stationary, concave, increasing and continuous utility of per capita consumption (see (U. 1), (U. 2), (4.1) and (7.0)). Section 2 contains some useful definitions and notation. The model is described in Section 3. In Section 4 we reformulate the model in per capita terms in order to simplify our notation and facilitate mathematical investigation of the model. The basic properties of the set of all feasible consumption plans starting from a given initial input vector are established in Section 5. Section 6 deals with stationary consumption plans. We show that, starting with a strictly positive input vector, we can approach any stationary consumption plan by a feasible one (see Corollary 6.3). In Section 7 we show that the class of all feasible plans starting from the same initial input vector contains an optimal plan (see Theorem 7.1). The last section deals with the problem of characterizing optimal plans, and is both of theoretical and computational significance. Sufficient and necessary conditions for optimality of a plan in the class of all plans starting from a given strictly positive initial input vector are given. Our conditions are the three classical ones: the gradient condition (see (8.15)), the intertemporal profit maximization principle (see (8.16)), and the vanishing in the limit of the present value of input (see (8.17)).
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