Problems in Production-Investment Planning Over Time

Abstract

IN A PREVIOUS PAPER [5] I attempted to develop the foundations for an integrated theory of production and investment in which the choice of best input-output technique, including the optimal stock levels of durable inputs, and the optimal replacement of durable inputs were simultaneously determined in a cost minimization model. Very little was done in that paper to develop the dynamics of the theory, which is the objective of the present paper. We imagine individual decision making units, which we call firms, whose input-output decisions are constrained by a convex function relating the stock and flow inputs to a producing process. This function involves both flow and physical stock inputs because the physical stock of the typical durable input to production directly influences the transformation function of flow inputs into flow outputs.2 A most important characteristic of such individual durable goods is that they can be assumed to be continuously variable in size when considering expansionary investment decisions but cannot be contracted in amount, except by shutting down individual physical units. Typical examples are boilers, buildings, pumps, and pipe lines. For example, a steam generating plant can install any size boiler, X, measured, say, in square feet of heating surface. But once a unit of size X0 is installed, disinvestment can occur only by removing the unit entirely from production. Once a unit has been installed, expansion can take place either by adding a parallel unit of appropriate size or discarding the existing facility and replacing it with a larger unit. Hence, the decision problem contains a curious mixture of continuity and lumpiness. The analysis will be confined to processes which require one current input and one capital equipment input.