Extensions of Arrow’s “Learning by Doing”

In a recent paper1 Arrow discusses the behavior of an economic model in which technological change is related to cumulated gross investment, or to the serial number of the machines used in production. Arrow uses a model of fixed proportions, and for a machine of specified vintage and serial number there is a fixed labor requirement for production. In the following paper we shall show that most of Arrow’s results can be extended to any homogeneous production function of the first degree with the type of technological change discussed by him. The production function shows ordinary convexity. Here we shall distinguish two types of production function according to the properties of the marginal product of labor with zero labor input. In type I the marginal product of labor with zero input is. finite and it is impossible to produce without labor (as an example we have the CES production function with elasticity of substitution smaller than 1). In type II the marginal product of labor with zero labor input is infinite, or it is possible to produce without labor input (as an example we have CES with elasticity of substitution greater than or equal to 1; if it is 1, we have of course the known Cobb-Douglas case). As we shall see, in type I, as in the fixed proportion case, there is discarding of capital. Capital is used up to the time when it is scrapped. We shall ignore physical wear and tear and assume that the economic life of machinery is shorter than its physical life. There is no difficulty, and none of the results would change, if we include an exponential force of mortality. In type II, where marginal productivity is infinite with zero labor input, there is of course no complete discarding.