1. So far we have seen that there is a short-run equilibrium corresponding to any given positive amounts of capital and labour. In the first half of this chapter, instead, the rate of real wages is exogenously fixed at some level, and the stock of capital and the number of workers are treated as variable. The equilibrium prices of the capital goods and the capital services, the activity levels of the two industries and the required amounts of capital and labour are found by solving inequalities (1)-(9) in the previous chapter with the real-wage rate specified at the given level. As will be recognized in the second half, this alternative line of argument, which may be referred to as the Growth Equilibrium method, will be more suitable for finding a long-run, steady growth equilibrium where outputs of all goods grow together at a rate equal to the growth rate of the labour force, all the prices remaining unchanged forever.
The argument proceeds in terms of the natural and the warranted rate of growth, the most fundamental concepts of growth economics originally due to Sir Roy Harrod. With no technological improvement the natural rate of growth may be equated to the rate of increase of the working population. On the other hand, to get the warranted rate, defined by Sir Roy as 'that over-all rate of advance which, if executed, will leave entrepreneurs in a state of mind in which they are prepared to carry on a similar advance', 1 some preliminary bulldozing is necessary; in fact, as is seen below, it is a concept that results from a combination of Samuelson's outer envelope of the factor-price frontiers and Kahn's inter-industrial multiplier.
We begin by elucidating the factor-price frontiers that give the correspondence between the real-wage rate and the rate of return on the capital good, where the latter is defined as the ratio of the rental (quasirent) that a unit of the capital good earns per annum to its price, r = q/p. Let us normalize prices in such a way that the price of the consumption good is unity; the normalized wage rate then gives the real-wage rate. As the price-cost inequalities contain three variables (i.e., the price of the capital good p, the price of capital service q and the real-wage rate w), two of them, say the first two, can be expressed as functions of the last. Hence we get a relationship between the rate of return of the capital good