1. As we have an enormous number of sectors in the actual world, it is of crucial importance to examine whether the main results so far confirmed in the test-tube will remain true outside it. In this chapter we return to the neo-classical model where the full employment of resources and labour is automatically attained and the Rule of Profitability and the Rule of Free Goods control behaviour of activities and prices. We shall complicate the previous two-good model so as to include many (say, n) kinds of commodities, and we will find that the neo-classical analysis of stability of long-run equilibrium will not necessarily be the same. The stability criterion may be different from model to model if the capital-good industry is divided into many sectors; though it will be shown that it is still valid in models with many consumption goods and one capital good. This is indeed an unhappy conclusion and means our two-sector model is inadequate to explain the actual process of capital accumulation. It may happen that in the actual world with many capital goods, the economy is getting further away from the state of long-run equilibrium, in spite of the process in our two-sector test-tube having displayed an easy passage towards a Silvery Equilibrium. Aggregation of various capital goods into one homogeneous stock of capital is a dangerous simplification; it may deform the economy in an intolerable way by forcing, for example, all instabilisers to disappear from the model altogether. An approach with many merits over the present Walrasian-type analysis will be offered in the next part; on this eve of the von Neumann Revolution in the theory of capital and growth a funeral march is played for evaluation of the neo-classical two-sector analysis.
Let us first consider a favourable case. We will show that the principles which we have been established by using the two-sector model would remain substantially unaffected even if we introduce many consumption goods. For the sake of efficiency of explanation, we take, in the following, the number of consumption goods as small as possible; we assume that each consumer is restricted to spend his income upon two sorts of goods. When expenditure is distributed between more than two goods, the argument will lose its simplicity with no addition to generality.
Let us continue to use the previous notation with the subscripts 1 and 2 indicating that the quantities or the prices at issue refer to those of the consumption good 1 and 2 respectively. For example, α1ι is the capital-input coefficient of the ιth process of the consumption-good industry 1,