The Contributions of Marx, Keynes and Kalecki
Appendix A: Formalization of Marx’s Schemes of Reproduction
The conditions for a balanced process of simple or expanded reproduction were determined by Marx through numerical examples. Here, we use a more general model that, although differing from Marx’s original analysis in several respects, retains the same basic features as Marx’s schemes.1 Let us consider an n-sector economy which produces consumer and capital goods (ﬁnal goods). Each good can be either consumed or used up as a means of production. Moreover, for simplicity, it is assumed that all the goods are basic commodities, that is that each good is used, directly or indirectly, to produce all the others (Sraffa, 1960). Wages are paid at the end of the period of production; the workers’ propensity to consume is equal to 1. There is no technical change and returns to scale are constant. Goods exchange at their prices of production (natural or normal prices).2 Let Yt be a positive (n × 1) vector, whose generic element yi,t (i = 1, 2, … , n) is the output of the i–th sector at t. The (n × n) matrix Xt is the matrix of inputs xij,t (i, j = 1, 2, … n).3 The price system of the economy is Ytpt = Xtpt (1 + rt) + Ltwt where pt is the vector of the prices of production; Lt is the vector of the quantities of labour employed in the n sectors; wt is the wage rate and rt is the uniform rate of proﬁts exogenously ﬁxed. It is assumed that the matrix Xt...
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