The “Real Wages-Rate of Interest Frontier”
Let us return to the “surplus” theories of interest. Further developments are required to introduce labour as an explicit input, and to express the wage rate in value (whatever the chosen numeraire), rather than as a vector of physical commodities.
This Sraffa did in his “third equations” (see Kurz and Gehrke 2006: 106-7). There are at least two good reasons to do so. The first is that there is no strict “recipe” for the sustenance of workers and their families; the second and more important reason is that the secular tendency for real wages has been to increase, and not to remain constant, and any conception of a radial, say, expansion of a given inventory of goods would violate the principle that the composition of consumption changes along with real income (Engel’s law). Sraffa’s “third equations” formalized an inverse relationship between the real wage and the rate of profits (interest) along Ricardian lines. A more complete analysis required further steps, including the introduction of a variety of methods of production amongst which the firms in the various industries can choose. This aspect, which of course was present in Ricardo, established also a link with the marginal theory of capital and interest. For Sraffa’s mathematical model (Sraffa 1960) has been the basis both of a revival of the Ricardian approach and of a critique of the marginalist theory of capital and interest.The original model assumed a finite number of potential processes in each industry, as in von Neumann’s model and in activity analysis. Also, successive developments normally kept this assumption (two classical reference books are Pasinetti 1977 and Kurz and Salvadori 1995). Nonetheless, for a neat comparison with the marginalist approaches, it is convenient here to assume a continuum of technical alternatives, as in Salvadori and Steedman (1985). This is not to downplay the non-differentiability case, but just to make some other and less obvious differences more transparent.
Let us describe the individual firm in terms of a twice differentiable unit cost function, so that the “marginal conditions” are automatically satisfied for each individual input in each firm.
Assuming identical firms in each industry, the unit cost function in industry j is cj((1 + i)p, w), homogeneous of degree one. By Shephard’s Lemma we have:
where (aj, lj) are, respectively, the cost minimising use of produced inputs and of labour per unit of output in industry j. The partial derivatives ∂αjZ∂((1 + i)pt) (t = 1, 2, ∙∙∙, n) and ∂ljZ∂w are no doubt (semi-)negative, since the cost function Hessian is negative semidefi- nite. This is the logical ground for downward sloping marginalist input demand curves. However, when (9) is satisfied, extra profit must be zero, implying also:
Equations (9) and (10) are the dual representation of the Wicksteed-Wicksell marginal productivity conditions corresponding to the individual firm, in a specific point of long- period equilibrium.
Now considering the comparative statics of such an equilibrium, we see at once that it is impossible to change just one price in (9) without violating (10): at least two prices must change and we can draw no conclusion on the input use-input price relation from the mere sign of the curly derivatives of marginalist partial equilibrium analysis (cf. Opocher and Steedman 2015: ch. 1.4).
The fact that some inputs are produced introduces further complications, because vector p in (10) cannot be simply “given”, unless we consider an “isolated” industry which imports its material inputs at given terms of trade. Equation (10) must hold for all industries. Therefore:
Since only relative prices matter in any genuine microeconomic argument, let us express all prices in terms of a real composite numeraire, s, so that:
Equations (9), (11) and (12) define the “real wage-rate of interest frontier”. It should be stressed that in our formulation (A, I) are continuous functions of relative prices.
Within the limits of the microeconomic assumptions made, no long-period theory of incomedistribution can violate it. Of course, it only determines a trade-off between the real wage and the rate of interest and not a pair of specific equilibrium values. Yet it is very useful in itself. First it permits a coherent theoretical analysis of the relation between input use and input prices both in each individual industry and in the economy as a whole. For let us consider a point on the frontier, however determined. How does a small increase, say, in the rate of interest affect input use per unit of output in each industry? The wage certainly falls both in numeraire and relative to each produced-input rental (1 + i)pt (t = 1, 2, ∙∙∙, n); but the latter may either increase or fall relative to each other. It follows that, even under differentiability assumptions, the industry-level use of an individual commodity per unit of output in (9) need not be inversely related to the rate of interest. Nor need its total value be inversely related, unless relative commodity prices in (11) are “frozen”. In the long period, to which distribution theories typically refer, we are left with no microeconomic foundation for a “demand for capital”, inversely related to the rate of interest. Things are different as far as labour use per unit of output is concerned: under differentiability assumptions, and if all pairs of inputs are Hicksian substitutes (that is, all the off-diagonal terms of the cost function Hessian are positive), then the industry-level input use per unit of output is in fact inversely related to the wage. It should be remarked, however, that the symmetry among inputs which characterized the upsurge of the marginal productivity theory of distribution is here broken without remedy. Moreover, it has been shown that, without the differentiability assumption, also this regularity would disappear (for a full-length discussion, the reader is referred to Opocher and Steedman 2015: chs 4, 5).
The Sraffian critique of the marginalist theory of distribution supports the view that, in a long-period perspective, the rate of profits (interest) and the real wage are hardly determined by a general principle of “factor demand and supply”: primary and produced inputs behave in different ways and they obey no general “law of demand” either at the firm/industry level or at the aggregate level. Rather, the theory of distribution should be “open ended” and admit some degrees of freedom.
The real wage(s)-rate of interest frontier has been used also in applications: a change in technology, in international prices and taxation can be analysed on the basis of proper modifications of equations (9) to (12) (see, for example, Steedman 1983; Mainwaring 1974; Metcalfe and Steedman 1971, respectively; see also Opocher and Steedman 2015: chs 7, 8). A positive (negative) shock ultimately increases (reduces) the maximum real wages and rate of interest that can be paid and determines a complex and often counterintuitive effect on relative prices. Also, it determines a redistribution of income. If we consider, then, the change in an individual real earning rate, it would be useful to distinguish between two components: one due to the income enhancing (reducing) effect of the shock, the other due to a redistribution effect. Using the frontier, we may conceptually distinguish between them: the first can be measured by a shift of the whole frontier, the other by a movement on it (see Opocher and Steedman 2015: ch. 8.1).