ABSTRACT
Production functions came back into advanced economic research with the advent of endogenous growth theory. The new start was made without any significant attempt to contradict the older debate about capital theory which started with Robinson (1953-54) and culminated in a series of papers rejecting and criticising Samuelson’s surrogate production function (Samuelson 1962). The debate had shown that a theoretically rigorous aggregation of capital and hence a logically stringent construction of the production function were impossible (Garegnani 1970, Harcourt 1972, Pasinetti 1966; more recently Cohen and Harcourt 2003), with critical implications for marginal productivity theory and even for intertemporal general equilibrium theory (Garegnani 2003, Schefold 1997, 2005). The Cambridge critique had been extended to empirical methods of estimating production functions by Anwar Shaikh (see Shaikh 1987), but these critiques did not prevent the extensive use of production functions both in the theory and in empirical work. The gap between the theoretical and empirical applications of the production functions on the one hand and the theoretical and empirical critiques on the other has never been bridged. With a few exceptions, marginal productivity theorists reject the critique without seriously trying to demonstrate its shortcomings, while the community of their opponents cannot explain how it is possible to erect a theoretical edifice as vast as the new growth theory on illogical foundations. One side regards the critique as irrelevant, the other cannot explain the apparent success of the prevailing theory. To confront the positions, a middle ground must be found for a better comparison of the relative merits of both. A mere empirical test could hardly be regarded as satisfactory. For we cannot verify; we can only fail to falsify a theoretical proposition if we follow Popper’s methodology in this context. We first need a theory of a less than fully rigorous construction of the surrogate production function for the confrontation, since the theoretically perfect justification of the aggregation underlying the production function cannot exist (to this extent the critique is irrefutable). Appropriate criteria to judge the validity of such an approach have to be developed. It may turn out that the construction would not be absurd, but not sufficiently correct to
serve its purpose. Or it may turn out to be hopeless. Or it may be adequate. The question is open. The name of the surrogate production function already suggested that its originator Samuelson (1962) had something less than perfect in mind. We return to the old debate in order to find out to what extent the criteria for a rigorous construction may be relaxed without falling into arbitrariness and in such a way that aggregation might be justified (wider issues of the critique for general equilibrium theory will be ignored here). The usual assumptions made for the construction of the surrogate production function are straightforward and will not be questioned: one deals with a closed economy, with a linear technology and constant returns to scale and single product industries in which one commodity is produced by means of other commodities, used as circulating capital, and by means of labour of uniform quality. There is no reason to generalise at this stage, since the introduction of heterogeneous labour, of fixed capital and joint production and of variable returns to scale do not render the existence of the surrogate production function more likely. The assumption of perfect competition should be retained, since monopoly control or other forms of imperfect competition would render the task of demonstrating the workings of the principle of marginal productivity more difficult. Even a set-theoretical description of technological alternatives does not eliminate the possibility of paradoxes of capital theory as long as strict convexity is not postulated, and strict convexity is an extremely problematic assumption (see Schefold 1976). Hence we assume a finite number of methods of production, available for the production in each industry in the form of a book of blueprints. Competition will then ensure that, at any given rate of profit, a certain combination of methods will be chosen, one in each industry, such that positive normal prices and a positive wage rate result, expressed in terms of a numéraire. The wage rate can then be drawn in function of the rate of profit for this combination of methods between a rate of profit equal to zero and a maximum rate of profit, and the ‘individual’ wage curve for this technique will be monotonically falling (see Han and Schefold (2006) for a more detailed description). If the choice of technique is repeated at each rate of profit, starting from zero, different individual wage curves will appear on the envelope of all possible wage curves, and the envelope will also be monotonically falling. Technical change is ‘piecemeal’ in that only one individual wage curve will be optimal in entire intervals, except at a finite number of switch points where generically only two wage curves intersect and where a change of technique generically takes place only in one industry, so that the two wage curves to the left and to the right of the switch point will have all other methods in all other industries in common. The intensity of capital and output per head changes discontinuously at the switch points (they can be represented geometrically for a given individual wage curve w(r), if the numéraire consists of the vector of output per head in the stationary state): output per head equals w(0) and capital per head k = (w(0) – w(r))/r. If many individual wage curves appear successively on the envelope, this envelope may be replaced by a smooth approximation, and each point on this
modified envelope may be thought to represent one individual technique represented by an individual wage curve. The surrogate production function is then defined by taking the tangent to this modified envelope (supposed to be convex to the origin): the slope of the tangent is equal to capital per head and the intersection of the tangent with the abscissa is equal to output per head (see Figure 9.1). If and only if the individual wage curves are linear, the construction is rigorous in that output per head and capital per head of techniques individually employed will be equal to those which we have just defined, and the paradoxes of capital theory (to be discussed below) will then be absent. However, the critique of the surrogate production functions starts from the observation that individual wage curves will in general not be linear and the envelope will not be necessarily convex to the origin; envelope ŵ(r) in Figure 9.1 provides an example. Output per head at r˜ is given by w˜ (0), where w˜ (r) is the individual wage curve tangent to ŵ(r) at r˜. The phenomenon which has attracted most attention is that of reswitching and reverse capital deepening: there may be switch points on the original envelope such that the intensity of capital does not fall with the rate of profit (reverse capital deepening), and the individual wage curve may have appeared on the envelope already at a lower rate of profit (reswitching). It is also possible that capital per head rises with the rate of profit in the industry where the switch of methods of production takes place (reversed substitution of labour) and, surprisingly, reverse capital deepening (the perverse change of aggregate capital per head) and reverse substitution of labour (a perverse change of capital per head at the industry level) need not go together in systems with more than two industries. Returns of processes seem to be frequent: a process which is used in one industry in one interval of the rate of profit is used again in another interval, but not in between. This is a generalisation of reverse capital deepening. It may be shown to imply large changes of relative prices and capital values and it demonstrates that processes cannot be classed as being inherently more or less capitalintensive, prior to their use in specific systems and at specific levels of distribution. Finally, there is likely to be a divergence between output per head and capital per head in the individual industry and the corresponding values which follow from the definition of the surrogate production function; this divergence is called declination and it is illustrated in Figure 9.1: output per head would be ŷ and k = tgα, if the individual wage curve ŵ(r) was linear, but since this is not the case there is the declination w˜ (0) – ŷ. Output per head equals ŷ according to the definition of the surrogate production function, but real output per head is w
̃ (0).
