On the Samuelson–Etula Master Function and the capital controversy

Abstract

The paper addresses the ambiguity that surrounds the conception of capital and its role in neoclassical price-and-distribution theory. The difficulties encountered in the various attempts to define the marginal product either of capital or of a capital good are recalled and the conclusion is drawn that neither concept appears theoretically sound. This historical reconstruction is combined with critical discussion of the recent attempt by Paul Samuelson to determine income distribution by means of the “Master Function”, a device previously developed and presented by Samuelson himself with Erkko Etula, and its “non-neoclassical” marginal products. Rather than the existence of a continuum of alternative technical possibilities, this construction assumes the simultaneous use of a discrete number of methods of production for the same commodity. Even though each technique employs the inputs in fixed proportions, the coexistence of various techniques permits the full employment of an arbitrarily given vector of input endowments. As is shown here, however, the coexistence of methods required for the differentiability of the Master Function can take place, if heterogenous capital goods are used in production, neither in the case with stationary relative prices nor in the non-stationary Arrow–Debreu framework.

Keywords:

• Capital
• capital goods
• marginal productivity
• Master Function
• Samuelson

Acknowledgements

Thanks are due to Fabio Petri and three anonimous referees for their comments and suggestions. As for the responsibility for all remaining errors, the usual caveat applies.

Notes

1 In particular, we refer to the notation introduced by Malinvaud (1953).

2 As the reader will have noticed, this is the notion of capital found, among others, in Marx with the money–commodities–money triad. A sum of money M, i.e. purchasing power, is initially turned into an amount (or vector) of commodities, C. This is done directly, in the case of merchants' capital, or indirectly, by buying the inputs that produce the commodities, in the case of industrial capital. The commodities are then turned back into a sum of money M′. This is because capital “is not spent, is merely advanced” (Marx 1909, Vol. 1, p. 166) and therefore returns to the capitalist augmented by the profit or “surplus-value”. See Marx (1909, Vol. 1., pp. 163–73).

3 In particular, such an assumption characterises Arrow--Debreu equilibrium models.

4 To give just one example, if the employment of labour is expressed – as it should be – in terms of labour hours, then an increase in the employment of labour brings about an increase in the amount of output, ceteris paribus. If it is instead expressed as the sum of the heights of all workers, then the relationship between labour employment and output is ambiguous, as no general conclusion can be drawn about the effect of an increase in workers' total height on output. Then, height is not a technical unit of measure for labour.

5 It is clear that if capital were an input, its technical unit of measure could be deduced directly from the observation of reality, as is the case for all true inputs.

6 This approach is adopted, for example, in Fratini (2013a) to study the effects of a change in the rate of interest on the supply of savings.

7 The neoclassical theory of distribution tends – at least in its initial formulations – to see wages and rents in the same terms as profits (interests). As a result, since profits appear in the same moment as outputs are sold, it is also assumed that wages and rents are paid in that moment.

8 It is worth noting that in these two cases, the transformation of capital into commodities does not have the same meaning as the Marxian M–C–M′. The C in Marx's expression is not, in fact, a vector of inputs but rather a vector of outputs that is sold for the amount of money M′. The Marxian transformation of M into C – and then of C into M′ – therefore requires no ad hoc assumptions and is decidedly general. On the contrary, the conversion of capital into a vector of means of production necessitates either assumption (i) or assumption (ii).

9 Capital goods can clearly be aggregated in various ways, including their weight and the quantities of labour they embody. The point is that the result of the aggregation cannot be regarded as an input or a factor of production. To be more precise, let us assume that there are many techniques, labelled θ = α, β, γ,…, each producing the same final output (consumption good) and denoted as y θ and k θ ∈ ℜ₊n, respectively, the net product and the vector of capital goods, both understood per unit of labour. The aggregation of capital goods consists in turning the vector k θ into a scalar s θ. In other words, it consists in finding a vector v ∈ ℜn such that v⋅k θ = s θ. This aggregation is, however, problematic in many respects.

First of all, given two techniques α and β, one of the following cases may very well happen: (i) s α = s β but y α ≠ y β, (ii) s α ≠ s β, but y α = y β or (iii) s α > s β, but y α < y β. It is clear in these cases that aggregation brings about a loss of relevant information about the relationship between inputs and output: s θ does not provide enough information to explain y θ. Second, if the price vector is used as vector v so that s θ = p⋅k θ, new problems arise. With r as the rate of interest, it is possible to have (iv) ds θ/dr > 0 and (v) sα > sβ if r = r′ and sα < sβ, if r = r″, with r′ ≠ r″. (It should be noted that (iv) is called “reverse capital deepening”, while (v) has no name.) In conclusion, it is thus impossible, in general, to say that one technique is more capital-intensive than another in anything other than tautological terms.For the problems arising from the aggregation of capital goods, see also the analysis presented in Zambelli (2004).

10 Just to give a reference, we can mention Kurz and Salvadori (1995, pp. 428–32).

11 Needless to say, an important role is played in this theory by the principle of decreasing marginal productivity. This implies, first, that the profit function is concave, so that the first-order condition, i.e. marginal equality, is necessary and sufficient for the maximisation of profits, and, second, that there is an inverse relationship between the employment of an input and its price, the quantities employed and the prices of the other inputs being constant.

12 On the weakness of this position, see in particular Pasinetti (1969), Garegnani (1984) and Fratini (2013b).

13 In Fratini (2010), for instance, the possibility of a monotonically decreasing schedule of the investment of capital associated with a non-monotonic behaviour of the curve of the (physical) net product per worker is shown.

14 See, for example, Hicks (1932) and Robertson (1931). For a reconstruction of this debate, see Trabucchi (2011).

15 As Samuelson himself states (1962, p. 196): “No alchemist can turn one capital good into another. [Capital good] [a]lpha needs labour to work with in a fixed proportion: more than its critical proportion of labour will yield nothing extra; take away either input, while holding the other input at the previously proper proportion, and you lose all the product that has resulted from the combined does of the two inputs”.

16 As has been insistently stressed by Garegnani (e.g. in 2007, p. 581--2 ) and Petri (2011, p. 381), the fact that marginal products are generally zero, hence capital goods' rental prices and possibly the real wage rate are zero, questions the plausibilty that income distribution is determined by factors' marginal products.

17 In particular, as Samuelson wrote (1966, p. 568), according to the “tale” told by Jevons and Böhm-Bawerk, an increase in the rate of interest should bring about the use of less “roundabout” or “mechanised” techniques, i.e. techniques that involve a smaller net product per unit of labour. Thanks to that debate, it is known that the very opposite may well occur.

18 There is, of course, no mathematical difficulty in doing this. It is possible to write a Cobb--Douglas or a CES production function y_t+1 = f(a_t) whose domain is the set of non-negative vectors of inputs a_t or a differentiable transformation function ϕ(a_t, b_t+1). These functions, however, overlook important aspects connected with the employment of capital goods in production, namely their complementarity and specialisation.

19 It is, in fact, known that the current mainstream macroeconomic theory is actually general equilibrium theory with some very restrictive assumptions imposed (e.g. just one agent, just one commodity,…).

20 Actually, Samuelson's and Etula's apparently “novel” attempt is not so new. The idea on which the MF is based can already be found in Samuelson (1959) and, particularly, in Kurz and Salvadori (1992, pp. 232–5). In this last contribution, Kurz and Salvadori have built a function that very much resembles the MF, whose marginal product of labour can be determined by means of the same mathematical tools used by Samuelson and Etula (i.e. the theory of linear programming). Kurz and Salvodori do not use their construction to determine income distribution by marginal-productivy theory; however, they use it to provide a formal explanantion of Ricardo's view of the capital-accumulation process in an economy that produces corn by means of labour and land only, along the lines of Passinetti (1960).

However, both in Samuelson (1959) and in Kurz and Salvadory (1992), the analysis is restricted to economies that do not employ capital goods in production. Kurz and Salvadori are very clear “that with heterogeneous capital goods no production function can be constructed” (p. 232). We shall see in the follwing section the reasons for this.

21 In fact, the authors offer no description of the working of the market for inputs and the equality of supply and demand appears to be an assumption rather than the result of the market mechanism.

22 Of course, in a capitalist economy, nobody consciously maximises total output (there is no “Central Planner”). However, as it should be clearer when we examine P1's dual problem below, D1, maximisation of corn production is the counterpart of the employment of cost-minimising methods of production, which is in turn the outcome of profit maximisation.

23 Apart from the non-negative constraints (P1.3(3) $A\times y \left(\omega \right)=\left[\begin{array}{cc}{L}^{\left(a\right)}& L^{\left(b\right)}\\ T^{\left(a\right)}& T^{\left(b\right)}\end{array}\right] \left[\begin{array}{c}{y}_a\left(\omega \right)\\ y_b\left(\omega \right)\end{array}\right]=\left[\begin{array}{c}L\\ T\end{array}\right]$(3) ).

24 In the general case, if $n$ is the number of methods and $m$ the number of constraints, with $n>m$, the vector of activity levels will have at most $m$ positive components.

25 In their first development of the MF, Samuelson and Etula (2006) referred exclusively to the case with just one kind of capital good. In that framework, as will become clear later, the problems we intend to show in the present paper cannot arise. For this reason, we consider directly the other case, with two kinds of capital goods, addressed by Samuelson in his (2007) article.

26 Once again, we follow Samuelson, who derives the MF by maximising corn gross production rather than corn net production. Under their hypotheses, however, both procedures are equivalent. The reason is that if the employment of corn capital is exogenously given as is the case in Samuelson's argument (because the endowment of corn is given and must be fully employed), and the endowment of corn must be reproduced by the stationary assumption, then gross and net outputs differ by a constant and the maximisation of gross corn production is equivalent to the maximisation of net corn.

27 In Section 3, fn. 14, we have seen that Samuelson (1962) himself was very aware of this fact when he built his Surrogate Production Function.

28 To see more clearly the problems raised by the non-fulfilment of condition (14(14) ${1+i^{*}\equiv \sigma }_1=\frac{{\sigma }_2}{\pi }$(14) ), consider the following example: let $i_1$ and $i_2$ be the (net) rates of return on the supply prices of $K_1$ and $K_2$, respectively. Since corn is the numéraire, condition ${\sigma }_1=1+i_1$ holds for the case of corn and condition $\frac{{\sigma }_2}{\pi }=1+i_2$ for the case of iron. Let us further define $p_1^D$ and $p_2^D$ as the demand, or selling, prices of capital goods 1 and 2, respectively. These are the maximum prices investors are willing to pay to buy K₁ and K₂ and must be equal, in equilibrium, to the present value of the sum of the future yields of each capital good. Investors will be indiferent between buying K₁ or K₂ as long as the return on demand prices is the same. Now, consider the case where the solution to system (13(13) $B^t\times p\equiv \left[\begin{array}{cccc}{L}^{(c)}& K_1^{(c)}& K_2^{(c)}& 0\\ L^{(d)}& K_1^{(d)}& K_2^{(d)}& 0\\ L^{(e)}& K_1^{(e)}& K_2^{(e)}& F^{(e)}\\ L^{(f)}& K_1^{(f)}& K_2^{(f)}& F^{(f)}\end{array}\right]\times \left[\begin{array}{c}w\\ {\sigma }_1\\ {\sigma }_2\\ -\pi \end{array}\right]=\left[\begin{array}{c}{Q}^{(c)}\\ Q^{(d)}\\ 0\\ 0\end{array}\right]$(13) ) implies $i_1>i_2$. In this situation, for K₁, demand and supply prices coincide, namely 1$=p_1^D=\frac{{\sigma }_1}{1+i_1}$. For capital good 2, on the other hand, its demand price must satisfy the following condition: $p_2^D\le \frac{{\sigma }_2}{1+i_1}.$ Otherwise, nobody will be willing to invest in this capital good. But given that $i_1>i_2$, this means that $p_2^D<\pi$. In other words, the fact that $i_1>i_2$ implies that the price at which investors are willing to buy K₂ is not sufficient to cover production costs, and hence the capital good will not be reproduced in the following period, violating the assumption of stationariness.The implication is that condition (14(14) ${1+i^{*}\equiv \sigma }_1=\frac{{\sigma }_2}{\pi }$(14) ) should be added to allow the reproduction of both capital goods.

29 For a detailed discussion about the difference between commodities' own rates of interest and the returns on the supply prices of capital goods, see Garegnani (2003, Appendix II [A]).

30 See also Samuelson (2007, p. 259), where the author claims that a system of equations like (13(13) $B^t\times p\equiv \left[\begin{array}{cccc}{L}^{(c)}& K_1^{(c)}& K_2^{(c)}& 0\\ L^{(d)}& K_1^{(d)}& K_2^{(d)}& 0\\ L^{(e)}& K_1^{(e)}& K_2^{(e)}& F^{(e)}\\ L^{(f)}& K_1^{(f)}& K_2^{(f)}& F^{(f)}\end{array}\right]\times \left[\begin{array}{c}w\\ {\sigma }_1\\ {\sigma }_2\\ -\pi \end{array}\right]=\left[\begin{array}{c}{Q}^{(c)}\\ Q^{(d)}\\ 0\\ 0\end{array}\right]$(13) ) is both “necessary and sufficient for characterising competitive distribution equilibrium”.

31 Our analysis, thus, seems to confirm a conjecture made by Opocher (2008), who “doubted” (p. 109) that the MF could be actually extended to cover the case of heterogeneous capital.

32 Fratini (2013a) provides a discussion of the set of hypotheses required in order to claim that reswitching is a possible source of equilibrium instability. Moreover, it is shown in Fratini (2007) for an overlapping generation model that some multiple equilibria are due to reswitching. Reswitching appears to be impossible, however, in the case of differentiable production functions, although the point is not crystal clear (see also Hatta, 1976).