On The Empirical Verification Of The Cambridge Capital Controversy

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Robert Vienneau
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My consistent position is that Sraffa and his followers, besides recovering an alternate approach to value and distribution found in classical economics and Marx, demonstrated the logical invalidity of marginalist economics. Empirical results are irrelevant to questions of logical validity.

Wage curves, as constructed from input-output matrices, are rational functions with the numerator and denominator both being some high order polynomial functions. I would have liked to see some more wobbles in those constructed empirically and more examples of reswitching and capital-reversing. Nevertheless, the finding that frontiers are close to linear functions, with only a few switch points, is not consistent with an emphasis on widespread marginal adjustments. It is more consistent with Marx's theory of value and Joan Robinson's understanding of technical change, in which the question of the choice of technique at a given moment in time is, at most, a secondary concern. Schefold's recent work (Schefold 2013, Schefold 2016, Götz and Schefold 2020) with random matrices is of interest here for trying to explain the empirical posts.

I have written about empirical results before. In this post I concentrate on Zambelli (2018) as the most recent, most extensive empirical examination of input-output matrices. See also the comments on Zambelli's work in Götz and Schefold (2020).

Increased computer power and more complete consistent national income and product accounts (NIPAs) has supported empirical research. If I recall correctly, Ochoa (1987) looks at wage curves as based on input-output matrices from different times. He looks for pairs that intersect more than onec.

In looking at such a pair, however, many more wage curves are available. One can construct input-output matrices, with one process for each industry, where the processes are not all from one matrix but combine processes among industries from the different matrices. Han and Schefold (2006) take this approach.

But this is not all. One need not limit oneself with processes from pairs of wage curves. One should look at the full range of techniques, where the process for each industry might be from any input-output matrix in your database. Zambelli (2018), in following this approach, uses an algorithm that he and his colleagues cleverly constructed to select the wage curves on the frontier, thereby keeping the combinatorial explosion in this approach somewhat under control.

Ideally, one would like internationally consistent classifications of industries in make and use tables and Leontief input-output matrices that include joint production. If the latter is not available, which it usually not, one needs consistent approximations for single-production. Since make and use tables, and the resulting Leontief input-output tables are typically price data, one needs price indices for industries or commodities. At what level of aggregation do some industries only appear in some tables? Many more questions arise here that are probably beyond me.

What supposedly 'perverse' phenomena should one look for in techniques formed out of empirical input-output matrices? I suggest instances of the reswitching of techniques, capital reversing, the reverse substition of labor, and the recurrence of processes in individual industries would be of interest. Reswitching on the frontier is sufficient, but not necessary for the occurrence of positive real Wicksell effects. I, like many others, define capital reversing (also known as reverse capital deepening) as equivalent to positive real Wicksell effects. Zambelli (2018), on the other hand, defines capital reversing to arise with positive real or price Wicksell effects.

I tend, in pointing out the invalidity of marginalist economics, to de-emphasize any concern with the direction of price Wicksell effects. As I understand it, the direction of price Wicksell effects is dependent on the selection of the numeraire. Also, I am aware of Burmeister's championing of Champernowne's chain index for capital. On the other hand, Baldone (1984) suggest this defense of mainstream economics fails. Fratini (2010) has an example with a continuous variation of techniques along the wage frontier and in which negative price Wicksell effects swamp positive real Wicksell effects, which I guess is a propos here.

I have been talking about wage frontiers and wage curves above. One can construct the aggregate production 'function', given the analysis of the choice of technique. In this analysis, one takes net output as of a given physical composition. It is convenient to take net output as the numeraire. The composition of capital goods varies at switch points, and their prices vary between switch points. At one point, though, Zambelli considers variations in the composition of capital goods between switch points, as I understand it. I relegate an explanation of what he is doing here to an appendix.

Zambelli (2018) is impressive empirical work. The failure of so-called neoclassical theory in 60 percent of the cases examined, as I understand it results, from a concentration on price Wicksell effects, which would not disconcert, for example, Burmeister. I also have difficulties with how Burmeister relates the aggregate production function to a problem of minimizing the value of aggregate capital.

I illustrate the construction of a production funcition as the solution of a maximization problem. A more general presentation would start with netput vectors and assume convexity. I briefly glanced at the appendix to chapter VI in Pasinetti (1977) in writing this.

For concreteness, suppose the managers of a firm have given quantities, x₁, x₂, and x₃, of three resources and know of four fixed-coefficient processes for producing a single commodity. The coefficicients of production for these four processes are:

(a_.j)^T = (a_{1, j}, a_{2, j}, a_{3, j}), j = 1, 2, 3, 4.

Let q_i, i = 1, 2, 3, 4, be the decision variables denoting how much output is produced with each process. Consider the linear following linear program (LP). Maximize output y:

y = q₁ + q₂ + q₃ + q₄

such that:

a_{1, 1}q₁ + a_{1, 2}q₂ + a_{1, 3}q₃ + a_{1, 4}q₄ ≤ x₁

a_{2, 1}q₁ + a_{2, 2}q₂ + a_{2, 3}q₃ + a_{2, 4}q₄ ≤ x₂

a_{3, 1}q₁ + a_{3, 2}q₂ + a_{3, 3}q₃ + a_{3, 4}q₄ ≤ x₃

q_i ≥ 0, i = 1, 2, 3, 4 = 1, 2, 3, 4.

The constraints express the condition that no more of a resource (also known as a factor of production) can be used than is given. Every process must be operated at a non-negative level. Let f express the solution of this LP as a function of factors of production:

y = f(x₁, x₂, x₃)

This is a discrete version of the production function for a given commodity. It has properties commonly assumed in marginalist economics. It exhibits constant returns to scale (CRS) and non-increasing marginal products. If one wanted to construct a production function differentiable everywhere, one could assume an uncountably infinite set of production processes.

I might as well write down the dual problem. It is to choose factor prices w₁, w₂, w₃ to minimize:

w₁x₁ + p₂x₂ + p₃x₃

such that:

a_{1, 1}w₁ + a_{2, 1}w₂ + a_{3, 1}w₃ ≥ 1

a_{1, 2}w₁ + a_{2, 2}w₂ + a_{3, 2}w₃ ≥ 1

a_{1, 3}w₁ + a_{2, 3}w₂ + a_{3, 3}w₃ ≥ 1

a_{1, 4}w₁ + a_{2, 4}w₂ + a_{3, 4}w₃ ≥ 1

w₁ ≥ 0, w₂ ≥ 0, w₃ ≥ 0

For a solution of these two LPs, the values of their objective functions are equal. Factor prices are such that output is completely distributed among the owners of the resources whose services are used in producing the given commodity. If a constraint in the dual is met with inequality, the corresponding decision variable in the primal LP is set to zero. That process is not operated. If a constraint in the primal LP is met with an inequality, that resource is in excess supply and its price is zero. Even though you see no derivatives above, this is an exposition of an aspect of the theory of marginal productivity.

All the parameters and variables in the primal LP are in physical units (for example, bushels, tons, person-years). It does not make much sense to me in an aggregate production function, with output and arguments in price terms, to maximize the value of output for a given value of capital or to minimize the value of capital for a given value of output. Nevertheless, that is what Zambelli does in Section 5.3 of his paper. I suppose he wanted to present a comprehensive empirical exploration of aggregate neoclassical theory, taking its illogic as given.

• Baldone, Salvatore. 1984. From surrogate to pseudo production functions. Cambridge Journal of Economics 8: 271-288.
• Burmeister, E. 1980. Capital Theory and Dynamics. Cambridge: Cambridge University Press
• Fratini, Saverio M. 2010. Reswitching and decreasing demand for capital. Metroeconomica 61 (4): 676-682.
• Han, Zonghie and Bertram Schefold. 2006. An empirical investigation of paradoxes: reswitching and reverse capital deepening in capital theory. Cambridge Journal of Economics 30: 737-765.
• Kersting, Götz and Bertram Schefold. 2020. Best techniques leave little room for substitution: a new critique of the production function. Centro Sraffa Working Paper n. 47.
• Ochoa, E. M. 1987. Is reswitching empirically relevant? US wage-profit-rate frontiers, 1947-1972. Economic Forum 16: 45-67.
• Pasinetti, Luigi L. 1977. Lectures on the Theory of Production New York: Columbia University Press.
• Schefold Bertram. 2013. Approximate surrogate production functions. Cambridge Journal of Economics 37 (5): 1161-1184.
• Schefold Bertram. 2016. Profits equal surplus value on average and the significance of this result for the Marxian theory of accumulation.. Cambridge Journal of Economics 40 (1): 165-199.
• Zambelli, Stefano. 2018. The aggregate production function is NOT neoclassical. Cambridge Journal of Economics 42: 383-426.