Direct And Indirect Methods, Axioms And Algorithms For The Choice Of The Technique

1.0 Introduction

Kurz and Salvadori (1995) explain prices of production with two methods of analysis: the direct method and the indirect method. The indirect method, for the circular capital case, involves the creation of the wage frontier, the most well-known diagram to come out of Sraffa (1960). The direct method characterizes a system of prices of production by axioms, while the indirect method suggests algorithms for finding cost-minimizing techniques.

2.0 The Direct Method

In both methods, the given technology is characterized as a set of processes. A process is specified as the quantities of inputs and the quantities of outputs for a given level of operation. In the case of reproducible natural resources, that is, land, the quantity available of each quality of land should be specified. I do not think that non-reproducible natural resources fit comfortably in this model; some work has been done on the corn-guano model exploring this.

I find it useful to assume constant returns to scale. I understand why Sraffa does not need this assumption in the first two sections of his book. He does not consider the choice of technique there or how a state can be reached in which the same rate of profits is obtained for each operated process. He considers which process is operated and the level of operation of each process as given. Sraffa explicitly says that his assumption, that no question of the returns to scale arises, does not apply to the last section, analyzing the choice of technique. I find it difficult to understand how the theory of joint production can be set out without considering the choice of technique.

Informally, the following six axioms characterize quantities and prices for a capitalist economy undergoing smooth reproduction:

1. The levels of operation of the processes comprising the technology are such that, after replacing commodities used up in production and those needed for accumulation at the given rate of growth, requirements for use can be satisfied by net output.
2. The levels of operation of the processes comprising the technology are such that one unit of labor is employed throughout the economy.
3. No pure economic profits can be made by operating any process. That is, for each process, the revenues obtained from operating it do not exceed its costs, including charges for the rate of profits, rent, and wages.
4. The price of the numeraire is unity.
5. The rule of free goods: The price of a good in excess supply (that is, with a level of production strictly exceeding its requirements for use) is zero.
6. The rule of non-operated processes: If the costs of operating a process, including charges for the rate of profits, strictly exceeds the revenues obtained from that process, it is not operated.

Kurz and Salvadori set out these axioms for specific models, along with non-negativity conditions. In models of rent, one discards the second axiom; scale matters. The most general model, I guess, is that of full joint production. What can be deduced from the axioms varies among these models.

3.0 The Indirect Method

One can set out the indirect method by applying combinatorics, in which one looks at all possible techniques that can be constructed from the processes comprising the technology. Discard those techniques that cannot satisfy requirements for use. In the circulating capital case, each technique yields a system of equations with one degree of freedom. These equations can be solved. So each technique has a wage curve, in which the wage is lower, the higher the rate of profits. All these curves can be graphed on the same graph, and the outer wage frontier shows which technique is cost-minimizing at any given wage or rate of profits.

How can a combinatorial explosion be avoided in this analysis? By applying certain algorithms which do not require one to consider every technique. Christian Bidard champions the Lemke algorithm for the case of fixed capital. As in the case of the simplex method for linear programs, one does not need to consider every feasible technique in finding the cost-minimizing technique for a given wage or rate of profits.

4.0 Questions

Different, but related questions, arise for the two methods. For the direct method, one asks, what conditions must technology satisfy such that a solution of quantities and prices exist? When is this solution unique? For the case of circulating capital, one looks at the Hawkins-Simon condition, the Perron-Frobenius theorem, and the misleadingly-named non-substitution theorem. In the case of general joint production, the special case in which the rate of profits is equal to the rate of growth is of interest. I suppose research questions might still revolve about how many of the nice properties of circulating capital carry over to models that are intermediate between this case and full joint production. For example, has any work been done on combining a fixed capital model with a model of extensive rent? Biao Huang recently found that revisiting the fixed capital model was worthy of research. Relaxing the assumption of free disposal is important for investigating environmental concerns.

For the indirect method, I ask on what platforms are these algorithms executing? When I apply numerical methods for finding fluke switch points, I am definitely not making any claims about how markets work. I think my application of linear programming and duality theory in my 2005 Manchester School article is another case of an external analyst examining an economic model. But sometimes, as in my 2017 ROPE article, an algorithm is described that can be executed by an abstract market. At least this seems to be Bidard's view. And these algorithms might even be able to be executed in parallel by, say, accountants working for different firms in different industries. What I would like to see, however, is how market structures, how bids and asks are resolved, for example, enters into these algorithms. Be that as it may, one can ask which algorithms converge. How fast? Is the point to which they converge unique and independent of the initial condition?

References

• Christian Bidard. 2004. Prices, Reproduction, Scarcity. Cambridge University Press.
• Heinz D. Kurz and Neri Salvadori (1995) Theory of Production: A Long-Period Analysis. Cambridge University Press.
• Keiran Sharpe. 1999. Note and comment: on Sraffa's price system. Cambridge Journal of Economics 23(1): 93-101.
• Piero Sraffa. 1960. The Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory. Cambridge University Press.
• K. Vela Velupillai. 2021. Definitions, assumptions, propositions and proofs in Sraffa's PCMC. In A Reflection on Sraffa's Revolution in Economic Theory (ed. by Ajit Sinha). Palgrave
• Ludwig Wittgenstein. Remarks on the Foundations of Mathematics, revised edition.
• J. E. Woods. 1990. The Production of Commodities: An Introduction to Sraffa. Macmillan.