Linear Programming, M-C-M, and C-M

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Robert Vienneau
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Consider typical Linear Programs (LPs) for formulating the theory of firm in classical and neoclassical economics. I claim that the classical theory can be formulated as M-C...P...C-M, and that the neoclassical theory of production is something like C...P...C-M.

The notation is from Marx. For Marx, simple commodity circulation is represented as C-M-C. A commodity is sold for money, and then that money is used to buy another commodity. An owner of a use value trades it for a more desired use value. The formula M-C-M characterizes capitalism. A capitalist buys commodities so as to later sell commodities to somehow obtain more money. The goal is the accumulation of capital, not the acquisition of commodities.

For the classical theory, I had a recent presentation here. This is the price-side of John Roemer's Reproducible Solution (RS). The question is what must prices be such that firms can be willing to choose to produce commodities such that capital goods are reproduced (perhaps on an expanded scale), so that the economy will continue.

For ease of exposition, I might as will assume all commodities are basic commodities (in Sraffa's sense) and that there is no choice of technique. I define the following variables:

• ω: A N-element column vector of commodities in existence at the start of the year.
• a₀: A N-element row vector of labor coefficients for each industry.
• A: A N x N Leontief matrix, with each column listing the coefficients of production for each industry.
• w: The wage.
• p: A N-element row vector of prices for each produced commodity.
• q: Decision variables. A N-element column vector of the quantity of each commodity to produce.
• r: Decision variable. The rate of profits.

Each firm begins with an inventory of produced commodities, after having sold those needed for consumption last year. A firm chooses quantities to produce, q, to:

Maximize {p - [p A + a₀w]} q

such that

p A q ≤ p ω

q_i ≥ 0, i = 1, 2, ..., N

The dual LP is to choose the rate of profits, r, to:

Minimize p ω r

such that:

p A(1 + r) + a₀w ≥ p

r ≥ 0

Some theorems from duality theorem are useful here. If a decision variable is positive in an optimal solution to the primal LP, the corresponding constraint is met with equality in the dual LP. In this simple presentation, where all commodities must be produced for the economy to be smoothly reproduced, all decision variables in the primal LP must be positive. Consequently, all constraints must be met with equality in the dual LP. That is, the dual LP provides a system of N equations in N + 2 price variables. An introduction of a choice of technique yields a justification of Kurz and Salvadori's direct method.

Commodities appear on the right-hand side of the constraint in the primal LP. And the decision variables are the commodities to be produced. But the constraint is that the value of the capital goods advanced be less than the value of the given inventory. Likewise, the capitalists are trying to maximize the increment of value. Realization problems are abstracted from here. One assumes that markets exist where one can trade inventory for more appropriate commodities for production plans, and likewise produced commodities can be sold. So the LP can be characterized as M-C-M'. It describes the wealth of society as "an immense accumulation of commodities."

For the neoclassical theory, you can look at an appendix in Pasinetti (1977). Neoclassical theory is about the allocation of given resources. I define the following variables:

• b: A M-element column vector of (unproduced?) factors of production available at the start of the year.
• A: A M x N Leontief matrix, with each column listing the coefficients of production for each process.
• p: A N-element row vector of prices for each produced commodity, with repeated prices for commodities with more than process for producing them.
• q: Decision variables. A N-element column vector of quantities to produce with each process.
• w: Decision variables. A M-element column vector of shadow prices.

The factors of production need have no relation to produced commodities. Don't think of seed corn and harvested corn. Think rather of various kinds of land, ores, and such-like for factors; and of consumer goods for produced goods. The managers of firms, in neoclassical theory, choose the levels, q, of operation of each process to:

Maximize p q

such that

A q ≤ b

q_i ≥ 0; i = 1, 2, ..., N

The dual LP is to choose shadow prices w

Minimize b^Tw

such that

A^Tw ≥ p^T

w_j ≥ 0; j = 1, 2, ..., M

If a process is not operated in the solution to the primal LP, its cost, at shadow prices, exceeds the price of its outputs. If a given resource has a positive shadow price in the solution to the dual LP, it will fully used in the primal LP. Or, if it is not fully used (its constraint in the primal LP is met with inequality) then its shadow price will be zero.

The neoclassical theory ends up with C-M as a description of produced commodities being sold on markets. Its starts with use-values, though. So I guess it can be represented at, a high-level, as C-M alone.

I have previously contrasted post-Sraffian price theory and the neoclassical theory of value. The former is about an analysis of what needs to be the case for the reproduction of society. The latter is about the allocation of scarce resources. This post has introduced another contrast. Post-Sraffian price theory applies to a capitalist society, in which the accumulation of monetary value is an end of itself. I am not sure what kind of society, if any, can be described by the neoclassical theory of production. I guess neoclassical economics makes a bit more sense when it is used to described a series of temporary equilibria strung together.

• Robert Dorfman, Paul A. Samuelson, and Robert M. Solow. 1958. Linear Programming and Economic Analysis
• Heinz Kurz and Neri Salvadori. 1995. The Theory of Production: A Long-Period Analysis.
• Luigi L. Pasinetti. 1977. Lectures on the Theory of Production
• John Roemer. 1979. Analytical Foundations of Marxian Economic Theory
• Robert L. Vienneau. 2005. On labour demand and equilibria of the firm. Manchester School: 73: 612-619.