Groucho Marxism

In 1937 the Hungarian mathematician and physicist Jon von Neumann published a paper entitled A Model of General Economic Equilibrium, in which he set out an abstract mathematical model of an economy. Von Neumann’s model is still used widely today, particularly by Marxist economists. In this blog post I will provide a brief summary. The model is based on two premises: (1) commodities are produced not only from ‘natural factors of production’ also from each other; and (2) there may be a different number of technically possible processes of production than types of commodities. The second premise is captured by assuming that there are m types of commodities which can be produced by n processes, where in general m ≠ n.

In each process j, quantities a_ij ≥ 0 are used up and quantities b_ij ≥ 0 are produced of commodity type i; let A = (a_ij) and B = (b_ij). In general, each commodity i will have an associated price p_i, and each process j will be used with a certain ‘intensity’ q_j. Given an mx1 price vector p = (p_i), the vector pA represents the total cost of operating the different processes and the vector pB represents the total revenue obtained from the different processes. Similarly, given a 1xn intensity vector q = (q_j), the vector Aq represents the total commodities of different types used up and the vector Bq represents the total commodities of different types produced. Von Neumann was concerned with the situation where the whole economy expands without change of structure, i.e. where the ratios between the intensities remains constant.

It is assumed that the matrices A and B are known. The unknowns are: the prices p = (p_i) ≥ 0 of the different commodity types; an ‘interest factor’ c = 1+r ≥ 0, where r is the interest rate or profit rate; the intensities q = (q_j) ≥ 0 of the different processes; and a ‘coefficient of expansion’ d = 1+g  ≥ 0, where g is the growth rate for the economy. These are related by the vector inequalities pB ≤ cpA and Bq ≥ dAq, and also by the equations p(B-cA)q = 0 and p(B-dA)q = 0. The first inequality says that it is impossible to produce more revenue at the next time step (pB) than what was spent at the previous time step multiplied by the interest factor (cpA). The second inequality says that it is impossible for the economy to consume more at the next time step (dAq) than was produced at the previous time step (Bq).

The equation p(B-cA)q = 0 says that if a process is unprofitable, it won’t be used and its intensity will therefore be zero. To see this, note that p(B-cA)q ≤ 0 for any intensity vector q. If p(B-cA)q < 0 then the economy will produce profits at a rate lower than the average rate for the economy, so by assumption we must have q = 0, which implies that p(B-cA)q = 0, a contradiction. Similarly, the equation p(B-dA)q = 0 says that if the economy consumes less of a commodity than was produced at the previous time step, the commodity becomes a free good and its price drops to zero. To see this, note that p(B-dA)q ≤ 0 for any price vector p. If p(B-dA)q < 0 then the economy consumes less than was produced at the previous time step, so by assumption we must have p = 0, which implies that p(B-dA)q = 0, a contradiction.

The inequalities pB ≤ cpA and Bq ≥ dAq can be restated as c = max_j{(pB)_j/(pA)_j} and d = min_i{(Bq)_i/(Aq)_i}, which shows that p uniquely determines c and q uniquely determines d. Von Neumann demonstrated that if there exist p and q satisfying the above equalities and equations, then c and d are both equal to pBq/pAq. In order to demonstrate that such solutions exist, von Neumann in addition assumed that A+B > 0, which implies that every process requires as an input or produces as an output some positive amount of every commodity. In a footnote to his paper, von Neumann also pointed out a link between his model and the theory of two-player zero-sum games, a theory that von Neumann himself was instrumental in developing. Let us now explore this link in a bit more detail.

Fix a number e > 0 and let C = B-eA. We can interpret C as a payoff matrix for zero-sum game between a minimizing player, who chooses a price vector p ≥ 0, and a maximizing player, who chooses an intensity vector q ≥ 0. The game determined by the matrix C is said to have a solution (p*,q*,v*) if (p*C)_j ≤ v* for each j and (Cq*)_i ≥ v* for each i; the number v* is then referred to as the value of the game. Von Neumann’s famous 1928 minimax theorem states that v* = min max pCq = max min pCq, where the minimum is taken over the price vector p and the maximum is taken over the intensity vector q. Moreover, if (p,q,c,d) is a solution to the problem defined above, then c = d, and if we set e = c = d then v* = 0, which implies that the game is fair.

In a Marxist interpretation, the ‘minimizing player’ may be identified with the working class and the ‘maximizing player’ with the ruling class. Of course, Marxists would argue that the game of capitalism is anything but fair, and is instead rigged in favour of the ruling class. This suggests that von Neumann’s model needs to be modified in order to fully capture the class relations of capitalism. I will leave this modification for a future blog post.