# Finding the Markets in the Math: Arbitrage and Optimization Theory

This is Chapter 10 from my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

One of the fundamental insights of mainstream neoclassical economics is the connection between competitive market prices and the Lagrange multipliers of optimization theory in mathematics.  Yet this insight has not been well developed.  In the standard theory of markets, competitive prices result from the equilibrium of supply and demand schedules.  But in a constrained optimization problem, there seems to be no mathematical version of supply and demand functions so that the Lagrange multipliers would be seen as equilibrium prices.  How can one “find the markets in the math” so that Lagrange multipliers will emerge as equilibrium market prices?

We argue that the solution to the “find the markets in the math” problem is to reconceptualize equilibrium as the absence of profitable arbitrage instead of the equating of supply and demand.  With each proposed solution to a classical constrained optimization problem, there is an associated market.  The maximand is one commodity, and each constraint provides another commodity on this market.  Given a marginal variation in one commodity, one can define the marginal change is any other given commodity so the market has a set of exchange rates between the commodities.  The usual necessary conditions for the proposed solution to solve the maximization problem are the same as the conditions for this mathematically defined “market” to be arbitrage-free.  The prices that emerge from the arbitrage-free system of exchange rates (normalized with the maximand as numeraire) are precisely the Lagrange multipliers.  We also show that the cofactors of a matrix describing the marginal variations can be taken as the prices (before being normalized) so the Lagrange multipliers can always be presented as ratios of cofactors.  The results about cofactors also allow an economic interpretation of inverse matrices and of Cramer’s Rule.

The basic mathematical result, which dates back to Augustin Cournot in 1838, is that:

there exists a system of prices for the commodities such that the given exchange rates are the price ratios if and only if the exchange rates are arbitrage-free (in the sense that they multiply to one around any circle).

This simple graph-theoretic theorem is known in its additive version as Kirchhoff’s Voltage Law:

there exists a system of potentials at the nodes of a circuit such that the voltages on the wires between the nodes are the potential differences if and only if the voltages sum to zero around any cycle.

Kirchhoff’s work was published after Cournot in 1847, so the result might be called “the Cournot-Kirchhoff law.”

This Cournot-Kirchhoff law has many applications outside of electrical circuit theory and economics.  For instance, the second law of thermodynamics can be formulated as the impossibility of a certain form of “heat arbitrage” between temperature reservoirs, and the “prices” that emerge in this case are the Kelvin absolute temperatures of the reservoirs.  Yet another application of the arbitrage framework is to probability theory.  Profitable arbitrage in the market for contingent commodities is called “making book.”  A person’s subjective probability judgments satisfy the laws of probability if they are “coherent” in the sense of not allowing book to be made against the person.  Thus arbitrage on the market for contingent commodities enforces the laws of probability.

The arbitrage interpretation of the first-order necessary conditions for classical optimization problems suggests a research program to extend the arbitrage reasoning to other parts of optimization theory (e.g., nonlinear and linear programming, calculus of variations, and optimal control theory).  In the appendix, we sketch the interpretation of the sufficient conditions for a classical optimum in terms of “arbitrage operating to eliminate its own possibility.”