Arbitrage Theory

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.

Click here to download the reprint from the SIAM Review (Society for Industrial and Applied Mathematics).