Finding the Markets in the Math

The paper mathematically develops the heuristic idea that the first-order necessary conditions for a classical constrained optimization problem are equivalent to a market being arbitrage-free – with the Lagrange multipliers being the arbitrage-free market prices. The arbitrage notions start with the multiplicative Kirchhoff’s Voltage Law and then generalize to matrix algebra. The basic result shows the normalized arbitrage-free Ā«market pricesĀ» (the Lagrange multipliers) resulting from a classical constrained optimization problem can always be obtained as a ratio of cofactors. The machinery also gives an economic interpretation of Cramer’s Rule as a competitive equilibrium condition. Thus one can “find the markets in the math” of a classical constrained optimization problem.

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