Money as Minimal Complexity

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Discussion Paper

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We consider mechanisms that provide traders the opportunity to exchange commodity i for commodity j , for certain ordered pairs ij . Given any connected graph G of opportunities, we show that there is a unique mechanism M G that satisfies some natural conditions of “fairness” and “convenience.” Let M ( m ) denote the class of mechanisms M G obtained by varying G on the commodity set {1, …, m }. We define the complexity of a mechanism M in M (m) to be a pair of integers τ( M ), π( M ) which represent the “time” required to exchange i for j and the “information” needed to determine the exchange ratio (each in the worst case scenario, across all i not equal to i ≠ j ). This induces a quasiorder \preceq on M ( m ) by the rule M \preceq M ’ if τ( M ) ≤ τ( M ’) and π( M ) ≤ π( M ’). We show that, for m > 3, there are precisely three \preceq-minimal mechanisms M G in M ( m ), where G corresponds to the star, cycle and complete graphs. The star mechanism has a distinguished commodity — the money — that serves as the sole medium of exchange and mediates trade between decentralized markets for the other commodities. Our main result is that, for any weights λ, μ > 0; the star mechanism is the unique minimizer of λτ( M ) + μπ ( M ) on M ( m ) for large enough m .

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