Graphical Exchange Mechanisms

Consider an exchange mechanism which accepts “diversified” offers of various commodities and redistributes everything it receives. We impose certain conditions of fairness and convenience on such a mechanism and show that it admits unique prices, which equalize the value of offers and returns for each individual. We next define the complexity of a mechanism in terms of certain integers τij , πij and ki that represent the time required to exchange i for j, the difficulty in determining the exchange ratio, and the dimension of the message space. We show that there are a finite number of minimally complex mechanisms, in each of which all trade is conducted through markets for commodity pairs. Finally we consider minimal mechanisms with smallest worst-case complexities τ = max τij and π = max πij . For m > 3 commodities, there are precisely three such mechanisms, one of which has a distinguished commodity – the money – that serves as the sole medium of exchange. As m → ∞ the money mechanism is the only one with bounded (π, τ). JEL Classification: C70, C72, C79, D44, D63, D82.