Pareto optima and exchange rates under risk neutrality: A note

In this note, we present a wealth model of a two-country economy where ffnancial assets and goods are traded. We consider the case where the agents are risk neutral, a very common assumption in ffnance in order to have explicit solutions for prices, and in particular in international ffnance for exchange rates using the Pareto optima. We show that the Pareto optima on the international good and capital markets are found to coincide with the net trade allocations. More notably, under a no- arbitrage condition on the international capital market, we can deffne an exchange rates system for which Purchasing Power Parity (PPP) holds. And if any Pareto optimum for the international capital market and its associated commodity prices clear the trade balance then the Uncovered Interest Rate Parity (UIRP) for the international capital market holds with this exchange rates system. When the international ffnancial market are complete, this condition is also sufficient for the trade balance with any Pareto allocation and its associated commodity prices. In this case, PPP on the international good market and UIRP for the international capital market are equivalent conditions. We show through an example of risk-neutral economy where a no-arbitrage condition for assets and PPP hold, but UIRP fails, that the only individually rational Pareto allocation clearing the international good market is no-trade. Finally, we recover Dumas [3] under risk aversion in a simpliffed two-country economy with a single ffnancial asset and no exchange of commodities and we prove that the only possible equilibrium (with transfers) is no-trade.