Some Mathematical Notes on the Quantity Theory of Money in an Open Economy

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THE RELATION between the quantity of money and the level of prices and incomes is generally treated by modern monetary theorists in terms of a "closed" economy, that is, without allowing for international trade as an integral part of the theory. In his "Monetary Analysis of Income Formation and Payments Problems,"1 J. J. Polak has shown how the existence of international trade, and the consequent possibility of balance of payments deficits, modifies in an essential way the theoretical analysis of the monetary system. The central element in his theory is that a given deficit in the balance of payments has cumulative effects on the monetary supply: it directly reduces the quantity of money until the consequent income and import changes are of sufficient size to correct the initial disequilibrium. The phenomenon is thus part of economic dynamics; that is, it relates the present state of certain economic variables to the rate of change of those and other variables. The use of mathematics seems to be unavoidable in analysis of this kind; indeed, the use of several alternative mathematical representations is occasionally of value in bringing out the specific role of the various relationships in the system. The first part of this paper reformulates Polak's system in terms of infinitesimal and continuous changes, thus replacing his difference equation by a differential equation. As a consequence, certain relations affecting liquidity that were implicit in his modelin the sense that they influenced movements within each period-have necessarily to be made explicit, so casting further light on the essential elements of the adjustment process. This generalization also allows another version of the model, of the kind used in the work of the Netherlands Bank, to be treated as a special case.