Direct and Indirect Trade Utility Functions

Abstract

In this paper the concept of an indirect trade utility function is introduced, its properties are developed, and its application to the theory of international trade and to the econometric estimation of export supply and import demand functions are discussed. The indirect trade utility function expresses the maximum level of utility a trading nation can attain, assuming the existence of a direct community utility function, as a function of a vector of prices for commodities, a vector of factor endowments and the balance of trade. As such it provides a summary of all the consumption and production decisions within a competitive economy. In Section 2 the indirect trade utility function is defined in terms of the ordinary indirect utility function and the gross national product (or variable profit) function. An extension of Roy's Identity is developed, allowing the easy generation of net export functions by differentiation. The fact that the derivatives of the indirect trade utility function with respect to the prices of commodities are proportional to the excess supply or net export functions implies that indirect utility is stationary at a closed equilibrium. It is then proved that the set of price vectors which minimize the indirect trade utility function coincides with the set of closed equilibrium price vectors. Section 3 specifies the properties of the indirect trade utility function implied by the model and then illustrates various applications in the theory of international trade. These include the welfare effects of divergence of prices from autarky levels and the characterization of optimal tariff structures. Section 4 briefly considers the extent to which the results of Section 2 can be extended to the case where there are many consumers. In Section 5 the relationship between the indirect trade utility function and the direct trade utility function of Meade, and between the latter and the direct utility function and the production possibility set are outlined. Section 6 is devoted to establishing rigorously the formal relationships between all these functions under quite general conditions. The results of this section serve to widen the application of duality theory, as surveyed by Diewert (1974), (1978b) for example, to the area of international trade theory. Some of the results on the direct trade utility function have previously been developed by Chipman (1970) in unpublished notes, but he did not consider the indirect trade utility function. In Section 7 the usefulness of the indirect trade utility function in generating functional forms for export and import demand functions and factor price functions suitable for the purposes of econometric estimation using data on prices, net exports and factor endowments is discussed. Section 8 concludes the paper with an indication of how the model may be reinterpreted to accommodate nontraded goods and variable factor supplies. While the discussion is focussed on a national economy trading with other nations in the world market, it should be clear that the model applies equally well to the individual consumer who undertakes marketable production. Examples include the self-employed businessman or farmer. The indirect trade utility function then indicates the maximum utility attainable given the prices of the commodities (marketable products and purchasable