Heckscher-Ohlin via Shephard Duality

Heckscher-Ohlin international-trade theory develops the theory of comparative advantage, by focussing on how factor endowments determine the direction of trade. A country having a high ratio of capital to labor endowment exports capital-intensive goods and imports labor-intensive goods. The natural modelling framework for the theory is a domestic-product primal and a domestic-income dual, in which the world price of outputs and the factor endowments are given, and one solves for the output quantities and the factor prices. Model production by Shephard's distance function. Rather than deriving the dual by adding and rearranging Fenchel inequalities, the Lagrangian sets the dual, by invoking the Shephard relationship between the unit isoquant and the unit isocost sets. The optimum value of the primal is domestic product, and the optimum value of the dual is domestic income. We analyze the special case of two sectors and two inputs, capital and labor. Fundamental arguments of Heckscher-Ohlin theory follow immediately. Compare two countries with identical technology but slightly different inputs. One country has a higher ratio of capital to labor than the other. This country will produce relatively more of the capital-intensive good and relatively less of the labor-intensive good, compared to the other country. Furthermore “factor-price equalization” applies: There is no economic incentive for capital or labor to flow between countries. The Stolper-Samuelson theorem asserts that imposing a tariff on the labor-intensive good raises the real wage. One models this idea by raising the output price of the labor-intensive good. For each sector, the capital/labor ratio rises. In each sector the marginal product of labor rises, so the real wage rises, measured in terms of either good. That there is no duality gap means that domestic product equals domestic income.