Fundamentals of Production Theory in International Trade. A Modern Approach Based on Theory of Duality

In terms of convex analysis the revenue function of a country with a given factor endowment may be seen as the support function of the production possibility set. At the same time this revenue function is the so called convex-conjugate of the indicator function of the production possibility set. The task of this paper is to apply duality results of this kind to sums of functions, where Rockafellar (1972) has shown that the operations of addition and the infimal convolution of convex functions are dual to each other. To be more concrete, we refer to the theory of international trade, where the factor endowment of each country is given and the factors of production are internationally immobile. If the country specific outputs sum up to a world production, what is the meaning of the appropriate dual problem? The answer will deal with the national and world-wide problem of revenue maximization. Moreover, we shall discuss the properties of an optimal commodity price vector in relation to the "dual" world output and to the national commodity supply. Similar problems will be analyzed on a national level, where the factors of production can easily be moved from one firm or sector to another. The second part of the paper draws the attention to the inverse production technology, namely the input correspondence. Here, the results based on convex revenue functions are applied to concave cost functions.