The Index-Theorem

Abstract

The index theorem, introduced to economics by Dierker [1972], allows to answer the question of uniqueness, which is a property of the excess demand considered globally, from the local structure of the excess demand around its potential equilibria. To get the idea of the index theorem consider the following graph of an excess demand for some commodity. Suppose there are just two commodities so that a zero excess demand of the commodity considered in the graph is by Walras Law already a competitive equilibrium. Now suppose furthermore that the graph is continuous and that it satisfies the boundary behaviour. Then, as has been proved in the previous section an equilibrium needs to exist. Note that in this case this conclusion can already be drawn from the intermediate value theorem. The astonishing feature of the Figure 4.1 is that we also get some idea on the number of equilibria. First of all, as Figure 4.1 shows, there may be a continuum of equilibria when the graph of the excess demand is flat at zero. This however is not a generic situation. On perturbing the characteristics of the economy one finds that the graph is transversal to the zero line. Hence generically equilibria are locally unique and since the set of equilibria is compact generically there is a finite number of those. Moreover suppose you do not know the graph globally but you only know how the graph cuts through the zero line. Then knowing that it will always cut from above shows that there is a unique equilibrium.