Knowledge that Transforms

The mathematical isomorphism between the multi-commodity price system and the Metzleric multi-region income system is not complete even in the wellbehaved case of gross substitutes (for the stable price system) and superior goods (for the stable income system). A difference arises because of the price system's characteristic of zero-degree homogeneity which imposes additional constraints upon the inverse. The inverse of the price system has diagonal elements dominating single off-diagonal elements in both the same row and the same column, while the inverse of the income system has diagonal elements which are necessarily dominant only over single elements in the same column. This leads to a whole class of comparative statics propositions for the price system which have no counterpart in the income system and to economic implications about our ability to aggregate sectors or markets for qualitative analysis. 1. THE LAWS OF COMPARATIVE STATICS AND HOMOGENEITY IT IS WELL KNOWN that there are formal structural similarities between economic models which encompass quite different subject matters. We cite, for example, the similarities between the multiple-commodity price system and the Metzleric multiple-region income system.' These similarities produce analogies between the laws of stability and comparative statics of the two systems which further our understanding of the basic unity of economic science. This unity derives from an underlying mathematical structure which is similar (but not identical). Consider a system in which all goods are gross substitutes, and a Metzleric system in which all income effects (including hoarding) are positive: then both systems are stable.2 Now consider three types of changes in (excess) demand and note the laws of comparative statics (proved in the final section) pertaining to each system. In the system it can be demonstrated3 that: 1 For a description of the Walrasian system see, for example, Hicks [6, Ch. V], Samuelson