The Aggregate Excess Demand Function and Other Aggregation Procedures

Two theorems are given; the first extends the Sonnenschein-Mantel-Debreu theorem characterizing aggregate demand functions from the set of n (greater than or equal to) 2 commodities to all 2 (superscipt n) minus (n+1) subsets of two or more commodities. The second theorem concerns spatial voting models for k (greater than or equal to) 2 candidates over a space of n issues. The relationships among the sincere election rankings of the candidates for all of the sets of 2 (superscript n) minus1 issues are given. Both theorems have the same kind of conclusion; anything can happen. By showing the mathematical reasons for theses results and by recalling some recent results from statistics, voting, and economics, it is argued that this "anything can happen" conclusion is the type one must anticipate from aggregation procedures; particularly processes of the type commonly used in economic models where the procedure is responsive to changes in agents' preferences, changes in data, etc.