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The Aggregate Excess Demand Function and Other Aggregation Procedures

  • Donald G. Saari
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    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.

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    File URL: http://www.kellogg.northwestern.edu/research/math/papers/908.pdf
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    Paper provided by Northwestern University, Center for Mathematical Studies in Economics and Management Science in its series Discussion Papers with number 908.

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    Date of creation: Oct 1990
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    Handle: RePEc:nwu:cmsems:908
    Contact details of provider: Postal: Center for Mathematical Studies in Economics and Management Science, Northwestern University, 580 Jacobs Center, 2001 Sheridan Road, Evanston, IL 60208-2014
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    1. Balasko, Yves, 1978. "The Transfer Problem and the Theory of Regular Economies," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 19(3), pages 687-94, October.
    2. Saari, Donald G & Simon, Carl P, 1978. "Effective Price Mechanisms," Econometrica, Econometric Society, vol. 46(5), pages 1097-1125, September.
    3. Saari, Donald G. & Williams, Steven R., 1986. "On the local convergence of economic mechanisms," Journal of Economic Theory, Elsevier, vol. 40(1), pages 152-167, October.
    4. Herbert E. Scarf, 1959. "Some Examples of Global Instability of the Competitive Equilibrium," Cowles Foundation Discussion Papers 79, Cowles Foundation for Research in Economics, Yale University.
    5. Mantel, Rolf R., 1974. "On the characterization of aggregate excess demand," Journal of Economic Theory, Elsevier, vol. 7(3), pages 348-353, March.
    6. Mas-Colell, Andreu, 1977. "On the equilibrium price set of an exchange economy," Journal of Mathematical Economics, Elsevier, vol. 4(2), pages 117-126, August.
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