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Aggregation and Market Demand: An Exterior Differential Calculus Viewpoint

  • P. A. Chiappori
  • I. Ekeland

We analyze under which conditions a given vector field can be disaggregated as a linear combination of gradients. This problem is typical of aggregation theory, as illustrated by the literature on the characterization of aggregate market demand and excess demand. We argue that exterior differential calculus provides very useful tools to address these problems. In particular, we show, using these techniques, that any analytic mapping in [open letter R][superscript n] satisfying Walras Law can be locally decomposed as the sum of n individual, utility-maximizing market demand functions. In addition, we show that the result holds for arbitrary (price-dependent) income distributions, and that the decomposition can be chosen such that it varies continuously with the mapping. Finally, when income distribution can be freely chosen, then decomposition requires only n/2 agents.

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Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 67 (1999)
Issue (Month): 6 (November)
Pages: 1435-1458

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Handle: RePEc:ecm:emetrp:v:67:y:1999:i:6:p:1435-1458
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  1. Martin Browning & Pierre-Andre Chiappori, 1994. "Efficient Intra-Household Allocations: a General Characterization and Empirical Tests," Department of Economics Working Papers 1994-02, McMaster University.
  2. Brown, Donald J & Matzkin, Rosa L, 1996. "Testable Restrictions on the Equilibrium Manifold," Econometrica, Econometric Society, vol. 64(6), pages 1249-62, November.
  3. Debreu, Gerard, 1974. "Excess demand functions," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 15-21, March.
  4. Russell, Thomas & Farris, Frank, 1993. "The geometric structure of some systems of demand equations," Journal of Mathematical Economics, Elsevier, vol. 22(4), pages 309-325.
  5. Hugo Sonnenschein, 1973. "The Utility Hypothesis and Market Demand Theory," Discussion Papers 51, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  6. McFadden, Daniel & Mas-Colell, Andreu & Mantel, Rolf & Richter, Marcel K., 1974. "A characterization of community excess demand functions," Journal of Economic Theory, Elsevier, vol. 9(4), pages 361-374, December.
  7. Mantel, Rolf R., 1976. "Homothetic preferences and community excess demand functions," Journal of Economic Theory, Elsevier, vol. 12(2), pages 197-201, April.
  8. Geanakoplos, J D & Polemarchakis, H M, 1980. "On the Disaggregation of Excess Demand Functions," Econometrica, Econometric Society, vol. 48(2), pages 315-31, March.
  9. Sonnenschein, Hugo, 1973. "Do Walras' identity and continuity characterize the class of community excess demand functions?," Journal of Economic Theory, Elsevier, vol. 6(4), pages 345-354, August.
  10. Diewert, W. E., 1977. "Generalized slutsky conditions for aggregate consumer demand functions," Journal of Economic Theory, Elsevier, vol. 15(2), pages 353-362, August.
  11. Mantel, Rolf R., 1974. "On the characterization of aggregate excess demand," Journal of Economic Theory, Elsevier, vol. 7(3), pages 348-353, March.
  12. Andreu, Jordi, 1982. "Rationalization of market demand on finite domains," Journal of Economic Theory, Elsevier, vol. 28(1), pages 201-204, October.
  13. Bottazzi, Jean-Marc & Hens, Thorsten, 1996. "Excess Demand Functions and Incomplete Markets," Journal of Economic Theory, Elsevier, vol. 68(1), pages 49-63, January.
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