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Approximate Versus Exact Equilibria

  • Felix Kubler
  • Karl Schmedders

This paper develops theoretical foundations for an error analysis of approximate equilibria in dynamic stochastic general equilibrium models with heterogeneous agents and incomplete financial markets. While there are several algorithms which compute prices and allocations for which agents' first order conditions are approximately satisfied (approximate equilibria), there are few results on how to interpret the errors in these candidate solutions and how to relate the computed allocations and prices to exact equilibrium allocations and prices. We give a simple example which illustrates that approximate equilibria might be very far from exact equilibria. We then interpret approximate equilibria as equilibria for close-by economies, that is, for economies with close-by individual endowments and preferences. We provide sufficient conditions which ensure that approximate equilibria are close to exact equilibria of close-by economies. We give a detailed discussion of the error analysis for two models which are commonly used in applications, and OLG model with stochastic production and an asset pricing model with infinitively lived agents. We illustrate the analysis with some numerical examples. In these examples, the derived bounds are at most one order of magnitude larger than maximal errors in Euler equations.

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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 1382.

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Date of creation: Dec 2003
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Handle: RePEc:nwu:cmsems:1382
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  1. Heaton, John & Lucas, Deborah J, 1996. "Evaluating the Effects of Incomplete Markets on Risk Sharing and Asset Pricing," Journal of Political Economy, University of Chicago Press, vol. 104(3), pages 443-87, June.
  2. Christopher A. Sims, 1989. "Solving nonlinear stochastic optimization and equilibrium problems backwards," Discussion Paper / Institute for Empirical Macroeconomics 15, Federal Reserve Bank of Minneapolis.
  3. Mailath, George J. & Postlewaite, Andrew & Samuelson, Larry, 2005. "Contemporaneous perfect epsilon-equilibria," Games and Economic Behavior, Elsevier, vol. 53(1), pages 126-140, October.
  4. Herbert E. Scarf, 1967. "On the Computation of Equilibrium Prices," Cowles Foundation Discussion Papers 232, Cowles Foundation for Research in Economics, Yale University.
  5. Ingram, Beth Fisher, 1990. "Equilibrium Modeling of Asset Prices: Rationality versus Rules of Thumb," Journal of Business & Economic Statistics, American Statistical Association, vol. 8(1), pages 115-25, January.
  6. Duffie, Darrell, et al, 1994. "Stationary Markov Equilibria," Econometrica, Econometric Society, vol. 62(4), pages 745-81, July.
  7. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-45, November.
  8. Manuel S. Santos, 2000. "Accuracy of Numerical Solutions using the Euler Equation Residuals," Econometrica, Econometric Society, vol. 68(6), pages 1377-1402, November.
  9. Postlewaite, Andrew & Schmeidler, David, 1981. "Approximate Walrasian Equilibria and Nearby Economies," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 22(1), pages 105-11, February.
  10. Hellwig, Martin F., 1982. "Rational expectations and the Markov property of temporary equilibrium processes," Journal of Mathematical Economics, Elsevier, vol. 9(1-2), pages 135-144, January.
  11. Rios-Rull, Jose-Victor, 1996. "Life-Cycle Economies and Aggregate Fluctuations," Review of Economic Studies, Wiley Blackwell, vol. 63(3), pages 465-89, July.
  12. Felix Kubler & Herakles Polemarchakis, 2004. "Stationary Markov equilibria for overlapping generations," Economic Theory, Springer, vol. 24(3), pages 623-643, October.
  13. Kam-Chau Wong & Marcel K. Richter, 1999. "Non-computability of competitive equilibrium," Economic Theory, Springer, vol. 14(1), pages 1-27.
  14. Wouter J. den Haan & Albert Marcet, 1993. "Accuracy in simulations," Economics Working Papers 42, Department of Economics and Business, Universitat Pompeu Fabra.
  15. Felix Kubler & Karl Schmedders, 2003. "Stationary Equilibria in Asset-Pricing Models with Incomplete Markets and Collateral," Econometrica, Econometric Society, vol. 71(6), pages 1767-1793, November.
  16. Manuel S. Santos & Jesus Vigo-Aguiar, 1998. "Analysis of a Numerical Dynamic Programming Algorithm Applied to Economic Models," Econometrica, Econometric Society, vol. 66(2), pages 409-426, March.
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