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The Harmonic Fisher Equation and the Inflationary Bias of Real Uncertainty




The classical Fisher equation asserts that in a nonstochastic economy, the inflation rate must equal the difference between the nominal and real interest rates. We extend this equation to a representative agent economy with real uncertainty in which the central bank sets the nominal rate of interest. The Fisher equation still holds, but with the rate of inflation replaced by the harmonic mean of the growth rate of money. Except for logarithmic utility, we show that on almost every path the long-run rate of inflation is strictly higher than it would be in the nonstochastic world obtained by replacing output with expected output in every period. If the central bank sets the nominal interest rate equal to the discount rate of the representative agent, then the long-run rate of inflation is positive (and the same) on almost every path. By contrast, the classical Fisher equation asserts that inflation should then be zero. In fact, no constant interest rate will stabilize prices, even if the economy is stationary with bounded i.d.d. shocks. The central bank must actively manage interest rates if it wants to keep prices bounded forever. However, not even an active central bank can keep prices exactly constant.

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  • Ioannis Karatzas & Martin Shubik & William D. Sudderth & John Geanakoplos, 2003. "The Harmonic Fisher Equation and the Inflationary Bias of Real Uncertainty," Cowles Foundation Discussion Papers 1424, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1424

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    References listed on IDEAS

    1. Chris Shannon & William R. Zame, 2002. "Quadratic Concavity and Determinacy of Equilibrium," Econometrica, Econometric Society, vol. 70(2), pages 631-662, March.
    2. Varian, Hal R, 1982. "The Nonparametric Approach to Demand Analysis," Econometrica, Econometric Society, vol. 50(4), pages 945-973, July.
    3. Donald J. Brown & Chris Shannon, 2000. "Uniqueness, Stability, and Comparative Statics in Rationalizable Walrasian Markets," Econometrica, Econometric Society, vol. 68(6), pages 1529-1540, November.
    4. Brown, Donald J & Matzkin, Rosa L, 1996. "Testable Restrictions on the Equilibrium Manifold," Econometrica, Econometric Society, vol. 64(6), pages 1249-1262, November.
    5. Shoven,John B. & Whalley,John, 1992. "Applying General Equilibrium," Cambridge Books, Cambridge University Press, number 9780521266550, May.
    6. Donald J. Brown & Caterina Calsamiglia, 2003. "The Strong Law of Demand," Cowles Foundation Discussion Papers 1399, Cowles Foundation for Research in Economics, Yale University.
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    Cited by:

    1. Orus, Juan & González, Manuel, 2004. "Inflation-Proof Credits and Financial Instruments. Making the Fisher Hypothesis a Reality," MPRA Paper 343, University Library of Munich, Germany.
    2. James R. Rhodes, 2006. "DEVOLUTION OF THE FISHER EQUATION: Rational Appreciation to Money Illusion," GRIPS Discussion Papers 07-05, National Graduate Institute for Policy Studies, revised Sep 2007.

    More about this item


    Inflation; equilibrium; Control; Interest rate; Central bank; Harmonic Fisher equation;

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • E41 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Demand for Money
    • E58 - Macroeconomics and Monetary Economics - - Monetary Policy, Central Banking, and the Supply of Money and Credit - - - Central Banks and Their Policies

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