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The Duality of Dynamics: Real-Time Indeterminacy and Notional-Time Instability

Author

Listed:
  • Katsuyuki Shibayama

Abstract

Applying Arrow and Hurwicz fs (1958a) classical notional-time (NT) stability analysis to dynamic general equilibrium models, we ask whether local Hamiltonian gradients, often interpreted as prices and shadow prices, provide correct signals to restore equilibrium following a deviation. Under some assumptions, a gduality h emerges between reduced real-time (RT) dynamics and the corresponding NT adjustment, both involving only dynamic variables. Let m denote RT indeterminacy (stable roots minus state variables), and n the number of unstable roots in this reduced NT adjustment (n . 0). We show that m and n have the same parity. Thus, in this reduced system, odd RT indeterminacy (e.g., m = 1) implies that the specified local NT adjustment has at least one unstable direction. Under concavity, boundary, and regularity conditions, we also establish global NT convergence for finite-horizon planner problems.

Suggested Citation

  • Katsuyuki Shibayama, 2026. "The Duality of Dynamics: Real-Time Indeterminacy and Notional-Time Instability," Studies in Economics 2603, School of Economics, University of Kent.
    • Handle: RePEc:ukc:ukcedp:2603
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    References listed on IDEAS

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    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • E3 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles
    • E5 - Macroeconomics and Monetary Economics - - Monetary Policy, Central Banking, and the Supply of Money and Credit

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