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Optimal control and the Fibonacci sequence

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Abstract

We bridge mathematical number theory with that of optimal control and show that a generalised Fibonacci sequence enters the control function of finite horizon dynamic optimisation problems with one state and one control variable. In particular, we show that the recursive expression describing the first-order approximation of the control function can be written in terms of a generalised Fibonacci sequence when restricting the final state to equal the steady state of the system. Further, by deriving the solution to this sequence, we are able to write the first-order approximation of optimal control explicitly. Our procedure is illustrated in an example often referred to as the Brock-Mirman economic growth model.

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  • Thomas von Brasch & Johan Byström & Lars Petter Lystad, 2012. "Optimal control and the Fibonacci sequence," Discussion Papers 674, Statistics Norway, Research Department.
  • Handle: RePEc:ssb:dispap:674
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    File URL: http://www.ssb.no/a/publikasjoner/pdf/DP/dp674.pdf
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    1. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, January.
    2. Magill, Michael J. P., 1977. "A local analysis of N-sector capital accumulation under uncertainty," Journal of Economic Theory, Elsevier, vol. 15(1), pages 211-219, June.
    3. Magill, Michael J. P., 1977. "Some new results on the local stability of the process of capital accumulation," Journal of Economic Theory, Elsevier, vol. 15(1), pages 174-210, June.
    4. Levine, Paul & Pearlman, Joseph & Pierse, Richard, 2008. "Linear-quadratic approximation, external habit and targeting rules," Journal of Economic Dynamics and Control, Elsevier, vol. 32(10), pages 3315-3349, October.
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    Keywords

    Fibonacci sequence; Golden ratio; Mathematical number theory; Optimal control.;

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling

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