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The Discrete Nerlove-Arrow Model: Explicit Solutions

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  • Patrice CASSAGNARD
  • Marc ARTZROUNI

Abstract

We use the optimality principle of dynamic programming to formulate a discrete version of the Nerlove-Arrow maximization problem. When the payoff function is concave we derive an explicit solution to the problem. If the time horizon is long enough there is a "transiently stationary" (turnpike) value for the optimal capital after which the capital must decay as the end of the time horizon approaches. If the time horizon is short the capital is left to decay after a first-period increase or decrease depending on the capital's initial value. Results are illustrated with the payoff function µK where K is the capital and 0 0. With this function, the solution is in closed form.

Suggested Citation

  • Patrice CASSAGNARD & Marc ARTZROUNI, 2010. "The Discrete Nerlove-Arrow Model: Explicit Solutions," Working Papers 2010-2011_3, CATT - UPPA - Université de Pau et des Pays de l'Adour, revised Nov 2010.
  • Handle: RePEc:tac:wpaper:2010-2011_3
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    File URL: http://gtl.univ-pau.fr/travaux/83F_2010_2011_3DocWcattDiscreteNerloveArrowModelMArtzrouniPCassagnard.pdf
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    References listed on IDEAS

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    1. Ulrich Doraszelski & Sarit Markovich, 2007. "Advertising dynamics and competitive advantage," RAND Journal of Economics, RAND Corporation, vol. 38(3), pages 557-592, September.
    2. Gustav Feichtinger & Richard F. Hartl & Suresh P. Sethi, 1994. "Dynamic Optimal Control Models in Advertising: Recent Developments," Management Science, INFORMS, vol. 40(2), pages 195-226, February.
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    More about this item

    Keywords

    Nerlove-Arrow; dynamic programming; optimization; turnpike;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • O21 - Economic Development, Innovation, Technological Change, and Growth - - Development Planning and Policy - - - Planning Models; Planning Policy

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