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An inventory model where commodity prices depend on a continuous time Markov chain

Author

Listed:
  • Presman, E.

    (Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow, Russia
    School of Economics, Lomonosov Moscow State University, Moscow, Russia)

  • Sonin, I.

    (Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow, Russia
    University of North Carolina at Charlotte, US)

Abstract

We present an inventory model where a manufacturer (firm) uses for "production" a "commodity" (resource), which is consumed with unit intensity. The price of the commodity follows a stochastic process, modelled by a continuous time Markov chain with a finite number of states and known transition rates. The firm can buy this commodity at the current price or use "stored'' one. The storage cost is proportional to the storage level. The goal of the firm is to minimize the total discounted performance cost. We prove the existence of an optimal strategy, which is defined by a vector of levels specifying the minimal commodity amount to keep for a given price. This is so called the "threshold" strategy. We also describe all situations when such a strategy is not unique. We present an algorithm to find an optimal strategy and the corresponding value functions. In typical optimization problems in continuous time involving Bellman (optimality) equation, a smooth pasting of the first derivatives of the value functions is used. A special feature of our model is that, in contrast to such situations, we have to prove and to use the continuity of the second derivatives. The inventory model described in our paper may have broader interpretation when resources are replaced by assets, consumption by demand, and storage costs by opportunity costs or transaction costs.

Suggested Citation

  • Presman, E. & Sonin, I., 2023. "An inventory model where commodity prices depend on a continuous time Markov chain," Journal of the New Economic Association, New Economic Association, vol. 59(2), pages 12-34.
    • Handle: RePEc:nea:journl:y:2023:i:59:p:12-34
    DOI: 10.31737/22212264_2023_2_12-34
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    References listed on IDEAS

    as
    1. Dirk Beyer & Feng Cheng & Suresh P. Sethi & Michael Taksar, 2010. "Markovian Demand Inventory Models," International Series in Operations Research and Management Science, Springer, number 978-0-387-71604-6, March.
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    More about this item

    Keywords

    inventory model; Markov chain; optimality equation;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D25 - Microeconomics - - Production and Organizations - - - Intertemporal Firm Choice: Investment, Capacity, and Financing
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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