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

E. Presman and I. Sonin
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E. Presman: Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow, Russia
I. Sonin: Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow, Russia

Journal of the New Economic Association, 2023, vol. 59, issue 2, 12-34

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.

Keywords: inventory model; Markov chain; optimality equation (search for similar items in EconPapers)
JEL-codes: C61 D25 D81 (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:nea:journl:y:2023:i:59:p:12-34

DOI: 10.31737/22212264_2023_2_12-34

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