Optimal Control and Equilibrium Behavior of Production-Inventory Systems

The relationship between commodity inventory and short-term price variations has received considerable attention, but the understanding has been limited to single-stage cross-sectional relation. In this paper, we aim to deepen our understanding of the inventory-price relationship in two dimensions: across time and across production stages. We first examine an individual firm controlling production and two stages of inventory under uncertain input and output prices and operating costs. We next establish and characterize the rational expectations equilibrium for an economy in which competitive production firms link a raw material market and a finished goods market, with uncertain and price-sensitive supply and demand. We characterize the dynamics of inventory, market price, and gross margin based on theoretical analysis, simulation, and empirical evidence from the petroleum industry. We find that inventory fluctuations lag behind price variations, and the length of the lags depend on how far the inventory is from the source of the supply or demand shocks. We also find that shocks are both dampened and delayed when propagating through the production stages, and that shocks have a prolonged effect on inventories and prices at both stages.