Abstract
We study a new kind of nonzero-sum stochastic differential game with mixed impulse/switching controls, motivated by strategic competition in commodity markets. A representative upstream firm produces a commodity that is used by a representative downstream firm to produce a final consumption good. Both firms can influence the price of the commodity. By shutting down or increasing generation capacities, the upstream firm influences the price with impulses. By switching (or not) to a substitute, the downstream firm influences the drift of the commodity price process. We study the resulting impulse-regime switching game between the two firms, focusing on explicit threshold-type equilibria. Remarkably, this class of games naturally gives rise to multiple potential Nash equilibria, which we obtain thanks to a verification-based approach. We exhibit three candidate types of equilibria depending on the ultimate number of switches by the downstream firm (zero, one or an infinite number of switches). We illustrate the diversification effect provided by vertical integration in the specific case of the crude oil market. Our analysis shows that the diversification gains strongly depend on the pass-through from the crude price to the gasoline price.
Highlights
Since Hotelling’s [20] seminal study of commodity prices, considerable efforts have been undertaken to understand the dynamics of the equilibrium price of commodities and in particular, its long-run properties
We are able to provide closed-form description of the dynamic equilibrium, offering precise quantitative insights regarding the producer and consumer roles and their equilibrium behavior. To emphasize the latter point, beyond several synthetic examples that illustrate and visualize our model features, we present a detailed case study of the diversification effect provided by vertical integration in the crude oil market circa 2019, viewed as a competition between crude oil producers and oil refiners that convert crude into gasoline and other consumer goods
As one example of comparative statistics that are possible in our model, we investigate the impact of volatility parameter σ of X on the equilibrium profits and behavior of X ∗
Summary
Since Hotelling’s [20] seminal study of commodity prices, considerable efforts have been undertaken to understand the dynamics of the equilibrium price of commodities and in particular, its long-run properties. We are able to provide closed-form description of the dynamic equilibrium, offering precise quantitative insights regarding the producer and consumer roles and their equilibrium behavior To emphasize the latter point, beyond several synthetic examples that illustrate and visualize our model features, we present a detailed case study of the diversification effect provided by vertical integration in the crude oil market circa 2019, viewed as a competition between crude oil producers and oil refiners that convert crude into gasoline and other consumer goods. We fitted the oil market to the generic type of equilibrium, considering that oil consumption experiences alternate phases of expansion when the price is low and contraction when the price is too high In this setting, we consider a small downstream firm asking herself whether she has an interest in getting more vertically integrated. As well as additional comparative statics, are delegated to Sect. 7
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