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The effect of inventories on the level and volatility of price is analyzed under alternative market structures. In a fairly general setting, it is shown that inventories have no effect on the average level of prices in a Cournot duopoly in the absence of depreciation. The effect of storage on price volatility is analyzed in a more restrictive, linear-quadratic setting in which it is found that imperfectly competitive producers make less use of inventories for smoothing random fluctuations than is efficient. The effect of market structure on price volatility depends on whether demand shocks or cost shocks are more important.

The purpose of this paper is to examine the use of inventories by imperfectly competitive producers in order to understand the effects of storage on both the level and the variability of price in oligopolistic markets. Imperfectly competitive firms will recognize the effects of their storage decisions on future outcomes, and so may alter their behavior from that of perfectly competitive firms. A particular concern in the analysis of inventories is the extent to which they smooth price fluctuations,1 so it is important to understand how imperfect competition affects the use of inventories for this purpose in addition to how it affects the price level. The complications caused by imperfect competition are easy to see. If production costs are convex, by reducing the amount of production required to meet a given level of sales, firms can reduce the marginal cost of sales in a given period by drawing down inventories. This is the basic idea behind models that examine the production smoothing usage of inventories in competitive markets.2 When producers are imperfectly competitive, these inventories will potentially have strategic value due to this ability to reduce the marginal cost of sales in future periods. In addition to any effect this strategic use of inventories may have on the price level, it is also possible that this consideration will also alter how firms respond to random disturbances, which will affect the use of inventories for price smoothing. The effect of storage on the equilibrium of a duopoly market has been examined in the context of a two period game in which duopolists can carry inventories into the second period. Allaz (1991) shows that the possibility of storage results in lower prices than is the case if storage were not possible. Producers wish to have lower costs than their rival in the second period and in the attempt to gain such an advantage invest in inventories which result in higher second period sales and lower price. Another paper that examines this issue is by Arvan (1985) who demonstrates that the equilibrium may be asymmetric when storage costs are linear, although he does not show what the effect on price is. Mitraille (2004) takes the analysis of Arvan further. He allows demand to vary over the two periods and shows that, for constant marginal production cost, the use of inventories will result in a lower second period price when demand is falling and have no effect when demand is rising. As these papers illustrate, a crucial determinant of the equilibrium in the two period game is how excess inventories are treated in the second period. Allaz assumes that they must be sold, while Arvan and Mitraille allow unsold inventories with no penalty. I focus on an infinite horizon in this paper in order to remove the effects of any assumption that must be made regarding inventories in the terminal period. Thille (2003) extends the analysis of strategic forward trading and storage to an infinite horizon. He shows that the strategic use forward trading is lessened by the possibility of storage. In the model of Thille (2003) there is no uncertainty, so it has no implications for price volatility. One result of that paper was to demonstrate that for a game with storage but without forward trading, the price level was the same as in a Cournot duopoly. The conditions under which this result is more generally true are examined in this paper. There are a couple of other papers that have examined storage in a long horizon duopoly. Kirman and Sobel (1974) analyze an infinite horizon, stochastic dynamic game in which firms simultaneously choose production and price at the beginning of a period. After these decisions are made, the stochastic quantity demanded is realized and the difference between demand and production is subtracted from or added to inventory. The timing of their game means that prices do not respond to the random demand shocks – market clearing is effected entirely by inventory movements. As such, the Kirman and Sobel model cannot be used to examine the implications of strategic storage for price volatility. Judd (1996) examines a long-horizon linear-quadratic game in which two firms compete by choosing both price and quantities, with inventory making up for any difference between supply and demand. His purpose is to allow for both price and quantity choice in a natural way in order to see whether the equilibrium will end up closer to the Bertrand or to the Cournot equilibrium. In order to answer this question, he solves the model numerically for parameter values that ensure inventory holdings are near zero, in order to abstract from any strategic use of inventory. Conversely, in this paper I am interested in allowing for the possibility that inventories are used strategically. Much of the other work in the industrial organization literature that considers inventories thinks of them as a device for providing commitment. Rotemberg and Saloner (1989) consider the usefulness of inventories to enforce collusive behavior in a repeated two-stage game. By holding stocks, firms relax their short-term capacity constraints and are able to increase sales by selling inventories to react to a deviation to the collusive equilibrium. Inventories thus represent a type of excess capacity that may be less costly than building more production capacity. Ware (1985) considers the use of inventories by an incumbent monopolist to deter entry. The stockpiling of inventory by the incumbent firm is a credible commitment to higher sales following entry. Saloner (1986) allows storage in a two-stage game in which production occurs prior to sales. Inventories reduce the commitment value of prior production and the extent to which they do so depends on the amount by which they depreciate. In the final stage, inventories are sold at an exogenous price, so there is no analysis of the holding of inventories for strategic purposes. Finally, inventories are implicit in the model of Saloner (1987) in which there are two production periods prior to the sales game. Production in the first period is committed to be sold in the following period. In each of these models, the two-stage nature of the game enhances the commitment value of the inventories. There is a significant body of work that examines imperfectly competitive storage (but not production) in commodity markets, for which price volatility is a particular concern. Newbery (1984) compares storage in a perfectly competitive market with that of a monopolist or a dominant firm. The focus is on the incentive to store to arbitrage price fluctuations that are caused by random harvests. The monopolist stores more than do firms in a perfectly competitive industry, resulting in less volatile prices when demand is linear. The results are reversed for constant elasticity demand. Other papers in this vein are Williams and Wright (1991), McLaren (1999), Rui and Miranda (1996), and Vedenov and Miranda (2001). A common feature of these models that differentiates them from the current paper is that the market power being analyzed is over the storage technology alone, not over production and storage. Production is in most cases simply modeled as a random harvest. The question being examined is the extent to which market structure affects the operation of arbitrage or speculation. Williams and Wright (1991) do discuss “monopoly over distribution and storage”, but do not analyze price volatility in this context. Examining market power over storage alone is somewhat limiting since it is likely harder to justify barriers to entry in storage than in production. Another difference between this literature and the present paper is the type of uncertainty examined. The uncertainty in the above models comes from uncertain production. In contrast, I allow for uncertainty in the form of both demand and cost shocks. Finally, there is a body of work that analyses price volatility under imperfect competition in the absence of inventories. Carlton (1986), Domberger and Fiebig (1993), and Slade (1991) provide empirical evidence that prices tend to be less volatile in more concentrated industries. Rotemberg and Saloner (1987) provide a theoretical model to show that a monopoly has less incentive to change price than a duopoly does when there are fixed costs to doing so. This results in less frequent price changes by the monopolist. In this paper I analyze the equilibrium of a model in which a duopoly produces a storable product and competes over an infinite time-horizon. I use a dynamic game in which firms choose levels of sales, production and inventory investment in each period to maximize the discounted sum of profits in the face of uncertain demand and costs. I compare the feedback Nash equilibrium of this game to that of a game in which storage is not possible in order to gauge the effects of storage on the price level. I then analyze a linear-quadratic version of the model comparing the non-cooperative equilibrium to the collusive and efficient outcomes in order to gauge the effects of market structure on price volatility for a storable good.

Two results regarding the use of inventories by imperfectly competitive producers have been demonstrated in this paper. First, in the absence of depreciation, sales differ from the perishable good case only when inventories are changed. The level of inventories does not affect the level of sales, and hence price. This result is in contrast to simpler, two-period models and is found to hold for fairly general demand and cost functions. Second, in a simple linear-quadratic setting, it is shown that imperfectly competitive producers under-utilize inventories. In the case where the source of uncertainty is primarily demand fluctuations, the under-utilization of inventories results in a less than efficient degree of price smoothing. Furthermore, as storage costs become linear, price volatility under imperfectly competitive market structure remains high, whereas under perfect competition it approaches zero. A conclusion that can be made is that there is no simple relationship between price volatility and market structure even in a linear-quadratic setting: less competitive market structures have relatively low price variance when uncertainty is primarily due to uncertain cost and relatively high price variance when uncertainty is primarily due to uncertain demand. The ambiguous effects of market structure on price volatility occur even though there is a fairly unambiguous ranking of storage by market structure. Total inventories per unit of production are lowest under perfect competition in all scenarios considered in this paper. This implies that there is no simple relationship between the amount of inventories carried and the volatility of prices. What matters more is the willingness to use the inventories in response to random events. In the case of a duopoly, more inventories are held than under perfect competition, but producers are less willing to use them.