Price and Inventory Dynamics in an Oligopoly Industry: A Framework for Commodity Markets

1 Introduction

This paper analyzes the interaction between price and inventory decisions in an oligopoly industry and its implications for the dynamics of prices. The developed framework is able to explain important features of prices such as price dispersion observed in retail industries and autocorrelation found in commodity markets.

Cross-sectional price dispersion is a common feature in many retail markets. Since Stigler’s (1961) seminal work price dispersion has usually been explained by consumer search costs. In contrast, Aguirregabiria (1999) shows numerically that retail inventories and lump-sum ordering cost can generate (S,s) dynamics of inventories. Under an (S,s) rule inventory moves between the target inventory level, S, and the order threshold, s, with s<S. Whenever the firm’s inventory level falls below the order threshold, a new order is placed such that the target inventory level S is attained. These (S,s) order policies can explain time variability of prices of supermarket chains. However, as in the model of Aguirregabiria (1999) monopolistic competition is analyzed price dispersion between different firms cannot be observed. This paper contributes to the described work in several ways. We extend the analysis of prices and inventories with lump-sum costs to address the question how oligopolistic competition affects the dynamics. With imperfect competition dispersion of prices can actually result. Additionally, the main focus of the paper by Aguirregabiria (1999) is an empirical analysis building on a numeric simulation. Thus, the author does not aim at theoretically proving the optimality of the considered inventory decision. ^[1] Consequently, our paper is the first to formally prove the optimality of (S,s) policy with endogenous prices.

In addition to Aguirregabiria (1999) who analyze optimal inventory decisions with lump-sum costs under monopolistic competition, Hall and Rust (2007) study these dynamics under perfect competition. Their paper extends the framework of Aguirregabiria (1999) in some ways but is otherwise limited to one decision variable as prices are taken as given. Hall and Rust (2007) show that in their perfect competition model the (S,s) policy is an optimal order strategy. ^[2] To the best of our knowledge, these works studying extreme cases of competition are by far the most elaborated theoretical papers investigating these decision problems. ^[3]

This paper extends the literature by characterizing an equilibrium in a model of price and inventory competition in oligopoly. The paper is unique as it formally proves the optimality of (S,s) policy with endogenous prices. Additionally, the developed model does not only incorporate pricing decisions into the framework but also allows oligopolistic firms to interact strategically.

Besides the mentioned literature less closely related studies of oligopolistic competition exists. Dutta and Sundaram (1992) and Dutta and Rustichini (1995) analyze a discrete choice stochastic duopoly game with lump-sum costs. In these frameworks, the one abstract decision variable affecting both firms’ payoffs cannot be interpreted as being related to inventory. Nevertheless, the optimality of an (S,s) policy can also be shown. More recently, Besanko and Doraszelski (2004) study decisions about prices and capacity. However, the main and important difference between inventory and capacity is that excess capacity is worthless while keeping inventory affects future competition. Hence, additional strategic effects due to kept stock are at place in a framework like ours where inventory decisions are considered. This is especially important when investigating oligopolistic competition. The developed approach also contributes to the theory and the field of estimation in recent literature like Ryan (2012) where it is assumed that firms play (S,s) investment in capacity policies. ^[4] In contrast, we prove that (S,s) strategies arise in equilibrium and derive a functional form for these policies.

The distinct characteristics of our model that is incorporating inventory and oligopoly in dynamic competition provide the most plausible framework for retail industries. Retail industries have become highly concentrated, i.e., in most categories like grocery, supermarkets, and office supplies just a handful of rivals compete locally. In the supermarket industry for example a small number of firms capture the majority of sales as supermarkets compete in tight regional oligopolies. Thus, this industry is a prime example of oligopoly. Besides, inventory costs are of major importance. Supermarkets invest in state of the art distribution systems to minimize storage and transportation costs. Hence, deciding the optimal inventory and store offer forms an important optimization problem for supermarket chains (see e.g. Beresteanu et al. 2006; Ellickson 2007; Chavas 2008).

In this work, we study the decision problem of a central store, i.e., its decision about retail prices and orders to suppliers, facing oligopolistic competition and taking into account the existence of lump-sum ordering cost. We develop a model of retail competition in which the impact of inventories on competition and prices can be evaluated. We analyze the characteristics of the optimal decision rule. We find that key factors for price fluctuations are lump-sum ordering costs and demand uncertainty. Lump-sum ordering cost generate (S,s) inventory behavior. Demand uncertainty creates a positive probability of excess demand, i.e., stockouts. The positive stockout probability has a negative effect on expected sales, which in turn creates substitutability between prices and inventories in the profit function such that in equilibrium prices depend negatively on the level of inventories. This results in a cyclical pattern of inventories and prices where prices decline significantly when an order is placed and consequently inventory reduction generates price increase. The pricing behavior in this model can generate cross-sectional price dispersion with cyclical patterns even without menu costs. These results are supported by empirical studies (e.g. Lach and Tsiddon 1992, 1996; Slade 1998, 1999; Pesendorfer 2002; Ellis 2009) as well as numerical simulations (e.g. Copeland, Dunn, and Hall 2011).

The developed model provides a very promising alternative for studying commodity markets. Commodity prices are extremely volatile and papers of the respective literature strand are concerned whether theory is capable of explaining the actual behavior of prices. The more recent literature on this topic (see for example Deaton and Laroque 1992, 1996; Pindyck 1994) builds on the supply and demand tradition (see e.g. Ghosh et al. 1987, for a review), but with explicit modeling of the behavior of competitive speculators who hold inventories of commodities in the expectation of making profits. ^[5] However, perfect competition and the absence of lump-sum ordering cost is always assumed in these papers. The studies are trying to explain extremely volatile prices as a result of exogenous shocks by modeling the behavior of competitive speculators holding inventories. Results are rather unsatisfying: In contrast to the models’ predictions, real price fluctuations are not randomly distributed over time and this autocorrelation cannot be explained by these types of models. In addition, some probably important characteristics of commodity markets are not captured in this literature. Studies of these characteristics (e.g. Carter and MacLaren 1997; Slade and Thille 2006) find that commodity markets are best described by oligopoly instead of perfect competition. Besides, lump-sum ordering cost are realistic in some markets. ^[6] Incorporating oligopoly competition and lump sum ordering costs could be important to study the dynamics of some commodity prices. The developed model does generate time dependent patterns of prices, which is apparently in line with empirical evidence. This is in contrast to the usual hypothesis that price fluctuations are the result of exogenous shocks and therefore randomly distributed over time. In summary, the framework presented in this paper provides an explanation for autocorrelation in commodity prices, which established theory cannot provide until now. Hence, this paper makes a contribution that is useful for both the computational theory and structural estimation realms of this literature.

The rest of the paper is organized as follows. Section 2 introduces the model and shows important characteristics of firms’ expected sales. Section 3 characterizes the optimal decision rules. Section 4 concludes while the Appendix contains the proofs of the results stated in the text.

2 The Model

Consider an oligopoly market where risk neutral firms, indexed by i∈{1,2,...,N}, sell differentiated storable products. Each firm sells a variety of the product. Firms compete in prices and they have uncertainty about temporary demand shocks. In the short run, firms cannot respond to these temporary shocks neither by changing prices nor by increasing supply, in case of excess demand. Firms do not face any delivery lags and cannot backlog unfilled orders. Thus, whenever demand exceeds quantity on hand, the residual unfilled demand is lost. Therefore, the quantity sold by firm i at period t is the minimum of supply and demand:

where yit is the quantity sold; sit is the level of inventories at the beginning of period t; qit represents new orders to suppliers during period t; and dit is consumers’ demand. Thus, the timing of events in every period is as follows: first, firms place orders that are fulfilled and set prices. Afterwards demand is realized and firms sell the minimum of demand and inventory on hand. Every period t a firm knows the levels of inventories of all the firms in the market, i.e., the vector st≡{s1t,s2t,...,sNt}. ^[7] Given this information, the firm decides on prices and new orders (pit,qit) to maximize its expected value Et(∑r=1∞βrΠi,t+r), where β∈(0,1) is the discount factor and Πit is the current profit of firm i at period t.

A firm’s current profit is equal to revenue minus ordering cost and inventory holding cost:

where ci is the unit ordering cost; ki is the fixed or lump-sum ordering cost; and hi is the inventory holding cost.The transition rule of inventories, i.e., state variables, is:

Firms have uncertainty about current demand. The demand of product i at period t is

Here, εit is a temporary and idiosyncratic demand shock that is independently and identically distributed over time with cumulative distribution function F(⋅) that is continuously differentiable on the Lebesgue measure. These shocks are unknown to firms when they decide prices and orders. Furthermore, dite is the expected demand that depends on the endogenous prices and the exogenous qualities of all products. The expected demand dite is a function of the prices of all firms such that it is strictly decreasing in the own price, strictly increasing in the prices of competitors, and the revenue function pidie is strictly concave in pi. By definition of expected demand, we have that E(exp{εit})=1. For technical reasons it is useful to assume that F(⋅) is such that the respective hazard rate h(⋅)=f(⋅)1−F(⋅) is smaller than one. ^[8] For examples and numerical exercises it may be useful to consider a logit demand model for the expected demand:

where {wi: i=1,2,...,N} are exogenous parameters that represent product qualities, and α is a parameter that represents the marginal utility of income. The logit demand model is convenient for the derivation and illustration of some future results, but it can be relaxed for all our results. ^[9]

3 Optimal Decision Rule

Let us now characterize the optimal decision rule for a firm in this game of oligopolistic competition.

In this section we will show that the (S,s) rule, where a new order is placed whenever the inventory level falls below the order threshold s such that the target level S is attained, is indeed the best response not only to an (S,s) rule but to any given strategy of the opponents. This, of course, implies that the equilibrium resulting from (S,s) strategies by all players is a MPE.

In order to represent the optimal decision rule of the oligopolists, it is convenient to represent the decision problem in terms of the variables pit and zit≡sit+qit. The variable zit represents the total supply of the product during period t. It is also useful to define the following “value” function which is independent of the firm’s own current inventory, i.e., the only state variable the firm can influence (however, it is not independent of the current state per se), and taking the other firms’ strategies in σ and so Fs−iσ as given:

such that

Given the function Qiσ, it is clear that an oligopolist chooses (zit,pit) as a best response to the other firms’ strategies in σ, i.e., other firms order and pricing decisions, to maximize Qiσ(zit,pit;s−it)−kI{zit>sit}. Making use of this “value” function Qiσ we can derive important characteristics of competition in prices and inventories:

The positive stockout probability has a negative effect on expected sales which in turn creates substitutability between prices and inventories in the profit function. This is the case as with low inventory optimal expected demand (under given demand uncertainty) is low and thus the optimal price is high.

Using σpσ(s) and σzσ(s) to represent the optimal response rules for p and z, respectively, we have

We define the optimal price as a function of current supply:

Since Qiσ is continuously differentiable and strictly concave in prices, pˉiσ(z;s−i) is implicitly defined by the first order condition ∂Qiσ(zi,pˉi;s−i)∂pˉi=0.

It is now possible to show that the best response to any strategy is an (S,s) rule:

wheresi∗σands_iσare scalars, withsi∗σ>s_iσ∀s−i, and the following definitions:

The proposition shows that consideration of oligopolistic competition does not affect the optimality of (S,s) inventory rules. ^[10] Fixed ordering costs generate infrequent orders. The upper band si∗σ is defined as the optimal order quantity when the firm has no inventory on hand, i.e., the optimal inventory level. The lower band s_iσ is the smallest value of inventory such that the desired order quantity is zero. This inventory level at which a new order is placed is decreasing in the lump-sum ordering costs k as Qiσ(si,pˉ(si); s−i) is a strictly k-concave function. Thus, the higher the lump-sum ordering costs the less often are orders placed.

The resulting order policy might appear to be a very natural and intuitive strategy. However, as shown in the appendix the value function is not concave such that in principle a much more complex decision rule could be optimal. Besides, oligopolistic competition assures that no additional assumption on prices like the “no expected loss condition” of Hall and Rust (2007) is necessary for the optimal trading strategy to be of the (S,s) form. ^[11] With endogenous prices, the optimality of the (S,s) rule is not limited but always fulfilled. Additionally, our model of oligopolistic competition with prices as decision variables of the firms allows for analyzing the dynamics of prices. The resulting (S,s) inventory behavior together with demand uncertainty generates cyclical patterns in prices. ^[12] The optimal price is a strictly decreasing function of a firm’s inventory on hand zi as the positive probability of stockouts creates strategic substitutability between prices and inventories. Thus, the price increases between two orders when the stock level decreases and it drops down when new orders are placed. This is the case as with low inventories the optimal expected demand is lower and hence the optimal price is higher. When the level of inventories decreases between two orders, the probability of stockout increases and so expected sales decrease and become more inelastic with respect to the price. Thus, the optimal price increases between two orders, and decreases when the elasticity of sales goes up as the result of positive orders.

The largest price increase occurs just after a positive order and the increments tend to be smaller when we approach to the next positive order. The reason for this behavior is that the cyclical path of prices generates a cyclical behavior in sales. The largest sales and, consequently, the largest stock reductions and price increases, occur just after a positive order.

4 Conclusion

We have shown that the best response not only to (S,s) strategies but to any strategy is an (S,s) rule. This result extends earlier findings of models without price competition (Hall and Rust 2007) and models without strategic competition (Aguirregabiria 1999) where fixed ordering costs generate infrequent orders. Thus, the (S,s) policy might appear to be a very robust strategy. However, it is not hard to change assumptions in ways that destroy its optimality. Additionally, with strategic competition in prices (S,s) inventory behavior together with demand uncertainty generates cyclical pattern in prices.

The model developed in this paper provides a very promising alternative for studying commodity markets. Observed autocorrelation in prices cannot not be explained by existing models of competitive speculators with the usual hypothesis that price fluctuations are the result of exogenous shocks and therefore randomly distributed over time. Incorporating oligopoly competition and lump sum ordering costs could be important to study the dynamics of commodity prices. The developed model accounts for important characteristics of commodity markets and thereby generates time dependent price fluctuations that are in principle in line with empirical evidence. Making use of the presented model, it should now be possible to relate findings to commodity price dynamics and show that lump-sum ordering cost and oligopoly competition can be important to explain extremely volatile prices and especially time dependencies in price fluctuations.

However, due to the relatively high complexity of the framework further research requires numerical experiments. By this means, other topics like precise reactions of firms on competitors’ orders provide scope for interesting studies. This important work is left for future research. Nevertheless, the results derived in this paper – in particular showing that (S,s) inventory policies arise in equilibrium – allow to restrict attention to according policies instead of having to consider all possible kinds of behavior. This can make it easier to compute equilibria to this model as well as to estimate these models.