On price competition in location-price models with spatially separated markets

In this paper we study price competition for two types of location-price models in which facility locations are set up and price decisions have to be made in order to maximise profit. We discuss the existence and determination of equilibrium prices in a general location space when facilities have different production costs. It is assumed that each price is bounded from below and demand for a single homogeneous product is price-inelastic. When facilities set mill prices, a price equilibrium rarely exists and necessary conditions for existence are obtained. In particular, when the location space is a tree network, we give a characterisation of the locations for which a unique equilibrium exists for two competitors. With spatial price discrimination, though equilibrium prices might not exist, it is shown that ε-equilibrium prices always exist for any locations of the facilities. A characterisation of ε-equilibrium is also given. Then the location-price problem is reduced to a location problem. A comparison of results with the two types of price determination is also presented. This work has been supported by the Ministry of Science and Technology of Spain under the research project BEC2002-01026, in part financed by the European Regional Development Fund (ERDF).     

Asymmetric dynamic price competition with uncertainty

We consider a dynamic setting-price duopoly model in which a dominant (leader) firm moves first and a subordinate (follower) firm moves second. We suppose that each firm has two different technologies, and uses one of them according to a certain probability distribution. The use of either one or the other technology affects the unitary production cost. We analyse the effect of the production costs uncertainty on the profits of the firms, for different values of the intercept demand parameters. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Repeated price competition with increasing marginal costs

We consider a repeated price setting game with firms facing increasing marginal costs and positive fixed costs. Besides setting prices, firms may decide to be not active. Since it is well known that there is no Nash-equilibrium in pure strategies in the stage game, we look for pure strategy equilibria in the repeated game and give a full characterization of all stationary symmetric equilibrium outcomes, supported by optimal penal codes (in pure strategies). [JEL-classification: D43, L13]     

Dynamic price competition with discrete customer choices

For many years, dynamic pricing has proven to be an effective tool to increase revenue in the airline and other service industries. Most studies, however, focused on monopolistic models and ignored the fact that nowadays consumers can easily compare prices on the Internet. In this paper, we develop a game-theoretic model to describe real-time dynamic price competition between firms that sell substitutable products. By assuming the real-time inventory levels of all firms are public information, we show the existence of Nash equilibrium. We then discuss how a firm can adapt if it knows only the initial – but not the real-time – inventory levels of its competitors. We compare a firm’s expected revenue under different information structures through numerical experiments.     

Coordinating pricing and production decisions in the presence of price competition

In this paper we study the effects of coordinating pricing and production decisions on the improvement of a firm’s position in a price-competitive environment. Assuming duopolistic market conditions, we use game-theoretic concepts and models to analyze two scenarios. A firm’s marketing and production departments may vertically coordinate their pricing and production quantity decisions and the two firms may horizontally compete for price-sensitive random demand. The two scenarios include (i) no coordination and (ii) coordination in both firms. We show that by coordinating their pricing and production decisions, competing firms can increase their profitability—especially when conditions are unfavourable (i.e., with smaller market sizes, higher unit costs and lower unit revenues). While it may be intuitive to expect that coordination will outperform non-coordination, our models provide a means for formalizing and quantifying the differences between the two policies.     

Rental price and rental duration under retail competition

Consider a retailer that rents products to customers for a pre-specified rental duration. By considering the dynamics of uncertain rental demand and return processes, we first present a base model that is intended to analyze the impact of rental duration on the stocking level, the rental price, and the retailer’s profit. Due to the complexity of the base model, we develop an approximation scheme to obtain tractable results. Also, we apply the base model to analyze a situation in which a retailer enters a revenue sharing agreement with a distributor. Moreover, we expand our base model to address the issue of competition in rental duration and rental price. The analysis of our competitive model in a duopolistic environment suggests that the market equilibrium depends on the market potential and the rental duration sensitivity. Furthermore, we establish conditions under which one firm will charge a lower rental price while the other firm will offer a longer rental duration in equilibrium.     

Matching and price competition: beyond symmetric linear costs

Bulow and Levin’s (2006) “Matching and Price Competition” studies a matching model in which hospitals compete for interns by offering wages. We relax the assumption of symmetric linear costs and compare the pricing equilibrium that results to the firm-optimal competitive equilibrium. With linear and asymmetric costs, competition in the pricing equilibrium may not be localized, but all other qualitative comparisons of Bulow and Levin (2006) hold. With non-linear and symmetric costs workers’ average utility in the pricing equilibrium may be higher than in the firm- optimal competitive equilibrium. With asymmetric and non-linear costs, firms need not choose scores from an interval in a pricing equilibrium, which may make competition even less localized.     

Measuring efficiency by a single price system

We propose a multi-criteria model to measure the relative levels of efficiency for a set of alternatives. Measurement is achieved by a single price system. This involves a marked difference between Data Envelopment Analysis (DEA) and the model herein. Neither the assumptions of the proposed model nor its mathematical solution are related to DEA. In contrast to DEA, the model allows for selecting a single efficiency optimum.     

Customer rebates and retailer incentives in the presence of competition and price discrimination

Promotions are important tools for matching supply and demand in many industries. In the United States automotive industry, promotions are frequently offered, which may be given directly to customers (rebates) or given to dealers (incentives) to stimulate demand. We analyze the performance of customer rebate and retailer incentive promotions under competition. We study a setting with two manufacturers making simultaneous pricing and promotion decisions, and with two price-discriminating retailers as Stackelberg followers making simultaneous order quantity decisions. In the benchmark case with no promotions, we characterize the equilibria in closed form. We find that retailer incentives can be used by manufacturers to simultaneously improve each of their profits but can potentially lead to lower retailer profits. When manufacturers use customer rebates, we show that a manufacturer is able to decrease the profit of her competitor while increasing her own profit, although she is also at risk for her competitor to use rebates in a similar fashion. Unlike the monopoly case where the manufacturers are always better off with retailer incentives, customer rebates can be more profitable under some cases in the presence of competition. Using numerical examples we generate insights on the manufacturers’ preference of promotions in different market settings.     

A note on price and quality competition between asymmetric firms

This paper analyzes the impact of asymmetry between firms on the outcome of price and quality competition from a microeconomic viewpoint. Consumers purchase a product based on not only its price but also its quality level; therefore, two firms compete in determining their prices and quality levels to maximize their profits. The asymmetry arises from the difference in consumers’ loyalty to each firm; that asymmetry then determines a character of differentiation between firms. Our purpose is to show how asymmetry influences competition under varying consumers’ price- and quality-sensitivity. In doing so, we extend earlier work in the area of price and quality competition. We show that in both the moderately quality-sensitive and price-sensitive markets, higher consumers’ sensitivity as well as lower consumers’ loyalty to any firm leads to intense competition, resulting in a decrease of both firms’ equilibrium profits. On the other hand, in highly quality-sensitive market, asymmetry compels the smaller firm to change its competitive strategy. In general, this is more beneficial to the larger firm, as the smaller firm’s profit tends to decline. In the worst case, the smaller firm is driven out of business under equilibrium.     

Autonomous and induced production learning under price and quality competition

Recent empirical studies indicate that improvements in product conformance quality exhibit learning-by-doing patterns. We address quality improvement in a competitive duopoly market for partially substitutable products characterized by levels of quality that are not necessarily identical. The products’ quality is described with a hazard rate that can be improved both by accumulating production experience (autonomous learning) and quality improvement efforts (induced learning). Given that defective items are fully reimbursable and the demands exhibit increasing returns to scale, we derive Nash equilibrium pricing and induced learning effort dynamic policies. We show that when the effectiveness of autonomous learning prevails over the effectiveness of efforts in induced learning, equilibrium prices gradually grow over time; the trend is quite the opposite when autonomous learning is less effective than induced learning.     

Matching and price competition: would personalized prices help?

We analyze the generalized deferred-acceptance algorithm when preferences are known with an error. This algorithm incorporates personalized salaries and is considered as a replacement for the current algorithm for National Resident Matching Program (NRMP). Maintaining Bulow and Levin’s (in Am Econ Rev 96(3):652–668, 2006) assumption on preferences, we show that an error in preferences of a worker propagates through the algorithm, leading to a change in the salary of every more productive worker. Thus, relatively small individual errors accumulate toward the top and may lead to highly distorted salaries for top workers the same way as mild compression translates into highly compressed salaries on the top in the Bulow and Levin study of the current NRMP algorithm.     

Equilibrium structures of two supply chains with price and displayed-quantity competition

This paper studies the equilibrium structure of two competing supply chains, each of which consists of one manufacturer and one retailer who faces the demand influenced by price and displayed quantity. Each chain has two structure options: integration or decentralization. Under linear demand, we present the optimal pricing/displayed quantity of all members in the two chains under possible structures: two integrated chains (II), two decentralized chains (DD), and one integrated chain and one decentralized chain (ID or DI). We then analyse the impact of the intensities of price and displayed-quantity competition on the equilibrium structure of two supply chains. The results show that both price and displayed-quantity competition intensities influence significantly the equilibrium structure. Moreover, under certain specific conditions, both price and displayed-quantity competition can have the two chains fall into the prisoner’s dilemma and play a game of chicken as well.     

The effect of competition on the efficiency of secondary schools in England

In this paper we calculate the technical efficiencies, based upon multiple outputs – school exam performance and attendance rates – of all secondary schools in England over the period 1993–1998. We then estimate models to examine the determinants of efficiency in a particular year, and the determinants of the change in efficiency over the period. Our results suggest that the greater the degree of competition between schools the more efficient they are. The strength of this effect has also increased over time which is consistent with the evolution of the quasi-market in secondary education. Competition is also found to be an important determinant of the change in efficiency over time. There is, however, some evidence of conditional convergence between schools.     

Cost efficiency in data envelopment analysis under the law of one price

To impose the law of one price (LoOP) restrictions, which state that all firms face the same input prices, Kuosmanen, Cherchye, and Sipiläinen (2006) developed the top-down and bottom-up approaches to maximizing the industry-level cost efficiency. However, the optimal input shadow prices generated by the above approaches need not be unique, which influences the distribution of the efficiency indices at the individual firm level. To solve this problem, in this paper, we developed a pair of two-level mathematical programming models to calculate the upper and lower bounds of cost efficiency for each firm in the case of non-unique LoOP prices while keeping the industry cost efficiency optimal. Furthermore, a base-enumerating algorithm is proposed to solve the lower bound models of the cost efficiency measure, which are bi-level linear programs and NP-hard problems. Lastly, a numerical example is used to demonstrate the proposed approach.     

On the equivalence of quantity competition and supply function competition with sunk costs

This paper considers competition in supply functions in a homogeneous goods market in the absence of cost or demand uncertainty. In order to commit to a supply schedule, firms are required to build sufficient capacity to produce any quantity that may be prescribed by that schedule. When the cost of extra capacity (given the level of sales) is strictly positive, any Nash equilibrium outcome of supply function competition is also a Nash equilibrium outcome of the corresponding Cournot game, and vice-versa. Conversely, when the cost-savings from reducing output (given the capacity level) are sufficiently small, any outcome of iterated elimination of weakly dominated strategies in the supply function game is also an outcome of the same process in Cournot, and vice-versa.     

A nonnegative extension of the affine demand function and equilibrium analysis for multiproduct price competition

We derive a nonnegative extension of the affine demand function for differentiated substitute products from the optimization problem facing a representative consumer whose utility function is quadratic. We show that the extended demand function reduces to a linear program. The linear program has a simple intuitive interpretation in terms of a shifted price vector. We prove the existence and uniqueness of the Bertrand equilibrium in oligopolies consisting of multiproduct firms under the proposed demand function. The equilibrium, available in closed form, coincides with that obtained when allowing negative demands.     

Supply quality management with wholesale price and revenue-sharing contracts under horizontal competition

In a number of industries (e.g., the airplane industry, aerospace industry, auto industry, or computer industry), certain suppliers essentially have a monopoly on the production technology for key components, and inevitably manufacturers in these industries have common suppliers. A key part of manufacturers’ work with suppliers concerns improving the quality of their respective products, which gives rise to a collaborative activity usually termed as “supply quality management”. When the manufacturers are competitors, they do not wish to see a common supplier dividing his involvement in quality improvement unequally between themselves and their rivals. However, as the suppliers collaborate with several manufacturers, it is highly questionable whether their efforts will be strictly equivalent for each manufacturer. In this paper, a non-cooperative dynamic game is formulated in which a single supplier collaborates with two manufacturers on design quality improvements for their respective products. The manufacturers compete for market demand both on price and design quality. The paper analyzes how each party should allocate resources for quality improvement over time. In order to take into account the potential coordinating power of the compensation scheme adopted in this type of decentralized setting, we compare the possible outcomes under a wholesale price contract and a revenue-sharing contract.     

Spatial mapping of price competition using logit-type market share models and store-level scanner-data

This paper proposes a methodology to obtain reliable spatial maps of price competition using store-level scanner data. Specifically, a procedure to obtain a symmetric matrix of similarities between brands considering their substitutability depending on price variations is proposed. The matrix is derived from a market response model where price cross-effects are split into two components. The first component accounts for the fact that price variation in one brand can have different effects to price variation in other brands (ie j → j′≠j′ → j). The second component accounts for the fact that the price of each brand can have different effects across competing brands (ie j → j′≠j → j^″). The matrix is obtained by imposing symmetry on this second component of price cross-effects. The parameterization of this symmetric matrix of similarities as the distances between the spatial representations of brands allows us to obtain the positioning maps. The proposed approach is illustrated through an empirical application.     

On the efficiency of Gaussian adaptation

Gaussian Adaptation (GA) is a stochastic process that adapts a Gaussian distribution to a region or set of feasible points in parameter space. As a result of the adaptation, GA becomes a maximum dispersion process extending the sampling over the largest possible volume in parameter space while keeping the probability of finding feasible points at a suitable level. For such a process, a general measure of efficiency is defined and an efficiency theorem is proved.