On pure-strategy Nash equilibria in price–quantity games

This paper examines the existence and characteristics of pure-strategy Nash equilibria in oligopoly models in which firms simultaneously set prices and quantities. Existence of a pure-strategy equilibrium is proved for a class of price–quantity games. If the demand function is continuous, then the equilibrium outcome is similar to that of a price-only model. With discontinuous demand and limited spillover, there are rationing equilibria in which combined production falls short of market demand. Moreover, there might again be an equilibrium reflecting the outcome of a price game. Competition in price and quantity thus yields Bertrand outcomes under a variety of market conditions.     

External spillovers, internal spillovers and the geography of production and innovation

We consider a three-location duopoly model such that (i) firms choose production and innovation locations before (Bertrand) competition takes place and (ii) there are internal and external knowledge spillovers. We show: (1) agglomerations where firms earn negative profits may exist when there are both external and internal knowledge spillovers; (2) greater external spillovers do not necessarily favor agglomeration; (3) decreasing communication costs tend to favor agglomeration; (4) there are exactly two types of agglomeration equilibria: either both firms innovate in the agglomeration, or there is an innovator and an imitator; and (5) if there is a location where both firms produce, then innovation must take place in this location.     

OLIGOPOLY GAMES AND THE COURNOT–BERTRAND MODEL: A SURVEY

Traditional oligopoly models hold that firms compete in the same strategic variable, output (Cournot) or price (Bertrand). Alternatively, a hybrid model allows some firms to compete in output and other firms to compete in price, also known as the Cournot–Bertrand model. When the choice of strategic variable is endogenous, the established dominant strategy is output competition. A growing body of work demonstrates, however, that the Cournot–Bertrand outcome can be a subgame‐perfect Nash equilibrium in the presence of market asymmetries. Observations of real‐world markets consistent with Cournot–Bertrand behavior bolster justification for the model and have stimulated an impressive and evolving literature on advances and applications. We lay out the roots of the Cournot–Bertrand model and explore a number of model developments. We categorize 12 primary models in the literature based on alternative assumptions. In particular, some authors consider when the timing of play as well as the choice of strategic variable are endogenous. Altogether, this research identifies when Cournot–Bertrand behavior can emerge in a dynamic setting and under alternative market conditions. We also review the Cournot–Bertrand model applications in the fields of international economics, industrial organization, labor, and public economics. We expect the literature to continue to expand in the future.     

Quantity precommitment and price-matching

We study the effects of price-matching in a capacity-constrained duopoly setting. We show that no firm does worse at any pure equilibrium under price-matching relative to Bertrand, but as capacity increases, one or both firms do better relative to Bertrand. If the firms choose their capacities simultaneously before making pricing decisions, then the effect of price-matching varies with the cost of capacity. Specifically, when the cost is “high” price-matching either (i) has no effect on the market price, i.e., the market price associated with the pure SPEs is the Cournot one, or (ii) weakly decreases the market price relative to Cournot. Furthermore, when the cost is “low” price-matching leads to a set of (pure) SPE prices that includes the Cournot price in the interior. Therefore, price-matching does not necessarily benefit the firms when firms select their capacities before competing in price.     

Hassle Costs: The Achilles' Heel of Price-Matching Guarantees*

We show that price-matching guarantees can facilitate monopoly pricing only if firms automatically match prices. If consumers must instead request refunds (thereby incurring hassle costs), we find that any increase in equilibrium prices due to firms' price-matching policies will be small; often, no price increase can be supported. In symmetric markets price-matching guarantees cannot support any rise in prices, even if hassle costs are arbitrarily small In asymmetric markets, higher prices can be supported, but the prices fall well short of maximizing joint profits. Our model can explain why some firms adopt price-matching guarantees while others do not.     

National pricing with local quality competition

We study the incentives of national retail chains to adopt national (uniform) prices across local markets that differ in size and competition intensity. In addition to price, the chains may also compete along a quality dimension, and quality is always set locally. We show that absent quality competition, the chains will never use national pricing. However, if quality competition is sufficiently strong there exist equilibria where at least one of the chains adopts national pricing. We also identify cases in which national pricing benefits (harms) all consumers, even in markets where such a pricing strategy leads to higher (lower) prices.     

Price Competition and Advertising Signals: Signaling by Competing Senders

Can price and advertising be used by vertically differentiated duopolists to signal qualities to consumers? We show that pure price separation is impossible if the vertical differentiation is small, while adding dissipative advertising ensures the existence of separating equilibria. Two simple, but nonstandard, equilibrium refinements are introduced to deal with the multisender nature of the game, and they are shown to produce a unique separating and a unique pooling profile. Pooling results in a zero‐profit Bertrand outcome. Separation gives strictly positive duopoly profits, and dissipative advertising is used by the high‐quality firm when products are sufficiently close substitutes. Finally, compared to the complete‐information benchmark, the separating prices of both firms are distorted upwards when the degree of vertical differentiation is large, and downwards when it is small.     

Bargains Followed by Bargains: When Switching Costs Make Markets More Competitive

In markets where consumers have switching costs and firms cannot price‐discriminate, firms have two conflicting strategies. A firm can either offer a low price to attract new consumers and build future market share or a firm can offer a high price to exploit the partial lock‐in of their existing consumers. This paper develops a theory of competition when overlapping generations of consumers have switching costs and firms produce differentiated products. Competition takes place over an infinite horizon with any number of firms. This paper shows that the relationship between the level of switching costs, firms' discount rate, and the number of firms determines whether firms offer low or high prices. Similar to previous duopoly studies, switching costs are likely to facilitate lower (higher) equilibrium prices when switching costs are small (large) or when a firm's discount rate is large (small). Unlike previous studies, this paper demonstrates that the number of firms also determines whether switching costs are pro‐ or anticompetitive, and with a sufficiently large (small) number of firms switching costs are pro‐ (anti‐) competitive.     

Note on the optimality of unemployment equilibria

An equilibrium concept for an economy with rigid prices has been given by Drèze (1975). He formulated a model where for some commodity, either the supplies or the demands are rationed. In this note we discuss ‘unemployment equilibria’, i.e., fixed price equilibria for which the quantity rationing affects the supplies only. It can be proved that there exists a set of unemployment equilibria. Therefore we consider the question whether it is possible to characterize the set of undominated unemployment equilibria.     

Price Equilibria in Pure Strategies for Homogeneous Oligopoly

For a homogeneous product oligopoly market, possibilities for pure strategy Nash equilibria in prices are studied. Consumers, who each nonstrategically purchase one unit up to a common reservation price, are hypothesized to be more concerned with large price differences (and therefore buy from the cheapest firm) than slightly different prices. For the duopoly case, existence, uniqueness, and characterization results are provided. Linear examples are given with 2 and n firms.     

Housing market dynamics: Evidence of mean reversion and downward rigidity

House prices often exhibit serial correlation and mean reversion. Using two large panel datasets, this paper analyzes the price dynamics in two significantly different types of markets, cyclical (or volatile) and non-cyclical (or tame), by applying an autoregressive mean reversion (ARMR) model. Our results show that cyclical markets have larger AR coefficients than non-cyclical markets. As a result, house prices in cyclical markets tend to have larger price cycles. We also find that the upward periods have larger AR coefficients than the downward periods. This demonstrates that house prices are likely to overshoot the equilibrium in appreciating markets while experiencing downward rigidity during periods of decline. The model developed in this paper can produce a forecast with rich house price dynamics across markets. Our results can also be used to determine how house prices in overvalued markets will ultimately adjust.     

Accreditation in a differentiated duopoly

Accreditation is increasingly important worldwide; however, some industries have higher accreditation rates than others. We suggest a duopoly model to discuss how market characteristics affect the incentive for firms to seek accreditation. The discussion relates to the effects accreditation might have on the costs and demands in the markets, the degree of product differentiation (addressing both substitutable and complementary goods), type of competition (various Cournot and Bertrand games), and welfare for society. It follows that markets with high accreditation rates are either characterized by fierce competition (price competition in substitutable goods) or by a high degree of coordination (complementarity).     

Spatial competition under uniform delivered pricing

A Hotelling-type model of spatial competition is considered, in which two firms compete in uniform delivered prices. First, it is shown that there exists no uniform delivered price–location equilibrium when the product sold by the firms is perfectly homogeneous andwhen consumers buy from the firm quoting the lower delivered price. Second, when the product is heterogeneous and when preferences are identically, independently Weibull-distributed with standard deviation μ, we prove that there exists a single uniform delivered price–location equilibrium iff μ≧1/8 times the transportation rate times the size of the market. In equilibrium, firms are located at the center of the market and charge the same uniform delivered price, which equals their average transportation cost, plus a mark-up of 2μ. Finally, we discuss how our result extends to the case of n firms and proceed to a comparison of equilibria under uniform mill and delivered pricing.     

Consumer Stockpiling and Price Competition in Differentiated Markets

In many storable-goods markets, firms are often aware that consumers may strategically adjust purchase timing in response to expected price dynamics. For example, in periods when prices are low, consumers stockpile for future consumption. This paper investigates the dynamic impact of consumer stockpiling on competing firms' strategic pricing decisions in differentiated markets. The necessity of equilibrium consumer storage for storable products is re-examined. It is shown that preference heterogeneity generates differential consumer stockpiling propensity, thereby intensifying future price competition. As a result, consumer storage may not necessarily arise as an equilibrium outcome. Economic forces are also investigated that may mitigate the competition-intensifying effect of consumer inventories and that, hence, may lead to equilibrium consumer storage.     

Intermediation in Search Markets

In markets, in which exchange requires costly search for trading partners, intermediaries can help to reduce the trading frictions. This intuition is modeled in a framework with heterogeneous agents, who have the choice between intermediated exchange and search accompanied by some bargaining procedure. The equilibria of such a game are characterized. In the case of a monopolistic intermediary, the tradeoff between the bid-ask spread and the costs of delay during private search determine the intermediary's clientele. In equilibrium the monopolist charges a positive spread. Traders with large gains from trade prefer to deal with him, whereas traders with relatively low gains from trade engage in search. In case of competition among intermediaries, the classical Bertrand result obtains, and bid and ask prices converge to the (unique) Walrasian equilibrium price. Thus, in the confines of the model, the Walrasian auctioneer of the market under consideration can be replaced by competing intermediaries. In addition a multiplicity of subgame perfect Nash equilibria emphasizes the coordination problems inherent in models of intermediation.     

Multiproduct price competition with heterogeneous consumers and nonconvex costs

This paper extends the oligopolistic model of price competition to environments with multiple goods, heterogeneous consumers, and arbitrary continuous cost functions. A Nash equilibrium in mixed strategies with an endogenous sharing rule is proven to exist. It is also shown that, in environments with fixed costs and constant marginal costs, all (symmetric and asymmetric) equilibria exhibit price dispersion across stores. Furthermore, the paper identifies scenarios in which prices will necessarily be random. In these markets, stores keep each other guessing because, given the fixed costs, they would incur a loss if their price strategies were anticipated and beaten by competitors. This is interpreted as an important economic feature that is possibly behind random price promotions such as weekly specials.     

Spatial discrimination: Bertrand vs. Cournot with asymmetric demands

This paper develops a barbell model a la Hwang and Mai [Hwang, H., and C.C. Mai, 1990, Effects of spatial price discrimination on output, welfare, and location, American Economic Review 80, 567–575.] with homogeneous product and asymmetric demands to compare prices, aggregate profits and social welfare between Cournot and Bertrand competition, and to analyze the firms' equilibrium locations. It focuses on the impacts of the spatial barrier generated from transport costs, and the market size effect resulting from asymmetric demands. It shows that the market-size effect is crucial in determining firms' locations under Cournot competition, but insignificant under Bertrand competition. Moreover, the equilibrium price of the large market and the aggregate profits are lower but the social welfare is higher under Cournot competition than under Bertrand competition if one of the markets is sufficiently large and the transport cost is high.     

Random encounters and information diffusion about product quality

This paper explores how social interactions among consumers shape markets. In a two-country model, consumers meet and exchange information about the quality of the goods. As information spreads, demand evolves, affecting the prices and quantities manufactured by profit-maximizing firms. We show that market prices with informational frictions reach the duopoly price with full information at the limit. However, this convergence can take different paths depending on the size asymmetry between countries. In particular, when the country producing the low-quality good is relatively large, the single market does not immediately turn into a duopoly and can be temporarily trapped in a situation of price instability where no Nash equilibrium in pure (but only in mixed) strategies exists and prices can fluctuate between their monopoly and duopoly levels. It follows that the classical price-reducing effects of international trade may take longer to appear. In view of an intense globalization process, understanding how social meetings affect market outcomes is critical for understanding the performance of international economic integration.     

Vertical Integration, Foreclosure, and profits in the Presence of Double Marginalization

Whether vertical integration between a downstream oligopolist and an upstream oligopolist is profitable for an integrated pair of firms is shown to depend on whether one means by this that profits increase no matter what other firms do, that all integrated firms are better off when all firms are integrated than when none are, or simply that no downstream-upstream pair of firms has an incentive to deviate from a situation where all firms are integrated. It is also shown to depend on the number of firms in each oligopoly and on the type of interaction that is assumed between firms that are integrated and firms that are not. In particular, it is shown that if no restriction is put on trade between integrated and nonintegrated firms, integrated firms may continue to purchase inputs from the nonintegrated upstream firms, with the goal of raising their downstream rivals' costs. Furthermore, even though firms are identical, asymmetric equilibria, where integrated and nonintegrated firms coexist, may actually arise as an outcome of the integration game.     

The revelation of information in strategic market games: A critique of rational expectations equilibrium

We criticize the R.E.E. approach to asymmetric information general equilibrium because it does not explain how information gets ‘into’ the prices. This leads to well-known paradoxes. We suggest a multiperiod game instead, where the flow of information into and out of prices is explicitly modeled. In our game Nash equilibria (N.E.) (1) generalize Walrasian equilibria to asymmetric information, (2) exist generically, (3) eliminate pure speculation, (4) allow prices to reveal information and markets to become more efficient over time, (5) are consistent with the weak efficient markets hypothesis that tracking past prices is not profitable, (6) yet always lead to higher utility for better informed agents (such as experts). Throughout the paper we use one concrete game. In the last section we prove that there is a broad range of games that would have the same properties.