Bertrand Equilibrium Research Articles

An Input-Output Model Building for Nine Regions of Japanese Economy Based on Microeconomic Foundation

Traditional input-output model has been built since primary work of W.Leontief; a sector production model plus a price determination model. This model is simplistic one which are aggregated by each economic element. However, in order to reflect tendency of model building in recent years, it is advocated that model should be built based on the micro economic theory, in which behaviors of all economic agents are deduced from optimization. Following this line, input-output modeling also should be done in a way of reflecting this tendency. Thus, we intend to develop input-output modeling for the nine interregional system of the Japanese economy based on the micro foundation. We are going to endogenize intermediate demand, consumption expenditure, labor demand, and price determination in interregional system; a)consumption expenditure is explained by Almost Ideal Demand system which comes from expenditure minimization, b)intermediate and labor demand are derived from generalized Ozaki cost function, and c)sector price is determined by the Bertrand equilibrium of oligopolistic market. We next incorporate these systems into interregional input-output model of nine regions, and it is estimated from every five years data of having eight time points from 1965 to 2000. In a final step of this paper, we show their model performance.

Existence of Bertrand equilibrium revisited

In this paper we reconsider existence of Bertrand equilibrium in a symmetric-cost, homogenous-product oligopoly. We prove the following main results. (a) If the cost function is strictly superadditive on [0, ∞) then there exists a pure strategy Bertrand equilibrium. Such Bertrand equilibria are necessarily non-unique. (b) If the cost function is strictly subadditive on [0, ∞) then there exists no Bertrand equilibrium, either in pure strategies or in mixed strategies.

Gauging Parallel Accommodating Conduct Concerns with the CPPI

The 2010 Merger Guidelines give greater prominence to the concept of parallel accommodating conduct. Parallel accommodating conduct (PAC) has a long history in oligopoly theory, dating back more than seventy years. It is a type of coordinated conduct that does not require an agreement. Instead, it involves a firm engaging in a certain conduct, with the expectation that one or more other firms will follow that same conduct. For example, PAC could involve two leading firms raising their prices in parallel – but without an agreement – over and above the prices determined by their unilateral pricing incentives. One firm would raise price above the Bertrand equilibrium level and the other firm would simply follow. We have formulated an index for gauging PAC concerns, which we call the “Coordinated Price Pressure Index,” or CPPI. The CPPI corresponds to the largest price increase that the two coordinating firms would be willing both to initiate and follow. A larger CPPI implies an incentive of the two firms for larger PAC price increases, which suggests more serious PAC concerns, ceteris paribus. We also have applied the CPPI to merger analysis. The impact of a merger on the magnitude of the potential parallel price increases is measured by the increase in the CPPI (or “Delta CPPI”).

Screening competition in mobile telephony†

This article presents a simple method for screening competition in differentiated products oligopoly with a small number of competitors. In many situations, estimation of price elasticities of demand may be impossible due to difficulties in defining demand or missing data on sales. However, even without information on price elasticities, in certain situations it is possible to test for the static noncooperative Nash–Bertrand equilibrium, which in the case of rejection, may be important screening information for antitrust authorities. The static noncooperative Nash–Bertrand equilibrium may be rejected when demand is linear and in the estimation of best-response functions, the coefficients on the competitors’ prices are statistically greater than 0.5. The application of this method is illustrated by the example of German mobile telephony using monthly data between January 1998 and December 2002. According to the estimation results, the observed prices in the segment of low-users cannot be the outcome of a static noncooperative Nash–Bertrand equilibrium.

Bertrand equilibrium with subadditive costs

We show here, in contrast to recent results, that if firms have different cost functions (that are strictly subadditive), such that the ‘monopoly breakeven prices’ are different, then in a homogeneous product duopoly there is always a Bertrand equilibrium (either in pure strategies or in mixed strategies).

Duopoly Pricing Under ‘Private Knowledge’ of Product Differentiation

This note studies price decisions in a duopoly industry where firms have private information over the degree of product differentiation (product-type). A Bayesian-Nash price solution is derived assuming firms maximize their ‘certainty-equivalent’ profit levels. The comparative-statics indicate that increased risk aversion over the rival’s product triggers price competition. Consequently, the results of the study suggest revealing information is a higher reward strategy than concealing information in situations where rivals have asymmetric information over product type. These findings contribute to the industrial economic literature by generalizing the Bertrand equilibrium in an asymmetric information game model.

Randomized strategy equilibrium in the action commitment game with costs of leading

We investigate a two-player action commitment game where one simultaneous-move and two sequential-move pure strategy equilibria exist when the cost of leading is zero, while the simultaneous-move outcome is not an equilibrium when the leading cost is small positive. We show that this discontinuity disappears if we consider randomized strategy equilibria. We investigate a price competition model and show that randomized strategy equilibria exist and any of them converges to the Bertrand equilibrium when the leading cost converges to zero.

Spatial price discrimination in the unidirectional Hotelling model with elastic demand

The unidirectional Hotelling model is extended to allow for elastic demand functions. A two-stage Bertrand-type model and a two-stage Cournot-type model are considered. If firms choose location and then set prices, agglomeration never arises; instead, if firms choose location and then set quantities, agglomeration arises at one endpoint of the segment when the transportation costs are low enough. Instead, when the transportation costs are high enough, a dispersed equilibrium arises in Cournot. The equilibrium distance between the firms is lower in Cournot than in Bertrand. When the transportation costs are high (low) the Bertrand equilibrium is welfare superior (inferior) to the Cournot equilibrium.

Oligopolistic Price Competition with Informed and Uninformed Buyers

The standard price competition of two or more players leads to Bertrand equilibrium in basic economic theory (if complete information is assumed, there are no capacity constraints, etc.). In reality, even on highly competitive Internet-based markets, the prices of seemingly undifferentiated goods (e.g. books and CDs on Amazon and similar e-shops) vary, although competition seems prima facie based on prices. I follow the literature that originated with Varian’s (1980) model, especially Kocas and Kiyak (2006), and analyze oligopolistic markets where buyers have reservation values drawn from a common distribution function rather than a single value (inelastic demand), as typically assumed in the models of Varian’s or Kocas and Kiyak’s type. The model presented in this paper is developed from the simplest symmetric set-up (uninformed buyers are assigned to sellers evenly) to the most complex asymmetric set-up with many competing sellers (uninformed buyers are distributed over sellers unevenly). The most complex set-up theoretically rationalizes the empirical findings of Kocas and Kiyak. In the equilibrium of my model, all sellers randomly choose prices from a non-trivial interval for (almost) every seller, while in Kocas and Kiyak’s theoretical model only two sellers randomize while others always offer the same price.

Existence Advertising, Price Competition and Asymmetric Market Structure

We examine a two-stage duopoly game in which firms advertise their existence to consumers in stage 1 and compete in prices in stage 2. Whenever the advertising technology generates positive overlap in customer bases, the equilibrium for the stage-1 game is asymmetric in that one firm chooses to remain small in comparison to its competitor. For a specific random advertising technology, we show that one firm will always be half as large as the other. No pure-strategy price equilibrium exists in the stage-2 game, and as long as there is some overlap in customer bases, the mixed-strategy price equilibrium does not converge to the Bertrand equilibrium.

Bertrand Competition in Markets with Fixed Costs

This paper provides necessary and sufficient conditions for the existence of a pure strategy Bertrand equilibrium in a model of price competition with fixed costs. It unveils an interesting and unexplored relationship between Bertrand competition and natural monopoly. That relationship points out that the non-subadditivity of the cost function at the output level corresponding to the oligopoly break-even price, denoted by D(pL(n)), is sufficient to guarantee that the market sustains a (not necessarily symmetric) Bertrand equilibrium in pure strategies with two or more firms supplying at least D(pL(n)). Conversely, the existence of a pure strategy equilibrium ensures that the cost function is not subadditive at every output greater than or equal to D(pL(n)).

99 Cent: Price Points in E-Commerce

This paper attempts to shed light on the pricing behaviour of firms or sellers in market places with price competition. In particular, it aims at testing the mechanism that lies at the heart of one of the most famous models in economics, namely the Bertrand model of competition. This model suggests that firms' profits will be zero as soon as there are at least two sellers in the market. The model is built on the idea that consumers will go to the cheapest shop, even if the differences in prices are infinitesimal. As a result, firms are tempted to undercut their opponents' price by a small amount and they do so over and over until they end up charging the cost at which they produce their products. However, the underlying assumption is at odds with the observation that consumers typically do not care very much about a single cent and the predicted outcome clearly goes against the fact that many prices end in 99c. If sellers really charged the price at which they buy or produce a good or a service, we should observe all possible price endings. Basu (2006) argues that the prevalence of 99 cent prices in shops can be explained by consumers who disregard the rightmost digits of the price. This bounded rational behaviour leads to an Bertrand equilibrium with positive mark-ups allowing firms to make at least small profits. We use data from an Austrian price comparison site, which bears the advantage that in such an environment other factors than price both can be observed online and play a minor role. Price comparison sites are therefore highly suited for our test. We find results highly compatible with Basu's theory. We can show that price points - in particular prices ending in 9 - are more frequent than other endings and have significant impact on consumer demand. This shows that the mechanism that was postulated by Bertrand is not in place. Consumers do not necessarily choose the cheapest price and they even less do so when the Euro digits of the price are the same (i.e. when the difference is small). Moreover, the nine-ending prices are sticky: neither the price-setter itself wants to change them nor the rivals do underbid these prices, if they represent the cheapest price on the market. This might either show that shops seem to believe that it is no use to only slightly undercut a rival's price if this doesn't trigger a change in the Euro, or it is an indication that there might be some sort of silent agreement to not undercut a price ending in 9. Our findings shed new light on the behaviour of consumers and firms, and propose that theoretical investigations on competitive behaviour should consider discrete state spaces, i.e. markets where the smallest plausible price change is by a Euro or at least by a full Cent.

Coordination and price competition in a duopoly common retailer supply chain

This paper analyzes the coordination and competition issues in a two-stage supply-chain distribution system where two vendors compete to sell differentiated products through a common retailer in the same market. The demand of a product not only depends on its own price, but also on the price of the other. Mathematical models have been developed to analyze the coordination issues under three different contexts: (i) price competition without channel coordination; (ii) price competition with channel coordination; and (iii) global coordination. It has been shown that under certain conditions, price competition through the dynamic process of price adjustment reaches the Nash–Bertrand equilibrium. Conditions have been derived for the Nash–Bertrand equilibrium to be dynamically stable. Further, it has been shown that duopoly competition can make consumers better-off or worse-off depending on the degree of product differentiation and the type of the product; while coordination enhances overall supply-chain profitability. The model is illustrated with suitable numerical examples.

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.

Predation Under Perfect Information

In an oligopoly configuration characterized by high barriers to (re-)entry, a finite horizon, perfect information about demand and costs and the presence of three identical firms, we show that two of them (the predators) can choose to charge an initial price that is so low that the third (the prey) decides to exit immediately, after which the predators can enjoy higher profits, even if they do not raise their price. Predatory prices are thus observed on the equilibrium path and the predators end up earning more than in the best Bertrand (or even, collusive) equilibrium with three firms.

Use of Pricing Schemes for Differentiating Information Goods

Information goods vendors offer different pricing schemes such as per user pricing and site licensing. Why do competing sellers adopt different pricing schemes for the same information good? Pricing schemes affect buyers' usage levels and thus the revenue generated from different segments of buyers. This can allow competing firms in a duopoly to differentiate themselves by offering different pricing schemes. Such strategic use of pricing schemes can allow undifferentiated sellers to earn substantial profits in a friction-free market for a commoditized information good. These conditions would otherwise lead to the Bertrand equilibrium and zero profits. We show that adopting asymmetric pricing schemes can be a Nash equilibrium for information goods with negligible marginal cost of production. We extend our model to the case of information goods that are horizontally differentiated and show that sellers will offer a single-pricing scheme that is different from competitors when the sellers are weakly differentiated. When the sellers are strongly differentiated, each seller will offer multiple pricing schemes. We show that it can be optimal for a seller to offer multiple pricing schemes—metered and flat fee pricing schemes, even in the absence of transactions costs.

Bertrand equilibrium with differentiated products considering fixed costs: Korean price regulation in the mobile telephone industry

In some industries such as telecommunications and electricity, the fixed costs are so high that competition is not sustainable without considering the fixed cost in pricing. The sustainability of competition is as important as enhancing competition when an industry is in the transitional period from being a monopoly owned or managed by the government to a competitive market structure driven by the market. A price competition model with asymmetric firms and product differentiation is considered in which firms compete with a normal profit constraint. The constraint is related to the sustainability of competition and can be realized through price regulation. With this constraint, firms gain only normal profit in equilibrium despite the asymmetry, and consumer surplus is maximized. The equilibrium is meaningful in defining the industry performance intended through the regulation. Moreover, the price regulation of the Korean mobile telephone market is considered to discuss any implications of this equilibrium.

Dynamic efficiency of Cournot and Bertrand competition: input versus output spillovers

We consider the efficiency of Cournot and Bertrand equilibria in a duopoly with substitutable goods where firms invest in process R&D that generates input spillovers. Under Cournot competition firms always invest more in R&D than under Bertrand competition. More importantly, Cournot competition yields lower prices than Bertrand competition when the R&D production process is efficient, when spillovers are substantial, and when goods are not too differentiated. The range of cases for which total surplus under Cournot competition exceeds that under Bertrand competition is even larger as competition over quantities always yields the largest producers’ surplus.

Equilibrios competitivos y de Bertrand, con y sin Diferenciación de Productos

In this paper we show that a homogeneous-product market with multiple Bertrand equilibria becomes a market with a single Bertrand equilibrium when we introduce a small degree of product differentiation. When differentiation tends to zero, that Bertrand equilibrium converges to the unique competitive equilibrium of the homogeneous-product market, which is in turn one of the multiple Bertrand equilibria for that market.

Computing welfare losses from data under imperfect competition with heterogeneous goods

We study the percentage of welfare losses (PWL) yielded by imperfect competition under product differentiation. When demand is linear, even if prices, outputs, costs and the number of firms can be observed, PWL is arbitrary in both Cournot and Bertrand equilibria. If in addition the elasticity of demand (resp. cross elasticity of demand) is known, we can calculate PWL in a Cournot (resp. Bertrand) equilibrium. When demand is isoelastic and there are many firms, PWL can be computed from prices, outputs, costs and the number of firms. We find that price–marginal cost margins and demand elasticities may influence PWL in a counterintuitive way. We also provide conditions under which PWL increases or decreases with concentration.