evaluating social welfare and redistributive effects of spanish personal income tax reform spain has recently concluded a process of wide ranging reform of its personal income tax irpf in force since 1992 the new irpf is applicable from 1999 onward the aim of this article is to analyse the implications of this tax reform for the distribution of personal income and additionally to provide a comparative evaluation in terms of social welfare of both taxes empirical analysis is performed by a simulation exercise employing the microdata contained in the institute of fiscal studies irpf panel of taxpayers the analysis shows that the new irpf induces a redistributive effect slightly lower than the old irpf the greater redistributive potential of the progressive structure of the new tax proves to be insufficient to compensate for the contrary effect caused by reduction in the level of tax liability however this new tax unambigously permits in an inequality adverse society a higher level of social welfare than that attained by the old tax

litigation and settlement in patent infringement cases a patent grants its owner the right to sue intruders that have been identified the patentholder must then supervise the market and react in case of infringement his reaction may be to go to court to settle or to accept entry we investigate how intensive the monitoring effort should be and how it will influence the entry decision in a simultaneous game even if the penalty paid by the infringer in case of a finding of liability is high the patentholder may prefer a settlement to a trial the likelihood of entry may increase with the penalty in sequential games regardless who plays first entry occurs comparatively less often than in the simultaneous game

the government procurement agreement implications of economic theory this paper analyzes the provisions of the new agreement on government procurement gpa drawing insights from trade theory and recent developments in the economics of information and law a central conclusion is that in a world where imperfectly informed procurers purchase from imperfectly competitive firms on behalf of imperfectly informed tax payers it is not easy to devise rules which would be optimal in all situations nevertheless the non discriminatory provisions of the gpa seem to approximate closely the rules which would maximize global welfare a significant benefit of the gpa is in helping to overcome national agency problems in procurement by creating mechanisms for reciprocal international monitoring supported by multilateral enforcement there is however scope for improvement first the gpa does not equip bidders for government contracts to vault over trade restrictions so the creation of genuine international competition for government procurement remains crucially dependent on the liberalization of trade secondly weaknesses remain in the enforcement mechanism these include the low level of compensation to a successful challenger the absence of restrictions on settlements and the lack of provision for challenge and review of bail outs the paper proposes certain improvements this abstract was borrowed from another version of this item

spatial competition and price discrimination with capacity constraints abstract we characterize mixed strategy equilibria when capacity constrained suppliers can charge location based prices to different customers we establish an equilibrium with prices that weakly increase in the costs of supplying a customer despite prices above costs and excess capacities each supplier exclusively serves its home market in equilibrium competition yields volatile market shares and an inefficient allocation of customers to firms even ex post cross supplies may restore efficiency only partly we show that consumers may benefit from price discrimination whereas the firms make the same profits as with uniform pricing we use our findings to discuss recent competition policy cases and provide hints for a more refined coordinated effects analysis

instability in indian agriculture a challenge to the green revolution technology abstract the green revolution technology succeeded in transforming india from a large food importer and large recipient of food aid in the 1950s and 1960s to a food secure country indeed during periods of the 1980s and 1990s india could occasionally export food however the green revolution technology may not have reduced the instability of agriculture during the last two decades this paper examines the factors responsible for instability in area yield and production for major crops in india from 1950 1951 to 2001 2002 this period is divided into a pre green revolution and post green revolution with each period analyzed separately as well as jointly further a similar analysis is done for rice yields for the top five producing states the results show that the sources of growth of crop production have been achieved primarily from yield increases especially since the introduction of the green revolution technology area cultivated did expand but this expansion was not rapid enough to generate the realized production changes results for the green revolution technology impact on instability are somewhat mixed the purest measure of crop instability coefficient of variation shows a decrease in production instability for wheat total cereals sugarcane and total pulses for all of india the remaining 10 crops and crop groups show an increase in production instability measures of dispersion for crop instability show similar results but the interpretation of these results is less transparent because these measures involve covariances between area and yields and other interactions effects overall many of the problems associated with production instability continue to persist and the challenge to find increased stability continues

congestion function specification and the publicness of local public goods abstract this paper is about the measurement of congestion within the median voter demand framework the model of congestion traditionally used in the literature and four others are compared and tested in a median voter framework the results show that a more flexible functional form which imposes fewer a priori restrictions on the data does at least as good a job of explaining variations in local government expenditures furthermore it is shown that measured publicness of local public goods depends entirely on the a priori restrictions imposed on the congestion function

capitalization and the theory of local public finance the tiebout literature is incomplete this paper argues because it has not fully accounted for the capitalization of local fiscal variables into house values the paper explains why capitalization arises why it persists in long run equilibrium how it affects both residential location and the outcome of local voting and why it interferes with the efficiency of a system of local governments the analysis which is based on the main tiebout assumptions plus a property tax combines a model of household bids in a housing market with a median voter model of local public service determination

price setting firms and the oligopolistic foundations of perfect competition theory fails to explain how prices are formed these observations lead us to study the theory of monopolistically competitive markets in which firms choose prices rather than quantities and then to ask whether such large markets approximate perfect competition price setting oligopolies can provide economists with a price formation story prices arise from the profit maximizing decisions of individual firms thus we are able to obtain a theory in which prices are picked by those economic actors who do so in reality although this is a matter of taste we believe that the use of prices instead of quantities as the strategic variables of firms can yield a better simple descriptive model of oligopoly that captures the essential features of many markets this motivation then leads to the question of which price setting model should be used for the study of the oligopolistic foundations of perfect competition in the original model formulated by joseph bertrand 1883 firms have constant marginal and average costs so that any one firm can supply the whole market this assumption leads to the conclusion that the presence of two or more price setting firms suffices to yield perfectly competitive outcomes be

call center outsourcing contract analysis and choice this paper considers a call center outsourcing contract analysis and choice problem faced by a contractor and a service provider the service provider receives an uncertain call volume over multiple periods and is considering outsourcing all or part of these calls to a contractor each call brings in a fixed revenue to the service provider answering calls requires having service capacity thus implicit in the outsourcing decision is a capacity decision insufficient capacity implies that calls cannot be answered which in turn means there will be a revenue loss faced with a choice between a volume based and a capacity based contract offered by a contractor that has pricing power the service provider determines optimal capacity levels the optimal price and capacity of the contractor together with the optimal capacity of the service provider determine optimal profits of each party under the two contracts being considered this paper characterizes optimal capacity levels and partially characterizes optimal pricing decisions under each contract the impact of demand variability and the economic parameters on contract choice are explored through numerical examples it is shown that no contract type is universally preferred and that operating environments as well as cost revenue structures have an important effect

bertrand competition for inputs and walrasian outcomes market making by merchants who obtain stock from suppliers and resell it to con sumers is modeled as a two stage pricing game with winner take all c ompetition for the inputs in contrast to fixed capacity models the re is a unique subgame perfect nash equilibrium spne which is walr asian for elastic demand and non walrasian for inelastic demand alte rnatively when merchants first sell forward contracts to consumers a nd then compete for supplies the unique spne is always walrasian th us the author has an equilibrium model in which walrasian price aris es not from the benevolent actions of a fictitious auctioneer but fr om optimal price setting behavior of merchants copyright 1988 by american economic association

monitoring the monitor an incentive structure for a financial intermediary this paper studies financial intermediation i e delegated monitoring in a costly state verification model there are a finite number of agents thus the intermediary cannot fully diversify its portfolio and is subject to default risk the role of the intermediary is to satisfy simultaneously the different portfolio preferences of borrowers and lenders two questions arise when a delegated monitor is subject to non trivial default risk a what arrangement solves the problem of monitoring the monitor b what intermediary portfolio accomplishes optimal asset transformation between borrowers and lenders unlike previous delegated monitoring studies the law of large numbers is not sufficient to obtain our results instead we appeal to a stronger results the large deviation principle which establishes that convergence in the law of large numbers is exponential

sources of increased instability in indian and u s cereal production recent growth in indian and u s cereal production has been accompanied by a more than proportional increase in the standard deviation of production this study applies variance decomposition procedures to state data on crop production to analyze the sources of this increased instability it is found that production has become significantly more covariate between states and crops largely because of increased yield variability and a loss in offsetting patterns of variation between crop yields in different states these changes may be associated with more variable prices with higher yielding technologies and with a narrowing of the genetic base

bundling decisions in procurement auctions with sequential tasks abstract this paper investigates the principal s bundling decision during a procurement auction for a project consisting of two sequential tasks in which task externality exists and information arrives sequentially we show that although increasing the number of bidders in the market for the second task always tilts the principal s choice toward unbundling increasing the number of consortiums that can perform both tasks tilts the principal s preference toward bundling if the externality is negative

coordinating investment production and subcontracting we value the option of subcontracting to improve financial performance and system coordination by analyzing a competitive stochastic investment game with recourse the manufacturer and subcontractor decide separately on their capacity investment levels then demand uncertainty is resolved and both parties have the option to subcontract when deciding on their production and sales we analyze and present outsourcing conditions for three contract types 1 price only contracts where an ex ante transfer price is set for each unit supplied by the subcontractor 2 incomplete contracts where both parties negotiate over the subcontracting transfer and 3 state dependent price only and incomplete contracts for which we show an equivalence result while subcontracting with these three contract types can coordinate production decisions in the supply system only state dependent contracts can eliminate all decentralization costs and coordinate capacity investment decisions the minimally sufficient price only contract that coordinates our supply chain specifies transfer prices for a small number 6 in our model of contingent scenarios our game theoretic model allows the analysis of the role of transfer prices and of the bargaining power of buyer and supplier we find that sometimes firms may be better off leaving some contract parameters unspecified ex ante and agreeing to negotiate ex post also a price focused strategy for managing subcontractors can backfire because a lower transfer price may decrease the manufacturer s profit finally as with financial options the option value of subcontracting increases as markets are more volatile or more negatively correlated

bertrand competition with subcontracting we investigate how the possibility of subsequently subcontracting production to each other influences rivals initial competition for a contract or a market as a two stage game in its first stage the two firms engage in price competition to supply a contract or a market in the second stage the firms may subcontract production to each other it is supposed that the firms produce the identical product with the same strictly convex cost function the incentive for subcontracting comesfrom the strictly convex production costs a firm is obliged to supply the entire quantity demanded at its quoted price our analysis discloses that if the winner of the game s first stage determines the terms of the subcontract in its second stage there exists a unique subgame perfect nash equilibrium spne in pure strategies in which the firms bid the same price in the first stage and both receive zero profits on the other hand if the loser of the game s first stage sets the terms of the subcontract in the second stage there exists a unique spne in pure strategies in which the firms bid the same price in the first stage and both receive positive profits the presence of the possibility of subcontracting supports a unique spne in pure strategies even though no actual subcontracting may occur the spne price is below the socially optimal price in the first case and is above it in the second case we also consider other modes of sharing the gains from subcontracting between the two firms such as the nash bargaining solution

plea bargaining and prosecutorial discretion a model of plea bargaining with asymmetric information is presented the prosecutor s private information consists of the strength of the case while the defendant s private information consists of h is guilt or innocence in equilibrium a fraction of cases are dismis sed because they are too likely to involve an innocent defendant in the remaining cases the prosecutor s offer of a sentence in exchange for a plea of guilty reveals the strength of the case it is shown t hat unlimited prosecutorial discretion may be socially disadvantageou s since it carries with it the requirement of sequential rationality copyright 1988 by american economic association

stability and competitive equilibrium in trading networks we introduce a model in which agents in a network can trade via bilateral contracts we find that when continuous transfers are allowed and utilities are quasi linear the full substitutability of preferences is sufficient to guarantee the existence of stable outcomes for any underlying network structure furthermore the set of stable outcomes is essentially equivalent to the set of competitive equilibria and all stable outcomes are in the core and are efficient by contrast for any domain of preferences strictly larger than that of full substitutability the existence of stable outcomes and competitive equilibria cannot be guaranteed

do government policies distort relative factor prices evidence from colombia developing countries have a variety of governmental and trade policies which are intended to affect the return to capital in an estimation of the return to capital in colombia an attempt is made to account for taxes both direct and indirect governmental subsidies and trade taxes and subsidies the economic income that accrues to colombia s capital stock is estimated by estimating the growth of the capital stock and the net cash flows generated by that capital additionally the average annual effective rate of protection to the manufacturing sector is estimated then using these effective rates of protection a test is made to determine if in fact protectionism affects the return to capital results reveal that there is a significant positive relationship between trade protection and the rate of return to capital in colombia furthermore support is found for a stopler samuelson effect of higher prices in the labourintensive agricultural sector leading to decline in the return for capital therefore

outsourcing vertical integration and price vs quantity competition abstract we show that standard conclusions about duopoly competition can be reversed when the production of key inputs is outsourced to a vertically integrated retail competitor with upstream market power under such outsourcing bertrand competition can produce higher prices higher industry profit lower consumer surplus and lower total surplus than cournot competition in addition to limiting the intensity of retail competition bertrand competition can limit the extent of wholesale competition by reducing the incentive of retail providers to produce key inputs themselves

weak cartels and collusion proof auctions abstract we study private value auctions in which bidders may collude without using side payments in our model bidders coordinate their actions to achieve an outcome that interim pareto dominates the noncooperative outcome we characterize auctions that are collusion proof in the sense that no such coordination opportunities exist and show that the efficient and revenue maximizing auctions are not collusion proof unless all bidders exhibit a concave distribution of valuations we then solve for revenue maximizing collusion proof auctions if distributions of valuations are symmetric and single peaked the optimal selling mechanism is a standard auction with a minimum bid followed by sequential negotiation in case no bidder bids above the minimum bid

did installed base give an incumbent any measurable advantages in federal computer procurement this research examines the relative strength and significance of the status of incumbent contractor in federal computer procurement one finding as expected is that an agency is likely to acquire a system from an incumbent vendor another finding perhaps more interesting is that the in compatibility between a buyer s installed base and a potential system also influences the vendor choice a result that may be the first econometric measurement of the competitive effects of incompatibility an illustration of this thesis comes from ibm s experience new evidence shows however that ibm s apparent disadvantage with government agencies is largely due to incompatibilities in ibm s product line

the existence of mixed strategy equilibria in a price setting oligopoly with convex costs this paper shows that a mixed strategy equilibrium will exist in an industry producing a homogeneous product with perfectly informed consumers if firms are identical and have weakly convex costs this generalises an earlier result by dasgupta and maskin

stochastic trend weather and us corn yield variability regional corn yield models incorporating stochastic trends prices and weather variables are estimated hypothesis tests suggest yield variability has increased because of an increase in error variance and an increase in weather related effects decomposition of the error variance shows much of its increase is due to an increased correlaion between regional yields

from policy aims and small farm characteristics to farm science needs this paper examines the agricultural science needs of small farms the green revolution successfully negotiated two tightropes agricultural productivity increases were larger than the fall in prices that resulted from producing more food and output per hectare rose faster than output per worker thus increasing the demand for labor the current challenge is to generate technology that helps poor farmers in low potential areas since research capacity is increasingly private incentives to produce the needed technology are needed perhaps through contracts to produce varieties with specific characteristics development of seeds needs complementary work on water for which economists plant scientists and engineers need to pool their skills

litigation settlement and collusion private enforcement of regulatory policy is a significant feature of many government sponsored contests such as procurements although private enforcement is supposed to promote social welfare we show that competitors can use it to achieve collusive outcomes in a noncooperative duopoly setting we show that the threat of litigation and the possibility of settlement can dramatically affect ex ante competition in the relevant market essentially the settlement process provides a legal mechanism for the exchange of side payments while the possibility of a court decision provides the plaintiff with a credible threat against the defendant so as to avert cheating the result does not require repeated play ex ante contracts or other commitment devices in the federal procurement context we show that our results are robust to alterations in the court remedy bargaining power of the litigants and many other factors

the convention on biological diversity and agriculture implications and unresolved debates1 summary the convention on biological diversity addresses two controversies that surround plant genetic resources pgrs one debate has been over property rights governing pgrs and the distribution of benefits from their use the second has been over the adequacy of measures to maintain crop genetic diversity this paper examines how these two debates are linked and reviews previous multilateral attempts to address them the convention signals wider international acceptance of stricter property rights over pgrs and the need for multilateral assistance for pgr conservation however current proposals to implement the convention appear too limited in scope to achieve their stated conservation objectives while several areas of controversy remain these include debates over i intellectual property protection for biological inventions ii control over pgrs in international gene banks iii the international biosafety protocol regulating biotechnology iv mechanisms to finance pgr conservation and v the allocation of funds between in situ and ex situ conservation q 1998 elsevier science ltd all rights reserved

optimal taxation and optimal tax systems the theory of optimal taxation has for the pas two decades been the reigning normative approach of taxation this paper argues that in its current state optimal tax theory is incomplete as a guide to action concerning many critical issues in tax policy it is incomplete because it has not yet come to terms with taxation as a system of coercively collecting revenues from individuals who will tend to resist the coercive nature of collection taxes implies that the resource cost of implementing a tax system have been and will continue to be a critical determinant of appropriate tax policy the paper first presents the three cornerstone propositions of optimal tax theory and then it discusses the influence of these propositions on recent tax policy developments it concludes by sketching an alternative to optimal taxation called the theory of optimal tax systems which embraces the insights of optimal taxation but also considers the technology of raising taxes and the constraints placed upon tax policy by that technology the optimal tax systems perspective is shown to shed light on the choice of tax instruments the problem of tax evasion and the appropriate tax treatment of capital income this abstract was borrowed from another version of this item

how to license intangible property we examine the optimal licensing strategy of a research lab selling to firms who are product market competitors we consider an independent lab as well as a research joint venture we show that 1 demands are interdependent and hence the standard price mechanism is not the profit maximizing licensing strategy 2 the seller s incentives to develop the innovation may be excessive 3 the seller s incentives to disseminate the innovation typically are too low 4 larger ventures are less likely to develop the innovation and more likely to restrict its dissemination in those cases where development occurs and 5 a downstream firm that is not a member of the research venture is worse off as a result of the innovation

plant breeding and poverty can transgenic seeds replicate the green revolution as a source of gains for the poor 1 abstract improved farm technology helps all main groups of the poor small farmers farmworkers other low wage labour when it raises labour value productivity but raises land and or water value productivity faster and cuts staples prices but raises smallholders total factor productivity faster from 1965 the green revolution walked these two tightropes largely by luck though targeting bigger piles of rice and wheat it cut poverty through consumption nutrition smallholder income employment risk reduction and ecological sustainability yet large areas were left out and from 1985 progress slowed in the new environment for research and agriculture how can transgenics revive and spread poverty reduction what has been the evidence so far what determines whether new varieties have traits conducive to poverty reduction who owns the research or what crop science is

physician behavioural responses to variation in marginal income tax rates longitudinal evidence the longitudinal component of the young physicians surveys 1987 1991 is used to examine the impact of changes in marginal tax rates on key indicators of physician behaviour following the federal tax reform act of 1986 many states marginal income tax rates changed this variation is used to estimate the responsiveness of work hours hmo contracting and practice size to changes in taxes

partial outsourcing monitoring cost and market structure we investigate firms outsourcing decisions when production requires a large number of inputs the novelty of our approach is that it provides a testable framework to characterize the equilibrium fraction of outsourced inputs we demonstrate that intensified competition in a cournot market for the final good typically enlarges the set of outsourced components relative to those produced in house the proportions of outsourced inputs are found to be strategic substitutes independently of whether firms compete with respect to quantities or prices in the market for the final good jel classification d20 l22 sous traitance partielle cout de monitarisation et structure de marche l auteur examine les decisions de sous traitance des entreprises quand la production reclame un grand nombre d intrants la nouveaute de l approche proposee est qu elle fournit un cadre conceptuel testable pour definir en equilibre la fraction des intrants obtenue en sous traitance on montre qu une concurrence plus intense pour le produit final dans un marchea la cournot accroit l ensemble des intrants obtenus en sous traitance par rapport a l ensemble de ceux qui sont produit a l interne il s avere que les proportions des divers intrants obtenus en sous traitance sont des substituts que les firmes se concurrencent par les prix ou les quantites dans le marche du produit final

impact of new agricultural technology on the instability of foodgrain production and yield data analysis for bangladesh and its districts abstract recent studies claim that the green revolution has increased foodgrain production and yield variability using bangladeshi national and regional data this paper provides intertemporal and cross sectional evidence to the contrary intertemporal data indicate that on the whole the relative variability of foodgrain production and yield has fallen with the adoption of green revolution technologies cross sectional regional data further support this view using various proxies for technological change the results indicate that regions with higher rates of adoption of green revolution technologies have lower relative variability of foodgrain production and yield

an analysis of the impact of climate change on crop yields and yield variability this paper develops an econometric model of stochastic production functions to quantify the impacts of climatic variables on the mean variance and covariance of crop yields the estimates of the production function parameters and their elasticities are utilized to analyse the impacts of the projected climate change on agriculture the results show that the climate change will have modest effects on the mean crop yields but will significantly reduce the variance and covariance for most of the crops considered the results have implications for allocations of agricultural land among crops and for crop production mix

preemptive investment with resalable capacity we study an entry game under perfect information with the salient feature that capacity is resalable we show that an inability to precommit to final output can help rather than hurt in contrast to other analyses of limited commitment e g maskin tirole 1988 which do not specify the sources of or reasons for limits to commitment we assume that such limits arise from the resalable nature of capacity and are led naturally to a consideration of additional strategic variables that can improve the incumbent s position in our model resalability increases the complexity of the incumbent s precommitment problem but it also furnishes her with an additional source of power in our model the price at which to resell capacity while a capacity resale market enables capacity to be shared and can help anticompetitive behavior the very existence of a resale market can generate an incentive for a much larger initial level of investment to the point where the overall equilibrium outcome can be more competitive

auctions for split award contracts the buyer of a homogeneous input divides his input requirements into two contracts that are awarded to different suppliers he uses a sequential second price auction to award a primary and a secondary contract with a fixed number of suppliers the buyer pays a higher expected price than with a sole source auction the premium paid to the winner of the secondary contract must also be paid to the winner of the primary contract as an opportunity cost when entry is endogenous we identify the conditions under which a secondary contract can increase the number of suppliers and lower the expected price

optimal local taxation and french municipal tax distortions this article presents both theoretical and empirical findings in the field of optimal local taxation i e neutral in locational decision this topic should throw light on the question of tax policy and tax reform we extend wildasin s 1987 model including mobile capital in this way we include his marginal cost of congestion taxation rule relating to mobile households proposition 1 this extension provides us with a new rule proposition 2 the optimal share out of taxes among household residents and firms in municipalities to illustrate these results we discuss the french municipal tax system properties and we pick out its main distortions

auction design and favoritism abstract the theory of auctions has ignored the fact that often auction designers not the principal design auctions in a multi attribute auction the auction designer may bias his subjective evaluation of quality or distort the relative weights of the various attributes to favor a specific bidder an ancient concern in the procurement of weapons in the auctioning of government contracts and in the purchase of electricity by regulated power companies the paper analyzes the steps to be taken to reduce the possibility of favoritism it is first shown that in the absence of favoritism quality differentials among firms are more likely to be ignored if the auction designer has imperfect information about the firm s costs second if the auction designer may collude with only one bidder the other bidders should be chosen if they are at least as efficient as the former bidder and no hard information about quality differentials is released by the auction designer that would justify fair discrimination in favor of the former bidder last if the auction designer can collude with any bidder the optimal auction tends to a symmetric auction in which quality differentials are ignored the possibility of favoritism reduces the auction designer s discretion and makes the selection process focus on non manipulable monetary dimensions of bids

lock in and the costs of switching mainframe computer vendors what do buyers see this paper closely studies the historical experiences of computer users faced with incompatibility problems one key point throughout the discussion is that operating system compatibilities and application software were the principal source of switching costs the larger point here is that vendor specificity is partially a choice variable for the buyer it influences many facets of an organization as well as management and employee behavior foresighted users anticipate that daily decisions regarding programming practices and equipment maintenance influence the costs of switching during a later vendor decision buyers take a wide variety of actions both simple and complex most of the paper is concerned with documenting and analyzing how buyers change investments collect information manipulate bidding procedures and change managements practices either in anticipation of or in response to incompatibility problems copyright 1997 by oxford university press

curbing agency problems in the procurement process by protest oversight we study a model in which a potential bidder in a government procurement may challenge its exclusion from the procurement before a quasi judicial board in the case of a sole source procurement the excluded vendor does not know whether the decision was justified in terms of expected surplus or alternatively was due to an agency problem we explain the occurrence of i equilibrium protests ii deterrence of inefficient sole sourcing iii overdeterrence the choice of a competitive procurement when sole source would be appropriate iv buyoff settlements which preserve inappropriate sole source procurement and v fedmail settlements which accompany appropriate sole source procurements our normative analysis addresses recent legislative initiatives to reform the protest process

bertrand competition with intermediation abstract we consider a duopolistic bertrand competition setting in which competing firms can turn into intermediaries the intermediation option allows firms to take advantage of the rival firm s low price we then give conditions for the existence of equilibrium