Preemptive entry in differentiated product markets

Summary. Models of spatial competition are typically static, and exhibit multiple free-entry equilibria. Incumbent firms can earn rents in equilibrium because any potential entrant expects a significantly lower market share (since it must fit into a niche between incumbent firms) along with fiercer price competition. Previous research has usually concentrated on the zero-profit equilibrium, at which there is normally excessive entry, and so an entry tax would improve the allocation of resources. At the other extreme, the equilibrium with the greatest rent per firm normally entails insufficient entry, so an entry subsidy should be prescribed. A model of sequential firm entry (with an exogenous order of moves) resolves the multiplicity problem but raises a new difficulty: firms that enter earlier can expect higher spatial rents, and so firms prefer to be earlier in the entry order. This tension disappears when firms can compete for entry positions. We therefore suppose that firms can commit capital early to the market in order to lay claim to a particular location. This temporal competition dissipates spatial rents in equilibrium and justifies the sequential move structure. However, the policy implications are quite different once time is introduced. An atemporal analysis of the sequential entry process would prescribe an entry subsidy, but once proper account is taken of the entry dynamics, a tax may be preferable. Received: April 26, 1999; revised version: September 22, 1999     

Profitability of price and quantity strategies in a duopoly with vertical product differentiation

Using a model according to Mussa and Rosen (1978) and Bonanno and Haworth (1998) we consider a sub-game perfect equilibrium of a two-stage game in a duopolistic industry in which the products of the firms are vertically differentiated. In the industry, there are a high quality firm and a low quality firm. In the first stage of the game, the firms choose their strategic variables, price or quantity. In the second stage, they determine the levels of their strategic variables. We will show that, under an assumption about distribution of consumers' preference, we obtain the result that is similar to Singh and Vives (1984)' proposition (their Proposition 3) in the case of substitutes with nonlinear demand functions. That is, in the first stage of the game, a quantity strategy dominates a price strategy for both firms. Received: April 23, 1999; revised version: May 31, 2000     

Finiteness property in vertically differentiated markets: a note on locally increasing and decreasing returns

Summary. The results previously obtained on the finiteness property in vertically differentiated markets, with cost functions having increasing or decreasing returns, are extended to a much larger class of cost functions with local properties in a vicinity of the zero output. Moreover, existence results are provided to prove that the demonstrated properties have cases of application.Received: 11 June 2001, Revised: 18 February 2003, JEL Classification Numbers: L1, D43.Grateful thanks are due to Jacques-François Thisse, Jean-Jaskold Gabszewicz, an anonymous referee and especially to Jean-Marc Bonnisseau and Xavier Wauthy for very helpful suggestions and discussions. The revision took place partly when the author was visiting CERMSEM, Université de Paris-I, Paris then CORE, Louvain-La-Neuve, Belgium. Hospitality of both institutions is also acknowledged.     

The set of sunspot equilibria in economies with incomplete financial markets: variable asset prices

Summary. I consider the set of equilibria of two-period economies with S extrinsic states of nature in the second period and I assets with linearly independent nominal payoffs. Asset prices are variable. If the number of agents is greater than (S-I), the payoff matrix is in general position and S 2I, the set of equilibrium allocations generically (in utility function space) contains a smooth manifold of dimension (S-1). Moreover, the map from states o f nature to equilibrium allocations (restricted to this manifold) is one-to-one at each equilibrium. Received: February 23, 1998; revised version: June 1, 2000     

The evolution of debt and equity markets in economic development

Summary. As noted by Gurley and Shaw, there is a typical pattern of economic development in which the evolution of the financial system is an essential aspect of the growth process. We focus on one component of this evolution: the increasing importance of equity markets as an economy grows. We develop a growth model where capital accumulation is financed externally through a combination of debt and equity. We illustrate why equity market activity might grow – often very rapidly – as an economy develops. We also illustrate why access to equity markets may not be needed in the early stages of economic development. Received: December 30, 1997; revised version: May 26, 1998     

Boiteux's solution to the shifting-peak problem and the equilibrium price density in continuous time

Summary. Bewley's condition on production sets, imposed to ensure the existence of an equilibrium price density when is the commodity space, is weakened to allow applications to continuous-time problems, and especially to peak-load pricing when the users' utility and production functions are Mackey continuous. A general form for production sets with the required property is identified, and examples are given of technologies which meet the weakened but not the original condition: these include industrial use and storage of cyclically priced goods. This gives a framework for settling Boiteux's conjecture on the shifting-peak problem. To make clear the restriction implicit in Mackey continuity, we interpret it as interruptibility of demand; and we point out that, without this assumption, the equilibrium can feature pointed peaks with singular, instantaneous capacity charges. The general equilibrium results are supplemented by results for prices supporting individual consumer or producer optima. Received: February 16, 2000; revised version: July 7, 2001     

Valuation in infinite-horizon sequential markets with portfolio constraints

Summary. We develop a theory of valuation of assets in sequential markets over an infinite horizon and discuss implications of this theory for equilibrium under various portfolio constraints. We characterize a class of constraints under which sublinear valuation and a modified present value rule hold on the set of non-negative payoff streams in the absence of feasible arbitrage. We provide an example in which valuation is non-linear and the standard present value rule fails in incomplete markets. We show that linearity and countable additivity of valuation hold when markets are complete. We present a transversality constraint under which valuation is linear and countably additive on the set of all payoff streams regardless of whether markets are complete or incomplete. Received: March 9, 2000; revised version: February 13, 2001     

Secret information acquisition in Cournot markets

Summary. Two-stage game models of information acquisition in stochastic oligopolies require the assumption that firms observe the precision of information chosen by their competitors before determining quantities. This paper analyzes secret information acquisition as a one-stage game. Relative to the two-stage game firms are shown to acquire less information. Policy implications based on the two-stage game yield, therefore, too high taxes or too low subsidies for research activities. For the case of heterogeneous duopoly we briefly discuss comparative statics results. Received: August 9, 1999; revised version: May 31, 2000     

The multiple unit auction with variable supply

Summary. The theory of multiple unit auctions traditionally assumes that the offered quantity is fixed. I argue that this assumption is not appropriate for many applications because the seller may be able and willing to adjust the supply as a function of the bidding. In this paper I address this shortcoming by analyzing a multi-unit auction game between a monopolistic seller who can produce arbitrary quantities at constant unit cost, and oligopolistic bidders. I establish the existence of a subgame-perfect equilibrium for price discriminating and for uniform price auctions. I also show that bidders have an incentive to misreport their true demand in both auction formats, but they do that in different ways and for different reasons. Furthermore, both auction formats are inefficient, but there is no unambiguous ordering among them. Finally, the more competitive the bidders are, the more likely the seller is to prefer uniform pricing over price discrimination, yet increased competition among bidders may or may not enhance efficiency. Received: June 18, 1998; revised version: January 13, 1999     

On the efficiency of markets for managers

Summary. This paper examines the efficiency of the outside labor market in inducing optimal managerial behavior in the presence of learning. It shows that the incentives provided by the market can be more efficient than the original analysis of Holmstr?m [6] would suggest. Moreover, under a mild additional assumption, the existence of an -efficient equilibrium can be guaranteed if a manager is patient. This result supports Fama's [4] original idea that the outside labor market can be efficient in disciplining top managers. These results also suggest that the empirically documented low levels of explicit incentives for managers might be due to the presence of implicit incentives provided by the outside market. Received: March 18, 1997; revised version: April 19, 2000     

Incomplete markets: transverse financial structures

Summary. In a multiperiod economy with incomplete markets and assets with payoff depending on the price history (e.g., asset and derivatives), we show that in order to get endowment generic existence of an equilibrium it is not needed to alter settlement features such as when payments are made and when the asset is traded. This is non-trivial as each such characteristic introduces a non-generic subclass of financial instruments. We show essentially that expiry date payments are the only payments that one needs perturbing (if at all). For previous periods - the P&L discovery map - is the one relevant for wealth transfers. This map transfers wealth between one period and the next by associating to each portfolio next period potential profit and losses as a function of the revealed information at the node. All present values involved can in general - because of backward induction pricing structure - be appropriately controlled via expiry payoffs only. This enables us to extend two-period work and introduce Transverse Financial Structures for multiperiod economies, where one cannot identify the payoffs of financial instruments to the P&L discovery map (in other words we introduce some financial ingeneering for Transverse Financial Structures). We capitalize on that difference using unexploited “maturity payout degrees of freedom” and rolling back the uncertainty tree. As an application of this approach we prove a conjecture by Magill and Quinzii that commodity forward contracts lead to endowment generic existence of an equilibrium in a multiperiod set-up. Received: June 25, 1999; revised version: April 4, 2001     

Computable general equilibrium with financial markets

Summary. There are a wide variety of theoretical general equilibrium models with incomplete security markets. In this paper we give a general recipe for using homotopy algorithm to compute equilibria in these models. In many models, taxes, transaction-costs or other market frictions introduce the additional difficulty that equilibrium prices or choices (but not equilibrium allocations) may be undetermined. In order to demonstrate how these difficulties can be dealt with, we develop a globally convergent algorithm to compute equilibria in a model with cash-in-advance constraints, several goods and incomplete financial markets. Furthermore we describe how to implement the algorithm using a publicly available suite of subroutines for homotopy-pathfollowing. Received: October 1, 1999; revised version: December 16, 2000     

Competitive search markets for durable goods

Summary. This paper considers a dynamic version of Akerlof's (1970) lemons problem where buyers and sellers must engage in search to find a trading partner. We show that if goods are durable, the market itself may provide a natural sorting mechanism. In equilibrium, high-quality goods sell at a higher price than low-quality goods but also circulate longer. This accords with the common wisdom that sellers who want to sell fast may have to accept a lower price. We then compare the equilibrium outcomes under private information with those under complete information. Surprisingly, we find that for a large range of parameter values the quilibrium outcomes under the two information regimes coincide, despite the fact that circulation time is used to achieve separation. Received: August 24, 2000; revised version: October 24, 2000     

Disaggregation of excess demand and comparative statics with incomplete markets and nominal assets

Summary. We prove that locally, Walras' law and homogeneity characterize the structure of market excess demand functions when financial markets are incomplete and assets' returns are nominal. The method of proof is substantially different from all existing arguments as the properties of individual demand are also different. We show that this result has important implications and is part of a more general result that excess demand is an essentially arbitrary function not just of prices, but also of the exogenous parameters of the economy as asset returns, preferences, and endowments. Thus locally the equilibrium manifold, relating equilibrium prices to these parameters has also no structure. Received: September 17, 1996; revised version: November 7, 1997     

Effect of credible quality investment with Bertrand and Cournot competition

We show how credible revelation and ability to commit to quality choice effect equilibrium qualities and welfare when product market is either Bertrand or Cournot competition. We show that results depend on the type of competition but not generally on the cost of quality function. We show that with Bertrand competition, the equilibrium qualities are lower with credible commitment. Competition is moderated and producer surplus is higher and consumer surplus lower. With Cournot competition, higher quality will be better but lower quality will be worse with credible commitment. Consumer surplus is always greater with credible commitment and if cost does not increase too quickly with quality, producer surplus will also increase. Thus credible commitment is a collusive device with Bertrand competition but it can improve social welfare with Cournot competition. Received: February 8, 2000; revised version: February 14, 2002 RID="*" ID="*" The idea of this paper originated in the weekly workshops of Mordecai Kurz at Stanford. I am forever in debted to Mordecai and fellow students – Luis Cabral, Peter DeMarzo, John Hillas, Michihiro Kandori, Steve Langois, Patrick McAllister, Steve Sharpe, Peter Streufert, Steve Turnbull and Gyu-Ho Wang – for their criticism and encouragement. I also benefited from comments from Yi-Heng Chen, Jin-Li Hu, Kala Krishna, Jinji Naoto, Thomas J. Prusa, and Shyh-Fang Ueng at various later stages of this work. Last but not least, I am grateful for the detailed comments of the referee.     

Informational efficiency: ranking markets

Summary. The paper studies informational properties of three types of imperfectly competitive markets: a one-signal speculative market (OSS market) in which agents have only private information about the fundamental value (v) of the risky asset traded, a two-signal speculative market (TSS market) in which agents have private information about both v and the asset supply, and a market in which agents are endowed with both information about v and shares of the risky asset traded. In this last market (JA market), agents have joint activities: they trade for both speculative and hedging purposes. It is shown that (i) the JA market and the OSS market are the most and the least efficient, respectively, and (ii) the levels of informational efficiency in the three markets are inversely correlated with the intensities with which traders use their private information about the fundamental value of the asset. Received May 28, 1999; revised version: May 28, 1999     

A model of vertically differentiated education

We analyze the allocation of students who differ in their ability to two school types. We assume that there are peer effects (students learn more the better their classmates) and that the expected income after school also depends on the average productivity of peers. We derive the allocation under free school choice and decentralized determination of the academic level; we show that, compared to the social optimum, the level is suboptimal and too many students attend the better school type. A social planner can improve on the allocation under free school choice by prescribing a higher academic level.     

Generic inefficiency of equilibria in the general equilibrium model with incomplete asset markets and infinite time

Summary. We consider a Lucas asset-pricing model with heterogeneous agents, exogenous labor income, and a finite number of exogenous shocks. Although agents are infinitely lived, endowments and dividends are time-invariant functions of the exogenous shock alone and are thus restricted to lie in a finite-dimensional space; genericity analysis can be conducted on sets of zero Lebesgue measure. When financial markets are incomplete, that is, there are fewer financial securities than shocks, we show that generically in individual endowments all competitive equilibria are Pareto inefficient. Received: November 22, 1999; revised version: March 4, 2002 RID="*" ID="*" We are grateful to an anonymous referee for very insightful comments on earlier drafts.     

Winner-take-all price competition

Summary. We analyze an oligopoly model of homogeneous product price competition that allows for discontinuities in demand and/or costs. Conditions under which only zero profit equilibrium outcomes obtain in such settings are provided. We then illustrate through a series of examples that the conditions provided are “tight” in the sense that their relaxation leads to positive profit outcomes. Received: April 7, 2000; revised version: September 14, 2000     

A new look for Stackelberg-Cournot equilibria in oligopolistic markets

Summary. This note deals with Cournot type oligopolies in which the market clearing price occasionally may be non-unique. A Stackelberg leading producer is present. Given that setting we explore continuity properties of the followers' reaction and provide sufficient conditions for existence of equilibrium. Received: June 20, 2000; revised version: April 24, 2001