Price leadership in a homogeneous product market

The present paper reexamines the price-setting Stackelberg duopoly with asymmetric and strictly convex cost functions in a homogeneous product market. It demonstrates that in a generic environment, the higher-cost firm is likely to be the price leader. That is, leadership by such a firm is either (a) a unique equilibrium or (b) a payoff- and risk-dominant equilibrium in the observable delay game. Thus, while this paper complements and generalizes the findings of recent studies that indicate the possibility of the higher-cost firm’s leadership in homogeneous product markets, it also contrasts with the traditional literature that predicts the dominant-firm price leadership in various environments.     

Bargaining with revoking costs

A simple two stage bilateral bargaining game is analyzed. The players simultaneously demand shares of a unit size pie. If the demands add up to more than one, the players simultaneously choose whether to stick to their demand or accept the other?s offer. While both parties sticking to their offers leads to an impasse, accepting a lower share than the original demand is costly for each party. The set of pure strategy subgame perfect equilibria of the game is characterized for continuously differentiable payoff and cost functions, strictly increasing in the pie share and the amount conceded, respectively. Higher cost functions are shown to improve bargaining power. The limit equilibrium prediction of the model, as the cost functions are made arbitrarily high, selects a unique equilibrium in the Nash Demand Game that corresponds to a Proportional Bargaining Solution of Kalai (1977).     

Asymmetric technologies in contests

I consider a contest between two risk-neutral players over a common-value prize, in which one player has a linear cost-of-effort function and the other a strictly convex cost-of-effort function f. I show that if the value of the prize is above (below) a certain threshold level, then the equilibrium aggregate effort in this contest is larger (smaller) than in a contest in which both players are characterized by the strictly convex cost-of-effort function f, and smaller (larger) than in a contest in which both players are characterized by a linear cost-of-effort function. Therefore, in contrast to the general result in the literature, asymmetry in contests can increase competition.     

Duplicating and Pricing Contingent Claims in Incomplete Markets

The paper studies the problem of pricing contingent claims in the situation where the constraints imposed on an investor's portfolios are important. There are two types of rule of constraint: under a rigid rule, an investor must strictly limit his portfolios inside the constraint; under an elastic rule, an investor pays a penalty when the constraint is violated. The central problem of pricing a contingent claim is to determine the initial investment required to duplicate the contingent claim. The following results are obtained: (i) under elastic rules, the cost to duplicate a contingent claim exists and is unique;(ii) this cost depends nonlinearly and convexly on the contingent claim;(iii) the cost under rigid rules resulting from passing the penalty to infinity is also a nonlinear and convex function of the contingent claim. Owing to this nonlinearity, the cost of duplication may be or may not be the nonarbitrage price of the contingent claim; this depends on how the market organizes the production of contingent claims. The conclusion that the cost of duplication is a convex function of the contingent claim provides an explanation for why the service of providing contingent claims is often a highly profitable business. The main mathematical tool in the analysis is backward stochastic differential equations (BSDEs). In fact the cost to duplicate a contingent claim is the solution of a BSDE in which the contingent claim is the terminal value.     

Setting Price Controls While Facing Variable Or Uncertain Market Conditions

Price controls under variable or uncertain market conditions do not lead to market equilibrium. Under different assumptions for the rationing mechanism during shortages and surpluses, I find, assuming small market shocks, that the optimal regulated price can be related in a simple way to the relative slopes of the marginal benefit and marginal cost functions. In addition, if the consumption price may differ from the production price, then consumers should pay less than or equal to what producers receive, implying possibly a unit subsidy to market transactions.     

Bargaining foundation for ratio equilibrium in public‐good economies

We provide a bargaining foundation for the concept of ratio equilibrium in public‐good economies. We define a bargaining game of alternating offers, in which players bargain to determine their cost shares of public‐good production and a level of public good. We study the stationary subgame perfect equilibrium (SSPE) without delay of the bargaining game. We demonstrate that when the players are perfectly patient, they are indifferent between the equilibrium offers of all players. We also show that every SSPE without delay in which the ratios offered by all players are the same leads to a ratio equilibrium. In addition, we demonstrate that all equilibrium ratios are offered by the players at some SSPE without delay. We use these results to discuss the case when the assumption of perfectly patient players is relaxed and the cost of delay vanishes.     

Licensing cost-reducing innovations under supply function competition

In this paper, we study the licensing of cost-reducing innovations in a duopoly under supply function competition. We show that the innovator prefers fixed-fee licensing to no licensing if its cost advantage is not extremely large. Moreover, if its cost advantage is not extremely small, the innovator prefers fixed-fee licensing and mixed licensing to revenue-royalty licensing. This second result arises only because of our assumption that the cost functions are quadratic. We show that if the cost functions are linear and royalty payments are per unit of output, the innovator strictly prefers royalty licensing to fixed-fee licensing. We also show that in our model, consumer surplus is remarkably higher under any type of licensing than under no licensing. However, revenue-royalty licensing is slightly superior for consumers to the other two types of licensing and only so if the cost advantage of the innovator is sufficiently large.     

Differentiable Technology,the Curvature of the Profit Function,and the Response of Supply to Own-Price Changes

The author establishes a property of supply for a competitive firm: Assuming differentiability of the production frontier, linearly independent price vectors have disjoint image sets under the supply mapping. This property supports the main results. First, the author drew a simple proof of McFadden's proposition that differentiability of the production frontier is necessary and sufficient for strict quasiconvexity of the profit function. This proposition rules out linearity of the profit function on any subset of price domain, in turn ruling out supply and input- demand functions that are constant and have zero values for all price effects. Second, for discrete price changes, own-price effects in supply are strictly positive, assuming a differentiable technology. In this context, strict convexity of the production set is irrelevant. The author indicates implications for cost and demand theory.     

Quality Competition and Social Welfare in Markets with Partial Coverage: New Results*

We use a vertical product differentiation model under partial market coverage to study the social welfare optimum and duopoly equilibrium when convex costs of quality provision are either fixed or variable in terms of production. We show the following new results. First, under fixed costs, the social planner charges a uniform price for the single variant that just covers costs of quality provision. Like the duopoly equilibrium, this socially optimal pricing entails a partially uncovered market, but a smaller share of the market is served compared with the duopoly equilibrium. Second, for the variable cost case, it is socially optimal to provide both high‐ and low‐quality variants, but market shares need not be equal. This differs from the result in fully covered markets. Third, in the duopoly equilibrium, the quality spread is too wide under variable costs relative to the social optimum. Under fixed costs, the duopoly produces two variants, but quality is too low relative to the social optimum, which has only one variant.     

Differentiability of the cost functions is equivalent to strict quasiconcavity of the production functions

The assumption that cost functions are differentiable with respect to input prices is equivalent to the assumption that production functions are strictly quasiconcave.     

Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types

We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence–Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.     

ω‐Homothetic Preferences: Theory and Applications

This paper develops a new class of homothetic preferences which generate Marshallian demand curves for individual goods which can be concave, convex or linear in own price under the assumption that agents treat aggregate price indices as given (as in Dixit‐Stiglitz, 1977). The preferences are represented by a cost function which has two parameters: one determining the curvature of the Marshallian demand; the other determining the elasticity of demand when all prices are equal. The elasticity of demand varies with relative prices. Illustrative examples are given of Cournot duopoly and exchange rate pass‐through.     

Notes on optimal transportation

We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence–Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.     

THE DETERMINANTS OF THE QUANTITY‐QUALITY BALANCE IN MONOPOLY*

This paper describes how a monopolist manipulates the balance of quantity and quality in order to increase revenue when its customers treat quantity and quality as substitutes. This ‘skewing’ of quality depends on the characteristics of customer's demand for quality. Customers differ in demand for quality, because they differ in either (i) their preferences and/or (ii) their time cost per unit. The monopolist is constrained to supply the same quality of good to all customers. The price and quality per unit are described under the assumption the monopolist (i) profit maximises; (ii) maximises social welfare subject to a profit constraint. The determinants of the skewing of quantity and quality are found under third‐degree price discrimination and uniform pricing.     

Implementation with partial provability

We extend implementation theory by allowing the social choice function to depend on more than just the preferences of the agents and allowing agents to support their statements with hard evidence. We show that a simple condition on evidence is necessary for the implementation of a social choice function f when the preferences of the agents are state independent and sufficient for implementation for any preferences (including state dependent) with at least three agents if the social planner can perform small monetary transfers beyond those called for by f. If transfers can be large, f can be implemented in a game with perfect information when there are at least two players under a boundedness assumption. For both results, transfers only occur out of equilibrium. The use of evidence enables implementation which is robust in the sense that the planner needs little information about agents? preferences or beliefs and agents need little information about each others? preferences. Our results are robust to evidence forgery at any strictly positive cost.     

Endogenous market power

In this paper we develop a framework to study markets with heterogeneous atomic traders. The competitive model is augmented as we provide traders with correct beliefs about their price impacts to define equilibrium with endogenously determined market power and show that such equilibrium exists in economies with smooth utility and cost functions and is generically determinate. Traders? price impacts depend positively on the convexity of preferences or cost functions of the trading partners and are subject to mutual reinforcement. Compared to the competitive model, the volume of trade is reduced, and hence is Pareto inefficient. The price effects of non-competitive trading depend on the convexity of marginal utility or cost function.     

When does tiered wholesale pricing create an incentive to reduce retail prices?

Wholesale ‘ladder pricing’ involves setting the wholesale price retailers face as a nonlinear (generally increasing) function of price chosen by retailers. This form of wholesale pricing occurred recently in UK Telecoms, and the issue became extensively debated in the law courts. A major concern in deciding the merits of the case lay with the question of whether or not the introduction of tiered wholesale pricing created incentives for retailers to actually reduce their prices. This paper examines the incentive for the case where the wholesale tariff is a non-linear continuous differentiable function. It is shown that so long as the tariff is strictly increasing, convex, and positive only for retail prices greater than the maximum retailer marginal cost, then there is indeed an incentive to reduce price, whatever the actual gradient of the tariff schedule.     

On the existence of pure strategy Bertrand equilibrium

Summary The paper analyses the existence of pure strategy Nash equilibrium in price competition (or Bertrand equilibrium) in a homogeneous product market when costs are strictly convex and proves that if output is demand determined such equilibrium always exists. This paper also characterises such equilibria and shows that if firms are identical such equilibria are necessarily non-unique. However for firms with asymmetric costs it can be unique or non-unique.I am greatly indebted to Anjan Mukherji and Kunal Sengupta for this paper. I also express my profound gratitude to a referee of this journal for a very helpful set of suggestions.     

A note on an intertemporal cost function for a class of dynamic problems

Some properties of an intertemporal cost function for isoperimetric problems are analyzed. It is demonstrated that the cost function will be strictly decreasing in the length of the horizon T, and under certain conditions, strictly convex in T.     

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