Stability of supply function equilibria implications for daily versus hourly bids in a poolco market

We consider a supply function model of a poolco electricity market where demand varies significantly over a time horizon such as a day and also has a small responsiveness to price. Although there are equilibria yielding prices at peak that are close to Cournot prices, it is known that the wider the range of demand uncertainty the narrower the range of such supply function equilibria. Here we show that such equilibria are also typically unstable and consequently would be difficult to sustain in practice. This strengthens the results of Green and Newbery by ruling out many equilibria that have high prices. We demonstrate this result both theoretically under somewhat restrictive assumptions and also numerically using both a three-firm example system and a five-firm example system having generation capacity constraints. Hence, this reinforces the conclusion that the market outcome is significantly influenced by a requirement that offers into the poolco be consistent over the time horizon. This result contrasts with markets where bids can be changed on an hourly basis, where Cournot prices are possible outcomes. The stability analysis has important policy implications for the design of day-ahead electricity markets. The stability perspective also provides a narrowing of the equilibrium selection that strengthens empirical analysis.      

Equilibrium prices in the Bertrand and Cournot oligopolies

The equilibrium prices for the Bertrand and Cournot oligopolies with product differentiation are compared. If all firms have linear demand and cost functions, and if, in addition, the Jacobian matrix of the demand functions has a dominant negative diagonal, the Cournot equilibrium prices are not lower than the Bertrand ones. The general condition for the comparison of the Bertrand and Cournot equilibrium prices can be derived even if the nonlinearity is involved in the cost and/or demand functions.     

Kreps & Scheinkman with Product Differentiation

Kreps and Scheinkman (1983)’s celebrated result is that in a two-stage model of a market with homogeneous products in which firms noncooperatively pick capacities in the first stage and set prices in the second stage, the equilibrium outcome is that of a one-shot Cournot game. This note derives capacity best response functions for the first stage and extends the Kreps and Scheinkman result to the case of differentiated products.     

Capacity precommitment and price competition yield the Cournot outcome

We introduce a simple model of oligopolistic competition where firms first build capacity, and then, after observing the capacity decisions, choose a reservation price at which they are willing to supply their capacities. This model describes many markets more realistically than the model of Kreps and Scheinkman [Kreps, D., Scheinkman, J., 1983. Quantity precommitment and Bertrand competition yield Cournot outcomes. Bell J. Econ. 14, 326–337]. We show that in this new model every pure strategy equilibrium yields the Cournot outcome, and that the Cournot outcome can be sustained by a pure strategy subgame perfect equilibrium.     

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.     

Coalition-proof supply function equilibria in oligopoly

In an industry where firms compete via supply functions, the set of equilibrium outcomes is large. If decreasing supply functions are ruled out, this set is reduced significantly, but remains large. Specifically, the set of prices that can be sustained by supply function equilibria is the interval between the competitive price and the Cournot price. In sharp contrast, when the number of firms is above a threshold we identify (e.g., three if demand is linear), only the Cournot outcome can be sustained by a coalition-proof supply function equilibrium.     

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.     

Separation of ownership and labour: Welfare considerations

Conclusion Separation of ownership and labour is a very appealing way to avoid the free-rider problem in productive teams. But if productive activity of the owner is possible, then in many cases the equilibrium outcome of an enlarged partnership with equal shares Pareto-domi-nates that of a capitalistic firm where the owner is inactive.University of Mannheim. An earlier draft of this paper appeared as Discussion Paper No. 100 of the Sonderforschungsbereich 21 at the University of Bonn, September 1982. The paper was partly revised and completed while I was visiting scholar at Harvard University. I wish to thank Tsuneo Ishikawa, Andreu Mas-Colell, and an anonymous referee for valuable comments. Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.     

Price and Quantity Competition in a Differentiated Duopoly With Network Compatibility Effects

We consider the efficiency of price and quantity competition in a network products market, where we observe product compatibility with network externalities (hereafter, network compatibility effects). In particular, if network compatibility effects between firms are sufficiently asymmetric, the Cournot equilibrium is more efficient than the Bertrand equilibrium in terms of larger consumer, producer and total surpluses. Then, we consider an endogenous choice of the strategic variables, price and quantity. If the degree of network compatibility effects of the rival firm is larger (smaller) than the degree of product substitutability, then choosing prices (quantities) is a dominant strategy for the firm. Thus, if the network compatibility effects of both firms are larger (smaller), the Bertrand (Cournot) equilibrium arises. Furthermore, if the network compatibility effects between the firms are sufficiently asymmetric, the firm with a larger (smaller) network compatibility effect than a certain level of product substitutability chooses quantities (prices). In this case, the Cournot–Bertrand equilibrium arises, which is less (more) efficient than the Cournot equilibrium in terms of consumer (producer) surplus.     

Capacity Choices and Price Competition in Experimental Markets

We report on experimental duopoly markets with heterogeneous goods. In these markets, sellers first choose capacities and then prices. While capacities remain fixed for either five or ten periods, prices have to be chosen in every period. The experiments starts with two sets of exogenously predetermined capacities. Independently of the distribution of capacities is, a unique pure-strategy in prices is subgame perfect. In equilibrium, capacities should correspond to the Cournot prediction. Given capacities, price-setting behavior is in general consistent with the theory. Average capacities converge above the Cournot level. Capacities converge at the industry level but are somewhat dispersed. Sellers rarely manage to cooperate.     

Competition with Congestible Networks

We analyse competition between two network providers when the quality of each network depends negatively on the number of customers connected to that network. With respect to price competition we provide a sufficient condition for the existence of a unique pure strategy Nash equilibrium. Comparative statics show that as the congestion effect gets stronger quantities will decrease and prices increase, under both Bertrand and Cournot competition. In an example with endogenous capacities it turns out that equilibrium capacities are at first increasing and then decreasing in the strength of congestion. Furthermore, capacities are higher under Cournot competition. Welfare comparisons between Bertrand and Cournot competition are unambiguous for fixed capacities, but may turn around for endogenous capacities.     

Price versus quantity in a duopoly of vertical differentiation with loss-averse consumers

In a model à la Mussa and Rosen (1978) in which consumers are loss-averse, I check the robustness of the result obtained by Tanaka (2001). As he did, I find that the quantity contract is a dominant strategy for both firms. Thus, Cournot is the outcome in equilibrium. Finally, I find that loss aversion in general intensifies competition.     

Market crashes,speculation and learning in financial markets

A natural conjecture is that if agents’ beliefs are almost correct then equilibrium prices should be close to rational expectations prices. Sandroni (J Econ Theory 82:1–18, 1998) gives a counterexample in an economy with sunspots and complete markets. We extend Sandroni’s result by showing that the conjecture is generically true for economies with complete markets. We consider a standard General Equilibrium model with large but finite horizon and complete markets. We show that, for almost every such economy, if conditional beliefs eventually become correct along a path of events then equilibrium prices of assets traded along this path converge to rational expectations equilibria in the sup-norm. Moreover, we establish that, generically, there exist along any such path local diffeomorphisms between individual beliefs and equilibrium prices. I would like to thank C. Ewerhart and A. Kirman for their comments, as well as the seminar participants at the University of Minho, the General Equilibrium Workshop 2005 in Zurich, and the 15th Asian General Equilibrium Conference 2007 in Singapore. An anonymous referee also provided very valuable comments.     

Strategic interaction between futures and spot markets

We study an oligopolistic industry where firms are able to sell in a futures market at infinitely many moments prior to the spot market. A kind of Folk-theorem is established: any outcome between perfect competition and Cournot can be sustained in equilibrium. We then find that the Cournot outcome can be sustained by a renegotiation-proof equilibrium. However, this is not true for the competitive outcome. Furthermore, only the monopolistic outcome is renegotiation-proof if firms can buy and sell in the futures market. These results suggest, contrary to existing literature, that the introduction of futures markets may have an anti-competitive effect.     

A Two-stage Cournot Duopoly with Inventory Costs

This paper analyses the Cournot duopoly model which has two production periods before the market clears. As shown by Saloner (1987), if inventory costs are zero, many outcomes including both Cournot and Stackelberg outcomes are subgame perfect equilibrium outcomes. However, if small inventory costs are introduced, the Cournot outcome is no longer found in equilibrium and the equilibrium outcomes are only of Stackelberg-type. This suggests that more attention should be paid to the Stackelberg model than to the Cournot model.     

Macroeconomic effects of an indirect tax substitution

We study an indirect tax reform in a general equilibrium model with imperfect competition for both the Cournot and the Free entry equilibria. We show that it is possible to attain a positive balanced budget multiplier by means of a substitution of specific by ad valorem taxation. Moreover, although any tax substitution causes higher prices and the flow up of firms in the long-run, the Free entry equilibrium output can increase with respect to that of the Cournot equilibrium. Finally, in contrast with the partial equilibrium, welfare decreasing tax reforms are likely to occur even when the balanced budget multiplier is positive.      

Price normalization under imperfect competition

In general equilibrium models of imperfect competition the equilibria depend on how prices are normalized. This note shows that a price normalization preserves convexity properties if and only if prices are measured in terms of a fixed commodity bundle. I am grateful to an anonymous referee for helpful comments, inspiring a simplification in the proof of the main result, and the argument in the Remark.     

Duopoly competitions with capacity constrained input

This paper focuses on the duopoly substitutability product with an upstream input subjected to capacity constraints. The effects of capacity constraints are captured. Combining competition effect with constraint effect, some interesting conclusions are reached. First, the relationship between capacity constraints and firm size is addressed. We argue that the capacity constraints reduce market size difference and price difference under Cournot. Second, under the Stackelberg case, the existence of solution is proved, and Stackelberg competitions enlarge firm-size difference and price difference if the more efficient firm plays the leading position. When the weaker firm plays the leading position, the conclusions depend on the total capacity. Finally, under the Stackelberg case, when the stronger firm plays the leading position, the firm-size difference and price difference decrease with total input under capacity constraints, which is contrary to the conclusions under Cournot competitions.     

Bertrand vs. Cournot equilibrium with risk averse firms and cost uncertainty

Summary. In an oligopoly game with cost uncertainty and risk averse firms, we show that Bertrand and Cournot equilibrium have different convergence properties when the market is replicated. The Cournot equilibrium price converges to the competitive price. Under very typical and somewhat general conditions, the highest Bertrand equilibrium price converges to one higher than the competitive equilibrium. We also give examples to show how to compute the limit of the highest Bertrand equilibrium prices and illustrate the ideas of the proof. We explore conditions under which the supply curve is upward sloping, a useful condition for our results. Received: April 20, 2000; revised version: May 10, 2001