Theory and Application of Linear Supply Function Equilibrium in Electricity Markets

We consider a supply function equilibrium (SFE) model of interaction in an electricity market. We assume a linear demand function and consider a competitive fringe and several strategic players having capacity limits and affine marginal costs. The choice of SFE over Cournot equilibrium and other models and the choice of affine marginal costs is reviewed in the context of the existing literature. We assume that bid rules allow affine or piecewise affine non-decreasing supply functions by firms and extend results of Green and Rudkevitch concerning the linear SFE solution. An incentive compatibility result is proved. We also find that a piecewise affine SFE can be found easily for the case where there are non-negativity limits on generation. Upper capacity limits, however, pose problems and we propose an ad hoc approach. We apply the analysis to the England and Wales electricity market, considering the 1996 and 1999 divestitures. The piecewise affine SFE solutions generally provide better matches to the empirical data than previous analysis.     

Supply function equilibria: Step functions and continuous representations

In most electricity markets generators must submit step-function offers to a uniform price auction. These markets are often modelled as simpler pure-strategy Supply Function Equilibria (SFE) with continuous supply functions. Critics argue that the discreteness and discontinuity of the required steps drastically change Nash equilibria, invalidating predictions of the SFE model. We prove that there are sufficient conditions, offered quantities can be continuously varied, offered prices are selected from a finite set, and the density of the additive demand shock is not too steep, where the resulting stepped SFE converges to the continuous SFE as the number of steps increases, reconciling the apparently very disparate approaches to modelling electricity markets.     

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.     

Perfectly competitive markets as the limits of cournot markets

A perfectly competitive, partial equilibrium market for a single homogeneous good with a (bounded) continuum of infinitesimal firms is considered. Cost functions are essentially unrestricted and are allowed to vary smoothly across firms. A sequence (net) of Cournot markets (each with a finite number of firms) which converge smoothly to the perfectly competitive limit in terms of both the inverse demand functions and the distributioon of firm technologies is introduced and it is shown that all markets sufficiently far along the sequence have a Cournot equilibrium and all the Cournot equilibria converge to the perfectly competitive equilibrium of the limit market.     

Search cost,trading strategies and optimal market structure

The theory of Walrasian equilibrium yields a set of prices at which the aggregate competitive demand for each commodity equals its aggregate competitive supply. However, even at equilibrium prices the theory of competitive equilibrium does not explicitly offer explanation regarding the manner in which trades are actually executed. This paper considers a model where trade takes place in a decentralized fashion and examines in a dynamic game-theoretic framework, the role of social institution of money and markets in facilitating exchange. The steady state Nash equilibrium derived in the paper demonstrates how, depending on the level of transaction costs associated with a market setup (synonymously, trading posts to exchange possible pairs of goods) appropriate monetary trade emerges, which like a hub and spoke network (Starr and Stinchcombe, 1999) makes some markets non-functioning and in equilibrium only the markets having trade through the medium of exchange continue to exist. However, despite the obvious advantages of a market setup in reducing search costs, pure random search for a complementary trading partner (as considered by Ostroy and Starr, 1974; Kiyotaki and Wright , 1989; and others) prevails in many economies, especially, in many developing economies. This paper models this feature of developing economies by introducing differences in transaction costs across agents and shows why sustainable equilibria might exist exhibiting random search for certain commodities even in the presence of established markets.     

Nonlinear incentive provision in Walrasian markets: a Cournot convergence approach

The paper studies insurance with moral hazard in a system of contingent-claims markets. Insurance buyers are modelled as Cournot monopolists or oligopolists. The other agents condition their expectations on market prices, as in models of rational-expectations equilibrium with asymmetric information. Thereby they correctly anticipate accident probabilities corresponding to effort incentives induced by insurance buyers’ net trades. When there are many agents to share the insurance buyers’ risks, Cournot equilibrium outcomes are close to being second-best. In contrast, if insurance buyers are price takers, equilibria fail to exist or are bounded away from being second-best.     

Computing Cournot equilibrium through maximization over prices

This paper presents a characterization of internal Cournot equilibrium based on first-order conditions corresponding to profit maximization over prices. This characterization may yield significant computational advantage as demand functions need not be inverted and simple first-order conditions are obtained.     

Non-existence of competitive equilibria with dynamically inconsistent preferences

This paper shows the robust non-existence of competitive equilibria even in a simple three period representative agent economy with dynamically inconsistent preferences. We distinguish between a sophisticated and naive representative agent. Even when underlying preferences are monotone and convex, at given prices, we show by example that the induced preference of the sophisticated representative agent over choices in first-period markets is both non-convex and satiated. Even allowing for negative prices, the market-clearing allocation is not contained in the convex hull of demand. Finally, with a naive representative agent, we show that perfect foresight is incompatible with market clearing and individual optimization at given prices.     

TO REGULATE OR TO DEREGULATE? THE ROLE OF DOWNSTREAM COMPETITION IN UPSTREAM MONOPOLY VERTICALLY LINKED MARKETS

This paper attempts to cast light to the effect of monopoly regulation in Cournot markets compared to its effect in Bertrand markets. To this purpose, we use a simple model of a vertically linked market, where an upstream regulated natural monopoly is trading via two‐part tariff contracts with a downstream duopoly. Combining our results to those of the existing literature on deregulated markets, we argue that when the downstream competition is in prices, efficiency dictates regulating the monopoly with a marginal cost based pricing scheme. However, this type of regulation leads to significant welfare loss, when the downstream market is characterized by Cournot competition.     

Non-Walrasian equilibria and the law of one price

We present a market game which features multiple posts for each commodity. We use this framework to illustrate the idea that in non-Walrasian markets, where individual activities influence market clearing prices, there are equilibria where commodities are exchanged simultaneously in two posts at different prices, thus defying the ‘law of one price’. Such equilibria are compatible with an apparent arbitrage possibility, which dissipates whenever individuals try to take advantage of it.     

On persistent disequilibrium

Consider a standard model of a Walrasian economy where the time derivative of price change is a sign preserving function of excess demand. The only steady states of such a system are the Walrasian equilibria. For if the system is not at a Walrasian equilibrium then there must be excess demands or supplies in some market and thus those prices must be changing. Walras' law implies that all prices cannot change in the same direction, and thus relative prices must change.However, it seems that in real world economies there have been states of persistent disequilibrium. How can this be? How can there be stationary states of an economy that are not Walrasian equilibria?The answer presented in this paper goes something like this: the demands actually presented to the market, i.e., the demands that affect price movements, are not the Walrasian demands. Rather, these demands, the effective demands, are a function of both price and quantity signals. There is no reason why there cannot be equilibria of the effective demand system that are not equilibria of the Walrasian demand system.In this paper, I present a simple abstract model of effective demand systems and given a condition for this system to have non-Walrasian equilibria. I also present a simple example of this phenomena. Finally I discuss the real world implications of this model. Related concepts of effective demand are discussed in [A1H], [2], [3], [L5], and [L6].     

Withholding investments in energy only markets: can contracts make a difference?

Although there are mechanisms to control market power in the spot market, withholding investments can still increase profits and hamper adequate capacity expansion. We examine the effect on investment of one suggested approach to reducing market power, contracting longer term. We construct a stylized model of an energy-only market where two firms, each specializing in one technology, invest in a first stage, contract part of their production in the second stage and sell the rest in the spot market in the third stage. We compare this model to one of an energy-only market having two stages, investment and a spot market. We find cases where the contracts change neither capacity nor peak prices, where the foreclosing effect of one player blocking the other from contracts markets increases investments and reduces prices, and where the opportunity to foreclose the market can incentivize one firm to lower its investment and increase its pricing power to the detriment of consumers. The model relies on the simplest possible assumptions of imperfect competition (subgame perfect equilibria with Cournot agents). We illustrate the different outcomes in a numerical example with two load steps (peak and off-peak) where we change one parameter, the height of the off-peak time segment. We find cases with increased and decreased capacity as well as no change in capacity. Since there is no general characterization of the consequences of contracts in this simple example, there can be no characterization in more complicated models that contain the market structures included here, and regulators or competition authorities cannot rely on contracts to induce sufficient capacity expansion by reducing market power. One other market mechanism that has been proposed to induce investment, a capacity auction with predetermined capacity requirements, is a potential alternative to limit market power that deserves further exploration to determine the extent to which it can provide an adequate incentive to invest in the presence of market power.     

Randomized double auctions: gains from trade,trader roles,and price discovery

Experimental double-auction commodity markets are known to exhibit robust convergence to competitive equilibria under stable or cyclical supply and demand conditions, but little is known about their performance in truly random environments. We provide a comprehensive study of double auctions in a stochastic setting where the equilibrium prices, trading volumes and gains from trade are highly variable across periods, and with commodity traders who may buy or sell their goods depending on market conditions and their individual outcomes. We find that performance in this stochastic environment is sensitive to underlying market conditions. Efficiency is higher and convergence to the competitive equilibrium stronger when the potential gains from trade are high and when the equilibrium spans a wide range of quantities, implying a large number of marginal trades. Speculative re-trading is prevalent, especially among those who have little to gain under equilibrium pricing. Those with the largest expected gains typically earn far less than predicted, while those with little or no predicted earnings gain modestly from speculation, leading to some redistribution of gains from high to low expected earners. Excessive trading volumes are associated with negative efficiencies in markets with low gains from trade, but not in the high-gains markets, where zero-sum trading and re-trading appear to enforce efficiency and near-equilibrium pricing. Buyers earn more relative to their competitive equilibrium benchmark than sellers do. Introducing trader specialization leads to fewer trading errors and higher market efficiency, but it does not eliminate zero-sum trading and re-trading.

     

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.     

Coalition-proof Bertrand equilibria

Summary. This paper examines the coalition-proof Nash equilibria of a Bertrand model of price competition where firms supply all demand. When firms are asymmetric we prove existence and provide a sufficient condition for uniqueness. For symmetric firms, we show that an equilibrium is necessarily unique. We also examine whether this unique equilibrium outcome is implementable through a sequential move game where the firms take turns at announcing prices. Finally we examine the limiting property of such equilibria as the number of firms go to infinity.Received: 20 March 2002, Revised: 5 August 2003JEL Classification Numbers: D43, D41, L13.Correspondence to: Kunal SenguptaWe are deeply indebted to an anonymous referee for very helpful and incisive comments that led to substantial improvements in the paper. We also gratefully acknowledge the hospitality of the Department of Finance, Hong Kong University of Science and Technology where much of the work on this paper was carried out.     

PLATFORM COMPETITION WITH ENDOGENOUS HOMING

We consider two-sided markets in which consumers and firms endogenously determine whether they single-home, multi-home, or exit the market. We find that the competitive bottleneck allocation in which consumers single-home and firms multi-home is always an equilibrium. In addition, we find equilibria with multi-homing and single-homing on each side of the market. However, unlike the standard pricing result where the side that multi-homes faces higher prices, we find that lower prices coincide with multi-homing: agents find multi-homing more attractive when faced with lower prices. We also show that endogenous homing can induce straddle pricing which deters price undercutting between platforms.     

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.     

N‐FIRM OLIGOPOLY WITH GENERAL ISO‐ELASTIC DEMAND

In this note oligopoly with iso‐elastic demand is analysed. Unlike previous studies we consider general iso‐elastic demand rather than the case of unit elasticity. An n‐firm Nash‐Cournot equilibrium for the case of heterogeneous constant marginal costs is derived. The main result is a closed‐form solution that shows the dependency of the equilibrium on the elasticity of demand and the share of industry costs. The result has applications to a wide range of areas in oligopoly theory by allowing comparisons across markets with different elasticities of demand.     

Supply function equilibria of pay-as-bid auctions

This paper characterizes the Nash equilibrium in a pay-as-bid (discriminatory), divisible-good, procurement auction. Demand by the auctioneer is uncertain as in the supply function equilibrium model. A closed form expression is derived for a one shot game. Existence of an equilibrium is ensured if the hazard rate of the demand distribution is monotonically decreasing with respect to the shock outcome and sellers have non-decreasing marginal costs. Multiple equilibria can be ruled out for markets, for which the auctioneer’s demand exceeds suppliers’ capacity with a positive probability. The derived equilibrium can be used to model strategic bidding behavior in pay-as-bid electricity auctions, such as the balancing mechanism of United Kingdom. Offer curves and mark-ups of the derived equilibrium are compared to results for the SFE of a uniform-price auction.      

Multiplicity of Cournot Equilibria and Involuntary Unemployment

In an overlapping generations model with Cournot competition on the goods market it is shown that a continuum of stationary states and perfect foresight trajectories exists with unemployment at arbitrary low wages. Decisive for this is the influence that different forecast functions have on the objective demand curve, even though they are consistent with perfect foresight. With an example it is shown that simple adaptive and constant memory forecast rules generate such unemployment equilibria. The corresponding temporary equilibrium dynamics may display stable unemployment and unstable full employment equilibria.Journal of Economic LiteratureClassification Numbers: D43, D51, E24