Energy Economics 31 (2009) 38–47

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Energy Economics

j ourna l homepage: www.e lsev ie r.com/ locate /eneco

Cournot versus Supply Functions: What does the data tell us?

Bert Willems a,b,⁎, Ina Rumiantseva c, Hannes Weigt c

a TILEC CentER, Tilburg University, Tilburg, The Netherlandsb Energy Institute and CES, K.U.Leuven, Leuven, Belgiumc Faculty of Business and Economics, Chair of Energy Economics and Public Sector Management, Dresden University of Technology, Dresden, Germany

⁎ Corresponding author. P.O. Box 90153, room K308lands. Tel.:+31 13 466 2588.

E-mail addresses: mail@bertwillems.com (B. Willemina.rumiantseva@mailbox.tu-dresden.de (I. Rumiantsevahannes.weigt@tu-dresden.de (H. Weigt).

1 See Twomey et al. (2004) for an overview of the basianalysis tools suitable for electricity markets.

0140-9883/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.eneco.2008.08.004

a b s t r a c t

a r t i c l e i n f o

Article history:

This paper compares two Received 29 October 2007Received in revised form 13 August 2008Accepted 13 August 2008Available online 22 August 2008

JEL classification:L94L13C72D43

Keywords:Supply Function EquilibriumCournot competitionElectricity markets

popular models of oligopolistic electricity markets, Cournot and the SupplyFunction Equilibrium (SFE), and then tests which model best describes the observed market data. Usingidentical demand and supply specifications, both models are calibrated to the German electricity market byvarying the contract cover of firms. Our results show that each model explains almost the same fractions ofthe observed price variations. We therefore suggest using Cournot models for short-term analysis, since thesemodels can accommodate additional market details, such as network constraints, and the SFE model forlong-term analysis (e.g., the study of a merger) since it is less sensitive to the calibration parameters selected.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Within the last two decades, as electricity markets throughout theworld have undergone significant structural changes because ofliberalization, a demand for sophisticated market analyses hasemerged. Typically, such analyses are especially complex becausethe decentralized structure of the new markets increases theinteractions between market players, and factors such as regulatoryguidelines, market design (and redesign) and the particular nature ofelectricity itself must be considered. In addition, desire on the part ofregulators and politicians to determine market outcomes, interest insimulating expected market situations and modeling the behavior ofmarket players have fostered the development of novel approaches.

Since classical economic tools, such as the Herfindahl–HirschmanIndex (HHI), are largely unsuitable for electricity markets, specifictools and models must be designed (Borenstein et al., 1999).1 Ventosa

, NL-5000 LE Tilburg, Nether-

s),),

c characteristics and economic

l rights reserved.

et al. (2005) identify three basic trends: optimization models, equi-librium models and simulation models. This paper will focus on twoequilibriummodels for oligopolistic wholesale electricity markets: theCournot model and the Supply Function Equilibrium model (SFE).While standard Cournot models are simple to calculate, the resultsoften do not represent reasonable market outcomes. For realisticvalues of demand elasticities, prices are too high and output too low.In Cournot models for electricity markets, it is, therefore, usuallyassumed that a fixed percentage of sales are covered by forwardcontracts.2

Supply Function Equilibrium models (e.g. Klemperer and Meyer,1989) on the other hand are deemed to represent electricity marketsmore realistically because they assume that generators, instead of onesingle quantity, compete by bidding complete supply functions in anoligopolistic market with demand uncertainty. The SFE approach hasbeen used in several applications to analyze electricity markets sinceits first application by Green and Newbery (1992) for England andWales. Themajor drawbacks of SFEmodels are that they are difficult tocalculate, often have multiple equilibria, often give unstable solutionsand require strong simplifications with respect to market and coststructures.

2 Alternatively, one could assume that generators are vertically integrated with thretail sector for a portion of their activities.

e

39B. Willems et al. / Energy Economics 31 (2009) 38–47

Given the strengths and weaknesses of both approaches, we set out todiscover whether the two models offer similar predictions and rangesof feasible outcomes. We ask whether the added complexity of SFEmodels relative to Cournot models can be compensated for by morerobust, realistic predictions. In order to investigate this issue, wecalibrate the two models with an identical dataset from the Germanelectricity market, and then compare the results with the observedmarket clearing prices.

The remainder of our paper is structured as follows: In the nextsection, an overview of the theory of SFE and Cournot modeling isgiven. Section 3 describes the implementation of the models and theunderlying assumptions. Section 4 presents a dataset for the Germanmarket and the model calibration methods. Section 5 presents anddiscusses the simulation results. We find that Cournot approachesallow for more complex market models while SFE models are lesssensitive to adjustments. Section 6 closes with a summary and ourconclusions about the suitability of each approach for short- and long-term economic analysis and policy-making.

2. Literature review

2.1. Theoretical background

Twomajor trends have emerged to analyze oligopolistic electricitymarket outcomes, in particular, wholesale markets.3 Both approachesassume that companies are profit-maximizing, but differ in theassumption regarding the free choice variables and the behavior of theremaining market participants. We now discuss these differences.

In both models the profit function of the firms is equal to therevenue minus generation costs:

p qi−ci qið Þ ð1Þwith qi the output of firm i, p the price of electricity, and ci(.) theproduction cost of the firm i. For this paper we assume that the inversedemand function depends on total production ∑

jqj, the time period t

and a random error component ξ. Hence we have the following.

p ¼ p ∑jqj; t; �

!ð2Þ

Generally speaking, a firm i can submit a bid function to the powerexchange that represents its willingness to supply a specific quantity qat a given price p, a given time period t and in cases when the firmsobserve the demand shock before bidding, the error term ξ.

qi ¼ Bi p; t; �ð Þ ð3Þ

The market clearing condition determines the production quan-tities and prices in each time period t and for each demand shock ξ;that is, they are determined jointly by Eqs. (2) and (3).

Cournot and SFEmodels make different assumptions regarding thestrategy space and the information set of the bidding firms. In aCournot equilibrium the demand realization (t, ξ) is known by the firmbefore bidding. The firms therefore bid the following:

qi ¼ BCi t; �ð Þ ð4Þ

It follows from Eq. (4) that the firm might bid a high quantityduring peak hours and a low quantity during off-peak hours. For agiven time period t and demand shock ξ each firm maximizes profitsby setting production quantities and sales qi knowing that the marketprice is a result of its own output and the output of its competitors q− i.

3 Another classic approach is the Bertrand model. As this is seldom used inelectricity markets, given the difficulty of presenting capacity constraints, we do notconsider it in detail.

The Cournot approach yields a direct outcome in terms of price andquantities for a given demand realization. Cournot models generallyprovide a unique Nash equilibrium.

In the SFE models (Klemperer and Meyer, 1989), firms cannotcondition their bids on the demand realization (t, ξ). It is assumed thatthe firms do not know the size of the demand shock ξ, and are notallowed to submit a different bid for different time periods t.4 Thebidding function depends solely on the price:

q ¼ BSFEi pð Þ ð5Þ

Each firmmaximizes (expected) profits by bidding a supply curveBiSFE(p), assuming that the supply function of its competitors B−i

SFE(p)remains fixed. Hence, the quantity produced by the competitorsdepends on the market price and, thus, indirectly on the outputdecision of the firm itself. In order to solve the SFE model, it isassumed that the stochastic demand shock and the time period shiftthe demand function horizontally:

p ¼ p ∑jqj−�−f tð Þ� � ð6Þ

Under these assumptions the (stochastic) optimization problemthat each firm must solve can be rewritten as a differential equation.Typically, a range of feasible equilibrium supply functions is foundwhen solving a set of differential equations, one for each firm. It can beshown that every SFE supply function lies between the Cournot andthe Bertrand solutions for any realized demand shock. Delgado andMoreno (2004) show, however, that only the least-competitiveequilibrium is coalition proof when the number of firms is sufficientlylarge. A serious drawback of SFE models is that they require simplifiedassumptions of the markets' supply structure to obtain feasiblesolutions.

Whether a more realistic representation of the bidding process isto neglect the price dependency as in the Cournot model, or to neglectthe market state dependency as in the SFE model is unclear a priori.

Note that if firms would have perfect information in the SFE model(as in the Cournot model), then both models would produce identicalresults. With perfect information, firms can coordinate on anyequilibrium for which the price is above the competitive price andbelow the Cournot price. As firms coordinate on the highest priceequilibrium, they will play the Cournot equilibrium.

2.2. Use of Cournot and SFE models in electricity markets

2.2.1. Cournot modelsAlthough Cournot models are ubiquitous, they often overestimate

observed market prices and underestimate market quantities. Sincethe model outcome is based only on quantity competition, the resultsare highly sensitive to assumptions about demand elasticity: inequilibrium, firm i sets its output such that its markup is proportionalto its market share si and inversely proportional to the demandelasticity ε of the total market.

p−cVip

¼ −sie

ð7Þ

Given that most electricity markets have few oligopolistic firms (siis large) and low, short-term demand elasticities, markups areaccordingly very high.

Following Allaz and Vila (1993), we allow for forward contractsthat can be used to predict more realistic outcomes. Firms can bothsell energy in a spotmarket and a certain amount of their supply in the

4 One of the referees correctly noted that upon learning the demand realization,firms do not have an incentive to “unilaterally” alter their bidding strategies. However,firms may choose to jointly deviate to a more profitable equilibrium. Hence, onceinformation is revealed, the SFE equilibrium is no longer coalitional proof.

40 B. Willems et al. / Energy Economics 31 (2009) 38–47

forward market. In a two-stage game, the oligopolists first determinethe quantity to be sold in the forward market before entering the spotmarket and playing a Cournot game. Thus, forward sales reduce theoligopolists' available quantity in the spot market, resulting in a morecompetitive market. When using a single-stage game, the impact offorward contracts can be simulated by accounting for the contractvolume Fi in the profit function:

p ∑jqj� �

qi−Fið Þ−ci qið Þ ð8Þ

This results in a reduced markup on marginal costs since thecontracting factor fi=Fi /qi must be considered:

p−cVip

¼ sie

1−fið Þ ð9Þ

Varying the contracting factor generates a “bundle” of feasibleCournot solutions that resemble SFE outcomes.

The role of forward contracts, when comparing Cournot resultswith real market outcomes, is demonstrated by Bushnell et al. (2008).They look at California, PJM and New England markets, comparingprice data of the power exchanges with competitive model outcomes,a standard Cournot model and a Cournot model using contract coveras an approximation for vertical arrangements. They conclude thatneglecting the contract cover yields results that vastly exceedobserved market prices. Ellersdorfer (2005) analyzes the competi-tiveness of the German electricity market using a multi-regional two-stage Cournot model. He shows the extent to which cross-bordernetwork extensions and increased forward capacities enhancecompetition and decrease market power.

Cournot approaches are often preferred when technical character-istics such as network constraints (voltage stability, loop flows) orgeneration characteristics (start-up costs, ramping constraints, unitcommitment) must be considered. The impact of congestion onmarket prices and market power has been analyzed in several studies.Smeers and Wei (1997) use variational inequalities to describe theCournot model. Willems (2002) discusses the assumptions necessaryto include transmission constraints in Cournot models. Neuhoff et al.(2005) summarize different characteristics of Cournot networkmodels, and show that although all models predict the sameoutcomes, they vary with respect to assumptions about market designand expectations of generators in cases of competitive markets.

A more general overview of Cournot models used to analyzemarket power issues appears in Bushnell et al. (1999). Other reviewsof electricity market models with respect to network issues are givenin Day et al. (2002) and Ventosa et al. (2005).

2.2.2. SFE modelsIn general, SFE models are frequently used in determining market

power issues. Bolle (1992) makes a theoretical application toelectricity markets by analyzing the possibility of tacit collusionwhen bidding in supply functions. He concludes that if firmscoordinate on bidding the highest feasible supply function, a decreasein market concentration does not necessarily result in convergence ofaggregated profits to zero. Green and Newbery (1992) present anempirical analysis for England and Wales using symmetric players.They compare the duopoly of National Power and PowerGen with ahypothetical five firm oligopoly, concluding that the latter results in arange of supply functions closer to marginal costs.

Baldick and Hogan (2002) set up anothermodel of the England andWales markets that incorporates price caps, capacity constraints, andvaries the time horizon by which supply functions must remain fixed.They show that the latter characteristic has a large impact on marketcompetitiveness. Evans and Green (2005) simulate the same marketfor the period of April 1997 to March 2004 and calculate the SupplyFunction Equilibrium taking into account the firms' asymmetricity.They approximate the competitive pressure in the market with ‘an

equivalent number of symmetric firms’ based on HHI and concludethat the change to a decentralizedmarket has no impact on short-termprices while reductions in concentration do have impacts. Sioshansiand Oren (2007) study the Texas balancing market, using SupplyFunction Equilibria with capacity constraints. They find that the largerfirms more or less behave according to the SFE for incremental bids.Theyargue that SFEmodelswith capacity constraints are an interestingtool to study balancing markets since demand is very inelastic, andhence the supply elasticity of competitors is a major component indetermining the elasticity of the residual demand of the firms. TheseSFE models with capacity constraints are also useful because theyreflect the actual bidding behavior of the firms, i.e. “hockey stickbidding”: firms bid (too) low for low levels of supply and have a steepsupply function for greater levels of supply.

Several theoretical contributions have extended Klemperer andMeyer (1989) by incorporating typical market characteristics. Holm-berg (2005) considers the problem of asymmetric companies bysimplifying the supply structure to constant marginal costs. He showsthat under this setup there is a unique SFE that is piece-wisesymmetric. Anderson and Hu (2005) propose a numerical approachusing piece-wise linear supply functions and a discretization of thedemand distribution. They show that the approach also has goodconvergence behavior in models with capacity constraints. Holmberg(in press) studies capacity constraints on generation units and showsthat a unique, symmetric SFE exists with symmetric producers,inelastic demand, price cap and capacity constraints. Green (1996),Rudkevich (2005) and Baldick et al. (2004) develop the theory of linearsupply functions. These functions are easier to solve, can also be used inasymmetric games and generally give stable and unique equilibria, butdo not account for capacity constraints. Boisseleau et al. (2004) use apiece-wise linear supply function and describe a solution algorithmthat obtains an equilibrium even when capacity constraints exist.

2.3. Comparing the models

Only a few authors have compared the equilibria in both models.Related to our paper is a recent work by Vives (2007), who studies theproperties of two auction mechanisms: one where firms bid supplyfunctions and one where they bid à la Cournot. Firms are assumed tohave private information about their costs. Solving for a Bayesianequilibrium in linear supply functions, Vives (2007) shows that supplyfunctions aggregate the dispersed information of the players, whilethe Cournot model does not. Hence, Cournot games might be sociallyless efficient, since the dispersed information is not used effectively.Our paper differs since we are less interested in understanding theproperties of auction mechanisms. In our model, firms have perfectinformation about their costs, firms sell forward contracts and supplyfunctions are not restricted to linear bid functions.

Hu et al. (2004) use a bi-level game to model markets for deliveryof electrical power on looped transmission networks with the focus onthe function of the ISO. Within this analysis they compare supplyfunction and Cournot equilibria and show that in case of transmissioncongestion, SFE need not be bounded from above by Cournotequilibria as is the case for unconstrained networks. They concludethat when congestion is present, Cournot gamesmay bemore efficientthan supply function bidding.

Ciarreta and Gutierrez-Hita (2006) undertake a theoretical analysisof collusion in repeated oligopoly games, using a supergame modelthat is designed both for supply function and quantity competition.They show that depending on the number of rivals and the slope of themarket demand, collusion is easier to sustain under supply functionrather than under quantity competition. An experimental approach byBrandts et al. (2008) includes the impact of forward contracts onelectricity market outcomes with differences due to Cournot andsupply function competition. They show that the theoretical outcomesof SFE models (mainly their feasibility range between Cournot and

Fig. 1. Price distribution on the EEX.

6 An earlier version of this paper also considered load-following contracts. For thesecontracts, a specified fraction of the sales is contracted forward for each realization ofthe demand shock: Fik=ϕiqik. Such load-following contracts are typically signed

41B. Willems et al. / Energy Economics 31 (2009) 38–47

marginal costs) can be reproduced by experimental economics.Furthermore, they find that introducing a forwardmarket significantlylowers prices for both types of competition.

3. Model formulation

The Cournot and SFE supply models are found by simultaneouslysolving a set of equations describing the market equilibrium fordifferent demand realizations k. In order to compare the results, weuse identical assumptions with respect to the demand and supplystructure of the market: the market demand is assumed to be linearwhereas the marginal cost curve is a cubic function based on theactual power plant costs (see Section 4).

The demand equation describes for demand realization k, how thedemand Dk

o that strategic players (oligopolies) face depends on ademand shock Δk and the price pk:

DOk ¼ α−γd pk−Δk ð10Þ

with k=1,…, K the index of the aggregate demand shock, both due tothe stochastic demand component and the deterministic timecomponent: Δk=ξ+ f(t) with ΔkbΔk +1, Δ1=0 and ΔK=α.

The energy balance equation describes that for all demandrealizations k, demand should equal total supply:

DOk ¼ SOk ð11Þ

where SOk ¼ ∑i¼1

nqik.

The marginal cost equation relates the output of firm i with themarginal cost of that production plant given by:

cik ¼ λi0 þ λi1qik þ λi2q2ik þ λi3q3ik ð12Þ

The continuity Eq. (13) imposes continuity of the supply function. Itdescribes the relation between the slope of the supply function β, theproduction levels and the price level of the firms. Continuity implies thatthe arc–slopeof the supply functioncanbewritten as theweighted sumofthe slopes at the twoendpoints of the interval. That is, the followingholds:

qikþ1−qik ¼ pkþ1−pkð Þ 1−�ikð Þβik þ �ikβikþ1� � ð13Þ

with 0bξikb1. This formulation specifies a piece-wise linear supplyfunction.5

5 This formulation, based on Anderson and Hu (2005), rewrites their Eqs. (13) and(19) by eliminating p̃ ik.

The pricing equation describes the first order conditions of eachplayer i for each demand shock k. It requires that a player's marginalrevenue and marginal cost be equal:

qik−Fikð ÞdpRik

dqik¼ pk−cik ð14Þ

Fik is the amount of contracts signed in equilibrium by firm i in

realization k, and dpRikdqik

is the slope of its inverse residual demand

function. We allow the firms to sign fixed-capacity contracts specifiedas a quantity (in MW) which is independent of the demand shock k:Fik = fi. A typical fixed contract is a base load contract (the firm commitsto sell forward a fixed amount of energy).6

The pricing equation differs for the two models. In the Cournotequilibrium, each player assumes the production of the other playersas given, and therefore the slope of the residual inverse demandfunction depends only on the slope of the demand function (1=γ).

Firm i signs fi fixed contracts and the pricing equation becomes:

qik−fi ¼ pk−cikð Þγ ð15Þ

By contrast, for the SFE model the slope of the residual demandfunction depends on the slope of the demand function and the slope ofthe supply functions of the competitors. With the assumed fixedcontract cover fi the pricing equation becomes:

qik−fi ¼ pk−cikð Þ ∑j≠iβjk þ γ

!ð16Þ

A Cournot equilibrium is a solution of Eqs. (10)–(12) and (15). A SFEis a solution of Eqs. (10)–(13) and (16). The equilibrium is described asthe price and demand level for realization k, Dk, pk and for each firm ia description of production, marginal costs and a slope of the demandfunction βik, qik, cik. The solution of these equations is not straightfor-ward, given the non-linearities in Eqs. (13) and (16).

The model is solved using the COIN-IPOPT solver in GAMS(Wächter and Biegler, 2006).7 Once the equilibrium quantities are

between a generator and a retailer. In this case, the generator does not sign a contractfor a fixed quantity, but promises to fulfill all of the retailer's demand.

7 The CONOPT solver which was used by Anderson and Hu (2005) did not alwaysconverge.

Fig. 2. Competitive supply functions.

10 We decided not to incorporate production capacity constraints on the grounds that

42 B. Willems et al. / Energy Economics 31 (2009) 38–47

determined we verify that the second order conditions of theoptimization problem are satisfied for each player (see Appendix).

4. Data

4.1. An oligopolistic market: Germany

For this paper, we selected Germany because its oligopolisticmarket structure lends itself to valid representations for both Cournotand SFE models. The German market consist of two large firms (E.ONand RWE) owning about 50% of generation capacity, two smaller firms(Vattenfall and EnBW) each with about 15% of the market and acompetitive “fringe” that acts as a price-taker.8 Germany is a winterpeaking system with a large share of nuclear and coal units and asignificant share of wind production in the North and East. Thetransmission grid is well-connected with the rest of Europe, and thecountry is a net exporter of electrical energy.

We study the winter period since strategic behavior is more likelyto occur when capacity is scarce. Since the models assume that theunderlying cost structures remain constant for the duration of thebidding period, we restrict our analysis to January and February 2006.Most of Germany's electricity is traded bilaterally, but voluntarypower exchanges selling standardized products are gaining favor. Themain price index for Germany is the day ahead price at the EuropeanEnergy Exchange (EEX). Fig. 1: shows the price distribution during theobservation period. Within the sample period, we consider four ‘timeperiods’: January peak and off-peak hours, and February peak and off-peak hours.9 For each time period the observed demand, windproduction and net import amounts, as well as prices are used toestimate the corresponding demand function for Germany (SeeSection 4.3).

4.2. Approximation of the cost functions of the generators

The marginal costs of generator i are described by a function ci(q),which accounts for each player's generation portfolio. Generationcapacities and ownership are obtained from public sources, mainlyVGE (2006). More than 300 power plants are considered, totaling100 GW of fossil and hydro generation. Wind, biomass and solarcapacities are not considered within the firm's generation portfolios.Plant capacities are decreased by seasonal availability factors follow-

8 Due to the characteristics of electricity markets, small companies can have marketpower potentials (Sioshansi and Oren, 2007). Neglecting the strategic behavior of thefringe may lead to price underestimations.

9 In our sample we classify all hours between 8 am and 8 pm as peak hours; theothers are off-peak.

ing Hoster (1996). Using a type-specific algorithm based on Schröter(2004) with construction year as proxy, we calculate a plant-specificefficiency to derive marginal costs. Fuel prices are taken from Bafa(2006) and resemble average monthly cross-border prices for gas, oiland coal. We include the price of CO2-emission allowances in the costestimate based on fuel type and plant efficiency. Allowance prices aretaken from the EEX. The marginal cost functions are estimated forJanuary and February respectively.

The marginal cost functions of the generators are simplified to acubic function10

~ci qð Þ ¼ λi0 þ λi1qþ λi2q2 þ λi3q3 ð17Þwhere the parameters of the function are found by minimizing theweighted squared difference of the parameterized function and thetrue cost function subject to the condition that marginal cost shouldbe upward sloping. The following optimization problem is solved

minλ0 ;::;λ3

∫ ~ci qð Þ−ci qð Þð Þ2dF c qð Þð Þs:t: ~cVi qð Þz0

ð18Þ

with F(∙) the cumulative distribution function of the prices in the EEX.Fig. 2 shows the approximate cubic cost function and the originalstepwise marginal cost function for Germany's four largest playersaveraged for January and February.

4.3. The demand function faced by the oligopolists

The final demand D for energy is served bywind production QW, byimports QI, by production of the four German oligopolists Qi and bythe fringe generator QF.

D ¼ QW þ QF þ QI þ ∑4

i¼1Qi ð19Þ

We assume that the final demand for electricity D does not dependon the price, but varies through time t and has a random componentD=αt

D+εtD. Wind production may also vary through time and have arandom component QW=αt

W+εtW.

the theoretical literature on supply function equilibria with capacity constraints is notyet sufficiently developed, especially for situations with multiple asymmetric firms.For instance Holmberg (2008) shows that with symmetric players, inelastic demand, aprice cap and (binding) capacity constraints there exists a unique, symmetricequilibrium. Delgado (2006) shows that there may be multiple equilibria in oligopolymodels with capacity constraints and that it is no longer evident that the firms willcoordinate on the Cournot equilibrium.

Table 1Results of the regression analysis

Contractcover

Var Correctionterm τ

Var Std DevError

R2 Number ofobservations

Fig. 3. Aggregate supply function of the oligopolists and the fringe.

43B. Willems et al. / Energy Economics 31 (2009) 38–47

The production level of the fringe generators depends on the pricethey can obtain for their output. The supply of the fringe is determinedby the inverse of its marginal cost function.

QF pð Þ ¼ MC−1 pð Þ ð20Þ

We approximate the marginal cost function of the fringe linearly,and rewrite the supply function as

QF pð Þ ¼ αF þ γFp ð21Þ

We use two approaches to determine the parameters αF and γF. Inthe first, we use aweighted least squares regression (the “weights” arederived from the price density function of the EEX prices). Thisapproach typically underestimates the marginal cost for large pricesbecause no capacity constraint is considered. In the second approach,we assume that the fringe is always producing at full capacity (γF=0)which underestimates marginal costs for low prices. The fringemarginal cost function is derived using information on monthly fuelprices and generation capacities. We obtain distinct estimates forJanuary and February.

German imports are determined by the difference of the price inGermany and the neighboring regions. If the price in Germany is highrelative to the price in neighboring regions, imports increase. Weestimate imports by regressing the following equation:

QIt ¼ γIpt−∑jγjpjt þ ∑

zβzδzt þ eIt ð22Þ

where pt is the price in Germany, pjt is the price in border country jand δzt is a vector of time dummies (day of week) for all hours t in therelevant period.11 We use a two-stage least squares estimator toaddress the endogeneity of the German price p with respect toimports. For instruments, we use the total demand level in GermanyDt and German wind production Qt

W. As explained in Bushnell et al.(2008), the demand level is a valid instrument for import levels sincedemand is inelastic in the short-term, and does not depend on theprice level in Germany. The import elasticities are estimated for eachof the four periods. Note that we do not explicitly model cross-bordercapacity constraints. The results of the regression can be found in theAppendix.

11 Hourly price data was taken from the Netherlands, France, Austria, Poland,Sweden, East Denmark and West Denmark.

Combining Eqs. (19)–(22), we can rewrite the residual demandfunction for the oligopolists DO:

DOt pð Þ ¼ αO

t − γF þ γIð Þpt þ eOt ð23Þ

Our four German oligopolists, therefore, face an elastic demandfunction due to import elasticity and the supply of the fringegenerators (γF+γI). An increase in the German electricity prices willincrease the import levels and the production by the fringe generators.In the model, the demand shock αt and the random component εt foreach time period t are transferred into a constant demand interceptαO and a positive shock Δk for a specified set of demand realizations k(see Eq. (10)). We choose the intercept of the demand level such thatwhen the shock is zero, 98% of the observations in the Germanmarketare below the demand function (2% of the outliers are not considered).

The estimation results show that demand is more elastic duringoff-peak hours and in January, and less so during peak hours and inFebruary. The higher demand elasticity likely reflects the fact thatcross-border transmission lines are less congested in off-peak hours.This increases the possibility of foreign supply and flattens thedemand function for locally produced electricity.

5. Results of the models

Combining the price data of the German power exchange, demand,imports and wind data, Fig. 3 shows the aggregate supply function ofGerman thermal production during January and February 2006. Theaim of this paper is to test whether a Cournot model or a SFE approachis capable of explaining this observed aggregated supply functionconsidering the cost of the firms and the strategic behavior of the fourlargest generation firms.

The Cournot model produces a single equilibrium given the demandrealization and the contract cover. The SFE model produces a bundle ofequilibria for a given contract cover. Which equilibrium firms choose

Cournot 49.8% 0.234 – – 9.41 0.85SFE 27.4% 0.664 – – 9.31 0.85Cournot 50.4% 0.311 0.064 0.001 9.39 0.85 1361SFE 25.5% 0.876 −0.106 0.001 9.26 0.85

Fig. 4. Unique Cournot outcome and bundle of SFE outcomes at the calibrated contract cover.

44 B. Willems et al. / Energy Economics 31 (2009) 38–47

depends on how the firms coordinate. We assume that the firms co-ordinate on the least-competitive SFE equilibrium, i.e. the equilibriumwhere prices are highest.12

The outcome of both models depends on the total amount ofcontracts signed by the firms. We use the contract coverage as acalibration parameter. The contract coverage fi is specified relative tothe total installed (and available) capacity for each firm (qimax)irrespective of the demand level:

fi ¼ /i qmaxi ð24Þ

with ϕi the contract coverage of firm i in percentage.No reliable data exists on the long-term commitments and

contracting positions of German electricity firms. Only about 20% ofelectrical power is traded on the EEX spot market. The uncontractedposition of firms may, however, differ significantly from this numberbecause: (i) an unknown amount of energy is traded in short-termOTC markets; (ii) the net position of each firm may be significantlysmaller than the gross level of trade in the market; and (iii) onlycontracts that specify a fixed price reduce market power, whereascontracts indexed on the spot price have no effect. We thereforeconclude that the contract cover that calibrates the model should notbe interpreted as the actual amount of forward contracts signed by thefirms. Furthermore, the model neglects important factors such asstart-up times, capacities and network constraints. The calibrationparameter may be affected by some of these effects as well.

To test numerically which model predicts the market outcomesmore realistically, we use the observed price-demand results at theEEX spot market as a benchmark. To compare the models' prices withthe observed market prices, we conduct a non-linear least squaresregression:

PObst ¼ ~

Pt /ð Þ þ e ð25Þ

12 This approach is also found in most other analyses that use the SFE model forpolicy studies. (e.g. Green and Newbery, 1992; also see Delgado and Moreno, 2004).

with ptObs the observed price on the market, P̃t the prediction of the

model and ε as error term. If the error term is normally distributed,this estimator of the contract cover will converge to the true value ofthe regression. For the regression we minimize the (squared) errorbetween the prices we observe and the prices predicted by the model:

min/

PObst −

~Pt /ð Þ

� �2ð26Þ

The model predictions P̃t are calculated for the same demandshockΔk as observed in themarket defined by Eq. (10).13 Bymeasuringgoodness-of-fit in this manner, we measure only the errors in thesupply side of themodel that are conditional on the demand functionsbeing defined correctly.

The first two lines of Table 1 show the results of the regressionanalyses. The optimal contract coverage for the Cournot model is onewhere firms contract 50% of their installed capacity. This is about twicethat for SFE. At the optimal contract cover, the standarddeviation of theerror term in the regression is about 9.41 EUR/MWh for the Cournotmodel and 9.31 EUR/MWh for the SFE model. We note that bothmodels perform equally well in predicting the market outcome.

An alternative measure of the goodness-of-fit is the R-squaredterm. It is a (relative) measure of how much of the variation inobserved prices is explained by the model:

R2 ¼ 1−Vmodel

Vdatað27Þ

with Vmodel the variance of the error term of the regression and Vdata

the variance of the observed market prices. Assuming that the error isnormally distributed, we can calculate a confidence interval forcontract covers expressed as a variance of the estimate. The highervariance of contract cover for the SFE model reflects the fact that this

13 This means that the observed price PtObs, the observed quantity produced by the

oligopolists QtO,Obs, the price prediction, P̃t and the quantity prediction D̃t

O are such thatQtObs− Q̃tObs=γ(ptObs− p̃t).

Table 2Results of the regression analysis for different periods, elastic fringe

Contractcover

Var Correctionterm τ

Var Std DevError

R2 Number ofobservations

Cournot peak 46.4 0.458 −0.033 0.001 11.26 0.77 727Cournot off-peak 56.8 0.851 0.056 0.003 5.53 0.84 631SFE peak 23.0 0.739 −0.120 0.001 10.18 0.81 727SFE off-peak 41.9 4.130 −0.180 0.003 6.03 0.81 631

45B. Willems et al. / Energy Economics 31 (2009) 38–47

model is less sensitive to changes in the contract cover, while theresults of the Cournot model depend more heavily on the contractcover.

Fig. 4 shows the Cournot solution and the “bundle” of SFE solutionsfor the contract positions found in the regression. Simply stated,Cournot firms sign contracts for 49.8% of their installed generationcapacity, while SFE firms contract for 27.4%. We only consider the SFEsolution with the highest price in our analysis, thus neglecting theremainder of the bundle. In the mid-load range both models producethe same outcome. However, during peak and off-peak load, the SFEsolution is greater than the Cournot's. In general the Cournot solutionfollows a more linear trend whereas the SFE solution is convex.

Since both models neglect start-up and capacity restrictions, weadjust the regression analysis by introducing a constant term τdepending on the difference between marginal fuel costs andmarginal costs and including start-up and capacity restrictions:

PObst ¼ ~

Pt /ð Þ þ τ MCmodt −MCstartup

t

� �þ e ð28Þ

with MCmod the marginal costs as used in both models and MCstartup thecorresponding marginal costs including start-up costs and restrictions.14

If the start-up adjusted marginal costs better reflect the real coststhan the marginal costs used in the model, we should observe anegative τ. As the start-up adjusted costs are on average larger thanthe modeled marginal costs, the price cost markup of the firms isslightly smaller and we, therefore, expect the model to predict a lesseramount of fixed contracts to be signed.

The last two lines of Table 1 show the results of the regressionincluding the term reflecting start-up costs. In the SFE model, theimpact of the marginal cost has the expected sign, but is relativelysmall in absolute terms.15 Correcting for the start-up costs, thecontract cover has decreased. In the Cournot model, the parameter hasthe wrong sign, but remains very small in absolute terms. In bothmodels the optimal contract coverage only varies slightly with anincrease in the according variance; the R-squared value remainsconstant. Thus, the fit of the models cannot be increased.

In order to test which approach performs better under varyingassumptions we conduct a series of robustness tests. The aim of theserobustness checks is to test whether the relative performance of theCournot and the SFE model depend on the particular assumptions wemake.

• We allow for different contract coverage during peak and off-peakperiods, vary the marginal generation costs on a monthly basis andestimate different import elasticities for the four peak and off-peakperiods.

• For the behavior of the fringe, we consider the two cases explainedin Section 4.3: one where the fringe has an elastic supply functionand one where the fringe always produces at full capacity. Theelastic supply function is on average the best representation ofthe supply of the fringe generator, but neglects capacity constraints.The inelastic supply function may be a better representation of the

14 The hourly values for MCstartup are obtained from Hirschhausen and Weigt (inpress).15 In the SFE model, firms are unable to make bids conditional on the time period,and are therefore unable to reflect start-up costs in their bids.

supply by the fringe during peak periods when capacity constraintsplay a larger role.

• We introduce load-following contracts in the Cournot simulationsallowing for extra flexibility in the contracting options of the firms.

We first assume an elastic fringe (Table 2). In this model, we observethat the contract coverage is higher during off-peak periods for bothapproaches. Likewise the variance during off-peak is higher—particu-larly for the SFE model—indicating that the results are less sensitive tothe chosen coverage. However, the Cournot model still yields a higheroptimal contract cover. Surprisingly, the R-squared values decreasecompared to the average base case, which does not allow for contractlevels that are differentiated according to the time of day. The reason isthat the R-squared is a relative measure of the quality of fit based on theunderlying sample data, and not an absolute measure. Another reasonmaking it hard to compare theR-squaredmeasures is thatwe drop 2% ofthe outliers in the peak and 2% of the off-peak hours (this is differentfrom dropping 2% over the total sample).

Furthermore,weobserve that theoff-peakoutcomeof theoligopolisticmodels is only slightly better than themarginal cost curvewhichhas anR-squared of 0.71. However, during peak hours, the marginal cost curve is abad predictor of market results (R=0.05), indicating that during peakhours it is more important to consider strategic company behavior. Theimpact of start-up and capacity restrictions has the expected sign inmostcases, although the coefficient is relatively small.

Whenwe fix the fringe output to its maximumgeneration capacity,the resulting residual demand function for the oligopolists will besteeper in both peak and off-peak. However, during off-peak theresidual demand level is lower since the fringe will satisfy a largerfraction of the total demand (see Table 3). We observe that duringpeak times the optimal contract cover increases for both approachesand the variance drops. In the SFE model the decreased demandelasticity results in doubling the contracting. During off-peak theoptimal contract level decreases.

The impact of start-up and capacity restrictions reverses when wefix the fringe output. However, the impact remains below 20% of themarginal cost difference (MCmod−MCstartup). Hence we conclude thatthe prices thatwe observe are onlymarginally driven by start-up costs.

Comparing both fringe assumptions, the optimal model is probablyone where the inelastic fringe is used during peak hours—the productioncapacity is binding—and the elastic fringe during off-peak hours—theproduction capacity is not binding. In that case a similar amount ofcontracts will be signed during peak and off-peak periods, implying thatduring off-peak hours, a larger fraction of total demand is covered bycontracts. The Cournot model shows a coverage of 10 to 15% higher andcorrections for start-up and capacity constraints have the expected sign inthree out of four cases.

As a last robustness check, we analyze the impact of load-followingcontracts for the Cournot approach.16 The regressions show that theoptimal amount of these contracts is equal to 0%, an indication that themodel can be finely calibrated during both peak and off-peak byrelying only on fixed contracts.

6. Conclusion

This paper compares the classical Cournot model with the SFEapproach to test whether the higher complexity of SFE results in abetter representation of strategic market outcomes. Both models aretested using the dataset of Germany's electricity market and the sameassumptions regarding demand and generation. The results are thencompared to observed market outcomes. We calibrate the model bychanging the amount of fixed-capacity contracts that firms sign andthe behavior of the fringe.

16 Load-following contracts can be understood as a representation of verticallyintegrated generators that aim to satisfy the demand of their customers.

Table 3Results of the regression analysis for different periods, inelastic fringe

Contractcover

Var Correctionterm τ

Var Std DevError

R2 Number ofobservations

Cournot peak 53.7 0.071 −0.025 0.001 10.48 0.80 726Cournot off-peak 47.7 0.227 0.234 0.001 5.10 0.86 629SFE peak 42.2 0.141 0.048 0.001 12.13 0.73 726SFE off-peak 32.8 1.111 0.103 0.001 5.23 0.85 629

46 B. Willems et al. / Energy Economics 31 (2009) 38–47

The results indicate that the Cournot approach can be well-calibrated to the observed market outcomes by assuming a relativelyhigh level of fixed-capacity contracts. For the SFE model, the best fit isfound when firms sign fewer contracts. Using the R-squaredcoefficient of a non-linear least squares regression as a measure, thecalibrated SFE and Cournot models perform almost equally well, i.e.,they explain the same percentage of the price variation in the market.We conclude therefore that the SFE model does not significantlyoutperform the Cournot model as a tool to study the Germanelectricity market. However, the SFE models rely less on calibrationparameters, and appear to give more robust, “realistic” predictions.

To solve the models numerically, especially the SFE model, wemake several simplifying assumptions with respect to the genera-tion and demand data. These assumptions may bias the quantitativeand qualitative results of our models. For example, the linearizationof import and fringe behavior can lead to a general overestimationof demand elasticity especially for high-demand periods, resultingin wholesale prices that are too low. The neglect of start-up andramping issues leads to an overestimation of costs during off-peakperiods. The general assumption of continuous supply function maylead to an underestimation of generation costs close to peakcapacity. We conduct a robustness check, testing for different fringebehavior, analyzing the impact of start-up costs and splitting thesample in different time periods. We find that the results do notdiffer significantly for both models.

We do not know whether our results extend to other electricitymarkets, but we conjecture that the difference between the twomodelsbecomes less pronounced as markets become less concentrated andmore dependent on imports. We observe that the SFE may give betterresults for markets that are less import-dependent and more concen-trated than Germany and that therefore have a lower demand elasticity.

Given the limited flexibility of SFE approaches to incorporatetechnical characteristics (unit commitment, start-up costs and net-work issues), we suggest that Cournot models should be the preferredoption when electricity market characteristics must be modelled intechnical detail. Thus, Cournot models are aptly suited for the study ofmarket rules or congestion allocation mechanisms. However, whenlong-term aspects come into play, e.g., in a merger study, SFE modelsmay become more relevant because they are less sensitive tocalibration parameters. Furthermore, for long-term simulations, onecannot assume that contract positions are exogenous, increasing thecomplexity of Cournot models.

We foresee that both models, Cournot and SFE, will continue to beused for practical purposes, with each being further tailored to aspecific range of applications. We hope that this paper will assistpolicy-makers, regulators and industry players to evaluate theadvantages and disadvantages of the two models.

Acknowledgments

We thank Lapo Filistrucchi, Tobias Klein, Karsten Neuhoff, AmritaRay Chaudhuri, Ramteen Sioshansi, Christian von Hirschhausen andparticipants of seminars in Tilburg University, the IAEE conference inFlorence and the IDEI conference in Toulouse, for their comments andsuggestions. Two anonymous referees have provided insightfulcomments and suggestions to improve the paper.

Appendix A

Table A12-SLS regression with demand and wind input as instruments (see Eq. (22))

Januarypeak

Februarypeak

Januaryoff-peak

Februaryoff-peak

January andFebruary

Coef.

t Coef. t Coef. t Coef. t Coef. t

EEX(Germany)

0.09

3.34 0.06 5.23 0.40 2.52 0.14 4.03 0.13 5.81

Day_1

1.49 1.77 0.50 1.19 0.62 1.18 0.26 0.81 1.00 3.13 Day_2 1.22 1.51 0.08 0.21 0.26 0.41 0.04 0.13 0.77 2.47 Day_3 0.72 0.91 −0.03 −0.08 0.77 1.37 0.37 1.19 0.73 2.34 Day_4 0.01 0.01 0.40 1.05 0.73 1.09 −0.13 −0.37 0.37 1.02 Day_5 −0.36 −0.44 0.67 1.78 1.06 1.95 0.52 1.57 0.55 1.70 Day_6 0.64 0.97 0.18 0.65 0.08 0.15 −0.15 −0.46 0.34 1.16 Netherlands 0.00 −0.25 −0.02 −5.71 −0.14 −2.65 −0.05 −2.57 −0.02 −3.78 France −0.06 −3.26 −0.02 −5.15 −0.06 −3.14 −0.05 −4.88 −0.04 −6.95 Austria −0.02 −0.9 0.00 −0.22 −0.14 −1.72 −0.03 −1.31 −0.06 −3.29 Poland 0.00 −0.43 −0.01 −0.89 0.01 0.77 0.03 2.24 −0.01 −1.28 DKeast −0.02 −1.92 −0.01 −2.89 0.03 1.39 −0.02 −1.48 −0.01 −2.81 DKwest −0.05 −1.75 0.02 2.12 −0.10 −2.00 0.01 0.23 −0.01 −1.12 Sweden 0.05 2.72 −0.01 −0.94 −0.01 −0.19 −0.03 −0.79 0.02 1.77 _cons −3.28 −6.57 −3.25 −13.97 −2.89 −6.31 −3.00 −11.38 −3.12 −13.97

Table A2Coefficients of the cubic marginal cost functions (see Eq. (17))

January

February January and February

Firm 1

λ0 −2.369 −2.916 −2.614 λ1 7.308 6.881 7.075 λ2 −0.720 −0.577 −0.643 λ3 0.031 0.024 0.028

Firm 2

λ0 −20.674 −20.697 −20.411 λ1 13.842 13.576 13.613 λ2 −1.200 −1.108 −1.143 λ3 0.037 0.033 0.034

Firm 3

λ0 −58.565 −51.792 −60.969 λ1 40.472 37.287 41.759 λ2 −5.431 −4.873 −5.529 λ3 0.244 0.218 0.245

Firm 4

λ0 −5.726 −5.975 −6.035 λ1 11.549 10.852 11.482 λ2 −1.675 −1.267 −1.534 λ3 0.158 0.125 0.145

Table A3Values for fringe supply function (see Eq. (21))

January

February January and February

Elastic fringe

αF 8.439 4.856 7.610 γF 0.077 0.140 0.094

Inelastic fringe

αF 17.930 17.930 17.930 γF 0.000 0.000 0.000

Appendix B. Verifying second order conditions

The second order conditions of firm k's optimization problem are(locally) satisfied if

qik−Fikð Þ d2pRikdq2ik

!þ 2

dpRikdqik

!−dcikdqik

b0 ð29Þ

with dcikdqik

the slope of the marginal cost function, dpRikdqik

and d2pRik

dq2ik

the first

and second order derivative of the inverse residual demand functionof firm i.

47B. Willems et al. / Energy Economics 31 (2009) 38–47

The condition can easily be checked numerically with the fol-lowing formulas:

dcikdqik

¼ cikþ1−cik−1qikþ1−qik−1

ð30Þ

dpRikdqik

¼ ∑j≠iβjk þ γ

!−1

ð31Þ

d2pRikdq2ik

¼ −∑j≠i βjkþ1−βjk−1� �pikþ1−pik−1

dpRikdqik

!3

ð32Þ

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