1

Wind Energy Can Mitigate Market Power

by

Ori Ben-Moshe The Unit of Energy Engineering, Faculty of Engineering Sciences, Ben-Gurion University of the Negev, Israel,

oribenm@post.bgu.ac.il

Ofir D. Rubin Department of Public Policy & Administration, Guilford Glazer Faculty of Business & Management, Ben-Gurion

University of the Negev, P.O. Box 653, Beer-Sheva, 84105, Israel, +972-8-6472597, orubin@som.bgu.ac.il

Abstract

A rich body of literature suggests that there is an inverse relationship between wind power penetration rate

and electricity market prices, yet it is unclear whether these observations are generalizable. Therefore, in this

paper we seek to characterize analytically market conditions that give rise to this inverse relationship. To this

purpose, we expand a theoretical framework to facilitate flexibility in modelling the structure of the electric

industry with respect to the degree of market concentration and diversification in the ownership of wind

power capacity. The analytical results and their attendant numerical illustrations indicate that it is not always

that wind energy depresses electricity prices. This is likely to occur when the number of firms is large enough,

the ownership of wind energy is sufficiently diversified, or most often a combination of the two. Importantly,

our study defines the circumstances in which the question who invests in wind power capacity is crucial for

market prices as it may generate incorrect technology-specific signals for new investments.

.

Keywords: Deregulated electricity markets; Oligopoly pricing; Wind energy

JEL: D4, L13, L94, Q4

2

1. Introduction

There is a large and ever growing body of literature indicating that as wind power penetration rate increases

electricity prices go down (e.g. Olsina et al. 2007; Botterud et al. 2010; Green & Vasilakos 2010; Traber &

Kemfert 2011; Woo, et al. 2011; Moreno, Lopez & Garcia-Alvarez 2012; Nielsen, Sorknaes & Ostergaard

2011; Pereira & Saraiva 2013) and even become negative in extreme cases of high wind power supply and

low realized demand (Nicolosi 2010; Brandstatt, Bruekreeft & Jahnke 2011). Two intertwined factors explain

this phenomenon: public support for investments in wind power capacity (and other renewables), which

enables wind generators to compete, and the fact that once wind capacity is installed, its marginal generation

costs are negligible compared to those of conventional units. Yet, in an economic environment characterized

by imperfect competition, strategic generation firms (GFs) are able to delay or, even to some extent, to exploit

the availability of wind power to maximize their profits. For example, Twomey and Neuoff (2010) show that

generators of conventional energy with market power can manipulate prices according to real-time conditions,

i.e., they elevate prices when they sell power to the market and to depress them when they need to buy power

back. Green & Vasilakos (2010) provide empirical support for this claim by showing that for a higher market

concentration level, ceteris paribus, conventional generators gain higher revenues than wind power producers.

These two studies (and those assuming perfect competition) postulate that renewable energy is introduced to

the market by new price-taker firms (aka fringe capacity). Basically, there is no reason to believe that

conventional generators will not consider diversifying their generation portfolio by adding sources of

renewable energy. To the best of our knowledge, the only two studies looking at market concentration from

the viewpoint of renewable energy production are that of Reichenbach and Requate (2012) and Rubin and

Babcock (2013). Reichenbach and Requate (2012) account for market power from two sources—producers of

electricity from fossil fuels and producers of renewable energy equipment. In their model, too, the producers

of renewable energy are considered as fringe capacity disengaged from the generator firms employing

conventional capacity. Rubin and Babcock (2013) examined the impact of the pricing method for wind energy

on market prices. They showed that in the presence of imperfect competition the pricing of wind energy in the

market increases firms' ability to extract oversized day-ahead premiums. In the case of market-independent

feed-in-tariff for wind energy, market power is reduced as wind power capacity expands. Diversified

ownership of wind power capacity has been simulated only for the two extreme cases where wind is fully

diversified (i.e., fringe capacity) or wind is entirely owned by strategic GFs. The authors reported that

complete diversification has only limited ability to mitigate market power. However, they do not supply

analytical results for the continuous range in which coupling market power with the ownership type of wind

power capacity may create a composite effect on average electricity market prices and average prices received

by producers of wind energy.

The observed changes in prices have a crucial implication in the long-run: Because market prices constitute

signals for new investments in power generation capacity, it is essential to verify that they are accurate. Only

if market prices reflect true technology-specific needs for installation of power generation, will the correct

generation mix be installed and at the right time. The important role of prices for new generation is widely

acknowledged (Newbery 2013). In the event that markets do not perform well, signals are inaccurate and

allocation of energy resources is inefficient (e.g., Hiroux and Saguan, 2010; Vandezande et al., 210; Weigt et

al., 2010 Traber & Kemfert 2011; Pereira and Saraiva 2013 Olsina et al. 2007). In the current paper, we

investigate the sensitivity of signals for investments in wind energy and conventional power plants to market

structure and particularly to the ownership type of wind power capacity.

Because it is realistic that large GFs with market power would consider investing in renewable capacity, it is

imperative to account for it when analyzing the impact of renewable energy on deregulated markets. From a

theoretical point of view, it is not clear whether the observed inverse relation between wind energy penetration

rate and market prices is generalizable and if not whether it would be beneficial to characterize the conditions

3

under which wind energy has dampening effect on electricity market prices. For this purpose, we extend the

theoretical framework of Twomey & Neuhoff (2010) to account for a Cournot oligopoly type of electric sector

with varying degrees of diversification of the ownership of wind power capacity. Based on the extended

model, we generate several analytical results that enable us to generalize outcomes with regard to the

integration of wind energy into deregulated electricity markets and their impact on electricity prices. We also

present numerical examples to examine how particular set of assumptions regarding installed wind power

capacity (penetration rate and variance) and market structure (firms' number and diversified ownership of

wind energy) affects the ability to exercise market power. This paper is thus the first attempt to generalize the

relation between wind energy penetration rate and market prices in deregulated electricity markets.

The remainder of the paper is organized as follows. In section 2, we introduce the extended model of

oligopolistic competition, followed by analytical results in section 3. A numerical example illustrates our

results in section 4. In section 5, we conclude our findings and outline the policy implications of this study.

2. Model

We extend the model of Twomey & Neuhoff (2010) to examine whether an increase in wind energy

penetration rate will necessarily bring market prices down. The original model investigated the linkage

between wind energy and market prices in perfect competition, monopoly, and duopoly market regimes. The

chosen theoretical framework is suitable for our purposes for various reasons. First, similar to almost all

deregulated electricity markets, in this model the marginal unit produced by the marginal generator determines

the realized market price. Second, the model takes into account forward contracting, which is the main

financial instrument for trading power in deregulated electricity markets today. Finally, it provides a closed-

form formula for examining the impact of wind energy characteristics in the modelled region, which in turn

makes the analytical inference easier. We expand this theoretical framework in two ways. First, we develop

the model to account for an oligopoly market, which will be useful to describe concentrated industries with

more than two GFs. Second, in Twomey & Neuhoff (2010), the ownership of wind power capacity is

introduced to the market by price-taker firms. However, we allow a degree of freedom in the ownership type

of wind power capacity, namely, in our model the strategic GFs are allowed to own and operate wind farms.

Again, this extension reflects a reality in which GFs often hold a portfolio mix of power generation

technologies.

In the model, trading takes place in two stages. In the first stage, forward contracts are signed, and in the

second quantities are balanced against realized demand in the spot market.

Consider the following inverse real-time demand function for electricity

(1)

where p is the spot market price, is demand intercept, is realized demand and is the demand slope.

Assume that electricity can be supplied only by two types of generator, conventional and wind, denoted by

and , respectively. Further, assume that the intermittent nature of wind power output is given by (i.e.,

deviation from average wind output). Because the power system has to be balanced in all times, we know that

the following must hold

(2)

Plugging equation (2) into the inverse demand function, we obtain

4

(3)

We assume the regular quadratic cost function of the conventional generator

( ) (4)

where and are costs parameters, is the overall quantity of electricity generated by conventional units,

and denotes the number of conventional generator firms. Taking the first derivative, we obtain the marginal

costs of electricity production

( ) (5)

Notice that the marginal costs of oligopolies increase with output by the factor . This parameterization is

useful as it fixes the industry marginal generation costs for all values of , which in turn allows us to separate

the effect of market power from this of industry scale.

We denote the share of wind energy in the market by , and

as the share of wind energy owned by

each generation firm. Finally, and stand for the number of forward contracts and their price received by

generator .

Adding up the payoff in spot and forward markets, we can write expected profits of generator as follows

[ ]

[ (∑

)]

(6)

Taking first derivative with respect to generation output,

(∑

)

(7)

The first order condition (FOC) is

∑

(8)

Next, we find the expression for ∑ . To this end, we solve FOC for all generators

[ ]

[ ]

[ ]

(9)

summing up the equations

5

∑

[ ∑

∑

]

(10)

and rearranging,

∑

∑

(11)

We use equation (11) in the FOC to describe explicitly the quantity of electricity produced by generator :

(12)

[ ] [ ] ∑

[ ]

Similarly, substituting ∑ for in the inverse demand function, we obtain the expression for spot

market price

[ ] ∑

(13)

Assuming , the expected spot price, denoted by is

[ ] ∑

(14)

Finally, the realized market price can be written as:

(15)

The expected profits of wind energy producers are

[ ] [ ] [(

) ]

(16)

Dividing equation (16) by , we are able to characterize the average price obtained by the wind energy

producers

(17)

This result shows that the average price received by wind energy producers is lower than the average market

price of electricity. The reason for this outcome is embedded in the fundamentals of the electricity market

6

design and the characteristics of wind energy production. A high supply of wind energy pushes the prices

down, because units of wind energy replacing conventional generation units push marginal costs down.

Therefore, on average, wind energy compensation is lower than the average market price. Moreover, it is clear

from equation (17) that in regions with a higher variance of wind the price gap will be larger, leaving the wind

energy producer with a relatively lower payoff compared with that of the conventional generator.

Forward contracts

Here, we present the case of an exogenous volume of contracts, as this is a more realistic case. The

development of endogenous contracts is presented in Appendix 1. The transition to a deregulated electricity

market regime requires that the regulator makes sure that generators do not withhold capacity to manipulate

market prices. In addition to potential welfare losses, this behavior may impede the reliability of the system

and postpone required investments. There are several possible policy instruments to regulate market

participation. Among them is the so called must-offer provision that ensures the deployment of generation

capacity and ex-post price analysis in which the market administrator investigates whether the GFs submit

workable bids into the market. Overall, generators are obliged to make their entire capacity available at all

times, except during periods of scheduled maintenances and outages, and it is the responsibility of the system

operator to ensure that they do not withhold capacity to maximize profits (Bushnell, 2005). In line with this,

an unbounded Cournot modeling approach for the number of forward contracts is inadequate for

characterizing electricity market equilibrium (see for examples results generated by Twomey & Neuhoff 2010

and Rubin & Babcock 2011).

For simplicity, we assume that is a regulation policy for the percentage of energy to be contracted in

advance. Therefore, the volume of contracts to be signed by generator is [ ]. We plug this

regulation policy into equation (8), and then, by symmetry, we can express the expected quantity to be

produced by generator as:

[ ]

[ (

) ] [ ] [ ]

[ ]

[ (

) ] [ ]

[ ]

[ (

) ] [ ]

[ ]

(18)

Solving for [ ], we obtain

[ ] (

)

(19)

Plugging this result back into the regulation policy, we find that the number of forward contracts in trade is

(

)

(20)

7

Finally, we use this term to express the ratio between the average price received by wind energy producer and

the average electricity market price. This price ratio is of great importance in the analysis, as it governs the

relation between market incentives given for investments in new capacity of wind power and conventional

plants.

(21)

The characterized oligopoly equilibrium described in this section generates several useful analytical results,

which we discuss below.

3. Analytical results

We aim to explore how electricity prices vary according to the characteristics of market structure and wind

energy capacity installed in the region under investigation. The analysis is carried out by assessments about

how model parameters representing the number of firms in the market, ownership structure of wind power

capacity, and wind energy penetration rate impact the quantity and prices of electricity.

First, we examine the effect on market performances of ownership of wind energy capacity by strategic GFs.

If ownership type of wind energy capacity has an effect on the ability to exercise market power, we expect to

see relatively lower production level of conventional units and higher prices, which represent typical results in

a Cournot game.

Proposition 1: The average production level (market price) decreases (increases) in the share of wind energy

capacity owned by strategic GFs.

Proof:

We take the first derivative of the expected production w.r.t. , that is:

[ ]

(22)

since are all positive, , we know that [ ]

. This result confirms that as the

ownership of wind energy capacity becomes more centralized, the production level decreases. Next, we look

at the derivative of the expected market price with respect to :

(23)

Because

ג

[ ]

, and by equation (22), we know that

[ ]

is negative, we therefore conclude that

. Q.E.D

8

Next, we investigate what happens to the price ratio of the average price received by wind energy producer to

the average market price with regard to diversification of ownership of wind energy capacity. If capacity-

specific incentives are sensitive to ownership type, it is obvious that the question who invests in wind power

capacity will have distorting effects on market performances and the ability of prices to reflect true economic

conditions in the markets.

Proposition 2: The ratio between the average price of wind energy and average market price increases in the

share of wind energy owned by strategic GFs.

Proof:

Taking the first derivative of (21) w.r.t. we obtain:

⁄

(24)

{

[ ] [

]

( )

}

We know from equation (23) that

; therefore we find that

⁄

. QED.

This implies that the gap between the average market price and wind energy price decreases when the

ownership of wind energy is relatively more centralized. When strategic GFs own wind energy they have an

incentive to keep its price higher. This statement is a generalization of the situation modeled by Twomey &

Neuhoff (2010), where they show that when ownership is fully diversified (i.e. wind energy owned only by

price-taker firms), the gap is substantial.

Our next task is to assess the impact of wind penetration rate on market prices. While theory and empirical

studies show that prices decrease in wind power penetration rate, this has not been considered yet with respect

to the type of ownership of wind power capacity. To account for ownership type, we first examine the sign of

. However, as we show below, we find that the sign of this derivative is uncertain.

(25)

First, we compute the impact of wind energy supply on the number of contracts

( )

(26)

We plug this term back and rearrange the equation to give

9

{

( )

}

(27)

This result demonstrates that the impact of wind energy on market prices is not definite. The sign of equation

(27) depends on the magnitudes of the parameters describing the market. This finding takes us to the next

proposition.

Proposition 3: The impact of wind penetration rate on market prices depends on the share of wind energy

capacity owned by strategic GFs.

Proof:

We start by identifying critical values of in equation (27). Characterizing for the case where

yields the following expression

(28)

We verify that is unique by taking the second derivative

(

)

{

}

(29)

Since and (

)

, we know that if an interior solution exists it is unique. Q.E.D.

This result suggests that there is a threshold for the degree of wind energy holdings by GFs, which determines

the direction for prices to change when more wind energy is installed in the region. Notice that interior

solution is not guaranteed here. For relatively large number of firms, it might be that the impact of wind

penetration rate on prices will be always negative due to a high degree of competition.1

Lastly, we examine whether our findings are sensitive to the characteristics of regional wind energy. This is a

meaningful question, as the variability of wind resources changes significantly from one region to another.

Proposition 4: The price ratio decreases in the variability of wind energy supply, namely, the gap between

average wind energy price and average market price increases with the variability of wind energy supply

Proof:

We take the first derivative of the price ratio w.r.t. the variance of wind energy

1 . Corner solution for can be easily computed by solving equation (28) for . One can show that the

solutions are ( ) √

. The positive solution of this equation defines the

minimal number of firms that guarantees that electricity prices decline when additional wind capacity is installed.

11

(

⁄ )

{

}

(30)

as this result is immediate. Q.E.D.

Our final analytical result suggests a higher variance of regional wind energy will alter the price ratio in favor

of the conventional units. A higher intermittent supply will have a dampening effect on wind energy revenues.

In times of higher wind supply, market prices and wind energy prices decrease for two reasons. First, marginal

generation costs are lower as wind replaces fossil fuels inputs consumed by conventional units, which are

characterized by convex cost function. Second, the residual demand is relatively lower, and therefore GFs

compete more aggressively, which in turn, reduces their ability to maintain high prices. In times of lower wind

energy supply, the opposite holds true: average prices are higher and the share of power generated by wind

energy is smaller.

4. Numerical examples

In this section, we demonstrate our analytical findings with regard to how wind energy characteristics and

market structure determine the electricity market outcome. The chosen parameter values are identical to these

in the study by Twomey & Neuhoff (2010), i.e., we assume the following demand function

and we model the generation cost by the equation . Installed wind power capacity for the base

case scenario is set at , which represents approximately 30% of electricity demand and is uniformly

distributed between 0 and 22. Our extended model is utilized to investigate market performances subject to

various numbers of GFs. The parameter is used to simulate different diversification levels in the ownership

of wind power capacity. Finally, we assume a regulation policy that demands conventional generators to

schedule 90% of their production level in advance (i.e., ), and we also assume that the residual amount

is traded in real-time.

First, we investigate the impact of the number of firms and the diversification in the ownership of wind energy

on electricity prices (Figure 1). We plot average electricity prices against the number of GFs and we use the

extreme cases of fully diversified and no diversification in the ownership of wind power capacity (i.e.,

and , respectively).

11

Figure 1 provides several lessons. First, as the number of GFs increases, average prices in all cases decline.

This trend is due only to changes in the ability of GFs to exercise market power as a result of more aggressive

competition for residual demand. The featured concave decline in the average price with respect to is an

expected result in a Cournot competition. Second, when GFs own all wind energy both average prices and

wind energy prices are higher than in the cases of fully diversified wind energy ownership. Third, the gap

between average market prices and the average price received by wind energy producer is higher in the case

of diversified wind. This is because GFs that own wind energy have the incentive and market power to keep

wind energy prices higher relative to the fully diversified case. Our simulation implies that in the case of a

market with more than 50 firms, ceteris paribus, prices decline to the point that the type of ownership of wind

power capacity no longer plays a role in pricing. We ran many simulations and found the same trend

irrespective of the number of firms. In the followings section, we show why examining only the two extreme

cases, represented by and may be incomplete in analyzing the impact of diversifying the

ownerships of wind power. We continue our illustrations with .

We investigate trends along the path of increasing wind penetration rate up to 30%. In Figure 2, we depict the

quantity produced by the conventional GFs as a function of wind penetration rate and wind energy ownership.

The result here is straightforward, showing a monotonic effect of the two factors. First, higher wind power

capacity means less power generated by conventional units. Second, we see that when installed wind capacity

is higher, wind ownership type plays a vital role in production decisions. In a relatively less diversified

regime, conventional production level is lower thereby allowing GFs to enjoy higher prices for their wind

energy. For example, in a 30% wind energy market ( ), the difference between conventional

productions levels for and is about 8%.

1 2 3 5 10 50 100

15

30

45

60

75

90

105

120

135

150

number of GFs (M)

Av

era

ge

Pri

ce

s

Po, =0

Pw, =0

Po, =1

Pw, =1

Figure 1 : Average prices , number of GFs and wind ownership

12

The effect of these two factors on pricing is more complex. We know from the theoretical part of this study

that the effect on prices depends on whether the diversification level is below or above the threshold defined

in equation (28). Plugging in the parameters in our numerical example into this equation yields the critical

value . Indeed, we see in Figure 3 that above (below) the ownership concentration level of 0.6 prices

increase (decrease) in relation to wind energy output. Moreover, it can easily be verified that for and

, the computed thresholds are 0.2 and 0.3, respectively. Finally, when is employed in the

analysis, reaches a corner solution for which prices necessarily decline when more wind capacity is

installed in the experimental region.

Figure 2: Average GFs output, Average wind energy output and wind ownership

13

Next, we examine the price ratio between the price received by wind energy producers and the average price

of electricity as a function of wind energy output and the percentage of wind energy owned by GFs. The price

ratio is telling, as it reveals the average market value of 1 MWh produced by a wind farm in relation to that

produced by a conventional unit. Figure 4 illustrates the analytical results of the previous section: for

relatively more diversified wind industries this ratio is lower. For example, in the case of complete

diversification, for a wind energy output rising from 7.5% to 30%, the price ratio changes from 0.957 to 0.791

(i.e., a difference of 21%). As increases, the difference between the two figures decreases until the point

that, for , the two price ratio figures are only 3.5% apart. Figure 4 shows that the relative price received

for wind energy may be reduced by installing more wind power capacity, diversifying the ownership of wind

power or a combination of the two.

Figure 3: Average electricity price , average wind energy

output and wind ownership

14

Lastly, we examine the impact of the degree of wind energy intermittency on the ratio of wind energy price to

the average market price along the path of wind power capacity expansion. We observe that the relative price

of wind energy declines with wind energy variance faster than the average market price (figure 5). For

example, for the case of , wind energy output of 30%, and assuming a 50% lower variance, we obtain an

increase of 2% in the price ratio in favor of wind energy producers (

vs.

). Similarly,

for , we obtain an increase of 11% (

vs.

). This result emphasizes how

diversification of ownership of wind energy defines its market value. When strategic firms do not own wind

energy, they do not have an incentive to push its price up; hence its intermittent supply is better reflected in its

price.

Figure 4: The ratio between wind energy price and average market price, average wind energy

output and percentage of wind capacity owned by GFs

15

(a)

(b)

5. Conclusions

Policy justifications for public support for renewable energy are many. Among them are scarcity of fossil fuel

resources, environmental concerns, and climate change mitigation, energy independency, and the financial

need to diversify portfolio mix of electricity production thereby reducing exposure to volatility in energy

prices. In this study, we add to these motivations another feature of renewable energy — the potential to

reduce market power in concentrated electric sectors. Indeed, previous studies have shown that electricity

prices in deregulated electricity markets decrease in wind power penetration rate. We investigate this

phenomenon theoretically and discuss the economic environments that give rise for the observed trend. Our

theoretical framework is an extension of a model of Twomey and Neuhoff (2010). We generalized the

question in hand in two ways, we develop the model further for oligopoly market and we account for the

reality that GFs are able to possess both conventional units and wind power capacity. In fact, we find that

price decrease is not certain. It depends on number of factors, which jointly determine the economic

conditions of the market in question.

Our main results are: First, in relatively concentrated markets, conventional GFs, which also own wind energy

farms, can raise their revenues from these farms although they cannot control their energy supply. This is

because GFs are able to manipulate market prices while maximizing joint profits from their conventional and

wind power generators. Second, we show that the impact of wind energy penetration rate on market prices is

not straightforward; it depends on wind energy ownership type. In particular, for a relatively low level of wind

(a)

Figure 5: Ratio between wind energy price and average market price, average wind energy output,

and wind energy variance. (a) 𝜸 ; (b) 𝜸

16

energy diversification (i.e. conventional GFs own most wind power capacity), prices may actually increase

with higher wind energy penetration rate. More precisely, in our numerical example with 5 GFs, holdings of

more than 60% of wind power capacity in the region by these strategic firms increases market prices while in

the opposite case, we get the familiar result of price decrease. Third, we find that the ratio of average price

received by wind energy producers to the average market price received by conventional generators reveals a

picture that does not benefit the formers because average prices for wind energy are systematically lower.

Finally, we examine the impact of wind variability on the price ratio of wind and conventional power

producers. We find that the relative price of wind energy decreases with the variance of wind energy. This is a

plausible result as the market value of uncertain supply should be lower. Moreover, the impact of wind

variability generates even a stronger result if ownership of wind power capacity is diversified. This finding is

of great importance as timely investments in new generation capacity rely on market capability to generate

accurate signals. If signals for developers of wind farms and conventional power plants are distorted by

market power (or other market failures), it is expected that investors will make suboptimal decisions, leading

to an inefficient power generation portfolio. This paper suggests that policy that seeks to diversify the

ownership of wind power capacity, as this industry continues to expand, may promote competition and

enhance technology-specific signals for new capacity investments.

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Acknowledgment

This research was supported by a Grant from the GIF, the German-Israeli Foundation for Scientific Research

and Development.

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Model symbols

demand intercept

demand slope

realized market demand

conventional generation output of generator i

spot price of electricity

forward contract volume of generator i

price of forward contract

average wind output

deviation from average wind output

marginal costs conventional generator

intercept of MC

slope of MC

expected average spot price

number of conventional generator firms

share of wind output owned by conventional generators

19

Appendix 1

Endogenous number of contracts

Conventional generator’s overall profits is given by

[ ] [ ]

Under risk neutrality and rational expectations , therefore

[ ] [

] [

]

The first order condition w.r.t. the number of contract is

[ ]

(

)

Index the three parts by

(

)

Notice that

1.

2.

[ ]

[ ]

Therefore

(

) (

[ ] )

( [ ]

)

where [ ] .

Next,

(

)

{ [ ] ∑

} (

[ ]

)

21

where

rearranging, we get

(

)

[ ]{ ∑

}

=

{ ∑

} {

[ ] }

[ ∑

]

{

}

Summing up the three parts , , , we obtain

[ ]{ ∑ }

[ ∑

]

Plugging in K and dividing by b

{ [ ]}

∑ {[ ] }

{[ ] }

{ [ ]

}

Employing symmetry, ∑

{ [ ]} { }

{ }

{ [ ]}

21

Rearranging and plugging in

, we get that the number of contracts traded by each conventional

generation firm is

{ }