数理解析研究所講究録
第 1557 巻 2007 年 187-210

187

Electricity markets analysis and design*
Alexander Vasin1,3, Polina Vasina2.

1Lomonosov Moscow State University
2Dorodnicyn Computing Center of the RAS

3New Economic Schoot Moscow
Abstract.
This paper considers Nash equilibria of the unique-price supply ffinction auction for ahomogeneous good. We discuss different estimates and indices of the market power with respect

to an electricity market and show that standard criteria of the market competitiveness are too softfor this market. We obtain the more strict conditions that provide a sufficiently small deviationof the market price$\theta om$

the Walrasian price. The second part studies the problem of multiple

Nash equilibria for the network supply hnction auction in the electricity market. We show that
under typical parameters of the market, its equilibria may be approximated by the equilibria ofthe market without transmission losses. This result permits to reduce the number of possible

equilibria and simplifies the analysis ofthe market.

1. Introduction.
An important economic tendency of the last 30 years was the development of markets forelectricity and natural gas in several countries. The creation of a market includes forming ofseveral private generating companies, and determination of the market mechanism for theirinteraction with consumers. In many existing wholesale markets, the most important part of thismechanism is a regular supply hnction or a double auction that determines the market price andthe production volume for each company. Typically this auction is organized as a unique priceauction (though some studies show that Vickrey auction might be the more efficient form of theinteraction, see Vasin, Vasina (2005).
This work was supported by the NES research program, the RFBR, grant 02-01-00610, and by the grant
1815.2003.1 of President of Russian Federation. We thank Fuad Aleskerov for useful comments.

188
Creation of the market structure concerns the following problem. On one hand, in order toreduce the market power and prevent a large increase of the market price over the competitive
equilibrium price, it seems reasonable to split th generation sector into many small companies.
$e$

On the other hand, the scale effect and the reliability of the electricity supply (that is very

important for Russia) require creation of sufficiently large generating companies. Thus an

important question is what minimal degree of the splitting provides the sufficiently small
deviation ofthe market price ffom the competitive equilibrium price.
Our previous study (Vasin and Vasina, 2005) shows that stable rational behavior of agents atthe supply ffinction auction corresponds to the Coumot equilibrium outcome. So the question

about splitting implies the following theoretical problems.
The first one is evaluation of the Cournot price deviation$\hslash om$

the Walrasian price under

given market structure and available information on the parameters of the market. It is important
to discuss the known indices of the market competitiveness (in particular, Concentration ratioand Herfindahl-Hircshmanindex, see Hircshman, 1963, Tirole, 1997) in context of such

evaluation. In Section 3 we obtain an estimate of deviation of the Cournot price ffom theWalrasian price depending on the demand elasticity and the share of the largest company in themarket. We also discuss standard criteria of the market competitiveness related to Concentration

ratio and Herfindahl-Hircshman index and show that they are too soft for the electricity market.
We obtain the more strict conditions that provide a sufficiently small deviation of the market

price$fi\cdot om$

the Walrasian price.

Another important problem relates to the network structure of electricity and gas markets.

Below we show that in context ofthe imperfect competition study, the losses under transmission

are not so important since the loss coeffcient usually does not exceed 0.1. However,transmission capacity constraints essentially influence the properties of the market in manycases. Our study (2005) shows that, even for the simplest network market with two nodes, there
exist 5 possible variants of the Nash Equilibrium (NE below), moreover, 3 NE may coexist

189
under some parameters. This makes the analysis of the market a complicated problem. Below wedevelop an approach to reduce the number of possible equilibria under consideration. We
employ two ideas. First, we show that some equilibria are incompatible, and provide a simple

rule that distinguishes one of three variants as a possible $NE$ under given parameters of themarket. Then we show that an equilibrium of any market with losses may be approximated bysome equilibrium of the similar market without losses. Thus we reduce to 3 the number ofpossible variants of $NE$ for a two-node market and show that at most two $NE$ may coexist for amarket without losses. We also give an example where two $NE$ coexist, so in general it is
impossible to improve the result.

2. Survey ofliterature.
The problem of imperfect competition in the markets for homogeneous goods (gas,electricity etc) is widely discussed in the literature. For the empirical investigation see Sykes andRobinson (1987). The corresponding theoretical models consider a local market without network
structure. Static one-period models (Baldick et al. (2000), Green (1992), Klemperer and Mayer(1989)) describe a sealed bid unique-price auction as a normal form game and characterize its

Nash equilibria. The latter paper studies a model of competition via arbitrary supply ffinctions set
by producers. For a given demand function they show that for any price above the Walrasian one

there exists the corresponding Nash equilibrium. Green and Newbery (1992) consider asymmetric duopoly with linear supply and demand functions and obtain the explicit expressions

for computation of the Nash equilibrium. Baldick et al. (2000) generalize their result for an
asymmetric oligopoly. Abolmasov and Kolodin (2002) and Dyakova (2003) apply this approach

for a study of the electricity markets in two Russian regions. They use affine approximations of
the actual supply functions.

Let us note that the assumption on the affine structure of supply hnctions does notcorrespond neither to the actual cost structure of generating companies, nor to the rules of supply

functions auctions. Typically every producer can make a bid corresponding to the non-

190
decreasing step supply function. The project of the Russian wholesale electricity market permits

up to 3 steps in a bid of one firm for each hour (see The Model of the Russian Wholesale
Market).

The step structure of a bid approximately corresponds to the actual structure ofvariable costs of generating companies. Usually every such company owns several generatorswith limited capacities and approximately fixed marginal costs. The main part of these costs is

the hel costs.

Our previous paper Vasin, Vasina (2005) studies properties of Nash equilibria for the supply

function auction, where a bid is a non-decreasing step function. We start with investigation ofthe
local market. We show that there exists a unique Nash equilibrium in the Coumot model for any

non-increasing demand Mctlon with the non-decreasing demand elasticity under mild
assumptions on the demand asymptotics as the price tends to infinity. We develop a descriptive

method for computation of the Coumot outcome under any affine demand $\Phi nction$ and piece-

wise constant marginal costs of producers. In the general case, we obtain an explicit upper
estimate of the deviation of the Coumot outcome from the Walrasian outcome proceeding ffom
the demand elasticity and the maximal share of one producer in th total supply at the Walrasian$e$price.

Amir (1996) and Amir and Lambson (2000) study existence and uniqueness of the Nashequilibrium in the Cournot model for logconvex and logconcave inverse demand ffinctions.
(Note that $D^{-- 1}(v)$ is concave (convex) if $p|D^{l}(p)|$ increases (decreases) in $p.$ ) Thus, the first

property is stronger than increasing of the demand elasticity while the second may hold or not
hold in our case. A typical example of the demand ffinction with increasing elasticity that does
not meet the both properties is the demand for a necessary good with the low elasticity for low

prices and the high elasticity for high prices, such that consumers prefer some substitute.

Vasin and Vasina (2005) consider also a model where the market price is determined fromthe balance of the demand and the actual supply of the sealed bid auction and producers set

arbitrary non-decreasing step supply functions as their strategies. We show that, besides the191
Cournot outcome, there exist other Nash equilibria. For any such equilibrium the cut price lies
between the Walrasian price and the Coumot price. Vice versa, for any price between the

Walrasian price and the Coumot price, there exists th corresponding equilibrium. However, we$e$show that only the Nash equilibrium corresponding to the Coumot outcome is stable with respect
to some adaptive dynamics ofproducers’ strategies under general conditions.

This result echoes Moreno and Ubeda (2002) who obtained a similar proposition for a twostage model where at the first stage producers choose production capacities, and at the second
stage they compete by setting the reservation prices. The difference is that in our model the

Coumot type equilibrium always exists under fixed production capacities since the agents set the
production volum es as well as the reservation prices.

Our results differ from Klemperer and Meyer (1989) who study competition with arbitrary
supply functions reported by producers. Under similar conditions, they obtain an infinite set ofNash equilibria corresponding to all prices above the Walrasian price. Our constraint that permits

only non-decreasin step functions is reasonable in context of studying electricity markets. The$g$step structure of the supply function is typical for generating companies and corresponds to theactual rules and the projects ofthe markets in different countries (see Hogan, 1998).
The second part of Vasin and Vasina (2005) considers a simple network market-the market
with two nodes. As above, each local market is characterized by the demand $\Phi nction$ and thefinite set of producers with non-decreasing marginal costs. For every producer his strategy is areported supply hnction that determines his supply of the good depending on the price. The

markets are connected by a transmitting line with fixed share of losses and transmission capacity.

Under given strategies of producers, the network administrator first computes the cut prices forthe separated markets. If the ratio of the prices is sufficiently close to one then transmission is
unprofitable with account of the loss. In this case, the outcome is determined by the cut prices forthe isolated markets. Otherwise the network administrator sets the flow to the market with thehigher cut price (for instance market 2). This flow reduces the supply and increases the cut price192
at the market 1. Simultaneously it increases the supply and reduces the cut price at the market 2.
If the transmitted volume does not exceed the transmission capacity, the network administrator

determines this volume so that the ratio of the final cut prices corresponds to the loss coefficient.
Otherwise, the administrator sets the volume to be equal to the transmission capacity. Thus, heacts as if perfectly competitive intermediaries transmit the good from one market to the other. It

is easy to show that such strategy maximizes the total welfare if the reported supply hnctions
correspond to the actual costs.
We consider Coumot competition model for this market. Our study shows that there exist
three possible types of Nash equilibrium: 1) an equilibrium with zero flow between the markets

and the ratio of the prices close to 1; such equilibrium is determined as if there are two separated
markets; 2) an equilibrium with a positive flow and the ratio of the prices corresponding to theloss coeffcient; 3) an equilibrium with a positive flow equal to the transmission capacity and theratio ofthe prices exceeding the loss coefficient.
Proceeding Rom the first order condition, we defme local equilibria of each type and show
how to compute them. Then we study under what conditions the local equilibrium is a real Nashequilibrium. For the market with constant marginal costs and affine demand functions, we

determine the set ofNash equilibria depending on the parameters. One interesting finding is that,in the symmetric case with equal parameters ofthe local markets and a small loss coefficient, thelocal equilibrium corresponding to the isolated markets is not a Nash equilibrium, but there exist
two asymmetric Nash equilibria with a positive flow ofthe good.
Then we consider a standard network auction of supply functions (with unique nodal prices)and generalize the results obtained for the local auction: stable Nash equilibria correspond to the

Coumot outcomes.

3. Evaluation of the market power and Cournot competition.
According to the previous results, the expected outcome ofthe unique-price supply hnction

auction under rational behavior of agents corresponds to the Cournot equilibrium. Hence it is193
reasonable to consider deviation ofthe Coumot price$p^{t}$

from the Walrasian price$\tilde{p}$as ameasure of the market inefficiency related to the market power of the agents. Below we obtainan estimate ofthis deviation depending on the demand elasticity and the share of the largest
company in the market. We also discuss the known market indices with respect to analysis ofthe
supply function auction at the electricity market.

Consider a market with a homogenous good and a finite set of producers$a$is characterized by his cost ffinction$v\in[0,V^{a}]$, where$C^{a}(v)$. Each producer

with the non-decreasing marginal cost foris his production capacity. The precise form of$V^{a}$$A$$C^{a}(v)$is his private

information. The practically important case is where the marginal cost is a step ffinction:$C^{a}(0)=0,$ $C^{a’}(v)=c_{j}^{a}$for$v \in(\sum_{/=0}^{j- 1}V^{a},\sum_{=0}^{i}V^{a}),$$V_{0}^{a}=0,$$i=1,..,m,$ $\sum_{\iota}^{m}V^{a}=V^{a}$. Consumers’

behavior is characterized by the demand $\Phi nctionD(p)$ , which is continuously differentiable,
decreases in, tends to

$p$

$0$

as

$p$

tends to infinity, and is known to all agents.

Recall basic definitions.
Combination$(v^{a},a\sim\in A)$

ofproduction volumes is a Walrasian equilibrium (WE) andWalrasian price of the local market if, for any $a,$$\sum_{a}v^{a}\sim=D(\tilde{p})$$\tilde{p}$

is a

def
$\sim v^{a}\in S^{a}(\tilde{p})=Arg\max_{v^{a}}(v^{a}\tilde{p}-C^{a}(v^{a}))$,.

Note. The theoretical supplyfunction $S^{a}(p)$ determines the (generally non-unique) optimal

production volume ofthe firm$a$under a given price$p$. Formally, it is a non-decreasing closed

upper semi-continuous point-set mapping with convex values. A trivial result is that the unique
Walrasian price exists under the specified assumptions on the demand function.
For a game $\Gamma=<A,X^{a},f^{a}(x),x\in X,a\in A>$ with the set of players

strategies and the payofffunction$f^{a}$

for each player

$a$

$A$

, the set$X^{a}$of, strategy combination $x^{n}=(x^{a^{*}},a\in A)$

194
is a Nash equilibrium (NE) if $f^{o}(x^{*})\geq f^{a}(x^{*}||x^{a})$ for any $a,$$x^{a}\in X^{a}$. Existence of NE forthe models under consideration is established below.
Cournot competition. Consider a model of Coumot competition for the given market. Then astrategy of each producer$a$is his production volume $v^{a}\in[0,V^{a}]$ . Producers set these values

simultaneously. Let $\vec{v}=(v^{a},a\in A)$ denote a strategy combination. The market priceequalizes the demand with the actual supply:

producer

determines his profit$a$$p( \overline{v})=D^{-1}(\sum_{o\in A}v^{a})$$f^{a}(\overline{v})=v^{a}p(varrow)-C^{a}(v^{a})$model corresponds to the normal form game$p(\overline{v})$. The payoff function of. Thus, the interaction in the Coumot$\Gamma_{C}=\{A,[0,V^{a}],f^{a}(\overline{v}),\overline{v}\in\bigotimes_{a\in\Lambda}[0,V^{a}],a\in A\}$,

where $[0,V^{a}]$ is a set of strategies $a\in A$ .

Combinationin the game
Let$\Gamma_{C}$of production volumes is a Cournot equilibrium (CE) if it is a NE$(v^{a^{l}},a\in A)$.

$(v^{a*},a\in\Lambda)$

denote Nash equilibrium production volumes and$p=D^{-1}( \sum_{aeA}v^{a}")$be thecorresponding price. A necessary and sufficient condition for this collection to be a Nashequilibrium is that, for any$a$,$\# D(p)-D(p)+v^{\theta})p-C^{a}(D(p)-D(p)+v^{a^{*}})\}$$p^{*}\in\Lambda rgp\in[D^{-1}(\begin{array}{l}m\sum_{b\cdot a}v^{b}+V^{o}\end{array}),)]$

Then the F.O.C. for Nash equilibrium is$v^{a^{r}}\in(p-C^{a’}(v^{a}))|D$‘ $(p^{r})|$$v^{a^{l}}=0$

where

if

$C^{a’}(0)\geq p^{\iota}$$C^{a’}(v)=[C_{-}^{a’}(v),C_{+}^{a^{1}}(v)]$

In particular,$C_{+}^{a’}(V^{a})=\infty$.

, for any

$a$

s.t.

$C^{a}$ ‘(0) $<p$,in the break points ofthe marginal cost function.

(1)
(2)

195
Combination $(p,v^{a^{l}},a\in A)$ is called a local Coumot equilibrium if it meets the necessary
conditions (1), (2).
Let us define the Coumot supply Rnction$S_{C}^{a}(p)$of a producer

for $p>0$ as a solution of$a$the system (1),(2). This function determines the optimal production volume of producer

is a Coumot equilibrium price. The function is uniquely defined for any cost function

$a$

if

$C^{a}$$p$. In

particular, consider the case with piece-wise linear cost functions and affne demand function$D(p)= \max$ ( , D-dp). Then$O$

$S_{C}^{a}(p)=\{\begin{array}{ll}0, p<c_{I}^{a},(p c_{1}^{a})d if (p-c_{1}^{a})d<r_{1}^{a},V_{1}^{a} if (p-c_{2}^{a})d<\nabla_{1}^{a}<(p-c_{1}^{a})d,(p c_{2}^{a})d if V_{1}^{a}\leq(p-c_{2}^{a})d\leq V_{1}^{a}+V_{2}^{0} , .V_{1}^{a} V_{2}^{g} if (p-c_{2}^{a})d<V_{1}^{g}+V_{2}^{a}<(p-c_{3}^{a})d,V^{a} if (p-c_{m}^{a})d>V^{a}.\end{array}$Figure 1 shows a typical form of this hnction. The Coumot priceequation$\sum_{a}S_{C}^{a}(p^{t})=D(p^{n})$$c_{1}$’$p$is determined by the.

$c_{2}$$p$$c_{3}$

Our paper Vasin, Vasina (2005) provides the following estimate of the Coumot pricedeviation

from

the

Walrasianprice,

proceeding

from

the

demandelasticity$e(p)=p|D’(p)|/D(p)$ and the maximal share of one firm in the total production at the Walrasianequilibrium.

196
Proposition 1. Let,$\tilde{p}/p\geq 1-1/(\overline{e}n)$$e(p)\geq\overline{e}$for any$p\geq\tilde{p}$,$\max_{a}S^{a+}(\tilde{p})/S^{+}(\tilde{p})\leq 1/m$$\sum_{a}v^{a}/D(\tilde{p})\geq(1-1/(\overline{e}n))^{\overline{e}}$, and $en>1$ . Then.

The given upper bound of deviation from the Walrasian price may be inconvenient forpractical use since the shares of firms and the demand elasticity at the Walrasian price are

typically unobservable, while the actual values under rational behavior of agents correspond to
the Coumot outcome. Below we focus on the case where $D(p)$ is linear in the practicallyimportant interval ofprices.
Proposition 2. Let the maximal share of one firm in the total production at the Cournot

equilibrium meet inequality$e=^{dp}/l(\overline{D}-dp)^{meet}$$\frac{p}{\tilde{p}}-1\leq\frac{1}{(e^{l}n-1)}$

and the demand elasticity at this price$\max_{a}v/a\sum_{be\Lambda}\mathcal{V}^{b}\leq/_{n}^{1}$condition $ne>1$ . Then

.

(3).

This estimate becomes a strict equality for a symmetric oligopoly with a ftxed marginal cost
$c=\tilde{p}$and unbinding capacity constraints. This case also gives the maximal possible loss of thetotal welfare:$w/_{\tilde{w}}=1-(1- \frac{D(p)}{D(c)})^{2}=1-\frac{1}{(n+1)^{2}}$Proof. Consider a fixed profile.

$(v^{a},a\in A)$

of production volumes at the Coumot

equilibrium. First, we determine the maximal deliation of the Walrasian price

Cournot price$p$,$S_{-}^{a}(p-v^{f}/d)\leq v^{a^{*}}$$\overline{p}=p-v^{\overline{a}}/d$,from theunder this profile. Note the following relation betmeen the Coumot andWalrasian supply functions: for any$a$$\tilde{p}$,$p$

and $a,$$S_{C}^{a}(p)>v’$.

$S_{-}^{a}(p-S_{C}^{a}(p)/d)\leq S_{C}^{a}(p)$Hence,$S_{-}(\overline{p})\leq S_{C}(p)=D(p)<D(\overline{p})$for. Thus. In particular, for any$\overline{a}=\arg\max_{a}v^{a}$$\tilde{p}\geq\overline{p}$. Moreover,andotherwi se$\tilde{p}=\overline{p}$

under

197
sufficiently

large$S_{+}^{\overline{a}}(\overline{p})$

and

,

the

maximal$p/ \tilde{p}-1=\frac{v^{\overline{a}}/d}{p-v^{\overline{a}}/d}=\frac{1/n}{dp/D(p)-1/n}=\frac{1}{e(p)n-1}.$

deviation

Q.E.D.

Note. Another interesting example that provides the same ratio between the Coumot and theWalrasian prices is where a large firm with the market share $/1n$ a fixed marginal cost
’

$c$

and

unbounded capacity interacts with the competitive fringe with lower costs and maximal capacity$V_{F}=(1- \frac{1}{n})D(p)$

.

Then

$d(p-\tilde{p})=(\overline{D}-dp^{l}/)n$$\tilde{p}=c$,

the

Coumot

price

meets

the

same

condition

The lower bound for deviation of the Coumot price from theWalrasian price under given conditions is$0$. Figure 2 provide the corresponding example.
$s$

Figure 2.
$V$

Now consider the following regulation problem. Assume that under transition to the

deregulated market a state regulated monopoly that provided electricity for some region issplitted in$n$companies with the same constant marginal co $stc^{a}=c$ . How large should be

$n$

in

order to prevent the increase of the market price more than 500/0 of the cost? It depends on the

demand elasticity. Consider a moderate estimate of this value for the electricity market:$e(p)=0.2$. Proceeding from proposition 2, we obtain $n=15$ . Even if we take the more198
favorable value $e(c)=0.2$ then we obtain $n= \frac{c}{(p-c)e(c)}-1=9$ . Thus a standard assumptionthat the 20% barrier for the large company provides a sufficiently competitive market (see, for$st$instance, Dyakova, 2003) seems to fail in this case.

Now consider another popular measure of the market competitiveness –the Herfindahl-

Hircshman index (Hirschman, $1963$ )of company$a$$HHI= \sum_{a}(y_{a}\cdot 100)^{2}$

where $y_{a}=^{V}/a_{V}$ is the market share.

Proposition 3. Under any fixed $HHI=^{10}/4n$ and market elasticity $e(p)=e$

for theCournot outcome, such that $e\sqrt{n}>$ ], the ratio of the Walrasian and the Cournot prices meetsthe inequality:(4)$p/*\tilde{p}^{-1\leq(e\sqrt{n}-1)^{-1}}$

This estimate becomes a strict equality in the case where a large firm with the market share$/\iota_{\sqrt{n}}$, a fxed marginal cost$c$and unbounded capacity interacts with the competitive fiinge

with lower costs and maximal capacity $V_{F}=(1- \frac{1}{\sqrt{n}})D(p)$ .
Under the symmetric oligopoly with the same $HHI$, the price deviation meets (3) as anequality.

Proof. Under the given HHI, the share of one firm in the total production does not exceed$n^{-1/2}$. So the inequality (4) follows from Proposition 2. The latter two statements of the

proposition relate to the examples considered in the note and proposition 2 respectively.
Consider again the regulation problem. Now, the question is: how low should be the $HHI$ inorder to prevent the increase of the market price more than 50% of the cost? The US govemment

agencies propose (see Report of Office of Economic, 2000) that $HHI\leq 1000(n\geq 10)$ meansthat no firn obtains the market power, and the market is sufficiently competitive. Consider

199
favorable demand electricity $e(c)=0.2$ . Under the symmetric oligopoly, we obtain the desirable
result.

in

However,the$p/_{c}-1= \frac{1}{(\sqrt{n}+1)e(c)}>1$

case

of a

large

firm

with

the

competitive

fringe,, that is, the market price exceed the marginal cost more than twotimes.
Note2. The value $e(c)=0.2$ means that $e(p)\geq 0.5$ in the latter example. For the demandwith the constant elasticity $e(p)=0.2$ , the result would be essentially worser.

4. Empirical study.
In this section we compute NE of the unique-price auction for several variants of theelectricity market in the Central economic region of Russia. The paper by Dyakova (2003) basedon the data from the RAO UES provides the following values of marginal costs and production
capacities of the generating companies in this region. (See Table 1.)
Table 1.

Generator

Marginal$(Rub/mwth)$$Mosener_{-}zo$:$Rosener_{-}goatom$

GC1:

GC2:

:

cost

Capacity(bln kwth per year)

200

GC3:

We consider several demand functions $D(p)=N-\wp$ corresponding to the consumption
and the price in 2000:

For the local market model, we find the Coumot outcome and Vickrey outcome for two
variants ofthe market structure:a)

5 independent companies,b) 3 independent companies (Mosenergo, Rosenergoatom and UGC including all the rest
generators).

For each slope ratio$\gamma$, we evaluate th deviations of the Coumot price from the Walrasian$e$price.
Table 2. Walrasian$(\tilde{p})$and Coumot $(p)$ prices for the electricity market in the Central

economic region of Russia. The cases with 5 and 3 generating companies.

Our results show that deviation of the Coumot price from the Walrasian price strongly
depends on the slope ofthe demand curve and practically interesting values $\gamma=0.1-0.2,$ .

201
5. Network markets and the problem of multiple Cournot equilibria.
Consider two local markets connected by a transmitting line. Every local market $l=1,2$ ischaracterized by the finite set$A^{l}$of producers, $|A^{l}|=n_{l}$ , the cost functions $C^{a}(v),a\in A^{l}$ , anddemand function $D^{l}(p)$ , in the same way as the local market in Section 3: each cost hnction isa private information of agent$a$, the demand function and other market parameters are acommon knowledge. Let $k\in(O,1)$ be the loss coefficient that shows the share of the lost good(in particular, the electric power) under transmission from one market to the other, $Q$ is the

maximal amount of the transmitted good. Consider Coumot competition in this model. Theneach producer sets $v^{a}\in[0,V^{a}]$ . The interaction is as follows.

1. Each firm finds out its cost function.
2.

Simultaneously and independently each firm reports the auctioneer its strategy.

3. For a given strategy combination nodal cut-off prices$C\sim l$and transmitted volume

$q$

are

determined as follows.
Let$\overline{c}^{l}(v),l=1,2$$\lambda=(1-k)^{-1}$. If, denote the cut-off prices for isolated markets, such that$\lambda^{-\downarrow}\leq\overline{c}^{2}(\overline{v})/\overline{c}^{1}(\overline{v})\leq\lambda$

that is, the markets stay isolated. Ifthen $q=0$ , the final prices are$\overline{c}^{2}(\overline{v})/\overline{c}^{1}(\overline{v})>\lambda$

then

$q$

$D^{l}(c \lrcorner)=\sum_{a\in\Lambda}v_{l^{q}}$$c^{l}\sim(\overline{v})=\overline{c}^{l}(\overline{v}),$,$l=1,2$ ,(the transmitted volume fiom

market 1 to market 2) is a solution of the system(5)$D^{2}(c^{2}\sim)=a^{\mathcal{V}^{a}+\overline{q};}\Lambda$$D(c^{1})= \sum_{A^{1}}v^{a}-\overline{q}$$c^{2}\sim=A_{C}^{\sim 1}$, until;(6)$\overline{q}>Q$.

The unique solution of the system exists because the involved functions are monotonous andcontinuous. If$\overline{q}>Q$

then $q=Q,$$C\sim i$are determined Rom (5),(6) with$\overline{q}=Q,$$\sim c^{2}>A_{C}^{\sim 1}$

. The

202
capacity constraint is binding in this case. The case$\overline{c}^{1}(\overline{v})/\overline{c}^{2}(\overline{v})>\lambda$

is treated in the symmetricway. This section aims to study these 3 types ofNash equilibria of the auction.
a). The first-order conditions for the type a) outcome with prices$p_{1}^{r},$$s.t$$p_{2}^{l}$.

$\lambda^{-1}<p_{2}^{l}/p_{1}^{l}<\lambda$,(7)

to be a Coumot equilibrium are quite similar to the conditions (1), (2) for the local market:$v^{a}’\in(p_{i}-C^{a’}(v^{a*}))|D^{j_{1}}(p_{i}^{l})|$$v^{a*}=0$

where

if

$C^{a\prime}(0)\geq p_{i}^{l}$$C^{a}$ ‘ $(v)=[C_{-}^{a\prime}(v),C_{+}^{a\prime}(v)]$, for any $a\in A^{j}$ s.t. $C^{a_{1}}(0)<p_{i}$,in the break points ofthe marginal cost hnction.

Besides that,$\sum_{A^{i}}v^{a*}=D(p_{j}^{l})$,$l=1,2$ .

b). For the type b) outcome with
$q\in(0,Q),$$\Phi i=p_{2}^{s}$,the first-order conditions of the Coumot equilibrium are obtained in a similar way. Note that, forany small change of the strategy$v^{a*}$, producer

stays in the market with the demand$a\in A^{1}$function$D^{1}(p_{1}( \overline{v}))+\lambda(D^{2}(\psi_{1}(\overline{v}))-\sum_{A^{2}}v^{a})$where the price$p_{1}(\overline{v})$meets equation$\sum_{A^{1}}v^{a}=D^{1}(p_{1})+\lambda(D^{2}(\phi_{1})-\sum_{A^{2}}v^{a}))$

.

Thus,
$v^{a*}\in(p_{1}-C^{a_{l}}(v^{a*}))|D^{1_{\dagger}}(p_{1})+\lambda^{2}D^{2\iota}(\psi_{1}^{l})|$for any$a\in A^{1}$

$st$

.

$C^{a*}(0)<p_{1}^{l},$ $v^{a_{*}}=0$

if$C^{a\uparrow}(0)\geq pi$

Similarly, producers in the market 2 face the demand.

$D^{2}( \psi_{1})+1/\lambda(D^{1}(p_{1})-\sum_{A^{1}}v^{a})$

, and

203
$\mathcal{V}^{a*}\in(\lambda pi-C^{a_{1}}(v^{a*}))|D^{2\uparrow}(\psi_{1}^{l})+D^{1_{1}}(p])/\lambda^{2}|$for any$a\in A^{2}$

st.

$C^{a_{\dagger}}(0)<p_{2}^{l},$$v^{a*}=0$

if$C^{a\uparrow}(0)\geq p_{2}^{t}$.

c). Finally, ifthe capacity constraint is binding$q=Q$,$\phi_{1}^{l}<p_{2}^{l}$,then the F.O.C. are$s$$v^{a}\in(p_{i}^{l}-C^{a^{1}}(v^{a^{*}}))|D^{j}$ ‘ $(p_{j})|$$v^{a^{t}}=0$

if$C^{a’}(0)\geq p_{i}^{l}$, for any

$a\in A^{j}$s.t. $C^{a’}(0)<p_{j}^{l}$,$\sum_{\Lambda^{1}}v^{a^{l}}=D(p_{1}^{t})+Q$$\sum_{A^{2}}v^{a}=D(p_{2}^{l})-\lambda Q$.

Thus, even in the simplest variant of the network market there exist 5 possible local

equilibria. The following proposition shows that some of them are incompatible, and reduces theset of variants for examination.
Proposition 4. Underfixedparameters of the two-node market, only one local equilibrium of
the types a) or c) may exist. More precisely, if the Cournot equilibrium prices$\overline{p}_{j},$$i=1,2$ ,forseparated markets meet condition (7) then only a local equilibrium of the type a) exists in themarket, otherwise if

market$j$$\overline{p}_{i}>\Phi_{j}^{-}$

then only a local equilibrium of the type c) with the flowfrom theto the market imay exist (but does not necesserely exist), $i=1,2,$$j=\{1,2\}\backslash i$.

Proof. Consider a system that determines the price for a local equilibrium of the type c)$s$with the flow ffom 1 to 2:$\sum_{A^{1}}S_{c}^{a1}(\overline{\overline{p}}_{1})=D^{1}(\overline{\overline{p}}_{1})+Q,\sum_{A^{2}}S_{c}^{a2}(p_{2}=)=D^{2}(\overline{\overline{p}}_{2})-Q,$$p_{2}>\phi_{1}==$.

Let $p_{i}(Q)=,$ $i=1,2$ , denote a solution of the equations for any $Q\geq 0$ . Note that
$i=1,2,$ $p_{1}=(Q)\uparrow Q,$ $p_{2}(Q)=\downarrow Q$. If the equilibrium a) exists then(8)$\overline{p}_{i}=\overline{\overline{p}}_{i}(0)$$p_{2}=(Q)<\overline{p}_{2}<\Phi_{1}^{-}<\overline{\Phi}_{1}(Q)$

,

for

204
any $Q>0$ , so the system (8) is incompatible and, similarly, a symmetric equilibrium with theflow Rom 2 to 1 does not exist. Otherwise if$\overline{p}_{2}>\Phi_{1}^{-}$

then a solution of (8) exists if

$Q$

is

sufficiently small, that is $p_{2}=(Q)>\phi_{1}^{=}(Q)$ . A solution to the symmetric system does not exist in
this case.

$0$This result simplifies analysis of the market. However, our paper (2005) shows that a typicalcase is where three different local equilibria of the types a) and b) exist, some of them are trueNE, the other are not stable with respect to the large change of the production volume by someagent. As the number of local markets in the network increases, the number of possible local

equilibria of the network market grows with the exponentical rate. The problem of their careful
computation and analysis for the actual markets eems to be hopeless.
$s$

This section aims to develop an altemative approach to analysis of such markets. Since theloss coefficients in the actual networks are usually less then 0.1, we show that NE of a market
with the losses may be approximated by the NE of the similar market without losses and evaluate

the error. Then we consider a problem of the Coumot equilibrium search and analysis for a

network market without losses.
Below we study these issues for the market with affine demand functions and constantmarginal costs. We also assume that capacity constraints in production are not binding. Formally

we assume that$D^{t}(p)=\overline{D}_{l}-dp,$$C^{a}(v)=c_{j}v$

for $a\in A^{j},i=1,2$ . First let us show that thedeviation of the type b) equilibrium for the market with losses from a similar equilibrium for themarket without losses smoothly depends on the loss coefficient and is small for a typical value$k\leq 0.1$

. Under a flow from market 1 to market 2, the equilibrium price$pi$

at the market 1 meetsequation$\sum_{A_{1}}(pi-c_{1})(d_{1}+\lambda^{2}d_{2})+\sum_{A_{2}}(\phi_{1}^{l}-c_{2})(d_{1}+\lambda d_{2})=D^{1}(p_{1})+\lambda D^{2}(\Phi i)$

The market without losses corresponds toofan implicit function,$\lambda=1$.

. According to the theorem on the derivative

205

Thus

for

$| \frac{dp_{1}}{d\lambda}|/p_{1}|_{\lambda=1}\leq\frac{2d_{2}|A_{1}|+(d_{1}+d_{2})|A_{2}|}{(d_{1}+d_{2})\Downarrow A_{1}|+|A_{2}|-1)}\approx 1$$d_{1}=d_{2}$;$\frac{p_{1}(1)-p_{1}(\lambda)}{p_{1}(1)}\approx\lambda-1\leq 0.1^{\cdot}$

undertypical values ofthe loss coefficient.
A similar evaluation holds for an equilibrium of the type c) with the binding transmission

capacity constraint. However, an equilibrium of the type a) (with separated markets) does notexist under $k=0$ , while for $k>0$ a local equilibrium of this type may exist and essentiallydiffer ffom any equilibrium of the network market without losses. For instance, consider asymmetric oligopoly with equal parameters for the both local markets. Then the Coumot pricefor each separated local market isthe united market with $k=0$ is$\overline{p}=c+\frac{(\overline{D}-dc)}{d(m+1)}$$p=c+ \frac{(\overline{D}-dc)}{d(2m+1)}$.

, where $m=|A_{1}|=|A_{2}|$ The Coumot price for, that is, $\overline{p}-c\approx 2(p-c)$ .

However, the local equilibrium with separated markets is not a true Nash equilibrium undertypical value of the electricity market parameters! Let us prove this proposition.
$s$

The conditions for the local equilibrium of the type a) take the form:$a\in A^{j},$$p_{j}^{*}-c_{j}= \frac{\simeq D_{i}}{d(n_{j}+1)}$

optimal price$\overline{p}_{1}$for$a\in A^{1}$

where$\approx D_{i}=\overline{D}_{i}-c_{l}d$. Hence$v^{a^{*}}=v_{j}^{l}=(p_{i}^{l}-c_{l})d$$f^{a}(\overline{v}^{l})=(p_{i}^{l}-c_{l})^{2}d,$$a\in A^{j}$

,

. The

in the joint market meets equation$\overline{p}_{1}-c_{1}=\frac{\overline{D}_{1}+\lambda\overline{D}_{2}-c_{1}(1+\lambda^{2^{*}})d-(n_{1}-1)v_{1}-An_{2}v_{2}^{*}}{2d(1+\lambda^{2})}$.

Condition (7) for existence of the local equilibrium and the conditions of its instability under
the optimal deviation ofagent$a\in A^{1}$take the form:$\lambda>p_{2}/p[>\lambda^{-1}\Leftrightarrow\lambda\geq(\frac{=D_{2}}{d(n_{2}+1)}+c_{2})/(\frac{=D_{1}}{d(n_{1}+1)}+c_{1})>\lambda^{-1}$

;

(9)

206
(10)

$f_{1}( \overline{v})<f_{1}(\overline{v}^{r}\Vert\overline{v}_{1})\Leftrightarrow\frac{2D_{1}^{\sqrt{1+\lambda^{2}}}--}{d(n_{1}+1)}<(’\lambda_{\mathcal{L}_{2}}-\lambda^{2}c_{1}+\frac{2Dl=}{d(n_{1}+1)}+\frac{\lambda D_{2}=}{d(n_{2}+1)})$

Let us show that condition (10) holds under typical parameters of an electricity market, so
the local equilibrium of the type a) (with separated local markets) is not a true NE. Proceedingfrom (9),(10) it suffices to check the following inequality:$\frac{2\overline{\overline{D_{1}}}(\sqrt{1+\lambda^{2}}-1)}{d(n_{1}+1)}<\frac{\overline{\overline{D}_{1}}}{d(n_{1}+1)}+c_{1}(1-\lambda^{2})$$1< \frac{1}{e(c_{1})(n_{1}+1)}(\frac{3-2^{\sqrt{1+\lambda^{2}}}}{\lambda^{2}-1})$

or

.

(11)

Consider several typical values of the loss coefficient.
Proposition 5. For the loss $coeff\iota cientk=0.1$ , the separated equilibrium does not exist if$(n+1)e(c)<0.08$ or ne $(p)<1.08$ , where$k=0.05$$p= \frac{c+(\overline{D}-dc)}{cd(n+1)}$

is the local equilibrium price. For

this proposition holds if $(n+1)e(c)<0.9$ or ne $(p)<1.9$ . In particular, $\iota fe(p)\leq 0.1$

then the equilibrium does not existfor $n<19$ .

Proof

immediately

followsfrominequality

(11)

and

relations$p=c(1+ \frac{1}{(n+1)e(c)})=\frac{c}{(1-1/ne(p))}o$

Proceeding ffom the given results we expect that in a general case the analysis of Coumot

equilibria for a network market may be reduced to the analysis of a similar market without

losses. For a two-node market without losses, we may consider only local equilibria of the typesb) and c), and there exist at most one equilibrium of each type. It would be nice to show that only

one Coumot equilibrium exists in this market in a general case. The following example showsthat, unfortunately, this is not true.

Consider conditions for local equilibria b) with the united market and c) with the binding

constraint and a flow from market 2 to market 1. Let$\overline{p}$and$(\overline{p}_{1},\overline{p}_{2})denote$the corresponding

207
prices,$S_{C}^{i,12}$, and$S_{C}^{i}$be the Coumot supply functions at the market$i$under the united market

and under the binding constraint respectively. Then
$S_{C}^{1,12}(\overline{p})+S_{C}^{2,12}(\overline{p})=D^{1}(\overline{p})+D^{2}(\overline{p}),$$|S_{C}^{2,12}(\overline{p})-D^{2}(\overline{p})|\leq Q$$S_{C}^{2}(\overline{p}_{2})-D^{2}(\overline{p}_{2})=Q=D^{1}(\overline{p}_{1})-S_{C}^{1}(\overline{p}_{1}),\overline{p}_{1}>\overline{p}_{2}$,.

First, let us show that these conditions are incompatible if, that is, the$S_{\dot{C}}^{112}(\overline{p})=S_{C}^{1}(\overline{p})$production capacity of the importing market is fully employed at the price$S_{C}^{2,12}(\overline{p})\geq S_{C}^{2}(\overline{p})$$\overline{p}\geq\overline{p}_{1}$and$S_{C}^{2,12}(\overline{p})-D^{2}(\overline{p})\leq Q$

then$\overline{p}\leq\overline{p}_{2}$under the given condition. This contradicts to. Moreover,$\overline{p}_{1}>\overline{p}_{2}$1, and$D^{1}(p)=\overline{D}^{1}-pd$$D^{1}(\overline{p})-S_{\dot{C}}^{112}(\overline{p})\leq Q$, so$S_{C}^{2,12}(\overline{p})=S_{c}^{2}(\overline{p}_{2})=S_{m\cdot x}^{2}$

producers and fixed marginal costs$m$. Since.

However, the both local equilibria may exist in the case where

Consider a symmetric oligopoly with$\overline{p}$$c^{a}\equiv c$.

at the market. Then the equilibrium price and the profit of each firm at theequilibrium c) are$\overline{p}_{1}=\frac{\overline{D}_{1}-Q+mdc}{d(m+1)},$$Pr_{1}=d(\overline{p}_{1}-c)^{2}$

,

(12)

and similar values for the equilibrium b) are$\overline{p}=\frac{\overline{D}_{1}+2mdc+d\check{p}_{2}}{2d(m+1)},$

where$\tilde{p}_{2}$$Pr=2d[(\overline{DI}\tilde{p}22d2^{-C}$(13)

is the competitive equilibrium and the Coumot price for the separated market 2:$\overline{D}_{2}-\Gamma p_{2}=S_{\max}^{2}$. So at each equilibrium the exporting market 2 supplies the same volume$S_{m\cdot x}^{2}$.

At the equilibrium c), each firm at the market 1 produces less than at the equilibrium b). So theprice is higher and the transmission capacity constraint is binding in the former local
equilibrium. By a sufficiently large increase of the production volume, a firm in the market 1

may reduce the price and make the constraint unbinding (see Fig. 3).

208
Figure 3.

Fig. 3. The inverse demandfunction under fixed strategies$v^{b}=\overline{v}^{a}$for$b\in A^{1}\backslash \{a\},$$v^{A_{2}}=S_{m\alpha}^{2}$.

The demand function in this case is$D(p)= \overline{D_{1}}+\overline{D}_{2}-S_{m*x}^{2}-2dp-\frac{m-1}{m+1}(\overline{D_{1}}-Q-cd)$

the maximal profit for

$a$

,

is

$Pr=(\overline{D_{1}}+\overline{D}_{2}-\frac{m-1}{m+1}(\overline{D_{1}}-Q-cd)-2cd)^{2}\oint d$, where$\overline{D}_{2e}=\overline{D}_{2}-S_{m*}^{2},$.

As to the local quilibrium b), it may be unstable with respect to sufficiently large decrease$\dot{e}$of the production volume by some agent$a\in A^{1}$, such that the transmission constraint becomesbinding. The demand function and the maximal profit in this case are
一$\hat{D}(p)=\overline{D_{1}}-Q-\frac{m-1}{m+1}(+\tilde{p}_{2}d-2cd)$

一$/2d-pd$ ,$/2d-cd]^{2}/Ld$ .

$\hat{P}r=[\overline{D_{1}}-Q-\frac{m-1}{m+1}(+\tilde{p}_{2}d-2cd)$

The following proposition summarizes the results of our study.
Proposition 6. For the given two-node market, two Cournot equilibria exist ifand only if the

profit values determined by (10) $-(13)$ meet inequalities $Pr\geq Pr,$

$Pr_{1}\geq\hat{P}r$

An equivalent system is:$\sqrt{2}(\overline{D_{1}}-Q-cd)\geq(\overline{D_{1}}+\frac{\overline{D}_{2\iota}(m+1)}{2}+\frac{(m-1)}{2}Q-\frac{cd}{2}(m+3)$

ノ

;

.

209
$( \overline{D_{1}}+\overline{D}_{2e}-2cd)\geq\sqrt{2}(\overline{D_{1}}-\overline{D}_{2e}\frac{(m-])}{2}-\frac{Q_{12}(m+1)}{2}+\frac{cd}{2}(m-3))$

Consider the following example. Let

following system for

$Q$

and

$m$

$\overline{D}_{2e}=1,\overline{D_{1}}=2.5,$

particular,for$c_{1}=1$. Then we obtain the:$\sqrt{2}(\frac{m+1}{2}I^{Q\geq 1.5(\sqrt{2}}-1)\geq Q(\frac{m-1}{2}+\sqrt{2})$In$d=1,$

.

any

odd

.
$m_{1}=2k+1$ ,weobtainaninterval

1. $S(\sqrt{2}-1)/(k+\sqrt{2})\geq Q\geq 1.5(\sqrt{2}-1)/(k+1)\sqrt{2}$ where two Cournot equilibria exist in themarket.

210
References
1. Abolmassov A., Kolodin D., (2002). “Market Power and Power Markets: Structural
Problems of Russian Wholesale Electricity Market’ $//EERC$ final report.
2. Amir, , (1996). “Coumot Oligopoly and the Theory of Supermodular Games’ $//Games$
and Economic Behavior, vo1.15, pp. 132-148.
$R$

3. Amir, R., and V. Lambson, (2000). “On the Effects of Entry in Coumot Markets’ //Review ofEconomic Studies, vol. 67, pp. 235-254.
4. Baldick,R., Grant, R., Kahn E., (2000). “Linear Supply Function Equilibrium:
Generalizations, Application, and Limitations’ $//POWER$ Working Paper PWP-078.
University of Califomia Energy Institute, August.

5. Dyakova, Yu., (2003). “Modelling ofthe Russian Wholesale Electricity Matrket. Masterthe $sis’//New$ Economic school, Moscow.

6. Green, R., (1996). “Increasing Competition in the British Electricity Spot Market’ //Joumal of Industrial Economics, 44(2), 205-216.
7. Hogan, W. W., (1998). “Competitive Electricity Market Design: a Wholesale Primer’ //Harvard University.

8. Green,

and D. Newbery, (1992). “Competition in the British Electricity Spot Market’ //Joumal of Political Economy, vol 100(5), pp. 929-953.
$R$

9. Klemperer, P. and M. Meyer, (1989). “Supply Function Equilibria in Oligopoly under
Uncertainty’ $//Econometrica$, vol. 57 (6), pp. 1243-1277.
10. Moreno, D. and L. Ubeda, (2001). “Capacity precommitment and price competition
yield the Coumot outcome’ $//unpublished$ manuscript.
11. Vasin A., Vasina P., (2005). “Models of Supply Functions Competition with Application
to th Network Auctions’ $//Final$ Research Report on the EERC Project R03-1011, site$e$eerc.ru

12. Hircshman A. O., (1963). “The potemity of an index’ $//American$ Economic Preview,$54(5),$ $761- 762$.

13. Tirole J., (1997). “The theory of Industrial $Org_{\bm{t}}ization’//MIT$ Press.
14. RepoIt of Office ofEconomic, Electricity and Natural Gas Analysis, Office ofPolicy
U.S. Department of Energy “Horizontal Market Power in Restructured Electricity
Markets”, Washington, DC, March 2000.
15. Sykes, A. and Robinson, C., (1987). “Current Choices: Good Ways and Bad to Privatize
Electricity’ $//Policy$ Study no 87. London: Centre Policy Studies.

16. The Model ofthe Russian Wholesale Market RAO UES Drafl Document. Version 2.2.