Economics Education and Research Consortium
Working Paper Series
ISSN 1561-2422
No 05/03
Models of supply functions competition
with application to the network auctions
Alexander Vasin
Polina Vasina
This project (No. 03-101) was supported
by the Economics Education and Research Consortium
All opinions expressed here are those of the authors
and not those of the Economics Education and Research Consortium
Research dissemination by the EERC may include views on policy,
but the EERC itself takes no institutional policy positions
Research area: Enterprises and Product Markets
JEL Classification: D44
VASIN A.A., VASINA P.A. Models of supply functions competition with application to the network auctions. —
Moscow: EERC, 2005.
This paper studies different auctions of supply functions in a local market and a simple network market of a homogeneous good with two nodes and a fixed transmission loss per unit of the good. We study problems of existence, uniqueness and computation of Nash equilibria for these models. We also obtain the estimate of Nash equilibria deviation from
the Walrasian equilibrium for each variant. We consider the problem of optimal auction organization from the point of
view of the social welfare maximization.
Keywords. Russia, supply function auction, Cournot, Vickrey, Russian electricity market.
Acknowledgements. The authors thank Richard Ericson and Michael Alexeev for helpful comments.
Alexander Vasin
Polina Vasina
Faculty of Computational Mathematics and Cybernetics
Moscow State University
Vorobiovy Gory, 119899, Moscow, Russia
Tel.: +7 (095) 939 24 91
Fax: +7 (095) 939 25 96
E-mail: [email protected]
 A.A. Vasin, P.A. Vasina 2005
CONTENTS
1. INTRODUCTION
1.1. Policy context of the study
1.2. Statement of the research problem
2. LITERATURE REVIEW
3. MODEL SPECIFICATION AND ESTIMATION RESULTS
3.1. A local market
3.2. A network market with two nodes
4
4
5
8
10
10
23
4. CONCLUSION
34
APPENDICES
36
A1. Mathematical proofs
A2. Empirical study
A3. Table of notations
REFERENCES
36
40
43
45
Economics Education and Research Consortium: Russia and CIS
4
1. INTRODUCTION
1.1. Policy context of the study
When the network market is designed, there are usually several variants of the market structure under consideration. For instance, several variants of the split of the government company CEGB
were discussed before deregulation of the electricity market in the UK. Besides, there exist different
ways of determination of auction outcome on the basis of participants' bids. In such situations it is
important to have an opportunity to evaluate for each variant a possible decrease of the welfare in
comparison with the competitive equilibrium due to the imperfect competition.
There exist two directions of the literature related to this problem. The first direction develops the
methods for computation of the competitive equilibrium for models of the network markets. In
practice these methods are used for auction outcome calculation. This approach assumes that participants' bids correspond to their actual production costs. However it does not take into account the
market power and strategic behavior of the agents. Meanwhile, the empirical studies show that producers, in order to increase the market price, often send bids that do not correspond to the actual
costs. The mentioned models do not provide a possibility to estimate the expected deviation of the
market price from the Walrasian price and the corresponding reduction of the social welfare.
The second direction of the literature studies game-theoretic models of competition via supply
functions for a local market, where the payment is at the cut-off price. This literature describes the
structure and properties of the Nash equilibrium in two cases: 1) when producers may propose arbitrary supply functions; 2) when producers use only linear supply functions. Meanwhile, the literature does not consider a practically important case where the set of producer's strategies is limited
with non-decreasing step supply functions. Such functions correspond (in the first-order approximation) to the actual costs of the producers of the electric power and also to the current rules of the
daily auction in Russia and some other countries. The problem of the Nash equilibrium computation
and study is not solved for network markets. No efficient methods for evaluation of the expected
deviation of the market price from the Walrasian price do exist for such markets. Let us note that
even small amounts of the good passing through the network may essentially change the equilibrium prices. Hence the forecasts and estimates obtained from the local model may be invalid for the
network market.
The analysis of concrete electricity markets shows that the cut-off price at the Nash equilibrium of
the supply function auction may significantly exceed the competitive equilibrium price. In this
contest the alternative methods of auction outcome determination seem to be of interest. The literature considers a Vickrey auction with reserve prices, menu auctions and common agency games.
For the former type it is shown that reporting actual information on the agent's type turns out to be a
dominating strategy. Meanwhile these models were not adapted to the market under consideration.
Also there was no analysis with respect to the concrete markets.
Economics Education and Research Consortium: Russia and CIS
5
The specifics of the electricity market is that the production costs and the maximal capacity of each
generator are rather precisely known to the auctioneer at the time of the auction. At the same time
the actual capacity can by essentially lower than the maximal one due to breakdowns, repairs and
other uncertain factors. The known models that study the problem of optimal auction organization
do not take these specifics into account.
The object of this research is to construct game-theoretic models for different kinds of supply functions auctions for the local and network markets and to find methods for computation of the Nash
equilibria for each variant. On this base, to obtain estimates for the expected deviation of the auction price from the Walrasian price depending on the characteristics of the market and the type of an
auction; to evaluate the reduction of the social welfare with respect to competitive equilibrium; to
develop recommendations on the optimal electricity market organization.
Note that we consider the Walrasian price as a benchmark since it corresponds to the maximum of
the welfare function and is rather close to the average wholesale price in the last years: for instance,
in 2000 the latter price exceeded the former one less than 70% of the Walrasian price.
1.2. Statement of the research problem
This report provides new findings on existence and properties of the Nash equilibrium for the Cournot oligopoly, a model of competition via supply functions, Vickrey auction with reserve prices and
its modification taking into account the common knowledge on producers' costs. In every case, the
underlying market includes a fixed finite number of producers that are heterogeneous in production
capacities and non-decreasing marginal costs of production. Consumers do not play any active role
in the models. Their behavior is characterized by the demand function that is the common knowledge.
Below we start with investigation of the local market. We show that there exists a unique Nash
equilibrium in the Cournot model for any non-increasing demand function 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 Cournot outcome under any affine demand
function and piece-wise constant marginal costs of producers. In the general case, we obtain an explicit upper estimate of the deviation of the Cournot outcome from the Walrasian outcome proceeding from the demand elasticity and the maximal share of one producer in the total supply at the
Walrasian price.
Amir (1996) and Amir, Lambson (2000) study existence and uniqueness of the Nash equilibrium in
the Cournot model for logconvex and logconcave inverse demand functions. (Note that D −1 (v) is
concave (convex) if p | D′( 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 function 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.
Economics Education and Research Consortium: Russia and CIS
6
Then we consider a model where the market price is determined from the balance of the demand
and the actual supply of the sealed bid auction, where producers set arbitrary non-decreasing step
supply functions as their strategies. We show that, besides the Cournot outcome, there exist other
Nash equilibria. For any such equilibrium the cut price lies between the Walrasian price and the
Cournot price. Vice versa, for any price between the Walrasian price and the Cournot price, there
exists the corresponding equilibrium. However, we show that only the Nash equilibrium corresponding to the Cournot outcome is stable with respect to some adaptive dynamics of producers'
strategies under general conditions.
This result echoes Moreno and Ubeda (2002) who obtained a similar proposition for a two-stage
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 Cournot type
equilibrium always exists under fixed production capacities since the agents set the production volumes 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 of Nash equilibria corresponding to all prices above the Walrasian price. Our constraint that permits only nondecreasing step functions is reasonable in context of studying electricity markets. The step structure
of the supply function is typical for generating companies and corresponds to the actual rules and
the projects of the markets in different countries (see Hogan, 1998).
The estimates of the Cournot outcome deviation from competitive equilibrium as well as the results
of calculations for the concrete market show that market price in the supply function auction can
essentially (3–5 times) exceed the Walrasian price under the current market organization. Thus, investigation of alternative variants of auction organization is of great theoretical and practical interest. Below we consider Vickrey auction with reserve prices. In such auction the cut-off price and
production volumes are determined in the same way as in the standard supply function auction.
However, the good obtained from a producer is paid at the reserve prices. The marginal price is a
minimum of the marginal cost of the same output for other producers and the marginal reserve price
of this output for consumers. The marginal cost is calculated on the basis of reported supply functions, but in this case reporting the actual costs and production capacities is a weakly dominating
strategy. In absence of information on production costs the guaranteed value of total profit reaches
its maximum at the corresponding Nash equilibrium.
Our results generalize the results of Ausubel and Cramton (1999) who studied Vickrey auction on
trading a divisible good. In their model the players are consumers. Moreover, we show that the
specified outcome corresponds to the so-called truthful equilibrium for the menu auction introduced in the paper by Bernheim and Whinston (1986), see also Bolle (2004). At this equilibrium
each producer obtains the profit equal to the increase of the total welfare of all participants of the
auction due to his participation in the auction. However, the construction of this equilibrium in
the specified papers needs the complete information on consumers' reserve prices (in our case, on
production costs). In framework of the Vickrey auction, the equilibrium in dominant strategies
Economics Education and Research Consortium: Russia and CIS
7
realizes this outcome under any actual cost functions and private information of each participant
on his function.
Our calculations for the Central Economic Region of Russia show that Vickrey auction price for
consumers exceeds the Walrasian price only 1.5 times (to compare with 3.5–5 times for the standard auction). However, such increase seems to be also rather essential. Besides, there exists reasonable arguing that participants of the auction typically would not reveal their actual costs, that is,
the specified equilibrium in dominant strategies is not realized (see Rothkopf et al., 1990). The
main argument is that reporting the actual costs gives an advantage to the auctioneer (and also to
other economic partners) in the further interactions with this producer.
The situation differs significantly if the marginal costs and the maximal capacity of each generator
are a common knowledge, and the uncertainty relates to a decrease of capacities due to breakdowns
and repairs. In this case current information on the working capacities is weakly correlated with the
future state, and the specified argument against revealing the actual costs turns out to be invalid.
Moreover, the common information may be used for redistribution of the total income in favor of
consumers. We specify the rule for calculation of reserve prices with account of such information.
This rule provides the maximal guaranteed value of the total profit of consumers. Under this rule,
reporting the actual producer's characteristics stays his dominant strategy, and the total welfare still
reaches the maximum.
The second part of this study considers a simple network market — the market with two nodes. As
above, each local market is characterized by the demand function and the finite set of producers
with non-decreasing marginal costs. For every producer his strategy is a reported supply function
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 for the 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 for isolated markets. Otherwise the
network administrator sets the flow to the market with the higher cut price (for instance market 2).
This flow reduces the supply and increases the cut price 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, he acts 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 functions correspond to the actual costs.
First we consider the Cournot 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 the loss coefficient; 3) an equilibrium with a positive flow equal to the transmission capacity and the ratio of
the prices exceeding the loss coefficient.
Economics Education and Research Consortium: Russia and CIS
8
Proceeding from the first order condition, we define local equilibria of each type and show how to
compute them. Then we study under what conditions the local equilibrium is a real Nash equilibrium. For the market with constant marginal costs and affine demand functions, we determine the
set of Nash equilibria depending on the parameters. One interesting finding is that, in the symmetric
case with equal parameters of the local markets and a small loss coefficient, the local equilibrium
corresponding to the isolated markets is not a Nash equilibrium, but there exist two asymmetric
Nash equilibria with a positive flow of the good.
Then we consider a standard network auction of supply functions (the first price auction) and generalize the results obtained for the local auction: for any Nash equilibrium, the market prices lie
between the Walrasian prices and Cournot prices. The inverse statement is also valid.
We also describe two variants of the network Vickrey auction with reserve prices: 1) without common information on costs and 2) with common information on marginal costs and maximal generation capacities. In each case we specify methods of computation of reserve prices and prove the optimality of the considered variant, that is: the supply function corresponding to the actual costs turns
out to be a dominating strategy for each participant, the maximum of the total welfare and the
maximal guaranteed value of consumers' welfare under uncertainty are realized at the corresponding
Nash equilibrium
2. LITERATURE REVIEW
One branch of literature related to this project considers a problem of competitive equilibrium
computation for a network market under given demand and supply functions of agents (McCabe
et al., 1989; Hogan, 1995). For a gas network auction the problem of total welfare maximization
reduces to a problem of linear programming, meanwhile the solution of a dual problem determines the balance prices in all network nodes. For electricity market the total welfare optimization problem generally turns out to be nonlinear due to specifics of network restrictions. Hogan
(1995) does not take in to account this specific. Kulish (2002) proposes correct problem formalization and linearizes it.
This branch does not consider a problem of imperfect competition and impacts of the market power
in network markets. For the empirical investigation see Sykes and Robinson (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 auction as a normal form game and characterize its Nash equilibria. The latter paper studied a model of
competition via arbitrary supply functions set by producers. For a given demand function they
showed that for any price above the Walrasian one there exists the corresponding Nash equilibrium.
Green and Newbery (1992) considered a symmetric duopoly with linear supply and demand functions and obtained the explicit expressions for computation of the Nash equilibrium. Baldick et al.
(2000) generalized their result for asymmetric oligopoly. Abolmasov and Kolodin (2002) and
Economics Education and Research Consortium: Russia and CIS
9
Dyakova (2003) apply this approach for a study of 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 functions does not correspond neither to the actual cost structure of generating companies, no to the rules of supply functions auctions. Typically every producer can make a bid corresponding to the non-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 of variable costs of generating companies.
Usually every such company owns several generators with limited capacities and fixed marginal
costs. The main part of these costs is the fuel costs.
Vasin et al. (2003) study properties of Nash equilibria for the supply function auction, where a bid
is a step non-decreasing function. The paper assumes that the marginal cost of any producer is fixed
and his production capacity is limited. The demand function monotonously decreases in the price
and is a common knowledge. The paper claims that the number of Nash equilibria is less or equal to
n + 1 , where n is a number of producers. However, this is not true. Below we show that for any
price between the Walrasian price and the Cournot price there exists the corresponding Nash equilibrium. The rest results are valid. The Nash equilibrium prices lie between the Walrasian price and
the Cournot price of the market. If the marginal costs do not exceed the Walrasian price then the
equilibrium corresponding to the Cournot outcome is stable and for some class of adaptive dynamics all paths converges to this outcome. Vasin et al. (2003) obtain also an estimate of the Cournot
outcome deviation from the competitive equilibrium. This result permits to evaluate a possible reduction of the social welfare depending on the minimal elasticity of the demand and the maximal
share of one producer in the Walrasian production volume. Below the present report generalizes
these results for local markets with non-decreasing marginal costs and for network auctions.
Vickrey auction with reserve prices is an alternative to the standard auction of supply functions.
This type of an auction was first proposed and investigated for selling an indivisible good in the
form of a second price auction (Vickrey, 1961). Its advantage is an existence of the equilibrium in
dominant strategies revealing the actual reserve prices by consumers. Thus, the good goes to that
one who values it most, that is the total welfare of participants reaches its maximum. The disadvantage is that the income of the seller may be very low. The terms of the auction promote collusion
between bidders.
Ausubel and Cramton (1999) developed a variant of Vickrey auction with reserve prices for selling
a divisible good. Reserve prices restrict a possibility of collusion and increase the seller's income.
Below we describe Vickrey auction for trading a homogeneous good with reserve prices that reflect
consumers' preferences. It possesses the specified positive properties and permits to get the better
result in comparison with a standard supply function auction in the sense of the total welfare as well
as in the sense of the good price for consumers. We also study relation between this auction and
menu auction proposed by Bernheim and Whinston (1986). For trading a divisible good, the terms
of the auction are as follows. Each player-consumer sets the demand function that specifies the
Economics Education and Research Consortium: Russia and CIS
10
amount of the good he is ready to buy depending on prices set by the auctioneer for him and other
consumers. The auctioneer determines these prices and the production volume by maximizing his
profit that is the difference between the sum of consumers' payments and the production cost. Each
consumer aims to maximize the difference of the utility of the good and the payment for it. In the
set of subgame perfect equilibria of this game, Bernheim and Whinston distinguish the truthful
equilibrium constructed as follows. Each player accepts the proposition by the auctioneer if it provides him a fixed profit value, otherwise his demand is equal to zero. Bolle (2004) notes that only
such equilibria are trembling-hand perfect. He reveals the connection between the truthful equilibrium and solution of the Team Selection Problem. Bolle proves that the consumer gain at the truthful equilibrium is equal to his contribution to the total welfare, that is the difference between the
characteristic function values for coalition of all participants and for the coalition of all with the exception of the specified consumer if this function is concave.
Below we prove a similar result for the supply function auction and show that its characteristic
function is concave. Moreover, the truthful equilibrium outcome coincides with the specified solution outcome of Vickrey auction with reserve prices. In contrast to construction by Bernheim and
Whinston, the latter solution does not require common information on production costs.
These results are also of interest in context of studying different communication mechanisms in
"several principals — one agency" type (see Peters, 2001; Martimort and Stole, 2002). These papers
show that, for any communication mechanism between auction participants and the auctioneer, for
any subgame perfect equilibrium there exist a menu auction and a SPE of this auction with the same
outcome. Thus, more complicated communication mechanisms do not enlarge the set of SPEoutcomes with respect to menu auctions.
Rothkoptf et al. (1990) consider theoretical problems and practical results of Vickrey auction. They
note that the analysis of this auction as a one-step game is misleading since the auction typically
repeats. Meanwhile, revealing of information on the actual participant's costs gives the auctioneer a
possibility to reduce the participant's gain in the future, for instance by introducing a specially
formed fictitious bid. We consider a modification of Vickrey auction where the marginal costs and
maximal capacities of generators are a common knowledge, and the uncertainty relates to capacity
decreases due to breakdowns and repairs. In this case the arguments against revealing the actual parameters seem to be weaker.
3. MODEL SPECIFICATION AND ESTIMATION RESULTS
3.1. A local market
Consider a market with a homogenous good and a finite set of producers A . Each producer a is
characterized by his cost function C a (v) with the non-decreasing marginal cost for v ∈ [0,V a ],
where V a is his production capacity. The precise form of C a (v) is his private information. The
11
Economics Education and Research Consortium: Russia and CIS
practically important case is where the marginal cost is a step function: C a (0) = 0, C a ' (v) = cia for
v ∈ (Vi−a1 ,Vi a ), i = 1, ..., m, V0a = 0, Vma = V a . Consumers' behavior is characterized by the demand
function D( p ) , which is continuously differentiable, decreases in p , tends to 0 as p tends to infinity, and is known to all agents.
Below we study and compare several variants of the supply function auction for this market: Cournot competition, the standard (first-price) auction, Vickrey auction with reserve prices, and, finally,
a general scheme for all these auctions is as follows.
Stage 1. Each participant a ∈ A finds out his cost function C a (v) . In the menu auction these functions are common knowledge.
Stage 2. Simultaneously and independently participants report to the auctioneer their bids (reported
supply functions R a ( p ) ), where p = ( p a , a ∈ A) is a price vector proposed to participants, R a ( p )
is a production volume that participant a is ready to supply. The set of permitted bids {R a (⋅)} is
specific for each auction. In the Cournot auction the volumes are fixed. In standard and Vickrey
auctions, R a depends only on p a and is an arbitrary non-decreasing step function with a finite
number of steps.
Stage 3. The auctioneer chooses vector p * by some rule. In the menu auction he realizes the
maximum of his paid function. In our case it is
p* → max( F (∑ R a ( p )) − ∑ p a R a ( p )) ,
p
a
(***)
a
where
V
F (V ) = ∫ D −1 (v)dv
0
is a total benefit of consumers due to obtained volume V of the good. In the Cournot competition
and the standard supply function auction price p a = p is the same for all producers. If equalizes the
total reported supply and the demand for the good. The price also maximizes a similar payoff function. For Vickrey auction the rule of choosing p * does not comply with this scheme: each producer
is payoff for his good at reserve prices determined by the demand and the bids of other producers.
Thus in every case, the interaction among producers may be considered as a normal form game
Γ = A, X a , f a ( x), x ∈ X , a ∈ A where a strategy x a of producer a is his bid, or reported supply
function, and the payoff function f a determines his profit depending on the strategy combination
x = ( x a , a ∈ A) ∈ X = ⊗ X a . A standard assumption is that, in such interaction, rational individuals
a
play a strategy combination that is a Nash equilibrium of the game Γ . Below we shall find Nash
equilibria for each variant and compare them with the Walrasian equilibrium of the market.
12
Economics Education and Research Consortium: Russia and CIS
Recall basic definitions.
Combination (v a , a ∈ A) of production volumes is a Walrasian equilibrium (WE) and p is a Waldef
rasian price of the local market if, for any a , v a ∈ S a ( p ) = Arg max(v a p − C a (v a )) ,
v
a
∑ v a = D( p ) .
a
Note. The theoretical supply function S a ( p ) determines the (generally non-unique) optimal production volume of the 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 Γ = < A, X a , f a ( x ), x ∈ X , a ∈ A > , strategy combination x* = ( x a* , a ∈ A) is Nash
equilibrium (NE) if f a ( x* ) ≥ f a ( x* || x a ) for any a , R a . Existence of NE for the models under
consideration is established below.
Cournot competition. Consider a model of Cournot competition for this market. Then a strategy of
each producer a is his production volume v a ∈ [0, V a ] . Producers set these values simultaneously.
G
G
Let v = (v a , a ∈ A) denote a strategy combination. The market price p (v ) equalizes the demand
G
with the actual supply: p(v ) = D −1 ( ∑ v a ). The payoff function of producer a determines his profit
a∈A
G
G
f a (v ) = v a p(v ) − C a (v a ). Thus, the interaction in the Cournot model corresponds to the normal
G G
form game ΓС = A,[0, V a ], f a (v ), v ∈ ⊗ [0, V a ], a ∈ A , where [0, V a ] is a set of strategies a ∈ A .
a∈A
Combination (v a* , a ∈ A) of production volumes is a Cournot equilibrium (CE) if it is a NE in the
game ΓC .
Let (v a *, a ∈ A) denote Nash equilibrium production volumes and p* = D −1 ( ∑ v a *) be the correa∈A
sponding price. A necessary and sufficient condition for this collection to be a Nash equilibrium is
that, for any a ,
p* ∈ Arg
{( D ( p ) − D ( p ) + v ) p − C ( D ( p ) − D ( p ) + v )}.
*
max





p∈ D −1  ∑ v b* +V a , D −1  ∑ vb*  




 b≠a

 b≠a
 

a*
a
*
a*
Then the F.O.C. for Nash equilibrium is
v a* ∈ ( p* − C a ' (v a* )) | D '( p* ) | , for any a s.t. C a '(0) < p* ,
(1)
v a* = 0 if C a ' (0) ≥ p* ,
(2)
where C a ' (v) = [C−a ' (v), C+a ' (v)] in the break points of the marginal cost function.
13
Economics Education and Research Consortium: Russia and CIS
In particular, C+a ' (V a ) = ∞ .
Combination ( p* , v a* , a ∈ A) is called a local Cournot equilibrium if it meets the necessary conditions (1), (2).
Let us define the Cournot supply function SCa ( p ) of a producer a for p > 0 as a solution of the
system (1), (2). This function determines the optimal production volume of producer a if p is a
Cournot equilibrium price. The function is uniquely defined for any cost function C a . In particular,
consider the case with piece-wise linear cost functions and affine demand function
D( p ) = max(0, D − dp ) . Then
0, p < c1a ,

a
a
a
( p − c1 )d if ( p − c1 )d < V1 ,
 a
a
a
a
V1 if ( p − c2 )d < V1 < ( p − c1 )d ,

SCa ( p) = ( p − c2a ) d if V1a ≤ ( p − c2a ) d ≤ V1a + V2a ,
 a
a
a
a
a
a
V1 + V2 if ( p − c2 )d < V1 + V2 < ( p − c3 ) d ,
....................
 a
V if ( p − cma )d > V a .

Fig. 1 shows a typical form of this function. The Cournot price p* is determined by the equation
∑ SCa ( p* ) = D( p* ) .
a
V
V1 + V2
V1
c1
c2
c3
p
Fig. 1
For the considered linear case, it is obvious that the price is unique. In the general case, the following proposition provides the condition for the unique Cournot price.
def
Proposition 1.1. Let demand function D( p ) and demand elasticity e( p ) = pD′( p ) / D ( p) meet one
of the following conditions:
14
Economics Education and Research Consortium: Russia and CIS
a) D( p ) > 0 and e( p ) ↑ p for p ∈ ( p , M ), D( p) = 0 for p ≥ M ,
b) D( p ) > 0 and e( p ) ↑ p for p ≥ p , lim e( p ) = L > 1/ n , where n is the total number of producp →∞
ers in the market.
Then there exists a unique Nash equilibrium in the game ΓС .
See Appendix for the proof. The idea is that p * meets the F.O.C. for Nash equilibrium iff it is a
solution of the equation
def
F ( p) =
∑ Sca ( p)
D( p ) = 1 .
a
Under given conditions, the function F ( p ) is continuous and monotonous in the interval (0, M ) .
Let
S a ( p ) = Arg max(
pv a − C a (v a )), S a + ( p ) = max S a ( p ), S ( p ) = ∑ S a ( p ) .
a
v
a
Now, we evaluate the deviation of the Cournot outcome from the Walrasian equilibrium proceeding
from the demand elasticity and the maximal share of one firm in the total production at the Walrasian equilibrium.
Proposition 1.2. Let e( p) ≥ e for any p ≥ p , max S a + ( p ) / S ( p ) ≤ 1/ m , and em > 1 . Then
a
p / p* ≥ 1 − 1/(em),
∑ v a* / D( p ) ≥ (1 − 1/(em))e .
(3)
a
Note. Non-decreasing demand elasticity seems to be a reasonable property of the demand of any
homogeneous group of consumers: as the price increases, the money become more and more important with respect to the good. However, a heterogeneous population may lack this property. Consider two groups of consumers with the different fixed demand elasticities. Then it is easy to check
that the total demand elasticity decreases in the price. On the other hand, the condition on demand
elasticity is crucial for uniqueness of Nash equilibrium. In particular, in the given example two
Nash equilibria may exist. A similar situation occurs in the network market with two nodes connected by the line with a fixed transmitting loss (see below).
The supply function auction. Consider the following closed bid auction: every producer a ∈ A
simultaneously sends to the auctioneer his reported supply (r-supply) function R a ( p) that determines the amount of the good this producer is ready to sell at price p, p ≥ 0 . Below we assume that
R a ( p) is a non-decreasing step function with a limited number of steps. So this not a usual function
but a point-set mapping: at any jump point its value is a stretch, and it obtains the same properties as
a theoretical supply function.
The combination of r-supply functions determines the total r-supply
R( p ) = ∑ R a ( p)
a
15
Economics Education and Research Consortium: Russia and CIS
and the cut price c ( R a , a ∈ A) that meets condition D(c ) ∈ R (c ) . Proceeding from the properties of
the demand function, the cut price is uniquely determined for any non-zero r-supply, as well as the
Walrasian price in the market model. (Below we sometimes omit dependence of c on the strategy
combination.)
In order to define the payoff functions, we should consider two cases. Let
def
def
R + ( p ) = sup R( p ), R − ( p ) = inf R( p ) .
If R a + (c~ ) = D(c~ ) then each producer sells the reported volume R a+ (c~ ) at the cut price. Otherwise,
first each producer sells R a− (c~ ) , and then the residual demand D(c~ ) − R− (c~ ) is distributed among
producers with R a + (c~ ) > Ra − (c~ ) according to some rationing rule.
Under a given rationing rule, the profit of producer b ∈ A is determined as follows:
f b( R a (.), a ∈ A) = c ( R a , a ∈ A)v b ( R a, a ∈ A) − C b (v b( R a , a ∈ A)) ,
where v b( R a, a ∈ A) ∈ [ R b−(c ), R b +(c )] is the final demand for his production. Thus, we have defined
the normal form game Γ S that corresponds to the sealed bid auction. Our first task is to study its
Nash equilibria and evaluate their deviations from the Walrasian equilibrium.
Note that there are three possible types of Nash equilibria for Γ S : a) those Nash equilibria for
which R + (c~ ) = D(c~ ) (Nash equilibria without rationing), b) those for which D(c) ∈ ( R − (c), R + (c))
(Nash equilibria with rationing), c) those for which D(c ) = R − (c) < R + (c ) (Nash equilibria with a
barrier, see Fig. 2).
a
c
b
Fig. 2
Proposition 1.3. Let the market meet conditions of Proposition 1.1.
a) For every Nash equilibrium without rationing, the production volumes correspond to the local
Cournot equilibrium. Vice versa, if (v a , a ∈ A) is a Cournot equilibrium, then the corresponding
Nash equilibrium exists in Γ S .
Economics Education and Research Consortium: Russia and CIS
16
b) If ( R a , a ∈ A) is a Nash equilibrium such that D(c ) ∈ ( R − (c ), R + (c)) , then there exists at most
one producer b ∈ A such that R b− (c ) < S b− (c ) (so v a ∈ S a (c ) for any a ≠ b ); the cut price lies in
the interval [ p , p* ] .
c) For any Nash equilibrium of the type c), the cut price lies in the interval [ p , p* ] . Vise versa, for
any p ∈ [ p , p* ) there exists a Nash equilibrium ( R a , a ∈ A) such that c ( R a , a ∈ A) = p .
This proposition corrects our result in Vasin et al. (2003), where we considered only equilibria of
the types a) and b) and found the finite set of such equilibria.
Let us note that every Nash equilibrium of the types b and c is unstable in some sense and, probably, cannot occur as an outcome of the competition via supply functions. In any such equilibrium,
the excessive supply at price c creates a barrier that makes it unprofitable for any player to increase
the cut price by reducing his supply level in the neighborhood of the price c . However, keeping this
barrier is unprofitable. Reduction of R a (c ) till v a for any a ∈ A does not change profits of any
players if other strategies are fixed. Moreover, as soon as the barrier is sufficiently small some
player finds it profitable to reduce his supply function and thus increase the profits of other players.
Indeed, under zero barrier the optimal supply of player a is determined by his Cournot supply
G
function. Since c < p* , v a ( R) > SCa (c ) for some a . Under conditions of Proposition 1.1, the profit
G
function decreases in v a in some vicinity of v a ( R) . Since the profit functions continuously depend
on the barrier value, it is profitable for the player a to reduce his supply (and thus increase c ) when
the barrier is sufficiently small.
Moreover, let us show that, under certain conditions, an adjustment of the supply functions leads to
the Cournot outcome. Assume that each player adjusts only the production volume v a , setting the
supply function
0, p < d a ,
R ( p) =  a
a
v , p ≥ d ,
a
with a fixed reservation cost d a ∈ [0, p ] . Consider a continuous-time adjustment process where, for
every a , the rate of change of v a is proportional to the derivative of the payoff function:
dv a dt = α∂f a (v(t )) ∂v a if v a < V a or ∂f a ∂v a < 0 , otherwise dv a dt = 0 .
(*)
Proposition 1.4. If the demand elasticity does not decrease and there exists a Cournot equilibrium
then for any initial point (v0a , a ∈ A) such that v0a ∈ (0, V a ] and
D −1 (∑ v0a ) > max d a ,
a
a
the solution of (*) converges to the Cournot outcome (v a* , a ∈ A) .
17
Economics Education and Research Consortium: Russia and CIS
Vickrey auction with reserve prices. The optimal auction procedure under uncertainty on
producers' parameters. The Nash equilibrium price in the standard supply function auction may
substantially exceed the competitive equilibrium price due to imperfect competition (see Proposition 1.2, 1.4 and Empirical Study below). Consumers' welfare, and also the total welfare decrease in
this case. One known alternative for the standard auction is Vickrey auction with reserve prices.
Below we specify the rules of such auction for the market under consideration. The procedure is of
special interest since this auction turns out to be the optimal way of trading under uncertainty on
producers' cost function (see Proposition 1.6).
Stage 1. Each producer independently sets and informs the auctioneer on his strategy — a reported
supply function R a ( p ) . He determines this function by specifying values p1a ≤ p2a ≤ ... ≤ pma ,
v1a ≤ v2a ≤ ... ≤ vma . Such bid means that producer a precommits to supply any required amount
v ≤ vla if the price p a of the good set by the auctioneer for him is not less than pla , l = 1, ..., m . For
each producer, p0a = 0, v0a = 0 .
Stage 2. The auctioneer determines the total reported supply function
R + ( p ) = ∑ R a + ( p ) = ∑ vla( a , p ) , p ≥ 0 ,
a
a
where l (a, p) = max{l | pla ≤ p} (Fig. 3).
R3(p)
R(p)
R1(p)
R2(p)
p12
p11 p13
p22
p12
p23 p33 p13
p32
Fig. 3
Stage 3. The auctioneer determines the cut-off price c that equalizes the demand and the total reported supply (see Fig. 4). Formally, D(c ) ∈ R (c ) .
Stage 4. The auctioneer determines production volumes v a , a ∈ A . If c ≠ pla , l = 1,..., m , then
v a = R a (c ) = vla( a ,c ) . For any producer which included c in his bid, v a lies in the interval
[ R−a (c ), R+a (c )] . That is, if pla = c then vla−1 ≤ v a ≤ vla . A typical case is that there is only one such
producer. Then he covers the whole residual demand at the cut-off price v a − vla−1 = D(c ) − R− (c ) .
18
Economics Education and Research Consortium: Russia and CIS
Otherwise the precise values are specified by some rationing rule. In particular, under proportional
rationing
vla( a ,c ) − vla( a ,c )−1
v a − R−a (c )
.
=
D(c ) − R− (c ) R+ (c ) − R− (c )
R(p)
D(p)
c
p
Fig. 4
Thus, the residual demand at the price c is distributed proportionally to the bids at this price.
Note that the set of strategies {R a ( p ), p ≥ 0} of each participant, the rule for determination of the
cut-off price c ( R a ( p), a ∈ A) and the output volumes stay the same as for the standard supply function auction.
Stage 5. For every producer a , the auctioneer determines reserve prices and the total payment for
his good. The payoff to producer a is calculated as follows. The marginal reserve price for additional volume dv of the good under output v a ∈ [0, v a ] is equal to
min{( R A\ a ) −1 (
∑
v b + v a ), D −1 (v b + v a )} .
b∈A \{a}
Here the first function points out the marginal cost of this volume that would be paid if we exclude
participant a from the auction. This cost is determined by the reported supply functions of other
players:
R A\ a ( p ) =
∑
R a ( p) .
b∈A\ a
The second function determines the reserve price that consumers are ready to pay for this volume.
Thus, the profit of agent a is
G
G
G
(*)
f a ( R) = I a ( R) − C a (v a ( R)) ,
where
a
G R ( c )
a
I ( R) = ∫ min{( R A\ a )−1 ( R A\ a (c ) + v a ), D −1 ( R A\ a (c ) + v a )}dv .
0
19
Economics Education and Research Consortium: Russia and CIS
Fig. 5 illustrates this definition. The profit corresponds to the shadowed square.
S a ( p) + R A \ a (c )
R a ( p ) + R A \ a (c )
R A\ a ( p )
c ( A \ a)
c
p
Fig. 5
Under such payment, the profit of producer a is equal to the surplus of the total welfare of all
agents (producers and consumers) related to his participation in the auction (see proposition 1.7).
Stage 6. The auctioneer determines the maximal price pV ( R a , a ∈ A) of the good, its amount and
cost for every consumer. Since the demand function is the sum of demands D b ( p ) of consumers
b ∈ B , each of them obtains the amount vˆb = Db (c ) . Since D b is a monotonously decreasing func-
tion until D b ( p ) = 0 , inverse function ( Db ) −1 (v) determines the marginal reserve prices of the
good for consumer b in interval [0, vˆb ] .
It is possible to distribute the total payment among the consumers in different ways. Consider the
following variant taking into account reserve prices of consumers and at the same time minimizing
a maximal price that they pay for the good. Consumer b buys the good at the maximal price pV
until it exceeds his marginal reserve price. The rest of the amount vˆb he buys out at his reserve
prices. Thus, the total cost of the good for consumer b is
Db ( c )
C b ( pV ) = pV Db ( pV ) +
∫
( D b )−1 (v)dv .
D b ( pV )
It is equal to the shadowed square at Fig. 6.
The maximal price pV equalizes the total cost of the good for consumers and the total payment to
producers:
G
∑ C b ( pV ) = ∑ I a ( R) .
b∈B
a∈A
20
Economics Education and Research Consortium: Russia and CIS
Db ( p )
c
pV
p
Fig. 6
Since each cost function monotonously increases in pV , the unique solution of the latter equation
may be obtained by a standard computational method.
In Appendix ii we compare price pV , Cournot and Walrasian prices for one variant of the electricity market in the Central Economic Region of Russia.
The given procedure determines a normal form game ΓV corresponding to the Vickrey auction with
reserve prices. Players are producers a ∈ A .
Proposition 1.5. For any a , strategy R a ( p ) ≡ S a ( p ) is weakly dominant in the game ΓV . At the
Nash equilibrium ( S a , a ∈ A) production volumes correspond to the competitive equilibrium and
the profit of player a is W ( A) − W ( A \ a) where W ( K ) is the maximal total welfare of producers
and consumers if only producers a ∈ K take part in the auction.
The known result is that this maximal welfare is obtained at the competitive equilibrium of the market with the set K of producers. The welfare value is equal to the square bounded by the demand
D( p ) and supply S K ( p ) =
∑ S a ( p) curves and the price axe (see Fig. 7).
a∈K
Thus for Vickrey auction, Nash equilibrium in dominant strategies corresponds to Walrasian equilibrium in the sense of production volumes. However, the payment to each firm exceeds her income
at the Walrasian equilibrium as the reserve prices according to (*) are grater than the Walrasian
price.
Let us show that under complete uncertainty about cost function C a (v) the specified payment rule
(*) is optimal in the following sense: this is a minimal payment that provides the maximal total welfare as a result of the auction under any value of C a (v) . Let us note that the rule (*) can be reformulated in the following way. The marginal reserve price r a (v a ) for an additional volume dv un-
21
Economics Education and Research Consortium: Russia and CIS
der production volume v a is determined by condition
D(r a ) ∈ R A\ a (r a ) + v a .
(')
S K ( p)
D( p)
W (K )
p ( K )
p
Fig. 7
The total payment for the output v a is equal to
va
a
I =
∫r
a
(v)dv .
(")
0
Proposition 1.6. The rule (', ") determines the minimal payment function under which, for any cost
function, the optimal production volume of the firm a is equal to the supply function value S a of
this firm at the competitive price.
Let us compare Vickrey auction with reserve prices and a menu auction discussed by Bernhaim,
Whinston (1986) and Bolle (2004). These papers assume that the cost functions of all participants
are common knowledge. In contrast to the studied auctions, a reported supply function ( R a ( p )) of
producer a may depend on price vector p = ( p a , a ∈ A) proposed to participants.
Under given strategies ( R a ( p ), a ∈ A) , the auctioneer chooses vector p * that realizes the maximum
of the total welfare of consumers:
p* → max( F (∑ R a ( p )) − ∑ p a R a ( p )) ,
p
a
a
where
V
F (V ) = ∫ D −1 (v)dv
0
(***)
22
Economics Education and Research Consortium: Russia and CIS
is a total benefit of consumers due to obtained volume V of the good.
The menu auction corresponds to the normal form game with payoff functions of players a ∈ A
f a ( R a , a ∈ A, p ( R a , a ∈ A)) = p a (⋅) R a ( p (⋅)) − C a ( R a ( p (⋅))) .
A strategy combination ( R a * ( p ), a ∈ A) along with p * form the truthful equilibrium, if it is the
subgame perfect equilibrium (in particular, (***) holds) and for any p ≠ p * either
def
f a ( R a *, a ∈ A, p ) = f a ( R a *, a ∈ A, p*) = W a ,
or
max(
p a v a − C a (v a )) < W a и R a *( p ) = 0 .
a
v
BW and Bolle argue for selection of the truthful equilibria from the set of SPE proceeding from the
coalition-proofness and the Trembling Hand Perfectness principles.
Proposition 1.7. The profit values of producers at the truthful equilibrium are wa = W ( A) − W ( A \ a)
where characteristic function W ( K ) is determined according to Proposition 1.5.
Thus, at the truthful equilibrium each firm realizes the optimal production volume S a ( p ) corresponding to the Walrasian equilibrium and obtains the payment W ( A) − W ( A \ a) + C a ( S a ( p )) . The
outcome, obviously, coincides with the equilibrium outcome of Vickrey auction with reserve prices.
However, a construction of the corresponding supply function R a *(⋅) requires the exact knowledge
of all participants' costs by each firm.
Vickrey auction with incomplete information on cost functions. Consider now the case where
for each participant a and for each generator i marginal costs cia and maximal capacities ViMa are a
common knowledge. Then the optimal auction procedure differs from the previous case in two aspects. The auctioneer restricts the set of possible bids in accordance with obtained information, accepting from player a only bids corresponding to the specified values of cia and some
via ≤ ViMa , i = 1, ..., m . Besides, he takes into account this information in computation of reserved
prices used for determination of the auction outcome. As in the previous variants, the production
volumes are determined by accepted bids as v a = R a (c ( R b , b ∈ A)), a ∈ A . The payment to firm a
for the good is calculated according to (") on the base of reserve prices but these prices are reduced
in comparison with (') with account of the given information. Let us describe the algorithm for
minimal reserve price r a (v) calculation. This function is determined by the reserve price r a (v) for
a standard Vickrey auction specified by (‘) and marginal cost function cMa (v) corresponding to
(cia ,ViMa , i = 1, ..., m(a )) .
Economics Education and Research Consortium: Russia and CIS
23
Stage 1. Let us find i1 = max{i | cia ≤ r a (0) = c ( R b , b ∈ A \ a)} . Let
V1 = ∑ ViMa , r a (v) = cMa (V1 − v) ,
i ≤i1
until inequality cMa − (V1 − v) > r+a (v) does not hold or v = V1 . In the first case let us define v1 as a
minimal volume for which the specified inequality holds.
Stage l . For a given value vl −1 let il = max{i | cia ≤ r+a (vl −1 )} ,
Vl = ∑ ViMa ,
i ≤il
r a (vl −1 + ∆v) = c a (Vl − ∆v) , until c−a (Vl − ∆v) > r+a (vl −1 + ∆v) does not hold or ∆v = Vl . In the latter
case the algorithm finishes its work. In the former case let us define vl as a minimal value of
vl −1 + ∆v such that the specified inequality realizes, and go to stage l + 1 .
The proposed algorithm calculates the maximal marginal cost allowing the firm a to produce the
volume dv under available information on her costs and the fact of selling this volume in the auction under given bids of other players.
Proposition 1.8. Let the payment to each firm for the supplied volume be calculated according to (")
with substitution of r a (v) for reserve price r a (v) . Then for any Vi a ≤ ViMa for any player a the
strategy R a = S a is weakly dominant, and the maximum of the total welfare is reached at the corresponding Nash equilibrium. The reserve price r a (v) is minimal among reserve prices providing the
specified property.
3.2. A network market with two nodes
Consider two local markets connected by a transmitting line. Every local market l = 1, 2 is characterized by the finite set Al of producers, | Al |= nl , the cost functions C a (v), a ∈ Al , and demand
function Dl ( p ), in the same way as the local market in p.I.: each cost function is a private information of agent a , demand function and other market parameters are a common knowledge. Let
k ∈ (0, L) 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. Agents' strategies ( R a ( p), a ∈ Al ) — r-supply functions — are defined as in p.I.
In this section we consider two variants of the market organization: a standard network supply
function auction and Vickrey network auction with reserve prices. In the both cases producers are
the active players. Each firm aims to maximize its profit. Consumers play a passive role. Each type
of auction is studied from the point of view of the total welfare and consumer prices for the good.
The interaction is as follows.
24
Economics Education and Research Consortium: Russia and CIS
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 l and transmitted volume q are determined as follows.
Let
c l ( R ), l = 1, 2
denote
the
cut-of
prices
for
isolated
markets,
λ = (1 − k ) −1 .
If
λ −1 ≤ c 2 ( R ) / c 1 ( R ) ≤ λ then q = 0 , c l ( R ) = c l ( R ), l = 1, 2 , that is, the markets stay isolated. If
c 2 ( R ) / c 1 ( R ) > λ then q (the transmitted volume from market 1 to market 2) is a solution of the
system
D 2 (c 2 ) ∈ ∑ R a (c 2 ) + q ;
(4)
D1 (c1 ) ∈ ∑ R a (c1 ) − q ;
(5)
A
2
A1
c 2 = λ c1 , until q > Q .
The unique solution of the system exists because the involved mappings are monotonous and
closed. If q > Q then q = Q , c i are determined from (4), (5) with q = Q , c 2 > λ c1 . The capacity
constraint is binding in this case. The case c 1 ( R ) / c 2 ( R ) > λ is treated in the symmetric way.
Finally, prices for consumers and payments to the firms are determined. In the standard auction this
values for a given node correspond to the cut-off price determined according to p.3. For the Vickrey
auction the payment is determined proceeding from reserve prices. A method for their computation
see below.
This section aims to determine computational methods for Nash equilibria of the both types of auction. We shall also characterize these equilibria in the sense of the total welfare and consumer gain.
Network supply function auction. First consider the Cournot competition in this model. Then each
producer sets R a ( p ) ≡ v a . The first-order conditions for the first type outcome with prices
p1* , p2* s.t. λ −1 < p2* p1* < λ ,
(6)
to be a Cournot equilibrium are quite similar to the conditions (1), (2) for the local market:
v a * ∈ ( pi* − C a ' (v a *)) | Di '( pi* ) | , for any a ∈ Ai s.t. C a '(0) < pi* ,
(7)
v a * = 0 if C a '(0) ≥ pi* ,
(8)
where C a '(v) = [C−a '(v), C+a '(v)] in the break points of the marginal cost function.
25
Economics Education and Research Consortium: Russia and CIS
Besides that,
∑ v a * = D( pi* ),
i = 1, 2 .
(9)
Ai
For the second-type outcome with
q ∈ (0, Q), λ p1* = p2* ,
(10)
the first-order conditions of the Cournot equilibrium are obtained in a similar way. Note that, for
any small change of the strategy v a * , producer a ∈ A1 stays in the market with the demand function
D1 ( p1 (v )) + λ ( D 2 (λ p1 (v )) − ∑ v a ) ,
A2
where the price p1 (v ) meets equation
∑ v a = D1 ( p1 ) + λ ( D 2 (λ p1 ) − ∑ v a )) .
A1
A2
Thus,
v a * ∈ ( p1* − C a '(v a *)) | D1 '( p1* ) + λ 2 D 2 '(λ p1* ) |
(11)
for any
a ∈ A1 s.t. C a *(0) < p1* , v a * = 0 if
C a '(0) ≥ p1* .
(12)
Similarly, producers in the market 2 face the demand
D 2 (λ p1 ) + 1 λ ( D1 ( p1 ) − ∑ v a ) ,
A1
and
v a * ∈ (λ p1* − C a '(v a *)) | D 2 '(λ p1* ) + D1 '( p1* ) / λ 2 |
(13)
for any
a ∈ A2
s.t. C a '(0) < p2* , v a * = 0 if
C a '(0) ≥ p2* .
(14)
Finally, if the capacity constraint is binding
q = Q, λ p1* < p2* ,
(15)
v a* ∈ ( pi* − C a ' (v a* )) | Di '( pi* ) | , for any a ∈ Ai s.t. C a ' (0) < pi* ,
(16)
v a* = 0 if C a ' (0) ≥ pi* ,
(17)
then the F.O.C.s are
Economics Education and Research Consortium: Russia and CIS
26
∑ v a* = D( p1* ) + Q ,
(18)
∑ v a* = D( p2* ) − λQ .
(19)
1
A
A2
Proposition 2.1. Every Nash equilibrium in the Cournot competition for the two-node market belongs to one of the given three types, that is, meets either conditions (6)–(9), or conditions
(10)–(14), or conditions (15)–(19).
This proposition is not trivial. There exist two other kinds of situations that could be Nash equilibria
in this model. One type meets conditions
q = 0, λ p1* = p2* ,
( p1* − C a ' (v a* )) | D1 '( p1* ) | ≤ v a* ≤ ( p1* − C a′ (v a* )) | D1′ ( p1* ) + λ 2 D 2′ (λ p1* ) , a ∈ A1 ,
(')
( p2* − C a ' (v a* ) | D 2 '( p2* ) | ≤ v a* ≤ (λ p1* − C a′ (v a* )) | D 2′ (λ p1* ) + D1′ ( p1* ) / λ 2 | , a ∈ A2 .
('')
or the symmetric relations, the other meets q = Q, λ p1* = p2* , and the same inequalities ('), ('') for
v a* . However, for the former type, deviation (in some direction) is profitable for producers a ∈ A1
in the market 1, and for the latter type, deviation is profitable for producers a ∈ A2 in the market 2
(see the proof).
Proposition 2.2. Let demand elasticities el ( p), l = 1, 2 of the separated markets and demand elasticity ( D1 ( p ) + λ D 2 (λ p )) in the joint market increase in p . Then there exist at most one local equilibrium of each type. For any such equilibrium the estimate holds: p l pl * ≥ 1 − 1 (nl e) where p l is the
competitive equilibrium price in the market l , e is the lower bound of the demand elasticity for
p ≥ p l , nl is a maximal share of one producer at the competitive equilibrium in the corresponding
market; for the joint market
def
S ( p1 ) =
∑ S a ( p1 ) + ∑ λ S a (λ p1 ) , S a ( p1 ) / S ( p1 ) ≤ 1n
A1
for a ∈ A1 ,
A2
λ S a (λ p1 ) / S ( p1 ) ≤
1
n
for a ∈ A2 .
Return to the supply function auction with non-decreasing step bid functions. The following proposition generalizes our results obtained for the local market.
Proposition 2.3. If NE ( R a , a ∈ A) is of the type i ∈ {1, 2, 3} then the price pl ( R a , a ∈ A), l = 1, 2 ,
lies between the Walrasian price p l and the price pl * at the local Cournot equilibrium of this type.
For any equilibrium without rationing the outcome correspond to the local Cournot equilibrium.
Nash equilibria of the the other types are unstable under stochastic perturbations of players' strategies.
27
Economics Education and Research Consortium: Russia and CIS
Thus, the expected outcome of the auction corresponds to the Cournot equilibrium.
The given conditions are necessary but not sufficient for the strategy combination to be a Nash
equilibrium. In contrast to the local market model studied above, even concavity of the demand
functions in the both markets together with meeting the F.O.C.s do not guarantee that a player cannot gain by large deviation from his strategy.
Let us establish necessary and sufficient conditions for every type of the local equilibrium to be a
true Nash equilibrium.
First, consider a local equilibrium of the type 1 (with zero flow between the markets). Under a sufficiently large increase of production volume by player a , the price in the market 1 reduces to the
level p1 = p2* λ . The further increasing of the volume permits the player to sell his production also
at the market 2. Under a sufficiently large volume, the price may fall below p̂ˆ1 such that
λ ( D 2 (λ pˆˆ1 ) − ∑ vb* ) = Q ,
A2
and the transmission capacity constraint becomes binding. Fig. 8 shows the demand function for
this producer.
D
D1 ( p1 ) −
vb* + Q
∑
1
A \a
1
D ( p1 ) −
v b* + λ ( D 2 (λ p1 ) − ∑ v b* )
∑
1
2
A \a
A
D1 ( p1 ) −
vb*
∑
1
A \a
p̂ˆ1
p2* λ
p1*
p1
Fig. 8
Note that, according to Proposition 1.1, v a* is the optimal strategy of agent a for the demand function
D1 ( p1 ) − ∑ vb* .
A1 \ a
28
Economics Education and Research Consortium: Russia and CIS
His optimal strategy under the demand
D1 ( p1 ) − ∑ vb* + λ ( D 2 (λ p1 ) − ∑ vb* )
A1 \ a
A2
corresponds to the price p1 s.t.
D1 ( p1 ) − ∑ vb* + λ ( D 2 (λ p1 ) − ∑ vb* ) = v a ∈ ( p1 − C a′ (v a )) | D1′ ( p1 ) + λ 2 D 2′ (λ p1 ) | .
A1 \ a
(20)
A2
Since the left-hand side meets the conditions of Proposition 1.1, the optimal values v a , p1 are
uniquely determined by (20).
Below we assume that the demand functions D1 ( p ) , D 2 ( p ) meet the conditions of Proposition 1.1:
either a) D( p ) > 0 and eD( p ) ↑ p for p ∈ ( p , M ), D( p) = 0 for p ≥ M or b) D( p ) > 0 and
eD( p) ↑ p for p ≥ p , lim e( p ) = L > 1/ m , where m is the total number of producers in the marp →∞
ket. The elasticity of the function D1 ( p ) + λ D 2 (λ p ) increases whenever D1 ( p ) > 0 and
λ D 2 (λ p) > 0 . Let p̂ˆ 2 meet condition
λ ( D1 (λ pˆˆ 2 ) − ∑ vb* ) = Q .
A1
Proposition 2.4. The local equilibrium point of the type a) (that meets conditions (5–8)) is not a
Nash equilibrium if and only if for some producer a ∈ A1 the optimal price p1 under demand
D1 ( p1 ) − ∑ vb* + λ ( D 2 (λ p1 ) − ∑ vb* )
A1 \ a
A2
is less than p2* λ and the payoff under this price exceeds f a (v * ) or, for some producer a ∈ A2 the
optimal price p2 under demand
D 2 ( p2 ) −
∑ vb* + λ ( D1 (λ p2 ) − ∑ vb* )
A2 \ a
A1
lies in interval ( pˆˆ 2 , p2* λ ) and the payoff under this price exceeds f a (v * ) .
Now, consider the local equilibrium of the type b). In order to avoid confusion, let p1 , v denote the
local equilibrium price and production volumes for this case, while pi* , i = 1, 2 , be the Nash equilibrium prices for isolated markets. In this case the demand function for producer a ∈ A1 is shown at
the Fig. 9 and is similar to the one in the Fig. 8.
The difference is that the price p̂2 is determined by equation
D 2 ( pˆ 2 ) = ∑ v b ,
A2
29
Economics Education and Research Consortium: Russia and CIS
D
D1 ( p1 ) −
∑ v b + λ ( D 2 (λ p1 ) − ∑ v b )
A1 \ a
A2
D1 ( p1 ) −
p1
∑ v b
A1 \ a
p̂2 λ
p1
Fig. 9
and p1 < pˆ 2 / λ . Reducing his supply, producer a ∈ A1 can increase the market price to pˆ 2 / λ . Under the further reduction, the market splits. The optimal price p1 for the demand function
D1 ( p1 ) − ∑ v b
A1 \ a
meets relation
D1 ( p1 ) − ∑ v b = v a ∈ ( p1 − C a′ (v a )) | D1′ ( p1 ) | .
A1 \ a
Proposition 2.5. A local equilibrium point of the type b) (that meets conditions (9–13)) is not a
Nash equilibrium if and only if for some producer a ∈ A1 the optimal price p1 under demand
D1 ( p1 ) − ∑ v b
A1 \ a
is less than p̂2 λ and the payoff under this price exceeds f a (v ) .
Note that for a ∈ A2 it is unprofitable to deviate from the local equilibrium of the type b) with a
positive flow from market 1 to market 2. Under fixed strategies of other players the demand function looks like in Fig. 10.
By sufficiently large increase of the production volume, agent a can decrease the price and split the
market in some cases. However, his payoff will decrease in this case because the demand function
for p < λ pˆ lies below the demand curve for the joint market.
For the type c) of local equilibria, the situation is symmetric in some sense. In order to avoid confusion, let pi* (Q), i = 1, 2 , and v a* (Q), a ∈ A1 ∪ A2 , denote equilibrium prices and production volumes
in this case. They meet conditions (15)–(19). The demand function for producer a ∈ A2 looks in
Fig. 11.
30
Economics Education and Research Consortium: Russia and CIS
D
D 2 ( p2 ) −
v b
∑
2
A \a
D 2 ( p2 ) −
λ p̂1
∑ vb + λ −1 (∑ vb − D1 (λ −1 p2 ))
A2 \ a
A1
p 2 = λ p1
p2
Fig. 10
D
D 2 ( p2 ) −
vb* (Q)
∑
2
A \a
D 2 ( p2 ) −
vb* (Q) + λ −1 (∑ vb* (Q) − D1 (λ −1 p2 ))
∑
2
1
A \a
A
D 2 ( p2 ) − λ Q −
λ p1* (Q)
p2* (Q)
∑ vb* (Q)
A2 \ a
p2
Fig. 11
Let p2 denote the optimal price for the joint market:
D 2 ( p2 ) −
∑ vb* (Q) + λ −1 (∑ vb* (Q) − D1 (λ −1 p2 )) = v a ∈ ( p2 − C a′ (v a )) | D 2′ ( p2 ) + λ −2 D1′ (λ −1 p2 ) | ,
A2 \ a
A1
D1 ( p1 (Q )) = ∑ v b* (Q ) .
A1
31
Economics Education and Research Consortium: Russia and CIS
Proposition 2.6. The local equilibrium point of the type c) (that meets conditions (15–19)) is not a
Nash equilibrium if and only if for some producer a ∈ A2 the optimal price p2 lies in interval
(λ p1 (Q ), λ p1* (Q)) and the payoff under this price exceeds f a (v * (Q)) .
As to the players from the market 1, under certain conditions some of them can unite the markets by
reduction of v a . However, this is never profitable for the agent. The situation is symmetric in some
sense to the case described by Fig. 8.
An important issue is whether several Nash equilibria may exist in this model and what is the
structure of the Nash equilibria set depending on parameters of the model. Below we study this issue for the market with affine demand functions and constant marginal costs. We also assume that
capacity constraints in production and transmission are not binding.
Formally we assume that D i ( p ) = D1 − dp , C a (v) = ci v for a ∈ Ai , i = 1, 2 . The conditions for the
local equilibrium of the type a) (with q = 0 ) take the form: v a* = vi* = ( pi* − ci )d , a ∈ Ai ,
pi* − ci =
Di
d (ni + 1)
where Di = Di − ci d . Hence f a (v * ) = ( pi* − ci ) 2 d , a ∈ Ai . According to (20), the optimal price p1
for a ∈ A1 in the joint market meets equation
p1 − c1 =
D1 + λ D2 − c1 (1 + λ 2 )d − (n1 − 1)v1* − λ n2 v2*
.
2d (1 + λ 2 )
Condition (6) for existence of the local equilibrium and the conditions of its instability under the
optimal deviation of agent a ∈ A1 take the form:
λ > p2 p1 > λ −1 ⇔ λ > (
p1 < p2* λ ⇔ λ (c1 (2 + λ 2 ) + λ c2 +
f1 (v * ) < f1 (v * || v1 ) ⇔
D2
+ c2 )
d (n2 + 1)
(
D1
+ c1 ) > λ −1 ,
d (n1 + 1)
2 D1
λ D2
D2
+
) < (1 + λ 2 )(
+ c2 ) ,
d ( n1 + 1) d ( n2 + 1)
d ( n2 + 1)
2 D1 1 + λ 2
2 D1
λ D2
< (λ c2 − λ 2 c1 +
+
).
d (n1 + 1)
d (n1 + 1) d (n2 + 1)
(21)
(22)
The conditions (21', 22') of instability under deviation of agent a ∈ A2 are the same, the only difference is in the change of indices of the markets 1 and 2. In order to simplify the study, assume that
c1 and c2 are negligible with respect to the values
def
D i =
Di
, i = 1, 2 . (Then pi* ≈ D i .)
d (ni + 1)
32
Economics Education and Research Consortium: Russia and CIS
Then
(6) ⇔ λ > D1 D 2 > λ −1 ,
(21) ∩ (22) ⇔ D 2 > D1
2λ
,
1+ 1+ λ2
(21') ∩ (22 ') ⇔ D1 > D 2
2λ
1+ 1+ λ2
.
Fig. 12 describes different areas of the parameters' plane distinguished by these relations.
ln( D1 D 2 )
y = ln λ
ln 2
III
ln
y = ln
1+ 2
2
2λ
1+ 1+ λ2
I
ln
ln1.4
2
II
ln 2
ln λ
1+ 2
IV
ln
1
2
Fig. 12
Proposition 2.7. The local equilibrium of the type a) is Nash equilibrium if the parameters of the
model lie in the area II in Figure 12. For parameters from the area III the equilibrium is unstable
with respect to the deviation of player a ∈ A1 , for parameters from IV — with respect to the deviation of player a ∈ A2 , for parameters in the area I — with respect to deviation of any player
a ∈ A1 ∪ A2 . Note that if
2λ
1+ 1+ λ2
<1
(in particular, for λ < 1.3 ), Nash equilibrium of the type a) does not exist.
Now, let us examine, under what conditions the local equilibrium of the type b) with a flow from
market 1 to market 2 is a Nash equilibrium. Under the previous assumptions on the parameters of
33
Economics Education and Research Consortium: Russia and CIS
the model,
v a = v1 = p1d (1 + λ 2 ), a ∈ A1 ; v a = v2 = λ p1d (1 λ 2 + 1), a ∈ A2 ;
p1 = ( D1 + λ D2 ) [(n1 + n2 + 1)d (1 + λ 2 )] = p 2 λ .
The local equilibrium exists iff there is a positive flow from market 1 to market 2:
λ 2 n2 − 1 n1
n1v1 > D1 − dp1 ⇔ λ D2 > D1 (1 +
)
1 + λ 2 + 1 n1
(23)
Thus, the equilibrium always exists if D 2 D1 ≥ λ . Moreover, under reasonable values of parameters
( λ ≤ 1.2, ni ≥ 3 ), the boundary of the set is close to D 2 D1 ≥ λ −1 . Under deviation of agent a ∈ A1 ,
the optimal price for him is
p1 = ( D1 − (n1 − 1)v1 ) (2d ) ,
the payoff under this price is p12 d . The conditions for the profitable deviation are (see Proposition 2.2):
p1 < pˆ 2 λ ⇔ D1 (λ 2 (1 +
1
2n2
2
)+
) > λ D 2 (2 + λ 2 (1 −
)) ,
n2 + 1 n2 + 1
n1 − 1
f a (v ) < f a (v || v a ) ⇔ D1 > ( D1 + λ D2 )(
n1 − 1
2
+
).
n1 + n2 + 1 1 + λ 2
(24)
(25)
It is easy to see that for λ ≤ 1.7 the latter inequality never holds. Thus, the local equilibrium of the
type b) is Nash equilibrium in this area.
Proposition 2.8. For any combination of parameters, there exists a local equilibrium of the type b)
(either with a flow form 1 to 2, or with a flow from 2 to 1, or the both of them), and at least one of
them is Nash equilibrium. Under symmetric parameter values ( n1 = n2 , D1 = D2 ) and λ ≤ 1.3 , the
symmetric local equilibrium of the type a) is not a Nash equilibrium while there exist two asymmetric Nash equilibria of the type b).
It is interesting that all producers lose in their gains in the latter asymmetric equilibrium with respect to the symmetric local equilibrium, and those who export their production to the other market
lose more than producers in that market.
Network vickrey auction with reserve prices. Under any r-supply functions ( R a , a ∈ A) , cut-off
prices, production volumes and the transmitted volume are determined in the same way as in the
standard network auction. Note that under strategies S a , a ∈ A corresponding to the actual costs
these values are the same as at the competitive equilibrium of the network market. They provide the
maximal total welfare. The payment to agent a ∈ Al for his good is determined, proceeding from
Economics Education and Research Consortium: Russia and CIS
34
reserve prices, as follows. Consider the market without player a ∈ A1 . Compute the cut-off price
and set r a (0) = c ( Rb , b ∈ A \ a ) . For any production volume v reserve price r a (v) is equal to the
cut-off price c1 ( R A\ a , R a ) under strategy R a ( p) ≡ v of player a and under given r-supply functions
of other players. If under v = 0 the transmitted volume is zero then the reserve price is determined
for the separated market 1 until the equality λ r a (v) = c 2 ( R A ) takes place. Under the further growth
of v the reserve price is determined for the joint market with a demand function D1 ( p ) + λ D 2 (λ p ) ,
and since the transmitted volume is equal to the transmission capacity the price is determined for
separated market 1 with demand function D1 ( p ) + Q .
A method for computation of the price r a (v) for a ∈ A2 is similar. Under growth of production
volume v the market can pass from the binding transmission capacity constraint to the joint market.
The maximal volume is determined proceeding from the equality of the reserve price to the marginal cost.
Proposition 2.9. For any a , strategy R a ( p ) ≡ S a ( p ) is weakly dominant in the game ΓV . At the
Nash equilibrium ( S a , a ∈ A) , production volumes correspond to the competitive equilibrium and
the profit of the player a is W ( A) − W ( A \ a) where W ( K ) is the maximal total welfare of producers and consumers if only producers a ∈ K take part in the auction. The specified reserve price is
the minimal one among those providing the maximal total welfare under any cost functions.
Note 1. If the marginal cost functions and maximal capacities of all generators are a common
knowledge then it is possible to compute the lower reserve prices. The computational algorithm is
similar to the one specified above (see Proposition 1.8). The only difference is that the initial reserve price r a (v) is determined as in this section. If the payments to producers turn out to be too
low than it is possible to use some intermediate reserve price.
Note 2. The scheme of the network Vickrey auction and the method of computation of reserve
prices may be generalized for any network auction with given loss coefficients (or transmission
costs) and transmission capacities on each arc. In particular, network gas auctions enter the category. The generalization problem for network auctions in electricity markets is more difficult because of the specifics of the alternating current, but the main idea of the reserve prices computation
stays the same in this case.
4. CONCLUSION
In this paper, we have studied the Cournot oligopoly in the local market and in the simple network
market with two nodes. For the local market, we have proved existence of the unique Cournot equilibrium for a demand function with non-decreasing elasticity. We have shown that the upper estimate of the difference between the Cournot price and the Walrasian price is proportional to the
Economics Education and Research Consortium: Russia and CIS
35
maximal share of one firm in the total production volume at the competitive equilibrium, and it is
inversely proportional to the demand elasticity at the Walrasian price. We obtain a descriptive
method for determination of the Cournot outcome under an affine function and the non-decreasing
piece-wise constant marginal costs of producers.
For the network market, we determined 3 possible types of local equilibria and obtained for each
type the constructive necessary and sufficient condition to be a true Nash equilibrium. For a quasisymmetric model, where the markets differ only in the numbers of producers, we have studied the
Nash equilibrium set depending on parameters of the model. One important conclusion is that multiplicity of local equilibria and, moreover, Nash equilibria is typical for the network market.
For the local market, we have also studied the competition of producers via supply functions in the
case where only non-decreasing step functions are permitted for the agents, as it is at the electricity
market in Russia. We have shown that one of the Nash equilibria corresponds to the Cournot outcome. For the other equilibria, the cut prices lie between the Walrasian and the Cournot prices.
Moreover, we have shown that only the Cournot-type Nash equilibrium is stable with respect to
some class of adaptive dynamics for this model. Thus, the estimate of the deviation from the Walrasian price obtained for the Cournot oligopoly is also valid for this market. Our computations for the
data on the Central economic region of Russia show that, under a typical demand elasticity, the
price for the oligopoly with 5 companies may be several times grater than the competitive equilibrium price
Taking into account this prospect, a reasonable alternative to the standard auction is Vickrey auction with reserve prices. In such auction the payment to each company is made at reserve prices that
are calculated on the base of the demand function and the bids of other firms. Our report describes a
mechanism of this auction and a method for its outcome calculation for a local market and a simple
network market with two nodes. The advantage of this auction is the maximization of the total welfare of participants (producers and consumers) under individually rational behavior (corresponding
to the Nash equilibrium in dominant strategies). Our computations for the data on the market in the
Central economic region of Russia show that under typical values of electricity demand elasticity,
the expected price for consumers in such auction would be essentially lower than at the Cournot
equilibrium for the supply function auction (see Table in Appendix ii). An additional decrease of
the price for consumers may be obtained if computation of the reserve prices takes into account information on marginal costs and maximal capacities of generators. Our report provides the corresponding method.
36
Economics Education and Research Consortium: Russia and CIS
APPENDICES
A1. Mathematical proofs
Proof of Proposition 1.1. Consider the function
def
F ( p) =
∑ SCa ( p)
D( p) = 1 .
(A1)
a
Under given conditions a), the function F ( p ) is continuous and monotonous in the interval (0, M ) .
It tends to 0 if p tends to 0 , and it tends to ∞ if p tends to M . (Assume from the contrary that
lim
p→ M
D′( p )
= B <∞.
D( p )
Then
p D′( p )
= MB( p ) ,
D( p )
where B( p) → B as p → M ,
dD
dp
= D( p) , ln D( p) + ∞ ≤ 2 MB(ln M − ln p) , that is impossible.)
D
p
Hence, there exists a unique solution of equation (A1). Under condition b), the only difference in
the arguing is that F ( p ) is defined and increasing for p ∈ (0, ∞) and lim F ( p ) = mL > 1 .
p →∞
Let p * be a solution of (A1). Then the corresponding combination v a* = SCa ( p* ), a ∈ A , is a Nash
equilibrium. Indeed, the payoff function
ϕ a ( p) = ( D( p) − ∑ vb* ) p − C a ( D( p) − ∑ vb* )
b≠a
b≠a
is unimodal in p since

C a′ ( D ( p ) − ∑ vb* ) 


b≠a
ϕ ′a ( p) = D( p) 1 − e( p)(1 −
)
p




decreases in p because D ( p ) ↓ , e( p) ↑ , C a′ ↓ and p ↑ p .
Proof of Proposition 1.2. Here we give the proof for the case where the marginal cost of production is constant for v a ≤ V a .
37
Economics Education and Research Consortium: Russia and CIS
Consider first the symmetric case c a = c, V a = V ,
a ∈ A , and the demand function with a fixed
elasticity D( p ) = K / p e , p ≥ p . Then, proceeding from (1), p* = c /(1 − (em)−1 ) and
v a* ≡ v* =
if
v* ≤ V ,
otherwise
K (1 − (em)−1 )e
mc e
1
v* = V , p* = ( K / mV ) e .
Note
that
1
p = max(c, ( K / mV ) e ),
v = min(V , K /(mc e )) . Thus, 1 ≥ p / p* ≥ 1 − (em) −1 and v * / v ≥ (1 − (em)−1 )e if v* ≤ V , otherwise
p = p*, v = v* = V .
Now, consider the symmetric case under condition e( p) ≥ e for p > p . Then the Nash equilibrium
price meets condition

 V
c 
1 = m  min 
, e( p )(1 − )   .
p 
 D( p )

Since
p
D( p ) = D( p ) exp(− ∫ (e( p ) / p )dp ) ,
p
the right-hand side is never less than the right-hand side under the constant demand elasticity.
Hence p* ≤ p *(e), v* ≥ v *(e) , where p *(e) and v *(e) are the Nash equilibrium values under
e( p) ≡ e for p ≥ p .
Next, let
V 1 ≥ V 2 ≥ ... ≥ V m , V 1 ≤ ∑ V a / n ,
(A2)
a
other conditions be the same. From (A1), the Nash equilibrium price p* is a unique solution of
equation
D( p ) =
∑
{
}
V a + v( p)m( p ) , where v( p) = ( p − c) D′( p) , m( p) = max a |V a ≥ v( p) .
a >m( p )
We may consider the right-hand side and the solution p* as function of
def
(
)
V = V a, a∈ A .
Then p* (V ) reaches its maximum under condition (A2) when this side reaches its minimum. It is
easy to see that, under a fixed total volume
∑V a
and for any p > c , the minimum is at the point
a
V 1 = V 2 = ... = V n = ∑ V a / n, V a = 0 for any a = n + 1, ..., m.
a
38
Economics Education and Research Consortium: Russia and CIS
Thus, the Nash equilibrium price under condition (A2) is less or equal to the Nash equilibrium price
for the symmetric case with n firms, and the total production volume is not less than the volume for
this case. On the other hand,
p ≥ c, D( p ) ≤ ∑ V a ,
a
as in the symmetric case. Thus, the estimates (3) hold under condition (A2).
Finally, consider the case with different production costs. Let c1 ≤ c 2 ≤ ... ≤ p ≤ c k +1 ≤ ... ≤ c m . The
Nash equilibrium price meets equation (A1) and does not exceed the solution of equation
(
)
D( p ) = ∑ min V a , D′( p ) ( p − c k ) ,
a ≤k
(A3)
since the right-hand side is less or equal to the right-hand side of (A1). But this equation corresponds to the oligopoly with
c 1 = c 2 ≤ ... ≤ c k = c k , V a = V a , a = 1, ..., k , ∑V a =
a
∑
Va .
a∈A+ ( p )
Thus, the estimates (3) follow from the previous case.
Now, consider the general case with convex cost functions C a (v), a ∈ A . Consider the market with
the same A and D( p ) , production capacities Vˆ a = S a + ( p ) and marginal costs cˆ a (v a ) ≡ p for
v a ≤ Vˆ a , a ∈ A . Let ( pˆ * , v a* , a ∈ A) denote the Cournot outcome for this market. Then the estimates
(3) hold, according to the previous proof, and Walrasian price p is the same. Proceeding from the
definition of the Cournot supply function (see (1), (2)), SˆCa ( p) ≤ SCa ( p) for any a, p . Hence
p* ≤ pˆ * , v a* ≥ vˆ a* , a ∈ A .
Proof of Proposition 1.3. Let b be a rationed producer with R b− (c ) < S b− (c ) .
Then vb = D(c ) − R − (c ) + Rb− (c ) , that is, b obtains the whole residual demand at the price c , otherwise any sufficiently small decrease of ci (b ) = c permits to increase the sales volume and the
profit of this producer. Hence v a = S a − (c ) and c ≥ p (since S − ( p ) < D( p ) ).
Let us show that c ≤ p * . Since ( R a , a ∈ A) is a Nash equilibrium, the function
p( D( p) − ∑ v a ) − C b ( D( p) − ∑ v a )
a ≠b
a ≠b
reaches its maximum in the interval p ∈ [0, c ] at the point c . The F.O.C. for this is
vb = D(c ) − ∑ v a ≥ SCb (c ) .
a ≠b
39
Economics Education and Research Consortium: Russia and CIS
For a ≠ b v a = S a − (c ) ≥ SCa (c ) . Hence c ≤ p * .
c) Consider the equilibrium of this type. Arguing similar to the case b) justifies that
v a ∈ [ SCa (c ), S a + (c )] . Hence c ∈ [ p , p*].
Inversely, for any p ∈ [ p , p*] , determine v a , a ∈ A such that v a ∈ [ SCa , S a + ] and
∑ v a = D( p ) .
a
Let
v a , p < p ,
R a ( p) = 
 M , p ≥ p.
Then, under any sufficiently large M , this strategy combination is a Nash equilibrium with the cut
price p .
Proof of Proposition 1.4. Since f a (v) = v a p(v) − C (v a ) then
∂f a ∂v a = p(v) − C ′(v a ) + v a D′( p (v)) , dv a dt = α [ p (v) − C ′(v a ) + v a D′( p (v))] ,
and
dp dt = ( D −1 (∑ v a ))′t = −α
a
[∑ | D′( p ) | ( p(v) − C ′(v a )) − D( p)]
a
( D′( p )) 2
= −α
∑ SCa ( p) − D( p)
a
( D′( p )) 2
.
According to the profit of Proposition 1.1, the difference in the numerator is negative for p < p*
and positive for p > p* . So dp dt > 0 if p < p* , and dp dt < 0 if p > p* . So p(t ) tends to p* as
t tends to ∞ . This implies the convergence of v a (t ) to v a* for any a ∈ A .
Proof of Proposition 1.5. The idea of the proof. For a it is profitable to increase the production
volume v until the cut-off price c (v) (such that D(c (v)) ∈ R A\ a (c (v a )) + v a ), a reserve price
D(c (v)) exceeds marginal costs C a′ (v) . Thus the optimal volume is equal to S a (c ( R A\ a , S a )) for
any R A\ a . Under given volume the gain does not depend on R a ( p) (see Pic.), so S a (⋅) is a weakly
dominant strategy. For the set ( S a , a ∈ A) an outcome v ( S a , a ∈ A) corresponds to competitive
equilibrium, and the gain of player a is equal to W ( A) − W ( A \ a) .
Proof of Proposition 1.6. Let the proposition be proved for the volumes v a ≤ v . Consider a firm
with fixed marginal costs c a = r a (v ) under v a ≤ v + dv = V a . The firm would produce the volume
v + dv only if the marginal price for an additional volume dv is greater than r a (v ) .
40
Economics Education and Research Consortium: Russia and CIS
Proof of Proposition 2.1. Let us show that a point (v a* , a ∈ A) that meets ('), ('') is not a Nash
equilibrium. Let agent a reduce v a* . Then p1 increases and the market splits. If the left inequality
is strict then a small reduction of v a* increases his payoff. Other wise the right inequality is strict.
Consider a small increase of v a* . Then producer a is in the joint market, and the optimal production volume corresponds to the right-hand side. Thus, his payoff increases with v a . The case q = Q ,
λ p1* = p2* is quite similar for producer a ∈ A2 .
Proof of Proposition 2.2. According to the proof of Proposition 1.1, the payoff to the agent a ∈ A1
under fixed strategies of other agents is a unimodal function of price p1 in the intervals p1 < p2* / λ
and p1 > p2* / λ , and
f a ( p1* ) > f a ( p2* / λ ) .
If p1 > p2* / λ then the payoff function increases in the first interval and p1* is a unique maximum.
Otherwise there are two local maximums, and p1* is a global maximum iff
f a ( p1* ) > f a ( p1 ) .
Proofs of propositions 2.3, 2.4 are similar.
A2. Empirical study
In this section we compute NE of the SF auction and Vickrey auction for several variants of the
electricity market in the Central economic region of Russia. The paper by Dyakova based on 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.)
We consider several demand functions D ( p ) = N − γ p corresponding to the consumption in 2000:
γ
0.1
0.2
0.4
0.6
N
279.9
316.1
388.4
460.7
For the local market model, we find the Cournot outcome and Vickrey outcome for two variants of
the market structure:
a) 5 independent companies,
b) 3 independent companies (Mosenergo, Rosenergoatom and UGC including all the rest generators).
For each slope ratio γ , we evaluate the deviations of the NE prices from the Walrasian prices.
41
Economics Education and Research Consortium: Russia and CIS
Table 1
Marginal cost (Rub/mwth)
Capacity
(bln kwth per year)
G1
0
5
G2
75
10
G3
80
10
G4
85
25
G5
90
10
G6
100
5
G7
165
10
Rosenergoatom
12.5
125.4
1
0
16
2
60
2
3
112
3
4
125
2
5
150
16
6
200
2
7
255
2
8
340
10
1
95
2.5
2
110
2.5
3
120
4
4
128
13
5
135
6
6
145
2
7
162
15
1
0
3.5
2
100
2.5
3
120
21
4
150
3.5
5
170
4.5
6
200
4.5
7
215
3
Generator
Mosenergo
GC1
GC2
GC3
42
Economics Education and Research Consortium: Russia and CIS
Table 2. Walrasian, Cournot and Vickrey prices for the electricity market in the Central economic region of Russia. The
cases with 5 and 3 generating companies
γ
p
p5* p
p3* p
pV 5 p
0.1
135
4.24
5.65
1.59
0.2
150
2.45
3.10
1.49
0.4
172.5
1.56
1.87
1.49
0.6
219.67
1.15
1.34
1.30
p — Walrasian price;
p * — Cournot price;
pV — Vickrey price.
Thus, for practically interesting values γ = 0.1 − 0.2 , Vickrey auction is essentially better with respect to consumers than a standard supply function auction.
We compare our results with the results based on affine approximations of the supply functions of
the companies. Papers Abolmasov and Kolodin (2002), Dyakova (2003), Baldick (2000) use this
approach in order to evaluate the expected deviations of the NE prices from the Walrasian prices.
Computation of deviation of the NE from the Walrasian equilibrium based on affine approximations
of the supply functions. Abolmasov and Kolodin(2002), and Dyakova (2003) used the following
method. For every company a = 1, ..., 5 , the supply function affine approximation
S La ( p ) = ka ( p − ca ) was determined by the least squares method. The Walrasian price for supply
function affine approximation was determined by condition
D( p Lm ) =
∑ S Aa ( p Lm ) .
A( m )
The NE supply functions
a = 1, ..., 5 for m = 5,
a ∈{1, 2, 6} for m = 3,
a
S NE
m ( p ) = β a ( p − ca ) ,
were determined according to the results by Baldick et al. (2000) (see also Abolmasov and Kolodin,
2002): coefficients β a meet the system
β a = (1 − β a / ka )(γ + ∑ β j ) ,
a ∈ A(m) .
j≠a
The NE price under the optimal supply functions was determined by the condition
D( p NE m ) =
∑ S Aa ( p NE m ) .
A( m )
43
Economics Education and Research Consortium: Russia and CIS
According to Dyakova's results, deviation of p NE 3 from p was about 50% while deviation of
p NE 5 from p was about 20–25%, for any γ ∈ (0.1, 1) . On this base she concluded that 5 companies are enough to provide a sufficient level of competition at the electricity market in Central economic region of Russia. Our results show that deviation of the Cournot price from Walrasian price
strongly depends on the slope of the demand curve and essentially exceeds these bounds under
γ ∈ (0.1, 0.2) .
A3. Table of notations
CE — Cournot equilibrium;
NE — Nash equilibrium;
SPE — subgame perfect equilibrium;
WE — Walrasian equilibrium;
V a — production capacity of producer a ;
C a (v) — cost function of producer a , v ∈ [0, V a ] ;
f
a
— payoff (profit) function of producer a ;
p — market price;
D( p) — demand function;
e( p) — demand elasticity;
S a ( p) — theoretical supply function of producer a , S ( p ) = ∑ S a ( p ) ;
a
~
p — Walrasian price;
v~ a — Walrasian production volume of producer a ;
p * — Cournot price;
v a * — Cournot production volume of producer a ;
S Ca ( p ) — Cournot supply function of producer a , S C ( p ) = ∑ S Ca ( p ) ;
a
S + = max S , S − = min S for any set S ;
G
R a ( p) — reported supply function of producer a , R = ( R a , a ∈ A) , R( p ) = ∑ R a ( p ) ;
a
c~ — cut price;
Economics Education and Research Consortium: Russia and CIS
44
r a (v) — marginal reserve price for an additional production of producer a under product volume v ;
r a (v) — the same under complete information on marginal costs and maximal production capacities of generators;
pV — NE price at Vickrey auction;
A l — set of producers in the market l , l = 1, 2 ;
nl — number of producers in the market l ;
k — loss coefficient of connecting line, λ = (1 − k ) −1 ;
Q — transmitting capacity of connecting line.
45
Economics Education and Research Consortium: Russia and CIS
REFERENCES
Baldick, Grant, and Kahn (2000) Linear Supply Function Equilibrium: Generalizations, Application, and Limitations,
POWER Working Paper PWP-078 (University of California Energy Institute) August.
Berheim, B.D., Whinston, M.D. (1986) Menu Auctions, Resource Allocation, and Economic Influence, Quarterly Journal of Economics, 1–31.
Martimort D., Stole L. (2002) The revelation and delegation principles in common agency games", Econometrica 70
(4), 1659–1673.
Bolle F. (2004) On Unique Equilibria of Menu Auction (Unpublished manuscript).
Green R., Newbery D. (1992) Competition in the British Electricity Spot Market, Journal of Political Economy 100 (5),
929–953.
Green R. (1996) Increasing Competition in the British Electricity Spot Market, Journal of Industrial Economics 44 (2),
205–216.
Henney A. (1987) Privatise Power: Restructuring the Electricity Supply Industry, Policy Study No. 83 (London: Center
Policy Studies).
Hogan W.W. (1998) Competitive Electricity Market Design: a Wholesale Primer (Harvard University).
Hogan W. (1995) Electricity Transmission and Merging Competition: Why the FERC’s Mega-NOPR Falls Short, Public Utilities Fortnightly 133 (13), 32–36.
Klemperer P., Mayer M. (1989) Supply Function Equilibria in Oligopoly under Uncertainty, Econometrica 57 (6),,
1243–1277.
Lawrence M.A., Cramton P. (1999) Vickrey Auctions with Reserve Princing (Unpublished manuscript).
McCabe K.A., Rassenti S.J., Smith V.L. (1989) Desighning 'Smart' Computer-Assisted Markets (an Experimental Auction for Gas Networks), Journal of Political Economy 5, 259–283 (North-Holland).
Peters M. (2001) Common agency and the revelation principle, Econometrica 69 (5), 1349–1372.
New Russian Energy Strategy up to 2020, Russian Government Proposal No. 389-p.
Rothkoft M., Teisberg T., Kahn E. (1990) Why are Vickrey Auctions Rate?, Journal of Political Economy 98, 94–109.
Sykes A., Colin R. (1987) Current Choices: Good Ways and Bad to Privatize Electricity, Policy Study No. 87 (London:
Centre Policy Studies).
The Model of the Russian Wholesale Market RAO UES Draft Document, Version 2.2.
Vasin A.A, Durakovich N. Vasina P.A. (2003) Cournot Equilibrium and Competition via Supply Functions (Forthcoming).
Аболмасов А., Колодин Д. (2002) Конкурентный рынок или создание монополий: структурные проблемы российского оптового рынка электроэнергии, ERRC final report.
Дьякова Ю.И., (2003) Моделирование оптового рынка электроэнергии в России (Российская экономическая
школа, Дипломная работа).
Кулиш И.В., Моделирование ценообразования на оптовом рынке электроэнергии при наличии договоров прямого платежа (Московский Государственный Университет имени М.И. Ломоносова, Факультет Вычислительной
Математики и Кибернетики, Кафедра Исследования Операций, Дипломная работа).