International Conference
”The European Electricity Market EEM-04”
September 20-22, 2004, Lodz, Poland
Proceedings Volume, pp. 135-142
67-75
THE EQUILIBRIUM
MARKET
MODELS
Agnieszka Wyłomańska
Wroclaw University of Technology
Wroclaw (Poland)
Abstract
We consider electricity producers playing on
the energy market. We model an oligopoly
market facing uncertain demand, where each
firm chooses as its strategy a supply function
relating quantity to price. Such strategy allows a
producer to adapt better to the uncertain
environment than either setting a fixed price or
a fixed quantity. We apply model of supply
function competition in oligopoly under demand
uncertainty, developed by Klemperer and Meyer
[5]. We assume that the uncertain demand is a
function of the price and the stochastic shock.
We describe the shock by PARMA system
(periodic ARMA model).
Our aim in this paper is to provide as
realistic as possible model of oligopolistic
competition in single period game. Our belief is
that the model we apply better represents
oligopoly behaviour than standard Bertrand and
Cournot models.
1. INTRODUCTION
1.1. Supply function
Economists have long debated whether
it makes more sense to think of firms as
choosing prices or quantities as strategic
variables. In the world with uncertainty a
firm may not want to commit to either of
these simple types of strategy, nor can all
IN
OLIGOPOLY
ELECTRICITY
Magdalena Borgosz-Koczwara
Institute of Power Systems Automation
Wroclaw (Poland)
decisions be deferred until the resolution of
the uncertainty. A lot of decision
concerning size and structure of the
organisation must be made in advance.
These decisions implicitly determine a
supply function that relates the quantity the
firm will sell to the price the market will
bear. Such a supply function allows the
firm to adapt better to changing conditions
than does the simply commitment to a fixed
price (Cournot) or quantity (Bertrand).
Any equilibrium concept requires some
story about how firms adjust to uncertainty.
For example, in a stochastic Cournot
game, firms must adjust their prices to the
realisation of demand, and in a stochastic
Bertrand game, firms must adjust their
quantities. Only in supply function
equilibrium do firms adjust to the
uncertainty in optimal manner given their
competitors’ behaviour - with stochastic
Cournot or Bertrand, firms would wish to
alter their behaviour after learning
something about demand [5].
1.2. PARMA models
To calculate supply function in the
presence of demand uncertainty first we
have to describe exogenous shock. The
shock may be connected with seasonality
and some exogenous economic conditions
(e.g. rapid economic growth). Therefore we
applied PARMA model (periodic ARMA
model) to describe the shock.
Series with periodic correlation should
not be modelled with the seasonal
autoregressive moving-average (SARMA)
class. Because SARMA models, contrary
to their name, are actually stationary
models with large (in absolute value)
autocovariances at lags that are multiples
of the period. A flexible class of shortmemory
models
that
have
the
autocovariance periodicities is the class of
periodic autoregressive moving-average
models
(PARMA).
Despite
their
applicability, prediction and likelihood
evaluation methods for PARMA models
remains relatively unexplored, especially
when compared with their stationary
autoregressive moving-average (ARMA)
counterparts.
If X n  is modelled by PARMA(p,q)
model that means the sequence fulfils the
following condition:
Xn 
p
 bi (n) X n  i
i 1

q 1
 a j ( n) n  j ,
(1)
j 0
where the coefficients are periodic in n with
the same period T and the innovations are
independent with zero mean. If p=1 then
for
| P || b(1)b(2)...b(T ) | 1
the solution of the system is periodically
correlated that means:
E (( X n  EX n )( X m  EX m )) 
 E (( X n  T  EX n  T )( X m  T  EX m  T )).
Time series with periodic correlation
naturally arise in climatology, hydrology,
signal processing and economics. Periodic
moving-average processes do not appear
to be useful in economics. Hence, the
econometrics analysis of periodically
correlated time series concentrates on
PAR models:
Xn 
p
 bi (n) X n  i
i 1
 n ,
(2)
where the coefficients and the innovations
have the same properties like in the
general model [3].
1.3. Oligopoly
We consider noncooperative oligopoly
models, for which we have to make
following assumption [2]:
 consumers are price takers,
 all
firms
produce
homogenous
products,
 there is no entry into the market,
 firms collectively have market power
and take own price or quantity
decisions independently.
Three of the best-known oligopoly
models are the Cournot, Bertrand and
Stackelberg models. In the Cournot and
Stackelberg models firms set output levels
(quantity production), whereas in the
Bertrand model firms set prices. In the
Cournot and Bertrand models all firms play
at the same time, whereas in the
Stackelberg model one firm set its quantity
production before the others. These
differences in the action result in different
equilibrium.
A lot of researches were made to show
which equilibrium is the best for the
specified market. But apparently such
simply strategies very often don’t confirm
the expectations. That is why we decided
to apply supply function equilibrium, which
relates the quantity the firm will sell to the
price the market will bear.
In the world with exogenous demand
uncertainty a firm has a set of profit
maximising points (one for each realisation
of uncertainty). It is proved in paper [5] that
in this setting, a firm can generally achieve
higher expected profits by committing to a
supply function than by committing to a
fixed price or quantity, because supply
function allows better adaptation to the
uncertainty.
1.4.
 Demand
 and cost function
P*
In our model we apply supply function
equilibrium in oligopoly under demand
uncertainty. Therefore we give some
economic basis to further consideration.
The generalised demand function lists
variables that influence demand, for
example: price of product, prices of related
goods, expectation of price changes,
tastes
and
preferences,
advertising
expenditures, income per capita. For use in
managerial decision-making, the demand
function must be made explicit. The
relation between quantity and each
demand-determining variable must be
clearly and explicitly specified.
Q  a1 X 1  a 2 X 2  a3 X 3  a 4 X 4
P
(3)
The demand function specifies the
relation between the quantity demanded
and all variables that determine demand.
The demand curve is a part of the demand
function that express the relation between
the prices charged for a product and the
quantity demanded, holding constant the
effect of all other variables. To examine the
relation between prices and the quantity
demanded we must hold constant other
variables.
One of the most important features of
the price elasticity concept is that it
provides a useful summary measure of the
effect of a price change on revenues. A
good estimate of price elasticity makes it
possible to make accurate estimates of the
effect of price changes on total revenue.
For decision-making purposes, three
specific ranges of price elasticity have
been identified: elastic demand (ε < -1),
unitary elasticity (ε = -1) and inelastic
demand (ε > -1) [4]. The horizontal line,
which is shown in Figure 1, describes the
product that is completely sensitive to the
price (elasticity    ). Other lines show
different ranges of price elasticity.
ε < -1
ε = -1
ε > -1
Q
Figure 1. Price elasticity of demand
Cases when ε =   or ε = 0 are rare in
real world, but the monopolies that sell
necessities such as pharmaceuticals enjoy
relatively inelastic demand, whereas firms
in highly competitive markets such as
grocery retailing face highly elastic demand
curves [4]. According to elasticity of the
demand, which is connected with market
structure, we analysed energy market and
we assumed that the price elasticity of
demand should be about 0,5.
All linear demand curves are subject to
varying elasticities at different points on the
curve, see Figure 2. Unitary elasticity is the
point where the effect of a price change is
exactly offset by the effect of a change in
quantity demanded and the firm achieves
the highest total revenues. Because each
firm tends to achieve the highest revenue
therefore it always operate near the unitary
elasticity.
P
Elastic range
ε < -1
Unitary elasticity
ε = -1
Inelastic range
ε > -1
Q
Figure 2. Elasticities along demand curve.
The most prevalent nonlinear model for
cost estimation is the quadratic cost
function [4], which can be written as
follows:
TC  b0  b1Q  b2 Q 2 
 bi X i
 TFC
TC (total cost) refers to relevant costs
during a typical observation period; Q is the
quantity of output produced during that
period; Xi designates all other independent
variables whose cost effects the analyst
wants to account for; and TFC means total
fixed cost.
A quadratic total cost function produces
a linear marginal cost function and a Ushaped average total cost function.
Figure 3 illustrates the typical average total
cost and marginal cost curves associated
with a quadratic total cost function.
The TC function can be written in
different way as follows:
TC  a  (Q  Q  ) 2  b
Q* is constant value and coefficients a and
b are grater than zero. Because the total
cost function is U-shaped so the minimum
of the function is in Q*.
We consider the market where the
supply function is dependent on the market
price and the shock, which is a solution of
PARMA system. The strategy of each firm
is a supply function. In the analysis we
assume:
p - the market price,
{ X n } - the shock,
D ( p, X n ) - demand function,
S i n ( p ) - supply function of i-th company.
Therefore the expected profit of i-th
company at the moment n is given by:
wi n  E{p[D(p,X n )  S j n (p)] 
 C(D(p,X n )  S j n (p))},
where E{G} means the expected value of
random variable G. The maximum profit for
the i-th company is achieved by the price p*
which fulfils the conditions:
(i )
dw i n 
(p )  0
dp
ii 
d 2 wi n
( p )  0
2
dp
i  1,2.
The first condition has the following form:
E{D( p  , X n )  S j n ( p )} 
C
[ p   E ' {C ( D( p  , X n )  S j n ( p  ))}] 
[
ATC
dE{D( p , X n )} dE{S j n ( p  )}

]  0.
dp
dp
MC
And the second condition is:
b1
[ p   E ' {C (D( p  , X n )  S
Q*
Q
Figure 3. Average total cost curve
[
d 2 E{D( p  , X n )}
dp 2


j
n
d 2 E{S
[ 2  E' ' {C ( D( p , X n )  S
[
2. SUPPLY FUNCTION EQUILIBRIUM IN
DOUPOLY
(4)
( p  ))}] 
j
dp
j
n
( p  )}
]
2
n
( p  ))}] 
dE{D( p  , X n )} dE{S j n ( p  )}

]  0.
dp
dp
If we assume the equilibrium on the
power market is symmetric and the cost of
production for each company is
the same. In that case we have
S 1 n ( p)  S 2 n ( p )  S n ( p) . Moreover the
process {Xn} is modelled by PARMA
process with innovations of zero mean and
unit variance. The first condition is given
by:
E{S n ( p  )}  [ p   E ' {C (S n ( p  ))}] 
[
dD( p , X n ) dE{S n ( p  )}

]  0.
dp
dp
Therefore the function E{Sn(p)} fulfils the
difference equation:
dE{S n ( p)}

dp
E{S n ( p)}
dE{D( p , X n )}

p  E ' {C (S n ( p))}
dp
(5)
[ p  E ' {C ( S n ( p))}] 
d 2 E{D( p, X n )}
dp 2

Example 2. We assume the industry
demand is D( p, X n )   mp  X n 2 p ( m  0
) and {Xn} is modelled by PAR(2) which is
given
by
the
equation
X n  b1 (n) X n  1  b2 (n) X n  2   n
(the
coefficients are periodic in n with the
period T), the production cost is a
2
function C (q)  c (q  q s ) / 2  u , where
c, q s , u  0 . In that case for each n
equation (5) has the form:
dE{S n ( p )}

dp
E{S n ( p)}
 m  E{X 2 n }.
p  c( E{S n ( p)}  q s )
and it fulfils the condition:
[
The solution of the above difference
equation is a random variable Sn such that
its expected value (which it taken
by the distribution of Xn) equals
1
4m
( m  m 2 
)( p  cq s ) also is not
2
c
dependent on n and on the coefficients of
PARMA model.
d 2 E{S n ( p )}
]
dp 2
For fixed n we obtain the solution of the
differential equation (7):
[ 2  E' ' {C ( S n ( p ))}] 
[
E{S n ( p )} 
dE{D( p , X n )} dE{S n ( p )}

]  0.
dp
dp
Example 1. We assume the industry
demand is D ( p, X n )   mp  X n ( m  0 )
and {Xn} is modelled by PAR(2)
which is given by the equation
X n  b1 (n) X n  1  b2 (n) X n  2   n
(the
coefficient are periodic in n with the period
T), the production cost is a function
C (q )  c( q  q s ) 2 / 2  u , where c, q s , u  0
(see [5]). In that case for each n equation
(5) has the form:
dE{S n ( p}}
E{S n ( p )}

m
dp
p  c(E{S n ( p )}  qs )
(7)
(6)
1
[ m  E{ X 2 n } 
2
(m  E{ X 2 n }) 2 
4(m  E{X 2 n })
]
c
(8)
 ( p  cq s ).
Therefore in this case the expected
value of supply function at the moment n is
a function dependent on the coefficient of
the PAR model becuase the second
moment of the shock is dependent on PAR
coefficients.
3. SUPPLY FUNCTION EQUILIBRIUM IN
OLIGOPOLY
We assume there are k independent
firms, which sell the power on the electricity
market. The strategy of i-th firm is its
supply function. In that case the profit of
i-th company at n is given by:
w n  E{ p[D( p, X n ) 
i

S
j
n
( p )] 
j i
C (D( p, X n )   S j n ( p))}.
j i
If we assume that there exists the
symmetric equilibrium on the market: each
of the firms has the same supply function
and the marginal cost equals:
wi n  E{ p[D( p, X n )  ( k  1)S n ( p )] 
C (D ( p, X n )  (k  1)S n ( p ))},
where S n ( p)  S j n ( p) , j  1,2,..k . The
first condition has the form:
E{S n ( p  )}  [ p   E ' {C (S n ( p  ))}] 
[

dD( p , X n )
dE{S n ( p )}
 ( k  1)
]0
dp
dp
and it is equivalent to the following
dE{S n ( p)}
E{S n ( p )}


dp
( p  E ' {C (S n ( p ))})(k  1)
1 dE{D( p, X n )}
.
k 1
dp
(9)
The second condition has the form:
[ p  E ' {C ( S n ( p))}] 
[
d 2 E{D( p, X n )}
dp 2
 (k  1)
d 2 E{S n ( p )}
]
dp 2
[ 2  E' ' {C ( S n ( p ))}] 
[
dE{D( p , X n )}
dE{S n ( p )}
 (k  1)
]  0.
dp
dp
Example 3. We consider the same
assumptions as in Example 2 where the
demand function and the marginal cost
function are given respectively by:
D( p , X n )   pm  X 2 n p
1
C (q )  c (q  q s ) 2  u
2
m0
c, q s , u  0.
{Xn} is modelled by PAR(2) which
is
given
by
the
equation
X n  b1 (n) X n  1  b2 (n) X n  2   n
(the
coefficients are periodic in n with the
period T).
We assume there are k independent
companies on the market and there exists
the symmetric equilibrium with supply
function S n ( p ) . In that case the expected
value of the supply function fulfils the
difference equation:
dE{Sn ( p)}
E{S n ( p)}


dp
( p  cE{S n ( p)}  qs )(k  1)
1
(m  E{ X 2 n })
k 1
(10)
The solution of the differential equation has
the following form:
E{Sn ( p )} 
( k  2) 2
p  cq s
[2  k  (m  E{X 2 n }) 
2(k  1)
2
 (m  E{ X n }) 2 
c2
4. CALCULATIONS
2 k( m  E{ X 2 n })
]
c
We based our calculations on demand
and price data since January 2003 to
August 2003.
First we consider Example 1 and we
assume the following:
D ( p, X n )   mp  X n
X n  b1 (n) X n  1  b2 (n) X n  2   n
C (q )  c (q  q s ) 2 / 2  u ,
where m = 0.4655, c = 0.05, qs = 213.0413,
u = 73, qs is a mean of the demand data, u
is a minimum of the price data and m is
calculated on the price and demand data.
In this case the expected value of the
supply function is a linear function of p (see
Figure 4) and it equals:
1
4m
( m  m 2 
)( p  cqs ) 
2
c
2.8273p+30.1170.
E{S n ( p )} 
900
800
700
supply function
600
500
where the coefficients m, c, qs, u are the
same as in the previous example.
In the analysis we take the period T=7
(7 days). The Yule-Walker method of the
PAR coefficients estimation gives the
result:
Table 1. The PAR coefficients
n
1
2
3
4
5
6
7
b1(n)
1.1181
1.2949
1.2971
1.1417
1.3556
1.2270
1.1473
b2(n)
-0.2223
-0.3772
-0.3459
-0.2533
-0.4065
-0.2952
-0.2542
400
In this case the expected value of the
supply function for a fixed n has the
following form:
300
200
100
0
0
50
100
150
p
200
250
300
E{S n ( p )} 
Figure 4. The supply function.
1
( 0.4655  E{X 2 n } 
2
(0 .4655  E{ X 2 n }) 2 
Figure 5. Total cost function.
 ( p  10.6521)
1400
In the Table 2 we presents the expected
value of the supply function for three
selected values of n. To obtain the second
moment of the shock we use 100
realisation of Xn for each n.
1200
cost function
1000
800
600
Table 2. The expected value of the supply
function
400
200
0
4(0.4655  E{ X 2 n })
)
0 .05
0
50
100
150
200
q
250
300
350
400
In Example 2 we have the following
assumptions:
2
D ( p, X n )  mp  X n p ,
X n  b1 (n) X n  1  b2 ( n) X n  2   n
2
C (q )  c( q  q s ) / 2  u ,
n (in days)
7
14
360
1.
The expected value
of supply function
2.8103p+29.9351
2.8105p+29.9374
2.8199p+29.9394
According to Example 3 we consider the
market with k producers. We assume that
4500
k=2
k=3
k=10
k=30
the marginal cost
4000
3500
supply function
3000
2500
2000
1500
1000
500
0
0
20
40
60
80
100
p
120
140
160
180
200
Figure 6. The supply functions for k
producers and the marginal
cost.
4500
k=2
k=3
k=10
k=30
the marginal cost
4000
3500
3000
supply function
the demand and cost functions are the
same as in the previous example.
On Figure 6 we present how the
expected value of the supply function
depends on the number of players. In the
analysis we take n=7. For comparison, on
Figure 7 we show the expected value of
supply function in case without the shock.
2500
2000
1500
1000
500
0
0
20
40
60
80
100
p
120
140
160
180
200
Figure 7. The supply functions for k
producers and the marginal
cost in case without the shock.
5. CONCLUSIONS
We provided three supply functions
equilibrium for duopoly and oligopoly
market. In case of duopoly we obtained two
supply functions equilibrium given by
solutions of differential equations (6) and
(7). When there are a lot of producers on
the market the supply function equilibrium
is given by a solution of equation (10).
We used the mathematical and
economical theory presented in section 2
and 3 to analyse the real price and demand
data. The supply function equilibrium
independent on n was obtained in case
without the shock. Next we analysed the
demand curve, which is a function of the
shock and estimated the PAR coefficients
for the real price and demand data, see
Table 1. In this case the supply function
increases in time, see Table 2. We
analysed the market with k producers to
compare the results and presented them
on Figure 6 and 7.
Moreover, in the case of k producers the
supply function increases in the number of
players. The marginal cost, which is a
function of p, exceeds the supply function
either in the case with shock or without
shock. The slope of the supply function
with k   tends to the slope of the
marginal cost curve.
6. REFERENCES
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Weron R., Wyłomańska A.: On
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Carlton D.W., Perloff J.M.: Modern
Industrial Organization. Second Edition.
HarperCollins CollegePublishers, New
York 1994.
Ghysels E., Osborn D.R.: The
econometric Analysis of Seasonal Time
Series, Cambridge University Press,
Cambridge 2001.
Hirschey M., Pappas J.L., Whigham D.:
Managerial
Economics.
European
Edition. The Dyren Press, London
1995.
Klemperer P.D., Meyer M.A.: Supply
function equilibria in oligopoly under
uncertainty. Econometrica, vol. 57, no.
6, November 1989, pp. 1243-1277.
Lund R., Basawa I.V.: Recursive
prediction and likelihood evaluation for
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Probability in Mathematical Statistics to
appear.
Agnieszka Wyłomańska
was born in 1978 in Nowa Ruda, Poland. She
received M.Sc. degree in mathematics science
from the Wroclaw University of Technology. At
present she is taking PhD dissertation. Her
areas of interest include ARMA models with
varying coefficients.
Mailing address:
Agnieszka Wyłomańska
Wroclaw University of Technology
Institute of Mathematics
14 Janiszewskiego St., 50-370 Wroclaw
POLAND
Phone: (+48)(71) 320-31-83
e-mail: [email protected]
Magdalena Borgosz-Koczwara
was born in 1972 in Swidnica, Poland. She
received M.Sc. degree from the Wroclaw
University of Technology. Presently she is
taking PhD dissertation. Her area of interest
concern application of game theory to energy
market.
Mailing address:
Magdalena Borgosz-Koczwara
Institute of Power Systems Automation
1 Wystawowa St., 51-618 Wroclaw
POLAND
Phone: (+48)(71) 348-42-21
e-mail: [email protected]