7/28/2019 Cournot and perfect comptency in markets
1/22
THE UNIVERSITY OF OKLAHOMA
FINAL REPORT
ECE 5973 - POWER SYSTEMS AND MARKET OPERATIONS
DR. JOHN JIANG
7/28/2019 Cournot and perfect comptency in markets
2/22
1
Cournot&PerfectCompetition
applicationandcomparisonin
ElectricityMarket
Osman Bayindir
Diego Rodrguez
7/28/2019 Cournot and perfect comptency in markets
3/22
2
Contents
1. Introduction ......................................................................................................................................... 3
2. Cournot and Perfect Market Description ......................................................................................... 4
a. Perfect Competition ..................................................................................................................... 4b. Cournot .......................................................................................................................................... 5c.
Comparison between Perfect Market and Cournot
................................................................. 9
3. Electricity Market Study ................................................................................................................... 10
a. Model............................................................................................................................................ 10b. Perfect Competition Case .......................................................................................................... 14c. Cournot Model Case ................................................................................................................... 15
4. Result Analysis .................................................................................................................................. 17
5. Conclusions ........................................................................................................................................ 19
6. References ......................................................................................................................................... 20
7. Appendix ............................................................................................................................................ 21
7/28/2019 Cournot and perfect comptency in markets
4/22
3
1. Introduction
Many years ago since Adam Smith formulated the idea of how economy works; the
conduct of the markets was based on the concept of perfect competition. That was a
model which made strong assumptions for the dynamical behavior of the society. This
model looked to maximize consumer and producer surplus and at the same time the
social welfare. However, the model of perfect competency cannot apply in all the
scenarios where the number of competitors is small. In that case, a more precise modelwho reflects the real dynamics in the market [1, 2, 3] is necessary. Some of the
proposed solutions to represent those cases are Cournot, Stackelberg, etc. game theory
models.
In this document we compare the perfect competency and Cournots model. To achieve
this goal the structure of the document is divided in four sections. First, the paper
makes the definition for perfect competency and Cournots model. Second, we apply
each model to a simple power system of three competitors. Third, analysis results forthat power system case are made. And finally conclusions are presented.
7/28/2019 Cournot and perfect comptency in markets
5/22
4
2. Cournot and Perfect Market Description
Game theory and behavioral economics are elements that daily accompany the market
agents. In the given circumstances, a lot of alternatives are shown up; one of them
being the oligopolies, whose popularity in the free market economics appears to be
more common as the years pass. The oligopolies are described by a diversity of models
among which three models are prominent, Cournot, Stackelberg and Bertrand. In this
document the first one is studied. This will be described later and represented by an
example.
Simultaneously the definition, implications, characteristics and situations of the real life
will be analyzed in a perfect competition dynamic which has narrow links with the
development in productive processes aimed to be worked out in high efficiency
environments.
a. Perfect Competition
To point out this document will start with the ideal case or the case of perfect
competition where the market requires a series of conditions which are only achievable
with mass consumer products. Those conditions are:
An elevated number of agents which demand and offer that act exclusively asprice takers, this means that there is no space for speculation.
No input or output barriers. High mobility of goods and factors The information is free, public and priceless. No asymmetry. Market whose center is a homogeneous product. There are no transactions or intermediation costs.
Under these conditions, the dealers cannot impose a price with the goal of maximizing
their benefits since their portion of the market is atomistic. In this sense if a company
wishes to impose a superior price than the markets, it will be taken out of the
competition or it must adjust its price to the newly given. In the last case, all the other
competitors will keep their market prices which are inferior to the wanted price by that
company. On a different side if the company wishes to use a lower price than the
7/28/2019 Cournot and perfect comptency in markets
6/22
5
market with the idea of reducing the number of competitors, it will end losing again.
Because as its good is produced by a great number of companies the new price will not
have an impact at its competitors. As a result if the price is smaller than the market
price the costs will be superior than its benefits leading to a new adjust at the selling
price in the market.
In a second matter the fact of free information obtaining (in the consumers and the
producers sides) does not create extra cost, resulting in the knowledge of rentable
opportunities to raise their prices, reduce their prices, substitute a product, etc.
On the other hand, the fact that there are no input barriers creates an automatic
regulation in the market. As an example if in a given moment the public sees exorbitant
returns in an activity that does not require a huge amount of capital, they will tend to
invest in this activity. The substantial raise in the number of competitors will reduce the
market price, with more competitors there will be more competition making the price tobe lowered and stabilized. In this manner there cannot only be observed the result in
the freedom of barriers but also the regulation of the prices and the market efficiency.
It is said that this type of markets is efficient when the production of goods is made at
the lowest possible cost, in a space where there are only demanders that accept to pay
the cost and the production at appropriated quantities. This fact makes that both the
producer and the consumer reach their maximum happiness. This fact results in the
social welfare maximization, which will be used in the next chapter to compare the final
result of having a market with perfect competition and one with the Cournot model.
b. Cournot
In first place, the basic characteristics of an oligopoly model will be mentioned, where
unlike the perfect competition model- the corporative decisions circle around the
behavior given by the rival companies. This behavior creates markets where the
entrance barriers are significant.
As part of this oligopolistic model the Cournots model is found, it can be identified by
particular characteristics, which are:
The inversion decisions and the production of the companies that are part of themarket segment depend on what it is done or not by their competitors. As a
7/28/2019 Cournot and perfect comptency in markets
7/22
6
result, the unattended demand left by the last named will be seen as a chanceand will be immediately satisfied.
As mentioned previously, there are entrance barriers which difficult theparticipation of a higher number of participants in the offer side.
Despite that the decision of rising or reducing the production depends on thebehavior of the companies that take part of the competence, the final productiondecision is independent. (i.e. not incentivized by the government)
Market power from small number players. They decide the number of items toproduce which will affect the final selling price in the market.
The price is supported by the markets demand to the product, in the case of theelectric market, it only would have to take into account the price in the spot thateliminates every excess of offer or demand. In the case of the forward contractsprice which in general is the most, a common agreement between the twoentities is dealt.
Once its postulates are defined, it is possible to see the implications of the model. The
model establishes that in the case of having two companies (case which can be
generalized to represent an oligopoly), the selling price will not be constant and will
depend on the number of commodities which are produced, resulting in the next
demand function1:
= 300
Where Q represents the total market demand, that is denoted byQ = q + q. The 300
represents the price to the public that demands the good. Additionally, the production
costs (raw matter transformation) in this case are assumed constant and equal to 30.
According to this, the revenue that will be obtained can be represented as:
= 30
As said previously, the first term is the demand function and allows estimating the sales
(billing) of the company in a unitary scale, to this production cost is subtracted; and
finally multiplied by the quantity of the produced units by the company to get the final
profit for one of the companies.
= 300 ( + ) 30
In the same sense the profit for the second company will be:
= 300 ( + ) 30
1This is an illustrative example to show how Cournot model works. The function was taken as given.
7/28/2019 Cournot and perfect comptency in markets
8/22
7
As the objective in companies with a profit goal is to maximize the revenue2, the
derivate of the pair of expressions is taken and equaled to zero:
In the case of :
() =
300 ( + ) 30
0 =
270
=270
2
Now in the case of :
=270
2
These new equalities are named reaction functions and represent the production of an
enterprise, given the performance of its counterpart in offer terms.
The behavior derived from the previous expressions is the one of a straight line with a
negative slope, which indicates that as a company produces less quantity of its product,the other can cover the rest as shown in the next graph, Figure 1:
2That differs from the perfect competition in which companies tend to minimize the cost, since the number of
products can not affect the price. In that condition, it is said that the companies are price takers but this will be
explain in the next section Perfect Competition
7/28/2019 Cournot and perfect comptency in markets
9/22
8
Figure 1 Reaction curves in Cournot model. Small oligopoly two companies
As a final result, it is generally said that industries associated with this type of model
achieve high profit margins, result that is evident in the case of the financial sector and
oil, not so much in the field of power generation and transmission where rates are
strictly regulated for most of the population; and only allowed variations and high
volatility in spot market where trading a small portion of the total contract.
7/28/2019 Cournot and perfect comptency in markets
10/22
9
c. Comparison between Perfect Market and Cournot
Market TypePerfect
Competition Cournot Model
Structure
#Companies
Huge Small
Kind ofentry
Free Barrier
ProductType
Homogeneous Homogeneous
Company
PowerMarket
No Yes
Conduct
PriceStrategy
No Interdependency
ProductionStrategy
Independent Interdependency
Results
Benefits Normal Excessive
Efficiency Good Inefficient
OptimizationProblem
Minimize Cost Maximize Profit
Table 1 Perfect Competition and Cournot Market Comparison
7/28/2019 Cournot and perfect comptency in markets
11/22
10
3. Electricity Market Study
a. Model
In this section we consider three different thermal power plant generators and
we set an electricity market respect to their models which we define. The objective of
producers in this electricity market is maximizing their profits. Essentially, this is the
motivation which alters their behaviors or incentives to be able to exist in the market.
As we know the profit function is the surplus remaining after total costs are deducted
from total revenue. Basically, profit shows the benefit of the company after its all costs
and loses.
Revenue:
Revenue is the total amount of money received by the company for goods sold orservices provided during a certain time period. In this model total revenue is electricity
price times the amount of electricity sold.
We denoted revenues of generators by Ri;
33
22
11
QPR
QPR
QPR
e
e
e
=
=
=
where eP is the electricity price.
Electricity Price:
Electricity price is obtained by the inverse demand curve of electricity market showed inFigure 2.
)( 321 QQQPe ++=
where1Q , 2Q , 3Q are outputs of three thermal power plant generators.
Costvenueofit = RePr
iei QPvenue =Re
7/28/2019 Cournot and perfect comptency in markets
12/22
11
Figure 2: Demand curve of electricity market
Cost:
In general, quadratic model has been accepted to represent cost function in electricity
market [4], which describes the fuel cost as a quadratic function of the generation.
We have the fuel-cost functions of three generators of thermal power plants [5], whichare denoted by Ci;
111
2
111cQbQaC ++=
222
2
222cQbQaC ++=
333
2
333cQbQaC ++=
Marginal Cost:
Marginal cost is the increase or decrease in the total cost of a production run for
making one additional unit of an item. In electricity market that means it is the
additional cost which responds to one more MW electricity generation. We draw the
marginal cost of each generator which is showed in Figure 3.
7/28/2019 Cournot and perfect comptency in markets
13/22
12
Figure 3: Marginal cost for three different generators
Supply Curve & Demand Curve:
According to our equations we set a supply and demand curve of this electricity marketwhich is showed in Figure 4. Demand curve shows the inverse relation betweenelectricity price and total output of generators. That means when the price goes up thetotal quantity goes down and vice versa.
Supply curve has the same direction with quantities. We draw supply curve by addingup all marginal cost of generators horizontally (Appendix). It has been seen from Figure4 supply curve is ascending with price. There are three different slopes on the curve
because we have three different marginal cost functions.Equilibrium:
The profits are maximized at the equilibrium where the marginal cost of each generatorgroup equals to its marginal revenue. In Figure 4 we have equilibrium at the point inwhich the total demand is 10333 MW which is also equal to total supply and theelectricity price is 44.61$.
7/28/2019 Cournot and perfect comptency in markets
14/22
13
}3,2,1{=
=
i
MRMCii
;
Figure 4: Market Equilibrium
Consumer Surplus:
Consumer surplus is a measure of the economic welfare enjoyed by consumers.Basically, it shows the amount of consumers who are happy. Consumer surplus is the
area between equilibrium price and demand curve.
Producer Surplus:
Producer Surplus is a measure of the economic welfare enjoyed by firms or producers.Basically, it shows the amount of the happiness of producers. Producer surplus is thearea between equilibrium price and supply curve.
7/28/2019 Cournot and perfect comptency in markets
15/22
14
Social Welfare:
In our study we define social welfare as the amount of happiness of consumers andproducers. We show it by summation of consumer surplus and producer surplus. Social
welfare is the area between demand curve and supply curve.
SurplusProducer+SurplusConsumer=WelfareSocial
b. Perfect Competition Case
In the perfect competition marginal cost is equal to price which is also equal to
marginal revenue.
{1,2,3}
i i eMC MR P
i
= =
=
;
)( 321 QQQPe ++=
We get three equations with three unknowns;
)(2 321111 QQQbQa ++=+
)(2 321222 QQQbQa ++=+
)(2 321333 QQQbQa ++=+
After solving this problem we get
1 4913.241 MWQ =
2 3258.827 MWQ =
3 2155.885 MWQ =
The total amount of outputs is MW10328321 =++ QQQ .
7/28/2019 Cournot and perfect comptency in markets
16/22
15
Figure 4 represents the perfect equilibrium. It has been seen the total of the optimum
outputs of generators, which are calculated above, are same with the total outputs in
Figure 4. The electricity price, which is corresponding to our total output, is $44.61.
According to Figure 3 we can get consumer surplus, producer surplus and also social
welfare in the case of perfect competition.
Consumer Surplus 389677
Producer Surplus 203061
Social welfare 592738
c. Cournot Model Case
In oligopolistic Cournot model each firms behave respect to other competitors
behaviors. Each power plant chooses its optimal outputs to maximize its profit. Cost
functions of generators are defined which depend on fuel cost. In case of any changes
in their total cost according to fuel price they must keep their profits maximized. In case
of any change of the equilibrium, in which each marginal cost is equal to marginal
revenue, generators will keep this equality constant.
=
=
=
33
22
11
Remax
Remax
Remax
3
2
1
Costvenueprofit
Costvenueprofit
Costvenueprofit
Q
Q
Q
This is a multi-objective optimization problem; generators maximize their profits
simultaneously.
We solve this optimization problem by using first order condition. After taking
derivatives of these three equations we get three equations with three unknowns.
}3,2,1{=
=
i
MRMC ii
7/28/2019 Cournot and perfect comptency in markets
17/22
16
)(2 3211111 QQQQbQa +++=+
)(2 3212222 QQQQbQa +++=+
)(2 3213333 QQQQbQa +++=+
After solving this problem we get optimum values.
1Q =3503.477 MW
2Q =2767.005 MW
3Q =2098.940 MW
The total amount of outputs is 1 2 3 8370 MWQ Q Q+ + =
The electricity price, which is corresponding to our total output, is 58.9$ showed inFigure 5.
According to Cournot equilibrium showed in Figure 5 we can get consumer surplus,
producer surplus and also social welfare.
Consumer Surplus 255703
Producer Surplus 315215
Social welfare 570918
Figure 5: Consumer surplus, producer surplus and social welfare in Cournots
model
7/28/2019 Cournot and perfect comptency in markets
18/22
17
4. Result Analysis
It has been seen from perfect competition and Cournot case analysis; the total
output of perfect competition case is larger than the total output of Cournot case
showed in Figure 6. As a result of these optimum outputs different market equilibriumpoints is obtained for the perfect case and the Cournot case that are shown in Figure 4
and Figure 5 respectively. According to these equilibrium points, consumer surplus,
producer surplus and social welfare results are demonstrated in Figure 7. Consequently,
consumer surplus and total social welfare in perfect competition case are bigger than
that in the Cournot model case. But, producer surplus in perfect competition case is
smaller than that in Cournot model case.
Figure 6: Comparison between Cournot and Perfect Competition. Output
power at each generator
3503.477
2767.005
2098.94
4913.241
3258.827
2155.885
0
1000
2000
3000
4000
5000
6000
Output of Generator
1 (Q1)
Output of Generator
2 (Q2)
Output of Generator
3 (Q3)
Comparison of generator outputs between
Cournot & Perfect
Cournot
Perfect
7/28/2019 Cournot and perfect comptency in markets
19/22
18
Figure 7: Comparison between Cournot and Perfect Competition. Consumer
surplus, producer surplus and social welfare
In order to understand why this is so, it is necessary to understand what perfect
competition is. In economic theory, perfect competition describes markets such that
firms do not have that much power to be able to change or set the price of the product.
Because of the condition for perfect competition firms have to keep their marginal costs
equal to market price that has been already fixed. Participants control only their costs
to reach maximum profit. Basically perfect competition is an approximation whichserves as a benchmark against which to measure real-life and imperfectly competitive
markets.
On the contrary, in the case of the Cournot model participants are able to set the
price of homogeneous product by changing their outputs. Firms compete on the
amount of output they will produce, which they decide at the same time. Producers
obtain profit in Cournot case more than that in perfect competition case. Therefore it is
obvious Cournot model case is provide with more benefits to producers, but it reduces
the number of happy consumers in the electricity market. On the other hand, the
increasing amount of producer surplus is less than the decreasing amount of consumer
surplus. Because of this, social welfare of Cournot model case is less than that in
perfect competition case. This difference is defined as social welfare loss in market. It
is shown by red color in Figure 5.
255703
315215
570918
389389
203061
592738
0
100000
200000
300000
400000
500000
600000
700000
Consumer Surplus Producer Surplus Social Welfare
Comparison of CS,PS and SW between
Cournot & Perfect
Cournet
Perfect
7/28/2019 Cournot and perfect comptency in markets
20/22
19
5. ConclusionsAfter applying Cournot model and perfect competition model on electricity market in
which there are three power plant generators, we compared the results of these two
cases and we obtain:
1.The total output of three generators in Cournot model case is less thanthat in the perfect competition case. Therefore electricity price in Cournot
model is higher than the price in perfect competition case.
2.The consumer surplus in Cournot model case is less than in perfectcompetition case.
3.The producer surplus in Cournot model case is larger than that in perfectcompetition case.
4.The amount of increase of producer surplus is less than the amount ofdecrease of consumer surplus in Cournot model case. Thus social welfare
in Cournot model case is smaller than that in perfect competition case.5. Perfectly competitive market is a benchmark which helps to understand
the behaviors and responds in Cournot model.
7/28/2019 Cournot and perfect comptency in markets
21/22
20
6. References
[1] Ryan, S.M.; Downward, A.; Philpott, A.B.; Zakeri, G.; , "Welfare Effects of Expansions in
Equilibrium Models of an Electricity Market With Fuel Network," Power Systems, IEEE
Transactions on , vol.25, no.3, pp.1337-1349, Aug. 2010
[2] Cunningham, L.B.; Baldick, R.; Baughman, M.L.; , "An empirical study of applied game
theory: transmission constrained Cournot behavior," Power Systems, IEEE Transactions on ,
vol.17, no.1, pp.166-172, Feb 2002
[3] Valenzuela, J.; Mazumdar, M.; , "Cournot Prices Considering Generator Availability and
Demand Uncertainty," Power Systems, IEEE Transactions on , vol.22, no.1, pp.116-125, Feb.
2007
[4] Yoshikawa, M.; Toshida, N.; Nakajima, H.; Harada, Y.; Tsurugai, M.; Nakata, Y.; , "On-line
economic load dispatch based on fuel cost dynamics," Power Systems, IEEE Transactions on ,vol.12, no.1, pp.315-320, Feb 1997
[5] Hadi Saadat. Power System Analysis. Shaniwar Peth, India, 2009
7/28/2019 Cournot and perfect comptency in markets
22/22
21
7. AppendixMATLAB CODE
a1=input('enter first slope offset = ');a2=input('enter second slope offset = ');
a3=input('enter third slope offset = ');b1=input('enter first constant offset = ');b2=input('enter second constant offset = ');b3=input('enter third constant offset = ');q1=input('enter first gen. max offset = ');q2=input('enter second gen. max offset = ');q3=input('enter third gen. max offset = ');Q1=[0:q1];Q2=[0:q2];Q3=[0:q3];MC1=a1*Q1+b1;MC2=a2*Q2+b2;
MC3=a3*Q3+b3;f_t=[MC1 MC2 MC3];sft=sort(f_t);figure;plot(sft)hold onxlabel('Total output(MW)')ylabel('Price ($/MWh)')title('Supply Curve & Inverse Demand Curve')q=120;p=0.0073;
Pe=q-p*(Q1+Q2+Q3);plot((Q1+Q2+Q3),Pe)figureplot(Q1,MC1)xlabel('Output(MW)')ylabel('Price ($/MWh)')title('Marginal Cost Function of First Generator')axis([0 2000 0 40])figureplot(Q2,MC2)xlabel('Output(MW)')ylabel('Price ($/MWh)')
title('Marginal Cost Function of Second Generator')axis([0 2000 0 40])figureplot(Q3,MC3)xlabel('Output(MW)')ylabel('Price ($/MWh)')title('Marginal Cost Function of Third Generator')