Cournot and perfect comptency in markets

7/28/2019 Cournot and perfect comptency in markets

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THE UNIVERSITY OF OKLAHOMA

FINAL REPORT

ECE 5973 - POWER SYSTEMS AND MARKET OPERATIONS

DR. JOHN JIANG


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Cournot&PerfectCompetition

applicationandcomparisonin

ElectricityMarket

Osman Bayindir

Diego Rodrguez


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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


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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.


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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


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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


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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.


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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


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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.


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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


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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;

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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


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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.


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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$.


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}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.


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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 .


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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.

=

=

=

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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


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)(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


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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


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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


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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.


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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


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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')

Cournot and perfect comptency in markets

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