Optimization With More Than One Variable II

8/10/2019 Optimization With More Than One Variable II

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EC115 - Methods of Economic AnalysisSpring Term, Lecture 5Optimization with more than one variable:

Economic applications

Renshaw - Chapter 15

University of Essex - Department of Economics

Week 20

Domenico Tabasso (University of Essex - Department of Economics)

Lecture 5 - Spring Term Week 20 1 / 31
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This weeks topics

Introduction: Review of a Single-Product CompetitiveFirm;

Case of a Multi-Product Competitive Firm;

Duopoly: The Cournot Model;

Duopoly: The Stackelberg Model.


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Introduction: Review of a Single-Product Competitive Firm

The concepts ofmaximum,minimumandsaddlepoints

of a function have several applications in economics.For example, consider the case of a group ofshareholdersof a firm that would like to maximize theprofits of their firm:

=TR TC

Assume this firm produces a single product and

operates in acompetitive market. This implies that ittakes the price of the good it produces, P, as given.Hence the only instrument they can use to achieve theirgoal is output, Q.


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Translation into Maths

The problem of the shareholders can be described by

maxQ

(Q), where(Q) =TR(Q) TC(Q).

where TR(Q) =P Q is the total revenue of the firm

and TC(Q)is the total cost of the firm (both asfunctions of output).

What is the level ofQthat maximizes profits?

When an additional unit of output brings in morerevenue than it costs to produce then the firm willincrease production. At this point total profits areincreasing!

Domenico Tabasso (University of Essex - Department of Economics)Lecture 5 - Spring Term Week 20 4 / 31
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The firm will stop increasing production when the extrarevenue from producing one more unit of output equalsits cost of producing that extra unit.

Suppose we let Q

denote the level of output at whichthe firm reaches this equality.

Why not produce more? If the firm did so, total profitswould start decreasing!

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The first order condition

This logic is summarized in the profit maximizing

condition:MR(Q) =MC(Q)

Note that since:

MR(Q) = dTR(Q)dQ

and MC(Q) = dTC(Q)dQ

,

the profit maximizing condition can be re-written as:

d(Q)dQ

= dTR(Q

)dQ

dTC(Q

)dQ

=0.

This expression is just the first order condition formaximizing(Q) with respect to Q.

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The second order condition

To be sure that Q is effectively a maximum and not a

minimum we have to verify that the profit function isconcaveat Q:

d2(Q)

dQ2


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Multi-product competitive firm

Now suppose that the firm is able to produce two

different goods, Q1 and Q2, each of which is sold in acompetitive market at per unit prices ofP1 and P2respectively. Again, the shareholders seek to maximize

profits.The profit function is given by:

(Q1, Q2) =TR1(Q1) + TR2(Q2) TC(Q1, Q2).

Now assume that the total cost is given by:

TC(Q) =2Q21 + 2Q1Q2 + Q22 .

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Hence the profits of this firm can be written as:

(Q1, Q2) =P1Q1 + P2Q2 2Q21 2Q1Q2 Q

22

where P1 and P2 are positive constants.

What the shareholders would like to know are the valuesofQ1 andQ2 the firm has to produce to maximizeprofits maxQ1,Q2 (Q1, Q2). Thus they need to solve:

maxQ1,Q2 (Q1, Q2) = (P1Q1 + P2Q2 2Q21 2Q1Q2Q

22 )

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The second order conditions are that at(Q1=Q

1 , Q2=Q

2 ):

2(Q1, Q2)

Q21


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Lets solve it

For simplicity assume that P1=10 and P2=8 such

that

(Q1, Q2) =10Q1 + 8Q2 2Q21 2Q1Q2 Q

22

What are the values ofQ1

and Q2

which make(Q1, Q2)as big as possible?

We start by solving the following first order conditions:

(Q1, Q

2)

Q1=0 10 4Q1 2Q2 =0

(Q1, Q2)

Q2=0 8 2Q2 2Q

1 =0

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This is a system of two linear equations!

The solution is:

Q1 =1 and Q

2 =3.

To verify this:

10 4Q1 2Q2 = 10 4 6=0

8 2Q2 2Q

1 = 8 2 6=0.

These are thecandidatepoints for a maximum.

The next step is to obtain the second partial and crosspartial derivatives and use them to check the secondorder conditions.

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The second order conditions

Here:

2(Q1, Q2)

Q21= 4


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We can conclude that the shareholders want the firm toproduceoneunit ofgood 1andthreeunits ofgood 2at the going prices.

The maximum profit the firm can make is then:

(Q1 , Q

2 ) =10 1 + 8 3 2 12 2 1 3 32

=10 + 24 2 6 9=34 17=17.

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Duopoly: An Introduction

We define as aduopolya situation in which only two

firms are present in a certain market and sell the samegood;

The firms have two options: they can competeor they

cancollude

;We will only focus on the competitive case.In this respect we will focus on two different models:

1 Duopoly la Cournot;2 Duopoly la Stackelberg.

NOTE:These models are not explained in the book from

Renshaw. In the CMR you will find some extra material on

duopoly. This material is 100% part of the course!

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Duopoly: The Cournot Model

In the Cournot model two identicalfirms (A and B)compete choosing the quantity so that they can bothmaximize their profits.

It is important to note that A and B choose their quantitiessimultaneously. So basically firm A observes what firm Bproduces and chosees its optimal answer (in terms ofquantity).

At the same timefirm B does the same: observes thequantity chosen by A and optimally reacts to this quantity.


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The Cournot Model - The reaction curve, 1

Imagine the market demand is given by Q=f(p).Imagine that for some reasons A decides to produce

qA=100. Hence B now faces a residualdemand

Q=f(p) 100.

But now B can act as a monopolist and choose thequantity qBsuch such that MR=MC.

Of course this quantity depends on qA: Had A chosenqA=200, the residual demand for B would have beendifferent and so the optimal quantity qB


Th C M d l Th i 2
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P

MR1MR2MR3

MC

q1q1q1q1

q2

q2

q1q1 q1 q1

q2=0


Th C M d l Th i 3
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So for each quantity chosen by A, theres a bestresponseqBwhich depends on (it is a function of) qA.

The same is true for A, which observes the behaviour offirm B and chooses its optimal qA as a reaction to Bs

choice.We can then graph two differentreaction curves: Onefor the reactions of A with respect to Bs choices and

one for Bs reactions to As choices.The equilibrium is found when both firms choose theiroptimal quantity as a reaction to the other firm choice.


Th C M d l Th i 4
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qB

Reaction

Function of

Firm A

qu r um

*BReaction

Function of

Firm B

qAq*A


Th C t M d l T l ti i t M th 1
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The Cournot Model: Translation into Math, 1

Suppose the market demand for the good is: Q=100 p,

so the inverse demand will bep

=100 Q

, whereQ=qA + qB.Each firm has identical production costs:

TCi(qi) =c qi, i=A, B

If we focus on firm A, we know it can only provide themarket with the quantity not already provided by B, so theprofit function for A will be:

A=qA p c qA

A=qA (100 qA qB) c qA


Th C t M d l T l ti i t M th 2
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The same situation holds for B, so the two firms have tosimultaneously solve the following problems:

maxqA

A=qA (100 qA qB) c qA

maxqB

B=qB (100 qA qB) c qB

where firm ican only maximize with respect to quantity i,

taking the quantity produced by the rival firm as given (asa constant).


The Cournot Model: Translation into Math 3
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The first order conditions are:

For firm A:AqA

=0 100 2qA qB c=0

For firm B:

BqB

=0 100 2qB qA c=0

which imply:

q

A=100 q

B

c

2 (1)

and

qB=100 qA c

2

(2)


The Cournot Model: Translation into Math 4
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Equations (1) and (2) above are exactly the equations that

define the reaction curves: We simultaneously observeqA=F(qB) and qB=F(qA).

Solving (1) and (2) simultaneously means solving a system

of 2 linear equations in 2 unknowns, qA andqB.

The solutions are:

q

A=q

B=

100 c

3 (3)

A=

B=(100 c)2

9 (4)


The Cournot Model: Some additional notes
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The Cournot Model: Some additional notes

We just found qA= qB and A= B. This is not by chance!Since the two firms are identical, they face the same demand and

they have the same cost function, in equilibrium they MUSTproduce the same quantities and have the same profits.

Of course this result would not be true if we had assumed somekind of asymmetries between the two firms (different cost

functions, production boundaries, different demands and so on).Note that the profits for the two firms would be higher if theycould collude, i.e. act as a monopolist and equally share theresulting monopolist profit.

All the results hold on the base of simultaneous competition onthe quantity. Competition on price would have led us toward aduopoly la Bertrand (that we dont study), while by relaxingthe assumption of simultaneity we end up in a framework laStackelberg.


Duopoly: The Stackelberg Model
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Duopoly: The Stackelberg Model

As already said, in the Stackelberg model the firms do notchoose their quantity simultaneously butsequentiallyFirst one firm (say A) chooses its quantity, then the other(say B) tries to maximize its profits taking intoconsideration the residual demand. The fact that firm A

can choose its quantity first gives to A the so-called

first-mover advantage. This means that A will be able tomake more profits than B.


The Stackelberg Model: Translation into Math 1
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The Stackelberg Model: Translation into Math, 1

Timing:

A market demand is observed;

1 Firm A chooses quantity qA so as to maximize itsprofits;

2 Firm B observes the residual demand and chooses qB soas to maximize its profits, taking qA as given.

The standard way to solve a sequential game is bybackward induction, i.e. we start from period 2 and goback in the time line.


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Assume the same inverse demand function as before:

p=100

Q.In the last period firm B problem is:

maxqB

B=qB p c qB (5)

maxqB

B=qB (100 qA qB) c qB (6)

where qA is the quantity chosen by A in the previousperiod. FOC:

BqB

=0 100 qA 2qB c=0 (7)


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Solving eq. 7we obtain:

qsB=100 q

A c

2

But note that when firm A chooses its quantity alreadyknows that B will try to maximize its profits, so A already

knows that qs

Bwill be the outcome of the max. process wejust outlined. Hence we can now show that in period 1,firm A problem is:

maxqA

A= qA p c qA

= qA (100 qA qsB) c qA

= qA

100 qA

100 qA c

2

c qA


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maxqA

A=qA 100 qA + c2

c qAFOC:

AqA

=0 50 + qA c

2=0

which implies that the optimal quantity for A is now:

qsA=50 c

2

which can be substituted into the previous expression forqsB in order to obtain:

qsb=25 c

4Domenico Tabasso (University of Essex - Department of Economics)Lecture 5 - Spring Term Week 20 31 / 31

The Stackelberg Model: Conclusions
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The Stackelberg Model: Conclusions

Note that, as long as cqsB

and thatsA >

sB

(check this last result holds). These outcomes are due to

the first mover advantage: as A moves first it is able toexploit this advantage in terms of quantity produced andhence profits.

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Optimization With More Than One Variable II