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.
Domenico Tabasso (University of Essex - Department of Economics)
Lecture 5 - Spring Term Week 20 2 / 31
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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.
Domenico Tabasso (University of Essex - Department of Economics)
Lecture 5 - Spring Term Week 20 3 / 31
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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!
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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 .
Domenico Tabasso (University of Essex - Department of Economics)Lecture 5 - Spring Term Week 20 8 / 31
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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 )
Domenico Tabasso (University of Essex - Department of Economics)Lecture 5 - Spring Term Week 20 9 / 31
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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
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The Cournot Model - The reaction curve, 2
P
MR1MR2MR3
MC
q1q1q1q1
q2
q2
q1q1 q1 q1
q2=0
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The Cournot Model - The reaction curve, 3
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.
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The Cournot Model - The reaction curve, 4
qB
Reaction
Function of
Firm A
qu r um
*BReaction
Function of
Firm B
qAq*A
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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
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Th C t M d l T l ti i t M th 2
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The Cournot Model: Translation into Math, 2
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).
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The Cournot Model: Translation into Math 3
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The Cournot Model: Translation into Math, 3
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)
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The Cournot Model: Translation into Math 4
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The Cournot Model: Translation into Math, 4
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)
Domenico Tabasso (University of Essex - Department of Economics)Lecture 5 - Spring Term Week 20 25 / 31
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.
Domenico Tabasso (University of Essex - Department of Economics)Lecture 5 - Spring Term Week 20 26 / 31
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.
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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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The Stackelberg Model: Translation into Math 2
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The Stackelberg Model: Translation into Math, 2
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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The Stackelberg Model: Translation into Math 3
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The Stackelberg Model: Translation into Math, 3
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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The Stackelberg Model: Translation into Math 4
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The Stackelberg Model: Translation into Math, 4
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
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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.
Domenico Tabasso (University of Essex - Department of Economics)Lecture 5 - Spring Term Week 20 32 / 31
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