firm theory and duality

8/10/2019 Firm Theory and Duality

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EC115 - Methods of Economic AnalysisSpring Term, Lecture 9Constrained Maximisation III:

Firm’s Theory and Duality

Renshaw - Chapter 16

University of Essex - Department of Economics

Week 24

Domenico Tabasso (Uni versity of Essex - Depart ment of Economics)

Lecture 9 - Spring Term Week 24 1 / 39

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Introduction

In the last lectures we have focused on how to maximize autility function under a budget constraint.We have outlined two methods:

Total Differentiation (⇒ MRS and Budget Constraint)

Direct Substitution.The next (and nal!) question that we want to answer inthis course are:

Is there any other method for nding a max under aconstraint?How about the theory of rms?.



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Introduction - 2

With respect to the rst question, the answer is:Yes!, There is third method and it is called the Lagrangemethod.We will study this technique next week as last topic of thecourse.

Today we will focus on the second question: how do rmsbehave under a constraint?.



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Firms Behaviour

Differently from consumer theory, in which the aim isalways the maximization of the utility under the budgetconstraint, for the rms the aim can be different. Inparticular we will distinguish between:

1 Maximization of the prots or of the output?

Domenico Tabasso (Uni versity of Essex - Depart ment of Economics)Lecture 9 - Spring Term Week 24 4 / 39

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Firms Behaviour

Differently from consumer theory, in which the aim isalways the maximization of the utility under the budgetconstraint, for the rms the aim can be different. Inparticular we will distinguish between:

1 Maximization of the prots or of the output?2 Maximization or minimization of the costs?


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Maximization of the Prots

In the most standard case, the aim of the rm is the

maximization of the prots. This case is rather easy tohandle.the maximization of the prots, in fact, implies themaximization of the Total Revenues minus the Total Costs,under the constraint of a certain production level.But this problem can be simply written as:

maxK ,L

Π(K , L) = p f (K , L) − wL− rK

where p is the price of the good sold by the rm, f (K , L) isthe production function, w is the wage paid to the workers

and r the cost of capital.Domenico Tabasso (Uni versity of Essex - Depart ment of Economics)Lecture 9 - Spring Term Week 24 5 / 39

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Maximization of the Prots

Evidently the prot function already contains the constraintin itself, so the maximization is straightforward and followsthe usual steps:

1 Obtain the rst order conditions;2 Solve the FOCs and obtain the critical values of K and

L, respectively K ∗ and L∗;3

Check the critical values satisfy the second orderconditions for a max.


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Maximization of the Output

Sometimes a rm can aim at maximizing the output andnot the prots (think about a rm which is owned by thestate).In this case the problem becomes:

maxK ,L

Q = f (K , L)

s.t. TC = rK + wL.

The objective function is the production function;The constraint is given by the total cost function (TC).


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This maximization problem can be solved following the twmethods studied with respect to the consumer theory. Inparticular, by using the total differential approach we aimat building up the following system:

Slope of the Isoquant (≡ MRTS ) = Slope of the IsocostEquation of the total Cost Function

where the similarities with the consumer’s theory areevident.


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So given the following problem:maxK ,L

Q = f (K , L) ;

s.t. TC = wL + rK .

where TC is just a number, by xing the output at a certainconstant level Q 0 and using total differentiation we have

dQ 0 = ∂ Q ∂ K dK + ∂ Q ∂ L dL.


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Maximization of the OutputSince dQ 0 = 0 the last equation can be rewritten as

∂ Q ∂ K

dK + ∂ Q ∂ L

dL = 0

which we can rearrange to obtain:

dK dL

= ∂ Q /∂ L∂ Q /∂ K

where dK dL is dened as the Marginal Rate of Technical

Substitution (or Marginal Rate of Substitution inproduction) which tells us the amount of capital that canbe released when the one more unit of labour is employed

and the output is constant.Domenico Tabasso (Uni versity of Essex - Depart ment of Economics)Lecture 9 - Spring Term Week 24 10 / 39

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So our system is given by:

dK

dL≡

∂ Q /∂ L

∂ Q /∂ K =

w

r

TC=wL+rK

and by solving the system we can nd K ∗

, L∗

.


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Maximization of the Output - Example

A rm’s production function is given by Q = K 1/ 3L1/ 3.Find K ∗, L∗, Q ∗ given TC = 2000 and w = r = 1.Solution:

MRTS ? =⇒ ∂ Q /∂ L∂ Q /∂ K

So:

MRTS =

13 K 1/ 3L− 2/ 3

13 K − 2/ 3L1/ 3 =

K L


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Maximization of the Output - Example

Setting the system we have:

K L

= w r

2000=wL+rK

which we can solve easily and obtainK ∗ = 1000, L∗ = 1000 which imply Q ∗ = 100.


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Cost Minimization

In the theory of rm we may also encounter a differentproblem: A rm wants to reach a given output trying tominimize the costs.Formally, the problem can be rewritten as:

minK ,L

TC = wL + rK ;

s.t.Q = f (K , L)

and can be interpreted as the complementary problem tothose studied so far. In particular now TC is unknown,while Q = Q 0 is given and equal to a constant.


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Cost MinimizationHow can we nd the optimal K ∗, L∗ that satisfy ourminimization problem?

Y

TC1

TC*

OptimumTC0

y*

X

x*


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Cost Minimization

Well, rst of all we can notice that at the optimum twocondition are satised:

Slope of the Isoquant (≡ MRTS ) = Slope of the Isocost (Again!)Equation of the Production Function (the constraint!)

So the system looks very similar to the one of themaximization problem. Is the solution going to be similaras well?


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Cost Minimization - Example

In order to understand our problem let’s start with anexample.A rm wants to nd the cost minimizing combination of capital and labour for producing Q = 100 given that

w = r = 1. The rm has the following productionfunction:Q = K 1/ 3L1/ 3.

The slope of the Total Cost function is simply given byw r = 1 while the MRTS can be obtained through totaldifferentiation of the Production Constraint and is againequal to K

L


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Cost Minimization - Example

So the system is K

L = 1100 = K 1/ 3L1/ 3

From the rst equation we obtain K = L which substitutedin the second equation implies

100 = K 2/ 3 =⇒ K = 1003/ 2 =⇒ K ∗ = 1000

and given that K = L =⇒ L∗ = 1000


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Summary of results

Problem:

maxK ,L

Q = K 1/ 3L1/ 3

s .t . 2000 = K + L

Solution:K ∗ = 1000, L∗ = 1000 and Q ∗ = 100


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Summary of results

Problem:

maxK ,L

Q = K 1/ 3L1/ 3

s .t . 2000 = K + L

Solution:K ∗ = 1000, L∗ = 1000 and Q ∗ = 100Problem:

minK ,L

TC = K + L

s .t . 100 = K 1/ 3L1/ 3

Solution:K ∗ = 1000, L∗ = 1000 and TC ∗ = 2000


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Duality

So the two problems lead to the same solution. Is this just a coincidence???


D lit

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Duality


Of course no!


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Duality


Of course no!Can we obtain something similar with respect toconsumer’s theory?


D alit

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Duality

Consider the following standard problem for a consumer:

maxx ,y

U (x , y ) = x α y β

s .t .m = p x x + p y y

We have studied this problem several times and we knowthat in order to solve it we have to nd the solution for thefollowing system:

MRS= − p x

P y

m = p x x + p y y


Duality

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Duality

Y

OptimumU1

y*

U*

X0 U0

*


Duality

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Duality

We know that in this case MRS = −αy β x and last week we

saw that the solution is given by:

x ∗ = αm

(α + β )p x

y ∗ = β m

(α + β )p y

Note that x ∗

and y ∗

are functions themselves! They arefunctions of m, p x , p y .


Duality

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Duality

So for any different combination of m, p x , p y we’ll observedifferent values of x and y that satisfy the maximizationproblem.We dene:

x = x (m, p x , p y ) y = y (m, p x , p y )

as the Marshallian demand functions of x and y , or simplythe demand functions .


Duality

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Duality

Remember that we started from a Cobb-Douglas utilityfunction. More generally we could have started from:

U = U (x , y )

But now we know that x is going to be x = x (m, p x , p y )and y is going to be y = y (m, p x , p y ), so that our utilityfunction can be expressed as:

U = U (x (m, p x , p y ), y (m, p x , p y )) = V (m, p x , p y )We call V = V (m, p x , p y ) the Indirect Utility Function


Duality

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DualitySo in our previous example:

U (x , y ) = x α y

β

and we obtained:

x = αm

(α + β )p x

y = β m

(α + β )p y

So the indirect utility function is:

V = V (m, p x , p y ) = αm

(α + β )p x

α β m(α + β )p y

β


Applying Duality to Consumer Theory

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Applying Duality to Consumer Theory

Consider now the following problem for a consumer:

minx ,y

p x x + p y y

s .t . U 0 = f (x , y ) = x α y β

What is the minimum expenditure the consumer mustmake to achieve a determine level of utility?

That is, given a utility level u 0, what is the choice of x and y such that the consumer reaches this level of utility at the minimum cost?


Duality

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Duality

Y

TC1

TC*

OptimumTC0

y*

X

x*


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Using the tangency condition method we have to solve tofollowing system

MRS= − p x

P y

x α y β = U 0

where evidently the rst equation is identical to the one wehave with the maximization of the utility problem, whiclethe constraint is now represented by the utility level.


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The MRS is still given by MRS = − α y β x so we have:

−α y β x

= −p x

p y ⇒ y =

β α

p x

p y x

U 0 = x α y

β ⇒ U 0 = x

α+ β β α

p x

p y

β

x ∗ = U 1

α + β0

αβ

p y

p x

βα + β

; y ∗ = U 1

α + β0

β α

p x

p y

αα + β


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Substituting hx and hy in our original objective function weobtain:

E = p x hx (p x , p y , U ) + p y hy (p x , p y , U )

The function E = E (p x , p y , U ) is called the Expenditure

Function.By construction:

hx (p x , p y , U ) = ∂ E ∂ p x ; hy (p x , p y , U ) =

∂ E ∂ p y


Duality

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y

In equilibrium: E (p x , p y , U ) = m (1)

This result leads to an important implication known asduality:

1 Solving eq.(1) for U we obtain an expression for theindirect utility function

2 Solving V (p x , p y , E ) = U for E gives us theexpenditure function.


Duality - Exercise

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y

Consider the following consumer problem:max

x ,y U (x , y ) = x

13 y

13

s .t .m = p x x + p y y

1 Find the indirect utility function;2 Find the expenditure function;3 Find the Hicksian demands for x and y


Solution

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In order to nd the indirect utility function we rst have toget the (Marshallian) demands for x and y .So as usual we have to solve:

MRS= − p x

P y

m = p x x + p y y

In this case: y

x =− p x

P y

m = p x x + p y y


Solution

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The solution of the system is (prove it yourself!):

x ∗ = m2p x

; y ∗ = m2p y

So, we have just obtained x and y as functions of (m, p x , p y ). Substituting these in the utility function weget the indirect utility.


Solution

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V = V (m, p x , p y ) = m2p x

1/ 3 m2p y

1/ 3= m2 / 3

(2p x )1 / 3 (2p y )1 / 3 .

How about the Expenditure function? Well, we have two

ways to get it:1 Minimize the expenditure under the utility constraint;


Solution

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V = V (m, p x , p y ) = m2p x

1/ 3 m2p y

1/ 3= m2 / 3

(2p x )1 / 3 (2p y )1 / 3 .

How about the Expenditure function? Well, we have two

ways to get it:1 Minimize the expenditure under the utility constraint;

2 Use Duality!


Solution

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Duality implies m = E so we can substitute E instead of mwithin the indirect utility function and solveU = V (E , p x , p y ):

U = E 2 / 3

(2p x )1 / 3 (2p y )1 / 3 .

Solving this for E we get the expenditure function:

E = 2(p x p y )12 U

32 .


Solution

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Finally, given the obtained expression for the ExpenditureFunction we can get the Hicksian demand functions withtwo simple derivatives:

hx = ∂ E ∂ p x = ( p x p y )−

12 U

32 ;

hy = ∂ E ∂ p y

= ( p x p y )−12 U

32 ;


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firm theory and duality

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