module 6: theory of constrained...

John Riley Summer 2013

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Module 6:

Theory of Constrained maximization

A. Economics of Constrained Maximization 2

B. Preferences and Demand 8

C. Application: Market demand with identical homothetic preferences 10

D. Answers to some of the exercises 14


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A. Economics of Constrained Maximization

Consider the following maximization problem with n variables and a single constraint (in

addition to the non-negativity constraints).

{ ( ) | , ( ) }n

xMax f x x g x b . (A.1)

The standard way of solving this problem is to write down the Lagrangian of this optimization

problem

( , ) ( ) ( ( ))L x f x b g x

and then set the partial derivatives of the Lagrangian equal to zero.

( , ) ( ) ( ) 0j j j

L f gx x x

x x x

, 1,...,j n .

In problem solving it is often helpful to rewrite these n equations as follows:

1

1

... n

n

ff

xx

g g

x x

.

For example in the basic consumer budget constrained maximization problem:

{ ( ) | , }n

xMax U x x p x I

the necessary conditions are

1 1

1 1...

n n

U U

p x p x

.

Note that 1

jp is the number of units that the consumer can purchases with an additional dollar

and j

U

x

is the marginal utility of an additional unit. Thus

1

jp j

U

x

is the marginal utility of


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spending an additional dollar on commodity j. If * 0x then the marginal utility per dollar

must be the same for all commodities.

Exercise A.1

1

( ) ln , 0n

j j

j

U x x

(a) Show that the necessary conditions can be written as follows:

1

1 1

... n

n np x p x

jx (b) Solve for each as a function of , then substitute back into the budget constraint to show

*

1

( )j

j n

jj

j

Ix

p

that

Corner solutions

Actually the “Lagrange Method” is only correct if no non-negativity constraint is binding. If for

some j * 0jx , the solution is often called a “corner solution” and the Lagrange method needs to

be modified.

The goal here is to help you understand why the Lagrange method works and how it must be

modified to deal with corner solutions. Consider the constraint ( )g x b . We can think of the

parameter b as the number of units of some resource that are available for use and ( )g x as the

resource requirements for the plan x.

As economists, it is natural to think about what the solution would be if there were a price 0p

for the resource (measured in units of the maximand) and the agent could either purchase

additional units or sell unused units at the price p.

If ( )g x b the agent purchases an additional ( )g x b units and so the total payoff is

( ) ( ( ) )L f x p g x b .


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If ( ) 0b g x the agent sells the unused units of the resource and so the total payoff is

( ) ( ( ))L f x p b g x . (A.2)

Of course these expressions are the same. I find it best to stick always to the same formulation

and use expression (A.2). Note that the Lagrangian is written in the form ( ) ( )L f x ph x

where ( ) ( ) 0h x b g x .

It turns out that under very weak assumptions, we can find a price 0p such that the solution of

the solution of the constrained maximization problem *x and the payoff are the same as the

solution of the unconstrained maximization problem. (This is the Kuhn-Tucker Theorem.)

Since the price p is a virtual price rather than a market price it is called a “shadow price” and is

typically written as a Greek letter. Mathematicians call it the shadow price a Lagrange

multiplier.

Case (i) Constraint not binding at the maximum

If the constraint is not binding, then *x solves { ( ) | 0}

xMax f x x . Also *( ) 0g x b . Then for

the payoff for the unconstrained maximization problem (A.2) to be the same as the payoff in

(A.1) 0p .

We can summarize this as follows:

*( ) 0b g x and *( ( )) 0p b g x (A.3)

Case (ii) Constraint binding at the maximum

Note from (A.2) that in this case

*( )L f x

Therefore the payoff in the unconstrained problem is the same as in the constrained problem.


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Next consider the marginal payoff as jx changes

* *( ) ( )j j j

L f gx p x

x x x

. (A.4)

If this is strictly positive, then the payoff can be increased by increasing jx . Therefore a

necessary condition for the unconstrained profit maximization is

* *( ) ( ) 0j j j

L f gx p x

x x x

.

Suppose that * 0jx . If 0j

L

x

profit can be increased by lowering jx . It follows that if * 0jx

then

* *( ) ( ) 0j j j

L f gx p x

x x x

.

We can summarize the necessary conditions as follows.

* *( ) ( ) 0j j j

L f gx p x

x x x

and * * * *[ ( ) ( )] 0j j

j j j

L f gx x x p x

x x x

(A.5)

What if * 0jx ? Note that j

L

x

cannot be strictly positive. However if the marginal payoff to

increasing jx is either zero or strictly less than zero then * 0jx is optimal.

Practical problem solving

How do you know if there is a “corner solution” where * 0jx for at least one j? This is usually a

trial and error process.

Step 1: Assume that the solution is not a corner solution.


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There are n necessary conditions and 1 constraint to solve. If there is such a solution great! If not,

the solution must be a corner solution

Step 2: If the solution is a corner solution

Ask what is wrong? Suppose 0ix and all the other variables are positive. Then guess that

* 0ix . There are now 1n necessary conditions that are equalities and one constraint. And

there n variables. So solve again. With luck you will have the solution.

Exercise A.2: Consumer choice with a modified Cobb-Douglas utility function

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1

( ) ln( ), 0j j

j

U x x

1 1 2 2 3 3p p p The consumer has income I. The price vector is p where

* 0x * * *

1 1 1 2 2 2 3 3 3( ) ( ) ( )p x p x p x (a) Show that if then .

*

3 0x *

1x *

2x(b) Hence show that if then and must also be strictly positive.

* 0x (c) Show that if income is sufficiently high then .

*

3x(d) What is the critical income below which is zero?

The complete solution to Exercise A.2 for different income levels is depicted below. At

sufficiently low income levels the consumer moves along the 1x axis as income increases. At

intermediate income levels the consumer purchases both commodity 1 and commodity 2. At high

income levels all three commodities are consumed.


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Exercise A.3: Consumer choice with a modified Cobb-Douglas utility function

2

1

( ) ln( ), , 0j j j

j

U x x

The consumer has income I. The price vector is p .

* 0x *x(a) Show that if income is sufficiently high then and solve for .

1 2

1 1 2 2p p

Suppose henceforth that .

(b) Which commodity will not be consumed if income is sufficiently low?

(c) What is the critical income below which only one commodity is consumed?

1x 2x(d) Depict the Income Expansion Path in a neat figure with on the horizontal and on the

vertical axis.

Exercise A.4: Consumer choice with “quasi-linear” preferences

Income expansion path


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A consumer has quasi-linear preferences if they can be represented by a utility function which is

1/2

1 2( ) 20U x x x linear in some commodity. Consider the quasi-linear utility function .A

81I (1,1)p consumer has income . The price vector is

* 0x (a) Show that there is no consumption bundle satisfying the necessary conditions.

(b) Solve for the utility maximizing consumption bundle.

(c) For what income levels is there a corner solution?

(b) Preferences and demand

Economic theorists usually make enough assumptions about preferences to ensure that a

consumer’s ranking of all the bundles of commodities in some set nX can be represented by

a continuous utility function ( )U x . For any consumption bundle y let ( )P y be the set of

bundles preferred to y . We include bundles over which the consumer is indifferent so

( ) { | ( ) ( )}nP y x X U x U y .

Aside: We can think of ( )P y as a mapping from each consumption bundle y into a “preferred

set” ( )P y . Set valued mappings are called correspondences. For each bundle y there is a

corresponding set ( )P y .


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Exercise B.1: Preferred sets

(1,2)y 1x 2x ( )P y(a) For depict in a neat figure (with and on the axes) the preferred set if (i)

1 2( ) 3ln 6lnU x x x 3 6

1 2( )U x x x ( )U x x 2, x 0 (ii) (iii) where and .1

(b) In case (ii) prove using methods that an undergraduate economics student could follow that

there is a unique solution to the following maximization problem.

1 1 2 2{ ( ) | , 0}x

Max U x p x p x I x .

( )U x 1/3( ) ( )V x U xHint: Is maximizing the same as maximizing ?

Convex combinations

Weighted averages of two vectors of the same dimension are called convex combinations. That

is, for any 0 1, nx x the vector 0 1(1 ) , 0 1x x x is a convex combination of 0x

and 1x .

Convex and strictly convex preferences

Consider any 0 1,x x X , such that 1 0( ) ( )U x U x . Preferences are convex if for every convex

combination x, 0( ) ( )U x U x . Preferences are strictly convex if for every convex

combination x, 0( ) ( )U x U x .

1 Consider vectors

0 1, nx x . If 1 0 , 1,...,j jx x j n then we write

1 0x x . If in addition 1 0

j jx x for

some j then we write 1 0x x . If the inequality is strict for all j we write

1 0x x


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Exercise B.2: Convex preferred set

( )U x x , nx 0 ( )P yIf where and prove that the preferred set is convex. Is it

strictly convex?

Proposition B.1: If preferences over the set nX are strictly convex, then for any price

vector 0p and income I, there is a unique solution ( , )x p I to the following consumer choice

problem.

{ ( ) | }x

Max U x p x I .

Exercise B.3: Prove this proposition

0x 1xHINT: Prove by the method of contradiction. That is suppose that and are both solutions

and obtain a contradiction.

C. Application: Market demand with identical homothetic preferences

Suppose that a consumer is indifferent between the consumption bundles 0x and

1x . There is no

reason to think that the consumer will be indifferent between the bundles 0 02y x and 1 12y x

None-the-less, economists often use this simplifying assumption, especially when dealing with

economies at the market level (as in macroeconomics.)

Definition: Homothetic Preferences

Preferences are homothetic if for any 0x and

1x , such that 0 1( ) ( )U x U x it follows that for all

0 0 1( ) ( )U x U x .


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Exercise C.1: Homothetic utility functions

(a) Show that the following utility functions are all homothetic

1 2

1 2( ) , 0U x x x 1

( ) lnn

j j

j

U x x

0 (i) (ii) where

1/2 1/2

1 2( ) 4U x x x 1/2 1/2 2

1 2( ) ( 4 )U x x x 1 1 1

1 2( ) ( 4 )U x x x (iii) (iv) (v) .

0 ( ) ( )U x U x (b) Under what additional restrictions (if any) is it also true that for any ,

Market demand with identical homothetic preferences

Suppose preferences are strictly convex so that there is a unique solution ( , )x p I to the

consumer’s choice problem { ( ) | }x

Max U x p x I .

The figure below depicts a case in which demand for commodity 2 on the vertical axis rises

faster than demand for commodity 1 as income rises,


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Proposition C.1: If preferences are strictly convex and homothetic, then demand rises

proportionally with income. That is, ( , ) ( ,1)x p I Ix p

Exercise C.2: Suppose that preferences are strictly convex. Provide a very short formal proof of

the following statements

( ( , )) ( ( ,1))U x p I U Ix p1

( ( ,1)) ( ( , ))U x p U x p II

(i) (ii)

Hint: The first inequality is depicted in the figure above.

I Exercise C.3: Appeal to (ii) and the definition of a homothetic utility function (with ) to

obtain a new inequality. Then appeal to (i) to prove Proposition 2.

Exercise C.4: Income elasticity of demand

Show that if preferences are strictly convex and homothetic then the income elasticity of demand

( , ) ( , )j

j

j

xIx I p I

x I

is equal to 1.

Exercise C.5: Aggregation of demands

Suppose that two consumers have the same strictly convex homothetic preferences and hence the

( , )h hx x p Isame demand function .

(a) Appeal to Proposition C.1 to show that

1 2 1 2( , ) ( , ) ( , )x p I x p I x p I I

(b) Generalize to the H consumer case.

1

( , ) ( , )H

h

h

x p I x p I

where 1

Hh

h

I I

.


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The representative consumer

The last exercise is important. It establishes that in the case of identical homothetic preferences,

market demand is independent of the distribution of income across individuals.

It follows that in such a case we can think of the market demand as being the demand of a single

“representative” individual with all of the income.


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D. Answers to exercises

Exercise A1:

As 0jx ln jx thus * 0x .

Necessary conditions: Equate marginal utility per dollar

1 1

1 1...

n n

U U

p x p x

For the Cobb-Douglas (logarithmic) utility function

1 2

1 1 2 2

... n

n np x p x p x

.

Then 1 j

j

j

xp

.

1 1 1

1 1n n nj

j j j j

j j jj

p x p x p Ip

.

Therefore

1

1n

j

j

I

and so *

1

( )j

j n

jj

j

Ix

p

Exercise A.2

Assume * 0x .


1 1 3 3

1 1...

U U

p x p x

Therefore


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1 1 1 2 2 2 3 3 3

1 1 1

( ) ( ) ( )p x p x p x

.

Hence

1 1 1 1 2 2 2 2 3 3 3 3

1p x p p x p p x p

Substituting into the budget constraint

3p x p I p .

Therefore

13( )I p

13( )j j j jp x p I p

13( )j j j jp x I p p

Since 1 1 2 2 3 3p p p it follows that

1 1 2 2 3 3p x p x p x

Therefore expenditure on commodity 3 is lowest.

Note that it is positive if and only if zero if

13 33

( ) 0I p p .

That is

3 33I p p .

Exercise A.3:

Assume * 0x .



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1 1 2 2

1 1U U

p x p x

Therefore

1 2

1 1 1 2 2 2( ) ( )p x p x

.

Hence

11 1 1 1p x p

and 2

2 2 2 2p x p

Substituting into the budget constraint

1 2p x p I p

.

Therefore

1 2

1 I p

1 2

1 1( )( )

j j j j

j

p x pI p

1 2

1( )( )

j j j j

j j

p x pI p

The right hand side must be positive for * 0x . Since 1 1 2 2

1 2

p p

it follows that we need to

ensure that the right hand side is positive for commodity 1, that is,

1 1

1 2 1

1( )( ) 0

pI p

Thus puts a lower bound on income.

For income in this range


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2 2 2 2 1 1 1 1

2 1

p x p p x p

.

Rearranging,

2 1 1 1 1 2 22

2 1 1 2

[ ( )]p x p p

xp

Thus the income expansion path is linear above the critical income level.

Exercise A4:

1/2

1 2 1 1 2 220 (81 )L x x p x p x where (1,1)p

1

1

1L

px

, 1/2

2 2

2

10L

x px

,

Step 1:

Necessary conditions with 0x .

1

1

1 0L

px

, 1/2

2 2

2

10 0L

x px

.

Solving:

212

2

100( ) 100p

xp

.

But then 1 1 2 2 19 0p x I p x .

Step 2: Step 1 suggests that *

1 0x

Necessary conditions

1

1

1 0L

px

, 1/2

2 2

2

10 0L

x px

.

If *

1 0x then *

2 81x (budget constraint)


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1/2

2

10(81) 0L

x

. So 10

9 .

Note that

1

1

11 0

9

Lp

x

.

Thus the necessary conditions are satisfied.

Section B: Answers

Exercise B.1: Preferred sets

(a) Partial answer

Along the boundary of ( )P y ,

1 23ln 6ln 3ln1 6ln 2 6ln 2x x

Dividing by 3,

1 2ln 2ln 2ln 2 ln 4x x

Equivalently,

2

1 2ln ln 4x x

Therefore

2

1 2 4x x and so 2

2

1

4x

x

Taking the square root,

2 1/2

1

2x

x .


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The graph of this function is readily mapped. You should confirm that the boundary curve (the

“indifference curve” through (1,2)y ) approaches an axis as 1 0x and 1x .

Partial answer to (b)

A mathematician would note that if there is any income left over the consumption bundle cannot

be maximizing since the maximand is strictly increasing. Then the budget constraint holds with

equality and so it is possible to solve for 2x as a linear function of 1x and substitute for 2x in the

utility function 1 2 1 1( , ) ( , ( ))U x x U x f x . Now you have one variable maximization problem.

If 1/3( ) ( )V x U x is maximized then ( )U x is also maximized since ( )V x is a strictly increasing

function of ( )U x . This observation makes the differentiation easier.

With 3 6

1 2( )U x x x the consumer has a utility of zero if consumption if either commodity is zero

and has a strictly positive utility if 0x so we know that the solution is not a corner solution.

Then the marginal utility per dollar must be equal for the two commodities.

1 1 2 2

1 1U U

p x p x

2 6

1 2

1

3U

x xx

and 3 5

1 2

2

6U

x xx

.

Therefore

2 6 3 5

1 2 1 2

1 2

3 6x x x x

p p .

Cross-multiplying,

2 6 3 5

2 1 2 1 1 23 6p x x p x x , that is, 2 2 1 12p x p x .

Substitute into the budget constraint to show that 11 1 3

p x I and 22 2 3

p x I .


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Section C: Answers

Exercise C.2

Given strict convexity the is a unique maximizing consumption bundle ( , )x p I . Since ( ,1)x p is

optimal with income 1. Note that ( ,1)Ix p costs I thus it is feasible with income I. By definition

( , )x p I is optimal thus

( ( , )) ( ( ,1))U x p I U Ix p . (*)

Since ( , )x p I costs I, 1

( , )x p II

costs 1 dollar. Since ( ,1)x p is optimal with income 1 it follows

that

1( ( ,1)) ( ( , ))U x p U x p I

I (**)

Exercise C.3

If U is homothetic it follows from (**) that for any 0

( ( ,1)) ( ( , ))U x p U x p II

.

Setting I

( ( ,1)) ( ( , ))U Ix p U x p I .

Appealing to (*)

( ( ,1)) ( ( , )) ( ( ,1))U Ix p U x p I U Ix p .

Therefore

( ( ,1)) ( ( , ))U Ix p U x p I .

Since there is a unique optimal consumption choice it follows that

( , ) ( ,1)x p I Ix p .


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Thus consumption rises proportionally with income.

Exercise C.5: Aggregation of demands with identical homothetic preferences.

Consumer h with income hI has consumption vector ( , ) ( ,1)h hx p I I x p .

Summing over consumers aggregate demand is

1 1

( ) ( , ) [ ] ( ,1)H H

h h

h h

x p x p I I x p

A single consumer with income 1

HR h

h

I I

has demand

1

( , ) ( ,1) [ ] ( ,1)H

R R h

h

x p I I x p I x p

.

Therefore ( ) ( , )Rx p x p I

module 6: theory of constrained...

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