Utility Maximization and Choice

8/22/2019 Utility Maximization and Choice

1/40

Chapter 4

UTILITY MAXIMIZATION

AND CHOICE

Copyright 2002 by South-Western, a division of Thomson Learning. All rights reserved.

MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONS

EIGHTH EDITION

WALTER NICHOLSON


2/40

Complaints about Economic

Approach No real individuals make the kinds of

lightning calculations required for utility

maximization

The utility-maximization model predicts

many aspects of behavior even thoughno one carries around a computer withhis utility function programmed into it


3/40

Complaints about Economic

Approach The economic model of choice is

extremely selfish because no one has

solely self-centered goals

Nothing in the utility-maximization

model prevents individuals from derivingsatisfaction from doing good


4/40

Optimization Principle

To maximize utility, given a fixed amountof income to spend, an individual will buythe goods and services:

that exhaust his or her total income

for which the psychic rate of trade-offbetween any goods (the MRS) is equal tothe rate at which goods can be traded forone another in the marketplace


5/40

A Numerical Illustration Assume that the individuals MRS = 1

He is willing to trade one unit ofXfor oneunit ofY

Suppose the price ofX= $2 and theprice ofY= $1

The individual can be made better off

Trade 1 unit ofXfor 2 units ofYin themarketplace


6/40

The Budget Constraint Assume that an individual has Idollars

to allocate between goodXand good Y

PXX+ PYYI

Quantity ofX

Quantity ofY The individual can affordto choose only combinationsofXand Yin the shadedtriangleYP

I

If all income is spenton Y, this is the amountofYthat can be purchased

XPI

If all income is spentonX, this is the amountofXthat can be purchased


7/40

First-Order Conditions for a

Maximum We can add the individuals utility mapto show the utility-maximization process

Quantity ofX

Quantity ofY

U1

A

The individual can do better than pointAby reallocating his budget

U3

CThe individual cannot have point Cbecause income is not large enough

U2

B

Point B is the point of utilitymaximization


8/40

First-Order Conditions for a

Maximum Utility is maximized where the indifferencecurve is tangent to the budget constraint

Quantity ofX

Quantity ofY

U2

B

constraintbudgetofslopeY

X

P

P

constant

curveceindifferenofslope

UdX

dY

MRSdX

dY

P

P

UY

X

constant

-


9/40

Second-Order Conditions for a

Maximum The tangency rule is only necessary but

not sufficient unless we assume that MRS

is diminishing ifMRS is diminishing, then indifference curves

are strictly convex

IfMRS is not diminishing, then we mustcheck second-order conditions to ensurethat we are at a maximum


10/40

Second-Order Conditions for a

Maximum The tangency rule is only necessary butnot sufficient unless we assume that MRS

is diminishing

Quantity ofX

Quantity ofY

U1

B

U2A

There is a tangency at pointA,but the individual can reach a higherlevel of utility at point B


11/40

Corner Solutions In some situations, individuals preferences

may be such that they can maximize utilityby choosing to consume only one of the

goods

Quantity ofX

Quantity ofY U2U1 U3

A

Utility is maximized at pointA

At pointA, the indifference curveis not tangent to the budget constraint


12/40

The n-Good Case

The individuals objective is to maximize

utility = U(X1,X2,,Xn)

subject to the budget constraint

I= P1X1 + P2X2++ PnXn

Set up the Lagrangian:L = U(X1,X2,,Xn) + (I-P1X1- P2X2--PnXn)


13/40

The n-Good Case First-order conditions for an interior

maximum:

L/X1 = U/X1 - P1 = 0

L/X2 = U/X2 - P2 = 0

L/Xn = U/Xn - Pn = 0

L/ = I- P1X1 - P2X2 - - PnXn = 0


14/40

Implications of First-Order

Conditions For any two goods,

j

i

j

i

P

P

XU

XU

/

/

This implies that at the optimal

allocation of income

j

iji

P

PXXMRS )for(


15/40

Interpreting the Lagrangian

Multiplier

is the marginal utility of an extra dollarof consumption expenditure

the marginal utility of income

n

n

P

XU

P

XU

P

XU

/...

//

2

2

1

1

n

XXX

P

MU

P

MU

P

MUn ...

21

21


16/40

Interpreting the Lagrangian

Multiplier For every good that an individual buys,

the price of that good represents his

evaluation of the utility of the last unitconsumed

how much the consumer is willing to payfor the last unit

iX

i

MUP


17/40

Corner Solutions When corner solutions are involved, the

first-order conditions must be modified:

L/Xi = U/Xi - Pi 0 (i= 1,,n)

IfL/Xi = U/Xi - Pi < 0thenXi = 0

This means that

iXi

i

MUXUP /

Any good whose price exceeds its marginalvalue to the consumer will not be purchased


18/40

Cobb-Douglas Demand

Functions Cobb-Douglas utility function:U(X,Y) =XY

Setting up the Lagrangian:L =XY + (I - PXX- PYY)

First-order conditions:

L/X= X-1

Y

- PX= 0L/Y= XY-1 - PY= 0

L/ = I- PXX- PYY= 0


19/40

Cobb-Douglas Demand

Functions First-order conditions imply:

Y/X= PX/PY

Since + = 1:

PYY= (/)PXX= [(1- )/]PXX

Substituting into the budget constraint:

I= PXX+ [(1- )/]PXX= (1/)PXX


20/40

Cobb-Douglas Demand

Functions Solving forXyields

Solving forYyieldsX

PX

I*

YP

YI

*

The individual will allocate percent ofhis income to goodXand percent ofhis income to good Y


21/40

Cobb-Douglas Demand

Functions The Cobb-Douglas utility function is

limited in its ability to explain actual

consumption behavior the share of income devoted to particular

goods often changes in response tochanging economic conditions

A more general functional form might bemore useful in explaining consumptiondecisions


22/40

CES Demand Assume that = 0.5

U(X,Y) =X0.5 + Y0.5

Setting up the Lagrangian:L =X0.5 + Y0.5 + (I - PXX- PYY)

First-order conditions:

L/X= 0.5X-0.5

- PX= 0L/Y= 0.5Y-0.5 - PY= 0

L/ = I- PXX- PYY= 0


23/40

CES Demand This means that

(Y/X)0.5 = Px/PY

Substituting into the budget constraint,we can solve for the demand functions:

]1[

*

Y

X

X

P

PP

X

I

]1[

*

X

Y

Y

P

PP

Y

I


24/40

CES Demand

In these demand functions, the share ofincome spent on eitherXorYis not a

constant depends on the ratio of the two prices

The higher is the relative price ofX(orY), the smaller will be the share ofincome spent onX(orY)


25/40

CES Demand

If = -1,

U(X,Y) =X-1 + Y-1

First-order conditions imply thatY/X= (PX/PY)

0.5

The demand functions are

]1[

* 5.0

X

Y

X

P

PP

XI

]1[

* 5.0

Y

X

Y

P

PP

Y

I


26/40

CES Demand

The elasticity of substitution () is equalto 1/(1-)

when = 0.5, = 2

when = -1, = 0.5

Because substitutability has declined,these demand functions are less

responsive to changes in relative prices The CES allows us to illustrate a wide

variety of possible relationships


27/40

Indirect Utility Function It is often possible to manipulate first-

order conditions to solve for optimalvalues ofX1,X2,,Xn

These optimal values will depend on theprices of all goods and income

X*n =Xn(P1,P2,,Pn, I)

X*1 =X1(P1,P2,,Pn,I)

X*2 =X2(P1,P2,,Pn,I)


28/40

Indirect Utility Function We can use the optimal values of theXs

to find the indirect utility function

maximum utility = U(X*1

,X*2

,,X*n

)

Substituting for eachX*i we getmaximum utility = V(P1,P2,,Pn,I)

The optimal level of utility will dependindirectly on prices and income

If either prices or income were to change,the maximum possible utility will change


29/40

Indirect Utility in the Cobb-

Douglas IfU=X0.5Y0.5, we know that

xPX 2

I

*YP

Y2

I

*

Substituting into the utility function, we get

5050

5050

222..

..

utilitymaximumYXYXPPPP

III


30/40

Expenditure Minimization

Dual minimization problem for utilitymaximization

allocating income in such a way as to achievea given level of utility with the minimalexpenditure

this means that the goal and the constraint

have been reversed


31/40

Expenditure level E2 provides just enough to reach U1

Expenditure Minimization

Quantity ofX

Quantity ofY

U1

Expenditure level E1 is too small to achieve U1

Expenditure level E3 will allow theindividual to reach U1 but is not theminimal expenditure required to do so

A

PointA is the solution to the dual problem


32/40

Expenditure Minimization The individuals problem is to choose

X1,X2,,Xn to minimize

E= P1X

1+ P

2X

2++P

nX

n

subject to the constraint

U1 = U(X1,X2,,Un)

The optimal amounts ofX1,X2,,Xn willdepend on the prices of the goods andthe required utility level


33/40

Expenditure Function The expenditure function shows the

minimal expenditures necessary toachieve a given utility level for a particular

set of pricesminimal expenditures = E(P1,P2,,Pn,U)

The expenditure function and the indirect

utility function are inversely related both depend on market prices but involve

different constraints


34/40

Expenditure Function from

the Cobb-Douglas Minimize E= PXX+ PYYsubject to

U=X0.5Y0.5 where U is the utility target

The Lagrangian expression isL = PXX+ PYY+ (U -X

0.5Y0.5)

First-order conditions are

L/X= PX- 0.5X-0.5Y0.5 = 0

L/Y= PY- 0.5X0.5Y-0.5 = 0

L/ = U-X0.5Y0.5= 0


35/40


the Cobb-Douglas These first-order conditions imply that

PXX= PYY

Substituting into the expenditure function:

E= PXX*+ PYY*= 2PXX*

Solving for optimal values ofX* and Y*:

XP

EX

2*

YP

EY

2*


36/40


the Cobb-Douglas Substituting into the utility function, we

can get the indirect utility function

5050

5050

222 ..

..

'YXYXPP

E

P

E

P

EU

So the expenditure function becomesE = 2UPX

0.5PY0.5


37/40

Important Points to Note:

To reach a constrained maximum, anindividual should:

spend all available income

choose a commodity bundle such that theMRS between any two goods is equal to theratio of the goods prices

the individual will equate the ratios of marginal utilityto price for every good that is actually consumed


38/40


Tangency conditions are only first-orderconditions

the individuals indifference map must exhibit

diminishing MRS the utility function must be strictly quasi-

concave

Tangency conditions must also be modifiedto allow for corner solutions

ratio of marginal utility to price will be lower forgoods that are not purchased


39/40


The individuals optimal choices implicitly

depend on the parameters of his budgetconstraint

choices observed will be implicit functions ofprices and income

utility will also be an indirect function of prices

and income


40/40


The dual problem to the constrained utility-maximization problem is to minimize theexpenditure required to reach a given utility

target yields the same optimal solution as the primary

problem

leads to expenditure functions in whichspending is a function of the utility target andprices

Documents

Utility Maximization and Choice