Chapter 4

UTILITY MAXIMIZATION

AND CHOICE

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

MICROECONOMIC THEORY

BASIC PRINCIPLES AND EXTENSIONS

EIGHTH EDITION

WALTER NICHOLSON

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 though no one carries around a computer with his utility function programmed into it

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 deriving satisfaction from “doing good”

Optimization Principle• To maximize utility, given a fixed amount

of income to spend, an individual will buy the goods and services:

– that exhaust his or her total income– for which the psychic rate of trade-off

between any goods (the MRS) is equal to the rate at which goods can be traded for one another in the marketplace

A Numerical Illustration• Assume that the individual’s MRS = 1

– He is willing to trade one unit of X for one unit of Y

• Suppose the price of X = $2 and the price of Y = $1

• The individual can be made better off– Trade 1 unit of X for 2 units of Y in the

marketplace

The Budget Constraint• Assume that an individual has I dollars

to allocate between good X and good Y

PXX + PYY I

Quantity of X

Quantity of Y The individual can afford

to choose only combinations

of X and Y in the shaded

triangleYP

IIf all income is spent

on Y, this is the amount

of Y that can be purchased

XP

I

If all income is spent

on X, this is the amount

of X that can be purchased

First-Order Conditions for a Maximum

• We can add the individual’s utility map to show the utility-maximization process

Quantity of X

Quantity of Y

U1

A

The individual can do better than point A

by reallocating his budget

U3

C

The individual cannot have point C

because income is not large enough

U2

B

Point B is the point of utility

maximization

First-Order Conditions for a Maximum

• Utility is maximized where the indifference curve is tangent to the budget constraint

Quantity of X

Quantity of Y

U2

B

constraint budget of slopeY

X

P

P

constant

curve ceindifferen of slope

UdX

dY

MRSdX

dY

P

P

UY

X constant

-

Second-Order Conditions for a Maximum

• The tangency rule is only necessary but not sufficient unless we assume that MRS is diminishing– if MRS is diminishing, then indifference curves

are strictly convex• If MRS is not diminishing, then we must

check second-order conditions to ensure that we are at a maximum

Second-Order Conditions for a Maximum

• The tangency rule is only necessary but not sufficient unless we assume that MRS is diminishing

Quantity of X

Quantity of Y

U1

B

U2

A

There is a tangency at point A,

but the individual can reach a higher

level of utility at point B

Corner Solutions• In some situations, individuals’ preferences

may be such that they can maximize utility by choosing to consume only one of the goods

Quantity of X

Quantity of Y U2U1 U3

A

Utility is maximized at point A

At point A, the indifference curve

is not tangent to the budget constraint

The n-Good Case

• The individual’s 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)

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

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 (

Interpreting the Lagrangian Multiplier

• is the marginal utility of an extra dollar of 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

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 unit consumed– how much the consumer is willing to pay

for the last unit

iX

i

MUP

Corner Solutions• When corner solutions are involved, the

first-order conditions must be modified:

L/Xi = U/Xi - Pi 0 (i = 1,…,n)

• If L/Xi = U/Xi - Pi < 0 then Xi = 0

• This means that

iXii

MUXUP

/

–Any good whose price exceeds its marginal value to the consumer will not be

purchased

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-1Y - PX = 0

L/Y = XY-1 - PY = 0

L/ = I - PXX - PYY = 0

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

Cobb-Douglas Demand Functions

• Solving for X yields

•Solving for Y yields

XPX

I*

YPY

I*

•The individual will allocate percent of his income to good X and

percent of his income to good Y

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 to changing economic conditions

• A more general functional form might be more useful in explaining consumption decisions

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 = 0

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

L/ = I - PXX - PYY = 0

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 PP

PX

I

]1[*

X

Y

Y PP

PY

I

CES Demand

• In these demand functions, the share of income spent on either X or Y is not a constant– depends on the ratio of the two prices

• The higher is the relative price of X (or Y), the smaller will be the share of income spent on X (or Y)

CES Demand• If = -1,

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

• First-order conditions imply that

Y/X = (PX/PY)0.5

• The demand functions are

]1[

* 5.0

X

YX P

PP

XI

]1[

* 5.0

Y

XY P

PP

YI

CES Demand• The elasticity of substitution () is equal

to 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

Indirect Utility Function• It is often possible to manipulate first-

order conditions to solve for optimal values of X1,X2,…,Xn

• These optimal values will depend on the prices 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)

Indirect Utility Function• We can use the optimal values of the Xs

to find the indirect utility function

maximum utility = U(X*1,X*2,…,X*n)

• Substituting for each X*i we get

maximum utility = V(P1,P2,…,Pn,I)• The optimal level of utility will depend

indirectly on prices and income– If either prices or income were to change,

the maximum possible utility will change

Indirect Utility in the Cobb-Douglas

• If U = X0.5Y0.5, we know that

xPX

2

I*

YPY

2

I*

•Substituting into the utility function, we get

5050

5050

222 ..

..

utility maximumYXYX PPPP

III

Expenditure Minimization

• Dual minimization problem for utility maximization– allocating income in such a way as to achieve

a given level of utility with the minimal expenditure

– this means that the goal and the constraint have been reversed

Expenditure level E2 provides just enough to reach U1

Expenditure Minimization

Quantity of X

Quantity of Y

U1

Expenditure level E1 is too small to achieve U1

Expenditure level E3 will allow the

individual to reach U1 but is not the

minimal expenditure required to do so

A

• Point A is the solution to the dual problem

Expenditure Minimization• The individual’s problem is to choose

X1,X2,…,Xn to minimize

E = P1X1 + P2X2 +…+PnXn

subject to the constraint

U1 = U(X1,X2,…,Un)

• The optimal amounts of X1,X2,…,Xn will depend on the prices of the goods and the required utility level

Expenditure Function• The expenditure function shows the

minimal expenditures necessary to achieve a given utility level for a particular set of prices

minimal 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

Expenditure Function from the Cobb-Douglas

• Minimize E = PXX + PYY subject to U’=X0.5Y0.5 where U’ is the utility target

• The Lagrangian expression is

L = PXX + PYY + (U’ - X0.5Y0.5)

• First-order conditions areL/X = PX - 0.5X-0.5Y0.5 = 0

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

L/ = U’ - X0.5Y0.5 = 0

Expenditure Function from the Cobb-Douglas

• These first-order conditions imply that

PXX = PYY

• Substituting into the expenditure function:

E = PXX* + PYY* = 2PXX*

Solving for optimal values of X* and Y*:

XP

EX

2*

YP

EY

2*

Expenditure Function from the Cobb-Douglas

• Substituting into the utility function, we can get the indirect utility function

5050

5050

222 ..

..

'YXYX PP

E

P

E

P

EU

•So the expenditure function becomes

E = 2U’PX0.5

PY0.5

Important Points to Note:• To reach a constrained maximum, an

individual should:– spend all available income– choose a commodity bundle such that the

MRS between any two goods is equal to the ratio of the goods’ prices• the individual will equate the ratios of marginal utility

to price for every good that is actually consumed

Important Points to Note:• Tangency conditions are only first-order

conditions– the individual’s indifference map must exhibit

diminishing MRS– the utility function must be strictly quasi-

concave• Tangency conditions must also be modified

to allow for corner solutions– ratio of marginal utility to price will be lower for

goods that are not purchased

Important Points to Note:• The individual’s optimal choices implicitly

depend on the parameters of his budget constraint– choices observed will be implicit functions of

prices and income– utility will also be an indirect function of prices

and income

Important Points to Note:• The dual problem to the constrained utility-

maximization problem is to minimize the expenditure required to reach a given utility target– yields the same optimal solution as the primary

problem– leads to expenditure functions in which

spending is a function of the utility target and prices