Utility Maximization and Choice
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Transcript of Utility Maximization and Choice
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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(
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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 ofX* and Y*:
XP
EX
2*
YP
EY
2*
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Expenditure Function from
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
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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
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Important Points to Note:
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
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Important Points to Note:
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
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Important Points to Note:
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