Chapter 4
UTILITY MAXIMIZATION
AND CHOICE
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.
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 Y
I
PY
If all income is spent
on Y, this is the amount
of Y that can be purchased
The individual can afford
to choose only combinations
of X and Y in the shaded
triangle
If all income is spent
on X, this is the amount
of X that can be purchased
I
PX
Quantity of X
First-Order Conditions for a
Maximum
• We can add the individual’s utility map
to show the utility-maximization process
The individual can do better than point A
by reallocating his budget
Quantity of Y
A
The individual cannot have point C
because income is not large enough
C
B
U3
U2
U1
Point B is the point of utility
maximization
Quantity of X
First-Order Conditions for a
Maximum
• Utility is maximized where the indifference
curve is tangent to the budget constraint
slope of budget constraint  
Quantity of Y
PX
PY
slope of indifferen ce curve 
dY
dX
B
U2
PX
dY
PY
dX
Quantity of X
U  constant
 MRS
U  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 Y
There is a tangency at point A,
but the individual can reach a higher
level of utility at point B
B
A
U2
U1
Quantity of X
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 Y
At point A, the indifference curve
is not tangent to the budget constraint
U1 U2 U3
Utility is maximized at point A
A
Quantity of X
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,
U / X i Pi

U / X j Pj
• This implies that at the optimal
allocation of income
Pi
MRS ( X i for X j ) 
Pj
Interpreting the Lagrangian
Multiplier
U / X1 U / X 2
U / X n


 ... 
P1
P2
Pn

MU X1
P1

MU X 2
P2
 ... 
MU X n
Pn
•  is the marginal utility of an extra dollar
of consumption expenditure
– the marginal utility of income
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
Pi 
MU X i

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
U / X i MU X i
Pi 



– 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) = XY
• Setting up the Lagrangian:
L = XY + (I - PXX - PYY)
• First-order conditions:
L/X = X-1Y - PX = 0
L/Y = XY-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
I
X* 
PX
• Solving for Y yields
I
Y* 
PY
• 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:
X* 
I
PX
PX [1  ]
PY
I
Y* 
PY
PY [1  ]
PX
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
X* 
I
 PY
PX [1  
 PX
0.5

 ]

Y* 
I
 PX
PY [1  
 PY
0.5

 ]

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 firstorder conditions to solve for optimal
values of X1,X2,…,Xn
• These optimal values will depend on the
prices of all goods and income
X*1 = X1(P1,P2,…,Pn,I)
X*2 = X2(P1,P2,…,Pn,I)
•
•
•
X*n = Xn(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 CobbDouglas
• If U = X0.5Y0.5, we know that
I
X* 
2Px
I
Y* 
2PY
• Substituting into the utility function, we get
 I
maximum utility  
 2PX



0.5
 I

 2PY



0.5

I
2PX0.5PY0.5
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 Minimization
• Point A is the solution to the dual problem
Expenditure level E2 provides just enough to reach U1
Quantity of Y
Expenditure level E3 will allow the
individual to reach U1 but is not the
minimal expenditure required to do so
A
Expenditure level E1 is too small to achieve U1
U1
Quantity of X
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 are
L/X = PX - 0.5X-0.5Y0.5 = 0
L/Y = PY - 0.5X0.5Y-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*:
E
X* 
2PX
E
Y* 
2PY
Expenditure Function from
the Cobb-Douglas
• Substituting into the utility function, we
can get the indirect utility function
 E
U '  
 2PX



0.5
 E

 2PY



0.5
E

0.5 0.5
2PX PY
• So the expenditure function becomes
E = 2U’PX0.5PY0.5
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 quasiconcave
• 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 utilitymaximization 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