Chapter 4
Utility Maximization
and Choice
Consumer Behavior
● Theory
of consumer behavior
Description of how consumers allocate incomes
among different goods and services to maximize
their well-being.
Consumer behavior is best understood in these
distinct steps:
1. Consumer Preferences
2. Utility
3. Budget Constraints
4. Consumer Choices
Key Definitions
Budget Set:
• The set of baskets that are affordable
Budget Constraint:
• The set of baskets that the consumer may
purchase given the limits of the available
income.
Budget Line:
• The set of baskets that one can purchase
when spending all available income.
3
The Budget Constraint
Assume only two goods available: X and Y
• Price of good x: Px ; Price of good y: Py
• Income: I
Total expenditure on basket (X,Y): PxX + PyY
The Basket is Affordable if total expenditure
does not exceed total Income:
PXX + PYY ≤ I
4
The Budget Constraint
• Assume that an individual has I amount of
income 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
A Budget Constraint Example
Two goods available: X and Y
I = $10
Px = $1
Py = $2
All income spent on X →
I/Px : units of X bought
All income spent on Y →
I/Py : units of Y bought
Budget Line 1:
1X + 2Y = 10
Or
Y = 5 – X/2
Slope of Budget Line
= -Px/Py
= -1/2
6
A Budget Constraint Example
Y
I/PY= 5
Budget line = BL1
A
•
•C
-PX/PY = -1/2
B
•
I/PX = 10
X
7
Budget Constraint
•
•
•
•
•
Shifting of Budget line:
1) Change in income
2) Change in price of good X
3) Change in price of good Y
4) Change in the price of both good X and Y.
Budget Constraint
• Location of budget line shows what the
income level is.
• Increase in Income will shift the budget line
to the right.
– More of each product becomes affordable
• Decrease in Income will shift the budget line
to the left.
– less of each product becomes affordable
9
A Budget Constraint Example
Y
Shift of a budget line
I = $12
PX = $1
PY = $2
If income rises, the budget line shifts parallel
to the right (shifts out)
6 Y = 6 - X/2 …. BL2
5
If income falls, the budget line shifts parallel
to the left (shifts in)
BL2
BL1
10
12 X
10
A Budget Constraint Example
Y
Rotation of a budget line
If the price of X rises, the budget
line gets steeper and the
horizontal intercept shifts in
I = $10
PX = $1
BL1 PY = $3
6
5
If the price of X falls, the budget
line gets flatter and the
horizontal intercept shifts out
Y = 3.33 - X/3 …. BL2
3.3
3
BL2
10
X
11
Consumer Choice
Assume:
 Only non-negative quantities
"Rational” choice:
The consumer chooses the basket
that maximizes his satisfaction given
the constraint that his budget imposes.
Consumer’s Problem:
Max U(X,Y)
Subject to: PxX + PyY < I
12
Constrained Consumer Choice
• Consumers maximize their well-being (utility) subject
to their budget constraint.
• The highest indifference curve attainable given the
budget is the consumer’s optimal bundle.
• When the optimal bundle occurs at a point of
tangency between the indifference curve and budget
line, this is called an interior solution / Interior
optimum.
Interior Optimum
Interior Optimum: The optimal consumption basket is at
a point where the indifference curve is just tangent to
the budget line.
A tangent: to a function is a straight line that has the
same slope as the function…therefore….
MRSx,y = MUx/MUy = Px/Py
“The rate at which the consumer would be willing to
exchange X for Y is the same as the rate at which they
are exchanged in the marketplace.”
14
Interior Consumer Optimum
Y
B
•
Preference Direction
•
Optimal Choice (interior solution)
IC
C
0
•
BL
X
15
Interior Consumer Optimum
• Utility is maximized where the indifference
curve is tangent to the budget constraint
Quantity of y
slope of budget constraint  
B
px
py
slope of indifference curve 
U2
dy
dx
U  constant
p x dy

 MRS
p y dx U  constant
Quantity of x
Interior Consumer Optimum
17
Interior Consumer Optimum: Assumptions
•U
(X,Y) = XY and MUx = Y while MUy = X
• I = $1,000
• P = $50 and P = $200
• Suppose Basket A contains (X=4, Y=4)
• Suppose Basket B contains (X=10, Y=2.5)
• Question:
• Is either basket the optimal choice for the consumer?
Basket A:
MRSx,y = MUx/MUy = Y/X = 4/4 = 1
Slope of budget line = -Px/Py = -1/4
Basket B:
MRSx,y = MUx/MUy = Y/X = 1/4
18
Interior Consumer Optimum
Y
50X + 200Y = I
2.5
•
0
10
U = 25
Chapter Four
X
19
Contained Optimization
What are the equations that the
optimal consumption basket must
fulfill if we want to represent the
consumer’s choice among three goods
or more than three goods?
20
Optimization Principle
• To maximize utility, given a fixed amount of
income, an individual will buy the goods and
services:
– that exhaust total income
– for which 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
– 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
Second Order Conditions for a Maximum
• The tangency rule is 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, we must check secondorder conditions to ensure that we are at a
maximum
SOCs for a Maximum
• The tangency rule is only a necessary condition
– we need MRS to be 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
• Individuals may maximize utility by choosing to
consume only one of the goods
Quantity of y
U1 U2
At point A, the indifference curve
is not tangent to the budget
constraint
U3
Utility is maximized at point A
A
Quantity of x
Y , Quantity
r
Consumer Maximization: Corner
Solution
Utility is maximized at point e
25
At point e, the
indifference curve
is not tangent to the
budget constraint
e
I3
I2
Budget line
I1
50
X, Quantity
Example for quasilinear corner
solution
• David is considering his purchases of food (x)
and clothing (y).
• He has a utility function U(x,y) = xy + 10x.
• His income is I = 10.
• He faces a price of Px = $1 and a price of
clothing Py = $2.
• What is David’s optimal basket.
Constrained Consumer Choice with Quasilinear
Preferences
If the relative price of one good is too high and preferences are
quasilinear, the indifference curve will not be tangent to the budget
line and the consumer’s optimal bundle occurs at a corner solution.
Constrained Consumer Choice with
Perfect Substitutes
• With perfect substitutes, if the marginal rate of
substitution does not equal the marginal rate of
transformations, then the consumer’s optimal bundle
occurs at a corner solution, bundle b.
Example for perfect substitutes
• Sara views chocolate and vanilla ice cream as
perfect substitute.
• She likes both and is willing to trade one scoop of
chocolate for two scoops of vanilla ice cream.
• Q. if the price of a scoop of chocolate ice cream is
(Pc) is three times the price of vanilla (Pv), will
Sara buy both types of ice cream?
• If not which will she buy?
Rational Choice and Marginal Utility
• 1) If MUx/Px > MUz/Pz,
– consume more of good x.
• 2) If MUy/Py > MUz/Pz,
– consume more of good y.
Constrained Consumer Choice with
Perfect Complements
• The optimal bundle is on the budget line and at the right
angle (i.e. vertex) of an indifference curve.
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
• FOCs 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
Interpreting the Lagrangian
Multiplier
U / x1 U / x 2
U / x n


 ... 
p1
p2
pn

MU x1
p1

MU x2
p2
 ... 
MU xn
pn
•  is the marginal utility of an extra dollar of
consumption expenditure
– the marginal utility of income
Interpreting the Lagrangian
Multiplier
• At the margin, the price of a good represents
the consumer’s 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
• A corner solution means that the first-order
conditions must be modified:
ℒ/xi = U/xi - pi  0 (i = 1,…,n)
• And If
ℒ/xi = U/xi - pi  0,
then xi = 0
• This means that
U / x i MU xi
pi 



– any good whose price exceeds its marginal value
to the consumer will not be purchased
Cobb-Douglas Demand Functions
With constant utility
• Cobb-Douglas utility function:
U(x,y) = xy
• Setting up the Lagrangian:
ℒ = xy + (I - pxx - pyy)
• Let us assume that  +  = 1
• FOCs:
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
Example of Cobb – Douglas Utility
• Suppose a person has a utility function given
as
U = x0.8y0.2
• Given that Px = 5 and Py = 3 and Income = 75.
• Calculate the amount of utility the consumer
derives.
• Suppose there is 1% increase in income, what
will be the changed utility.
CES Demand
• To illustrate cases in which budget
shares are responsive to economic
circumstances,
• 3 specific examples of CES Function.
1) Assume that  = 0.5
2) If  = -1
3) If  = -
CES Demand
• 1) Assume that  = 0.5
U(x,y) = x0.5 + y0.5
• Setting up the Lagrangian:
ℒ = x0.5 + y0.5 + (I - pxx - pyy)
• FOCs:
ℒ/x = 0.5x -0.5 - px = 0
ℒ/y = 0.5y -0.5 - py = 0
ℒ/ = I - pxx - pyy = 0
•This means that
(y/x)0.5 = px/py
CES Demand
• Substituting into the budget constraint, we can
solve for the demand functions
I
I
x* 
y* 
px
py
px [1  ]
py [1 
]
py
px
• 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, the smaller
will be the share of income spent on x
CES Demand
• 2) If  = -1,
U(x,y) = -x -1 - y -1 = -(1/x) – (1/y)
Though Utility function sounds strange the marginal utilities
are positive and diminishing.
• First-order conditions imply that
y/x = (px/py)0.5
• The demand functions are
x* 
I

p
px 1   y

 px



0 .5



y* 
I
 p
py 1   x
  py





0 .5




• These demand functions are less price responsive.
• As Px ↑, the individual cuts back only modestly on good x, so
the spending on x ↑.
CES Demand
• 3) If  = -,
U(x,y) = Min(x,4y)
• The person will choose only combinations for
which x = 4y
• This means that
I = pxx + pyy = pxx + py(x/4)
I = (px + 0.25py)x
CES Demand
• Hence, the demand functions are
I
x* 
px  0.25 py
I
y* 
4 p x  py
• Share of the person’s budget to good x ↑as Px ↑.
• Why?
• X and y must be consumed in fixed
proportion.
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 be
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 x’s to
find the indirect utility function
maximum utility = U(x*1,x*2,…,x*n)
maximum utility = V(p1,p2,…,pn,I)
• The optimal level of utility will depend
indirectly on prices and income
The Lump Sum Principle
• Taxes on an individual’s general purchasing
power are superior to taxes on a specific good
– an income tax allows the individual to decide
freely how to allocate remaining income
– a tax on a specific good will reduce an individual’s
purchasing power and distort his choices
The Lump Sum Principle
• A tax on good x would shift the utilitymaximizing choice from point A to point B
Quantity of y
B
A
U1
U2
Quantity of x
The Lump Sum Principle
• An income tax that collected the same amount
would shift the budget constraint to I’
Quantity of y
Utility is maximized now
at point C on U3
I’
A
B
C
U3 U1
U2
Quantity of x
The Lump Sum Principle
• 1) If the utility function is Cobb-Douglas with 
=  = 0.5, we know that
I
x* 
2 px
I
y* 
2 py
• The indirect utility function is
V ( px , py , I )  (x*) (y*)
0.5
0.5

I
2 px0.5 py0.5
The Lump Sum Principle
• If a tax of $1 was imposed on good x
– the individual will purchase x* = 2
– indirect utility will fall from 2 to 1.41
• An equal-revenue tax will reduce income to $6
– indirect utility will fall from 2 to 1.5
The Lump Sum Principle
• 2) If the utility function is fixed proportions
with U = Min(x,4y), we know that
I
x* 
px  0.25 py
I
y* 
4 p x  py
• The indirect utility function is
I
V ( px , py , I )  Min( x *,4 y *)  x* 
px  0.25 py
4
I
 4y * 

4 px  py px  0.25 py
The Lump Sum Principle
• If a tax of $1 was imposed on good x
– indirect utility will fall from 4 to 8/3
• An equal-revenue tax will reduce income to
$16/3
– indirect utility will fall from 4 to 8/3
• Since preferences are rigid, the tax on x does
not distort choices
Expenditure Minimization
• Dual minimization problem for utility
maximization
–allocate income to achieve a given level
of utility with the minimal expenditure
• the goal and the constraint have been
reversed
Expenditure Minimization
• Point A is the solution to the dual problem
E2 provides just enough to reach U1
Quantity of y
E3
E3 will allow the individual to reach U1
but is not the minimal expenditure
required to do so
E2
E1
A
E1 is too small to achieve U1
U1
Quantity of x
E1, E2, E3, are different expenditure level
Expenditure Minimization
• The individual’s problem is to choose
x1,x2,…,xn to minimize
total expenditures = E = p1x1 + p2x2 +…+ pnxn
subject to the constraint
utility = Ū = U(x1,x2,…,xn)
• 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 related
–both depend on market prices but involve
different constraints
Expenditure Minimization with Calculus
• Minimize expenditure, E, subject to the constraint
of holding utility constant:
• The solution of this problem, the expenditure
function, shows the minimum expenditure necessary
to achieve a specified utility level for a given set of
prices:
Duality
The mirror image of the original (primal)
constrained optimization problem is called the
dual problem.
Min PxX + PyY
(X,Y) subject to: U(X,Y) = U*
where: U* is a target level of utility.
If U* is the level of utility that solves the primal
problem, then an interior optimum, if it exists, of
the dual problem also solves the primal problem.
Optimal Choice
Y
Example: Expenditure Minimization
Optimal
Choice
(interior
solution)
•
U = U*
Decreases in
expenditure level
0
PXX + PYY = E*
X
63
Expenditure Function
• Two ways to compute expenditure function.
• 1) Cobb- Douglas case
• The indirect utility function in the two-good
V ( p x , py , I ) 
I
2 px0.5 py0.5
• If we interchange the role of utility and
income (expenditure), we will have the
expenditure function
E(px,py,U) = 2px0.5py0.5U
Expenditure Function
• 2) Fixed – Proportions case
• The indirect utility function is
I
V ( p x , py , I ) 
px  0.25 py
• If we again switch the role of utility and
expenditures, we will have the expenditure
function
E(px,py,U) = (px + 0.25py)U