Utility Maximization
References:
1. Snyder and Nicholson (10th ed), Chapter 4
2. Varian, Chapter 7
Utility maximization =
to maximize utility, given a fixed amount of income to spend an
individual will buy those quantities of goods that exhaust his or her
total income
Marginal Rate of Substitution = MRS = the rate at which the goods can
be traded one for the other in the market place
The two-good Case: A Graphical Analysis
Budget constraint
Quantity of y
px x  p y y  I
No more than I can be spent on the two goods
I / py
I  px x  p y y
Slope =
I / px

px
py
Quantity of x
FOC for maximum
Quantity of y
Slope of budget constraint = I  px x  p y y
px

py
= MRS  
dy
dx U U1
Y*
U
X*
Quantity of x
Intuitive result =
Utility is maximum when all income is spent and MRS = the ratio of the
prices of the goods
Second order condition --> what is the use of it? --> use the figure
FIRST ORDER CONDITION WITH LAGRANGIAN MULTIPLIER
............................... (page 119)
Implication of FOC:

MRS 
U / xi
p
 i
U / x j p j
 Interpreting lagrangian multiplier:

U / xi MU xi

pi
pi
--> at utility
maximizing point, each good purchased should yield the same
marginal utility per dolar spend on that good
 Or we can rewrite as:
pi 
MU xi

Example of Utility maximization:
1.
2.
U ( x, y )  x1/ 3 y 2 / 3
Income  100
U ( x, y )  x1/ 2  y1/ 2
Income  I
Indirect Utility Function
Indirect utility = U ( x *i )  V ( pi , I )
 Gives maximum utility as a function of p and I
 The optimal level of utility depends indirectly on the prices of the
goods and the individual income
Quantity of y
I / py
Quantity of y
I  px x  p y y
I / px
I  px x  p y y
I / py
Quantity of x
I / px
The Lumpsum Principles
Example (Excise tax and income tax):
Suppose that the consumer budget constraint:
The effect of an excise tax: ( p1  t ) x1  p2 x2  m
p1 x1  p2 x2  m
Suppose that revenue collected by the tax = tx1*
For the same tax revenue, the income tax should be imposed on
consumer’s income, hence: p1 x1  p2 x2  m  tx1*
Evaluate which tax is worse off for the consumer
Solution:
Quantity of x2
I / px
Quantity of x1
Quantity of x
The lumpsum principle suggests:
 An income tax leaves the individual free to decide how to allocate
their income
 Taxes on specific goods will change a person’s purchasing power and
distor his or her maximizing choices because of the artificial prices
incorporated in such schemes.
 The income grants to low-income people will raise utility more than
will a similar amount of money spent subsidizing specific goods
Expenditure Minimization
Expenditure Function (The inverse of the indirect utility
function)
Expenditure function:
e(p, u) = min px
Such that: u(x)  u
Expenditure function = relates income and utility
= gives the minimum cost of achieving
a fixed level of utility
Properties of the expenditure function:
(1)
(2)
e(p,u) is nondecreasing in p.
e(p,u) is homogenouse of degree 1 in p
(3)
e(p,u) is concave in p.
(4)
e(p,u)is continous in p, for p>> 0.
(5)
If h(p,u) is the expenditure-minimizing bundle
necessary to achieve utility level u at prices p, then:
hi ( p, u ) 
e( p, u )
pi
for i= 1,…,k
 HICKSIAN DEMAND
FUNCTION
(assuming: derivatives exist and pi>0)
Hicksian demand function:
hi ( p, u )
Marshallian demand function:
demand function
 not directly observable
x ( p, m)
 ordinary market
Some Important Identities
UTILITY MAX PROBLEM:
v(p,m*) = max u(x)
such that: px m*
EXPENDITURE MIN PROBLEM:
e(p,u*) = min px
such that: u(x) u*
FOUR IMPORTANT IDENTITIES:
(1)
e(p, v(p,m))  m
The minimum expenditure necessary to reach utility
v(p,m) is m.
(2)
v(p, e(p,u))  u
The maximum utility from income e(p,u) is u
(3)
xi(p,m)  hi (p, v(p,m))
The Marshallian demand at income m is the same as the
Hicksian demand at utility v(p,m).
(4)
hi (p,u) xi(p,e(p,u))
The Hicksian demand at utility u is the same as the
Marshallian demand at income e(p,u)
Identity (4) shows that the Hicksian demand function—the
solution to the expenditure minimization problem—is equal to
the Marshallian demand function at an appropriate level of
income.
Roy’s Identity
If x(p,m) is the Marshallian demand function, then:
v( p, m)
pi
xi ( p, m)  
v( p, m)
m
for i=1, …, k
Exercises:
1. Mr. Odde Ball enjoys commodities x and y according to the
utility function:
U ( x, y )  x 2  y 2
Maximize Mr Ball’s utility, if
px  3; p y  4
Income  50
2. On a given evening, Mr P enjoys the consumption of cigars
(c) and brandy (b) according to the following function:
U (c, b)  20c  c 2  18b  3b2
How many cigar and brandy did he consume that evening?
(There is no cost limitation for him)
If his doctors advised him to limit the total amount of
brandy and cigars consumed to 5. How many glasses of
brandy and cigars will he consume under this
circumstances?
The Money Metric Utility Functions
Consider:
 some prices p; and
 some given bundle of goods x.
We can ask the following question: how much money would a given
consumer need at the prices p to be as well off as he could be by
consuming the bundle of goods x?
Min pz
Such that u(z) u(x)
Figure: Direct money metric utility function
The money metric utility function (= known as the "minimum income function)
gives the minimum expenditure at prices p necessary to purchase a
bundle at least as good as x.
An alternative definition (minimum income function):
m( p, x)  e( p, u ( x))
Indirect money metric utility function:
 ( p; q, m)  e( p, v(q, m))
Figure:
Example (The Cobb-Douglas Utility Function):