Updated: 16 October 2007

ECON 501
Nicholson Chapter 4, Part 2
ANOTHER LOOK AT COBB-DOUGLAS UTILITY FUNCTION
U
C-D
(X,Y) = X
α
Y
β assume α + β = 1 and ST: I = P
X
*X + P
Y
*Y
Setting up the Lagrangian expression:
L = U(X
α
Y
β
) + λ(I – P
X
*X – P
Y
*Y)
Then, let’s set all partial derivatives to zero and find solution, satisfying the constraint:
(1)
∂L/∂X = αX α-1
Y
β
– λ P
X
= 0
(2)
∂L/∂Y = βX α
Y
β-1
– λ P
Y
= 0
(3)
∂L/∂λ = I – P
X
*X – P
Y
*Y = 0


Let’s take a ratio of (1)/(2):

X

X
 
1
Y

Y

 
1



P
X
P
Y then
*
Y
X

P
X
P
Y where:
P
Y
Y

β

* P
X
X then


*
X
X
 
1
Y

Y

 
1

P
X
P
Y then
α and β are constants; P
X
and P
Y
are also constants; only X and Y are variable (MRS depends only on Y/X)
(4) P
Y
Y

1 -
α

P
X
X let’s substitute (4) into (3): I – P
X
X –
1 α

P
X
X = 0
I = P
X
X*

1
1 -
α

Let’s look at good Y:
P
Y
Y

β
α
* P
X
X then or let’s substitute (5) into (3):
I =
1 -
β
β
* P
Y
Y + P
Y
Y
Y = β(I/P
Y
)
I = P
X
X * (1/α)
P
Y
Y

β
1
 β
* P
X
X or or I = P
Y
Y * (
1 -
β
β

1 ) = = P
Y
Y * (
1
β
) identical result as for good X.
X = α(I/P
X
) or (5) P
X
X

1 -
β
β
* P
Y
Y then:
1

CES UTILITY FUNCTIONS
The only problem with Cobb-Douglas functions is that they don’t include “cross-price” effect of the other goods on the demand for the good in question. That is why the use of Cobb-Douglas functions is limited, and constant elasticity functions (CES) are more common. CES are more close to reality; demand for a good depends on the ratio of prices.
U
CES
(X,Y) = X
δ
+ Y
δ assume
δ
is same for X and Y; also ST: I = P
X
*X + P
Y
*Y
We will consider the specific CES function where,
δ = 0.5
U
CES
(X, Y) = X
0.5
+ Y
0.5
Setting up the Lagrangian expression:
L = (X
0.5
+ Y
0.5
) + λ (I – P
X
*X – P
Y
*Y)
Then, let’s set all partial derivatives to zero and find solution, satisfying the constraint:
(1)
∂L/∂X = 0.5X
-0.5
– λP
X
= 0
(2)
∂L/∂Y = 0.5Y
-0.5
– λP
Y
= 0
(3)
∂L/∂λ = I – P
X
X + P
Y
Y = 0 from (1) and (2):
0.5X

0.5
0.5Y

0.5

P
X
P
Y
Y
X



P
X
P
Y


2 then or
0 .
5
0 .
5
*
X

0 .
5
Y

0 .
5
(P
Y
) 2

P
X
P
Y
Y=(P
X
) 2 X then then
P
Y
(P
Y
Y
X
Y) = (P
0 .
5
X
)

P
X
P
Y
2 X and P
Y
=
P
X
2
X
P
Y
Y
, substituting into (3) we have I – P
X
X –
P
X
2
XY
P
Y
Y
= 0 then:
I – P
X
X –
P
2
X
X
P
Y
= 0 then I – P
X
X

 1

P
X
P
Y

 = 0 now, solving for X*
We have, X* =
I
P
X

 1

P
X
P
Y

 and in a similar fashion we can derive demand function for Y
Y* =
I
P
Y

 1

P
Y
P
X


2

This is a demand function derived from a CES function. Quantity demanded is a function of the ratio of prices. Share of spending on one good is not constant, as contrasted to the Cobb-Douglas function.
We can look at the share of expenditure on a good: (P
X
X/I)
Since I – P
X
X

 1

P
X
P
Y

 = 0 then I = P
X
X

 1

P
X
P
Y


P
X
X
I


 1

1
P
X
P
Y


Share of expenditure on a good is not constant; it depends on the price ratio of
P
X
P
Y
. In this specific case as P x
is increased the share spent on X will fall. Demand for X is very responsive to price change. Elasticity of Substitution,
 
1
1
 
for Cobb-Douglas,

= 0,

= 1.
In the above case,
 
1
1

0 .
5
= 2.
A CES Function with less substitutability ,
U (X, Y) = –X
–1
– Y
–1
Y
In this case from the first order conditions we have
X



P
X
P
Y


0 .
5
and demand functions,
X X =I/ P
X


1



P
Y
P
X


0 .
5


Y
Y
= I/ P
Y


1



P
X
P
Y


0 .
5


These functions are much less price responsive as the exponents on P
X
and P
Y
are only 0.5 with

= -1 and

=
1 -
1
(-1)

1
2
3

The Fixed Proportions Utility Function
(elasticity of substitution equal to zero)
Suppose we have a utility function where X and Y must be consumed in fixed proportions. For example 1 unit of Y is consumed with 4 units of X.
The utility function U(X,Y) = min (X, 4Y)
Person will consume goods so that,
X = 4 Y Y = X / 4
I = Px X + Py Y = Px X + Py X/4
= (Px + 0.25 Py) X
The demand function for X is therefore,
X * =
I
Px

0 .
25 Py
Similarly I = Px X + Py Y = (Px 4Y + Py Y)
=(4 Px + Py) Y
Y * =
I
4 Px

Py
INDIRECT UTILITY FUNCTIONS
Sometimes, it is more useful to express a utility function not in terms of quantity of good(s) but in terms of given prices and income. Such a function is called “indirect utility function”.
Utility = U(X
1
,
Utility = V(P
1
,
X
2
, X
3,
…,X
P
2
, P
3,
…, P
N
N
)
, I)
- maximizes utility by consuming more quantity of goods with given income and prices
- targets a certain level of utility by minimizing total expenditure with fixed prices
The optimum values of either X* or Y* for Cobb-Douglas or CES functions depend on the prices of goods and income available. Because of individual’s desire to maximize utility, given a budget
4

constraint, the optimal level of utility ultimately will depend on the prices of the goods being consumed and on the individual’s income. If we have multiple goods, then:
X
1
* = X
1
(P
1
, P
2
, P
3,
…,P
N
, I) = χ
1
X
2
* = X
2
(P
1
, P
2
, P
3,
…,P
N
, I) = χ
2 where χ i
is demand function for good X i
. . . .
X
N
* = X
N
(P
1
, P
2
, P
3,
…,P
N
, I) = χ
N
Let’s substitute each optimal value of good X i
into our utility function:
Utility = U(χ
1
,
χ
2
, χ
3,
…,χ
N
) or
Utility = U[(X
1
(P
1
, P
2
, P
3,
…,P
N
, I), X
2
(P
1
, P
2
, P
3,
…,P
N
, I), … ,X
N
(P
1
, P
2
, P
3,
…,P
N
, I)] or
V(P
1
, P
2
, P
3
, …, P
N
, I)
Ex: δ = 0.5
Assume:
U(X,Y) = X
0.5
Y
0.5
P
X
= 1 and P
Y
= 4 and I = 8
(1) X* = I/2P
X and (2) Y* = I/2P
Y let’s substitute (1) and (2) into U(X,Y)
Maximize U(X*, Y*) = (X*)
0.5
(Y*)
0.5
=


I
2P
X


0.5


I
2P
Y


0.5
NO quantities, only prices and income
Let’s insert income and prices given by the example I = 8, P
X
= 1 and P
Y
= 4.
U(X,Y) =


I
2P
X


0.5


I
2P
Y


0.5
=
8
2 * 1
0.5
8
2 * 4
0.5
=
8
2
0.5
8
8
0.5

4
1
0.5

2
=2
1
NOTE: This example continues from previous lecture, and this result is the same utility level as calculated before. Thus, indirect utility function gives the same result as direct utility function, using a different way of calculation.
PRACTICAL USE OF INDIRECT UTILITY FUNCTIONS
Ex: Which tax would be preferred by the government given a task to collect a certain amount of tax revenues and desire to minimize distortions created by tax in economy? In other words,
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the government must collect a certain amount of taxes with least possible negative affect on the society’s utility function. Assume two taxes are available: tax on one good alone (or taxation of income) or a general tax on many goods. Both methods of tax collection must ultimately bring in the same amount of tax collections.
Let’s continue with our example.
Society Utility = U(X,Y) = X
0.5
Y
0.5
Assume: TR = 4 and
P
X
= 1 and P
Y
= 4
I = 8
Lump-Sum Taxation
Suppose, the government decides to impose a uniform income tax, which brings Tax Revenue of
2.4, leaving society with 5.6 of disposable income, i.e. I` = 5.6. Thus:
Maximize U(X,Y) = X
0.5
Y
0.5
=


I`
2P
X


0.5


I`
2P
Y


0.5
= Indirect Utility Function
U`(X,Y) =


I`
2P
X


0.5


I`
2P
Y


0.5
=
5.6
2 * 1
0.5
5.6
2 * 4
0.5
=
5.6
2
0.5
5.6
8
0.5

31.36
16
0.5

1 .
4
This means that society now has less utility, since U` < U.
Ad Valorem Taxation of One Good
How much good X alone should be taxed in order to bring 2.4 in tax revenues? Suppose, the government decides to impose a tax on one good only (import tax), which brings TR of 2.4, by raising the price of X from 1 to 2, i.e. tax is r=1.00 per unit of X (100%). Would such a tax rate be enough to collect TR of 2.4? The income is unchanged at 8. Thus:
If P
X
= 1.0
If P`
X
= 2.0 then: then:
X* = I/2P
X
= 8 / 2(1) = 4 at initial price
X`* = I/2P`
X
= 8 / 2(2) = 2 at initial price + tax
Consumer will purchase only 2 units of X at P`
X
, instead of 4 units at P
X
. What is tax collection revenue at this level of consumption?
TR = X*r or 2.0*1.00 = 2.0 which is below target level of TR
However, if r` = 1.50 and X``* = 1.6 then 1.5*1.6 = 2.4
This tax rate results in consumption of 1.6 units of X, but brings enough tax revenues. Let’s calculate the utility of society with this tax rate (P
X
`` = P
X
+ r` = 1+1.50 = 2.50):
6

Max U``(X,Y) =


I
2P``
X


0.5


I
2P
Y


0.5
=
8
2 * 2 .
5
0.5
8
2 * 4
0.5
=

8
5
0.5
8
8
0 .
5

1 .
265
This result means that welfare of society is lower with ad valorem tax on one good compared with lump-sum tax on income. Thus: U`` <U` < U
We can also estimate so-called “welfare cost” (or “economics loss”, “deadweight loss”, “excess burden” in literature), which means the cost of distortion introduced by government policies.
Welfare Cost = (U` – U``) = (1.4 – 1.265) = 0.135
There is a classical argument that a general income tax is better than a partial sales tax, but the issue of tax-base in each case must be considered. It is often the case that the base of income tax is not as broad as sales tax. VAT tax is more progressive than income tax in reality.
Fixed Proportion Case:
V( Px, Py, I ) =
Px
I

0 .
25 Py
E( Px, Py, U ) = U ( Px

0 .
25 Py )
If Px=1 Py=4 and U=4 Then,
E( Px, Py, U) = 4 ( 1 + 0.25 4) = 8.
Fixed proportions U (X, Y) = Min (X, Y)
= X *

Px
I

0 .
25 Py
= Y *

4 Px
I

Py
In this case indirect utility is given by V = (Px, Py, I ) = min (X*, 4Y*) = X *

Px
I

0 .
25 Py
= 4 Y *

4 Px
4 I

Py
and with Px = 1, Py= 4, and I = 8, indirect utility is given by V = 4= (8/2).
If price of X goes from $1 to $2, then the derived utility would be reduced from 4 to 8 / 3,
7

V’ = X* = 4Y* =
2
8

0 .
25 ( 4 )

8
3
= V '

8
1


8 / 3
0 .
25 ( 4 )

16 / 3
2

8
3
EXPENDITURE MINIMIZATION
How do we achieve a certain level of utility with minimum expenditure? It is the other side of the problem of the maximization of utility with given budget and prices. It is often called “dual” problem, i.e. minimization of expenditure with a given utility level is the dual to the problem of maximization of utility subject to budget constraint. I* is the minimum income that is able to give the consumer a level of satisfaction of U*. good Y
I
3
/P
Y
I *
2
/P
Y
I
1
/P
Y
I
0
/P
Y
U*
- E
0 -
E
1
- E
2
- E
3
O
0
/P
X 1
/P
X
* I
3
/P
X
1
, X
2,
..., X n
so as to minimize total expenditures = E = P
1
X
1
+ P
2
X
2
+... + P n
X n subject to the constraint utility = U
2
=
U(X
1
+ X
2
+... + X n
)
Expenditure Function shows the minimum expenditures necessary to achieve a given level of utility for a particular set of prices: minimal expenditures = E (P
1
, P
2
, ..., P n
, U).
Application to a Cobb-Douglas utility function with

= 0.5 and

= 0.5.
Minimize E = P
X
X + P
Y
Y ST: U = X
0.5
Y
0.5
L = P
X
*X + P
Y
*Y + λ(Ū – X 0.5
Y 0.5
) then:
8

(1)
∂L/∂X = P
X
– 0.5λ X -0.5
Y
0.5
= 0
(2)
∂L/∂Y = P
Y
– 0.5λ X
0.5
Y
-0.5
= 0
(3)
∂L/∂λ = Ū – X 0.5
Y
0.5
= 0 from (1) and (2):
0.5X

0.5
Y
0.5
0.5X
0.5
Y

0.5

P
X
P
Y then
Y
X

P
X
P
Y or (4) P
Substituting (4) into the expenditure function:
E = P
X
X + P
Y
Y = 2P
Y
Y*, solving for Y*, where
Y* =
1
2


E
P
Y

 and, similarly X* =
1
2


E
P
X


X
X = P
Y
Y
This is exactly the same result as we obtained before. Let’s look at the utility by substituting (5) into the utility function
Ū =


E
2P
X


0.5


E
2P
Y


0.5

E
2P
X
0.5
P
Y
0.5
Now we solve for the minimum level of expenditures that will yield a level of utility Ū given the values of P x
and P y
:
We have,
E = U

2P
X
0.5
P
Y
0.5

If Px=1, Py=4 and U=2 then,
E = 2(2) (1 0.5 4 0.5
) = 8
Question: What would happen to expenditure if we double both prices given that targeted utility of Ū=2. Prices initially are P x
= 1 and P y
= 4.
Assume: If P`
X
= 2 P`
Y
= 8
Ū` =2
Then: Ū =
E
2P
X
0.5
P
Y
0.5
satisfy same level of utility of 2. Therefore,
E
2 =
2(2)
0.5
(8 )
0 .
5
Requires E = 16 to
9

Answer: If prices are doubled, the minimum expenditure will be doubled also.
Comparison of Cobb-Douglas and Fixed Proportion Case
Cobb – Douglas Case :
Cobb – Douglas utility function U(X, Y) = X

Y

with

=

= 0.5,
X *

I
2 Px
and Y *

I
2 Py
So the indirect utility function in this case is
V( Px, Py, I ) = U (X*, Y*) = (X*)
0.5
(Y*)
0.5
=
I
2 P x
0 .
5
P y
0 .
5
E( Px, Py, U ) = 2 P x
0 .
5
P y
0 .
5
U
If Px=1 Py=4 and U=2 Then required minimal expenditures are can be calculated as $8
(=2 1
0.5
4
0.5
).
When Px=1, Py=4 and I=8 and by using the above indirect utility function we can calculate the derived utility as;
V

8
2 .
1 .
2

2
If price of X rise from $1 to $2, then the derived utility would decrease from 2 to 1.41,
V’=
I
2 P x
0 .
5
P y
0 .
5
8
=
2 .( 2 )
0 .
5
( 4 )
0 .
5

1 .
41
Compensating for a price change:
If the price of X rises from $1 to $2 and in order to keep derived utility as a constant:
For C-D case: Given expenditure function, E( Px, Py, U ) = 2 P x
0 .
5
P y
0 .
5
U
Where Px=2, Py=4, and U =2, amount of expenditure should increases from $8 to $11.31:
E= (2 2
0.5
4
0.5
2) =11.31
10

In this case, X

11 .
31
2 ( 2 )

2 .
83 and Y

11 .
31
2 ( 4 )

1 .
41
For Fixed Proportions Case: Given the expenditure function, E( Px, Py, U ) = U ( Px

0 .
25 Py )
Where Px=2, Py=4, and U =4, amount of expenditure should increases from $8 to $12:
E=4(2 +0.25(4)) =12.
In this case, X

12
2

0 .
25 ( 4 )

4 .
00 and Y

4 ( 2
12
)

4

1 .
00
The compensation needed to keep utility constant in the C-D case is less than in the fixed proportion case because in the C-D case consumers can substitute away from the consumption of
X toward Y when its price rise.
11