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Updated: 16 October 2007
ECON 501: MICROECONOMICS
Lecture 3
Topics to be covered:
a. Cobb-Douglas Utility Function
b. Ces Utility Function
c. Indirect Utility Function
d. Lumpsum Taxation
e. Ad valorem Taxation of one Good
f. Expenditure Minimization
ECON 501
Nicholson Chapter 4, Part 2
ANOTHER LOOK AT COBB-DOUGLAS UTILITY FUNCTION
UC-D(X,Y) = XαYβ
assume α + β = 1
and
ST:
I = PX*X + PY*Y
Setting up the Lagrangian expression:
L = U(XαYβ) + λ(I – PX*X – PY*Y)
Then, let’s set all partial derivatives to zero and find solution, satisfying the constraint:
(1)
∂L/∂X = αXα-1Yβ – λ PX = 0
(2)
∂L/∂Y = βXαYβ-1 – λ PY = 0
(3)
∂L/∂λ = I – PX*X – PY*Y = 0
Let’s take a ratio of (1)/(2):
X  1Y  PX
then

X  Y  1 PY
 Y PX
* 
where:
 X PY
 X  1Y  PX
*

 X  Y  1 PY
then
only X and Y are variable (MRS depends only on Y/X)
PY Y 
β

* PX X
(4) PY Y 
then
1- α

1- α
let’s substitute (4) into (3):
I – PXX –
 1- α 
I = PXX* 1 

 

I = PXX * (1/α)
then

PX X
PX X = 0
X = α(I/PX)
or
Let’s look at good Y:
PY Y 
β
* PX X
α
β
* PX X
1 β
or
PY Y 
or
I = PYY * (
or
(5) PX X 
1- β
* PY Y
β
let’s substitute (5) into (3):
I=
1- β
* PY Y + PYY
β
Y = β(I/PY)
1- β
1
 1 ) = = PYY * ( )
β
β
identical result as for good X.
1
then:
PREVIOUS LECTURE
α and β are constants; PX and PY are also constants;
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.
UCES(X,Y) = Xδ + Yδ
assume δ is same for X and Y;
also
ST: I = PX*X + PY*Y
We will consider the specific CES function where,
δ = 0.5
UCES(X, Y) = X0.5 + Y0.5
Setting up the Lagrangian expression:
L = (X0.5 + Y0.5) + λ (I – PX*X – PY*Y)
Then, let’s set all partial derivatives to zero and find solution, satisfying the constraint:
(1)
∂L/∂X = 0.5X-0.5 – λPX = 0
(2)
∂L/∂Y = 0.5Y-0.5 – λPY = 0
(3)
∂L/∂λ = I – PXX + PYY = 0
from (1) and (2):
0.5X 0.5 PX

0.5Y 0.5 PY
Y  PX 
 
X  PY 
and
0.5 X 0.5 PX
*

0.5 Y 0.5 PY
then
0.5

PX
PY
2
or
(PY)2 Y=(PX)2 X
PX2 X
PY =
, substituting into (3) we have
PY Y
PX2 X
I – PXX –
=0
PY
We have,
then
Y 
 
X
X* =
then
then
PY(PYY) = (PX)2X
PX2 XY
I – PXX –
= 0 then:
PYY
 P 
I – PXX 1  X  = 0
 PY 
now, solving for X*
I
 P 
PX 1  X 
 PY 
and in a similar fashion we can derive demand function for Y
Y* =
I

P 
PY 1  Y 
PX 

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: (PXX/I)
 P 
I – PXX 1  X  = 0
 PY 
Since
then
 P 
I = PXX 1  X 
 PY 
PX X
1

I
 PX 
1  
 PY 
Share of expenditure on a good is not constant; it depends on the price ratio of
PX
. In this
PY
specific case as Px is increased the share spent on X will fall. Demand for X is very responsive to
price change. Elasticity of Substitution,  
In the above case,  
1
for Cobb-Douglas,  = 0,  = 1.
1 
1
= 2.
1  0 .5
A CES Function with less substitutability,
U (X, Y) = –X–1 – Y–1
Y  PX 


In this case from the first order conditions we have
X  PY 
0. 5
and demand functions,
  P  0.5 
X =I/ PX 1   Y  
  PX  


X
  P  0.5 
Y = I/ PY 1   X  
  PY  


Y
These functions are much less price responsive as the exponents on PX and PY 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=4Y
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(X1, X2, X3,…,XN)
- maximizes utility by consuming more quantity of goods
with given income and prices
Utility = V(P1, P2, P3,…, PN, I)
- 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
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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:
X1* = X1(P1, P2, P3,…,PN, I) = χ1
where χi is demand function
X2* = X2(P1, P2, P3,…,PN, I) = χ2
for good Xi
.
.
.
.
XN* = XN(P1, P2, P3,…,PN, I) = χN
Let’s substitute each optimal value of good Xi into our utility function:
Utility = U(χ1, χ2, χ3,…,χN)
or
Utility = U[(X1(P1, P2, P3,…,PN, I), X2(P1, P2, P3,…,PN, I), … ,XN(P1, P2, P3,…,PN, I)]
or
V(P1, P2, P3, …, PN, I)
Ex:
δ = 0.5
U(X,Y) = X0.5Y0.5
Assume:
PX = 1 and
(1) X* = I/2PX
Maximize
and
PY = 4
and
let’s substitute (1) and (2) into U(X,Y)
(2) Y* = I/2PY
0.5
U(X*, Y*) = (X*)
0.5
(Y*)
 I 

= 
 2PX 
I=8
0.5
 I 


 2PY 
0.5
NO quantities, only prices
and income
Let’s insert income and prices given by the example I = 8, PX = 1 and PY = 4.
 I 

U(X,Y) = 
2P
 X
0.5
0.5
 I 
 8   8 
8 8

 = 
 
 =   
 2 *1   2 * 4 
 2 8
 2PY 
0.5
0.5
0.5
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) = X0.5Y0.5
Assume:
TR = 4
and
I=8
PX = 1 and
PY = 4
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:
0.5
Maximize
 I` 

U`(X,Y) = 
 2PX 
0.5
0.5
U(X,Y) = X Y
0.5
 I` 

= 
 2PX 
0.5
0.5
 I` 

 = Indirect Utility Function
 2PY 
 I` 
 5.6   5.6 
 5.6   5.6 

 = 
 
 =
 

 2 *1   2 * 4 
 2   8 
 2PY 
0.5
0.5
0.5
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 PX = 1.0
then:
X* = I/2PX = 8 / 2(1) = 4
If P`X = 2.0
then:
X`* = I/2P`X = 8 / 2(2) = 2
at initial price
at initial price + tax
Consumer will purchase only 2 units of X at P`X, instead of 4 units at PX. What is tax collection
revenue at this level of consumption?
TR = X*r
However, if
or
r` = 1.50
and
2.0*1.00 = 2.0
which is below target level of TR
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 (PX`` = PX + r` = 1+1.50 = 2.50):
6
Max
 I 

U``(X,Y) = 
 2P``X 
0.5
0.5
 I 
 8   8 
8 8

 = 
 
 =    
 2 * 2.5   2 * 4 
5 8
 2PY 
0.5
0.5
0.5
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 ) =
I
Px  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* 
I
Px  0.25Py
=
Y* 
I
4 Px  Py
In this case indirect utility is given by V = (Px, Py, I ) = min (X*, 4Y*) = X * 
= 4Y * 
I
Px  0.25Py
4I
and with Px = 1, Py= 4, and I = 8, indirect utility is given by V = 4= (8/2).
4 Px  Py
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* =
8
8
8 8/3
16 / 3 8
 = V '


2  0.25(4) 3
1  0.25(4)
2
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
I3/PY
I*2/PY
I1/PY
I0/PY
- E0
O
I /P
- E1
- E2
I*2/PX
I /P
- E3
U*
I /P
0 X
1 X
X
Formally the individual’s expenditure
minimization
problem is to3 choose
X1, X2, ..., Xgood
as to
n so X
minimize total expenditures = E = P1X1 + P2X2 +... + PnXn subject to the constraint utility = U2 =
U(X1 + X2 +... + Xn)
Expenditure Function shows the minimum expenditures necessary to achieve a given level of
utility for a particular set of prices: minimal expenditures = E (P1, P2, ..., Pn, U).
Application to a Cobb-Douglas utility function with  = 0.5 and  = 0.5.
Minimize
E = PXX + PYY
ST:
U = X0.5Y0.5
L = PX*X + PY*Y + λ(Ū – X0.5Y0.5)
8
then:
(1)
∂L/∂X = PX – 0.5λ X-0.5Y0.5 = 0
(2)
∂L/∂Y = PY – 0.5λ X0.5Y-0.5 = 0
(3)
∂L/∂λ = Ū – X0.5Y0.5 = 0
from (1) and (2):
0.5X 0.5 Y 0.5 PX

0.5X 0.5 Y 0.5 PY
Y PX

X PY
then
or
(4) PXX = PYY
Substituting (4) into the expenditure function:
E = PXX + PYY = 2PYY*, solving for Y*, where
Y* =
1 E 
1 E 
  and, similarly X* =  
2  PY 
2  PX 
This is exactly the same result as we obtained before. Let’s look at the utility by substituting (5)
into the utility function
 E 

Ū = 
 2PX 
0.5
 E 


 2PY 
0.5

E
2P P
0.5 0.5
X
Y
Now we solve for the minimum level of expenditures that will yield a level of utility Ū given the
values of Px and Py:
We have,
E = U 2PX0.5 PY0.5 
If Px=1, Py=4 and U=2 then,
E = 2(2) (10.5 40.5) = 8
Question:
What would happen to expenditure if we double both prices given that
targeted utility of Ū=2. Prices initially are Px = 1 and Py = 4.
Assume: If
P`X = 2
Then:
Ū=
Ū` =2
P`Y= 8
E
2P P
2=
0.5 0.5
X
Y
E
2(2) (8) 0.5
satisfy same level of utility of 2. Therefore,
9
0.5
Requires E = 16 to
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
So the indirect utility function in this case is
2 Py
V( Px, Py, I ) = U (X*, Y*) = (X*)0.5 (Y*)0.5 =
I
0.5
x
2 P Py0.5
E( Px, Py, U ) = 2Px0.5 Py0.5U
If Px=1 Py=4 and U=2 Then required minimal expenditures are can be calculated as $8
(=2 10.540.5).
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
2.1.2
If price of X rise from $1 to $2, then the derived utility would decrease from 2 to 1.41,
V’=
I
0.5
x
0.5
y
2P P
=
8
0.5
2.(2) (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 ) = 2Px0.5 Py0.5U
Where Px=2, Py=4, and U =2, amount of expenditure should increases from $8 to $11.31:
E= (2 20.540.5 2) =11.31
10
In this case, X 
11.31
11.31
 2.83 and Y 
 1.41
2(2)
2(4)
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
12
 4.00 and Y 
 1.00
2  0.25(4)
4(2)  4
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.
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