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Updated: 01.03.2006
ECON 501:
MICRO-ECONOMICS
Lecture 2
Topics to Be Covered:
a. Utility Function and MRS in a formal way
b. Homothetic Utility Function
c. Budget Constraint
d. Maximizing Utility Functions subject to a Budget
Constraint
e. Derivation of 1st order conditions for maximizing
utility with Cobb-Douglas Utility Functions and CES
(constant elasticity of substitution)
f. Derivation of Demand Functions
ECON 501
Nicholson Chapter 3, Part 2 and Chapter 4, Part 1
UTILITY FUNCTION AND MRS IN A FORMAL WAY
Utility functions can be written as a function of a series of goods:Utility = U(X1, X2, X3,..,XN)
Where XN are all goods and services available to the consumer with some items being even
“bads”, not “goods.”
Question: what is the marginal utility if we change one of these goods by 1 unit?
MUi = ∂U/∂Xi
MUi measures a change in utility due to a change in Xi, given all other variables are kept
constant. The value of MU depends on where the consumer is at, in terms of his consumption of
the mix of all other goods.
Let’s look at a utility with two variables, and see what is the change in the total utility due to
changes in the variables:
Total differential:
dU
=
(∂U/∂X)*dX + (∂U/∂Y)*dY
=
MUX*dX + MUY*dY
Let’s set it to zero:
MUX*dX + MUY*dY = 0
definition of indifference curve
- MUY*dY = MUX*dX
 dY
dX

U
MU X
 MRS
MU Y
1
Ex:
MUX = 4 utils and
MUY = 2 utils,
then:
MRS= -(dY/dX) = 4/2 = 2
Meaning that consumer needs 2Y in order to be as happy as having 1X. MU is measured in
“utils”, which is rather a subjective index, but it can be used because actual numbers are not
important, order is important.
DIMINISHING MARGINAL UTILITY
Diminishing MU is one of the fundamental assumptions in economics, implying that the
indifference curves are convex to the origin, having the shapes, as shown below:
good Y
O
good X
Convexity of indifference curves is independent of measure of utility. Parallel changes in income
may cause different change in utility. But all three situations will result in absolutely identical
demand function. Demand function does not depend on index, but it depends on the curvature of
the indifference curve.
I3
good Y
good Y
I2
I2
I1
I1
100
I0
I3
130
I0
80
95
65
60
40
40
O
O
good X
MU of income is constant
good Y
MU of income is increasing
I3
I2
I1
70
I0
63
55
40
O
good X
good X
MU of income is decreasing
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COBB-DOUGLAS FUNCTIONS
Douglas was a professor at the University of Chicago, and then he became a US-Senator in the
50s. Cobb was a student of his, and they noticed that a lot of consumer behavior can be estimated
by function: U = XαYβ where: α + β = 1 (but not always).
Ex:
a utility function with Y (kebabs) and X (soft drinks)
U = X0.5Y0.5
MUX = 0.5X-0.5Y0.5
MUY = 0.5X0.5Y-0.5
MRS =
MU X 0.5X -0.5 Y 0.5 Y
 dY



dX U MU Y 0.5X 0.5 Y -0.5 X
This means, that in this case MRS entirely depends on the ratio of the quantities of kebabs and
soft drinks consumed.
If Y = 10, and X = 5, then:
MRS = 10/5 =2
(to maintain the same level of satisfaction, we can give up 2X for 1Y)
If Y = 5, and X = 10, then:
MRS = 5/10 =1/2
(to maintain the same level of satisfaction, we can give up 1/2X for 1Y)
INDIFFERENCE CURVE MAP
For Cobb-Douglas type of indifference curve, map looks as in the figure below. They are a
convex set. Any straight line cutting through the curve at points A and B, will have all points
INSIDE the curve.
“Quazi-concave” function – implies that its curve is “U” shaped to the origin, and we don’t have
any straight line sections in the indifference curve.
good Y
U
A
B
O
good X
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“Perfect Substitutes” Goods Utility function
The graph of the indifference curve would be a straight line, if utility function is simply:
U = αX + βY
then,
MUX = α
MRS =
MU X 
 dY


dX U MU Y 
MUY = β
and
MRS is a constant number at any point of the curve
Perfect substitutes case should NOT be modeled with Cobb-Douglas function. Cobb-Douglas
function implies that MRS is a ratio of the proportions α and β, but not a constant number.
Ex:
Different vendors of drinking water in Famagusta are perfect substitutes, but bottled water
versus vendor water are NOT perfect substitutes. Empirically, the people of Famagusta
confirm this.
Perfect Complements
L shaped indifference curves
Quantity Y
Quantity X
These preferences for goods that go together. E.g. fishing poles and fishing line.
Extra amounts of one good beyond a point adds no additional utility.
Suppose a person prefers 100 meters of fishing line with one fishing pole or 200 meters of line
with 2 fishing poles.
If the number of fishing poles are given as X and the meters of fishing line are shown as Y then
utility is given by:
Utility = u(X, Y) = min (100X, Y)
1 fishing pole and 100 meters of line yields 100 units of utility because min (100, 100) = 100.
Also 2 fishing poles and 100 meters of also yields a utility of 100 because min (200, 100) = 100.
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Only if the number of fishing poles and the amount of line are both doubled will utility increase
to 200. More generally, neither of the two goods in the equation utility = min (aX, bY) will be in
excess only if aX = bY
Hence, Y/X = a/b which is the fixed proportional relationship between the two goods that must
occur if the choices are to be at the vertices of the indifference curve.
“HOMOTHETIC PREFERENCES”
Homothetic functions – is a class of utility functions, which have property that the MRS depends
only on the ratio of X/Y, or MRS = f(X/Y), and not on the total quantities of the two goods. What
do we mean by that?
If we draw an indifference map, and:
take straight lines from origin to north-east
each indifference curve has an identical slope along a given ray from the origin
good Y
SAME slope S’’
SAME slope S’
U2
U1
SAME slope S’’’
U0
O
good X
Let’s consider a Cobb-Douglas function: U = XαYβ
MUX = αXα-1 Yβ
and
MUY = βXαYβ-1
where: α + β = 1
then:
MU X X 1Y    Y 
 dY
MRS =


  
dX U MU Y X Y  -1   X 
Since α and β are constant, the MRS will depend only on the ratio X/Y, and not on how far a
particular point is located from the origin.
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For functions, involving many variables:
Utility = U(X1, X2, X3,…,XN)= X1β1 X2β2 X3β3…XNβN
MRS =
βi  X j 
 
β j  X i 
Quality differences among similar goods
How to model quality differences among similar goods?
Each feature of quality can be assigned a variable:
Utility = U(q, a1(q), a2(q),..., aN(q))
q – quantity of the good,
Where:
aN – quality of type N of the good
Addiction
Utility also can be modeled:
Utility = U (Xt, Yt, St)
Where: X – is addictive substance, and St =

X
t 1
t 1
Past history of consumption (St) determines the current satisfaction.
Implications: discourage children from smoking
raising price will have a bigger effect over time, since measured impact
will accumulate
Second Party Preferences
This is the issue of dependency of utility of one person with well-being of other person(s). How
do we feel if another person becomes better or worse: does a person’s utility increase or decrease
as a result?
Utility = U(Xi, Yi, Uj)
Where:
If
If
Uj – is the utility of another person,
U i
0
U j
U i
0
U j
then:
altruistic behavior, person feels good if another person is made better off
neutral behavior
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U i
0
U j
If
envy behavior, person feels good if another person is made worse off
Ex:
-
-
Provision of subsidized clean water to the poor. Case of Manila water supply where
the subsidy size was about 30% of per unit cost of water. There was a substantial
financial cost of subsidizing this water project. However, this project results in the
poor being healthier and allowing them to achieve more basic needs. The “good
feelings” that the rich get from seeing the poor better-off is estimated to be 8 times
higher than the cost of subsidy required.
Case of trade off between the level of taxation and degree of social protection. Higher
social security standards (Sweden) imply that taxes must be raised in order to finance
social security. US is somewhere in the middle on this scale as compared to other
countries.
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Nicholson Ch.4: Utility Maximization and Choice
INDIVIDUAL’S BUDGET CONSTRAINT
Fundamental assumptions:
1) consumers will try to maximize utility
2) all available money will be spent
3) choose a bundle of goods, which will make consumer happy
good Y
A – not all money is spent
I
Py
B – is not the maximum, consumer can be even
more happier
D
B
C
C – is the affordable point with the maximum
utility and all money spent
U2
U1
A
U0
D – not affordable
O
good X
I
Px
Px. X + Py.Y≤ I
I
Slope of budget constraint is -
I
Py
Px

Px
Py
8
 dY
Slope of U = dX
U
and
= MRS
I
Slope of budget line =
I
PY

PX
PX
PY
therefore:
Maximum utility is reached when the two slopes are equal. i.e. MRS 
RULE:
Px
Py
All income should be spent and the MRS should equal the relative prices of the
goods. This is called the first order conditions for utility maximization.
Ex:
Suppose:
MRS = MUX/MUY = 3
(means that at this point, satisfaction from
consumption of 1 more unit of X is 3 times higher
than consumption of 1 more unit of Y)
and assume:
PX = 8
PY = 2
and
PX/PY = 8/2 = 4
Question:
Answer:
therefore:
(since the prices are fixed, this means that X costs
you 4 times higher than Y)
How can the individual maximize his utility?
X is too expensive at the current level of consumption, and the consumer will
reduce his consumption of X until his MUX/MUY becomes equal to 4, which is the
current ratio of the prices (PX/PY).
If a consumer does not like a good offered at a certain price, he is free to choose not to consume
it. However, if the price of that good falls, then it is likely that some of this good will be
consumed, and as the price falls further – even more will be demanded.
Ex:
Mercedes-Benz Demand for an individual
U0 – initial preferences of the consumer (corner solution)
Y – units of Mercedes -Benz
and
X – units of all other goods
PY – price of Mercedes-Benz
and
PX – prices of all other goods
I – income is fixed
Assume:
Indifference Map
good Y
U0
Demand Curve
Price Y
U1
PY1+30%VAT
I/PY1
P Y0
I/PY0
P Y1
Y1
I/PY1+30%VAT
good X
O
Quantity Y
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Y0
X1
X0
O
Y1=1
Y0
In the initial position, the price of Y is too high and none is demanded, i.e. corner-solution. If
there is a recession, then it is likely that the prices of luxury goods will fall, meaning that P Y0 will
decrease to PY1. If the drop in the price is big enough, some of the good Y might be demanded.
However, if excise tax on the cars is raised to 30%, that will raise the effective price of Y to PY2
which is, in fact, higher that the initial price PY0. Again, a corner-solution is reached and none of
the good is consumed.
FIRST-ORDER CONDITIONS FOR UTILITY MAXIMIZATION
We need to put our analysis into algebra, which is helpful.
Let’s maximize: Max
Utility = U(X1, X2, X3, …, XN)
Subject to a budget constraint: I = P1*X1 + P2*X2 + P3*X3+…+ PN*XN)
[I – (P1*X1 + P2*X2 + P3*X3+…+ PN*XN)] = 0
which can be re-written as
Setting up the Lagrangian expression:
L = U(X1, X2, X3, ..., XN) + λ[I – (P1*X1 + P2*X2 + P3*X3+...+PN*XN)]
Then, let’s set all partial derivatives to zero and find solution, satisfying the constraint:
∂L/∂X1 = ∂U/∂X1 – λ P1 = 0……………(1)
∂L/∂X2 = ∂U/∂X2 – λ P2 = 0……………(2)
∂L/∂X3 = ∂U/∂X3 – λ P3 = 0
.
.
.
.
.
.
.
.
(n+1) model
n – is number of goods
∂L/∂XN = ∂U/∂XN – λPN= 0
∂L/∂λ = I – P1*X1 – P2*X2 – P3*X3– ... – PN*XN = 0
The equilibrium condition for all equations is necessary for a maximum. The first order condition
for X1 and X2 can be written as:
U
Dividing equation ½
U
X1

X 2
λP1 P1
MU X1

 MRS 
λP2 P2
MU X2
then:
U
From (1)
∂U/∂X1 = λP1
λ=
X1 MU X1

P1
P1
and
U
From (2)
∂U/∂X2 = λP2
λ=
X 2 MU X2

P2
P2
10
therefore:
MU X1
MU X2 MU XN
 

P1
P2
PN
This finding means that the marginal satisfaction from the last component of the basket of goods
consumed must be equal. In other words, each good consumed has the same marginal benefit per
unit of cost.
Normally, prices are fixed, but it could be useful to turn our result around and solve it for price:
U
U
X1 MU X1
X 2 MU X2
and
P1 

P2 





This means that given the fixed nature of prices, our marginal satisfaction is proportional to the
prices of goods consumed. Thus, if P1=2P2, then it means that MUX1 = 2MUX2.
“Corner Solution” Situation
If we have a corner solution, then one good is not consumed and the Lagrangian expression will
also change:
∂L/∂Xi = ∂U/∂Xi – λPi ≤ 0
if
(i = 1, 2, 3, ..., N)
∂L/∂Xi = ∂U/∂Xi – λPi < 0
then Pi > MUi/λ
and
Xi (consumed) =0
Here, λ represents the opportunity cost of money, and if the price is higher than the marginal
satisfaction from the good divided by value of money, then this good is not consumed.
UTILITY MAXIMIZATION WITH COBB–DOUGLAS FUNCTIONS
Properties of Cobb-Douglas functions are important to understand and we must be able to derive
them.
UC-D(X,Y) = XαYβ
assume α + β = 1
Subject to a budget constraint: I = PX*X + PY*Y
I – PX*X – PY*Y = 0
which can be re-written as
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
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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
α and β are constants; PX and PY are also constants;
only X and Y are variables (MRS depends only on Y/X)
or
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
(5)

PX X
PX X = 0
therefore:
X = α (I/PX)
Which can be interpreted in two ways:
-
X is the demand function and it depends on constant α, as well as the income of the
individual with given market price of good X.
If we transform (5) into form PXX = αI which tells us that the amount of expenditure
on good X at any time will be proportional of its share α in the total income.
Similarly PYY=
(1   )

PXX and substitute into (3)  Y   (
I
)
PY
Ex:
Assume:
X (soft drinks)
PX = 0.25
α =0.5
and
and
and
Y (kebabs)
PY = 1.00
β = 0.5
and
I = 2.00
UC-D(X,Y) = X0.5Y0.5
Subject to a budget constraint: 2 = 0.25X + 1Y
2 – 0.25X + 1Y = 0
which can be re-written as
Setting up the Lagrangian expression:
L = U(X0.5Y0.5) + λ(2 – 0.25X – 1Y)
Then, let’s set all partial derivatives to zero and find solution, satisfying the constraint:
∂L/∂X = 0.5X-0.5Y0.5 – λPX = 0
12
∂L/∂Y = 0.5X0.5Y-0.5 – λPY = 0
∂L/∂λ = 2 – 0.25X – 1Y = 0
0.5 X 0.5Y 0.5  0.25

1
0.5 X 0.5Y 0.5
then
0.5 X 0.5Y 0.5 0.25
*

0.5 X 0.5Y 0.5
1
then MRS=
Y 0.25

X
1
X = αI/PX = 0.5*2/0.25 = 4
Y = βI/PY = 0.5*2/1 = 1
UC-D(X,Y) = 40.510.5 = 2.00
Thus:
Utility
Also:
αXα-1Yβ – λPX = 0
= Income
then:
λ = (αXα-1Yβ)/PX = [0.5(4^-0.5)*(1^0.5)]/0.25 = [(0.5/2)*1]/0.25 = 0.25/0.25 = 1
MU X MU Y

meaning that there is the same level of marginal
PX
PY
satisfaction from consumption of either X or Y per unit of cost eg. Per $ when utility is at the
maximum.
Remember that  
Let’s change income available to the consumer by 0.1, so that becomes I = 2.1.
Since
λ=1
X = αI/PX = 0.5*2.1/0.25 = 4.2
Y = βI/PY = 0.5*2.1/1 = 1.05
Question:
Answer:
Does a change in the price of X affect the quantity of Y demanded?
NO. Simply because PYY/I=β is constant.
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