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Utility Maximization
Continued
July 5, 2005
Graphical Understanding

Normal Indifference
Curves
Downward Slope with
bend toward origin
Graphical
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Non-normal
Indifference Curves
Y & X Perfect Substitutes
Graphical
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Non-normal
Only X Yields Utility
Graphical
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Non-normal
X & & are perfect
complementary goods
Calculus caution
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When dealing with non-normal utility
functions the utility maximizing FOC
that MRS = Px/Py will not hold
Then you would use other techniques,
graphical or numerical, to check for
corner solution.
Cobb-Douglas
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Saturday Session we know that if
U(X,Y) = XaY(1-a) then X* = am/Px
m: income or budget (I)
Px: price of X
a: share of income devoted to X
Similarly for Y
Cobb-Douglas
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How is the demand for X related to the
price of X?
How is the demand for X related to
income?
How is the demand for X related to the
price of Y?
CES
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Example U(x,y) = (x.5+y.5)2
CES Demand
Eg: Y = IPx/Py(1/(Px+Py))
Let’s derive this in class
CES Demand | Px=5
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I=100 & I = 150
I=150
I=100
CES | I = 100
Px=10
Px=5
For CES Demand
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If the price of X goes up and the
demand for Y goes up, how are X and Y
related?
On exam could you show how the
demand for Y changes as the price of X
changes?
dY/dPx
When a price changes
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Aside: when all prices change (including
income) we should expect no real
change. Homogeneous of degree zero.
When one prices changes there is an
income effect and a substitution effect
of the price change.
Changes in income
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When income increases demand usually
increase, this defines a normal good.
∂X/∂I > 0
If income increases and demand
decreases, this defines an inferior good.
Normal goods
As income increase (decreases) the
demand for X increase (decreases)
Inferior good
As income increases the demand
for X decreases – so X is called
an inferior good
A change in Px
Here the price of X changes…the
budget line rotates about the
vertical intercept, m/Py.
The change in Px
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The change in the price of X yields two
points on the Marshallian or ordinary
demand function.
Almost always when Px increase the
quantity demand of X decreases and
vice versa.
So ∂X/∂Px < 0
But here, ∂X/∂Px > 0
This time the Marshallian or ordinary
demand function will have a positive
instead of a negative slope. Note that
this is similar to working with an
inferior good.
Decomposition

We want to be able to decompose the
effect of a change in price
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The income effect
The substitution effect
We also will explore Giffen’s paradox –
for goods exhibiting positively sloping
Marshallian demand functions.
Decomposition

There are two demand functions
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The Marshallian, or ordinary, demand
function.
The Hicksian, or income compensated
demand function.
Compensated Demand
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A compensated demand function is
designed to isolate the substitution
effect of a price change.
It isolates this effect by holding utility
constant.
X* = hx(Px, Py, U)
X = dx(Px, Py, I)
The indirect utility function
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When we solve the consumer
optimization problem, we arrive at
optimal values of X and Y | I, Px, and Py.
When we substitute these values of X
and Y into the utility function, we obtain
the indirect utility function.
The indirect utility function
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This function is called a value function.
It results from an optimization problem
and tells us the highest level of utility
than the consumer can reach.
For example if U = X1/2Y1/2 we know
V = (.5I/Px).5(.5I/Py).5 = .5I/Px.5Py.5
Indirect Utility
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V = 1/2I / (Px1/2Py1/2)
or
I = 2VPx1/2Py1/2
This represents the amount of income
required to achieve a level of utility, V,
which is the highest level of utility that
can be obtained.
I = 2VPx1/2Py1/2
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Let’s derive the expenditure function,
which is the “dual” of the utility max
problem.
We will see the minimum level of
expenditure required to reach a given
level of utility.
Minimize
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We want to minimize
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Subject to the utility constraint
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PxX + PyY
U = X1/2Y1/2
So we form
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L = PxX + PyY + λ(U- X1/2Y1/2)
Minimize Continued
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Let’s do this in class…
We will find
E = 2UPx1/2Py1/2
In other words the least amount of
money that is required to reach U is the
same as the highest level of U that can
be reached given I.
Hicksian Demand
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The compensated demand function is
obtained by taking the derivative of the
expenditure function wrt Px
∂E/∂Px = U(Py/Px)1/2
Let’s look at some simple examples
Ordinary & Compensated
In this example our utility function is: U = X.5Y.5. We
change the price of X from 5 to 10.
State
Px
Py
m
Mx
My
U
Hx
1
5
4
100
10
12.5
11.18033989
10
2
10
4
100
5
12.5
7.90569415
7.071067812