This preview shows pages 1–3. Sign up to view the full content.
Econ 604 Advanced Microeconomics
Davis
Spring 2005
16 February 2006
Reading.
Chapter 5 (pp. 116130) for today
Chapter 5 (pp. 130144) for next
time
Problems:
To collect:
Ch. 4:
4.2, 4.4, 4.6, 4.7
4.9
N
ext time: Ch. 5
5.1 5.2 5.4
Observe:
5.1 and 5.2 don’t fit into the
standard Lagrangian setup, because, in each case,
the products are perfect substitutes.
Below a given price level, the consumer will devote all income to only one product 5.1 is
a case of perfect substitutes. 5.2 pertains to perfect complements.
Lecture #5
REVIEW
IV. Utility Maximization and Choice
A. An Introductory Illustration. The two good case. (This is largely a graphical
representation)
1. The Budget Constraint
2. First Order Conditions for a Maximum
P
x
/P
y
=
U
x
/U
y
, which implies
U
x
/P
x
=
U
y
/P
y
Intuitively: Purchase goods until the MU per dollar spent on each good is the
same.
3. Second Order Conditions for a Maximum.
The intersection of first
order conditions will be a maximum if the quasiconvexity conditions are satisfied (that
is, for any order pairs on an indifference curve
U
, (x
o
, y
o
), (x
1
, y
1
), it is the case that
U(
α
x
o
+ (1
)x
1,
y
o
+ (1
)y
1
)>
U
This condition will be satisfied if the quasiconcavity condition in 2.107 is met.
It will
alternatively be satisfied if d
2
Y/dX
2
.
>0
Example, Suppose U* = XY.
Differentiating,
dU = YdX + XdY.
On a level curve, this expression equals zero, so
dY/dX = Y/X
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
What if we just took the derivative of the first order condition w.r.t. X?
That is,
d
2
Y/dX
2
=
Y/X
2
> 0.
Does this imply convexity in XY space?
No!
We must recognize that Y is
implicitly a function of X.
Rather, we have two (ultimately equivalent ways) to
check for the concavity of the indifference curve.
One is to totally differentiate U
twice, subject to the constraint that utility is constant.
We developed the
sufficient condition for concavity of the constrained utility function in equation
2.107
[U
2
2
U
11
2 U
1
U
2
U
12
+ U
1
2
U
22
]/ U
2
2
<0
For U = XY,
U
1
= Y,
U
2
= X,U
12
=
1
U
11
= 0,
U
22
= 0,
Inserting,
2YX/X
3
= 2Y/X
2
<0
(Note the sign is reversed relative to the standard
second order condition.
This is due to the way 2.107 is set up)
The alternative approach is to solve the expression YX = U directly for Y and
then differentiate twice.
For example, let U = U* a constant.
Then
Y = U*/X
dY/dX = U*/X
2
(with U* =XY,
dY/dX = Y/X)
d
2
Y/dX
2
=
2U*/X
3
=
2Y/X
2
<0
4. Corner Solutions
B. The nGood Case
1. First Order Conditions.
For n goods, an optimizing consumer purchases all products so that the MU per
dollar spent is equal for all goods.
2. Implications of First Order Conditions
3. Interpreting the LaGrangian Multiplier
C. Indirect Utility Function
We can express U(x
1
, x
2
,….x
n
) +
λ
(I –p
1
x
1
 … p
n
x
n
) as
U(x
1
(p
1
…p
n
,I),……. x
n
(p
1
…p
n
,I)
or
V((p
1
…p
n
,I)
The advantage of this is that indirect utility is expressed in terms of observables.
Thus, given, of course our assumptions about the functional form of utility) we can talk
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 02/23/2011 for the course ECON 604 taught by Professor Littleton during the Spring '10 term at Harvard.
 Spring '10
 Littleton
 Microeconomics, Econometrics

Click to edit the document details