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AEB 6182 Lecture I
Professor Charles Moss
1
Expected Utility
Lecture I
I.
Basic Utility
A.
A typical economic axiom is that economic agents (consumers, producers,
etc.) behave in a way that maximizes their expected utility.
The typical
formulation is:
( 29
Y
x
p
x
p
st
x
x
U
x
x
≤
+
2
2
1
1
2
1
,
,
max
2
1
where x
1
and x
2
are consumption goods and Y is monetary income.
In
decision making under risk, we are typically interested in the utility of income
U(Y).
How do these concepts relate?
B.
The linkage between these two concepts is the indirect utility function which
posits optimizing behavior by the economic agent.
Specifically, assuming an
CobbDouglas utility function the general utility maximization problem can
be rewritten as:
Y
x
p
x
p
st
x
x
x
x
≤
+
2
2
1
1
2
1
,
2
1
max
b
a
Due to the concavity of the utility function, the inequality can be replaced
with an equality.
The maximization problem can then be reformulated as a
Lagrangian:
( 29
2
2
1
1
2
1
x
p
x
p
Y
x
x
L


+
=
l
b
a
The first order conditions are then:
0
0
0
2
2
1
1
2
2
2
1
2
1
1
2
1
1
=


=
∂
∂
=

=
∂
∂
=

=
∂
∂
x
p
x
p
Y
L
p
x
x
x
x
L
p
x
x
x
x
L
l
l
b
l
a
b
a
b
a
Taking the ratio of the first two first order conditions yields
2
1
1
2
p
p
x
x
a
b
=
Substituting this result into the third first order condition yields the demand
for x
1
as a function of prices and income
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This note was uploaded on 07/15/2011 for the course AEB 6145 taught by Professor Moss during the Spring '11 term at University of Florida.
 Spring '11
 Moss

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