Expected Utility: Lecture X
Charles B. Moss
September 10, 2010
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
max
x
1
,x
2
U
(
x
1
, x
2
)
s
.
t
.p
1
x
1
+
p
2
x
2
≤
Y
(1)
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 gen
eral utility maximization problem can be rewritten as
max
x
1
,x
2
x
α
1
x
β
2
s
.
t
.p
1
x
1
+
p
2
x
2
≤
Y
(2)
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
L
=
x
α
1
x
β
2
+
λ
(
Y

p
1
x
1

p
2
x
2
)
(3)
The first order conditions are then
1
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AEB 6182 Agricultural Risk Analysis and Decision Making
Professor Charles B. Moss
Lecture X
Fall 2010
∂
L
∂
x
1
=
α
x
α
1
x
β
2
x
1

λ
p
1
= 0
∂
L
∂
x
2
=
β
x
α
1
x
β
2
x
2

λ
p
2
= 0
∂
L
∂λ
=
Y

p
1
x
1

p
2
x
2
= 0
(4)
Taking the ratio of the first two first order conditions yields
x
2
=
β
α
x
1
p
1
p
2
(5)
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 Fall '08
 Weldon
 Economics, Utility, p1, Professor Charles B. Moss

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