Chapter 12
NAME
Uncertainty
Introduction.
In Chapter 11, you learned some tricks that allow you to
use techniques you already know for studying intertemporal choice. Here
you will learn some similar tricks, so that you can use the same methods
to study risk taking, insurance, and gambling.
One of these new tricks is similar to the trick of treating commodi-
ties at diFerent dates as diFerent commodities.
This time, we invent
new commodities, which we call
contingent commodities
.I
fe
i
th
e
ro
ftw
o
events
A
or
B
could happen, then we de±ne one contingent commodity
as
consumption if A happens
and another contingent commodity as
con-
sumption if B happens.
The second trick is to ±nd a budget constraint
that correctly speci±es the set of contingent commodity bundles that a
consumer can aFord.
This chapter presents one other new idea, and that is the notion
of von Neumann-Morgenstern utility. A consumer’s willingness to take
various gambles and his willingness to buy insurance will be determined
by how he feels about various combinations of contingent commodities.
Often it is reasonable to assume that these preferences can be expressed
by a utility function that takes the special form known as
von Neumann-
Morgenstern utility
. The assumption that utility takes this form is called
the
expected utility hypothesis
.
If there are two events, 1 and 2 with
probabilities
π
1
and
π
2
, and if the contingent consumptions are
c
1
and
c
2
, then the von Neumann-Morgenstern utility function has the special
functional form,
U
(
c
1
,c
2
)=
π
1
u
(
c
1
)+
π
2
u
(
c
2
). The consumer’s behavior
is determined by maximizing this utility function subject to his budget
constraint.
Example:
You are thinking of betting on whether the Cincinnati Reds
will make it to the World Series this year. A local gambler will bet with
you at odds of 10 to 1 against the Reds. You think the probability that
the Reds will make it to the World Series is
π
=
.
2. If you don’t bet,
you are certain to have $1,000 to spend on consumption goods. Your
behavior satis±es the expected utility hypothesis and your von Neumann-
Morgenstern utility function is
π
1
√
c
1
+
π
2
√
c
2
.
The contingent commodities are
dollars if the Reds make the World
Series
and
dollars if the Reds don’t make the World Series
.L
e
t
c
W
be
your consumption contingent on the Reds making the World Series and
c
NW
be your consumption contingent on their not making the Series.
Betting on the Reds at odds of 10 to 1 means that if you bet $
x
on the
Reds, then if the Reds make it to the Series, you make a net gain of $10
x
,
but if they don’t, you have a net loss of $
x
. Since you had $1,000 before
betting, if you bet $
x
on the Reds and they made it to the Series, you
would have
c
W
=1
,
000 + 10
x
to spend on consumption. If you bet $
x
on the Reds and they didn’t make it to the Series, you would lose $
x
,