Northwestern University
Marciano Siniscalchi
Fall 2009
Econ 3310
RISK AND RISK AVERSION
1.
Introduction
The main reason why expectedutility theory is interesting is the phenomenon of
risk aversion
.
The key idea is that people are typically unwilling to take up “fair” bets, i.e. monetary lotteries
that yield zero in expectation.
This lecture develops these ideas further, and makes them precise. Doing so uncovers important
details, such as the fact that your willingness to accept a bet may depend on your wealth.
A typical pattern we shall follow throughout this course emerges here, as well:
we begin by
formalizing an intuitive idea, risk aversion, in terms of simple but precise behavioral properties.
Then, we seek a tractable characterization of these properties. Please keep this pattern in mind.
Terminology and notation
. Henceforth, we use the abbreviation EU for “expected utility”
and DM for “decision maker”. The set
X
of prizes (the values that random variables of interest
can take) will be some subset of
R
.
Also, and crucially, when we say that a preference relation
<
on a set
F
of random variable is
an
EU preference with Bernoulli utility
u
:
R
→
R
we mean that, for every
X, Y
∈ F
,
X
<
Y
if
and only if E[
u
(
X
)]
≥
E[
u
(
Y
)].
2.
Risk Aversion Defined
Recall that, in the St. Petersburg Paradox, a person who ranks random variables according to
their expected value would be willing to pay an infinite amount of money in order to receive the
“benefits” of the bet we described. Of course, most people would be willing to pay a relatively small
sum for the privilege of participating in that lottery; in particular, an expectedutility DM with
logarithmic utility is exactly indifferent between the certain sum $2 (equivalently: the degenerate
random variable that equals $2 with probability 1) and the St. Petersburg bet.
This leads to the following general idea:
an individual is
riskaverse
if, for any r.v.
X
, she
(weakly) prefers the certain quantity E[
X
] to
X
itself.
We can restate this a bit more formally as follows. Our main object of interest is a preference
relation
<
on a set
F
of random variables. It is tempting to write something like “E[
X
]
<
X
” to
mean that the certain quantity E[
X
] is at least as good as
X
, but this would be imprecise: E[
X
]
is a number, not a random variable! So, we need the following, standard notation: the symbol
δ
x
indicates the degenerate r.v. that takes up the value
x
∈
R
with probability one. Thus, to formalize
the assumption that the DM prefers E[
X
] to
X
, we can write
δ
E[
X
]
<
X
. This works!
It is useful to notice that, since
δ
x
is a r.v., we can compute its expected utility E[
u
(
δ
x
)];
however, since the probability distribution of
δ
x
is simply (
x,
1), we have E[
u
(
δ
x
)] =
u
(
x
). Be sure
you understand why this is the case.
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 Spring '09
 Marciano
 Economics, Utility, Convex function

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