Lecture 13: Risk Aversion and Expected Utility
Uncertainty over monetary outcomes
Let x denote a monetary outcome.
C is a subset of the real line, i.e. [a, b] f ú.
A lottery L is a cumulative distribution function F : ú 6 [0, 1].
Let f(x) be the density function associated with F(x).
The expected value of L is
Consumers’ preferences are represented by U : ‹ 6 ú.
By the expected utility theorem, there is an assignment of values
u(x) to monetary outcomes with the property that any F(A) can
be evaluated by a utility function U(A) of the form:
which we call the expected utility of F.
Note: by MWG convention,
U(A) is the vNM utility function defined over lotteries.
u(A) is the Bernoulli utility function defined over monetary
outcomes.
Risk Aversion and Utility
Definition: An individual is (weakly) risk averse if for any
lottery F(A), the degenerate lottery that places probability one on
the mean of F is (weakly) preferred to the lottery F itself.
If the individual is always indifferent between these two
lotteries, then we say the individual is risk neutral.
An individual is a risk lover if a degenerate lottery is never
preferred to the lottery F.
With a Bernoulli utility function representation of these
preferences, an individual is therefore risk averse if and only if:
for all F(A),
This is Jensen’s Inequality and is the defining property of a
concave function. Hence, risk aversion is equivalent to the
concavity of a Bernoulli utility function u(x). Therefore
strict concavity ] strict risk aversion
linearity ] risk neutrality
strict convexity ] risk loving
Certainty Equivalence
Definition: Given a Bernoulli utility function u(A), the certainty
equivalent of a lottery F(A), denoted c(F,u), is the quantity that
satisfies the following equation:
An individual would be exactly indifferent between a lottery
that placed probability one on the certainty equivalent and the
lottery F(A).
Risk Premium
Definition: Given a Bernoulli utility function u(A) and a lottery
F(A), the risk premium, denoted ñ(F,u), is the difference between
the mean of F and the certainty equivalent c(F,u):
Application: Risk Aversion and Insurance
A strictly risk-averse individual has initial wealth of w but faces
the possible loss of D dollars. This loss occurs with probability
ð.
This individual can buy insurance that costs q dollars per unit
and pays 1 dollar per unit if a loss occurs.
The individual is deciding how many units of insurance, á, she
wishes to buy. For a purchase of á units of insurance, the
individual faces the following set of monetary outcomes and the
corresponding lottery:
C = {w - áq, w - áq - D + á}
L = ((1 - ð), ð)
The expected wealth of the individual is:
EW = (1 - ð)(w - áq) + ð(w - áq - D + á)
= w - áq - ð(D - á).
The utility maximization problem, with Bernoulli utility
function u(A), is:
The FOC is:
-q(1 - ð) A uN(w - á*q) + ð(1 - q)uN(w + (1 - q)á* - D) = 0
assuming á* > 0.
Now, suppose that the price of insurance is actuarily fair, in the
sense that q = ð. Then the FOC becomes:
uN(w + (1 - q)á* - D) = uN(w - á*q)
Since uN is strictly decreasing by strict risk aversion, we must
have
w + (1 - q)á* - D = w - á*q
or equivalently
á* = D.
Proposition: If insurance offered is actuarily fair, a strictly risk
averse individual will choose full insurance.
What if insurance offered is not actuarily fair?
Measuring Risk Aversion
Local Risk Aversion
Definition: Given a twice-differentiable Bernoulli utility
function u(A), the Arrow-Pratt measure of absolute risk aversion
at x is defined as:
For two individuals, 1 and 2, with twice-differentiable, concave,
utility functions u1(A) and u2(A), respectively, person 2 is more
risk averse than person 1 at the level of income x iff
This measure allows us to compare attitudes towards risky
situations whose outcomes are absolute gains or losses from
current wealth x.
Note: Why not uO(x) as measure?
Note: Approximate relationship to ñ (for small gambles).
Global Risk Aversion
Given two twice-differentiable Bernoulli utility functions u1(A)
and u2(A), individual 2 is globally more risk averse than
individual 1 if and only if there exists a concave function ø(A)
such that
u2(x) = ø(u1(x)).
That is, u2(A) is a concave transform of u1(A).
Risk Premium and Certainty Equivalent
Consider two individuals with utility functions u1(A) and u2(A).
Individual 2 is more risk averse than individual 1 if and only if:
c(F, u2) < c(F, u1) for every lottery F(A).
Since ñ = EV - CE, equivalently individual 2 is more risk averse
than individual 1 when 2’s risk premium is higher:
ñ(F, u2) > ñ(F, u1) for every F(A).
Pratt’s Theorem:
The three previous measures of risk aversion are all equivalent,
given twice-differentiable utility functions.
Relative Risk Aversion
Definition: Given a twice-differentiable Bernoulli utility
function u(A), the coefficient of relative risk aversion at x is
defined as:
We can write it as follows:
Risk Aversion and Wealth
Definition: The Bernoulli utility function u(A) a exhibits
decreasing (constant) (increasing) absolute risk aversion if
rA (x,u) is a decreasing (constant) (increasing) function of x.
e.g. consider two different wealth levels w1 > w2.
The set of possible outcomes involves a monetary payment
x.
A person’s utility function u exhibits decreasing absolute
risk aversion (DARA) iff
rA (w1 + x, u) < rA (w2 + x, u).
Some useful specific utility functions
Consider set of utility functions with harmonic absolute risk
aversion (HARA).
Definition: A function displays HARA if the inverse of its
absolute risk aversion is linear in wealth.
Definition: Absolute risk tolerance T is the inverse of absolute
risk aversion.
T(x) = rA (x)-1 = -uN(x)/uO(x)
The HARA class of utility functions take the following spacial
form:
These functions are defined on the domain of x such that
We then have that
To ensure that uN > 0 and uO < 0, we need to have æ(1 - ã)ã-1 > 0.
The different coefficients related to the attitude toward risk are
thus equal to
and
3 Important Special Cases of HARA.
1)
Constant Absolute Risk Aversion (CARA)
rA is independent of x if ã 6 +4, with rA (x) = rA = 1/ç.
u(x) = - exp(-x/ç)/(1/ç)
(alternatively, usually represented as u(x) = -e-ëx, ë > 0)
2)
Constant Relative Risk Aversion (CRRA)
rR = ã if ç = 0. If choose æ so as to normalize uN(1) = 1,
then uN(x) = x-ã or
3)
Quadratic Utility Functions
Set ã = -1
Note: Only defined on x < ç
CARA, CRRA Utility Functions (From Gollier, 2001)
Estimating degree of risk aversion:
What is the share of your wealth that you are ready to pay to
escape the risk of gaining or losing a share á of it with equal
probability?