Lecture 13: Risk Aversion and Expected Utility

Lecture 13: Risk Aversion and Expected Utility

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

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:

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,

1 2

utility functions u (A) and u (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

1

Given two twice-differentiable Bernoulli utility functions u (A)

2

and u (A), individual 2 is globally more risk averse than

individual 1 if and only if there exists a concave function ø(A) such that

2 1

u (x) = ø(u (x)).

2 1

That is, u (A) is a concave transform of u (A).

Risk Premium and Certainty Equivalent

1 2

Consider two individuals with utility functions u (A) and u (A). Individual 2 is more risk averse than individual 1 if and only if:

2 1

Since ñ = EV - CE, equivalently individual 2 is more risk averse than individual 1 when 2’s risk premium is higher:

2 1

ñ(F, u ) > ñ(F, u ) 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:

Risk Aversion and Wealth

Definition: The Bernoulli utility function u(A) a exhibits decreasing (constant) (increasing) absolute risk aversion if

A

r (x,u) is a decreasing (constant) (increasing) function of x.

1 2

e.g. consider two different wealth levels w > w .

The set of possible outcomes involves a monetary payment x.

A person’s utility function u exhibits decreasing absolute risk aversion (DARA) iff

A 1 A 2

r (w + x, u) < r (w + 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.

A

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 > 0. -1

The different coefficients related to the attitude toward risk are thus equal to

3 Important Special Cases of HARA.

1) Constant Absolute Risk Aversion (CARA)

A A A

r is independent of x if ã 6 +4, with r (x) = r = 1/ç. u(x) = - exp(-x/ç)/(1/ç)

(alternatively, usually represented as u(x) = -e , ë > 0)-ëx

2) Constant Relative Risk Aversion (CRRA)

R

r = ã if ç = 0. If choose æ so as to normalize uN(1) = 1, then uN(x) = x or-ã

3) Quadratic Utility Functions Set ã = -1

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?