Lecture 13: Risk Aversion and Expected Utility

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1 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]. A lottery L is a cumulative distribution function F : [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 :. By the expected utility theorem, there is an assignment of values u(x) to monetary outcomes with the property that any F( ) can be evaluated by a utility function U( ) of the form: which we call the expected utility of F. Note: by MWG convention, U( ) is the vnm utility function defined over lotteries. u( ) is the Bernoulli utility function defined over monetary outcomes.

The expected value of L is Consumers preferences are represented by U :.

2 Risk Aversion and Utility Definition: An individual is (weakly) risk averse if for any lottery F( ), 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( ), 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

An individual is a risk lover if a degenerate lottery is never preferred to the lottery F.

3 Certainty Equivalence Definition: Given a Bernoulli utility function u( ), the certainty equivalent of a lottery F( ), 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( ). Risk Premium Definition: Given a Bernoulli utility function u( ) and a lottery F( ), 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.

Risk Premium Definition: Given a Bernoulli utility function u( ) and a lottery F( ), the risk premium, denoted ñ(f,u), is the difference between the mean of F and the

4 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( ), is: The FOC is: -q(1 - ð) u (w - á*q) + ð(1 - q)u (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: u (w + (1 - q)á* - D) = u (w - á*q)

wealth of the individual is: EW = (1 - ð)(w - áq) + ð(w - áq - D + á) = w - áq - ð(d - á).

5 Since u 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( ), 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 u 1( ) and u 2( ), respectively, person 2 is more risk averse than person 1 at the level of income x iff

Measuring Risk Aversion Local Risk Aversion Definition: Given a twice-differentiable Bernoulli utility function u( ), the Arrow-Pratt measure of absolute risk

6 This measure allows us to compare attitudes towards risky situations whose outcomes are absolute gains or losses from current wealth x. Note: Why not u (x) as measure? Note: Approximate relationship to ñ (for small gambles). Global Risk Aversion Given two twice-differentiable Bernoulli utility functions u 1( ) and u 2( ), individual 2 is globally more risk averse than individual 1 if and only if there exists a concave function ø( ) such that u 2(x) = ø(u 1(x)). That is, u 2( ) is a concave transform of u 1( ). Risk Premium and Certainty Equivalent Consider two individuals with utility functions u 1( ) and u 2( ). Individual 2 is more risk averse than individual 1 if and only if: c(f, u 2) < c(f, u 1) for every lottery F( ).

Global Risk Aversion Given two twice-differentiable Bernoulli utility functions u 1( ) and u 2( ), individual 2 is globally more risk averse than individual 1 if and only if there

7 Since ñ = EV - CE, equivalently individual 2 is more risk averse than individual 1 when 2 s risk premium is higher: ñ(f, u 2) > ñ(f, u 1) for every F( ). 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( ), the coefficient of relative risk aversion at x is defined as: We can write it as follows:

Pratt s Theorem: The three previous measures of risk aversion are all equivalent, given twice-differentiable utility

8 Risk Aversion and Wealth Definition: The Bernoulli utility function u( ) a exhibits decreasing (constant) (increasing) absolute risk aversion if r (x,u) is a decreasing (constant) (increasing) function of x. A e.g. consider two different wealth levels w 1 > w 2. The set of possible outcomes involves a monetary payment x. A person s utility function u exhibits decreasing absolute risk aversion (DARA) iff r A(w 1 + x, u) < r A(w 2 + 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. -1 T(x) = r A(x) = -u (x)/u (x)

A person s utility function u exhibits decreasing absolute risk aversion (DARA) iff r A(w 1 + x, u) < r A(w 2 + x, u).

9 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 -1 To ensure that u > 0 and u < 0, we need to have æ(1 - ã)ã > 0. The different coefficients related to the attitude toward risk are thus equal to and

that -1 To ensure that u > 0 and u < 0, we need to have æ(1 - ã)ã > 0.

10 3 Important Special Cases of HARA. 1) Constant Absolute Risk Aversion (CARA) r A is independent of x if ã +, with r A(x) = r A = 1/ç. u(x) = - exp(-x/ç)/(1/ç) -ëx (alternatively, usually represented as u(x) = -e, ë > 0) 2) Constant Relative Risk Aversion (CRRA) r R = ã if ç = 0. If choose æ so as to normalize u (1) = 1, -ã then u (x) = x or 3) Quadratic Utility Functions Set ã = -1 Note: Only defined on x < ç

u(x) = - exp(-x/ç)/(1/ç) -ëx (alternatively, usually represented as u(x) = -e, ë > 0) 2) Constant

11 CARA, CRRA Utility Functions (From Gollier, 2001)

12 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?

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