Microeconomic Theory III Spring 2009

Transcription

1 MIT OpenCourseWare Microeconomic Theory III Spring 2009 For information about citing these materials or our Terms of Use, visit:

2 MIT (2009) by Peter Eso Lectures 3-4: Applications of EU 1. Risk and Risk Attitudes 2. Stochastic Dominance 3. Applications to Insurance and Finance Read: MWG 6.C-6.D Solve: 6.C.2, 6.C.3, 6.C.17, {6.C.19 or 6.C.11}

3 St Petersburg Gamble Flip a fair coin repeatedly until the outcome becomes tails. If tails comes up for the first time on the n th try, get $2 n. How much should you be willing to pay for this gamble? Blaise Pascal and Pierre Fermat: Expected Value = n 1 (2 -n )2 n =. Not a good descriptive model as people typically pay $5-$10. Daniel Bernoulli and Gabriel Cramer (18 th century) suggested measuring the prize on a logarithmic scale. E.g., a 1% increase in the prize may correspond to a 0.01 increase in utility. Pay x such that ln(x) = n 1 (2 -n )ln(2 n ), which yields x = 4. We need to extend our EU framework to X with X = in order to evaluate monetary gambles in interesting applications Lectures 3-4, Page 2

4 Money Lotteries Denote amounts of money by x and a monetary lottery by a cumulative distribution function F: [0,1]. Preferences are defined on the set of all distributions over : D = {F: [0,1] F weakly, right-cont., F( )=1, F(- )=0}. Expected Utility representation: v(f) = u(x) df(x) = E F [u(x)]. Axiomatizations that yield EU for lotteries over a continuum of outcomes involve stronger versions of continuity. Continuity in prizes as well as probabilities, nothing surprising. Assume u is monotone increasing, continuous, and bounded. Why bounded? Imagine playing the St Petersburg Gamble with prize x n (=payment if tails come up on n th try) such that u(x n )>2 n Lectures 3-4, Page 3

5 Risk Aversion DEF: The preference relation on D exhibits risk aversion if F D: δ x df(x) F. The agent prefers to get the expected value of a lottery for sure. DEF: is strictly risk averse if the above preference relation is strict for any non-degenerate cdf F. DEF: The function u: is concave if x,y and λ [0,1], u(λx+(1-λ)y) λu(x) + (1-λ)u(y). DEF: u is strictly concave if the inequality is strict for λ (0,1). If u is twice continuously differentiable, then concavity of u is equivalent to u (x) 0 for all x. (Strict concavity: u (x) < 0, x.) Lectures 3-4, Page 4

6 Risk Aversion THM: A preference relation on D with vnm utility function u exhibits risk aversion if and only if u is concave. Risk aversion is equivalent to u(e F [x]) E F [u(x)]. This is Jensen s inequality, a defining property of concavity of u. DEF: The certainty equivalent of lottery F is CE(F,u) such that u(ce(f,u)) = E F [u(x)]. DEF: π(f,u) = E F [x] CE(F,u) is the risk premium of gamble F. THM: u exhibits risk aversion iff F D: E F [x] CE(F,u). Equivalently, u is risk averse iff F D: π(f,u) 0. Follows from Jensen s inequality and monotonicity of u Lectures 3-4, Page 5

7 Measuring Risk Aversion DEF: For a twice-differentiable vnm utility u, the Arrow-Pratt coefficient of absolute risk aversion is r A (x,u) = -u (x)/u (x). r A (x,u) is scale-free: If u 1 = au 2 + b, then r A (x,u 1 ) = r A (x,u 2 ). Examples: If u(x) = ln(x), then u (x) = 1/x and u (x) = -1/x 2, hence r A (x,ln) = 1/x. Decreasing Absolute Risk Aversion (DARA). If u(x) = -e -rx /r, then u (x) = e -rx and u (x) = -r e -rx, hence r A (x,exp) = r. Constant Absolute Risk Aversion (CARA) Lectures 3-4, Page 6

8 Measuring Risk Aversion Interpretation of the coefficient of absolute risk aversion, r A (x,u) : Consider a gamble ± ε with 50-50% chance at initial wealth x. Let the risk premium be π(ε), i.e., u(x-π(ε)) ½[u(x+ε) + u(x-ε)]. Differentiate in ε to get -u (x-π(ε)) π (ε) ½[u (x+ε) u (x-ε)]. Differentiate both sides again in ε: u (x-π(ε)) π (ε) 2 u (x-π(ε)) π (ε) ½[u (x+ε) + u (x-ε)]. This also holds at ε=0 by continuity. Using π(0)=π (0)=0, we get -u (x) π (0) = u (x), hence π (0) = r A (x,u). The coefficient of absolute risk aversion is (proportional to) the curvature of the risk premium for infinitesimal gambles Lectures 3-4, Page 7

9 Measuring Risk Aversion THM: Suppose that u 1 and u 2 are twice-differentiable, concave, strictly increasing utility functions on representing expectedutility preferences 1 and 2, respectively. The following conditions are equivalent: 1) For all x, r A (x,u 2 ) r A (x,u 1 ). (The coefficient of absolute risk aversion is greater under u 2 than u 1 for all wealth levels.) 2) There exists a strictly increasing, weakly concave function g such that u 2 = g(u 1 ). (u 2 is more concave than u 1.) DEF: u 2 is more risk averse than u 1 if either (1) or (2) hold Lectures 3-4, Page 8

10 Proof of the Theorem Since u 1 and u 2 are both strictly increasing, there exists a strictly increasing g such that u 2 (x) g(u 1 (x)). Indeed, g is twicedifferentiable because u 1 and u 2 both are. Differentiate the identity twice in x: u 2 (x) = g (u 1 (x)) u 1 (x), u 2 (x) = g (u 1 (x)) u 1 (x) 2 + g (u 1 (x)) u 1 (x). Dividing the latter equation by the former, we get u 2 (x)/u 2 (x) = g (u 1 (x))/g (u 1 (x)) u 1 (x) + u 1 (x)/u 1 (x). Since g > 0 and u 1 (x) > 0, we have u 2 (x)/u 2 (x) u 1 (x)/u 1 (x) if and only if g (u 1 (x)) Lectures 3-4, Page 9

11 Insurance Application A consumer with vnm utility u has initial wealth w and faces a loss of $1 with probability p. Insurance that pays $1 in case of loss costs q; he buys x units. Observation 1: If p = q (fair premium), then x = 1 (full insurance). His wealth is w xq 1+ x if the loss occurs, and w xq if it does not. In expectation, it is w p + (p-q)x = w p for all x. Setting x = 1 results in ex-post wealth w p for sure. Observation 2: If an agent (u 1 ) buys full insurance, then a more risk averse agent (u 2 = g(u 1 ), g concave) also buys full insurance. u 2 (w-q) = g(u 1 (w-q)) g(pu 1 (w-1)+(1-p)u 1 (w)) > pg(u 1 (w-1)) + (1-p)g(u 1 (w)) = pu 2 (w-1) + (1-p)u 2 (w) Lectures 3-4, Page 10

12 Optimal Portfolio Choice With initial wealth w, invest α [0,w] in a risky asset that pays a return z per each $ invested; z has cdf F on [0, ). Expected utility from investing is U(α) = 0 u(w+αz-α) df(z). Observation 1: It is optimal to invest α > 0 iff E[z] > 1. U (0) = 0 u (w)(z-1) df(z) = u (w)(e[z]-1) > 0. Hence the agent is better off setting α > 0 iff E[z] > 1. Observation 2: If agent with utility u 1 optimally invests α 1, then an agent with more risk averse u 2 (same w) optimally invests α 2 < α 1. [Proof using a common technique is detailed next.] Lectures 3-4, Page 11

13 Optimal Portfolio Choice For i=1,2, the FOC of optimal investing is: U i (α i ) 0 ui (w+α i z-α i )(z-1) df(z) = 0. By u 2 =g(u 1 ), U 2 (α 1 ) = 0 g (u1 (w+α 1 z-α 1 )) u 1 (w+α 1 z-α 1 ) (z-1) df(z). Since g (.) is decreasing in z, g (u 1 (w+α 1 z-α 1 )) < g (u 1 (w)) iff z > 1. Noting that g (.) and u 1 (.) are both positive, we can rewrite U 2 (α 1 )< z<1 g (u 1 (w)) u 1 (w+α 1 z-α 1 ) (z-1) df(z) + z>1 g (u 1 (w)) u 1 (w+α 1 z-α 1 ) (z-1) df(z) = g (u 1 (w)) 0 u1 (w+α 1 z-α 1 ) (z-1) df(z) = 0. As U 2 (α 1 ) < 0, the FOC for i=2 can only hold with α 2 < α Lectures 3-4, Page 12

14 DARA and CARA Utilities DEF: u exhibits Decreasing Absolute Risk Aversion if r A (x,u) is decreasing in x. DEF: Constant Absolute Risk Aversion: r A (x,u) constant in x. Defining property of u(x) = -e -rx /r. THM: u is DARA iff there exists an increasing, concave g such that for all w 1 > w 2 and all x, u(w 2 +x) = g(u(w 1 +x)). Define u i (x) = u(w i +x) for i=1,2. Clearly, u is DARA iff r A (x,u 2 ) r A (x,u 1 ) x. Apply the Theorem on Slide #8. Comparison of utility functions of the same agent at different wealth levels is analogous to comparison of different agents Lectures 3-4, Page 13

15 DARA / CARA Portfolio Choice Problem (Slide #11): With initial wealth w, invest α [0,w] in a risky asset resulting in expected utility U(α) = 0 u(w+αz-α) df(z). Observation 2 for Optimal Portfolio Choice: If the agent s utility function is DARA, then his optimal investment in the risky asset increases in his wealth level. CARA the optimal investment is constant in wealth. In economic models, DARA is plausible, CARA is not. (IARA, Increasing Absolute Risk Aversion, is patently silly, never assume it in applied work!) Due to its tractability, CARA utility coupled with normally distributed returns is the most widely used model in finance Lectures 3-4, Page 14

16 Prudence and Precaution DEF (Kimball): An agent is prudent if an uninsurable, zero-mean risk in his future income increases his optimal level of savings. Dynamic savings problem: Periods Set t = 0,1; income w 0, w 1 + z, E[z] = 0; savings s at t=0. s to maximize v(w 0 s) + E[u(w 1 +z+s)]. No-risk optimum: v (w 0 s * ) = u (w 1 +s * ). Optimal s under risk exceeds s * iff u (w 1 +s * ) E[u (w 1 +z+s * )]. THM: Prudence E[z]=0 implies u (x) E[u (x+z)], that is, -u exhibits risk aversion (-u is concave, or u 0). DEF: Precautionary premium for gamble z (cdf F) is ψ=ψ(x,u,f): u (x-ψ) = E[u (x+z)]. Risk premium associated with -u Lectures 3-4, Page 15

17 Prudence and DARA Recall coefficient of absolute risk aversion: r A (x,u) = -u (x)/u (x). DARA: r A (x,u) decreasing in x. THM: DARA agent is prudent. -u /u decreasing iff (u u u u )/u u 0. By u > 0, the inequality can only hold with u 0. DEF: Degree of absolute prudence is p A (x,u) = -u (x)/u (x). THM: u is DARA iff p A (x,u) r A (x,u) for all x. Differentiating r A (x,u) in x yields r A (x,u)/ x = r A (x,u)[r A (x,u) p A (x,u)] Lectures 3-4, Page 16

18 Prudence and DARA Suppose we add zero-mean gamble z (cdf F) to initial wealth x. Risk premium π(x,u,f) solves u(x π(x,u,f)) E F [u(x+z)]. π(x,u,f)/ x = {u (x-π(x,u,f)) E F [u (x+z)]} / E F [u (x+z)]. THM: u is DARA iff x,f: π(x,u,f)/ x 0, or u(x π) = E F [u(x+z)] u (x π) E F [u (x+z)]. Risk increases E[u ] more than an equivalent wealth reduction. Same is true considering compensating risk premium instead of equivalent risk premium: u(x) = E F [u(x+z+π)] u (x) E F [u (x+z+π)]. Even if a DARA agent s utility is compensated for the gamble he takes, his (expected) marginal utility remains too high Lectures 3-4, Page 17

19 Relative Risk Aversion DEF: Coefficient of relative risk aversion: r R (x,u) = -xu (x)/u (x). DEF: Constant / Decreasing / Increasing Relative Risk Aversion: r R (x,u) is constant / decreasing / increasing in x. CRRA(ρ) is u(x) = x 1-ρ /(1-ρ); it becomes u(x) = ln(x) as ρ 1. THM: u is DRRA iff for all x 1 > x 2, u 1 (t) u(tx 1 ) is an increasing, concave transformation of u 2 (t) u(tx 2 ). Observation 2 for Portfolio Choice Application: If the agent s utility function is DRRA (resp. CRRA), then the optimal proportion of his wealth invested in the risky asset is increasing (resp. constant) in his wealth level Lectures 3-4, Page 18

20 Calculation with Log-Utility Job market candidate Maurice has vnm utility u(x) = ln(x). He bets 1/7 of his current wealth that he will get a top-10 job. After talking to his advisor, he bravely triples his bet. Is this a sign that his confidence in a top-10 job increased dramatically? Denote his probability of winning (getting a top-10 job) by p. He bets x to maximize pln(w+x) + (1-p)ln(w-x). From the FOC, the optimum is attained at x = (2p-1)w. Before meeting his advisor he chose x/w = 1/7, hence his winning probability was p = 4/7. After the meeting x/w = 3/7, hence p = 5/7. His (estimated) chance of winning went up by only 14% Lectures 3-4, Page 19

21 The HARA Family Harmonic Absolute Risk Aversion: The inverse of the coefficient of absolute risk aversion (called risk tolerance) is linear in x. That is, u is HARA if 1/r A (x,u)= u (x)/u (x) is linear in x. HARA utility: u(x) = (η + x/ρ) 1-ρ such that η + x/ρ > 0. Calculate r A (x,u) = 1/(η + x/ρ), so indeed, 1/r A (x,u) is linear in x. Special cases: η=0 : CRRA(ρ), as seen before. η>0, ρ : CARA(1/η). ρ = -1: quadratic u, restrict x<η. Problem: IARA (silly) Lectures 3-4, Page 20

22 Stochastic Dominance Goal: Compare payoff distributions irrespective of the decision maker s preferences; assuming only that it admits an Expected Utility representation, and perhaps risk aversion. Assume that the support of all payoff distributions is bounded. Let the support of any lottery be [a,b] with a = 0 and b <. Two main concepts: First-order Stochastic Dominance: A payoff distribution is preferred by all monotonic Expected Utility preferences. Second-order Stochastic Dominance: A payoff distribution is preferred by all risk averse EU preferences Lectures 3-4, Page 21

23 FSD DEF: Payoff distribution F first-order stochastically dominates G if, for every weakly increasing u:, u(x)df(x) u(x)dg(x). THM: F first-order stochastically dominates G if and only if F(x) G(x) for all x. Define H(x) = F(x) G(x) for all x. Only if : Suppose towards contradiction that x * : H(x * ) > 0. Define u(x) = 1 {x>x*}. Then, u(x)dh(x) = H(x * ) < 0. If : Note u is differentiable a.e., assume it is everywhere. By Integration by Parts, u(x)dh(x) = [u(x)h(x)] 0 u (x)h(x)dx. The first term is zero, hence u(x)dh(x) 0 by u 0 and H Lectures 3-4, Page 22

24 MPR and MLR Stochastic Orders DEF: F dominates G in the Monotone Probability Ratio (MPR) sense if k(x) G(x)/F(x) is weakly decreasing in x. THM: MPR dominance implies FSD. k decreasing (MPR) implies x: k(x) 1 (FSD) as k(b) = 1. DEF: F dominates G in the Monotone Likelihood Ratio (MLR) sense if l(x) G (x)/f (x) is weakly decreasing. THM: MLR dominance implies MPR dominance. This is easiest to prove using supermodularity arguments; we skip the proof. See Athey (QJE, 2002) Lectures 3-4, Page 23

25 SSD DEF: For any pair of distributions F and G with the same mean, F second-order stochastically dominates G if for every weakly increasing, concave function u: u(x)df(x) u(x)dg(x). DEF: G is a mean-preserving spread of F if Z G = Z F + ε where Z G and Z F are the random variables whose cdf s are G and F respectively, and ε is an independent zero-mean random variable. THM: For F and G with the same mean, (i) (iii) are equivalent: (i) F second-order stochastically dominates G. (ii) G is a mean-preserving spread of F. (iii) t 0, 0 t G(x)dx 0 t F(x)dx Lectures 3-4, Page 24

26 SSD Example: G (dotted) is a mean-preserving spread of F (solid). The grey areas are equal because F and G have the same mean. 1 G F 0 x x Note that t 0, 0 t G(x)dx 0 t F(x)dx. The proof of (i) (iii) will be given in Recitation Lectures 3-4, Page 25

27 Risk and Portfolio Choice Does less risk (in the FSD, SSD, whatever sense) imply more investment in a risky asset by a risk averse decision maker? The answer is not simple (e.g., FSD is not enough). THM: In the Optimal Portfolio Choice Problem, a change in the distribution of the asset s return increases investment by a riskaverse investor if either one of the following conditions holds: (i) The change is FSD and r R (w,u) < 1. (ii) The change is SSD, u DARA, IRRA with r R (w,u) < 1 at w. (iii) The change is MLR. Condition (iii) is perhaps the most useful one; try to prove it! Lectures 3-4, Page 26