Attitudes Towards Risk 14.123 Microeconomic Theory III Muhamet Yildiz Model  C = R = wealth level  Lottery = cdf F (pdf f)  Utility function u : R→R  U(F) ≡ EF(u) ≡∫u(x)dF(x)  EF(x) ≡∫xdF(x) 1Attitudes Towards Risk DM is  risk averse if EF(u) ≤ u(EF(x)) (∀F)  strictly risk averse if EF(u)< u(EF(x)) (∀ “risky” F)  risk neutral if EF(u)= u(EF(x)) (∀F)  risk seeking if EF(u) ≥ u(EF(x)) (∀F) DM is  risk averse if u is concave  strictly risk averse if u is strictly concave  risk neutral if u is linear  risk seeking if u is convex Certainty Equivalence  CE(F) = u⁻¹(U(F))=u⁻¹(EF(u))  DM is  risk averse if CE(F) ≤ EF(x) for all F;  risk neutral if CE(F) = EF(x) for all F;  risk seeking if CE(F) ≥ EF(x) for all F.  Take DM1 and DM2 with u1 and u2.  DM1 is more risk averse than DM2  Ù u1 is more concave than u2,  Ù u1= φ◦ u2 for some concave function φ,  Ù CE1(F) ≡ u1⁻¹(EF(u1)) ≤ u2⁻¹(EF(u2)) ≡ CE2(F) 2Absolute Risk Aversion  absolute risk aversion: rA(x) = -u′′(x)/u′(x)  constant absolute risk aversion (CARA) u(x) =-e-αx  If x ~ N(μ,σ²), CE(F) = μ - ασ²/2  Fact: More risk aversion Ù higher absolute risk aversion everywhere  Fact: Decreasing absolute risk aversion (DARA) Ù ∀y>0, u2 with u2(x)≡u(x+y) is less risk averse Relative risk aversion:  relative risk aversion: rR(x) = -xu′′(x)/u′(x)  constant relative risk aversion (CRRA) u(x)=-x1-ρ/(1-ρ),  When ρ = 1, u(x) = log(x).  Fact: Decreasing relative risk aversion (DRRA) Ù ∀t>1, u2 with u2(x)≡u(tx) is less risk averse 3Application: Insurance  wealth w and a loss of $1 with probability p.  Insurance: pays $1 in case of loss costs q;  DM buys λ units of insurance.  Fact: If p = q (fair premium), then λ = 1 (full insurance). ■ Expected wealth w – p for all λ.  Fact: If DM1 buys full insurance, a more risk averse DM2 also buys full insurance.  CE2(λ) ≤ CE1(λ) ≤ CE1(1) = CE2(1). Application: 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,∞).  U(α) = ∫0 ∞ u(w+αz-α) dF(z); concave  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).  If agent with utility u1 optimally invests α1, then an agent with more risk averse u2 (same w) optimallyinvests α2 ≤ α1.  DARA ⇒ optimal α increases in w.  CARA ⇒ optimal α is constant in w.  CRRA (DRRA) ⇒ optimal α/w is constant (increasing) 4Optimal Portfolio Choice – Proof  u2=g(u1); g is concave; g’(u1(w)) = 1.  Ui(α) ≡∫ui(w+α(z-1))(z-1) dF(z)  U2’(α)- U1’(α) = ∫[u2(w+α(z-1))- u1(w+α(z-1))](z-1)dF(z) ≤ 0.  g’(u1(w+α1z-α1)) < g’(u1(w)) = 1 Ù z > 1.  u2(w+α(z-1)) < u1(w+α(z-1)) Ù z > 1.  α2 ≤ α1 Stochastic Dominance  Goal: Compare lotteries with minimal assumptions on preferences  Assume that the support of all payoff distributions is bounded. Support = [a,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. 5FSD  DEF: F first-order stochastically dominates G Ù F(x) ≤ G(x) for all x.  THM: F first-order stochastically dominates G Ù for every weakly increasing u: →, ∫u(x)dF(x) ≥∫u(x)dG(x). Proof:  “If:” for F(x*) > G(x*), define u = 1{x>x*}.  “Only if”: Assume F and G are strictly increasing and continuous on [a,b].  Define y(x) = F-1(G(x)); y(x) ≥ x for all x  ∫u(y)dF(y) = ∫u(y(x))dF(y(x)) = ∫u(y(x))dG(x) ≥∫u(x)dG(x) MPR and MLR Stochastic Orders  DEF: F dominates G in the Monotone ProbabilityRatio (MPR) sense if k(x) ≡ G(x)/F(x) is weaklydecreasing in x.  THM: MPR dominance implies FSD.  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. 6SSD  Assume: F and G has the same mean  DEF: F second-order stochastically dominates G Ù for every non-decreasing concave u, ∫u(x)dF(x) ≥∫u(x)dG(x).  DEF: G is a mean-preserving spread of F Ù y = x + ε for some x ~ F, y ~ G, and ε with E[ε|x] = 0.  THM: The following are equivalent:  F second-order stochastically dominates G.  G is a mean-preserving spread of F .  ∀t ≥ 0, ∫0 tG(x)dx ≥∫0 tF(x)dx. SSD  Example: G (dotted) is a mean-preservingspread of F (solid). 1 x0 b FG 7MIT OpenCourseWarehttp://ocw.mit.edu 14.123 Microeconomic Theory III Spring 2010 For information about citing these materials or our Terms of Use, visit: