麻省理工MIT(微观经济学)lec03_04_Risk and risk attitudes
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14.123 Microeconomic Theory III
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Blaise Pascal and Pierre Fermat: Expected Value = ∑n≥1 (2-n)2n = ∞.
• Not a good descriptive model as people typically pay $5-$10.
Daniel Bernoulli and Gabriel Cramer (18th 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.
14.123 Lectures 3-4, Page 3
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.)
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 xn (=payment if tails come up on nth try) such that u(xn)>2n.
14.123 Lectures 3-4, Page 4
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(EF[x]) ≥ EF[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)) = EF[u(x)]. • DEF: π(F,u) = EF[x] – CE(F,u) is the risk premium of gamble F. • THM: u exhibits risk aversion iff ∀F ∈ D: EF[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. ■
• Pay x such that ln(x) = ∑n≥1 (2-n)ln(2n), which yields x = 4. • We need to extend our EU framework to X with |X| = ∞ in order to evaluate monetary gambles in interesting applications.
14.123 Lectures 3-4, Page 6
Measuring Risk Aversion
• Interpretation of the coefficient of absolute risk aversion, rA(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) = rA(x,u).
MIT 14.123 (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
14.123 Lectures 3-4, Page 2
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) = EF[u(x)]. • Axiomatizations that yield EU for lotteries over a continuum of outcomes involve stronger versions of continuity.
Measuring Risk AFra Baidu bibliotekersion
• THM: Suppose that u1 and u2 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∈, rA(x,u2) ≥ rA(x,u1). (The coefficient of absolute risk aversion is greater under u2 than u1 for all wealth levels.) 2) There exists a strictly increasing, weakly concave function g such that u2 = g(u1). (u2 is “more concave” than u1.) • DEF: u2 is more risk averse than u1 if either (1) or (2) hold.
Solve: 6.C.2, 6.C.3, 6.C.17, {6.C.19 or 6.C.11}
St Petersburg Gamble
• Flip a fair coin repeatedly until the outcome becomes “tails”. If “tails” comes up for the first time on the nth try, get $2n. • How much should you be willing to pay for this gamble?
• The coefficient of absolute risk aversion is (proportional to) the curvature of the risk premium for infinitesimal gambles.
14.123 Lectures 3-4, Page 7
14.123 Lectures 3-4, Page 5
Measuring Risk Aversion
• DEF: For a twice-differentiable vNM utility u, the Arrow-Pratt coefficient of absolute risk aversion is rA(x,u) = -u”(x)/u’(x). • rA(x,u) is scale-free: If u1 = au2 + b, then rA(x,u1) = rA(x,u2) . • Examples: If u(x) = ln(x), then u’(x) = 1/x and u”(x) = -1/x2, hence rA(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 rA(x,exp) = r . Constant Absolute Risk Aversion (CARA).
14.123 Lectures 3-4, Page 8
Proof of the Theorem
■ Since u1 and u2 are both strictly increasing, there exists a strictly increasing g such that u2(x) ≡ g(u1(x)). Indeed, g is twicedifferentiable because u1 and u2 both are. Differentiate the identity twice in x:
14.123 Microeconomic Theory III
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Blaise Pascal and Pierre Fermat: Expected Value = ∑n≥1 (2-n)2n = ∞.
• Not a good descriptive model as people typically pay $5-$10.
Daniel Bernoulli and Gabriel Cramer (18th 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.
14.123 Lectures 3-4, Page 3
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.)
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 xn (=payment if tails come up on nth try) such that u(xn)>2n.
14.123 Lectures 3-4, Page 4
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(EF[x]) ≥ EF[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)) = EF[u(x)]. • DEF: π(F,u) = EF[x] – CE(F,u) is the risk premium of gamble F. • THM: u exhibits risk aversion iff ∀F ∈ D: EF[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. ■
• Pay x such that ln(x) = ∑n≥1 (2-n)ln(2n), which yields x = 4. • We need to extend our EU framework to X with |X| = ∞ in order to evaluate monetary gambles in interesting applications.
14.123 Lectures 3-4, Page 6
Measuring Risk Aversion
• Interpretation of the coefficient of absolute risk aversion, rA(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) = rA(x,u).
MIT 14.123 (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
14.123 Lectures 3-4, Page 2
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) = EF[u(x)]. • Axiomatizations that yield EU for lotteries over a continuum of outcomes involve stronger versions of continuity.
Measuring Risk AFra Baidu bibliotekersion
• THM: Suppose that u1 and u2 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∈, rA(x,u2) ≥ rA(x,u1). (The coefficient of absolute risk aversion is greater under u2 than u1 for all wealth levels.) 2) There exists a strictly increasing, weakly concave function g such that u2 = g(u1). (u2 is “more concave” than u1.) • DEF: u2 is more risk averse than u1 if either (1) or (2) hold.
Solve: 6.C.2, 6.C.3, 6.C.17, {6.C.19 or 6.C.11}
St Petersburg Gamble
• Flip a fair coin repeatedly until the outcome becomes “tails”. If “tails” comes up for the first time on the nth try, get $2n. • How much should you be willing to pay for this gamble?
• The coefficient of absolute risk aversion is (proportional to) the curvature of the risk premium for infinitesimal gambles.
14.123 Lectures 3-4, Page 7
14.123 Lectures 3-4, Page 5
Measuring Risk Aversion
• DEF: For a twice-differentiable vNM utility u, the Arrow-Pratt coefficient of absolute risk aversion is rA(x,u) = -u”(x)/u’(x). • rA(x,u) is scale-free: If u1 = au2 + b, then rA(x,u1) = rA(x,u2) . • Examples: If u(x) = ln(x), then u’(x) = 1/x and u”(x) = -1/x2, hence rA(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 rA(x,exp) = r . Constant Absolute Risk Aversion (CARA).
14.123 Lectures 3-4, Page 8
Proof of the Theorem
■ Since u1 and u2 are both strictly increasing, there exists a strictly increasing g such that u2(x) ≡ g(u1(x)). Indeed, g is twicedifferentiable because u1 and u2 both are. Differentiate the identity twice in x: