Lecture 8: choice under uncertainty
Lecture 8:choice under uncertainty
Content Lotteries and expected utility Risk aversion · Metric Subjective probability theory
Content • Lotteries and expected utility • Risk aversion • Metric • Subjective probability theory
Lotteries and expected utility A lottery:L=(P,…,)P≥0∑P=1 A compound lottery: L=(L 1k:1 va20∑P=1andL=(P…,P) A simplified lottery of(L, . Lk;a, ak) 1sL=ax1L1+…+ a,l or l=(P;…P) P=a,P1+…+a,PR See the fig
Lotteries and expected utility • A lottery: • A compound lottery: and . • A simplified lottery of is or See the fig 1 ( , , ) L P P = N 1 , 0 1 N i i i i P P = = 1 ( , , ) k k L P P k N = 1 1 ( , , ; , ) L L L = k k 1 , 0 1 N k i i k P = = 1 1 ( , , ; , ) L Lk k L L L = + + 1 1 k k 1 ( , ) L P P = k 1 1 k P P P n n k n = + +
Lotteries and expected utility The preference of lotteries Continuous: LL'L"e@ the set is closure {a∈[0,1]:aL+(1-a)L%L"}c[0,1 {a∈[0,1]:L"%aL+(1-a)L}c[0,1 Independence axiom aL+(1-a)L”%aL+(1-a)L”冷→L%L 连续性和不相关性
Lotteries and expected utility • The preference of lotteries: – Continuous: the set is closure – Independence axiom: L L L , , L { [0,1]: (1 ) } [0,1] + − L L L % { [0,1]: (1 ) } [0,1] + − L L L % L L L L L L + − + − (1 ) (1 ) % % 连续性和不相关性
Expected utility V N-M expected utility function U(D)=4P+…+RN Proposition 1: a utility function U: @->R is an expected utility function if and only if it's liner, that is vk L∈and(a,…a)>0.∑ar=1 We have:UC∑aL)=∑AaU(L)
Expected utility • v.N-M expected utility function: • Proposition1: a utility function is an expected utility function if and only if it’s liner, that is we have: 1 1 ( ) U L u P u P = + + N N U :L → 1 ( , ) 0, 1 k k i = k L and L 1 1 ( ) ( ) K K k k k k k k U L U L = = =
Expected utility Proposition 2: U: @- >R is the v N-M exp utility function of preference on o if and only if 3B>0 and Y, U(L)=BU(L)+r VLEO, U(L)is another vN-M expected utility function
Expected utility • Proposition 2: is the v.N-M exp. utility function of preference on if and only if , is another v.N-M expected utility function U :L → L = + 0 and , ( ) ( ) U L U L L L U L( )
Expected utility Proposition3: if the preference on g can be represented by an expected utility function, then satisfied independent axiom Proposition: (expection utility theorem) if the policymaker take a continuous and independent preference on @ then we can find a v N-M expected utility function to represent it See the fia
Expected utility • Proposition3: if the preference on can be represented by an expected utility function, then satisfied independent axiom. • Proposition4:(expection utility theorem) if the policymaker take a continuous and independent preference on , then we can find a v.N-M expected utility function to represent it. See the fig. L % L
Risk aversion A lottery with monetary payoffs continuous quantity of money x is a random variable Accumulated distribution function F:R>[0, 1 VN-M expected utility function U(F)=u(x) F(x) where u() is Bernoulli utility function u()is increasing, continuous and bounded
Risk aversion • A lottery with monetary payoffs : – continuous quantity of money is a random variable – Accumulated distribution function: – v.N-M expected utility function where is Bernoulli utility function. • is increasing, continuous and bounded. x F : [0,1] → U F u x dF x ( ) ( ) ( ) = u(.) u(.)
Risk aversion A risk aversion man: xdF(x) is as better at least as a lottery with F(x) Jenson' s inequality: u( xd F(x)2 u(x)d F(x) u()is concave or strictly concave if the man is(strictly) risk aversion See the fig
Risk aversion • A risk aversion man: is as better at least as a lottery with F(x) . • Jenson’s inequality: • u(.) is concave or strictly concave if the man is (strictly) risk aversion. See the fig. xdF x( ) u xdF x u x dF x ( ( )) ( ) ( )
Risk aversion Certainty equivalence: a risk premium c(Fu) make it indifferent with a lottery with FO. u(C(F, u)=u(x)dF(x) Probability premium: an extra probability over the impartial probability, (x,E, u) l(x)=(+x(x2E,u)(x+E)+(x-(x,E2)(x-E)
Risk aversion • Certainty equivalence: a risk premium c(F,u) make it indifferent with a lottery with F(.) . • Probability premium: an extra probability over the impartial probability, u c F u u x dF x ( ( , )) ( ) ( ) = 1 1 ( ) ( ( , , )) ( ) ( ( , , )) ( ) 2 2 u x x u u x x u u x = + + + − − ( , , ) x u