《高级微观经济学Advanced Microeconomics》PPT课件：Lecture 8 uncetainty and time

Expected utility Proposition 2: U: is the v N-M exp utility function of preference on if and only if 3B>0 and Y, U(L)=BU(L)+r VLEO U(L)is another V N-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 Proposition if the preference on e can be represented by an expected utility function, then satisfied independent axiom Proposition4: expection utility theorem) 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] V N-M expected utility function U(F)=u(x)dF(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 ( ( )) ( ) ( )   