Expected utility Proposition2:U:②→>9 is the v. N-Mexp utility function of preference on o if and only If 3B>0 and 1, 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 o 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 @r, then we can find a v N-M expected utility function to represent it See the fig
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: - >[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( xdF(x)2 u(x)dF(x) u( is concave or strictly concave if the man is strictly risk aversion See the fia
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 ( ( )) ( ) ( )