文档介绍：Lecture 8:uncetainty and time
Content
Lotteries and expected utility
Risk aversion
Metric
Subjective probability theory
Lotteries and expected utility
A lottery:
pound lottery:
and .
A simplified lottery of is or
See the fig
Lotteries and expected utility
The preference of lotteries:
Continuous:
Independence axiom:
Expected utility
-M expected utility function:
Proposition1: a utility function is an expected utility function if and only if it’s liner, that is we have:
Expected utility
Proposition 2: is the -M exp. utility function of preference on if and only if
, is another -M expected utility function
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 -M expected utility function to represent it. See the fig.
Risk aversion
A lottery with ary payoffs :
continuous quantity of money is a random variable
Accumulated distribution function:
-M expected utility function where is Bernoulli utility function.
is increasing, continuous and bounded.
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.
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,