Lecture 8 uncetainty and time
Lecture 8:uncetainty and time
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:1=(,…,)V,P20∑P=1 A compound lottery:L=(L1…,k;a1…ak) v.ak20∑P=1 and l=(P1,…,P) A simplified lottery of (L k212 sL=a1L1+…+ a.L. or L=(P2…) P=a1P+…+akhn n 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 o {a∈[O,1]:aL+(1-a)L%”}c[0,1 {a∈[0,]:L"%aL+(1-a)L}c[0,1 Independence axiom aL+(1-a)L"%aL+(1-a)”<L%L
Lotteries and expected utility • The preference of lotteries: – Continuous: – 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 ) % %
Expected utility v N-M expected utility function U(D=41B+…+lPR Proposition 1: a utility function U: @> is an expected utility function if and only if it's liner that is vk and (a k)>0 We have:U(∑Aa)=∑4aU(4)
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 = = =