Question

Calwlate Expected utility (ECU (w) and the utility of expected value (CEV) for a) U (w)=51n(w), state the relationship and classity the risk attitude; b) u (w)=562, state the relationship and classify to risk attitude, c) U (w) = 250-w, state the relationship and classity & risk attitude, A gamble bassed on fair coin toss which pays $250 if the coin lands and $20 it the coin lands tail (fair coin toss i e. probabity of heads is 50%= probability of tails is 50%)

Answer 1

The gamble pays $250 with probability 0.5 and pays $20 with probability 0.5.

Here expected wealth will be E(u) = summ( piwi) where pi is the probability of the respective state ( heads or tails) and wi is the corresponding wealth in that state.

E(w) = 0.5 (250) + 0.5 (20) = 125 + 10 = 135

Now, to calculate Expected utility, we do: E (U(w)) = (summ(piU(wi))) where U(wi) is the utility of wealth corresponding to a state.

And to calculate utility of expected wealth { U( E(w))} we just put the value of expected wealth in the expression for Utility.

A) U = 5lnw

E(U(w)) = 0.5( 5ln(250) ) + 0.5( 5ln(20))= 13.8 + 7.49 = 21.29

U(E(w)) = 5ln(135) = 24.53

Here, expected utility from gamble is lesser than utility of expected wealth . This means that the person is risk averse as per the definition.

B) U = 5(w)2

E(U(w)) = 0.5(5(250)^2) + 0.5(5(20)^2)= 156,250+1000 = 157250.

U(E(w)) = 5( 135)^2 = 36,450

Since expected utility is greater than utility of expected wealth . As per definition, person is a risk lover.

C) U = 250 - w

Now, E( U( w)) = 0.5( 250-250) + 0.5( 250-20)

= 117.45

U( E(w)) = 250-135 = 115

Here utility from expected wealth is lesser than expected utility of wealth. Therefore person is risk loving.