Question

a) A consumer derives utility from wealth according to function u(w) = ln w. He is offered
the opportunity to bet on the flip of a coin that has the probability pi of coming heads.
If he bets $x, he will have w + x if head comes up and w-x if tails comes up. Solve
for the optimal x as a function of pi. What is the optimal choice of x when pi = 1/2 ?

b) A consumer derives utility from wealth according to function u(w) = -1/w. He
is offered a gamble which gives him a wealth of w1 with probability p and w2 with
probability 1- p. What wealth would he need now to be just indifferent between
keeping his current wealth or accepting this gamble?

Answer 1

Solution:

a) Utility function: u(w) = ln w

Denoting pi by p, for ease of writing. Given the information, prayog function for the consumer is as follow:

Expected payoff or utility, M = p*ln(w + x) + (1-p)*ln (w - x)

We ought to solve for optimal x. So, optimizing (that is maximizing payoff) using the first order condition: = 0

= p/(w + x) + ((1-p)/(w - x))*(-1)

So, finding the FOC: p/(w+x) - (1-p)/(w-x) = 0

p(w-x) - (1-p)(w+x) = 0

pw - px - w - x + pw + px = 0

So, we have optimal x as a function of p, x*(p) = w(2p -1)

Then, with p = 1/2, we get optimal x* = w(2*(1/2) - 1) = 0

So, when probability of heads is 1/2, optimal choice would be not to bet anything.

b) Now, u(w) = -1/w

So, if the person has current wealth of w, his utility is (-1/w)

If the person takes up the gamble, the expected utility will be:

Expected utility, M = p*u(w1) + (1-p)*u(w2)

M = p*(-1/w1) + (1-p)*(-1/w2)

M = -p/w1 - (1-p)/w2

To be indifferent, the consumer's expected utility from gamble must equal the current wealth utility. So, we find value of w, such that

-1/w = -p/w1 - (1-p)/w2

So, on solving it we get

w = w1*w2/(w1 + p(w2 - w1))