Q$1.$ There is a risk$-$averse individual with the utility function $u(w)=lnw$ and endowment

$$40,000.$ The probability of getting an accident is $\frac{1}{4}$ and the loss is $10,000.$ Let's call the

gamble faced by the individual Gamble A$.$

A $($insurance$)$ company provides two contracts, GC $($good day contract$)$ and BC $($bad

day contract$)$ at prices $p_g$ per unit paying $$1$ dollar if there is no accident and $p_b$ per unit

paying $$1$ dollar if there is an accident.

a$.$ Represent Gamble A in a diagram where the good $($bad$)$ day wealth is represented on

a horizontal $($vertical$)$ axis. Find the expected utility of Gamble A and draw the indifference

curve passing through Gamble A$.$

b$.$ Given the fair prices $(p_g,p_b)=(\frac{3}{4},\frac{1}{4}),$ write down the budget line showing the possible

choices available to the individual and represent the budget line in the diagram.

c$.$ Would the individual purchase GC and$/$or BC$?$ To answer the question, write down

the maximization problem for the individual, and solve for the individual's optimal choice.

How many units of GC and$/$or BC she would purchase or sell?

In the same diagram, represent the individual's optimal choice. Find the Expected utility

at the optimal choice and draw the indifference curve passing through the choice.