Question

3. 10 marks total] Suppose that a farmer faces a random weather shock, which could result in a good or bad outcome. There is a 60 percent chance that the good outcome will arise, which yields $800 of consump- tion. In the event of a bad outcome, consumption will equal $200. As- sume that utility follows log form, such that U(Y) = ln(Y), where Y is consumption (a) 1 mark] Is the farmer risk averse? Why or why not? (b) 1 mark] Calculate the expected value of the random shock (c) [1 mark] Calculate the expected utility of the random shock (d) 1 mark] Calculate the utility of the expected value of the random shock (e) 1 mark] Calculate the certainty equivalent. (f) [1 mark] Calculate the cost of the risk. (g) [2 mark] Draw parts (b) - (f) in a graph 2 (h) 2 marks] Suppose that an insurance company offers a policy that will guarantee an income of $460. The premium for this policy is $115. Is the farmer willing to purchase this policy? Why or why not?

Answer 1

It's mandatory to answer only first four parts

A) as utility function is concave in shape, so agent is risk averse

dU/dY = 1/Y

d²U/d²Y = -1/Y² < 0

So concave utility function​​​​​

B) expected value = Pg*Cg + Pb*Cb

g : good state

b: bad state

EV = . 6*800 + . 4*200 = 480+80 = 560

C) EU = .6*U(800) + .4*U(200)

= . 6*ln(800) + . 4*ln(200)

= 4.01 + 2.12

= 6.13

.

d) U(EV) = U(560)

= ln(560)

= 6.33

.

e) for CE

U(CE) = EU = 6.13

ln(CE) = 6.13

CE = Antiln(6.13)

= 459.44