Question

Kelly is a farmer with zero wealth. She can either plant rice or cotton. If she plants cotton, Kelly earns an income of $1800 if the weather is GOOD, and $0 if the weather is BAD. If she plants rice, Kelly will earn an income of $1000 under both GOOD and BAD weather. The probability of GOOD weather is 0.7. The probability of BAD weather is 0.3. Kelly's utility function is u(c) = 31c, where cis the value of consumption. Use this information to fill out the following table for Kelly. Table 1 Plant Rice Plant Cotton Kelly's... Expected Value of Income Expected Utility Certainty Equivalent Risk Premium a. What type of risk preferences does Kelly have? Justify your answer. b. Which crop will Kelly choose to plant? Justify your answer.

Answer 1

a) Expected income = prob of GOOD *income + probability of BAD *income

Expected Utility = prob of GOOD *utility + probability of BAD * utility

For cotton:

Expected income = 0.7 *1800 + 0.3 *0 = 1260

Exepcted utility = 0.7 (3(1800)^1/2) + 0.3 (3(0)^1/2 = 94.87

Certainity equivalent (C) is the money that makes farmer indifferent between taking money and planting a crop i.e. utility of certainty equivalent should be equal to the expected utility.

3(C)^^1/2 = 94.87 => C = (94.87/3)² = 1000

Risk Premium = Expected Income - Certainity equivalent = 1260 - 1000 = 260

For Rice:

Expected income = 0.7 *1000 + 0.3 *1000 = 1000

Exepcted utility = 0.7 (3(1000)^1/2) + 0.3 (3(1000)^1/2) = 89.10

Certainity equivalent (C) is the money that makes Kelly indifferent between taking money and planting a crop i.e. utility of certainty equivalent should be equal to the expected utility.

3(C)^1/2 = 89.10 => C = (89.10/3)² = 882

Risk Premium = Expected Income - Certainity equivalent = 1000 - 882 = 118

	Plant Rice	Plant Cotton
Expected Income	1000	1260
Expected Utility	89.10	94.87
Certainity Equivalent	882	1000
Risk Premium	118	260

Risk Preferences are decided through Marginal utility of income. If marginal utility is decreasing then Kelly will be risk averse, if it would be increasing, Kelly will be risk-loving and in case of constant marginal utility, Kelly will be risk neutral

Given:

2 U = 3(0)1/2

Marginal utility is given by

MU ди де дз(с) 1/2 де

MU = -3/2(c)-1/2 < 0

Which implies a decreasing MU of income, hence, Kelly is risk averse.

Since Kelly is risk averse, She will prefer to plant rice as the risk is low in case of rice compared to Cotton, although the expected income and expected utility is high for cotton.