Chapter 5: Problem 8

As the owner of a family farm whose wealth is $\$ 250,000,$ you must choose between sitting this season out and investing last year's earnings $(\$ 200,000)$ in a safe money market fund paying 5.0 percent or planting summer corn. Planting costs $\$ 200,000,$ with a six-month time to harvest. If there is rain, planting summer corn will yield $\$ 500,000$ in revenues at harvest. If there is a drought, planting will yield $\$ 50,000$ in revenues. As a third choice, you can purchase AgriCorp drought-resistant summer corn at a cost of $\$ 250,000$ that will yield $\$ 500,000$ in revenues at harvest if there is rain, and $\$ 350,000$ in revenues if there is a drought. You are risk averse, and your preference for family wealth (W) is specified by the relationship $U(W)=\sqrt{W} .$ The probability of a summer drought is $0.30,$ while the probability of summer rain is 0.70 Which of the three options should you choose? Explain.

Short Answer

Expert verified

Based on the calculations, investing the earnings in a safe money market fund seems to be the best option as it has the highest utility value

Step by step solution

Calculate the expected utility for the safe money market fund

Investing in a safe money market fund, the wealth after 6 months will be $200,000*1.05=210,000$ dollars. The utility will be $U(W)=\sqrt{210,000}=458.26$

Calculate the expected utility for planting summer corn

The expected value of planting summer corn can be calculated using the formula $E[W]=P(Rain)*W(Rain)+P(Drought)*W(Drought)$. The probability of Rain is 0.7 and of Drought is 0.3, The wealth if it Rains is 500,000 - 200,000 (planting cost)= 300,000 and if it Drought is 50,000 - 200,000 (planting cost) = -150,000. Calculating the values we get $E[W]= 0.7*300,000+0.3*-150,000=195,000$. Then, the utility will be $U(W)=\sqrt{195,000}=441.68$

Calculate the expected utility for purchasing AgriCorp drought-resistant summer corn

The expected value of purchasing AgriCorp drought-resistant summer corn is again calculated using the formula $E[W]=P(Rain)*W(Rain)+P(Drought)*W(Drought)$. The wealth is now 500,000 - 250,000 (cost)= 250,000 for Rain and 350,000 - 250,000 (cost) = 100,000. Hence, $E[W] = 0.7*250,000+0.3*100,000 = 195,000$. The utility will be $ U(W)=\sqrt{195,000} = 441.68$