Question

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) = √W. The probability of 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

The first option of not planting the crop will be chosen.

Step-by-step solution

Explanation

The expected utility under the safe option is calculated below:

$$\begin{gathered} E\left[U\right]={\left(250,000+200,000\left(1+0.05\right)\right)}^{0.5} \\ ={\left(250,000+210,000\right)}^{0.5} \\ ={\left(460,000\right)}^{0.5} \\ =678.23 \end{gathered}$$

The expected utility under the corn plantation option is calculated below:

$$\begin{gathered} E\left(U\right)=0.7{\left(250000+\left(500000-200000\right)\right)}^{0.5}+0.3{\left(250000+\left(50000-200000\right)\right)}^{0.5} \\ =0.7{\left(250000+300000\right)}^{0.5}+0.3{\left(250000-150000\right)}^{0.5} \\ =0.7{\left(550000\right)}^{0.5}+0.3{\left(100000\right)}^{0.5} \\ =\left(0.7\times 741.62\right)+\left(0.3\times 316.23\right) \\ =519.134+94.869 \\ =614 \end{gathered}$$

The expected utility under the drought-resistant option is calculated below:

$$\begin{gathered} E\left(U\right)=0.7{\left(250000+\left(500000-250000\right)\right)}^{0.5}+0.3{\left(250000+\left(350000-200000\right)\right)}^{0.5} \\ =0.7{\left(250000+250000\right)}^{0.5}+0.3{\left(250000-100000\right)}^{0.5} \\ =0.7{\left(500000\right)}^{0.5}+0.3{\left(350000\right)}^{0.5} \\ =\left(0.7\times 707.11\right)+\left(0.3\times 591.61\right) \\ =494.977+177.483 \\ =672.46 \end{gathered}$$

The higher expected utility option will be chosen; thus, the first option will be chosen, i.e., not planting the corn.