Question

A farmer believes there is a \(50-50\) chance that the next growing season will be abnormally rainy. His expected utility function has the form \\[ \text { expected utility }=\frac{1}{2} \ln Y_{N R}+\frac{1}{2} \ln Y_{R} \\] where \(Y_{N R}\) and \(Y_{R}\) represent the farmer's income in the states of "normal rain" and "rainy," respectively. a. Suppose the farmer must choose between two crops that promise the following income prospects: $$\begin{array}{lcc} \text { Crop } & \boldsymbol{Y}_{N R} & \boldsymbol{Y}_{R} \\ \hline \text { Wheat } & \$ 28,000 & \$ 10,000 \\ \text { Corn } & 19,000 & 15,000 \end{array}$$ Which of the crops will he plant? b. Suppose the farmer can plant half his field with each crop. Would he choose to do so? Explain your result. c. What mix of wheat and corn would provide maximum expected utility to this farmer? d. Would wheat crop insurance, available to farmers who grow only wheat, which costs \(\$ 4000\) and pays off \(\$ 8000\) in the event of a rainy growing season, cause this farmer to change what he plants?

Short answer

Answer: Based on the income prospects, the farmer should plant corn, as it has a higher expected utility (8.8392) than wheat (8.6385). However, if the farmer decides to buy wheat crop insurance, the expected utility of insured wheat increases to 9.0009, making it a better choice than corn. In this case, the farmer would choose to plant insured wheat only.

Step-by-step solution

a. Determining which crop to plant

For the farmer to choose between wheat and corn, we need to calculate the expected utility for each. We will use the given function: \[ \text { expected utility }=\frac{1}{2} \ln Y_{N R}+\frac{1}{2} \ln Y_{R} \] For wheat: \[ \text { expected utility (wheat) } = \frac{1}{2} \ln 28000 + \frac{1}{2} \ln 10000 \] For corn: \[ \text { expected utility (corn) } = \frac{1}{2} \ln 19000 + \frac{1}{2} \ln 15000 \] Calculate the values: \[ \text { expected utility (wheat) } \approx 8.6385 \] \[ \text { expected utility (corn) } \approx 8.8392 \] Since the expected utility of corn is higher, the farmer should choose to plant corn.

b. Evaluating a mix of both crops

If the farmer plants half wheat and half corn, the incomes for each weather condition would be: \[ Y_{N R} = 0.5 * 28000 + 0.5 * 19000 = 23500 \] \[ Y_{R} = 0.5 * 10000 + 0.5 * 15000 = 12500 \] Now, we calculate the expected utility for this 50-50 mix: \[ \text { expected utility (50-50 mix) } = \frac{1}{2} \ln 23500 + \frac{1}{2} \ln 12500 \approx 8.8253 \] The expected utility of the 50-50 mix (8.8253) is less than that of corn (8.8392); thus, it would not be a better choice for the farmer.

c. Determining the optimal crop mix

Let \(x\) represent the proportion of the wheat crop, and \((1-x)\) represent the proportion of corn. Then, we get: \[ Y_{N R} = x * 28000 + (1-x) * 19000 \] \[ Y_{R} = x * 10000 + (1-x) * 15000 \] The expected utility function becomes: \[ f(x) = \frac{1}{2} \ln (28000x + 19000 - 9000x) + \frac{1}{2} \ln (10000x + 5000 - 5000x) \] To find the maximum, we need to find the critical points of this function by taking its derivative and setting it to zero: \[ f'(x) = \frac{d f(x)}{d x} = \frac{19000 - 9000}{2(28000x + 19000 - 9000x)} - \frac{5000 - 5000x}{2(10000x + 5000 - 5000x )} \] Solving \(f'(x) = 0\) for \(x\); we get that: \(x \approx 0.5963\). In other words, the optimal mix consists of about 59.63% wheat and 40.37% corn.

d. The impact of crop insurance

With wheat crop insurance, a cost of \(4000 will be deducted from the income in the normal rain scenario, and the payout of \)8000 will be added in the event of a rainy season: For insured wheat: \[ Y_{N R (Insured Wheat)} = 28000 - 4000 = 24000 \] \[ Y_{R (Insured Wheat)} = 10000 + 8000 = 18000 \] Calculate the expected utility for insured wheat: \[ \text { expected utility (insured wheat) } = \frac{1}{2} \ln 24000 + \frac{1}{2} \ln 18000 \approx 9.0009 \] Since the expected utility of insured wheat (9.0009) is higher than that of corn (8.8392), the farmer would now choose to plant insured wheat only.