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 \(U(W)=\sqrt{W} .\) 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

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 \( U(W)=\sqrt{195,000} = 441.68\)

Compare utilities

Compare the calculated utilities: 458.26, 441.68 and 441.68. The highest is that of the safe money market fund, hence this option should be chosen

Key concepts

Expected Utility

When faced with uncertain outcomes, individuals often rely on the concept of _expected utility_ to make decisions. This method combines the different possible outcomes of a decision with their respective probabilities, reflecting the decision maker's preferences.

In this scenario involving a family farm, expected utility helps weigh the benefits of different options: a safe money market investment vs. various agricultural investments with differing risks. The expected utility of each choice is calculated by:

• Multiplying the utility of each outcome by its probability
• Summing these products to determine the expected utility value

The higher the expected utility, the more preferred the option under analysis.
Consideration of expected utility highlights not just potential gains or losses, but also incorporates the likelihood of those outcomes happening. Thus, it aids in arriving at a decision aligning with the individual's risk preference, which is crucial for handling uncertainty effectively.

Risk Aversion

Risk aversion describes a decision maker's tendency to prefer certainty over uncertainty, generally opting for a guaranteed, lower-return option over a riskier, higher-return alternative. In our exercise, the concept expresses itself through the utility function:

Your utility function is specified as \(U(W) = \sqrt{W}\), representing a concave relationship. This indicates that as wealth increases, additional wealth leads to smaller increases in utility, revealing an attitude of caution or risk aversion.

When considering the farm options, a risk-averse individual will likely lean towards the safe money market fund due to its guaranteed stability versus the volatile outcomes in agriculture. Despite potentially higher revenues from planting corn, the discomfort of uncertainty can sway decision-making. This preference is vividly demonstrated by favoring lower expected utility values from investments with potential losses compared to the consistent return of the safer option.
Risk aversion plays a pivotal role in decision making, especially in volatile environments like farming, where financial security is crucial.

Probability

Probability forms the backbone of decision-making under uncertainty by quantifying the likelihood of various outcomes. In this exercise, probabilities are used to assess weather conditions influencing farming revenues: a 0.70 probability of rain vs. 0.30 probability of drought.

Understanding probability enables one to manage uncertainty by assigning numerical values to potential events. In our case:

• The probability of rain, important for a successful harvest.
• The probability of drought, impacting farming revenues adversely.

This knowledge aids in calculating expected utility, where each probability influences the expected outcome significantly.
Probability is essential for forecasting and planning, contributing to more informed, calculated decision-making in uncertain situations. It helps manage and balance the risk associated with each option, essential for determining the most viable investment path.