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Chapter 8: Problem 7

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 \\[ \begin{array}{c} \mathbf{1} \\ \text { expected utility }=-\boldsymbol{I} \boldsymbol{n} \boldsymbol{Y}_{N R}+-\boldsymbol{I} \boldsymbol{n} \boldsymbol{Y}_{R} \end{array} \\] 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: $$\begin{array}{lcr} \text { Crop } & Y_{H} & Y_{R} \\ \hline \text { Wheat } & \$ 28,000 & \$ 10,000 \\ \text { Corn } & 19,000 & 15,000 \end{array}$$ 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

Expert verified
Answer: The farmer should plant wheat because it provides the highest expected utility compared to corn and wheat with insurance. The expected utility of planting wheat is 9,000, which is greater than the expected utility of planting corn (2,000) and wheat with insurance (5,000).

Step by step solution

01

Calculate the Expected Utility for Each Crop

We are given the probability of a rainy season and the incomes for each crop under normal and rainy conditions. We can use the expected utility formula to compute the expected utilities for wheat and corn: $$ E_U(\text{Wheat}) = 0.5 \times (\text{Income}_\text{Normal Rain} - \text{Income}_\text{Rainy}) $$ $$ E_U(\text{Corn}) = 0.5 \times (\text{Income}_\text{Normal Rain} - \text{Income}_\text{Rainy}) $$ Plug in the numbers and calculate the expected utility for each crop: $$ E_U(\text{Wheat})= 0.5 \times (28000 - 10000) = 0.5 \times 18000 = 9000 $$ $$ E_U(\text{Corn}) = 0.5 \times (19000 - 15000) = 0.5 \times 4000 = 2000 $$
02

Determine Optimal Crop Choice

Compare the expected utilities of planting wheat and corn: $$ E_U(\text{Wheat}) > E_U(\text{Corn}) $$ Since the expected utility of planting wheat is higher than that of corn, the farmer should plant wheat.
03

Discuss the Effect of Planting a Mix of Both Crops

The problem states that the farmer can plant half his field with each crop. However, we cannot directly determine if this would be a better option without knowing the size of the field or the yield of each crop. This information is not provided in the problem. We can only say that if the yield per acre is constant, the expected utility of planting a mix of both crops will be the same as planting all wheat.
04

Explore the Impact of Crop Insurance on the Farmer's Decision

If the farmer buys insurance and plants only wheat, his income will be as follows: - Normal Rain: \(28000 - 4000 = 24000\) - Rainy: \(10000 + 8000 - 4000 = 14000\) Now, calculate the expected utility with insurance: $$ E_U(\text{Wheat with Insurance}) = 0.5 \times (24000 - 14000) = 0.5 \times 10000 = 5000 $$ Compare the expected utilities of planting wheat without insurance, wheat with insurance, and corn: $$ E_U(\text{Wheat}) > E_U(\text{Wheat with Insurance}) > E_U(\text{Corn}) $$ The farmer's decision will not change, as planting wheat without insurance still provides the highest expected utility.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Income Prospects
Income prospects refer to the potential future earnings an individual or business might expect under different conditions. For a farmer, as shown in our textbook exercise, prospects vary depending on whether the season is typically rainy or abnormally rainy. Calculating expected utility is a critical factor for decision-making, where the farmer assesses the income from planting wheat or corn in the context of both normal and rainy conditions.

Using the provided formula and annual income possibilities for each crop, we can identify which crop offers a better income prospect. This calculation takes into account both the probability of rainy conditions and the actual monetary difference, leading to the selection of the crop with the higher expected utility—highlighting the importance of income prospects in agricultural planning.
Optimal Crop Choice
The concept of optimal crop choice revolves around maximizing income or utility based on different scenarios. In our example, the farmer must decide between growing wheat or corn, considering the possibility of a rainy season. The step-by-step solution shows how to calculate and compare the expected utilities for each crop.

By comparing the expected outputs, the farmer can make an informed decision on which crop to grow based on potential income. It's important to realize that 'optimal' doesn't always mean highest income; instead, it means the best balance between risk and reward, which the expected utility calculation helps to determine.
Effect of Insurance on Decision Making
The decision-making process can be significantly influenced when insurance is available as an option. Insurance provides a safety net against potential losses, by offering compensation in the event of unfavorable conditions, such as an abnormally rainy season affecting crop yield. In the provided exercise, the availability of wheat crop insurance alters the income prospects for the rainy season scenario.

The calculated expected utility with and without insurance helps the farmer to see the numerical impact of the insurance policy. Despite the insurance seeming like a beneficial safety net, the results indicate that the optimal choice for the farmer remains to plant wheat without insurance, as this option still retains the highest expected utility.
Probability and Decision Making
Probability plays a fundamental role in decision-making processes, especially in scenarios involving uncertainty, like agriculture. In the textbook exercise, the farmer deals with a 50-50 chance of a normal or rainy season. This probability directly affects the expected utility, which is a quantified expectation based on potential outcomes.

The steps show how probability is combined with income data to calculate expected utility for both crops. Understanding and applying probability helps individuals like our farmer assess risks and make choices that align with their preferences for uncertainty and financial outcomes, thereby integrating a scientific approach into everyday decision-making.

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Most popular questions from this chapter

An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all of the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: Strategy 1: Take all 12 eggs in one trip. Strategy 2: Take two trips with 6 in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that on the average, 6 eggs will remain unbroken after the trip home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable? c. Could utility be improved further by taking more than two trips? How would this possi bility be affected if additional trips were costly?

Suppose there is a \(50-50\) chance that a risk-averse individual with a current wealth of \(\$ 20,000\) will contact a debilitating disease and suffer a loss of \(\$ 10,000\) a. Calculate the cost of actuarially fair insurance in this situation and use a utility-of-wealth graph (such as shown in Figure 8.1 ) to show that the individual will prefer fair insurance against this loss to accepting the gamble uninsured. b. Suppose two types of insurance policies were available: (1) A fair policy covering the complete loss. (2) A fair policy covering only half of any loss incurred. Calculate the cost of the second type of policy and show that the individual will generally regard it as inferior to the first.

Blue-eyed people are more likely to lose their expensive watches than are brown-eyed people. Specifically, there is an 80 percent probability that a blue-eyed individual will lose a \(\$ 1,000\) watch during a year, but only a 20 percent probability that a brown-eyed person will. Blue-eyed and brown-eyed people are equally represented in the population. a. If an insurance company assumes blue-eyed and brown-eyed people are equally likely to buy watch-loss insurance, what will the actuarially fair insurance premium be? b. If blue-eyed and brown-eyed people have logarithmic utility-of-wealth functions and cur rent wealths of \(\$ 10,000\) each, will these individuals buy watch insurance at the premium calculated in part (a)? c. Given your results from part (b), will the insurance premiums be correctly computed? What should the premium be? What will the utility for each type of person be? d. Suppose that an insurance company charged different premiums for blue-eyed and brown-eyed people. How would these individuals' maximum utilities compare to those computed in parts (b) and (c)? (This problem is an example of adverse selection in insurance.)

In Problem \(8.5,\) Ms. Fogg was quite willing to buy insurance against a 25 percent chance of losing \(\$ 1,000\) of her cash on her around-the-world trip. Suppose that people who buy such insurance tend to become more careless with their cash and that their probability of losing \(\$ 1,000\) rises to 30 percent. What is the actuarially fair insurance premium in this situation? Will Ms. Fogg buy insurance now? (Note: This problem and Problem 9.3 illustrate moral hazard.)

In some cases individuals may care about the date at which the uncertainty they face is resolved. Suppose, for example, that an individual knows that his or her consumption will be 10 units today \((Q)\) but that tomorrow's consumption \(\left(\mathrm{C}_{2}\right)\) will be either 10 or \(2.5,\) depending on whether a coin comes up heads or tails. Suppose also that the individual's utility function has the simple Cobb-Douglas form \\[U\left(Q, C_{2}\right)=V^{\wedge} Q.\\] a. If an individual cares only about the expected value of utility, will it matter whether the coin is flipped just before day 1 or just before day \(2 ?\) Explain. More generally, suppose that the individual's expected utility depends on the timing of the coin flip. Specifically, assume that \\[\text { expected utility }=\mathbf{f}^{\wedge}\left[\left(\mathrm{fi}\left\\{17\left(\mathrm{C}, \mathrm{C}_{2}\right)\right\\}\right) \text { ) } \text { ? } |\right.\\] where \(E_{x}\) represents expectations taken at the start of day \(1, E_{2}\) represents expectations at the start of day \(2,\) and \(a\) represents a parameter that indicates timing preferences. Show that if \(a=1,\) the individual is indifferent about when the coin is flipped. Show that if \(a=2,\) the individual will prefer early resolution of the uncertainty - that is, flipping the coin at the start of day 1 Show that if \(a=.5,\) the individual will prefer later resolution of the uncertainty (flipping at the start of day 2 ). Explain your results intuitively and indicate their relevance for information theory. (Note: This problem is an illustration of "resolution seeking" and "resolution-averse" behavior. See D. M. Kreps and E. L. Porteus, "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica [January \(1978]: 185-200\).
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