Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 1

A farmer's tomato crop is wilting, and he must decide whether to water it. If he waters the tomatoes, or if it rains, the crop will yield \(\$ 1,000\) in profits; but if the tomatoes get no water, they will yield only \(\$ 500 .\) Operation of the farmer's irrigation system costs \(\$ 100 .\) The farmer seeks to maximize expected profits from tomato sales. a. If the farmer believes there is a 50 percent chance of rain, should he water? b. What is the maximum amount the farmer would pay to get information from an itiner ant weather forecaster who can predict rain with 100 percent accuracy? c. How would your answer to part (b) change if the forecaster were only 75 percent accurate?

Short Answer

Expert verified
Answer: The farmer should water his tomato crop with a 50% chance of rain, as the expected profit from watering ($900) is higher than the expected profit without watering ($750). The maximum amount the farmer should pay for perfectly accurate weather information is $50, while he should pay up to $37.50 for 75% accurate information.

Step by step solution

01

Calculate expected profits with and without watering

To decide whether the farmer should water his crop or not, we need to compare the expected profits in each case. 1. Calculate the expected profit with watering: - Irrigation operating cost: $100 - Profit if it rains, or irrigation is used: $1,000 - Expected profit from watering: \(1,000 - \)100 = $900 2. Calculate the expected profits without watering: - Probability of rain: 50% or 0.5 - Profit if it rains: $1,000 - Profit if there's no rain: $500 - Expected profit without watering: (0.5 * \(1,000) + (0.5 * \)500) = $750
02

Compare expected profits to make a decision

Now, we'll compare the expected profits in both scenarios: - Expected profit with watering: $900 - Expected profit without watering: $750 Since the expected profit with watering is higher, the farmer should water his crop when there's a 50% chance of rain. b. Calculate the maximum amount the farmer would pay for accurate weather forecasting
03

Determine the value of perfect information

If the farmer can obtain perfectly accurate weather forecasting information, he can make the best decision every time, either irrigating when rain won't come or not irrigating when it will. With a 50% chance of rain, the optimal decision would yield an expected profit of (0.5 * \(1,000) + (0.5 * \)900) = $950 The value of perfect information is the difference between the expected profit with perfect information and the expected profit with current information: Value of perfect information = \(950 - \)900 = $50 The farmer should be willing to pay up to $50 for perfectly accurate information from the weather forecaster. c. Determine the maximum amount the farmer would pay for 75% accurate weather forecasting
04

Calculate expected profits with and without a 75% accurate forecast

With a 75% accurate weather forecaster, the farmer can choose the optimal action in 75% of the cases. This means he will make the correct decision 75% of the time. 1. Probability of correct decision: 75% or 0.75 2. Probability of incorrect decision: 25% or 0.25 Expected profit with a 75% accurate forecast: - (0.75 * \(950) + (0.25 * \)900) = $937.50 The value of the 75% accurate information is the difference between the expected profit with the 75% accurate forecast and the expected profit with current information: Value of 75% accurate information = \(937.50 - \)900 = $37.50 The farmer should be willing to pay up to $37.50 for 75% accurate information from the weather forecaster.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that if an individual's utility-of-wealth function is convex (rather than concave, as shown in Figure 8.1 ), he or she will prefer fair gambles to income certainty and may even be willing to accept somewhat unfair gambles. Do you believe this sort of risk-taking behavior is common? What factors might tend to limit its occurrence?

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?

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.

Suppose there are two types of workers, high-ability workers and low-ability workers. Workers' wages are determined by their ability- high ability workers earn \(\$ 50,000\) per year, low ability workers earn \(\$ 30,000 .\) Firms cannot measure workers' abilities but they can observe whether a worker has a high school diploma. Workers' utility depends on the difference between their wages and the costs they incur in obtaining a diploma. a. If the cost of obtaining a high school diploma is the same for high-ability and low-ability workers, can there be a separating equilibrium in this situation in which high-ability workers get high-wage jobs and low-ability workers get low wages? b. What is the maximum amount that a high-ability worker would pay to obtain a high school diploma? Why must a diploma cost more than this for a low-ability person if having a diploma is to permit employers to identify high-ability workers?

In deciding to park in an illegal place, any individual knows that the probability of getting a ticket is \(p\) and that the fine for receiving the ticket is / Suppose that all individuals are risk averse (that is, \(U^{\prime \prime}(W)<0\), where Wis the individual's wealth) Will a proportional increase in the probability of being caught or a proportional increase in the fine be a more effective deterrent to illegal parking? \([\text {Hint}\) : Use the Taylor \(\text { series approximation }\left.U(W-f)=U(W)-f U^{\prime}(W)+-U^{\prime \prime}(W) .\right]\)
See all solutions

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free