Question

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

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

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

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

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

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.

Key concepts

Probability in Decision Making

When making decisions, especially under uncertain conditions, probability can be extremely helpful. Let's look at the case of the farmer's watering decision. The probability of rain is a key factor. In this scenario, a 50% chance of rain was given. Probability helps us determine how likely it is that a particular outcome will occur.
By understanding these chances, the farmer can better assess the potential profits or losses associated with each decision. Here, the probability helps the farmer weigh the profit from watering versus not watering. If the probability of rain were higher or lower, the decision might change.
Understanding probability allows individuals to make informed decisions by evaluating the risk and potential rewards. Hence, using probability let's the farmer improve his decision-making by considering all possible outcomes and their likelihood. Some essential elements to consider include:

• Probability of a favorable outcome (rain leading to more profit)
• Probability of unfavorable outcomes (no rain leading to lesser profit)
• Using probability to determine expected value, which aids in decision making.

Cost-Benefit Analysis

Cost-Benefit Analysis (CBA) is a method used to weigh the benefits and costs of a particular decision. For the farmer, watering the crop versus not watering it requires a CBA. The benefits of watering are higher profits if conditions are right, while the cost is the operational expense of the irrigation system, which is $100.
In this situation, performing a cost-benefit analysis involves calculating the expected profits in dollars for each scenario. With watering, the potential for earning $1,000 minus the watering cost results in an expected profit of $900. Without watering, the expected profit, considering probability, is $750.
This comparison allows the farmer to see that watering provides a larger net benefit (or profit) of $150 more than not watering. CBA is a valuable tool for decision-making as it quantifies the potential benefits and costs, assisting in making choices that maximize profit or value.
The core steps of a Cost-Benefit Analysis are:

• Identify all potential costs and benefits.
• Quantify them in economic terms.
• Compare the total benefits and costs to determine the most profitable option.

Value of Information

Knowing the "Value of Information" is crucial in making smart choices. But what does it mean? Information has value if it can change the decision you make to achieve better outcomes. For the farmer, knowing the weather forecast with certainty has value. This value is the improvement in expected profits by using the additional information.
For perfect information about rain, the farmer finds out that the maximum value he can get is $950. This means if he had perfect weather prediction, he could potentially increase his profit by $50 compared to no extra information ($900). Thus, he should be willing to pay up to $50 for exact forecasts.
With a 75% accurate forecast, the expected additional profit is $937.50, which equates to a $37.50 increase. This tells the farmer the maximum amount he should be willing to pay for the forecasts considering the information's accuracy.
The calculation for the information value involves:

• Assess the benefits of having the information.
• Evaluate decisions that will change due to new information.
• Compute the difference in outcomes with and without information.
• Keep in mind that more accurate information often holds more value because it reduces uncertainty.