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: $$\begin{array}{lcc} \text { Crop } & Y_{N R} & Y_{R} \\ \hline \text { Wheat } & \$ 28,000 & \$ 10,000 \\ \text { Corn } & \$ 19,000 & \$ 15,000 \end{array}$$ 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-which is available to farmers who grow only wheat and which costs \(\$ 4,000\) and pays off \(\$ 8,000\) in the event of a rainy growing season-cause this farmer to change what he plants?

Risk assessment in economics plays a crucial role in decision-making, particularly in environments where uncertainty is prevalent. It involves evaluating the potential outcomes of different choices and quantifying the likelihood of various scenarios. In financial contexts, risk assessment often revolves around the anticipation of returns on investments and the potential for financial loss.

For instance, a farmer deliberating between planting wheat or corn is essentially performing a risk assessment. This involves anticipating the income generated under different weather conditions, which embody the inherent uncertainty of agricultural endeavours. The farmer utilizes the expected utility theory to determine the option that provides the optimal balance between risk and reward, aiming to maximize his expected utility given the unpredictable nature of the weather.

By quantifying the potential outcomes (e.g., income from crops) and their associated probabilities, the farmer can make an informed decision that takes into account not just potential gains, but also the risks involved in each agricultural choice. As seen in the exercise, the farmer estimates the expected utility for each crop choice by calculating the natural logarithm of the expected incomes under different precipitation scenarios, representing the likelihood of 'normal rain' and 'rainy' conditions respectively.

Decision-making under uncertainty is an inevitable part of both personal life and business. When dealing with uncertainty, individuals and organizations strive to make choices that will yield the best possible outcomes based on the information available. Economic models, like the expected utility theory, provide a framework for making consistent and rational decisions in uncertain environments.

An important aspect of decision-making under uncertainty is the consideration of all possible states of nature and their associated probabilities. In the case of the farmer from our exercise, a 50-50 chance of 'normal rain' and 'rainy' conditions represents two possible states of nature with equal probabilities. The decision to plant one type of crop over another is taken after careful comparison of the expected utilities for each option.

Furthermore, by considering diversification, as in the option of planting half wheat and half corn, the farmer mitigates risk. This strategy can lead to a more stable outcome across various scenarios, as demonstrated by the higher expected utility. The concept is similar to that employed by investors diversifying their portfolio to spread risk. The farmer's decision-making process, therefore, reflects a balance between potential reward and a tolerance for risk.

The utility function is a powerful tool in economics representing a consumer’s preference structure. It assigns a numerical value to every possible choice, reflecting the relative satisfaction or desirability of each outcome to the individual. In the context of expected utility theory, the utility function is crucial for representing how risk-averse or risk-neutral a person is.

In the exercise, the farmer’s expected utility function is depicted as a natural logarithm of income, which is a common form for representing risk-averse behavior—where a person prefers a certain outcome over a risky one, even if the expected values are the same. The concept of diminishing marginal utility applies here; as income increases, the additional satisfaction gained from the next unit of income reduces.

Therefore, when the farmer considers the various crop-planting scenarios and their potential incomes, his utility function guides the decision. The logarithmic form ensures that the relative differences in income are appropriately weighted, with substantial weight given to lower incomes—a reflection of the farmer's aversion to the risk of low income during an abnormally rainy season. When comparing the utility of different mixtures of wheat and corn, this utility function quantifies the farmer’s subjective experience of wealth and helps him aim for the highest expected satisfaction.