Question

Suppose that Natasha's utility function is given by \(u(I)=\sqrt{10 I},\) where \(I\) represents annual income in thousands of dollars. a. Is Natasha risk loving, risk neutral, or risk averse? Explain. b. Suppose that Natasha is currently earning an income of \(\$ 40,000(I=40)\) and can earn that income next year with certainty. She is offered a chance to take a new job that offers a .6 probability of earning \(\$ 44,000\) and a .4 probability of earning \(\$ 33,000\) Should she take the new job? c. \(\operatorname{In}(\mathrm{b}),\) would Natasha be willing to buy insurance to protect against the variable income associated with the new job? If so, how much would she be willing to pay for that insurance? (Hint: What is the risk premium?)

Short answer

Natasha is risk averse. She should take the new job offer because the expected utility of the new job is higher than her current job utility. Furthermore, she would like to buy an insurance against the new job's income risk and is willing to pay up to $800 for such insurance.

Step-by-step solution

Risk Attitude Analysis

To determine if Natasha is risk averse, risk neutral or risk loving, inspect the second derivative of her utility function. A risk averse individual has a decreasing marginal utility, hence the second derivative of their utility function will be negative. Since \(\frac{d^2}{dI^2}(\sqrt{10I}) = -\frac{5}{4I^{3/2}} < 0\) for all \(I > 0\), Natasha is risk averse.

Expected Utility Comparison

To determine if Natasha should take the new job, compare the expected utility of the new job with the utility of her current job. The expected utility of the new job is the sum of the utilities of possible outcomes each multiplied by their respective probabilities, which comes out to be \(0.6\sqrt{10(44)} + 0.4\sqrt{10(33)} \approx 28.2\). The utility of her current job is \(\sqrt{10(40)} = 20\). As the expected utility of the new job is higher than her current job, she should take the new job.

Insurance Decision & Risk Premium Calculation

To determine if Natasha would like to buy an insurance against the variability of the income from the new job, calculate her risk premium. Risk premium is the amount by which the expected income of the risky job exceeds the certain income Natasha is willing to accept instead. This certain income ([CI]) satisfies the equation: \( \sqrt{10[CI]} = 0.6\sqrt{10(44)} + 0.4\sqrt{10(33)} \). Solving this, we get CI = 39.2. Hence, the risk premium is \(40 - 39.2 = 0.8\). Natasha is willing to pay up to $800 for insurance against the income risk.

Key concepts

Utility Function

In microeconomics, understanding how individuals make decisions under uncertainty is crucial, and the utility function is a foundational concept in this context. A utility function represents a person's preference for different outcomes, typically associated with income or wealth. In our example, Natasha's utility function is given by u(I) = \(\sqrt{10I}\), where I represents her annual income in thousands of dollars.

This mathematical expression is not just a formula; it tells us how Natasha values each additional dollar. The square root implies diminishing marginal utility - the more she has, the less additional value she places on each extra dollar. By assessing the shape of the utility function, we can also infer risk attitudes, a topic we'll cover in the following section.

Risk Aversion

Risk aversion is a central theme in the study of economic behavior, especially when it comes to financial decisions under uncertain conditions. Individuals who are risk-averse prefer a sure thing over a gamble with the same expected value. It's by scrutinizing the second derivative of the utility function that we can divine one's attitude towards risk.

For Natasha, her utility function's second derivative is negative (\(\frac{d^2}{dI^2}(\sqrt{10I}) = -\frac{5}{4I^{3/2}}\)), confirming she is risk averse. This means she experiences a stronger emotional impact from a loss than pleasure from a gain of the same size, and it drives her choices when faced with uncertain income scenarios.

Expected Utility

The concept of expected utility is key when comparing uncertain prospects. It's the sum of the utilities of all possible outcomes, each weighted by their probability. In short, it quantifies the desirability of risky jobs, investments, or gambles.

In our case, the expected utility approach helps Natasha decide whether to change jobs by comparing her utility from a certain income to one from a risky job. The calculation \(0.6\sqrt{10(44)} + 0.4\sqrt{10(33)}\) compares the two utilities to see which job offers her the greater expected utility, and therefore which one she should prefer. The result suggests Natasha should indeed take the new job because it maximizes her expected utility, aligning with her personal preferences described by her utility function.

Insurance and Risk Premium

When faced with risky economic decisions, insurance can play a pivotal role for risk-averse individuals like Natasha. Insurance allows them to exchange a risky situation for a certain one, for a price. The risk premium is the maximum amount a risk-averse person is willing to pay to avoid the risk.

To calculate this for Natasha, we determine the certain income equivalent to the expected utility of the risky job. We find Natasha would need an assured income of \(39,200 to achieve the same utility as the risky job. However, her current salary is \)40,000. Thus, she would be willing to pay up to $800 (the risk premium) for insurance to guarantee her current level of utility. This presents a clear value for the insurance cost that Natasha, or any risk-averse individual, would find acceptable to protect against income fluctuations.