Question

Suppose that Natasha’s utility function is given by u(I) = √110I, 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. In (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

a. Natasha is risk-averse since double differentiation of the utility function is less than zero.

b. Natasha should not take the new job.

c. Yes, she would be willing to buy insurance. She would be willing to pay $198.

Step-by-step solution

Explanation for part (a)

Whether Natasha is risk-loving, risk-neutral, or risk-averse can be found from her utility function; the condition will be:

$$\begin{gathered} \mathrm{Risk}\mathrm{lover}: \\ U\text{'}\left(I\right)>0,U"\left(I\right)>0 \\ RiskNeutral: \\ U\text{'}\left(I\right)=0 \\ RiskAverter: \\ U\text{'}\left(I\right)>0,U"\left(I\right)<0 \end{gathered}$$

Let’s check for Natasha,

$$\begin{gathered} U\left(I\right)=\sqrt{10I} \\ ={10}^{0.5}{I}^{0.5} \\ U\text{'}I=0.5\times {10}^{0.5}\times I^{0.5}>0 \\ U"I=0.25\times {10}^{0.5}\times I^{-1.5}<0 \end{gathered}$$

Thus, Natasha is a risk averter.

Explanation for part (b)

Natasha’s utility from the current salary will be $\sqrt{10\left(40\right)}=\sqrt{400}=20$

The expected utility from the new job will be:

$$\begin{gathered} E\left[U\right]=\left(0.6\right)\sqrt{10\left(44\right)}+\left(0.4\right)\sqrt{10\left(33\right)} \\ =0.6\times 20.976+0.4\times 18.166 \\ =12.59+7.27 \\ =19.86 \end{gathered}$$

The expected utility is lower than the initial utility. As Natasha is a risk averter; she will not accept the new job.

Explanation for part (c)

If Natasha accepts, then being a risk averter, she will buy insurance to cover the risk.The risk premium is the amount that Natasha is willing to be paid to get the expected salary than the risky salary from the new job.

The expected salary from the new job will be

(0.6x44,000)+(0.4x33,000)=26,400+13,200=$39,600.

The expected utility from the new job is 19.86; thus, to keep the utility unchanged, the income has to be changed. Thus, with utility 19.86, the income will be:

$$\begin{gathered} u=\sqrt{10I} \\ 19.85=\sqrt{10I} \\ 394.0225=10I \\ I=39.40225 \\ I=\$39,402 \end{gathered}$$

Thus, to keep the utility at 19.85, then the income has to be $39,402.

Natasha will be willing to pay $198 (=39,600 – 39,402) as a premium to guarantee an income of $39,600.