Chapter 13, Problem 13.10P

a)

To determine

To calculate: The amount of wage rate and associated risk to get the job.

Explanation of Solution

Given

  $\begin{array}{l}Utility=\sqrt{wl}\\ where,w\text{ is the hourly wage rate}\\ \text{and }l\text{ is the number of hours worked}\end{array}$

(a).

Let u₁ denote the utility of the first job. Since the first job gives a constant work hour of 8 hours at an hourly wage of $50, the utility would be:

  $\begin{array}{l}Utility\left(u_1\right)=\sqrt{50\times 8}\\ Utility=\sqrt{400}\\ Utility=20\end{array}$

Let u₂ denote the utility of the risky job. Sometimes the salesman gets 4 hours of work with probability 0.5 and sometimes he gets 12 hours of work with probability 0.5, then the expected utility from the risky job is:

  $Utility\left(u_2\right)=0.5\sqrt{w\times 4}+0.5\sqrt{w\times 12}$

The risky job will be chosen ifu₁ > u₂

  $\begin{array}{l}0.5\sqrt{w\times 4}+0.5\sqrt{w\times 12}>20\\ \text{squaring both sides}\\ 0.25\times 4\times w+0.25\times 12\times w+2\times 0.25\times 4w\times \sqrt{3}>400\\ w+3w+2w\times 1.732>400\\ 7.464w>400\\ w>53.6\end{array}$

Hence, for an hourly wage rate higher than 53.6, the risky job will be taken.

b)

To determine

To show: Whether the imposition of proportional tax on income will impacts decisions regarding choice of job.

Explanation of Solution

If the proportional tax is imposed on both the risky job and a certain job, then results do not change as the expected utility difference remains the same. Let us check:

Let the proportional tax rate be, t. Utility in case of a certain job will be:

  $u_1=\sqrt{50\times 8\times \left(1-t\right)}$

Utility in case of the risky job will be:

  $u_2=0.5\sqrt{w\times l\times \left(1-t\right)}+0.5\sqrt{w\times l\times \left(1-t\right)}$

The risky job will be chosen if u₁ > u₂

  $\begin{array}{l}0.5\sqrt{w\times 4\times \left(1-t\right)}+0.5\sqrt{w\times 12\times \left(1-t\right)}>\sqrt{50\times 8\times \left(1-t\right)}\\ \text{squaring both sides}\\ 0.25\times 4\times w\times \left(1-t\right)+0.25\times 12\times w\times \left(1-t\right)+2\times 0.25\times 4w\times \left(1-t\right)\times \sqrt{3}>400\left(1-t\right)\\ \left(w+3w+2w\times 1.732\right)\times \left(1-t\right)>400\left(1-t\right)\\ \left(1-t\right)cancels\text{ out}\\ 7.464w>400\\ w>53.6\end{array}$

The wage rate for both jobs remains the same so, the person’s choice will not be impacted.

c)

To determine

To calculate: Let us calculate the utility to decide whether the decision will be impacted or not:

Explanation of Solution

Utility in case of constant working hours:

In this case, 300 will not be taxed and 100 (400-300) will be taxed at the rate of 50%

  $\begin{array}{l}Utility\left(u_1\right)=\sqrt{300+100\times 0.5}\\ Utility\left(u_1\right)=\sqrt{350}\end{array}$

Utility in case of random working hours:

In this case, also 300 will not be taxed and wages above 300 will be taxed at the rate of 50%

  $\begin{array}{l}Utility\left(u_2\right)=0.5\sqrt{300+\left(4w-300\right)0.5}+0.5\sqrt{300+\left(12w-300\right)0.5}\\ Utility\left(u_2\right)=0.5\sqrt{150+2w}+0.5\sqrt{150+6w}\\ \text{Now let us compare the wages:}\\ 0.5\sqrt{150+2w}+0.5\sqrt{150+6w}>\sqrt{350}\\ \text{Squaring both sides}\\ \text{0}\text{.25}\left(150+2w\right)+0.25\left(150+6w\right)+2\times 0.25\sqrt{22500+1200w+12w^2}>350\\ 0.25\left(300+8w+2\times 2\sqrt{5625+300w+3w^2}\right)>350\\ 300+8w+2\times 2\sqrt{5625+300w+3w^2}>1400\\ 4\sqrt{5625+300w+3w^2}>1100-8w\\ \sqrt{5625+300w+3w^2}>275-2w\\ 5625+300w+3w^2>{\left(275-2w\right)}^2\\ 5625+300w+3w^2>75625+4w^2-1100w\\ w^2-1400w+70000>0\end{array}$

Since, in this case, the equation of wages has changed the decision will also change.