<p><strong>Question:</strong></p>
<p>The utility from daily income <span class="math-container">$(x)$</span> of an individual is given by <span class="math-container">$$U(x) = 100x - 0.5x^2$$</span> The only source of income is his earnings. Hence, <span class="math-container">$x = wl$</span> where <span class="math-container">$w = \text{wage}$</span> and <span class="math-container">$l = \text{labour hours}$</span>. He has two job offers:</p>
<ol>
<li>Job <span class="math-container">$1:$</span> <span class="math-container">$8$</span>-hour day at wage <span class="math-container">$5$</span>/hr</li>
<li>Job <span class="math-container">$2:$</span> Work hours are random - with a mean of <span class="math-container">$8$</span> hrs and <a href="https://en.wikipedia.org/wiki/Standard_deviation" rel="nofollow noreferrer">SD</a> of <span class="math-container">$6$</span> hrs. Wage <span class="math-container">$w_0$</span> is promised.</li>
</ol>
<p>For what <span class="math-container">$w_0$</span> will the individual take up the second job?</p>
<p><strong>Book's solution:</strong></p>
<p><span class="math-container">$\mathbb{E}[U(x_{\text{job}_1})] = 3200$</span> and <span class="math-container">$\mathbb{E}[U(x_{\text{job}_2})] = \mathbb{E}[100w_0h - 0.5 (w_0h)^2] = 800w_0 - 0.5 w_0^2 (36 + 64) = 800w_0 - 50w_0^2$</span></p>
<p><strong>My attempt:</strong></p>
<p>Utility from the first job is <span class="math-container">$U(8 \cdot 5) = 3200$</span> and the expected utility from the second job is <span class="math-container">$\mathbb{E}[U(w_0h)]$</span> where <span class="math-container">$h$</span> is a random variable with <span class="math-container">$\mu_h = 8, SD(h) = 6$</span>. <span class="math-container">$h$</span> gives the number of labour hours worked for job <span class="math-container">$2$</span> on a particular day.</p>
<p><span class="math-container">\begin{align}
\mathbb{E}[U(w_0h)] &amp;= 100w_0 \mathbb{E}[h] - 0.5w_0^2 E[h^2] \\
&amp;= 100w_0 \mathbb{E}[h] - 0.5 w_0^2 (\text{Var}(h) + \mu_h^2) \\
&amp;= 100w_0 \mathbb{E}[h] - 50 w_0^2 
\end{align}</span></p>
<p>The requires <span class="math-container">$w_0$</span> will be given by one that satisfies <span class="math-container">$100w_0 \mathbb{E}[h] - 50 w_0^2 \geq 3200$</span> (or in particular, equality, to achieve indifference).</p>
<p><strong>How do I proceed?</strong></p>
<p>I think the book's solution assumes <span class="math-container">$\mathbb{E}(h) = 8$</span> but that isn't necessarily true. The distribution isn't mentioned and that would assume something like the normal distribution.</p>

CC BY-SA 4.0
2022-10-28T11:00:25.353Z
Why do you think it isn't necessarily true that $\mathbb{E}(h)=8$?  The specification of Job 2 states a mean of 8 work hours.
Adam Bailey
CC BY-SA 4.0
2022-10-28T12:30:54.427Z
Note from OP: $\mathbb{E}(h) = 8$. That solves the problem.
Grunt
<microeconomics>
Compensating wage differentials for risk
Grunt
# Compensating wage differentials for risk

(Asked by: username_0 on 2022-10-28)

**Question:**

The utility from daily income $(x)$ of an individual is given by $$U(x) = 100x - 0.5x^2$$ The only source of income is his earnings. Hence, $x = wl$ where $w = \text{wage}$ and $l = \text{labour hours}$. He has two job offers:

1. Job $1:$ $8$-hour day at wage $5$/hr
2. Job $2:$ Work hours are random - with a mean of $8$ hrs and [SD](https://en.wikipedia.org/wiki/Standard_deviation) of $6$ hrs. Wage $w\_0$ is promised.

For what $w\_0$ will the individual take up the second job?

**Book's solution:**

$\mathbb{E}[U(x\_{\text{job}\_1})] = 3200$ and $\mathbb{E}[U(x\_{\text{job}\_2})] = \mathbb{E}[100w\_0h - 0.5 (w\_0h)^2] = 800w\_0 - 0.5 w\_0^2 (36 + 64) = 800w\_0 - 50w\_0^2$

**My attempt:**

Utility from the first job is $U(8 \cdot 5) = 3200$ and the expected utility from the second job is $\mathbb{E}[U(w\_0h)]$ where $h$ is a random variable with $\mu\_h = 8, SD(h) = 6$. $h$ gives the number of labour hours worked for job $2$ on a particular day.

\begin{align}
\mathbb{E}[U(w\_0h)] &= 100w\_0 \mathbb{E}[h] - 0.5w\_0^2 E[h^2] \\
&= 100w\_0 \mathbb{E}[h] - 0.5 w\_0^2 (\text{Var}(h) + \mu\_h^2) \\
&= 100w\_0 \mathbb{E}[h] - 50 w\_0^2
\end{align}

The requires $w\_0$ will be given by one that satisfies $100w\_0 \mathbb{E}[h] - 50 w\_0^2 \geq 3200$ (or in particular, equality, to achieve indifference).

**How do I proceed?**

I think the book's solution assumes $\mathbb{E}(h) = 8$ but that isn't necessarily true. The distribution isn't mentioned and that would assume something like the normal distribution.

**Tags:** microeconomics
> **Comments:**
> **username_1**: Why do you think it isn't necessarily true that $\mathbb{E}(h)=8$?  The specification of Job 2 states a mean of 8 work hours.  
> **username_0**: Note from OP: $\mathbb{E}(h) = 8$. That solves the problem.  

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## 0 answers

