<p><span class="math-container">$W(4) = \max\left\{4+bW(4),\frac{1}{2}\left(4 - k + bW(4)\right) + \frac{1}{2}\left(16 - k + bW(16)\right)\right\}$</span></p>

<p><span class="math-container">$W(16) = \max\left\{16+bW(16),\frac{1}{2}\left(4 - k + bW(4)\right) + \frac{1}{2}\left(16 - k + bW(16)\right)\right\} = 16+bW(16)$</span></p>

<p>First solve for <span class="math-container">$W(16)$</span> to get <span class="math-container">$W(16) = \frac{16}{1-b}$</span>. Then substitute it in <span class="math-container">$W(4)$</span> to solve for <span class="math-container">$W(4)$</span> as a function of <span class="math-container">$k$</span> and <span class="math-container">$b$</span>.</p>

CC BY-SA 4.0
2020-01-09T07:37:36.380Z
Is this first part? Well, how can I show the second part? That’s, how I can show whether the policy function is optimal it not. Thank you so much for your great help. I don’t exactly  know this topic. Please expand your answer please. Thank you so so much 
studentp
CC BY-SA 4.0
2020-01-09T11:59:07.047Z
Dear @Amit , if you write answer a bit clearly, I will happy. I guess what you did is the first part.  And I don’t understand how to do the second part (optimality question).
studentp
CC BY-SA 4.0
2020-01-09T13:39:48.013Z
At wealth = 4, the individual is indifferent between searching and not searching if $\frac{4}{1-b} = 10 - k + \frac{10b}{1-b}$. For $k > 10 + \frac{10b - 4}{1-b}$, not searching is optimal and for $k  < 10 + \frac{10b - 4}{1-b}$ searching is optimal.
Amit
CC BY-SA 4.0
2020-01-09T17:06:59.230Z
Many thanks dear @Amit . You are the best! :)
studentp
CC BY-SA 4.0
2020-01-09T02:18:37.223Z
2020-01-09T02:18:37.223Z
Amit
<p>Let us assume that there is a labor who lives in discrete time universe and discounts future payoffs with the discount factor <span class="math-container">$b\in (0,1)$</span>. </p>

<p>And we assume that this labor is at period <span class="math-container">$t=0$</span> at first. </p>

<p>Wages are assumed to have a discrete support defined as <span class="math-container">$W=\{4,16\}$</span>. That’s, any wage offered by any company is either 4 or 16. </p>

<p>This labor has a job currently and the current wage is <span class="math-container">$w_t\in W$</span>. If she wants to search for another job at period <span class="math-container">$t$</span>, she has to incur a fixed search cost <span class="math-container">$k\in (0,4)$</span> per period. </p>

<p>If the labor wants to search for a job, either the labor takes the same wage as an offer from another company with prbability of <span class="math-container">$1/2$</span> or he takes the other wage level as an offer with probability of 1/2.</p>

<p>We Suppose that the labor consumes all of her wage income (I.e. net of the search cost) at any time <span class="math-container">$t$</span>, and let the period utility from consumption is linear.</p>

<p>My question are </p>

<p>(1)Firstly, after defining the state, control variables and bellman equation, I would like to do an intuition that simply gives that the labor does not do job search
in period <span class="math-container">$t$</span> if <span class="math-container">$w_t = 16$</span> and then I want to calculate the value of this ”no search” policy.</p>

<p>(2) Secondly, I would like to show whether the following policy is optimal. “Search in period <span class="math-container">$t$</span> when <span class="math-container">$w_t=4$</span> and do not search in period <span class="math-container">$t$</span> when <span class="math-container">$w_t=16$</span>. How can I explain why is this?</p>

<p>——</p>

<p>My solution attempt is </p>

<p>The labor wants to maximize <span class="math-container">$E_0\{ \sum_{t=0}^{\infty}b^t c_t\}$</span></p>

<p>And if the labor does not want to search for a job, then <span class="math-container">$c_t=w$</span> but if the labor want to search for a job, then <span class="math-container">$c_t=w-k$</span></p>

<p>Let <span class="math-container">$F(w)$</span> is i.i.d. distribution. </p>

<p>The bellman equation for searching for a job and don’t search for a job are given by </p>

<p><span class="math-container">$W(w)=w+bW(w)$</span></p>

<p><span class="math-container">$U= (w-k)+b\int_0^{\infty} \max \{U, W(w)\}dF(w)$</span></p>

<p>where <span class="math-container">$W(w)$</span> is payoff from accepting a wage <span class="math-container">$w$</span> and <span class="math-container">$U$</span> is payoff from searching a wage offer, earning <span class="math-container">$w-k$</span> and sampling again next period. </p>

<p>Then when I calculate the equations,I obtain that <span class="math-container">$W(w)=w/(1-b)$</span> which is strictly increasing in <span class="math-container">$w$</span>. So, the reservation wage <span class="math-container">$w_R$</span> such that <span class="math-container">$W(w_R)=U=w_R/(1-b)$</span>. Then the labor sohund accept if <span class="math-container">$w\ge w_R$</span> and doesn’t accept if <span class="math-container">$w&lt;w_R$</span>. </p>

<p>And I obtain that <span class="math-container">$w_R=(1-b)(w-k) b\int_0^{\infty} max\{w_R, w\}dF(w)$</span></p>

<p>I cannot proceed after that point. Please help me to do this question. Thanks a lot. </p>

<macroeconomics><microeconomics><labor-economics><self-study>
Job search theory in discrete time
studentp
# Job search theory in discrete time

(Asked by: username_0 on 2020-01-08)

Let us assume that there is a labor who lives in discrete time universe and discounts future payoffs with the discount factor $b\in (0,1)$.

And we assume that this labor is at period $t=0$ at first.

Wages are assumed to have a discrete support defined as $W=\{4,16\}$. That’s, any wage offered by any company is either 4 or 16.

This labor has a job currently and the current wage is $w\_t\in W$. If she wants to search for another job at period $t$, she has to incur a fixed search cost $k\in (0,4)$ per period.

If the labor wants to search for a job, either the labor takes the same wage as an offer from another company with prbability of $1/2$ or he takes the other wage level as an offer with probability of 1/2.

We Suppose that the labor consumes all of her wage income (I.e. net of the search cost) at any time $t$, and let the period utility from consumption is linear.

My question are

(1)Firstly, after defining the state, control variables and bellman equation, I would like to do an intuition that simply gives that the labor does not do job search
in period $t$ if $w\_t = 16$ and then I want to calculate the value of this ”no search” policy.

(2) Secondly, I would like to show whether the following policy is optimal. “Search in period $t$ when $w\_t=4$ and do not search in period $t$ when $w\_t=16$. How can I explain why is this?

——

My solution attempt is

The labor wants to maximize $E\_0\{ \sum\_{t=0}^{\infty}b^t c\_t\}$

And if the labor does not want to search for a job, then $c\_t=w$ but if the labor want to search for a job, then $c\_t=w-k$

Let $F(w)$ is i.i.d. distribution.

The bellman equation for searching for a job and don’t search for a job are given by

$W(w)=w+bW(w)$

$U= (w-k)+b\int\_0^{\infty} \max \{U, W(w)\}dF(w)$

where $W(w)$ is payoff from accepting a wage $w$ and $U$ is payoff from searching a wage offer, earning $w-k$ and sampling again next period.

Then when I calculate the equations,I obtain that $W(w)=w/(1-b)$ which is strictly increasing in $w$. So, the reservation wage $w\_R$ such that $W(w\_R)=U=w\_R/(1-b)$. Then the labor sohund accept if $w\ge w\_R$ and doesn’t accept if $w.

And I obtain that $w\_R=(1-b)(w-k) b\int\_0^{\infty} max\{w\_R, w\}dF(w)$

I cannot proceed after that point. Please help me to do this question. Thanks a lot.

**Tags:** macroeconomics, microeconomics, labor-economics, self-study

---

## 1 answers

### Answer by username_1 on 2020-01-09. Score: 1 | Accepted answer

$W(4) = \max\left\{4+bW(4),\frac{1}{2}\left(4 - k + bW(4)\right) + \frac{1}{2}\left(16 - k + bW(16)\right)\right\}$

$W(16) = \max\left\{16+bW(16),\frac{1}{2}\left(4 - k + bW(4)\right) + \frac{1}{2}\left(16 - k + bW(16)\right)\right\} = 16+bW(16)$

First solve for $W(16)$ to get $W(16) = \frac{16}{1-b}$. Then substitute it in $W(4)$ to solve for $W(4)$ as a function of $k$ and $b$.

> **Comments:**
> **username_0**: Is this first part? Well, how can I show the second part? That’s, how I can show whether the policy function is optimal it not. Thank you so much for your great help. I don’t exactly  know this topic. Please expand your answer please. Thank you so so much   
> **username_0**: Dear @Amit , if you write answer a bit clearly, I will happy. I guess what you did is the first part.  And I don’t understand how to do the second part (optimality question).  
> **username_1**: At wealth = 4, the individual is indifferent between searching and not searching if $\frac{4}{1-b} = 10 - k + \frac{10b}{1-b}$. For $k > 10 + \frac{10b - 4}{1-b}$, not searching is optimal and for $k  < 10 + \frac{10b - 4}{1-b}$ searching is optimal.  
> **username_0**: Many thanks dear @Amit . You are the best! :)  

---

