Job search theory in discrete time 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?

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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 $w&lt;w_R$. 

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

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

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 $b$.