1
$\begingroup$

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 $w<w_R$.

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.

$\endgroup$

1 Answer 1

1
$\begingroup$

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

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

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 $b$.

$\endgroup$
4
  • $\begingroup$ 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 😊 $\endgroup$
    – studentp
    Jan 9, 2020 at 7:37
  • $\begingroup$ 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). $\endgroup$
    – studentp
    Jan 9, 2020 at 11:59
  • 1
    $\begingroup$ 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. $\endgroup$
    – Amit
    Jan 9, 2020 at 13:39
  • 1
    $\begingroup$ Many thanks dear @Amit . You are the best! :) $\endgroup$
    – studentp
    Jan 9, 2020 at 17:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.