# 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.

And I obtain that $$w_R=(1-b)(w-k) b\int_0^{\infty} max\{w_R, w\}dF(w)$$

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

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

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 $$b$$.

• 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 😊
– 1190
Jan 9 '20 at 7:37
• 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).
– 1190
Jan 9 '20 at 11:59
• 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
Jan 9 '20 at 13:39
• Many thanks dear @Amit . You are the best! :)
– 1190
Jan 9 '20 at 17:06