### Simple Labor Search Model with Unemployment Benefits

Labor Search Model and Unemployment Benefits

Labor Search Model Review.

This is a multi-period model in which workers life forever. Each period, if unemployed, workers receive some wage offer . Workers won’t know what wage offer they’ll receive, but they do know that it’ll be between a minimum of zero and a max of , and that they have some distribution . The density function of wage offers is  When a worker accepts a wage offer they accept it for life. When they reject it, they receive unemployment benefit b (for b, you can also think of it as the value of not working; leisure has its benefits) and the workers waits until the next period to receive another wage offer.

As a quick aside to help understand , suppose that workers know they’ll receive next period one of three possible wages, \$50,000, \$70,000 or \$150,000.  But the 50K offer comes in at probability 0.25, the 70k comes in at probability 0.70, and the 150K offer at probability 0.05 (total probability must be 1).

f(w) therefore looks like

You’d really like to get the 150K wage offer, but it’s extremely unlikely, and each period you turn down a wage you’re turning down actual money in the bank.

To Reject a Wage Offer to Not Reject

Thus, when will a worker accept a wage offer? and when will they reject it? Quite simply, the worker will reject offers for which the present discounted value (PDV) of accepting it is less than the PDV of rejecting the offer. And accept offers for which the PDV of accepting the offer is greater than the PDV of rejecting it.

From the given equation above;

 This is the PDV of receiving the wage offer w-hat for the rest of one’s life. (Beta is the discounting rate, reflecting the fact that a dollar in your hand now is more valuable to you than a dollar in the future – think of beta as about 0.98) This is the PDV of rejecting the wage offer w-hat. It’s equal to b, unemployment benefits for rejecting period, plus some crazy integral. The crazy integral (that you won’t need to know too much about for the exam, so we’re told) is the PDV of all possible wage offers in the future. It’s times the discount rate, reflecting the fact that we won’t get another one of these wage offers until the next period.

Drawing this as a graph:

On the horizontal axis are wage offers w-hat. Wage offers come at workers each period between zero and the maximum w-bar. The vertical axis is just ‘value’ or ‘utility’.

The PDV of accepting the wage offer is upward sloping, reflecting the fact that the value of low wage offers is valued less than a high wage offer (duh).

The PDV of rejecting wage offers is a horizontal line because this value does not depend on the current period’s wage offer. It depends only the unemployment benefit b, and the expected value of future period’s wage offers.

Reservation Wage – There is therefore some wage offer  at the cut-off between reject and accepting wage offers.

• Reject: If the wage the job-seeker sees this period is less than the reservation wage  then the job-searcher rejects the offer & keeps searching.

• Accept: If the wage the job-seeker sees this period is above the reservation wage  then the job-searcher accepts the offer.

Unemployment benefit increases from  to , where

(b) & (c) With benefits from being unemployed,  increasing to , the reservation wage increase to . That means that job-seekers will only accept at the higher reservation wage. Assuming those per-period job offers come in from the same distribution of possible offers, that means that the typical job-searcher will have to wait longer to find work that is acceptable to them. Implying unemployment will be higher with the higher .

Exam time – get ready for a change in something else, perhaps beta, perhaps a minimum wage. (see your notes).