2.1 Endogenizing the job finding probability
The job-finding rate $f$ is not a fixed parameter but depends on how intensively firms
search for workers and how intensively workers search for jobs. Here, we will focus on
the worker's decision. When deciding how intensively to search for a job, the worker
compares the benefit of finding a job with the benefit of staying unemployed and cost of
searching for a job.
In this exercise, you will find the following formula useful:
$\sum_{t=0}^{\infty} aq^t = \frac{a}{1 - q}$ for $q \in (-1, 1)$.
We model the benefit of finding a job as the present discounted value of wages from the
job. A worker who finds a job starts with wage $w_0$. Future wages are discounted at interest
rate $R$. The wage stays constant over time but the worker faces the risk of losing the job,
expressed by the separation rate $\delta > 0$ (per period). Mathematically, you can model this
as future wages $w_t$ declining at a rate $\delta$ (recall the calculation of present discounted value
with wage growth $g$; instead, wages now grow at a negative rate $-\delta$).
Question 2.6 Denote $V$ the value of a job, i.e., the present discounted value of wages. Com-
pute $V$ as a function of $w_0$ and other parameters of the model.
