Handout Unemployment dynamics with population growth Jorge F. Chavez April 24, 2012

Unemployment dynamics with population growth
Jorge F. Chavez
April 24, 2012
This note presents a more general version of the simple model of unemployment dynamics that we studied in term 1. As before, let Lt denote the size of the labor force at time
t. Now we will allow entrants to this labor force (i.e. people that become the right age to
look for a job, or people that just finished their studies and therefore will start looking for
a job).
Suppose that Lt grows at a constant positive rate n ∈ (0, 1), that is Lt+1 = (1 + n)Lt
for all t. As before Lt is composed by those that have a job at time t, denoted Et , plus
those that are unemployed Ut in that period.
Let s and f denote the job separation rate and the job finding rate, correspondingly.
The assumption now is that each period, the new members of the labor force will find a
job with probability f , or which is equivalent, a proportion f of the additional workers will
become part of Et :
Et+1 = Et + f Ut − sEt + f ∆Lt+1
We can get an analogous law of motion for Ut :
Ut+1 = Ut + sEt − f Ut + (1 − f ) ∆Lt+1
where we are taking into account that a proportion (1 − f ) of the new workers are unable
to find a job.
Take (2) as base, divide it by Lt and define the unemployment rate as µt ≡ Ut /Lt :
= µt + s (1 − µt ) − f µt + (1 − f ) n
Then divide and multiply by both sides of (3) by Lt+1 to get µt+1 :
Ut+1 Lt+1
= µt+1 (1 + n) = µt + s (1 − µt ) − f µt + (1 − f ) n
Lt+1 Lt
If we had chose to work with (1) instead, the equation we would reach would be:
(1 − µt+1 ) (1 + n) = 1 − µt + f µt − s (1 − µt ) + f n
Finally, in steady-state µt = µt+1 = µ, therefore applying this condition we get:
s + (1 − f ) n
s+f +n
Note than in the special case in which n = 0 we are back in the case in which there was
not population growth and therefore the expression for the unemployment rate in steady
state is again µ = s+f