Unemployment cycles

E 0 =
E =
U =
w 2 + βλ(0)mE + βδU
1 − β(1 − δ)(1 − λ(0)m)
w + βδU
1 − β(1 − δ)
b
1 − β
which take into account that V = 0 due to free entry.
The firm’s objective is to choose a triple T, w 1 , w 2 to maximize the value of a low productivity job J T subject
to three constraints:
s.t.
max
T,w 1 ,w 2
J T
E T ≥ U
E 0 (0) ≥ E 0 (1)
E T −1 (1) ≥ E T −1 (0)
The first constraint states that the wages must be such that the worker is at least as well off taking the job as in
unemployment; the second states that after T periods, wages must be such that the worker weakly prefers not
to search; the third constraint ensures that the worker does not want to deviate from the strategy search until
period T (and not thereafter): If the worker prefers to search in period T − 1, he also prefers to search in all
periods t < T − 1 since in period T − 1 it is most tempting to not search due to the soon-to-be-expected wage
increase. The first two constraints will hold with equality, otherwise the firm would forgo profits. We recover w 2
from constraint 2 and, given w 2 , we recover w 1 from constraint 1. Last, we verify that constraint 3 holds for any
paramater constellation; it is slack.
We then evaluate the objective function J T at the wages and check its properties: Our simulations (available
upon request) reveal that it is either monotonic increasing or decreasing, i.e. T ∗ is at a corner. For most
parameter ranges, J T is increasing, always weakly below the value of the on-the-job search equilibrium J with
lim T →∞ J T = J. This implies that the deviation is not profitable; firms do not seek to discourage search by
backloading wages. For some paramater constellations (in particular, for unnaturally low β and high k), J T can
be decreasing with its maximum at T ∗ = 0. In this case, in which on-the-job search is costly and workers do not
value much future benefits, it is worth it for the firm to discourage search, and the firm would do so immediately
after hiring.