Unemployment cycles

specific types (previous subsection), with the only difference that π =
the fraction of high type vacancies.
v
v+v
is endogenous and denotes
The equilibrium wage in the terminal jobs is set such that the ‘losing” firm (which is the challenging
firm with low match-specific productivity), when competing for the worker, is indifferent between paying
that wage and opening a new vacancy. This implies
w 2 = w 2 ′ = w 2 = py
since this equally holds for any firm trying to poach the worker when there is at least one y firm
competing. When both firms that compete are of type y, then the wage is:
w 2 ′ = py.
Observe that all the wages in the terminal jobs are time invariant.
Then the equilibrium tightness can be written as:
θ(Ω) =
v
s(Ω) =
v
u + λ(Ω) [ γ + γ ].
We now derive the steady state equilibrium values where Ω is either 1 or 0. As before:
U = pb
r ,
and using this expression for U to solve for E 1 and E 1 we get:
E 1 = w 1(Ω) − Ωpk + λ(Ω)m(θ(Ω))((1 − π)E 2 + πE 2 ′)
r + λ(Ω)m(θ(Ω))
E 1 = w 1(Ω) − Ωpk + λ(Ω)m(θ(Ω))((1 − π)E 2 + πE 2 ′)
.
r + λ(Ω)m(θ(Ω))
Then solving for the terminal values using the value for unemployment and the equilibrium wages gives:
E 2 = E 2 ′ = E 2 =
E 2 ′ =
pb
py + δ
r
r + δ
pb
py + δ
r
r + δ
and similarly for the terminal job values:
J 2 = J 2 ′ = J 2 =
p(y − y)
r + δ
J 2 ′ = 0