Unemployment cycles

π(1) =
1
k(λ 0 + λ 1 )(δ(y − y)(λ 1 (y − b) + k(λ 0 + λ 1 )) + r(λ 1 y(b − y) + k(λ 0 + λ 1 )(y − y))) ×
{
b 3 (−δ)λ 2 1 + b 2 λ 1 (δk(2λ 0 + λ 1 ) + 2δλ 1 y + kr(λ 0 + λ 1 ))
+b ( −k(λ 0 + λ 1 )(kλ 0 (δ + r) + δλ 1 y) + δkλ 1 y(λ 1 − λ 0 ) − δλ 2 1y 2)
+k ( δ(k(λ 0 + λ 1 ) + λ 1 y)(y(λ 0 + λ 1 ) − y(λ 0 + 2λ 1 )) + r(λ 0 + λ 1 ) ( ky(λ 0 + λ 1 ) − ky(λ 0 + 2λ 1 ) − λ 1 y 2))}
where π(0) gives the highest value of π that is consistent with an equilibrium where no one searches actively and
π(1) is the lowest π that is consistent with an equilibrium where everyone searches actively.
Finally, to obtain the bounds in terms of primitive p, we use the zero profit conditions (either V = 0 or
V = 0) for both the equilibrium of active and non-active search and solve for p l and p h respectively
p l =
m(θ(1))u(y−b)
θ(1)(δ+r)(γ(λ 0+λ 1)+u) −
cy
km(θ(1))u
θ(1)(γ(λ +
0+λ 1)+u)(δ+m(θ(1))(λ 0+λ 1)+r)
m(θ(1)) 2 (1−π(1))u(λ 0+λ 1)(y−y)
θ(1)(δ+r)(γ(λ 0+λ 1)+u)(δ+m(θ(1))(λ 0+λ 1)+r)
cθ(0)y(δ + r)(γλ 0 + u)(δ + λ 0 m(θ(0)) + r)
p h = −
m(θ(0))u(bδ + bλ 0 m(θ(0)) + br − δy + λ 0 m(θ(0))π(0)y − λ 0 m(θ(0))π(0)y − λ 0 m(θ(0))y − ry)
which we evaluate at π(Ω), θ(Ω) as well as u, γ from above to obtain expressions that solely depend on parameters.
The expressions are involved but one can show via simulations that there exists a paramater range for which
p h > p l and π(1) > π(0). We have the following result (exact expressions available upon request).
Proposition 5 Let m(θ) = φ αθ
αθ+1 . Then there are multiple steady states if and only if p ∈ [p l, p h ]. The set
[p l , p h ] is non-empty for an open set of parameters.
Also in this set-up with permanent firm types, the strategic complementarity and thus multiplicity survives.
When both equilibria coexist, the active on-the-job search equilibrium is characterized by more search effort,
larger market tightness and a larger share of high-productivity vacancies v (with y). What gives rise to this
strategic complementarity between on-the-job search and (high-type) vacancies? Here workers are incentivized
to search actively, not only if tightness is high enough (as before) but also if the fraction of high productivity
vacancies is high enough. High type vacancies encourage on-the-job search because it offers workers the chance
to end up in a job where they extract the entire surplus (after having met a high productivity firm not only after
unemployment but, crucially, also after on-the-job search). In turn, firms are encouraged to not only post more
vacancies but especially more high type vacancies in the presence of active on-the-job search because on-the-job
search biases the pool of searcher towards the employed. This bias implies that high type vacancies match faster
(v cannot attract on-the-job searchers) and the match duration with employed searchers is longer than with
unemployed ones.