Unemployment cycles

We now check two possible deviations and derive conditions under which those deviations are not
individually rational: 1. when all are actively searching on-the-job, there is no deviation by a single
agent to stop active search:
( ) k(r + δ)
E(1|1) > E(0|1) ⇐⇒ m −1 < θ(1).
λ 1 (y − b))
2. when no worker is actively searching on-the-job, there is no deviation of a single agent to search:
( ) k(r + δ)
E(0|0) > E(1|0) ⇐⇒ θ(0) < m −1 .
λ 1 (y − b))
These two no-deviation conditions give rise to the following result.
Lemma 1 There exist multiple steady state equilibria if and only if
( ) k(r + δ)
θ(0) < m −1 < θ(1).
λ 1 (y − b))
All proofs of this section are in Appendix A. Under the condition that the value of market tightness is
not too extreme, there exists two pure strategy equilibria, one where all search actively and one where no
one actively searches on-the-job. Whenever the two pure strategy equilibria exist, there is also a mixed
strategy equilibrium where every agent actively searches on-the-job with probability ω = Ω ∈ (0, 1)
in every interval of time dt, i.e. workers randomize between the choice of search effort (see Appendix
A for formal statement). This is illustrated in Figure 2, where we plot the mutual best-responses of
workers’ search effort and firms’ vacancy posting (reflected by tightness). The workers’ best response to
tightness is an increasing step function whereas the firms’ best response of tightness to workers search
effort is an increasing function, indicating the strategic complementarity. Points A and C mark the
pure strategy steady states while B indicates the steady state equilibrium in workers’ mixed strategies.
For the remainder of the paper, we will focus attention on the two pure strategy steady states.
Of course, θ(Ω) is an endogenous object. Unfortunately, we cannot in general compute conditions
under which Lemma 1 is satisfied.
We do find a necessary and sufficient condition in terms of the
primitives of the model under a particular matching function, the telegraph matching function:
m(θ) = φ
αθ
αθ + 1 , (13)
where φ is the overall matching efficiency of the matching technology and α is a parameter that determines
curvature. 18 In much of what follows, we will use the telegraph for three reasons. First, it has all
18 This is also the matching function used in the money and search literature, where φ and α are set to one. There it is
interpreted as a matching process where buyers (money holders) and sellers are one population, and hence under uniform
random matching, the likelihood of meeting a buyer is proportional to the number of buyers in the total population of
buyers and sellers: m(θ) =
θ = b where θ = b is the ratio of buyers to sellers.
1+θ b+s s
13