Firm Dynamics, On-the-Job Search, and Labor Market Fluctuations

Proof of Proposition 4
The derivations of (28) and (29) are provided in the main text, so we focus on establishing existence here.
Consider first the limit as θ→0⁠, which implies λ→0⁠. It can be verified that Proposition 1 and its proof apply, and that the boundaries satisfy 0<ml<mh<∞ in this limit. Furthermore, from Proposition 2, the hiring region becomes degenerate, mu→mh⁠. Using Proposition 3, and applying L’Hôpital’s rule, the worker distribution G(m)→ln(m/ml)/ln(mh/ml)⁠. It follows from (28) and (29) that limθ→0UJC(θ)<L=limθ→0UBC(θ)⁠.
Now consider the limit as θ→∞⁠, which implies λ→∞⁠. It can be verified that the roots of the fundamental quadratic (21) satisfy γ1→−∞ and γ2→1/(1−α)⁠. Abel and Eberly (1996) prove that
0<φ(0)=γ1γ1−11ρ(0)<φ(1)=1ρ(0)<γ2γ2−11ρ(0)=φ(∞).
(A.33)
It follows that the solution to (A.4) satisfies G→∞⁠, with φ(G)→1/[αρ(0)] and φ(G−1)→1/ρ(0)⁠. Thus, the boundaries in (A.5) satisfy (1−ω1)ml→αω0⁠, and (1−ω1)mh→∞. Now, note from Proposition 3 that G(m)→0 for all m∈(ml,mh)⁠. Thus, g(ml)→0⁠, and ς→0⁠. Furthermore, since mh→∞⁠, it must be that ∫m1/(1−α)g(m)dm→∞⁠. It follows from (28) and (29) that limθ→∞UJC(θ)=L>0=limθ→∞UBC(θ)
Since all objects in (28) and (29) are continuous in λ⁠, and thereby θ⁠, it follows that there must exist at least one θ∈(0,∞) that satisfies (28) and (29). □
Proof of Lemma 2
(a) s→0⁠. To establish (i), simply note that Proposition 1 and its proof apply for all s≥0⁠. Property (ii) follows directly from Proposition 2: since δ(mh)→0 as s→0⁠, and since δ(m) is declining for m>mh for all s>0⁠, it follows that mu→mh⁠. Finally, (iii) emerges from Proposition 3 and application of L’Hôpital’s rule,
G(m)→(m/ml)1−ασ2/2sλ−1(mh/ml)1−ασ2/2sλ−1→(m/ml)1−ασ2/2sλln(m/ml)(mh/ml)1−ασ2/2sλln(mh/ml)→ln(m/ml)ln(mh/ml),ass→0.
(A.34)
(b) α→1⁠, holding fixed X and ˜σ2≡σ2/(1−α)⁠. For (i), note that, combining the latter with the aggregate stationarity condition (24), the fundamental quadratic (21) is
ρ(γ)=−12˜σ2(1−α)γ2−[−12˜σ2+(1−α)sλ]γ+r+sλ=0.
(A.35)
It follows that γ1→−2ρ(0)/˜σ2≡˜γ1⁠, and γ2→∞⁠, as α→1⁠. Thus, (A.3) and (A.4) become
ϑ(G)→{1forG≥1G1−˜γ1forG<1,andϑ(G)→1ρ(1)[1−ϑ(G)˜γ1].
(A.36)
The latter and (A.4) imply that the solution for G satisfies limα→1G>1⁠, and that therefore
limα→1φ(G)=1ρ(1)(1−1~γ1)>1ρ(1)(1−G˜γ1−1˜γ1)=limα→1φ(G−1).
(A.37)
It follows from (A.5) that 0<ml<mh<∞⁠. To verify (ii), note from (A.12) that δ′(m)→−∞ as α→1 for all m>mh⁠. (iii) follows from Proposition 3 and the definition of ˜σ⁠. □