Lumpy Investment, Sectoral Propagation, and Business Cycles

Proposition 1 Suppose that λj and δj are common across j . Suppose that sj,t is
a random variable which follows a uniform distribution independently across j. Then
m₀∕>∕N asymptotically follows a normal distribution with mean zero and variance μ(1-
μ). Let m be a positive integer. ∣W∣ conditional to ∣m₀∣ = m follows a distribution
function asymptotically as we take ξ → 1 first and then N → ∞:

Pr(∣W ∣ = w ∣ ∣m0∣ = m) = (m∕w)e^-φw (φw)^w-m∕(w - m)!          (19)

for w = m, m + 1, .... The tail of the probability function is approximated by:

Pr(∣W∣ = w ∣ ∣m₀∣ = m) ≈ (m(φe)^-m∕√2π)(φe^1-φ)^ww^-1.5          (20)

The unconditional distribution of W is symmetric.

Proof is deferred to Appendix A.1. The key to the proof is to embed the best
response dynamics in a branching process so that the recursivity of the branching
process becomes available. Let G(s) be the generating function of the total adjustment
W given the initial deviation from the time average level, m₀ = 1. Let x be the number
of sectors that adjust capital due to m₀, and F (s) be its generating function. Each
adjustment of x then has a chance to propagate in the next step just like the initial
adjustment m0 . Thus the total number of offsprings which are originated from each
of x follows G(s). Hence we obtain a functional equation G(s) = sF (G(s)), from
which we derive the distribution of W . A similar functional equation obtains for a
large class of models with features such as heterogeneous λ_i and δ_i , non-uniformly
distributed s_j⁰_,t, or non-constant returns to scale technology, as shown in Nirei (2003).
The functional equation characterizes the propagation distribution completely, because
all the moments can be derived from it.

Proposition 1 implies that the capital growth log kt+1 - log kt conditional to m0 is
approximated by a power distribution w^-1.5 truncated by an exponential distribution
that declines at rate 1 - φ. We can calculate moments when φ < 1. The capital
growth conditional to m0 = 1 has an asymptotic mean log λ∕(N (1 - φ)) and variance
(log λ∕N)²(2 - φ)∕(1 - φ)³. The variance of an unconditional capital growth rate is
calculated as ((logλ)²∕N)(μ(1 — μ)(1 — 2∕π)/(1 — φ)² + ^2μ(1 — μ)/(πN)/(1 — φ)³) by
approximating m0 by an integer random variable. This is a natural result as obtained
in usual models: a fraction μ of sectors are induced to adjust by the deterministic trend
in mean. The variance of the capital growth rate declines linearly in N , hence the law
of large numbers obtains. One notable difference is that the variance has a 1∕(1 — φ)³
term, which can be quite large when φ is close to one. In a continuously adjusting

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