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
m
0∕>∕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)(φe1-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, m
0 = 1. Let x be the number
of sectors that adjust capital due to m
0, 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 m
0 . 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 δi , 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 m
0 = 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 m
0 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

10



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