Lumpy Investment, Sectoral Propagation, and Business Cycles

term is equal to N log((1 —δ)∕g)/ log λ. The first term converges to m₁ by the argument
in the previous paragraph. Let us study m₁. By the assumption that sj_,t follows a
uniform distribution, we obtain Pr(s0 < 0) = μ. Then the number of producers who
adjust their capital at the first step, m₁, follows a binomial distribution Bin(N,μ). By
the central limit theorem, m₁∕-√^zN — VNμ asymptotically follows a normal distribution
with mean zero and variance μ(1 — μ). Combining these results, we obtain that m₀/y∕N
asymptotically follows a normal distribution with mean zero and variance μ(1 — μ).

Next we examine m_u conditional to m_u-1. We have Pr(j ∈ H_u|j ∈/
∪v=1,2,...,u-1Hv) = φ(log k^u — log k^u-1)/ log λ.     Thus mu follows Bin(N —

∑u=1 m_v, φ(log k^u — log k^u-1)/ log λ). This defines the stochastic process m_u completely.
As we let ξ → 1 and N → ∞, the binomial converges to a Poisson distribution with
an asymptotic mean φm_u-1 .

Since a Poisson distribution is infinitely divisible, the Poisson variable with mean
φm_u-1 is equivalent to a m_u-1 -times convolution of a Poisson variable with mean φ.
Thus the process mu is a branching process with a step random variable being a Poisson
with mean φ. Since φ ≤ 1, the process mu reaches 0 by a finite stopping time with
probability one. Thus the best response dynamics is a valid algorithm of equilibrium
selection. Let T denote the stopping time. Using the asymptotic formula, we have
W → ∑T=₁ mu. By using the property of a Poisson branching process (Kingman
(1993)), we obtain the infinitely divisible distribution for the accumulated sum W =
∑^t=₁ mu as in the proposition.

References

Bak, P., K. Chen, J. Scheinkman, and M. Woodford (1993): “Aggregate
fluctuations from independent sectoral shocks: Self-organized criticality in a model
of production and inventory dynamics,” Ricerche Economiche, 47, 3-30.

Bartelsman, E. J., and W. Gray (1996): “The NBER manufacturing productivity
database,” NBER Technical Working Paper, 205.

Caballero, R. J., and E. M. R. A. Engel (1991): “Dynamic (S,s) economies,”
Econometrica, 59, 1659-1686.

Caplin, A. S., and D. F. Spulber (1987): “Menu cost and the neutrality of money,”
Quarterly Journal of Economics, 102, 703-726.

Christiano, L. J., and T. J. Fitzgerald (1998): “The business cycles: It’s still a
puzzle,” Federal Reserve Bank of Chicago Economic Perspectives, 22(4), 56-83.

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