Lumpy Investment, Sectoral Propagation, and Business Cycles

lations more realistic with keeping the results of the paper unaltered. Yet it is not
obvious that a sectoral capital accumulation process incurs such degree of inflexibility.
This leads to the question as to whether the business cycle patterns still obtain if we
disaggregate the economy to the establishment level. Our analytics shows that the
large aggregate fluctuations can occur in principle regardless of the number of agents,
but a quantitative demonstration of the theoretical possibility is left open.

A Appendix

A.1 Proof of Proposition 1

The details of proof draw on Nirei (2003). Here we outline the proof. Let us rewrite
the best response dynamics for the investment in t. We use u to denote the step in the
dynamics and suppress t. Let k^u denote the mean capital defined by (11) with a profile
ku Define k*^u by the threshold formula (16) with k^υ except for u = 0 at which we
define k*⁰ = k*_t. Define s^u = (log k^u — log k*^u)/log λ. Then the dynamics of (k^u, s^u) is
written as follows.

kj⁰  =  kj,t(1 — δ)∕g                                                   (30)

s_j⁰  =  sj,t + (log k_j⁰_,t — log kj,t)/ log λ                                  (31)

f kj^uλ   if s^u < 0

k^u+1  =  J k^u∕λ if s^u > 1                                           (32)

[ k^u otherwise

s^u+1  =  s^u + (log k^u+1 — log k^u — log k^u+1 +log k^u)/log λ          (33)

We consider for u > 1 the case m₀ > 0. The case m₀ < 0 is proved symmetrically

by changing the sign of adjustments. W = 0 if m0 = 0. Define Hu as the set of j such
that log k_j^u+1 — log k_j^u = log λ. Define mu as the size of Hu . First we derive a formula
for N (log k^u+1 —log k^u). By definition, we have log k_j^u+1 = log k_j^u + log λ for u ∈ Hu and
log kj'⁺¹ = log k^u for u ∈ H_u. Let us define φ = α(ξ — 1)∕(ξα + 1 — α). Then the first
term of a Taylor expansion is ∑_j∙∈H_u (k^u∕k^u')^ψ log λ. All the terms after the second term
either contain φ or are of order 1/N. Also the series are absolutely convergent. We
have φ → 0 as ξ → 1. Hence we obtain N (log k^u+1 — log k^u) → ∑_j∙∈H_u log λ = m_u log λ
as ξ → 1 and N → ∞.

Next we examine m0 = N (log k¹ — log kt)∕log λ. We break m0 into two terms as
m0 = N (log k¹ — log k⁰)∕log λ + N (log k⁰ — log kt)∕log λ. The first term represents the
first step adjustments and the second term represents the depreciation. The second

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