Lumpy Investment, Sectoral Propagation, and Business Cycles

model, the variance is of order 1/(1 - φ)². We can regard the extra 1/(1 - φ) as the
contribution of the discrete propagation to the fluctuations.².

The fluctuation of the capital growth exhibits quite a different behavior, however,
when φ = 1. The distribution of W becomes a power law distribution. With the
exponent 0.5 (in a cumulative distribution), it is known that the distribution does not
have either mean or variance. That is, the sample moments diverge as the sample size
increases.

In fact, the variance of the capital growth rate ceases to depend on N when φ = 1.

Proposition 2 When φ = 1, the variance of the aggregate capital growth rate con-
verges to a non-zero constant as N → ∞. The limit variance is approximated by:

(log^λ)2 (μ(1^- μ) + ∖ 2μ(∣   μ) !9^π)) .                      (²1)

Proof: We concentrate on W/N which is the mean capital growth rate divided by
log λ. The unconditional variance Var(W/N) is decomposed as E(Var(W/N | m0)) +
Var(E(W/N | m₀)). Since W is symmetrically distributed and since |W | conditional
to m₀ follows the same distribution as the sum of m₀ number of |W | conditional to
m0 = 1, we have Var(W/N | m0) = |m0|Var(W/N | m0 = 1) and E(W/N | m0) =
m0E(W/N | m0 = 1) for an integer m0. We linearly interpolate this formula for
any real number m₀. Then, using the asymptotic distribution of m₀, we obtain that
Var(W/N) = y(2∕π)Nμ(1 - μ)Var(W∕N | mο = 1) + Nμ(1 - μ)(E(W/N | mο = 1))².
Next we derive the moments of W/N conditional to m0 = 1. At φ = 1, the distribution
of W becomes a pure power-law. Also by construction, W conditional to m0 = 1 only
takes integer values between 1 and N. Thus the distribution of W/N converges to
a continuous distribution in [0, 1] with keeping the power-law exponent. For a large
N , let us approximate the probability distribution of W conditional to m₀ = 1 by a
density function x^-1.5/(2(1 — 1∕√N)) for x ∈ [1, N]. Then the density function of W/N
is given by y^-1∙⁵/(2(VN — 1)) for y ∈ [1/N, 1]. Note that the distribution converges
to a delta function at zero only at the speed 1 /VN. Hence the mean and variance of
W/N conditional to m₀ = 1 are of order 1/VN. Combining with the previous result,
we obtain that the unconditional variance of W/N is of order N⁰ . More precisely we
obtain the following formula:

Var(W/N ) = !>1—f√π (√N —1/N )/(3(VN —1))+Nμ(1-μ)(1-1/^)²/(VN —1)2

(22)

²See Nirei (2003) for details.

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