|
lumpiness (log λ) | |||||
|
0.02 |
0.05 |
0.1 |
0.2 |
0.4 | |
|
periodicity 4 |
0.011 |
0.028 |
0.055 |
0.110 |
0.220 |
|
(Vμ) 6 |
0.010 |
0.024 |
0.049 |
0.098 |
0.195 |
|
8 |
0.009 |
0.022 |
0.044 |
0.089 |
0.178 |
Figure 1: Scale-free standard deviation of capital growth g
By taking a limit of N, we obtain our result. □
This result means that the growth rate fluctuation is scale-free, and that the law
of large numbers is broken. No matter how large the aggregative system is, a non-
linearity in an individual level can add up to an aggregate fluctuation. Consider the
case of lumpy investment behaviors. Cooper, Haltiwanger, and Power (1999) docu-
ments that, in the Longitudinal Research Database, the investment episodes in which
the investment-capital ratio exceeds 20% constitutes 20% of the plants and account
for 50% of gross investment. Considering that there are about 350, 000 plans in U.S.
manufacturing as they report, the aggregate fluctuation generated by the lumpy in-
vestment in individual level is negligible in a situation where the central limit theorem
holds. To the contrary, our result establishes a possibility of the aggregate fluctuations
via a stochastic propagation effect.
The formula (21) gives the standard deviation of growth rates as a function of λ and
μ. λ is the lumpiness parameter, and 1∕μ is interpreted as the periodicity of capital
oscillation in individual level. Some numerical examples are shown in Table 1. We
observe that the magnitude of lumpiness observed in data is large enough to generate
the fluctuations in aggregate production.
Our analytical results imply two things on the investment propagation. First, it
challenges the conventional view that the sectoral propagation does not add up to a
large aggregate fluctuation due to the law of large numbers effect. Our result shows
that, when the price response is rigid enough so that φ is close to one, the sectoral prop-
agation generates a significant fluctuation in aggregate level. Secondly, our result shows
that the large, non-degenerate investment fluctuation can occur endogenously in a de-
terministic environment. This implies that an interdependence of a small non-linearity
in a micro behavior may play a crucial role in aggregate investment fluctuations.
The propagation distribution derived here has an interesting link with other models
of non-linear dynamics in a network, such as the self-organized criticality or a perco-
lation in the Bethe lattice. These analytical connections are explored in Nirei (2003).
In this paper, let us move on to the next question of how this propagation effect may
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