Lumpy Investment, Sectoral Propagation, and Business Cycles



inaction region in kj,t with a lower bound k*t and an upper bound λjk*,t. Consider two
sequences of k
j,s which are identical except at t. Such sequences can be constructed by
assigning a positive investment at t
- 1 and zero investment at t in one sequence and
zero investment at t
- 1 and a positive investment at t in the other. Then the lower
bound of the inaction region is derived by solving for k*
t at which the two sequences
yield the same discounted profit. Namely, if k
j∙,t is strictly less than k*t, the producer
is better off by adjusting it upward rather than waiting. Let one sequence with zero
investment at t
1 and one tick of investment at t have kj,t = k*t, and let the other
sequence have k
j,t = λjk*,t. Then the both sequences have the same amount of capital
at t
1 and t + 1: kj,t-1 = (g∕(1 δj∙))k*t and (λj∙(1 δj∙)∕g)k*t. Solving for k*t which
equates the discounted profits of the two sequences, we obtain:

kj,t = Dj (w(1-a)/a(rt 1 + δj ))-(ξα+1-α)kt                     (13)

where Dj (^a^-1)/^a+1-a) 1)Do∕(λj 1))ξα+1-α.

Equation (13) expresses the feedback relation from the mean capital level kt to the
threshold for an individual capital level k
j,t . Note that the feedback effect on kj,t is non-
linear because of the threshold behavior. The mean capital level k
t affects the threshold
of the inaction region, but it may or may not induce the adjustment of k
j,t . Also note
that the effect on the inaction region is linear. This implies that, in the situation when
an individual capital adjustment occurs continuously (λ
j 1), the feedback effect
from the mean capital to an individual capital is linear. The linear feedback means
that the individual capital moves proportionally to the mean capital level. These two
observations are summarized as local inertia and global strategic complementarity of
the individual behavior. The individual capital is insensitive to a small perturbation in
the mean capital level, while it synchronizes with the mean capital if the perturbation
is large.

The global strategic complementarity is perfect in a sense that the percentage
changes coincide in an individual and mean capital. We will show shortly that this
perfect complementarity induces a large fluctuation in propagation of capital adjust-
ments. The perfect complementarity results from the constant returns to scale of the
technology. This point is shown as follows. Consider an identical economy as above
except for that the production incurs only capital as in Y
j = Aj1-θKjθ. Then the lower
bound of the inaction region of capital is shown to be proportional to ^
1/(1+^1^-1)).
The lower bound is linear in k
t only when the returns to scale is constant (θ = 1).
The strategic complementarity is less than or more than proportional depending on
whether the returns to scale is diminishing or increasing.

For simplicity, let us for a while focus on this feedback network of producers in



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