the product markets while abstracting the rest of the economy by assuming that the
equilibrium wage and interest rate only depends on the mean capital level. We come
back to the equilibrium price functionals in Section 4. Suppose that the equilibrium
wage and interest are approximated by constantly elastic functions of the mean capital
kt in the vicinity of arbitrary chosen levels (most naturally the time-average) of wage,
interest, and mean capital w,r,k:
(rt - 1 + δj)∕(k - 1 + δj) = (kt/k)θ (14)
Wt/W = (kt∕k)θw (15)
This is a partial equilibrium assumption when θr = θw = 0 in particular. By assuming
(14-15), the threshold (13) is simplified as:
⅛∕k) = (k<∕i)φ (16)
where k* is the threshold corresponding to (u, W5, u), and φ represents the strategic
complementarity between the individual and mean capital:
φ = 1 - (αθr + (1 - α)θw)(ξ - 1 + 1∕α) (17)
Note that φ is less than one. This implies that the strategic complementarity between
producers is decreased from the perfect complementarity due to the equilibrium re-
sponse of the wage and interest rate. The wage and interest rise in our approximation
when the production is higher than the time average level. This price response works
as a dampening factor in the investment propagation.
The equilibrium of the product markets is given by a capital profile which satisfies
kj,t ∈ [kj,t, λjk*,t]. This condition allows multiple equilibria in general. Here we employ
best response dynamics as an equilibrium selection algorithm. Suppose that a prede-
termined capital kj,t resides in the inaction region. The next period capital kj,t+1 only
decreases by depreciation and technology progress unless adjusted. In the first step of
the best response dynamics, the producers adjust capital by λj if their capital levels
go below k*t given kt. Note that, assuming δj∙ + g < λj∙, the adjustment never exceeds
λj . In the second step, kt is calculated by a new capital profile, and the producers
adjust their capital responding to the revised kt . We repeat this procedure until the
capital profile converges. The adjustments after the second step can be upward or
downward, depending on whether the first step upward adjustments by some produc-
ers weigh more or less than the inertial depreciation of overall capital. Let us formally
define the best response dynamics as follows. Set the initial point of the dynamics as