Ptj

0 12345678 Figure 17.1

reached. As illustrated in Fig. 17.1, the resulting path is neither the usual oscillatory type (not alternating between values above and below y in consecutive periods), nor the usual fluctuating type (not smooth); rather, it displays a sort of stepped fluctuation. As far as convergence is concerned, though, the decisive factor is really the R' term, which, like the ehl term in (15.24'), will dictate whether the stepped fluctuation is to be intensified or mitigated as t increases. In the present case, the fluctuation can be gradually narrowed down if and only if R < 1. Since R is by definition the absolute value of the conjugate complex roots (h ± vi), the condition for convergence is again that the characteristic roots be less than unity in absolute value.

To summarize: For all three cases of characteristic roots, the time path will converge to a (stationary or moving) intertemporal equilibrium—regardless of what the initial conditions may happen to be—if and only if the absolute value of every root is less than 1.

Example 8 Are the time paths (17.11) and (17.12) convergent? In (17.11) we have R = therefore the time path will converge to the stationary equilibrium (= 4). In (17.12), on the other hand, we have R = 4, so the time path will not converge to the equilibrium (= 0).

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