Introduction to Modern Economic Growth and thus c∗ not depend on the instantaneous utility function u (·) The form of the utility function only affects the transitional dynamics (which we will study next), but has no impact on steady states This is because the steady state is determined by the modified golden rule This result will not be true when there is technological change, however 8.5 Transitional Dynamics Next, we can determine the transitional dynamics of this model Recall that transitional dynamics in the basic Solow model were given by a single differential equation with an initial condition This is no longer the case, since the equilibrium is determined by two differential equations, repeated here for convenience: k˙ (t) = f (k (t)) − (n + δ)k (t) − c (t) (8.24) and c˙ (t) = (f (k (t)) − δ − ρ) c (t) εu (c (t)) Moreover, we have an initial condition k (0) > 0, also a boundary condition at (8.25) infinity, of the form Z t ảá lim k (t) exp − (f (k (s)) − δ − n) ds = t→∞ As we already discussed in the context of the q-theory of investment, this combination of an initial condition and a transversality condition is quite typical for economic optimal control problems where we are trying to pin down the behavior of both state and control variables This means that we will again use the notion of saddle-path stability introduced in Theorems 7.17 and 7.18 instead of those in Theorems 2.4, 2.5 and 2.6 In particular, the consumption level (or equivalently the costate variable µ) is the control variable, and its initial value c (0) (or equivalently µ (0)) is free It has to adjust so as to satisfy the transversality condition (the boundary condition at infinity) Since c (0) or µ (0) can jump to any value, we again need that there exists a one-dimensional curve (manifold) tending to the steady state In fact, as in the q-theory of investment, if there were more than one paths tending to the steady state, the equilibrium would be indeterminate, since there would be multiple values of c(0) that could be consistent with equilibrium 387