Dynamics of the perfect foresight model

C .D. Carroll Economics Letters 68 2000 67 –77 75

4. Dynamics of the perfect foresight model

Analysis of growth models often proceeds by linearizing the model around the steady-state. For the usual neoclassical model this involves linearizing the aggregate budget constraint and the difference equation for consumption. We derive here the difference equations for s and x under the assumption that the real interest rate is constant. This is the correct procedure in an endogenous growth model with a fixed rate of return to capital; the extension to the neoclassical production function would add a third equation to the system derived here, describing the evolution of the gross interest factor as derived from the standard neoclassical production function. The key step in obtaining the steady-state approximations is to find the difference equations that govern the evolution of x and s. Begin by defining s 5 c c and x 5 c h , and note that: t t t 21 t t t s c t t 21 ]] c h 5 38 t t h t c h t 21 t 21 ]] ]] c h 5 s 39 t t t h h t 21 t h t 21 ]]]]]] c h 5 s x 40 t t t t 21 1 2 lh 1 lc t 21 t 21 1 ]]]]] x 5 s x 41 t t t 21 1 2 l 1 lx t 21 c h Substituting in for u and u in the Euler equation gives: t t 2 r g r 21 2 r g r 21 2 r g r 21 c h 5 b c h R 1 1 2 l 1 glx 2 R bc h 1 2 l 1 gl x f g s d s d t t t 11 t 11 t 11 t 12 t 12 t 12 g r 21 2 r c h t 11 t 11 ]] ]] 1 5 b R 1 1 2 l 1 glx s d S D F F G t 11 c h t t g r 21 2 r c h t 12 t 12 ]] ]] 2 R b 1 2 l 1 glx 42 s d S D G F G t 12 c h t 11 t 11 and use the fact that h h 5 [1 2 l 1 lx ] [see Eqs. 39–41] to obtain: t 11 t t 2 r g r 21 1 5 s 1 2 l 1 lx b [R 1 1 2 l 1 glx s d t 11 t t 11 2 r g r 21 2 R bs 1 2 l 1 lx 1 2 l 1 glx ] 43 s d s d t 12 t 11 t 12 r g 12r s 1 2 l 1 lx b 2 R 2 1 2 l 1 glx s d t 11 t t 11 2 r g r 21 5 2 R bs 1 2 l 1 lx 1 2 l 1 glx s d s d t 12 t 11 t 12 r g 12r R 1 1 2 l 1 glx 2 s 1 2 l 1 lx b s d t 11 t 11 t 2 r ]]]]]]]]]]]]]]]] s 5 44 t 12 g r 21 R b 1 2 l 1 glx 1 2 l 1 lx s ds d t 12 t 11 76 C .D. Carroll Economics Letters 68 2000 67 –77 r g 12r 2 1 r R 1 1 2 l 1 glx 2 s 1 2 l 1 lx b s d t 11 t 11 t ]]]]]]]]]]]]]]]] s 5 45 F G t 12 g r 21 R b 1 2 l 1 glx 1 2 l 1 lx s ds d t 12 t 11 Eqs. 41 and 45 are difference equations for x and s which can be linearized or log-linearized around the steady-state values derived above to allow analysis of the near-steady-state behavior of the model.

5. Conclusions