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# lecture3 - Economics 202A Lecture Outline#3(version 1.0...

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Unformatted text preview: Economics 202A Lecture Outline #3 (version 1.0) Maurice Obstfeld Steady State of the Ramsey-Cass-Koopmans Model In the last few lectures we have seen how to set up the Ramsey-Cass- Koopmans Model in discrete time, and with an in&nite horizon. We have reviewed the solution method for (bivariate) di/erence equation systems and derived an exact solution to the model in a special case. I have not yet developed the main qualitative implications of the model however. These can perhaps be drawn out most conveniently by de&ning the model¡s steady state (or balanced growth allocation) and linearizing the model around that long-run destination. As we saw, when f 00 ( k ) = 0 as in the last lecture, the model has no steady state in c and k , but in the customary case with f 00 ( k ) < a well-de&ned steady state exists. It is instructive to examine its properties. Let & c and & k denote the steady-state values. Then they must satisfy the intertemporal Euler equation u (& c ) = & & 1 + f ¡ & k ¢ & ¡ £ u (& c ) ; which is equivalent to f ¡ & k ¢ = 1 & & & + ¡: (1) Intuitively, this states that the net marginal product of steady-state capital, f ¡ & k ¢ & ¡ , equals the rate of pure time preference. 1 Steady state values must also ensure that & k = f ¡ & k ¢ + (1 & ¡ ) & k & & c 1 + n ; or, solving for & c , that & c = f ¡ & k ¢ & ( n + ¡ ) & k: (2) 1 If & = 1 = (1 + ¡ ) ; then we call ¡ the rate of pure time preference. With this notation eq. (1) becomes f ¡ & k ¢ = ¡ + ¢: 1 Di/erentiation of eq. (2) shows that per capita consumption is maximized when the constant capital stock equals k & ; where f ( k & ) = n + &: (3) This is the &golden rule¡point where the marginal product of capital just equals replacement needs. Because the model assumed (1+ n ) ¡ < 1 , however, it follows that n < 1 & ¡ ¡ : Because f 00 ( k ) < , a comparison of (3) with (1) shows that & k < k & : Given the consumer optimization assumed in this model, there is no possi- bility of a dynamically ine¢ cient steady state with capital over-accumulation. This is one di/erence compared to the Solow model. An excellent exercise is to linearize this model in the neighborhood of & & c; & k ¡ and investigate its dynamic properties, in particular showing that the two chacteristics roots are, respectively, greater than and less than 1. The £rst question on Problem Set 2 involves an example like that one, so I will not pursue the linearization here. Instead, I will look in greater detail at the continuous-time version of this model, using that as a springboard to a discussion of optimal control theory....
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lecture3 - Economics 202A Lecture Outline#3(version 1.0...

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