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Unformatted text preview: Economics 202A Lecture Outline #3 (version 1.0) Maurice Obstfeld Steady State of the RamseyCassKoopmans Model In the last few lectures we have seen how to set up the RamseyCass 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 models steady state (or balanced growth allocation) and linearizing the model around that longrun 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 wellde&ned steady state exists. It is instructive to examine its properties. Let & c and & k denote the steadystate 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 steadystate 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 rulepoint 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 overaccumulation. 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 continuoustime version of this model, using that as a springboard to a discussion of optimal control theory....
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This note was uploaded on 02/28/2012 for the course ECON 202A taught by Professor Akerlof during the Fall '07 term at University of California, Berkeley.
 Fall '07
 AKERLOF
 Economics

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