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lecture2 - Economics 202A Lecture#2 Outline(version 1.4...

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Economics 202A Lecture #2 Outline (version 1.4) Maurice Obstfeld I have commented on the ad hoc nature of the saving behavior postulated by Solow. The next model assumes instead that people plan ahead in making saving decisions. One advantage of this assumption is that we can do welfare analysis of economic changes. The model delivers °normative± answers to questions such as, °How much should a country save?±In its various forms, the following model has many applications in macroeconomics and public ²nance beyond the analysis of growth. The Ramsey-Cass-Koopmans Model in Discrete Time I will initially develop this model in discrete time. Then I will go to the continuous-time limit to derive a mathematical framework comparable to the Solow model³s. This will also serve to illustrate the principles of optimal control theory , a very useful tool. There are many other approaches to the derivation, such as the one based on dynamic programming in my notes at http://www.econ.berkeley.edu/~obstfeld/ftp/perplexed/cts4a.pdf. Another possible source is Martin Weitzman³s book Income, Wealth, and the Maxi- mum Principle (Harvard University Press, 2003). Assumptions: ° There is a single composite good produced with the constant-returns production function for total output, Y = F ( K; N ) : Here, N is popu- lation, which I assume equal to the (fully employed) labor force. (Feel free to add labor-augmenting technical change as an exercise.) ° Population growth is N t = (1 + n ) N t ° 1 . ° A °generation±lives for a period t and maximizes U t = u ( c t ) + ° (1 + n ) U t +1 , where ° (1 + n ) < 1 and c t is the consumption of a representative family member on date t . The idea is that you care about your own consumption and the welfare of your 1 + n children. 1
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° Capital depreciates at the rate ± 2 [0 ; 1] . Because U t = u ( c t )+ ° (1+ n ) u ( c t +1 )+ ° 2 (1+ n ) 2 u ( c t +2 )+ ° 3 (1+ n ) 3 U t +3 , etc., we may assert that the generation born on date t = 0 maximizes U 0 = 1 X t =0 ° t (1 + n ) t u ( c t ) subject to K t +1 ± K t = F ( K t ; N t ) ± N t c t ± ±K t ; K t ² 0 ; K 0 given. Alternatively, we can express the constraints in the intensive form k t +1 = 1 1 + n [ f ( k t ) + (1 ± ± ) k t ± c t ] ; k t ² 0 ; k 0 given, where k ³ K=N; f ( k ) = F ( k; 1) : Ramsey looked at the case n = 0 and ° = 1 . The latter assumption may seem paradoxical from a mathematical point of view (isn³t the in²nite sum de²ning U 0 likely to be divergent then?), but a problem set will show how Ramsey handled it. One simpli²cation is to assume the Inada condition on consumption that lim c ! 0 u 0 ( c ) = 1 . In that case, we can forget about the interim nonnegativity constraints on the capital stock. We will never optimally get close to zero capital, because the marginal utility of consumption would be very high. It will be useful ²rst to solve the ²nite-horizon problem max f c t g T X t =0 ° t (1 + n ) t u ( c t ) subject to k t +1 = 1 1 + n [ f ( k t ) + (1 ± ± ) k t ± c t ] ; (1 + n ) k T +1 ² 0 ; k 0 given.
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