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Unformatted text preview: Macroeconomics 1. Master APE. 20092010. TD7 Prof. Xavier Ragot / T.A : Eric Monnet The rst exercise of the problem set is a take home exam due November, November 17 . Since we will correct it together in class (some of you will present their results), papers must be handed at the beginning of the course. If you cannot attend class this Tuesday ; send me your results by email before class ! Work with another people and give me one paper for two . 1 Convergence in the neoclassical model and the 'augmented Solow model'. This problem is based on Mankiw, Romer and Weil, 1992, (available on the web page of the course). It examines the derivation of the convergence equation you have derived in class, and studies the implications for the convergence hypothesis of the 'augmented Solow model'. Consider rst the traditional neoclassical model where output is given by Y t = ( A t L t ) α K 1 α t . The technology grows at rate x , the population at rate n , and the stock of capital depreciates at rate δ . There is a constant saving rate s . Thus we have in particular : ˙ K t = sY t δK t , where ˙ K t = dK t /dt 1 Write the variables in terms of `e ciency units of labor', that is y t = Y t / ( A t L t ) and k t = K t / ( A t L t ) . 2 Derive an expression of ˙ k t (= dk t /dt ) in function of k t 3 Derive the steady state value of k , denoted k * . Interpret. Then draw a phase diagram showing that the steady state is stable. 4 Derive the steady state level of income in e ciency units denoted y * . 5 Express ˙ y y in function of ˙ k k , and then show that ˙ y t y t = (1 α )( δ + x + n ) " y * y t α 1 α 1 # 1 Macroeconomics 1. Master APE. 20092010. TD7 Prof. Xavier Ragot / T.A : Eric Monnet Interpret this equation in terms of growth 'convergence'. 6 Recall that ˙ y y = d ( logy t ) /dt , and then use a Taylor expansion to nd an expres sion of d log y t dt in function of (log y t log y * t ) . What is the speed of convergence of output toward its steady state ? 7 (optional / bonus points) Solve the previous di erential equation (question 6) in order to obtain an expression of log [ y ( t ) /y (0)] , that will be more tractable to test. (cf p.423 in MRW's article) 8 Mankiw, Romer and Weil then introduce an augmented model adding human capital accumulation to the Solow model. What are the reasons for such a modi cation ? In the new model, output is a function of human capital, H , as well as of labor and physical capital, Y t = ( A t L t ) 1 α β K α t H β t where < α + β < 1 . The gross investment rates in the two types of capital are a fraction s k and s h of output, respectively. Both depreciates at the same rate....
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This note was uploaded on 01/12/2011 for the course ECO 010023 taught by Professor Mrraggillpol during the Fall '09 term at Paris Tech.
 Fall '09
 MrRaggillpol
 Macroeconomics

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