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Unformatted text preview: Macroeconomics 1. Master APE. 2009-2010. TD5 Prof. Xavier Ragot / T.A : Eric Monnet To work on these exercises, you can use your classnotes on overlapping gener- ation models, and Chapter 2 - part B of Romer's textbook for example. 1 Diamond Overlapping Generations Model Consider the Diamond overlapping generations model. L t individuals are born in period t and live for two periods, working and saving in the rst and living o capital in the second period. Assume population is growing at a constant rate, n , and technological progress occurs at exogenous rate g . Markets are competi- tive and labor and capital are paid their marginal products. There is no capital depreciation. Utility is logarithmic with individual discount rate > ρ >- 1 . U = log ( c t ) + 1 1 + ρ log ( c t +1 ) The production function is Cobb-Douglas , with output (Y), capital (K), labor(L), and 'knowledge' or 'e ectiveness of labor'(A) to which g applies : Y = F ( K,AL ) . Inada conditions apply. In intensive/per capita terms (dividing everything by AL) : y = f ( k t ) = k α t 1- Explain how to derive the intensive form of the production function and express both the marginal product of capital (r) and the marginal product of labor (w) in terms of f ( k t ) To do so we need a function with constant return to scale in its two arguments, such that 1 AL F ( K,AL ) = F ( K AL , 1) = f ( k ) . We check that for the Cobb Douglas function we get : f ( k ) = k α . Note that Inada conditions also apply to f . The real intensive interest rate is then equal the extensive rate and follow di- rectly from the derivative of the intensive production function (recall there is no 1 Macroeconomics 1. Master APE. 2009-2010. TD5 Prof. Xavier Ragot / T.A : Eric Monnet depreciation of capital in the model). r t = f ( k ) = αk α- 1 t From the de nition of the intensive production function we get F ( K,AL ) = ALf ( k ) with k = K/AL . Now we nd the partial derivative of the extensive production function in labour and thus obtain the real extensive wage. W = AL.f ( k )[- K/ ( AL ) 2 ] + Af ( k ) W = A.f ( k )- f ( k ) .K/L And since the intensive wage (w) equals (W/A) 1 w = f ( k )- kf ( k ) and w = (1- α ) k α 2- Determine the intertemporal budget constraint for each individual. Set up the consumers utility maximization problem and derive the equilibrium condition (Euler equation). Use that condition and the budget constraint to solve for rst period consumption, c t , and the savings rate (ie, the fraction of income saved), s t . in period 1 ( rst part of the life of an individual), the budget constraint for the representative agent is C t = W t- S t , where W t = A t w t and S t is savings that will be used as capital in the second part of the individual's life....
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