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TDsol5_6 - Macroeconomics 1 Master APE 2009-2010 TD4-5 Prof...

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Unformatted text preview: Macroeconomics 1. Master APE. 2009-2010. TD4-5 Prof. Xavier Ragot / T.A : Eric Monnet Solutions 1 Dynamic Programming Consider a problem of an in nitely-lived rm. Every period this rm uses physical capital, K t , and labor, L t , as inputs to produce nal output, Y t = F ( K t ,L t ) . The rm pays a fraction of its output to the workers in the form of wage payments and then decides how much to invest by choosing I t . In short, objective of the rm is : Max ( I t ,L t ) = ∞ X t =0 1 1 + r t [ F ( K t ,L t )- w t L t- I t ] The production function F ( K t ,L t ) is twice continously di erentiable, strictly in- creasing in both arguments, strictly concave, and satis es Inada conditions. r is a constant interest rate in the economy, while w t is the real wage rate. Every period t , the rm also faces a constraint : K t +1 = (1- δ ) K t + I t where δ is the depreciation rate of capital. K is given. 1- Interpret the objective of the rm and the budget constraint (transition equa- tion). What is the state variable in this problem ? What are the control variables ? The rm maximizes its pro t (production minus cost of labour and investment) intertemporaly under the constraint of the evolution of capital. The transition equation of capital gives you the state variable : K t . Decisions variables are I t and L t . As usual,using the transition equation,you can rewrite the program taking K t +1 as a control variable. 2- Write the Bellman equation for this problem. V ( K t ) = max L t ,K t +1 F ( K t ,L t )- w t L t- ( K t +1- (1- δ ) K t + 1 1 + r V ( K t +1 ) 1 Macroeconomics 1. Master APE. 2009-2010. TD4-5 Prof. Xavier Ragot / T.A : Eric Monnet 3- Derive the rst order conditions using the Bellman equation. Find and provide careful interpretation of the optimality conditions (for the choice of K and L) of the rm behavior. F L ( K t ,L t )- w t = 0- 1 + 1 1 + r V ( K t +1 ) = 0 and Envelope condition : V ( K t ) = F K ( K t ,L t ) + 1- δ Combining these conditions yields : F L ( K t ,L t ) = w t marginal product of labor is equal to wage rate on the optimal path. F K ( K t ,L t ) = r + δ marginal product of physical capital is equal to gross return on physical capital since physical capital will depreciate at rate δ in the process of production. 4- Write the intertemporal budget constraint and nd the transversality condition of this sequential problem. Provide an interpretation of this equation. Iterating the transition equation of capital until t + 1 , you get K t +1 (1- δ ) t = (1- δ ) K + t +1 X t =0 I t (1- δ ) t The transversality condition is lim →∞ K t +1 (1- δ ) t = 0 . This expression is equivalent to the general form of transversality condition in a Bellman program lim →∞ K t +1 V ( K t +1 ) (1 + r ) t +1 = 0 , which can be rewritten here (using FOC) lim →∞ K t +1 (1+ r ) t = 0 Thus, equivalence between the two conditions required that r =- δ when t tends toward in nity. This 2 Macroeconomics 1. Master APE. 2009-2010. TD4-5 Prof. Xavier Ragot / T.A : Eric MonnetProf....
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