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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 11

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online