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Macro2PS6sol

# Macro2PS6sol - Macroeconomics 2 Master APE 2009-2010 PS6...

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Unformatted text preview: Macroeconomics 2. Master APE. 2009-2010. PS6 Prof. Christian Haefke / T.A : Eric Monnet 1 Exercise 14 a) c ( t ) + p ( z ( t )) x ( t + 1) ≤ z ( t ) x ( t ) + p ( z ( t )) x ( t ) The right hand side of the constraint is the income of the household. The term z ( t ) x ( t ) is the amount of consumption goods delivered at time t by the claims on the tree and p ( z ( t ) x ( t ) is the market value of the claims. The left hand side is the expenditure of the household : he spends c(t) and reinvests the remaining to buy x(t+1) at the the current market price p(t). b) Given a stationary price function p(z), the payo relevant state variables for a household are her current claims on the tree, x, and the current state, z. The policy function is y = π ( x ) .Then we have : V ( x,z ) = max c> ,y> u ( c ) + βE ( V ( y,z ) | z ) s.t to c + p ( z ) y ≤ [ z + p ( z )] x . which can be rewritten, V ( x,z ) = max { u ( z + p ( z )) x- p ( z ) y ) + βE ( V ( y,z ) | z ) } where y ∈ [0 ,p ( z )- 1 ( z + p ( z ) x )] as an equivalent of c ≥ because c = ( z + p ( z )) x- p ( z ) y . c) Many ways to prove it : see your classnotes. One quick way is to use the result of question e) (below), that allows you to reduce the solutions of this problem to a compact set X = [0 , 1] , and so the constraint set is y ∈ [0 ,p ( z )- 1 ( z + p ( z ) x )] ∩ X . Then it is straightforward to show that the equivalent of Assumptions 1 and 2 for stochastic dy- namic programming (and so theorem 3 ) hold (cf handout on dynamic programming). Since U is strictly concave and the constraint set is convex, there is a unique solution to this problem (theorem 4). d) the FOC are p ( z ) u (( z + p ( z )) x- p ( z ) y ) = βE [ V y ( y,z ) | z ] ane the envelope condition V x = (( z + p ( z )) u ( c ( t )) Updating the envelope equation to the next period and substituting it into the FOC, we have p ( z ) u ( c ( t )) = βE (( z + p ( z )) u ( c ( t + 1) e) The market clearing condition for claims on tree is x ( t ) = 1 . This condition is su cient for 1 Macroeconomics 2. Master APE. 2009-2010. PS6 Prof. Christian Haefke / T.A : Eric Monnet market clearing since when each individual holds one unit of the tree at all times the aggre- gate holding of claims necessarily equates aggregate supply of claims, which is also one unit per...
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Macro2PS6sol - Macroeconomics 2 Master APE 2009-2010 PS6...

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