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Unformatted text preview: Macroeconomics 2. Master APE. 20092010. 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. 20092010. 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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 Spring '10
 J.Baran
 Macroeconomics, Probability theory, Markov chain, Eric Monnet, Prof. Christian Haefke, Master APE.

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