problem4sol - Solution to Last Problem Set Econ 210 Core...

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Unformatted text preview: Solution to Last Problem Set Econ 210 Core Macro Monika Piazzesi Stanford University October 21, 2010 1. Tree-cutting (a) The Bellman equation is ( ) = max { ( ( )) } Intuitively, the tree is growing and thus getting more and more valuable. We want to have a rule that says that we should keep the tree when the growth rate is larger than the interest rate and cut the tree when its growth rate is less than the interest rate. The rule would be +1 keep the tree +1 cut the tree Here are the assumptions: A1: (0) because the tree is growing A2: We need boundedness, so lets assume that the state space is bounded, so where is big A3: ( ) is increasing (a larger tree today will lead to an even larger tree tomorrow). A4: ( ) crosses only once: from above (except possibly zero.) This means that there exists a such that for all , we have ( ) and for all we have that ( ) 1 Under these assumptions, lets check that Blackwells su cient conditions hold: B1: monotonicity: If ( ) ( ) for all , then ( ( )) ( ( )) and max { ( ( )) } max { ( ( )) } B2: discounting: max { [ ( ( )) + ] } max { + [ ( ( )) + ] } = max { ( ( )) } + Hence, there exists a unique value function ( ) (a f xed point of the operator). ( ) is increasing: If ( ) ( 00 ) for all 00 , then ( ( )) ( ( 00 )) and max { ( ( )) } max { ( ( 00 )) } Hence limiting ( ) is also increasing. Proof of existence of a simple rule 1. Assume that for all we have that ( ) Then max { ( ( )) } max { ( ) } = max { ( ) } = ( ) = which means that once we start from some function satisfying all the assumptions, all the future iterations also stick to it. So does the f xed point. Hence, we have shown that for all ( ) . Since ( ) by de f nition, we can state that for all we have that ( ) = Hence, we should cut down the tree for all 2. Assume that for all we have that ( ) . Then max { ( ( )) } max { ( ) } = max { ( ) } = ( ) = 2 which means that once we start with some function satisfying the assumption all the future iterations also stick to it. So does the f xed point. Hence, we have shown that for allxed point....
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problem4sol - Solution to Last Problem Set Econ 210 Core...

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