ans4 - ECON 831 Answers Homework #4 Exercise 1:...

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Unformatted text preview: ECON 831 Answers Homework #4 Exercise 1: (Consumption-savings problem)- First, we consider the 1-period problem starting at T for any state w : V T ( w ) = max c ∈ [0 ,w ] √ c One has to maximize a strictly increasing function over a compact set. We easily deduce that the optimal choice is c = w . The associated value function is: V T ( w ) = √ w .- Second, we consider the 2-period problem starting at ( T- 1) for any state w : V T- 1 ( w ) = max c ∈ [0 ,w ] h √ c + p (1 + r )( w- c ) i FOC: 1 2 √ c- √ 1 + r 2 √ w- c = 0 ⇔ c = w 1 + (1 + r ) The associated value function writes: V T- 1 ( w ) = r w 2 + r + r (1 + r )( w- w 2 + r ) = p 1 + (1 + r ) √ w- Third, we consider the 3-period problem starting at ( T- 2) for any state w : V T- 2 ( w ) = max c ∈ [0 ,w ] h √ c + p 1 + (1 + r ) p (1 + r )( w- c ) i FOC: 1 2 √ c- p 1 + (1 + r ) √ 1 + r 2 √ w- c = 0 ⇔ c = w 1 + (1 + r ) + (1 + r ) 2 The associated value function writes: V T- 2 ( w ) = r w 1 + (1 + r ) + (1 + r ) 2 + r (1 + r )( w- w 1 + (1 + r ) + (1 + r ) 2 ) = p 1 + (1 + r ) + (1 + r ) 2 √ w 1- Guess: given the current state w , the optimal consumption and value function are, c T- t ( w ) = w ∑ t i =0 (1 + r ) i and V T- t ( w ) = √ w v u u t t X i =0 (1 + r ) i We prove it by recurrence. Suppose that c t +1 and V t +1 write as above; we need to show...
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This note was uploaded on 02/19/2010 for the course ECON 831 taught by Professor Antoine during the Fall '09 term at Simon Fraser.

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ans4 - ECON 831 Answers Homework #4 Exercise 1:...

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