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Unformatted text preview: ECO 475 PS3 Solution Yu LIU November 11, 2010 Question 1 (a) Consumer's Problem: Given r , max { c 1 ,c 2 (˜ y 2 ) ,a } u ( c 1 ) + β E u ( c 2 (˜ y 2 )) subject to c 1 + a = y 1 , c 2 (˜ y 2 ) = ˜ y 2 + (1 + r ) a, ∀ ˜ y 2 From FOCs and budget constraints, we can get the Euler equation u ( c 1 ) = β (1 + r ) E u ( c 2 (˜ y 2 )) Rewrite it with budget constraint u ( y 1 a ) = β (1 + r ) E u (˜ y 2 + (1 + r ) a ) (1) (b) From equation (1), using Implicit Function Theorem, we can get da dβ = (1 + r ) E u (˜ y 2 + (1 + r ) a ) β (1 + r ) 2 E u 00 (˜ y 2 + (1 + r ) a ) + u 00 ( y 1 a ) > The agents save more in the rst period if they become more patient. The reason is that they are more afraid of the uncertainty in the second period. (c) From equation (1), using Implicit Function Theorem, we can get da dr = β E u (˜ y 2 + (1 + r ) a ) + β (1 + r ) b E [ u 00 (˜ y 2 + (1 + r ) a )] β (1 + r ) 2 E u 00 (˜ y 2 + (1 + r ) a ) + u 00 ( y 1 a ) Q ⇐⇒ E u (˜ y 2 + (1 + r ) a ) + (1 + r ) a E [ u 00 (˜ y 2 + (1 + r ) a )] Q So the answer depends on the form of utility function and the nature of uncertainty. The reason generates from the balance between wealth e ect and substitution e ect. (d) Since u 000 > , then u is convex. From the de nition of mean preserving spread, E u (˜ y 2 + (1 + r ) a ) increases. Examining both side of equation (1), savings needs to be increased to balance the equation. The reason is that period 2 becomes more risky after a mean preserving spread. 1 Question 2 (a) Consumer's Problem: Given r , V ( a,y ) = max { c,a } u ( c ) + β E V ( a ,y ) subject to c + a 1 + r = y + a a ≥ a a ≥ a given where a ≥ a is the No Ponzi condition. From FOC, envelop condition, and budget constraint, we can get the Euler equation u ( c ) = β (1 + r ) E u ( c ) which is e γc = β (1 + r ) E e γc Let x = y + a. (2) To prove a = δx + b (3) for some constant δ and b , we need to prove that it satis es both the Euler equation and transversality condition....
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This note was uploaded on 09/06/2011 for the course ECO 475 taught by Professor Hong during the Fall '07 term at Rochester.
 Fall '07
 Hong

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