ps5_101_2011_sol

# ps5_101_2011_sol - ECON 101 SOLUTIONS TO PS 5 GARTH...

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ECON 101 SOLUTIONS TO PS 5 GARTH BAUGHMAN (1) Suppose Sheila receives an income stream of income ( I 0 ,I 1 ) over two periods. Her preferences over consumption in the two periods are described by the utility function U ( c 0 ,c 1 ) = log c 0 + 1 (1 + γ ) log c 1 . (a) Suppose Sheila can borrow and lend at the same interest rate r . Derive her demand functions for period-0 and period-1 consumption. Solution. The problem we need to solve is max c 0 ,c 1 U ( c 0 1 ) s.t c 0 (1 + r ) + c 1 = I 0 (1 + r ) + I 1 . Get ﬁrst order conditions: w.r.t c 0 : 1 c 0 + λ (1 + r ) = 0 w.r.t c 1 : 1 (1 + γ ) c 1 + λ = 0 w.r.t c 0 : c 0 (1 + r ) + c 1 = I 0 (1 + r ) + I 1 . The ﬁrst two give us our familiar MRS equals the price ratio, (1 + γ ) c 1 c 0 = (1 + r ) = c 0 = 1 + γ 1 + r c 1 and so the budget condition implies c 1 (1 + γ + 1) = (1 + r ) I 0 + I 1 This gives us demand for consumption in period 1 c * 1 ( r,I 0 1 ) = I 0 (1 + r ) + I 1 2 + γ . Substituting back above we get demand for consumption in period 0: c * 0 ( 0 1 ) = ± 1 + γ 2 + γ ²± I 0 (1 + r ) + I 1 1 + r ² . (b) Suppose that ﬁnancial transactions are subject to a ﬁnancial services tax, τ , so that if Sheila saves x , she receives x (1 + r (1 - τ )) in period 1, while borrowing x for period-0 consumption, requires her to repay x (1 + r (1 + τ )) in period 1. (i) What is her budget constraint? Illustrate in a diagram. 1

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2 GARTH BAUGHMAN Solution. The constraint is: c 1 = ( ( I 0 - c 0 )(1 + r (1 - τ )) + I 1 if c 0 I 0 ( I 0 - c 0 )(1 + r (1 + τ )) + I 1 if c 0 > I 0 . In a picture, this is (ii) Derive her demand functions period-0 and period-1 consumption. Solution. So, we have one of three cases. Either we have an interior solution on the upper part, an interior solution on the lower part, or a corner solution at the kink. We know which depending upon MRS at the kink. If MRS between c 0 and c 1 is greater than or equal to the price ratio, then we are on the lower piece. If MRS is smaller than or equal to the price ratio, then we are on the upper piece. If MRS is inbetween these two values then we are at the kink. Note, that MRS tells us rate we are willing to trade, so if MRS between c 0 and c 1 is greater than the price
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## This note was uploaded on 03/02/2011 for the course ECON 101 taught by Professor Dannicatambay during the Spring '08 term at UPenn.

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ps5_101_2011_sol - ECON 101 SOLUTIONS TO PS 5 GARTH...

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