finalmacro2sp04ans

# finalmacro2sp04ans - Econ 387L Macro II Spring 2004...

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Econ 387L: Macro II Spring 2004, University of Texas Instructor: Dean Corbae Answers - Final Exam Please answer each question on separate sheets. Remember to adjust the time you spend on each question in proportion to what they are worth. (1) 30 points. Consider the following two period problem where the government can tax capital gains and/or bequests to fund an exogenous amount of government expenditure. There is a unit measure of households (HHs) who make a portfolio choice at t =0 and a bequest decision at t =1 . Their preferences are given by u ( c )+ αu ( b ) where c 0 is consumption at t , b 0 are bequests for one’s children chosen at t , and α> 0 measures the relative importance of bequests vs own consumption. Assume. u 0 ( · ) > 0 , u 00 ( · ) < 0 , and u 0 (0) = . Each HH receives an endowment ω> 0 at t andatthattime must either make a storage decision in tax-free assets a 0 yielding gross return equal to 1 at t or a productive asset k 0 yielding R> 1 which is proportionately taxed at rate δ [0 , 1] in period t . Assume that if HHs are indifferent between storing in a or k, they save in k. Then at t , HHs choose how much to consume c 0 or leave bequests b 0 . At t , HH’s have to pay τb for any bequests they leave where τ [0 , 1] . Taxes are used to f nance government expenditure G at t . a) 20 points. Assume government announces a tax package ( δ,τ ) at t and can commit to it. De f ne a Ramsey equilibrium. Be explicit about the HH’s choice problem. What are the optimal capital gains δ taxes? How do bequests react to changes in τ ? Do HHs leave bequests? Are bequest taxes 0 or 100% with commitment? Answer: HH problem max c,k,a,b u ( c αu ( b ) s.t.a + k = ω c + b = R (1 δ ) k + a As long as R (1 δ )=1 or δ = R 1 R , HHs will choose k = ω and a . In that case, HHs solve max b u ( R (1 δ ) ω (1 + τ ) b αu ( b ) satisfy (1 + τ ) u 0 ( ω (1 + τ ) b )= αu 0 ( b ) (1) To see bequest taxes τ decrease bequests, use implicit function theorem: F ( τ,b ) (1 + τ ) u 0 ( ω (1 + τ ) b ) αu 0 ( b )=0 ⇐⇒ db = F τ F b db = u 0 ( c )+(1+ τ ) u 00 ( c ) b (1 + τ ) 2 u 00 ( c ) αu 00 ( b ) < 0 We also must check that the government budget constraint is satis f ed. To get a suf f cient 1

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condition invert (1) to yield ω (1 + τ ) b = u 0 1 μ α (1 + τ ) b and take τ =1 b ( τ =1)= ω £ u 0 1 ¡ α 2 ¢ +2 ¤ In that case ( R 1) ω + τb > ( R 1) ω + b ( τ =1) , since the max of τb could always be τ . (b) 10 points. Assume that there is no commitment on the part of the government with respect to its announced tax package. That is, after households choose their portfolio in t =0 , the government chooses taxes in t , and then households choose their bequests. De f ne and solve for a subgame perfect equilibrium. What are equilibrium ( δ,τ ) ? Compare tax revenues in part (a) and (b). Answer: If the government chooses ( ) after the household moves at t , then it will choose δ . Given this the household chooses k and a = ω.
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finalmacro2sp04ans - Econ 387L Macro II Spring 2004...

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