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ps10macro2sp08ans - Econ 387L MacroII Spring 2008...

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Econ 387L: MacroII Spring 2008, University of Texas at Austin Instructor : Dean Corbae Problem Set 10 Solution 1. τ t N t + q t B t +1 + δ t B t = B t + G t (1) 2. The Household’s problem is to solve: max T X t =0 β t u ( c t , N t ) (2) subject to c t + q t B t +1 + δ t B t = (1 τ t ) N t + B t r t 1 ,t D t + D t +1 (3) 3. A competitive Equilibrium is a policy sequence ( δ t , τ t ) , an allocation sequence ( c t , N t , D t , B t ) and a price system ( q t , r t ) such that: ( i ) Given prices and taxes, the allocations ( c t , N t , D t , B t ) solve the HH problem. ( ii ) Markets clear: C t + G t = N t D t +1 = 0 B s t +1 = B d t +1 ( iii ) The sequence of taxes ( δ t , τ t ) satis fi es: G t + (1 δ t ) B t = τ t N t + q t B t +1 t = 0 , 1 , 2 , ... 4. From the budget constraint of the HH we have c t + q t B t +1 + δ t B t = (1 τ t ) N t + B t r t 1 ,t D t + D t +1 c t +1 + q t +1 B t +2 + δ t +1 B t +1 = (1 τ t +1 ) N t +1 + B t +1 r t,t +1 D t +1 + D t +2 c t + q t 1 δ t +1 c t +1 q t 1 δ t +1 D t +2 + q t q t +1 1 δ t +1 B t +2 = 1
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= (1 τ t ) N t + q t 1 δ t +1 (1 τ t +1 ) N t +1 r t 1 ,t D t + B t (1 δ t ) + D t +1 1 r t,t +1 1 δ t +1 q t ¸ Hence, in equilibrium we should have: q t 1 δ t +1 = 1 r t,t +1 I.e., arbitrage implies that in equilibrium the rates of return on the two commodities must be the same. 5. A Ramsey Equilibrium is a policy sequence ( δ t , τ t ) , allocation rules ( c t ( δ t , τ t ) , N t ( δ t , τ t )) and price rules ( q t ( δ t , τ t ) , r t ( δ t , τ t )) such that: ( i ) The policy sequence ( δ t
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