hw-03-sol - Macroeconomic Theory I Viktor Tsyrennikov...

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Homework 3 Macroeconomic Theory I. Viktor Tsyrennikov Question 1. Government needs to finance a stream of government pur- chases g t > 0 , t . It can raise tax revenue T t in any period t up to the total income Y t > 1. Government seeks to minimize the welfare cost imposed on the economy: - summationdisplay t =0 β t 0 . 5 T 2 t . Government can borrow and lend at a constant interest rate R . a) Compute the optimal tax and borrowing plan { T t , D t } for the govern- ment that starts with no debt: D 0 = 0. Solution. Government’s budget constraint is: D t +1 = R t +1 ( D t + T t - g t ) . The system of FOCs is: T t : 0 = - β t T t + λ t R t +1 , D t +1 : 0 = - λ t + λ t +1 R t +2 . They can be combined to obtain the following Euler equation: T t = βR t +1 T t +1 . When R t = R, t then T t = ( βR ) t T 0 . Using the intertemporal budget constraint: summationdisplay t R t T t = T 0 / (1 - βR 2 ) = summationdisplay t R t g t G. So, T t = ( βR ) t (1 - β 1 R 2 ) G . Debt plan can be computed recursively using budget constraint.
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