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homework4 2010 solutions

# homework4 2010 solutions - ECN/APEC 7240 Spring 2010...

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ECN/APEC 7240 Spring 2010 Homework 4 Solutions 1) We augment the Consumption RBC Model with Cobb-Douglas production and CRRA utility by including government spending financed by lump-sum taxes. Let T t be the tax at t with τ t = ln T t = τ t 0 + τ t 1 . We assume that τ t 0 = τ 0 + gt and τ t 1 is an AR(1) process such that , 1 1 1 1 τ ε φτ τ + + + = t t t where φ ( - 1,1) and ε t τ is i.i.d and uncorrelated with ε t a . ζ = T 0 / Y 0 is exogenous. a) The Bellman equation for this model is ] , | ) ' , ' ), , ( ' , ' ( [ ) ( max ) , , , ( ' , T A T A A K K K V E C u T A K K V a a K C a β + = subject to . ) , ( ) , ( ' K A K R A K W T K C a a + = + + Note that the factor prices will still be . 1 ) , ( ) 1 ( ) , ( 1 1 δ α α α α - + = - = - - A A K A K R A K A K W Thus in equilibrium, where K a = K , the budget constraint implies the income-expenditure identity: . ) 1 ( ) , ( ) , ( 1 K A K K A K R A K W δ α α - + = + - K Y T K C ) 1 ( ' δ - + = + + . The Lagrangian for the household’s problem is ] ' ) , ( ) , ( [ ] , | ) ' , ' ), , ( ' , ' ( [ ) ( T K C K A K R A K W T A T A A K K K V E C u L a a a a - - - + + + = λ β . The first-order conditions are

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ECN/APEC 7240 Spring 2010 0 ) ( ' = - = λ C u C L 0 ] , | ) ' , ' ), , ( ' , ' ( [ ' = - = λ β T A T A A K K K V E K L a a K The Envelope Theorem implies ) , ( ) , , , ( A K R T A K K V a a K λ = . Combining these equations, we get the Euler equation )], ' ( ' ) ' , ' ( [ ) ( ' C u A K R E C u β = where we impose the equilibrium condition K = K a and K = K a . In time series form, this is )]. ( ' [ ) ( ' 1 1 + + = t t t t C u R E C u β b) For the variable X t , let X t 0 be the balanced growth path, so for an extrinsic variable X t 0 = X 0 G t and for an intrinsic variable X t 0 = X 0 , where G is the gross growth rate , 0 0 1 t t A A G + = where A 0 = 1. On a balanced growth path, we must have 0 0 0 0 0 ) 1 ( K Y T C GK δ - + = + + θ θ β - - = ) ( ) ( 0 0 0 GC R C δ α α - + = - 1 ) ( 1 0 0 K R . ) ( 0 0 α K Y = Thus θ β G R 1 0 - =
ECN/APEC 7240 Spring 2010 δ α - + = 1 0 0 0 K Y R δ β α δ α θ + - = + - = - 1 1 1 0 0 0 G R Y K . 1 1 1 ) 1 ( 1 1 0 0 0 0 0 0 ζ δ β δ α δ θ - + - + - - = - - - + = - G G Y T Y K G Y C For a variable X , let x = ln X , x t 0 = ln X t 0 , and x t 1 = x t - x t 0 . Also let g = ln G . Then β θ ln 0 - = g r δ α + 0 0 0 r Y K . 1 0 0 0 ζ δ δ α - + + - r g Y C c) Let us assume 1 1 1 1 t c t ca t ck t a k c τ η η η τ + + = 1 1 1 1 t y t ya t y t a k y τ η η η τ τ + + = . 1 1 1 1 1 t k t ka t kk t a k k τ η η η τ + + = + The production function implies t t t a k y ) 1 ( α α - + = . Since above we have Y 0 = ( K 0 ) α , y 0 = α k 0 . ] )[ 1 ( ] [ 1 1 0 1 0 t t t a gt k gt k y gt y + - + + + = + + α α This implies . ) 1 ( 1 1 1 t t t a k y α α - + = Thus we still have η yk = α , η ya = 1 - α , and now we also have η y τ = 0. Taxes have no

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ECN/APEC 7240 Spring 2010 direct effect on output here since output at t is entirely determined by K t and A t , which are predetermined at t . (This would not be the case if we endogenize labor, which will depend on taxes.) The income-expenditure identity can be written . 1 1 1
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homework4 2010 solutions - ECN/APEC 7240 Spring 2010...

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