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Unformatted text preview: ECN/APEC 7240 Spring 2010 Homework 4 Solutions 1) We augment the Consumption RBC Model with CobbDouglas production and CRRA utility by including government spending financed by lumpsum taxes. Let T t be the tax at t with t = ln T t = t + t 1 . We assume that t = + gt and t 1 is an AR(1) process such that , 1 1 1 1 + + + = t t t where ( 1,1) and t is i.i.d and uncorrelated with t a . = T / Y 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 + = 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  + = = A A K A K R A K A K W Thus in equilibrium, where K a = K , the budget constraint implies the incomeexpenditure identity: . ) 1 ( ) , ( ) , ( 1 K A K K A K R A K W  + = + K Y T K C ) 1 ( '  + = + + . The Lagrangian for the households 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 + + + = . The firstorder conditions are ECN/APEC 7240 Spring 2010 ) ( ' = = C u C L ] ,  ) ' , ' ), , ( ' , ' ( [ ' = = 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 = . Combining these equations, we get the Euler equation )], ' ( ' ) ' , ' ( [ ) ( ' C u A K R E C u = 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 b) For the variable X t , let X t be the balanced growth path, so for an extrinsic variable X t = X G t and for an intrinsic variable X t = X , where G is the gross growth rate , 1 t t A A G + = where A = 1. On a balanced growth path, we must have ) 1 ( K Y T C GK  + = + +  = ) ( ) ( GC R C  + = 1 ) ( 1 K R . ) ( K Y = Thus G R 1 = ECN/APEC 7240 Spring 2010  + = 1 K Y R + = + = 1 1 1 G R Y K . 1 1 1 ) 1 ( 1 1  + + = + = G G Y T Y K G Y C For a variable X , let x = ln X , x t = ln X t , and x t 1 = x t x t . Also let g = ln G . Then ln = g r + r Y K . 1  + + r g Y C c) Let us assume 1 1 1 1 t c t ca t ck t a k c + + = 1 1 1 1 t y t ya t y t a k y + + = ....
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This note was uploaded on 01/09/2011 for the course ECON 7230 taught by Professor Feigenbaum during the Spring '10 term at Utah Valley University.
 Spring '10
 Feigenbaum
 Macroeconomics, Utility

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