Handout 5 Key - + d ) k G ) 86 : 60 ( : 15) (192 : 45) ) c...

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Econ 3140 Fall 2009 Handout #5 with answers 1. (a) We have Y t = F ( K t; N t ) = AK 1 = 3 t N 2 = 3 t dividing both sides by N t y t = f ( k t ) = AK 1 = 3 t N 1 = 3 t ) f ( k t ) = 15 k 1 = 3 t , since k t = K t N t . (b) The condition to get k max : k max ) f ( k t ) = ( n + d ) k t ) 15 k 1 = 3 t = ( : 03 + : 12) k t ) k 2 = 3 t = 100 ) k t = (100) 3 = 2 ) k max = 1000 (c) f ( k max ) = 15 (1000) 1 = 3 = 150 (d) c ( k max ) = 0 , since all the output is used to replace the capital stock and then nothing is left to consume. (e) The condition to get k : k ) s f ( k t ) = ( n + d ) k t ) ( : 25) ± 15 k 1 = 3 t = ( : 03 + : 12) k t ) k 2 = 3 t = 25 ) k t = (25) 3 = 2 ) k = 125 (f) y ) f ( k ) = 15 ± (125) 1 = 3 ) y = 75 (g) c = f ( k ) ² ( n + d ) k ) 75 ² ( : 15) ± (125) ) c = 56 : 25 (h) The condition to get k G is given by: k G ) @c @k = 0 @c @k = 0 ) @f ( k t ) @k ² @ ( n + d ) k t @k = 0 ) 15 3 k ± 2 = 3 = 3 20 ) k 2 = 3 = 300 9 ) 300 9 ± 3 = 2 ) k G = 192 : 45 (i) y G ) f k G ± = 15 ± (192 : 45) 1 = 3 ) y G = 86 : 60 (j) c G = f k G ± ² ( n
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Unformatted text preview: + d ) k G ) 86 : 60 ( : 15) (192 : 45) ) c G = 57 : 73 (k) Since k < k & , that is, savings > amount of investment needed to keep k constant, so k rises until it reaches the steady-state value of the capital labor ratio k & = 125 . (l) Since k > k & , that is, savings < amount of investment needed to keep k constant, so k decreases until it reaches the steady-state value of the capital labor ratio k & = 125 . 1...
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