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Solutions_to_PS7_Fall2011

# Solutions_to_PS7_Fall2011 - Solutions to PS7 Econ 202A...

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˙ q ( t ) = rq ( t ) - π ( K ( t )) ˙ K ( t ) = f ( q ( t )) ( ¯ q, ¯ K ) bracketleftBigg ˙ q ( t ) ˙ K ( t ) bracketrightBigg = bracketleftBigg A B C 0 bracketrightBigg bracketleftBigg q ( t ) - ¯ q K ( t ) - ¯ K bracketrightBigg bracketleftBigg ˙ q ( t ) ˙ K ( t ) bracketrightBigg = bracketleftBigg ˙ q ( t ) ∂q ( t ) ˙ q ( t ) ∂K ( t ) ˙ K ( t ) ∂q ( t ) ˙ K ( t ) ∂K ( t ) bracketrightBigg | SS bracketleftBigg q ( t ) - ¯ q K ( t ) - ¯ K bracketrightBigg bracketleftBigg ˙ q ( t ) ˙ K ( t ) bracketrightBigg = bracketleftBigg r - π ( ¯ K ) f (1) 0 bracketrightBigg bracketleftBigg q ( t ) - ¯ q K ( t ) - ¯ K bracketrightBigg - π ( ¯ K ) > 0 π ( K ( t )) < 0 f ( q ( t )) > 0 A, B, C > 0 G bracketleftBigg r - π ( ¯ K ) f (1) 0 bracketrightBigg G G - λI det ( G - λI ) = 0 det bracketleftBigg r - λ - π ( ¯ K ) f (1) - λ bracketrightBigg = 0

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- ( r - λ ) λ + π ( ¯ K ) f (1) = 0 λ 2 - + π ( ¯ K ) f (1) = 0 λ 1 , 2 = r ± radicalBig r 2 - 4 π ( ¯ K ) f (1) 2 λ 1 = r + radicalBig r 2 - 4 π ( ¯ K ) f (1) 2 > 0 and λ 2 = r - radicalBig r 2 - 4 π ( ¯ K ) f (1) 2 < 0 r = r 2 - 4 π ( ¯ K ) f (1) > 0 . r < radicalBig r 2 - 4 π ( ¯ K ) f (1) λ 2 < 0 X = bracketleftBigg λ 1 f (1) λ 2 f (1) 1 1 bracketrightBigg X X 1 GX = Λ X = bracketleftBigg v 11 v 21 v 12 v 22 bracketrightBigg and Λ = bracketleftBigg λ 1 0 0 λ 2 bracketrightBigg v 12 = v 22 = 1 v 11 v 21 GX = X Λ bracketleftBigg r - π ( ¯ K ) f (1) 0 bracketrightBigg bracketleftBigg v 11 v 21 1 1 bracketrightBigg = bracketleftBigg v 11 v 21 1 1 bracketrightBigg bracketleftBigg λ 1 0 0 λ 2 bracketrightBigg bracketleftBigg rv 11 - π ( ¯ K ) rv 21 - π ( ¯ K ) f (1) v 11 f (1) v 21 bracketrightBigg = bracketleftBigg λ 1 v 11 λ 2 v 21 λ 1 λ 2 bracketrightBigg v 11 = λ 1 f (1) v 21 = λ 2 f (1) X = bracketleftBigg λ 1 f (1) λ 2 f (1) 1 1 bracketrightBigg
bracketleftBigg ˜ q ( t ) ˜ K ( t ) bracketrightBigg = X 1 bracketleftBigg q ( t ) - ¯ q K ( t ) - ¯ K bracketrightBigg ˙ ˜ q ( t ) ˙ ˜ K ( t ) = X 1 bracketleftBigg ˙ q ( t ) ˙ K ( t ) bracketrightBigg bracketleftBigg ˙ q ( t ) ˙ K ( t ) bracketrightBigg = G bracketleftBigg q ( t ) - ¯ q K ( t ) - ¯ K bracketrightBigg X 1 bracketleftBigg ˙ q ( t ) ˙ K ( t ) bracketrightBigg = X 1 G bracketleftBigg q ( t ) - ¯ q K ( t ) - ¯ K bracketrightBigg ˙ ˜ q ( t ) ˙ ˜ K ( t ) = Λ X 1 bracketleftBigg q ( t ) - ¯ q K ( t ) - ¯ K bracketrightBigg X 1 GX = Λ GX = Λ X 1 ˙ ˜ q ( t ) ˙ ˜ K ( t ) = bracketleftBigg λ 1 0 0 λ 2 bracketrightBigg bracketleftBigg ˜ q ( t ) ˜ K ( t ) bracketrightBigg ˙ ˜ q ( t ) ˙ ˜ K ( t ) = bracketleftBigg ˜ λ 1 q ( t ) ˜ λ 2 K ( t ) bracketrightBigg bracketleftBigg ˜ q ( t ) ˜ K ( t ) bracketrightBigg = bracketleftBigg ˜ q (0) e λ 1 t ˜ K (0) e λ 2 t bracketrightBigg X X bracketleftBigg ˜ q ( t ) ˜ K ( t ) bracketrightBigg = X bracketleftBigg ˜ q (0) e λ 1 t ˜ K (0) e λ 2 t bracketrightBigg bracketleftBigg q ( t ) - ¯ q K ( t ) - ¯ K bracketrightBigg = bracketleftBigg λ 1 f (1) λ 2 f (1) 1 1 bracketrightBigg bracketleftBigg ˜ q (0) e λ 1 t ˜ K (0) e λ 2 t bracketrightBigg

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bracketleftBigg q ( t ) - ¯ q K ( t ) - ¯ K bracketrightBigg = bracketleftBigg ˜ q (0) λ 1 f (1) e λ 1 t + λ 2 f (1) ˜ K (0) e λ 2 t ˜ q (0) e λ 1 t + ˜ K (0) e λ 2 t
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Solutions_to_PS7_Fall2011 - Solutions to PS7 Econ 202A...

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