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appendixL3Growth

# appendixL3Growth - Mathematical Appendix of Lecture 3 Part...

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Mathematical Appendix of Lecture 3 Part IIB, Paper 2: Growth Dr. Tiago Cavalcanti Michaelmas 2010 1. Recall that the fundamental equation of the capital stock in intensive form is: ˙ ˜ k ( t ) = s ˜ k ( t ) α - ( δ + n + g ) ˜ k ( t ) , given ˜ k (0) > 0 . (1) This is a non-linear differential equation in ˜ k . The stationary equilibrium is ˜ k * = [ s ( δ + n + g ) ] 1 1 - α . Define x ( t ) = ˜ k ( t ) 1 - α . Therefore: ˙ x ( t ) = (1 - α ) ˜ k ( t ) - α ˙ ˜ k ( t ) ˙ ˜ k ( t ) = ˙ x ( t ) (1 - α ) ˜ k ( t ) - α . (2) Using (2) into (1), we have that: ˙ x ( t ) = (1 - α ) s - (1 - α )( δ + n + g ) x ( t ) . (3) This is a standard linear first-order differential equation in x ( t ). The solution of this equation is the sum of a homogenous solution, x h ( t ), and a particular solution, x p ( t ). (If you are not sure about this, check any mathematical book on first-order differential equation - Alpha Chiang, “ Fundamental Methods of Mathematical Economics ”, for instance, contains material on this.) The particular solution is the one in which ˙ x ( t ) = 0, or x ( t ) = x p : ˙ x ( t ) = 0 x p = s ( δ + n + g ) . The homogenous solution solves: ˙ x ( t ) + (1 - α )( δ + n + g ) x ( t ) = 0 . 1

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The solution is x h ( t ) = Ce - (1 - α )( δ + n + g ) t , where C is a constant that should be determined. The general solution of the differ- ential equation: x
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