appendixL4Growth

# appendixL4Growth - α γ ˙ ˜ k t ⇒ ˙ ˜ k t = ˙ x t(1...

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Mathematical Appendix of Lecture 4 Part IIB, Paper 2: Growth Dr. Tiago Cavalcanti Michaelmas 2010 1. Recall that the fundamental system of equations of the growth model with human capital studied in lecture 4 is: ˙ ˜ k ( t ) = s K ˜ k ( t ) α ˜ h ( t ) γ - ( δ + n + g ) ˜ k ( t ) , given ˜ k (0) > 0 , (1) ˙ ˜ h ( t ) = s H ˜ k ( t ) α ˜ h ( t ) γ - ( δ + n + g ) ˜ h ( t ) , given ˜ h (0) > 0 . (2) Assuming that the returns on physical capital and human capital are the same, we have that: R K = r + δ = α ˜ y ˜ k , R H = r + δ = γ ˜ y ˜ h . Therefore: ˜ h = γ α ˜ k. Using this into equation (1) yields: ˙ ˜ k ( t ) = s K B ˜ k ( t ) α + γ - ( δ + n + g ) ˜ k ( t ) , (3) where B = ( γ α ) γ . This is a non-linear diﬀerential equation in ˜ k . Deﬁne x ( t ) = ˜ k ( t ) 1 - ( γ + α ) . Therefore: ˙ x ( t ) = (1 - ( α + γ )) ˜ k ( t )
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Unformatted text preview: -( α + γ ) ˙ ˜ k ( t ) ⇒ ˙ ˜ k ( t ) = ˙ x ( t ) (1-( α + γ )) ˜ k ( t )-( α + γ ) . (4) Using (4) into (3), we have that: ˙ x ( t ) = (1-( α + γ )) Bs-(1-( α + γ ))( δ + n + g ) x ( t ) . (5) This is a standard linear ﬁrst-order diﬀerential equation in x ( t ). See the mathematical appendix of lecture 3 to see an analogous solution to this equation. Notice that since ˜ h is proportional to ˜ k , they will approach the steady-state at the same rate. 1...
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