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Ih2_SL

# Ih2_SL - The Growth Model In Continuous Time Prof Lutz...

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The Growth Model In Continuous Time Prof. Lutz Hendricks August 6, 2009 L. Hendricks () Continuous time August 6, 2009 1 / 35

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Topics 1 Tools: 1 Solving models in continuous time (optimal control). 2 Phase diagrams. 2 Solow model. 3 Cass-Koopmans / neoclassical growth model. L. Hendricks () Continuous time August 6, 2009 2 / 35
Continuous Time vs. Discrete Time L. Hendricks () Continuous time August 6, 2009 3 / 35

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Continuous time So far, time was divided into discrete "periods." It is often more convenient to shrink the length of periods to 0. Di/erence equations then become di/erential equations. L. Hendricks () Continuous time August 6, 2009 4 ± 35
Continuous time Example: Law of motion for capital Discrete time: K t + 1 K t = I t δ K t (1) More generally: K t + Δ t K t = [ I t δ K t ] Δ t (2) Continuous time ( Δ t ! 0 ) : lim Δ t ! 0 K t + Δ t K t Δ t = ˙ K t = I t δ K t (3) L. Hendricks () Continuous time August 6, 2009 5 / 35

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Growth rates in continuous time g ( x ) = ˙ x x = d ln x dt (4) Growth rate rules: 1 g ( xy ) = g ( x ) + g ( y ) . 2 g ( x / y ) = g ( x ) g ( y ) . 3 g ( x α ) = α g ( x ) . 4 x ( t ) = e γ t = ) g ( x ) = γ . L. Hendricks () Continuous time August 6, 2009 6 / 35
Di/erential equations L. Hendricks () Continuous time August 6, 2009 7 ± 35

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Di/erential equations Take a function of time: x ( t ) = a + bt (5) There is another way of describing this function: Take the derivative: ˙ x ( t ) = dx ( t ) / dt = b (6) Fix x ( 0 ) = a . The two pieces of information (the derivative and x ( 0 ) ) completely describe x ( t ) . Only one function x ( t ) L. Hendricks () Continuous time August 6, 2009 8 ± 35
A di/erential equation (DE) is a function of the form ˙ x ( t ) = f ( x ( t ) , t ) (7) Higher order

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Ih2_SL - The Growth Model In Continuous Time Prof Lutz...

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