Dynamic Macroeconomic Modeling with Matlab

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4 Numerical Simulation of Macroeconomic Models and Perli, 1994, for the derivation) ˙k = Ak α h 1−α+γ u 1−α − c (52) ˙h = δ(1 − u)h (53) ˙c = σ −1 c(αAk α−1 h 1−α+γ u 1−α − ρ) (54) (γ − α)δ ˙u = u (1 − u) + α δ c − α k . (55) Balanced growth requires u, c/k as well as k α−1 h 1−α+γ to be constant. The latter requirement in turn demands (1 − α) ˙ k k = (1 − α + γ) ˙ h h . In this simple case the common balanced growth rate µ of k and c can be computed by solving the system under balanced growth assumptions: µ = 1 − α + γ (δ − ρ) (1 − α + γ)σ − γ Growth is balanced if the four variables of the system satisfy the three following equations: 1 − u = k α−1 h 1−α+γ = 1 − α (1 − ρ/δ) (56) (1 − α + γ)σ − γ c/k = ((γ − α)ψµ + δ)/α (57) σµ + ρ αA uα−1 where ψ := (1 − α)/(1 − α + γ). We construct a stationary system by employing scale- adjustment. The transformed variables are ke −µ t , he −ψµ t −µ t , ce and u. To avoid extra notation we continue to use the old designations of variables. The new, adjusted growth rates are reduced by the constants of adjustment, µ and ψµ, respectively. The growth rate of u remains unchanged. Therefore, the transformed system reads (58) ˙k = Ak α h 1−α+γ u 1−α − c − µk (59) ˙h = δ(1 − u)h − ψµh (60) ˙c = σ −1 c(αAk α−1 h 1−α+γ u 1−α − ρ) − µc (61) (γ − α)δ ˙u = u (1 − u) + α δ c − α k . (62)
4 Numerical Simulation of Macroeconomic Models Due to scale adjustment, balanced growth solutions represented by equations (56), (57) and (58) now turn into stationary points of system (59) - (62). 1 Therefore, the system exhibits a curve of stationary equilibria (see Figure 8). This means that a unique trajectory is given by initial values of physical and human capital. However, the final steady state, to which the economy converges, is not unique but depends on the initial values of physical and human capital. h e h 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 1.5 1 −ψµt CSE 0.5 Unscaled Variables Scaled Variables 0 0 0.5 1 1.5 2 2.5 3 k k e −µt Figure 8: Phase diagram of the Lucas model Numerical computation employing the relaxation algorithm requires the solution of the system of differential equations (59) - (62) with two initial conditions and two final conditions. The initial conditions are given by the initial values of state variables k(0) = k0, h(0) = h0. Fi- nal conditions should ensure convergence towards the center manifold given by equations (56), 1 It is straightforward to verify that the right hand sides of system (59) - (62) are linearly dependent, and that equations (56), (57) and (58) are the only solution. BGP
Page 1 and 2: Dynamic Macroeconomic Modeling with
Page 3 and 4: 1 Introduction 1 Introduction The a
Page 5 and 6: 2 An Outline of the Theory x(t) 1 0
Page 7 and 8: 2 An Outline of the Theory South-Ea
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Page 19 and 20: 4 Numerical Simulation of Macroecon
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Page 37 and 38: References References Ascher, U. M.
Page 39: References Solow, R. M. (1956): “

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