Dynamic Macroeconomic Modeling with Matlab

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4 Numerical Simulation of Macroeconomic Models (57) and (58). However, the relaxation algorithm can only incorporate two final boundary conditions. Therefore, we incorporate stationary conditions for the state variables, implicitly defined by ˙ k(∞) = 0 and ˙ h(∞) = 0. The set of valid parameter values has been investigated by Benhabib and Perli (1994). They define the subsets Θ1 = Θ2 = ρ(α − 1) (A,α,γ,δ,ρ,σ) > 0 0 < ρ < δ ∧ σ > 1 + (63) δ(1 − α + γ) δ(1 − α + γ) (A,α,γ,δ,ρ,σ) > 0 δ < ρ < ∧ α − 1 ρ(α − 1) σ < 1 + . (64) δ(1 − α + γ) For parameter sets originating from Θ1, the balanced growth path and trajectories converging towards the balanced growth path are unique. For parameter sets originating from Θ2, the balanced growth path is unique, but for local transitional dynamics two different cases occur. Θ2 can further be divided into Θ A 2 and Θ B 2 . If parameters from Θ B 2 are chosen, the balanced growth path is unstable. For parameters from Θ A 2 , the case of indeterminacy occurs, i.e. for given initial endowments of physical and human capital, there exists infinitely many trajectories converging towards the balanced growth path. We focus on the case of determinate adjustment. 4.3.2 Exercises 1. Solve adjustment dynamics for the Lucas model employing the parameter set A = 1, α = 0.3, δ = 0.1, γ = 0.3, σ = 1.5 and ρ = 0.05. Show that the economy converges towards different long-run values by simulating transitional dynamics for different initial conditions. 2. Plot the original (de-scaled) value of consumption for two different economies on the time interval [0, 30]. 3. Choose one particular steady state value (k ∗ ,h ∗ ) and compute transitional dynamics for 20 economies with (k0,h0) randomly chosen from [0.9k ∗ , 1.1k ∗ ] and [0.9h ∗ , 1.1h ∗ ]. Plot the phase diagram (k ∗ ,h ∗ ) showing the transition of these economies.
4 Numerical Simulation of Macroeconomic Models 4.4 The Romer (1990) Model 4.4.1 The model Model setup Households maximize intertemporal utility of consumption according to max c(t) ∞ Final output is produced in a competitive sector according to 0 Y = L 1−α−β H β Y c 1−σ 1 − σ e−ρt dt (65) A 0 x(i) α di (66) Noting the general symmetry of intermediate goods, x(i) = x, we can simplify to Y = L 1−α−β H β Y Axα Introducing capital as K := Ax the aggregate production function can be written as to Y = L 1−α−β H β Y A1−α K α Each type of intermediate good i is produced by a monopolistic competitive firm according (67) (68) x(i) = k(i) ∀iǫ[0,A] (69) where k(i) is capital employed by firm i. Hence, marginal costs for producing x(i) are given by r. Total capital demand is given by K = A Capital is supplied by households who own financial wealth with number of firms (blueprints) A and their value PA. 0 k(i)di (70) a = K + PAA (71) The R&D sector is competitive and produces blueprints according to ˙A = ηAHA η > 0, A(0) = A0 (72)
Page 1 and 2: Dynamic Macroeconomic Modeling with
Page 3 and 4: 1 Introduction 1 Introduction The a
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Page 37 and 38: References References Ascher, U. M.
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