Dynamic Macroeconomic Modeling with Matlab

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2 An Outline of the Theory c50 − c0 50 k100 − c50 50 c100 − c50 50 c50 − c0 = g , 2 k50 − k0 2 c100 − c50 = f , 2 k100 − k50 2 c100 − c50 = g , 2 k100 − k50 2 The set of equations is augmented by the initial condition k(0) = k0 To complete the list of equations we have to fix a final boundary condition. For example, we could force consumption to equal its steady state value after 100 periods: Solving the set of equations c(100) = c ∗ For solving the set of equations a Newton procedure is applied. The idea of the procedure can be seen in Figure 5. An initial guess for the root of the equation is needed. In a next step the function value for this guess is calculated. To update the initial guess the function is linearized and the root of the tangent is calculated. This root is the starting point for the next iteration. For multidimensional functions the algorithm works analogously. The procedure again needs an initial guess of the solution. The equations are evaluated for the initial guess and a multi-dimensional error function is calculated. In each iteration, the solution is updated (see Figure 4). If the error is sufficiently small, the iteration is aborted. Since the set of equations inherits a special structure from the time dependence, the set of equations can be solved using only moderate computer time. Time transformation The idea is to transform the “model-time” on the interval [0, 1]. Therefore, we define τ := νt/(1 + νt) (19) <strong>with</strong> a parameter ν > 0. Then, the differential equations have to be modified according to dx dx = f(t,x,y) ⇒ dt dτ = τ f ν(1−τ) ,x,y ν(1 − τ) 2 (20)
2 An Outline of the Theory Figure 5: The Newton Procedure The differential equations exhibit a singularity for τ = 1, but the algorithm does not need to evaluate for τ = 1 (meaning t = ∞). Hence, we can solve the problem for the full interval [0, ∞). The accuracy of the algorithm can be computed for the Ramsey model by employing a special parametrization that allows also for an analytical solution. The results are shown in table 1. 2.4.1 Related Literature The relaxation procedure and similar finite-difference procedures have already been employed in various fields of economics. Prominent examples comprise the solution of two point boundary value difference equations (e.g. Laffargue, 1990; Juillard et al., 1998), differential-difference equations (e.g. Boucekkine et al., 1997) as well as partial differential equations (e.g. Candler, 1999). There are also a few applications for continuous-time optimization models in the economics literature. For instance, Oulton (1993) and Robertson (1999) employ the relaxation routine
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Dynamic Macroeconomic Modeling with Matlab

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