Economic Growth Second Edition
Numerical Solutions of the Nonlinear System
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BarroSalaIMartin2004Chap1-2
Numerical Solutions of the Nonlinear System
We now assess the convergence properties of the model with a second approach, which uses numerical methods to solve the nonlinear system of differential equations. This approach avoids the approximation errors inherent in linearization of the model and provides accurate results for a given specification of the underlying parameters. The disadvantage is the absence of a closed-form solution. We have to generate a new set of answers for each specification of parameter values. 25. Equation (2.41) implies that the effects on β are unambiguously negative for α and positive for δ. Our numerical computations indicate that the effects of the other parameters are in the directions that we mentioned as long as the other parameters are restricted to a reasonable range. 114 Chapter 2 We can use numerical methods to obtain a global solution for the nonlinear system of differential equations. In the case of a Cobb–Douglas production function, the growth rates of ˆk and ˆc are given from equations (2.24) and (2.25) as γ ˆk ≡ ˙ˆk/ˆk = A · (ˆk) α−1 − (ˆc/ˆk) − (x + n + δ) (2.43) γ ˆc ≡ ˙ˆc/ˆc = (1/θ) · [α A · (ˆk) α−1 − (δ + ρ + θx)] (2.44) If we specified the values of the parameters ( A, α, x, n, δ, ρ, θ) and knew the relation between ˆc and ˆk along the path—that is, if we knew the policy function ˆc (ˆk)—then standard numerical methods for solving differential equations would allow us to solve out for the entire time paths of ˆk and ˆc. The appendix on mathematics shows how to use a procedure called the time-elimination method to derive the policy function numerically. (See also Mulligan and Sala-i-Martin, 1991). We assume now that we have already solved this part of the problem. Once we know the policy function, we can determine the paths of all the variables that we care about, including the convergence coefficient, defined by β = − d(γ ˆk )/d[log(ˆk)]. (In the Cobb–Douglas case, the convergence coefficient for ˆy is still the same as that for ˆk.) Figure 2.4 shows the relation between β and ˆk/ˆk ∗ when we use our benchmark parameter values ( δ = 0.05, x = 0.02, n = 0.01, ρ = 0.02), θ = 3, and α = 0.3 or 0.75. 26 For either setting of α, β is a decreasing function of ˆk/ˆk ∗ ; that is, the speed of convergence slows down as the economy approaches the steady state. 27 At the steady state, where ˆk /ˆk ∗ = 1, the values of β—0.082 if α = 0.3 and 0.015 if α = 0.75—are those implied by equation (2.41) for the log-linearization around the steady state. If ˆk /ˆk ∗ < 1, figure 2.4 indicates that β exceeds the values implied by equation (2.41). For example, if ˆk /ˆk ∗ = 0.5, β = 0.141 if α = 0.3 and 0.018 if α = 0.75. If ˆk/ˆk ∗ = 0.1, β = 0.474 if α = 0.3 and 0.026 if α = 0.75. Thus, if we use our preferred high value for the capital-share coefficient, α = 0.75, the convergence coefficient, β, remains between 1.5 percent and 3 percent for a broad range of ˆk /ˆk ∗ . This behavior accords with the empirical evidence discussed in chapters 11 and 12; we find there that convergence coefficients do not seem to exceed this range even for economies that are very far from their steady states. In contrast, if we assume α = 0.3, the model incorrectly predicts extremely high rates of convergence when ˆk is far below ˆk ∗ . Since the convergence speeds rise with the distance from the steady state, the durations of the transition are shorter than those implied by the linearized model. We can use the results on the time path of ˆk to compute the exact time that it takes to close a specified percentage 26. For a given value of ˆk /ˆk ∗ , the parameter A does not affect β in the Cobb–Douglas case. 27. This relation does not hold in general. In particular, β can rise with ˆk/ˆk ∗ if θ is very small and α is very large, for example, if θ = 0.5 and α = 0.95. Growth Models with Consumer Optimization 115 (a) (b) 0.30 0.75 0.75 kˆ kˆ 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 kˆ kˆ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.028 0.026 0.024 0.022 0.020 0.018 0.016 0.014 Download 0.79 Mb. Do'stlaringiz bilan baham: |
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