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2022 | OriginalPaper | Buchkapitel

5. Numerical Solution Methods

verfasst von : Alfonso Novales, Esther Fernández, Jesús Ruiz

Erschienen in: Economic Growth

Verlag: Springer Berlin Heidelberg

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Abstract

We start by considering the stochastic optimal growth model of Chap. 4, without taxes, explaining the construction of linear and log-linear approximations. Different solution methods are described: the Blanchard and Kahn approach, Uhlig’s method of undetermined coefficients, and Sims’ method based on an eigenvalue–eigenvector decomposition. We pay special attention to characterizing stability. We explain their practical implementation and discuss some of the results obtained. After that, we implement the same methods to solve the stochastic optimal growth model under different tax specifications and discuss some policy issues. The chapter closes with nonlinear solution methods, such as Marcet’s Parameterized Expectations and Projection methods. We apply them to the standard Cass–Koopmans growth model and provide MATLAB programs for their implementation.

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Fußnoten
1
There is no externality of any kind in this model, so welfare theorems apply, and the solution to the planner’s problem leads to the same allocation of resources as the competitive equilibrium. This, in turn, can be obtained as the solution to the representative agent problem.
 
2
The certainty-equivalence principle is also know as the separation principle, since it states that in linear-quadratic problems (optimization problems with quadratic objective functions subject to linear constraints) we can separate the control problem from the estimation problem. That implies that we need to solve just the deterministic version of the problem, to then add conditional expectations in the decision rules in front of any term involving future decision or control variables.
 
3
We are using the fact that, for \(\mu _{2}>1, \frac {1}{1-\mu _{2}B}=-\frac {1 }{\mu _{2}B-1}= -\frac {B^{-1}}{\mu _{2}-B^{-1}}= -\frac {B^{-1}}{\mu _{2}}\frac {1}{1-\frac {1}{\mu _{2}}B^{-1}} =-\frac {B^{-1}}{ \mu _{2}}\sum _{s=0}^{\infty }\frac {1}{\mu _{2}^{s}}B^{-s}= -\sum _{s=1}^{\infty }\frac {1}{\mu _{2}^{s}}B^{-s},\) so that \(\frac {1}{1-\mu _{2}B}X_{t}= -\sum _{s=1}^{\infty }\frac {1}{\mu _{2}^{s}}X_{t+s}.\)
 
4
Using the definitions of the α and β coefficients, the second degree equation in η kk can be written: \(\sigma \left [ \eta _{\textit {kk}}^{2}-\left ( 1+\frac {1}{\beta }+\frac {\left [ 1-(1-\delta )\beta \right ] \left ( 1-\alpha \right ) }{\sigma }\frac {c_{\textit {ss}}}{k_{\textit {ss}}}\right ) \eta _{\textit {kk}}+ \frac {1}{\beta }\right ] =0\) and an argument similar to that used in section Systems with a saddle path property in the Mathematical Appendix can be used to show that one of the roots is greater than 1∕β, while the other root is less than one.
 
5
Note that \(\theta _{t+1}=e^{\ln \theta _{t+1}},\) so that \(\frac {\partial \theta _{t+1}}{\partial \ln \theta _{t+1}}=e^{\ln \theta _{t+1}}=\theta _{t+1}.\)
 
6
A similar expression applies to any future expectation, the higher time index showing up in the definition of function g(.)  : E t g t+k = g t+k − ξ t+k, with E t ξ t+k = 0.
 
7
Later on we will introduce a different approach to compute impulse responses.
 
8
Notice that if we change the value of any structural parameter, these standard deviations would have to be estimated again.
 
9
Initially, the σ parameter may be set up to 1, but that might change in subsequent iterations, as explained below.
 
10
Let Ψi(x), Ψj(x) be two polynomials from a same family of basis functions. The two polynomials are said to be orthogonal to each other if there is a weighting function W(x) such that:
$$\displaystyle \begin{aligned} \int_{a}^{b}W(x)\Psi _{i}(x)\Psi _{j}(x){\textit{dx}}=0,\,\forall i\neq j.\end{aligned} $$
A family of polynomials is said to be orthogonal if any two polynomials in the family are orthogonal to each other. The weighting function that makes Chebychev polynomials orthogonal to each other, is: \(W(x)= \frac {1}{\sqrt { 1-x^{2}}}.\)
 
11
Stable solutions will always have control and state variables moving in a bounded space. The [a,b] interval can be chosen allowing for relatively wide fluctuations around steady-state. Violation of that assumed range by the numerical solution may point out to potential instability problems. Otherwise, the range can be widened and the solution algorithm implemented again.
 
12
Let us suppose that \(k_{\min }=0\) and \(k_{\max }=100,\) and that we choose d = 10. Chebychev nodes are then: 0.62, 5.45, 14.65, 27.30, 42.18, 57.82, 72.70, 85.35, 94.55 and 99.38.
 
13
According to Rivlin’s theorem, ‘Chebychev node polynomial interpolants are very nearly optimal polynomial approximants’.
 
14
Using Chebychev nodes and Chebychev polynomials is just one among the many alternative choices available. Chebychev nodes have been shown to provide a superior approximation than alternatives like equally-spaced nodes. Similarly, to compute the approximated decision rule, we could use monomials, splines or a family of orthogonal polynomials other than Chebychev polynomials.
 
15
Note the similarity, but also the differences, with respect to Chebychev polynomials.
 
16
In general, we compute μ by solving a system of equations of the form: ∫Ω W i(x)R(x; μ)dx = 0, i = 1,  2, …, d. In particular, Galerkin’s method consists on choosing as weights the basis polynomials: W(x) ≡ Ψ(x), which in this case will be the complete set of polynomials for (k, z).
 
17
This can be easily changed in the program.
 
18
This may be the most sensitive parte of the numerical algorithm for the projection method, and finding initial values can sometimes be tricky. It is strongly advisable to start solving simplified versions of the model, to gain some insight into appropriate initial values for the μ-coefficients .
 
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Metadaten
Titel
Numerical Solution Methods
verfasst von
Alfonso Novales
Esther Fernández
Jesús Ruiz
Copyright-Jahr
2022
Verlag
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-63982-5_5