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2022 | OriginalPaper | Buchkapitel

1. Introduction

verfasst von : Alfonso Novales, Esther Fernández, Jesús Ruiz

Erschienen in: Economic Growth

Verlag: Springer Berlin Heidelberg

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Abstract

Through the different chapters of the book, we will be presenting exogenous and endogenous, non-monetary and monetary growth models, discussing their main properties and learning how to obtain numerical solutions to them. Solutions take the form of a set of time series for the main variables in the model. Numerical solutions allow us to estimate the effects of alternative policies. After a review of some time series issues, we review in this chapter standard stochastic, dynamic structural models, as an introduction to model simulation. We then explain why economic growth models are very relevant to the analysis of policy questions. The chapter closes with an introduction to numerical solution methods.

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Fußnoten
1
That amounts to constructing the forecast by application of the conditional expectation operator to the analytical representation of the future value being predicted, where the conditional expectation is formed with respect to the sigma algebra of events known at time t.
 
2
This is the limit of a random variable, and an appropriate limit concept must be used. It suffices to say that the power of ρ going to zero justifies the zero limit for the product random variable.
 
3
That is, if the innovation ε t has zero variance.
 
4
When working with several variables, responses can be obtained for impulses in more than one variable. To make the size of the responses comparable, each innovation is supposed to take a value equal to its standard deviation, which may be quite different for different innovations.
 
5
Or significantly increasing the innovation variance. What are the differences between both cases in terms of the values taken by the process?
 
6
The two polynomials can be written as 1 − a 1 B − a 2 B 2 = (1 − B)(1 − λB), the second root being 1∕λ. The reader just needs to find the value of λ in each case.
 
7
We could have done otherwise, like starting the first order autoregressions at their mathematical expectation, and the second order autoregressions outside their expected values. The reader can experiment with these changes.
 
8
As it would be obtained by a profit maximizing competitive firm with a Cobb–Douglas technology, \(Y=a_{0}K^{a_{1}}L^{a_{2}},a_{1}+a_{2}\leq 1,\) represented in logs by the first relationship, with \(d_{0}=\ln (a_{0}a_{1}).\)
 
9
A system with as many equations as endogenous variables.
 
10
We assume here, for simplicity, that all random shocks are white noise. Extending the model to incorporate possible autoregressive structures for the shocks is straightforward.
 
11
If we denote by p i the i-th row of the s × q matrix P, then \( {\textit {Var}}(z_{i})=p_{i}^{\prime }\varSigma _{\varepsilon }p_{i},\) \({\textit {Var}}(z_{j})=p_{j}^{ \prime }\varSigma _{\varepsilon }p_{j},\) \({\textit {Cov}}(z_{i},z_{j})=p_{i}^{\prime }\varSigma _{\varepsilon }p_{j},\) and \({\textit {Corr}}(z_{i},z_{j})=\frac {p_{i}^{\prime }\varSigma _{\varepsilon }p_{j}}{\sqrt {p_{i}^{\prime }\varSigma _{\varepsilon }p_{i}}\sqrt {p_{j}^{\prime }\varSigma _{\varepsilon }p_{j}}},\) with Σ ε being the q × q variance–covariance matrix of vector ε.
 
12
If, for instance, C t, C t−1, and C t−2 appear in the model, both, C t and C t−1, will form part of vector z t, while C t−1 and C t−2 will be included in vector z t−1. The representation could also be extended easily to accommodate lagged innovation values.
 
13
Such equation is λ 2 − (a 2 + β 2)λ + β 2 = 0.
 
14
Which is known as Cholesky identification strategy, from the way how a factor decomposition of the variance–covariance matrix of the original innovations is used to produce the linear transformation of the system of equations.
 
15
Alternatively, we could have considered a process with some inertia for Government expenditures or even change the model to make the value of Government expenditures to be related to the past level of output, for instance.
 
16
The choice of the steady-state level as initial condition is arbitrary. However, in this stochastic version of the model that choice is as good as any other, since the economy is already going to experience fluctuations due to the stochastic component of government expenditures.
 
17
Notice the difference between computing relative volatility by the ratios of standard deviations and through the ratios of the coefficients of variation, the latter option being preferable.
 
18
The autocorrelation function is the sequence of values Corr(Y t, Y ts), for all s.
 
19
This suggests no evidence of residual autocorrelation, a potential source of misspecification in the consumption equation.
 
20
This is arbitrary. We should take an impulse of size equal to one standard deviation of the innovations estimated from actual time series data, since that is the likely single period fluctuation in each variable.
 
21
This is, in fact, very important, since the structure and implications of a model may significantly change by just a change in assumptions on the timing of decisions, the arrival of information, or the opening and closing of markets.
 
22
This modelling approach is now commonplace in macroeconomics. Dynamic models with microeconomic foundations for aggregate economies are often used in Public Finance, Monetary Theory, Labour Economics, or International Economics, as they are used in Growth Theory. The main difference for the latter is their focus on characterizing the main determinants of short- and long-run growth.
 
23
A standard result in intermediate microeconomics courses.
 
24
Of course, different utility functions could give raise to different functional forms for the way how current consumption relates to future consumption and interest rates.
 
25
With δ being the percent per-period depreciation rate of capital.
 
26
In consistency with the utility maximization problem above, we can either assume that there is a single consumer or household in the economy or interpret labor and capital stock in this equation in per-capita terms.
 
27
And also on conditional expectations of nonlinear functions of future state and decision variables, in the case of stochastic growth models, as we will see in the next paragraph.
 
28
Under endogenous prices, optimization problems solved by economic agents do not have a linear-quadratic structure, implying that their decision rules are nonlinear. Since these decisions are part of the system summarizing the model, that system ends up being nonlinear as well.
Sargent’s Macroeconomic Theory [34] contains a variety of partial equilibrium models in which, with exogenous prices, optimization problems have a linear-quadratic structure. In that simple setup, decision rules are linear functions.
 
29
Expectations of future variables or functions of variables appearing in a model need to be treated as new variables, so that a model that includes an explicit role for expectations is not complete without incorporating some kind of assumption on the way agents form their expectations. The assumptions on the expectations formation mechanism play the role of additional equations. They are a crucial part of a stochastic model, as important as the assumptions on the functional form of the utility function or the aggregate production function, and affect the model implications regarding the time behavior for the endogenous variables.
 
30
These expectation mechanisms are said to be backward-looking, since they are substituted by a function of past variables, agents’ views about the future not playing any role.
 
31
Alternative specifications for limited rationality, in which agents are assumed to form expectations that are partially rational, have been shown to be useful to explain some regularities in actual time series data.
 
32
Significant progress has already been done in dealing with agents’ heterogeneity [11, 31], although the representative agent framework is still predominant.
 
33
This summary is intended to provide an overview to readers unfamiliar with Growth Theory. We do not have any pretension of being fully comprehensive.
 
34
As mentioned, the model also had implications regarding the convergence of economies in terms of per-capita income, which developed a huge empirical literature aiming to test such implications that are still very much alive, now in reference to more sophisticated growth models that have been developed since then. Along this line of reasoning, Growth Theory would not be very different from other areas of economic theory that imply more or less tight restrictions among the joint behavior of variables, which can be reduced to parameter testing in relatively simple econometric models.
 
35
Kydland and Prescott [24] point out: “In other words, modern business cycle models are stochastic versions of neoclassical Growth Theory. And the fact that business cycle models do produce normal-looking fluctuations adds dramatically to our confidence in the neoclassical Growth Theory model—including the answers it provides to growth accounting and public finance questions.”
 
36
In any event, like in any other growth model, the relationships among per- capita variables emerging from the model will generally be nonlinear, and a linear econometric model might be too poor an approximation to them.
 
37
Constant returns to scale in the single cumulative input as a reason for positive long-term growth is the characteristic of the AK economy, introduced by Rebelo [30]. An explicit role for public capital as a productive input was proposed by Barro [2]. The model with a variety of intermediate goods is due to Spence [40], Dixit and Stiglitz [15], Ethier [17] and Romer [32, 33]. Uzawa [43], Lucas [26], and Caball é and Santos [6] assigned an explicit role to the stock of human capital in the production of the final good.
 
40
For a discussion of analytical solution methods for linear rational expectations models, see Whiteman [45].
 
41
We do not pretend these methods to be superior in any sense to those not covered in the chapter. They have been chosen because of their relative simplicity. An introduction to more complex, but possibly more exact methods, is also provided in that chapter.
 
42
Unless we work under the assumption of rational expectations, the model’s implications regarding the way agents’ expectations relate to state variables are generally hard to derive.
 
43
What is called a bubble equilibrium.
 
44
Welfare should be understood as the discounted time aggregate value of current and future utility. We are thinking here about a set of identical consumers, who live together forever, a usual assumption in growth models.
 
45
However, the appropriate approach to use frequency distributions from the alternative models to evaluate in probability terms (or in likelihood terms) their ability to fit the data is still very much open to discussion. And so it is the selection of statistics whose value in actual data should be replicated by the theoretical models considered.
 
46
Exogenous shocks could be modelled as shocks in productivity, as it is done often throughout the book.
 
47
Alternatively, we could consider the possibility of maintaining tax rates unchanged and finance the fluctuations in expenditures by debt management or money injections. Appropriate conditions guaranteeing long-run solvency would then have to be imposed, as it is discussed at different points in this textbook.
 
48
Of course, the type of results reached by Poole [28] in a static setup that “in the presence of supply shocks it is better to implement a monetary policy aimed to maintaining a given growth rate of money, while leaving interest rates to be determined in the market, the opposite being true if randomness enters mainly through the demand side” is another result typical from the type of analysis described in these sections.
 
49
And hence, in which per-capita variables display zero growth.
 
50
Macroeconometrics textbooks include some of these methods. As examples, see Canova [9] or De Jong and Dave [14].
 
51
Impulse responses and variance decompositions are statistics computed from vector autoregressive models. We will show how to compute them in actual and in simulated data, providing MATLAB programs to do so.
 
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Metadaten
Titel
Introduction
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_1