2022 | Buch

# Economic Growth

## Theory and Numerical Solution Methods

verfasst von: Prof. Dr. Alfonso Novales, Dr. Esther Fernández, Dr. Jesús Ruiz

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Texts in Business and Economics

2022 | Buch

verfasst von: Prof. Dr. Alfonso Novales, Dr. Esther Fernández, Dr. Jesús Ruiz

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Texts in Business and Economics

This is the third corrected and extended edition of a book on deterministic and stochastic Growth Theory and the computational methods needed to produce numerical solutions. Exogenous and endogenous growth, non-monetary and monetary models are thoroughly reviewed. Special attention is paid to the use of these models for fiscal and monetary policy analysis. Models under modern theories of the Business Cycle, New Keynesian Macroeconomics, and Dynamic Stochastic General Equilibrium models, can be all considered as special cases of economic growth models, and they can be analyzed by the theoretical and numerical procedures provided in the textbook.

Analytical discussions are presented in full detail. The book is self-contained and it is designed so that the student advances in the theoretical and the computational issues in parallel. Spreadsheets are used to solve simple examples. Matlab files are provided on an accompanying website to illustrate theoretical results from all chapters as well as to simulate the effects of economic policy interventions. The logical structure of these program files is described in "Numerical exercise"-type of sections, where the output of these programs is also interpreted. The third edition corrects a few typographical errors, includes two new and original chapters on frequentist and Bayesian estimation, and improves some notation.

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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.

Abstract

We present in this chapter the first growth model, introduced almost simultaneously by R. Solow and S. Swan in two different papers published in 1956. We show that, in the absence of technological growth, the economy does not experience long-run per capita growth. In aggregate terms, growth may come from population growth and from improvements in technology. We discuss the main properties of the model. A special steady-state, the Golden Rule, is introduced. We solve the continuous and discrete time versions of the deterministic model. We perform numerical exercises on the effects of changes in structural parameters and on characterizing situations of dynamic inefficiency. Finally, we explain how to obtain numerical solutions for the stochastic, discrete time version of the Solow–Swan growth model.

Abstract

We present the continuous time Cass–Koopmans model, characterizing the rate of capital accumulation that maximizes some social welfare criterion. Thus, we no longer consider a constant savings rate. We show the existence and stability of a unique optimal path. We discuss some numerical exercises on the long-run effects of changes in structural parameters, paying attention to the relevance of the different structural characteristics of the economy in characterizing the transition path between steady-states. The chapter closes with an economy with government. We examine the potential inefficiency of its competitive equilibrium with government, showing that the Ricardian doctrine, on the irrelevance of the financing tools used by the government, may not hold under some types of distortionary taxation.

Abstract

We consider in this chapter the discrete time version of some of the issues discussed in the previous chapter. We introduce a government in the economy and define and characterize the competitive equilibrium. The intertemporal government budget constraint, the relationship between the competitive equilibrium allocation and that of the benevolent planner mechanism, and the Ricardian doctrine, can be all analyzed in discrete time in a similar fashion as we have done in the continuous time version of the model. Dealing with all the details of the discrete time version of the Cass–Koopmans economy is very instructive in order to be able to formulate alternative, more complex growth models, as well as to perform policy analysis, as we do toward the end of the chapter.

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.

Abstract

The continuous and discrete time versions of the AK model are examined in detail, showing the absence of transition and the existence of a balanced growth path with all per capita variables growing at the same constant rate. We show that transitory policy interventions or structural changes in endogenous growth models have permanent effects. We analyze dynamic Laffer curves, a possibility that is specific of endogenous growth models. We explain how to obtain numerical solutions to the stochastic, discrete time version of the AK model. After that, we consider Barro’s version of the AK model that includes government expenditures, and we introduce the Jones and Manuelli variant of the AK model, describing the transitional dynamics, characterizing the stability conditions, and explaining how to compute numerical solutions.

We present in this chapter some additional mechanisms by which endogenous growth arises. We start with an economy in which technological progress shows up in the form of the number of varieties of products. After that, a model of technological diffusion between two countries, a leader in innovation and a follower, that adopts the innovations developed in the leading country. We then present a model economy with creative *destruction* à la Schumpeter in which growth is driven endogenously by attempts to improve the quality of existing goods through innovation. We close with an important model, that of a two-sector economy in which human and physical capital accumulate over time, and where time devoted to education plays an important role.

Abstract

In this chapter we characterize optimal growth in a monetary economy, using the steady-state optimality conditions to analyze alternative designs for monetary policy. We introduce the concept of optimal steady-state rate of inflation and characterize conditions under which Friedman’s prescription for a zero nominal interest rate can be optimal. Steady-state optimality conditions are used to characterize the feasible combinations of monetary and fiscal policies. A numerical exercise is used to evaluate the welfare implications of alternative monetary policies. In another exercise, the reader will see the existence of a Laffer curve in this economy. The chapter closes with a characterization of optimal monetary policy in a dynamic Ramsey model, which takes into consideration the short- as well as the long-run effects of a policy intervention.

Abstract

In Chap. 8 we characterized dynamic optimality conditions for monetary economies, but we only evaluated the steady-state effects of monetary policy. This chapter completes the analysis by characterizing the transitional dynamics of a monetary economy, as it moves from its initial condition to the steady-state, following a monetary policy intervention. We analyze the potential instability of public debt, an issue that restricts the set of feasible policies. We study both neoclassical and new Keynesian monetarist models, of the kind that are increasingly used at central banks around the world. We describe their theoretical foundations and characterize analytically the equilibrium conditions. After that, we present in detail the methodology to obtain numerical solutions to these models, which allows for the analysis of effects of different policy interventions.

Abstract

The chapter starts with the Generalized Method of Moments estimator, describing its main properties, and applying it to the estimation of an equilibrium asset pricing model. After that, we explain the implementation of the Maximum Likelihood estimator. The Kalman filter, the main tool for the numerical evaluation of the likelihood on the state-space representation of the model, is discussed in detail. We estimate cyclical and trend components in US GDP and the unemployment rate. Finally, we compute the ML estimator to the Hansen (Journal of Monetary Economics 16:309–327, 1985) model of indivisible labor. We explain MATLAB programs provided to estimate structural parameters and generate some interesting properties of Growth models, as impulse responses to supply and demand shocks, and the decomposition of the variance of forecast errors.

Abstract

The chapter starts with an introduction to Bayesian inference, and two applications examples in the context of regression models. After that, we introduce Markov Chain Monte Carlo Methods and provide a theoretical discussion of two families of such methods: Gibbs-sampling and Metropolis-Hastings algorithms. We estimate the parameters of a linear regression model using the Gibbs-sampling algorithm. Three applications of the Metropolis-Hastings algorithm are considered: random number generation from a Cauchy distribution; estimation of a GARCH(1,1) model, and estimation of a DSGE model which has been already estimated in Chap. 10 under a frequentist approach, so that the reader can compare the two different methodologies for the estimation of Growth models.

Abstract

This chapter presents the main mathematical tools used throughout the book: (1) solving the system of equations that characterize the solution to discrete time and continuous time deterministic control problems, with special attention to the transversality conditions that guarantee stable solutions, (2) solving first order difference equations with constant or time varying coefficients, (3) matrix algebra, with particular emphasis in the computation of eigenvalues and eigenvectors, since the Jordan decomposition of the transition matrix of a system of equations plays a central role in the numerical solution of the growth models considered in the book, (4) complex analysis, since the eigenvalues and eigenvectors of the transition matrices that come out of growth models are often complex numbers and vectors.