In Dynamic Macroeconomic Analysis: Theory and Policy in General Equilibrium, Altug, Chadha and Nolan, hereafter ‘ACN’, have undertaken the extremely difficult task of bringing the reader up to date on the vast literature that has developed in this key area of economics since such seminal books as Cooley's Frontiers Of Business Cycle Research (1995).

ACN tackle this formidable assignment by recruiting top scholars to write individual chapters. Each individual chapter is of exceptionally high quality because each scholar is a first-rate expert in the chapter's area.

The approach of the book is ‘quantitative theorising’, in the sense that each chapter presents not only recent developments in theory but also recent developments in compilation of the facts that confront the theory. Discrepancies between facts and theories are carefully discussed. Many of the chapters offer modifications to the baseline theory that make it do a better job of matching the facts.

I give the reader a quick overview of the contents of this book in a brief introduction to each of the chapters. Three chapters are discussed in more detail than the others in order to keep my foreword within standard space limits. But all of the chapters are equally exciting and have equal command on the reader's attention. I hope this somewhat unusual approach to writing a foreword will entice more readers to add this important book to their libraries. In addition to saying a few words about the chapters, I shall use the discussion of some of them to give some reflections about the field they cover as well as give some speculations and opinions about potentially fruitful future research.

Jim Pemberton (chapter 1) reviews dynamic life cycle consumption/savings models, the facts they are designed to explain, the struggles in attempting to modify the basic core model to fit these facts and other uses to which they are put. Particularly interesting is discussion of how minor modifications can lead to big and rather counterfactual predictions by some of the models. For example, adding a small safety net in one model caused the model's consumer to borrow too large an amount of money early on to be consistent with fact.

INTRODUCTION

Growth and fluctuations are sometimes referred to as the Twin Horns of Harrod because of two influential essays (see Harrod 1936, 1939). Both Hicks and Solow reacted to the ‘knife-edge path’ implied by these papers’ dynamics. Hicks emphasised the consequences of ‘falling off’ the edge. Solow gave a rationale, grounded in agents’ behaviour, that would keep the economy on this knife-edge path. Yet, the urge to study growth and fluctuations can be traced back, by way of Ricardo, Malthus and numerous other classical authors to the very birth of economics as a field. Given the vastness of the subject matter, one must set clear goals to what one hopes to achieve here.

This chapter is going to visit several schools that have attempted to blend the theory of economic fluctuations with the theory of growth. Both theories are interesting in their own right and are technically difficult. The early non-market approaches of the Hicksian accelerator and of the Goodwin predator–prey model are a good point of departure because of their intuitive nature. Hicks and Goodwin wrote at a time when an author could not rely on mathematical expertise to get published and in a style that borders, at times, on story telling. This chapter will start by reviewing their approaches, in section 2, as they have essentially disappeared from modern textbooks. Therein lies a deep vein of theory that, as we shall see, has influenced more modern approaches and still has the potential to enrich them further. Next, in section 3, we will review both real business cycle (RBC) theory and the deterministic dynamic system approach. RBC has established itself as the leading explanation for economic fluctuations and its paradigm when applied to growth is the model of King, Plosser and Rebelo (KPR) (1988a, 1988b). Anyone interested in working on the subject would be well advised to study that model. To help researchers, details of the rather intricate calculations in KPR are provided. Section 4 will evoke the problem of persistence in macroeconomic data. It will point to a convincing source for this phenomenon, namely aggregation. It will argue that persistence fogs our reading of economic relationships. It will call for the development of new tools to deal with the problem of persistence, in the absence of which no theory can be tested properly.

INTRODUCTION

The introduction of labour income into both deterministic, and more particularly stochastic, endogenous growth models has been somewhat problematical. The standard AK model of Barro (1990) and Rebelo (1991) assumes either explicitly or implicitly that labour income is introduced in the form of a return to human capital. Rebelo does so explicitly, by introducing human as well as physical capital in production. But Barro does so only implicitly, by assuming that capital in the AK technology is sufficiently broadly defined to be an amalgam of physical and human capital, which are assumed to be perfect substitutes in the production process. Neither of these procedures is entirely satisfactory. The assumption that the two forms of capital are perfectly interchangeable is obviously a polar one. Introducing current labour through human capital, which can be accumulated only gradually, ignores the short-run labour–leisure tradeoffs. As a consequence, taxes levied on labour income and consumption both operate as lump-sum taxes, thereby failing to capture the distortionary effects of these taxes on the growth rate of the economy; see Turnovsky (2000a).

The problem for stochastic growth models is even more acute. The solution procedure proposed by Merton's (1969, 1971) pioneering work involves explicitly solving the stochastic Bellman equation for the value function. This is a task that is tractable only under very restrictive assumptions, namely that output be generated as a linear function of current wealth (capital), thereby in effect, being represented by a stochastic AK technology. As a consequence, Merton's approach and the literature that it spawned basically restricted itself to income from assets and ignored labour income; see Eaton (1981), Gertler and Grinols (1982), Grinols and Turnovsky (1993, 1998), Obstfeld (1994), and Smith (1996). Indeed, presumably for this reason the most prominent area of application of these techniques has been to portfolio allocation problems in finance; see, e.g., Adler and Dumas (1983), Stulz (1981, 1983).

In this chapter we show how the equilibrium growth path can be easily obtained for both deterministic and stochastic economies in the case where the production function is of the Romer (1986) form, in which output is a linear homogeneous function of (i) private capital and (ii) labour supply expressed in efficiency units. The latter is measured as the product of labour with the average economy-wide stock of capital, which the individual agent takes as given, but which in equilibrium accumulates endogenously along with private capital.

INTRODUCTION

This chapter highlights some key topics in understanding the dynamic general equilibrium (DGE) behaviour of open economies. In line with the evolution of best practice in closed economy macroeconomic theory, DGE models are now the standard workhorse in the international macroeconomics literature. In addition, the incorporation of nominal rigidities and imperfect competition means that the current generation of open economy DGE models is also able to address the concerns of policy-makers regarding potential inefficiencies in adjusting to fundamental shocks. In this way, the ‘new open economy macroeconomics’ (NOEM) is a direct descendant of the traditional Mundell–Fleming–Dornbusch model (Rogoff 2001). While respecting this lineage, the microfounded nature of the new generation of models means that much more can be done in terms of providing a rigorous welfare evaluation of alternative policy regimes.

An open economy DGE model must contain a number of essential elements. Household preferences must be specified: this is more complex than in a closed economy model since the elasticity of substitution between home- and foreign-produced goods must be specified. This also applies to the specification of production functions since imported intermediate goods represent a potentially important linkage across economies. The international dimension of asset trade must also be specified, detailing whether home and foreign households share risks via state-contingent assets or just engage in bond trade or face even more restricted opportunities for international financial transactions. Of course, the form of nominal rigidities must also be determined (sluggishness in goods prices versus wages; the duration of rigidities): in an open economy, the researcher faces the problem of deciding the currency denomination of these sticky goods or factor prices. Finally, the nature of monetary and fiscal policies must be incorporated. Again, policy formation in an open economy involves extra dimensions in fixing the domestic policy response to foreign disturbances and evaluating whether there are gains to international policy coordination.

Lane (2001a) and Sarno (2001) provide broad surveys of this recent literature on ‘sticky’ DGE (NOEM) models that was initiated by the seminal ‘Redux’ model of Obstfeld and Rogoff (1995). Our strategy in this chapter is to focus on some key issues that are at the core of current research on open economy DGE modelling.

INTRODUCTION

During recent years economists have again been devoting attention to the issue of economic growth. In contrast to the neoclassical growth models derived in the Solow (1956)–Swan (1956) tradition, in which the steadystate rate of growth is given exogenously by technological and demographic factors, in the more recent literature the long-run growth rate is endogenously determined as the equilibrium outcome of the system; see, e.g., Barro (1990), Ireland (1994), Jones and Manuelli (1990), Lucas (1988), Rebelo (1991), Romer (1986) and Turnovsky (1996, 2000b). This is important, since it assigns a potentially significant role to fiscal policy as a determinant of long-run growth performance, something that is infeasible in the Solow–Swan framework. While the endogenous growth framework is not without its limitations, it provides an attractive and tractable approach to addressing issues pertaining to fiscal policy in an intertemporal context.

Most of the endogenous growth literature is based on perfect certainty. However, the endogenous growth framework can be easily extended to a stochastic context and thereby analyse issues relating to risk-taking and economic growth. The objective of this chapter is to construct such a stochastic growth model and to use it to analyse aspects of fiscal policy in the context of a stochastically growing economy. The formulation and solution of the problem employs continuous-time intertemporal optimising methods, rather than adopting the more familiar discrete-time approach. The main reason for this choice is that although continuous-time problems are tractable only under restrictive conditions, when these conditions are met, the solutions they yield are highly transparent, providing substantial insights into the characteristics of the equilibrium and the role of risk in its determination.

We should emphasise that our focus is on characterising the macroeconomic equilibrium, and to deriving its implications for macroeconomic policy-making – particularly fiscal policy – rather than on dwelling on the technical details of the solution procedures. At the same time, we should stress that the solutions themselves do involve substantial technical details and that the solutions to these stochastic growth models can be quite challenging.

A key assumption necessary to sustain a steady stochastic growth equilibrium is that all random disturbances are proportional to the current state of the economy, as represented by the capital stock or wealth.

I have written this book mainly for students who will need to apply maths in science or engineering courses. It is particularly designed to help the foundation or first year of such a course to run smoothly but it could also be useful to specialist maths students whose particular choice of A-level or pre-university course has meant that there are some gaps in the knowledge required as a basis for their University course. Because it starts by laying the basic groundwork of algebra it will also provide a bridge for students who have not studied maths for some time.

The book is written in such a way that students can use it to sort out any individual difficulties for themselves without needing help from their lecturers.

A message to students

I have made this book as much as possible as though I were talking directly to you about the topics which are in it, sorting out possible difficulties and encouraging your thoughts in return. I want to build up your knowledge and your courage at the same time so that you are able to go forward with confidence in your own ability to handle the techniques which you will need. For this reason, I don't just tell you things, but ask you questions as we go along to give you a chance to think for yourself how the next stage should go.

I have thoroughly revised all the ten chapters in the original edition, both making some changes due to comments from my readers and also checking for errors. I've also added a chapter on vectors which continues naturally from the present chapter on complex numbers.

I wrote the first version of this new chapter as an extension to the book's website (which is now at http://www.mathssurvivalguide.com) building up the pages there gradually. Their content was influenced by emails from visitors, often with particular problems with which they hoped for help. I've now extensively rewritten and rearranged this material. Writing in book form, it was possible to structure the content much more closely than on the Web so that it's easy to see the connections between the different areas and how results can be applied to later problems. The new chapter also has, of course, many practice exercises with complete solutions just as the earlier chapters have.

I'm once again very grateful to Rodie and Tony Sudbery and to David Olive for their helpful suggestions and comments. I must also thank all the people who emailed me, both with comments on the original ten chapters, and also with particular needs in using vectors which I've tried to fulfil here.

I hope that this two-way communication will continue. You can email me from the book's website if you would like to.