The conception of this book began in the autumn semester of 1990 when I undertook a course in Advanced Economic Theory for undergraduates at the University of Stirling. In this course we attempted to introduce students to dynamics and some of the more recent advances in economic theory. In looking at this material it was quite clear that phase diagrams, and what mathematicians would call qualitative differential equations, were becoming widespread in the economics literature. There is little doubt that in large part this was a result of the rational expectations revolution going on in economics. With a more explicit introduction of expectations into economic modelling, adjustment processes became the mainstay of many economic models. As such, there was a movement away from models just depicting comparative statics. The result was a more explicit statement of a model's dynamics, along with its comparative statics. A model's dynamics were explicitly spelled out, and in particular, vectors of forces indicating movements when the system was not in equilibrium. This led the way to solving dynamic systems by employing the theory of differential equations. Saddle paths soon entered many papers in economic theory. However, students found this material hard to follow, and it did not often use the type of mathematics they were taught in their quantitative courses. Furthermore, the material that was available was very scattered indeed.

Since the advent of generalised floating in 1973 there have been a number of exchange rate models, most of which are dynamic. In this chapter we shall extend our discussion of the open economy to such models. Besides having the characteristic of a flexible exchange rate they also have the essential feature that the price level is also flexible, at least in the long run. This is in marked contrast to chapter 12 in which the price level was fixed. The models are often referred to, therefore, as fix-price models and flex-price models, respectively.

The majority of the flex-price models begin with the model presented by Dornbusch (1976). Although the model emphasised overshooting, what it did do was provide an alternative modelling procedure from the Mundell–Fleming model that had dominated international macroeconomic discourse for many years. It must be stressed, however, that the model and its variants are very monetarist in nature. Although the Mundell–Fleming model assumed prices fixed, which some saw as totally inappropriate, the models in the present chapter assume full employment, and hence a constant level of real income. This too may seem quite inappropriate. Looked at from a modelling perspective, it allows us to concentrate on the relationship between the price level and the exchange rate. Of particular importance, therefore, in such models is purchasing power parity. It does, of course, keep the analysis to just two main variables.

The optimal control problem

Consider a fish stock which has some natural rate of growth and which is harvested. Too much harvesting could endanger the survival of the fish, too little and profits are forgone. Of course, harvesting takes place over time. The obvious question is: ‘what is the best harvesting rate, i.e., what is the optimal harvesting?’ The answer to this question requires an optimal path or trajectory to be identified. ‘Best’ itself requires us to specify a criterion by which to choose between alternative paths. Some policy implies there is a means to influence (control) the situation. If we take it that x(t) represents the state of the situation at time t and u(t) represents the control at time t, then the optimal control problem is to find a trajectory {x(t)} by choosing a set {u(t)} of controls so as to maximise or minimise some objective that has been set. There are a number of ways to solve such a control problem, of which the literature considers three:

1. (1) Calculus of variations

2. (2) Dynamic programming

3. (3) Maximum principle.

In this chapter we shall deal only with the third, which now is the dominant approach, especially in economics. This approach is based on the work of Pontryagin et al. (1962), and is therefore sometimes called the Pontryagin maximum principle.

I was very encouraged with the reception of the first edition, from both staff and students. Correspondence eliminated a number of errors and helped me to improve clarity. Some of the new sections are in response to communications I received.

The book has retained its basic structure, but there have been extensive revisions to the text. Part I, containing the mathematical background, has been considerably enhanced in all chapters. All chapters contain new material. This new material is largely in terms of the mathematical content, but there are some new economic examples to illustrate the mathematics. Chapter 1 contains a new section on dimensionality in economics, a much-neglected topic in my view. Chapter 3 on discrete systems has been extensively revised, with a more thorough discussion of the stability of discrete dynamical systems and an extended discussion of solving second-order difference equations. Chapter 5 also contains a more extensive discussion of discrete systems of equations, including a more thorough discussion of solving such systems. Direct solution methods using Mathematica and Maple are now provided in the main body of the text. Indirect solution methods using the Jordan form are new to this edition. There is also a more thorough treatment of the stability of discrete systems.

In this chapter we consider a renewable resource. Although we shall concentrate on fishing, the same basic analysis applies to any biological species that involves births and deaths. A fishery consists of a number of different characteristics and activities that are associated with fishing. The type of fish to be harvested and the type of vessels used are the first obvious characteristics and activities. Trawlers fishing for herring are somewhat different from pelagic whaling. In order to capture the nature of the problem we shall assume that there is just one type of fish in the region to be harvested and that the vessels used for harvesting are homogeneous and that harvesters have the same objective function.

Because fish reproduce, grow and die then they are a renewable resource. But one of the main characteristics of biological species is that for any given habitat there is a limit to what it can support. Of course, harvesting means removing fish from the stock of fish in the available habitat. Whether the stock is increasing, constant or decreasing, therefore, depends not only on the births and deaths but also on the quantity being harvested. The stock of fish at a moment of time denotes the total number of fish, and is referred to as the biomass. Although it is true that the biomass denotes fish of different sizes, different ages and different states of health, we ignore these facts and concentrate purely on the stock level of fish.

Introduction

The interest and emphasis in deterministic systems was a product of nineteenth-century classical determinism, most particularly expressed in the laws of Isaac Newton and the work of Laplace. As we pointed out in chapter 1, if a set of equations with specified initial conditions prescribes the evolution of a system uniquely with no external disturbances, then its behaviour is deterministic and it can describe a system for the indefinite future. In other words, it is fully predictable. This view has dominated economic thinking, with its full embodiment in neoclassical economics. Furthermore, such systems were believed to be ahistoretic. In other words, such systems were quite reversible and would return to their initial state if the variables were returned to their initial values. In such systems, history is irrelevant. More importantly from the point of view of economics, it means that the equilibrium of an economic system is not time-dependent.

Although the physical sciences could in large part undertake controlled experiments and so eliminate any random disturbances, this was far from true in economics. This led to the view that economic systems were subject to random shocks, which led to indeterminism. Economic systems were much less predictable. The random nature of time-series data led to the subject of econometrics. The subject matter of econometrics still adheres to the view that economic systems can be captured by deterministic components, which are then augmented by either additive or multiplicative error components.

Very few topics in the theory of the firm have been considered from a dynamic point of view. There has been some work on the dynamics of advertising and the topic of diffusion (see Shone 2001). One topic that has been considered is the stability of the Cournot solution in oligopoly. Even this topic, however, is rarely treated in intermediate microeconomics textbooks. We shall try to redress this balance in this chapter and consider in some detail the dynamics of oligopoly, both discrete and continuous versions.

To highlight a number of the issues discussed in the literature, we concentrate on a single simple example. We outline first the static result that is found in most intermediate textbooks. Here, however, we utilise the mathematical packages in order to derive the results and especially the graphical output. We then turn to the dynamics. From the very outset it is important to be clear on the dynamic assumptions made. In the spirit of Cournot (see Friedman 1983, Gandolfo 1997) in each time period each firm recalls the choices made by itself and other firms in the industry. Furthermore, each firm assumes that in time period t its rivals will choose the same output level they chose in time period t - 1, and chooses its own output so as to maximise its profits at time t. This is by no means the only dynamic specification. It assumes that output adjusts completely and instantaneously.

Introduction

Every student of economics is introduced to demand and supply and from then it becomes a major tool of analysis, both at the microeconomic and macroeconomic level. But the treatment is largely static, with the possible exception of the cobweb model. But even when teaching this subject to first-year students, there is something unsatisfactory about the textbook analysis. Consider the situation shown in figure 8.1, where D denotes the demand curve and S the supply curve. We have a single market and the analysis is partial, i.e., this is the only market under investigation. For simplicity we also assume that the demand and supply curves have conventional slopes and are linear.

Suppose the price is presently P0. What happens? The typical textbook argument is that there is excess demand at this price and so suppliers, noting they can sell all they wish, will raise the price. This process will continue until the market is in equilibrium and there is no longer excess demand. But what is going on during this process? At the price P0 do we assume that demand is not satisfied and that the quantity actually transacted is Q0, but that in the next period the price is higher? Or do we assume that these curves indicate market wishes on the part of demanders and suppliers and that such excess demand is a signal to the market that a better deal can be struck?

What this book is about

This is not a book on mathematics, nor is it a book on economics. It is true that the over-riding emphasis is on the economics, but the economics under review is specified very much in mathematical form. Our main concern is with dynamics and, most especially with phase diagrams, which have entered the economics literature in a major way since 1990. By their very nature, phase diagrams are a feature of dynamic systems.

But why have phase diagrams so dominated modern economics? Quite clearly it is because more emphasis is now placed on dynamics than in the past. Comparative statics dominated economics for a long time, and much of the teaching is still concerned with comparative statics. But the breakdown of many economies, especially under the pressure of high inflation, and the major influence of inflationary expectations, has directed attention to dynamics. By its very nature, dynamics involves time derivatives, dx/dt, where x is a continuous function of time, or difference equations, xt - xt-1 where time is considered in discrete units. This does not imply that these have not been considered or developed in the past. What has been the case is that they have been given only cursory treatment. The most distinguishing feature today is that dynamics is now taking a more central position.