In Chapters 4 and 6 we assumed that the time horizon T, the initial values of state variables, si0, and their terminal values siT were exogenously specified. Obviously these are very restrictive assumptions. In many economic problems we want to allow some of these values to be determined endogenously (subject to constraints). For example, the optimal consumption problem (Example 6.4.1) may be modified to allow the planner to select the value of the terminal stock s(T), subject only to some constraint such that s(T) may not be less than a certain lower bound s, or even to select the economy's doomsday T after which all activities cease. Obviously, when s(T) or T is not fixed, we need additional necessary conditions to determine the new unknown (s*(T) or T*); these conditions are called the transversality conditions.

We shall look at various cases, beginning with the simplest. Section 7.8 contains a general statement synthesizing the various transversality conditions. Because there are many kinds of boundary conditions, there are also many kinds of transversality conditions. This array of special cases sometimes appears formidable to students of optimal control theory. For this reason a summary table is provided in Section 7.10. The table lists various features of control problems, and for each one gives the associated transversality condition. If a problem has several of these features, all corresponding transversality conditions apply.

Each of the following sections presents one type of problem and derives the associated transversality condition.

Problems with two-state variables

Nearly all the models hitherto encountered in this book have contained a single state variable. (Exceptions are the models of Sections 8.1, 8.3, and 9.6.) We have relied very heavily on phase diagrams in shedding light on the optimal solution. When there are two state variables, however, the (state, costate) space is four-dimensional and cannot be represented straightforwardly. It must be understood that, given the usual regularity conditions, we have in the maximum principle a set of necessary and sufficient conditions for an optimum, whatever the size of the problem, and if all functional forms and other restrictions were fully specified, we could – possibly using numerical methods – provide an explicit solution to the problem. However, since most models of interest in economic theory involve some unspecified functional forms, an explicit solution is normally unobtainable. This is why phase diagrams are such a useful device for pulling together all the pieces of information contained in the maximum principle.

Since they fail us here, we must devise other means of synthesizing the information. Unfortunately, this is often quite difficult, and in many cases a complete characterization of the solution escapes us. This is not to say that we cannot offer a partial characterization of the solution. It is the aim of this section to illustrate what can indeed be done. First note that in the models of Sections 8.1 and 8.3, the analysis was reduced to a two-dimensional phase diagram. The reader is referred to those sections.

It was stated at the end of Chapter 4 and again in Section 6.1 that control variables are required only to be piecewise-continuous. This means that they can exhibit jump discontinuities at a finite number of dates along the horizon. These discontinuities in the control may in turn result in discontinuities for the time derivatives of the state and costate variables, but the state and costate variables are themselves piecewise-differentiate (i.e., there may exist a finite number of points where the left- and righthand- side derivatives differ from one another). We claimed that this feature greatly enlarged the variety of problems that optimal control theory could handle and proved our point with a simple example in Section 6.1. In electrical engineering such discontinuities result in the operation of various circuit switches; in economics this takes the form of policy switches.

In the examples studied so far we have restricted the Hamiltonian to be strictly concave in the controls and the optimal trajectories have been continuous. Difficulties arise when we deal with problems that are linear in the controls (or can be made so), at least over some ranges; in these cases there are often bounds on the control variables, either imposed exogenously or generated endogenously. Except for a few theorems on timeoptimal problems (see Pontryagin, 1962, pp. 120–4), there are no general results for dealing with these problems, so we have chosen to illustrate them with various examples.

As the range of problems tackled by economists expands, the curriculum of economics programs follows. Questions of choice in dynamic economic models are often an integral part of such programs. The most useful technique for dealing with these questions is optimal control theory. It was developed in the late 1950s as an outgrowth of the centuries-old calculus of variations, and it has been traditional to present an exposition of the latter as a preliminary to this more modern technique. Here we break with this tradition on the grounds that there is nothing to be learned from the calculus of variations that cannot be learned from optimal control theory, whereas the converse is not true. Our approach emphasizes the links between the methods of classical programming and those of optimal control theory. For this reason we begin with a thorough and lengthy exposition of static optimization techniques: unconstrained, equality-constrainted, and inequality-constrained problems (Chapter 1). After presenting some simple solution techniques for differential equations and their qualitative analysis through phase diagrams (Chapter 2), we proceed with a very short and informal chapter introducing various concepts related to optimization in dynamic models (Chapter 3). Chapter 4 describes the optimal control format for dynamic optimization problems and the core of its solution procedures, known as the maximum principle. We have attempted to make the reader's first encounter with a standard control problem as limpid as possible by relegating all complications to a later stage and emphasizing the links with the Lagrangean methods of static optimization.

In this chapter we deal with problems involving the choice of values for a finite number of variables in order to maximize some objective. Sometimes the values the variables may take are unrestricted; at other times they are restricted by equality constraints and also by inequality constraints. In the course of the presentation an important class of functions will emerge; they are called concave functions and are closely associated with “nice” maximum problems. They will be encountered throughout this book. For this reason we weave the concept of concavity of functions through the exposition of maximization problems. This is done to suit our purposes, but concave functions have other important properties in their own right.

The notation we use is fairly standard. If in doubt, the reader should refer to the appendix to this chapter, which also contains a reminder of the basic notions of multivariate calculus and some matrix algebra needed to follow the exposition.

Unconstrained optimization, concave and convex functions

In what follows we assume all functions to have continuous second-order derivatives, unless otherwise stated. Strictly speaking, all domains of definitions should be open subsets of the multidimensional real space so that no boundary problems arise.

Unconstrained maximization

Consider the problem of finding a set of values x1, x2, …, xn to maximize the function f(x1,…,xn). We often write this as

where x is understood to be an n-dimensional vector. We refer to the problem of (1.1) as an unconstrained maximum because no restrictions are placed on x.

In this chapter we present a first account of optimal control theory. The maximum principle is the central result of the theory. (It was originally developed by Pontryagin and his associates; see Pontryagin et al., 1962.) To help the reader become thoroughly acquainted with it, we proceed with the analysis of a simple case, without paying undue attention to some technical regularity conditions. (These and other matters will be dealt with in Chapter 6.)

A simple control problem

Consider a dynamic system – for instance, a moving spaceship or an economy. Some variables can be identified that describe the state of the system: they are called state variables – for instance, the distance of the spaceship from earth or the stock of goods present in the economy. The rate of change over time in the value of a state variable may depend on the value of that variable, time itself, or some other variables, which can be controlled at any time by the operator of the system. These other variables are called control variables – for instance, the pitch of the motor or the flow of goods consumed at any instant. The equations describing the rate of change in the state variables are usually differential equations, as discussed in Chapter 2. Once values are chosen for the control variables (at each date), the rates of change in the values of the state variables are thus determined at any time, and given the initial value for the state variables, so are all future values.

Introduction

In this chapter we consider the role of macroeconomic models in the forecasting process. The use of macroeconomic models has developed extensively since the 1960s. In the US private institutions have been able to obtain forecasts of the US economy based on the Wharton model since 1963 and for the UK the London Business School has been modelling the economy since 1966. Since that date the number of forecasts based on macroeconomic models has been the subject of considerable expansion and Fildes and Chrissanthaki (1988) suggest that over 100 agencies are involved in making macroeconomic forecasts for the UK. Many of these agencies utilise macroeconomic models for this purpose but, for commercial reasons, details of their forecasting methods are not published. In section 5.2 we examine the general nature of macroeconomic models, followed by the presentation of a simple illustrative model in section 5.3. In section 5.4 we demonstrate how forecasts are prepared using this simple model as an example and in section 5.5 consider the role of judgemental adjustments. Forecasts of the 1980–82 recession are examined in section 5.6 and the decomposition of forecast errors in section 5.7. In section 5.8 we discuss the accuracy of macroeconomic forecasts and our conclusions are presented in 5.9.

Nature of macroeconomic models

Essentially a macroeconomic model is an attempt to describe an economy. This description may be presented in verbal form, diagrammatic form or in the form of mathematical equations. Standard macroeconomic texts are firmly based in the first two approaches with some use of mathematics.

The choice between different techniques

This book has been concerned with the methods used in economic forecasting. For this final chapter we first consider the problem of choosing an appropriate technique. The remainder of the chapter examines current developments in forecasting methods (section 7.2), and the interaction between forecasting and government policy (section 7.3).

From our review of the benefits of combining forecasts in chapter 3 it is clear that in many circumstances no particular technique dominates all others. It should also be apparent that many of the studies which compare different methods of forecasting are not comprehensive in their coverage of these methods. Initially we will examine the general approaches to forecasting, with a view to seeing what sort of information is needed before they can be used, then we will review some evidence on what techniques are employed in practice and taught in universities and other institutions of higher education.

In chapter 1 we saw that there are various ways of classifying forecasting methods such as subjective versus model-based or causal versus non-causal. Here we will limit consideration to quantitative forecasts with particular reference to judgement, surveys, extrapolation procedures and econometric models. Starting with judgement and surveys, these require very little extra information since the respondents use their own methods of producing forecasts. In particular there is no need to have a past history of the event being forecast.

Introduction

Other things being equal, the best method for forecasting the future values of a given variable would be to build a structural econometric model employing the correct theory, estimate its parameters from an accurate data base, and employ this model to predict the future values of the variable of interest. Since, by construction, such a model embodies the correct economic theory, it must produce forecasts which are, a priori, superior to those derived from other methods. However, the practitioner is seldom in this ideal situation and it is not always possible to construct an econometric model. This is because first, the practitioner may be unclear as to what constitutes the appropriate economic theory. Thus, for example while the theory of the consumption function is relatively uncontroversial, the role of money is not. Second, reliable data on the values of the variables believed to be relevant for the model may not exist. For example, monthly wealth and national income figures are not published in the UK. Third, the cost of constructing and estimating an econometric model may be greater than the perceived benefits from such an exercise, so that a cheaper method of forecasting is sought.

In chapter 1 we discussed some extrapolation methods in which past values of a single series were smoothed to give forecasts. Here we outline some of the more complex procedures, generally known as time-series methods, for univariate forecasting and then discuss multivariate forecasting methods.