Introduction

Many properties of the linear system Ld will recur repeatedly and Chapters 5 and 6 survey some of those most commonly used. Several advantages accrue to this concentrated exposition. The various controllability and observability properties have a propensity for arising as technical conditions in a host of later policy analyses. Their explication now, prior to their use, avoids the inevitable digression their subsequent introduction will otherwise entail. But, just as importantly, their juxtaposition in these chapters emphasises similarities and differences, as well as basic theoretical significances, which are likely to go unremarked if these properties are introduced across later chapters when technical needs dictate.

There has been in the relevant economics literature a tendency to treat these various controllability and observability properties as technical conditions only, devoid of any intuitive significance. This is probably characteristic of initial interdisciplinary applications, but also partially reflects the representational differentiation of the linear system discussed in Section 4.4. This differentiation, by inhibiting access to the linear systems literature, has also inhibited the development of the conceptual significance of technical properties like controllability and observability. The primary motivation for Chapters 5 and 6 is accordingly expositional, and the work of system theorists such as Kalman, Ho and Narendra (1963), Desoer (1970), Wonham (1967), Hautus (1969,1970), and Zadeh and Desoer (1963) is employed extensively.

Chapter 5 concerns itself with the fundamental property of state controllability, and does so along three fronts.

Introduction

Under the conventional assumption of quadratic preferences, the flexible objective problem generates first-order conditions constituting a linear system – so bringing the analysis of the flexible objective problem closer to the analysis of the fixed objective problem than might otherwise appear.This correspondence of the fixed and flexible analyses induced by linearity is the central preoccupation of Chapter 3.

A heuristic explanation will help motivate the essential unity of the fixed and flexible objective approaches. A policymaker poses a fixed objective problem. Using the analyses of Chapters 1 and 2, he determines that global policy existence fails; and additionally that local policy existence in some neighbourhood of a specific fixed objective also fails. Accordingly, he specifies a quadratic loss metric centred on the desired, but unattainable, fixed objective, and minimises this to get as close to the desired fixed objective as the policy model constraint permits. In fact (recall Figure 1.4) with quadratic preferences, the ‘closest’ feasible fixed objective is a projection of the desired fixed objective onto the range space of the fixed objective policy model mapping. In particular, if the policymaker's preferences are neutral with respect to individual fixed objectives (that is, loss contours are circular, as in Figure 1.4), the ‘best’ or ‘closest’ feasible fixed objective is actually the orthogonal projection of the desired fixed objective onto the feasible objective space.

Introduction

The framework set out in earlier chapters has basically extended and embroidered the approach to the analysis of policy in Tinbergen's pioneering work (1963). Central to each chapter however has been the presumption that there is a ‘policy invariant’ law of motion to the system (Prescott (1977)). This assumption came in for strong criticism during the early seventies, for example Lucas (1972), and its relaxation has led to models as in Sargent and Wallace (1975) in which (monetary) policy had no impact: no (monetary) policy existed to achieve a set of defined objectives as the private sector always offset government policy because of policy induced variation in their expectations.

This proposition, that rational expectations (REs) in a model can cause a failure of policy existence, has not been quietly accepted by the profession and there have grown up a number of counter-examples designed to show that policy may still exist even when agents do anticipate policy correctly – Fischer (1979), Buiter (1979), and Taylor (1979a). What seems lacking from this literature though is an abstract investigation into the consequences of REs for the theory of policy – only Turnovsky (1977) and Aoki and Canzoneri (1979) having attempted it – and yet there is a pressing need for this in the face of a continued proliferation of special models reaching contradictory conclusions. Therefore, this chapter aims to provide a generalised treatment of static and dynamic fixed target policy in linear models with REs.

This book has as its basic objective a unified treatment of that area of economic theory now commonly known as, following the pioneering work of Tinbergen and Hansen in the 1950s, the theory of economic policy. It is a theoretical rather than an applied or econometric study. On the crucial assumptions of a certainly known and linear economic structure, the book focuses on the abstract theory of policy implied by a generic linear deterministic policy problem. This generic policy problem is defined as resulting from the interaction of a policy objective, representing some abstract policymaker's desires, with a policy model, representing the feasible outcomes of policy actions. By identifying a variety both of policy objectives and of policy models within the imposed limitations of certainty and linearity, it is possible to encompass and thus unify a variety of policy problems in a common perspective.

The procedure for attaining this perspective is to begin from the known territory of the static theory of policy. In the work of Tinbergen, Hansen and Theil, two broad policy objectives – the fixed objective and the flexible objective – are associated with a given linear static policy model. In some relevant region of the target space, the fixed objective specifies a particular configuration of targets as desired; whereas the flexible objective alternatively specifies a preference ordering over all target configurations in that region.

Introduction

Of ultimate concern to the theory of policy, as conceived in this book, is the operation of instruments on targets. For the dynamic theory of policy, the linear system representation interposes the system state between the instruments as inputs and the targets as outputs. Chapter 5 has just finished examining the operation of the instruments on the states. Chapter 6 now completes the link from instruments to targets by examining the operation of the states on the targets. Just as Chapter 5 associates various controllability properties with the ability of the instruments to affect the states, in similar fashion Chapter 6 associates various observability properties with the ability of the states to affect the targets.

But whereas state controllability is an existence property, state observability is a uniqueness property. The question to be studied is not, as in Chapter 5, whether the instruments can be adjusted intertemporally to effect an arbitrary state transfer; but rather, given that a particular intertemporal target transfer has occurred, is there a unique intertemporal state transfer responsible for this target transfer? Why such a uniqueness question is of recurring importance, and precisely how it pairs with the existence question, are topics to be resolved during this chapter.

Some heuristic motivation can be provided with the benefit of hindsight.

Introduction

Conceptually distinct from the stationarity objective is the target path objective, most simply described when defined in Section 4.3 as a consecutive sequence of arbitrary point objectives. In vivid contrast to the stationarity objective, the path objective ruthlessly discards any overt connections with the static policy heritage anduncompromisingly poses its own specific dynamic problem. Accordingly, in its technical analysis the Gordian decomposition of mappings characteristic of the stationarity analysis disappears and in further elaboration of another typical feature of dynamic analysis, as introduced in Chapter 5, emphasis on an appropriate nested, time-indexed sequence of linear mappings re-emerges.

As for any dynamic policy problem, so for the path problem a quartet of issues – existence, uniqueness, stability and design – awaits analysis. Existence issues will form the basis of this chapter, and Chapter 9 will then refer to the remaining trio of issues, but with primary emphasis on design. It should be noted that this chapter is intended as a specific sequel to the paper on path controllability by Preston and Sieper (1977). Sections 8.2–8.4 and Section 8.7 largely convey ideas and material from that earlier paper; Sections 8.5, 8.6 and 8.8 extend that analysis, as does Chapter 9.

Section 8.2 formulates the path existence problem and derives some immediate existence propositions by application of the linear mapping theory of Chapter 2. Because every path problem comprises a sequence of point problems, properties of the target point problem are necessarily crucial to the path problem: Section 8.3 therefore analyses this trivial or polar path problem.

Introduction

By direct analogy with the static theory of policy, dynamic policy problems arise from the interaction of a dynamic policy objective with a dynamic policy model. This chapter develops the analytical framework required for a dynamic theory of policy; and therefore does so by specifying a variety of policy models and of policy objectives. The actual interaction of policy model and policy objective will be studied in ensuing chapters.

As for the static theory of policy, the structural form and reduced form are conventional representations of thedynamic policy model. With economic structure fixed, certain, and linear, the choice between either of these representations is one of convenience; as for the static theory of policy the reduced form is typically preferred. Some added variety appears in the dynamic framework because of the introduction of the linear system representation, also known as the state space form, of the policy model. State space models have been widely employed in the dynamic policy literature in recent years, largely because of certain analytical advantages enjoyed by this representation over the orthodox reduced form representation of the policy model. For a unified dynamic theory of policy, a clear appreciation of the interrelationship between the reduced form representation and the linear system representation is imperative and this will be one of the objectives of Chapter 4.

While the policy model can thus be represented in various forms – the structural form, reduced form, or state space form – provided these are truly equivalent such representational variety cannot be responsible for generating varieties of dynamic policy problems.

A unified foundation for the linear theory of policy in both its static and dynamic guises has been the overriding objective assailed throughout this book. It is now appropriate, if chastening, to review briefly its salient features, to identify specific contributions intended to inhere in those foundations, and to gloss quickly over its known limitations.

Three fundamental issues fall for analysis in any policy context – existence, uniqueness and design. When the juxtaposition of a linear policy model with a policy objective generates a linear policy mapping from instrument space into target space, the theory of linear mappings affords a substantial resolution of this trio. To be singled out as cornerstones of the book's deployment of that analytical framework are the following results: Tinbergen's Theorem as generalised (Theorem 2.23); the existence/uniqueness duality; the fixed/flexible correspondence; the linear system representation; the Cayley–Hamilton theorem; and the structure algorithm.

Of these, the first three find employment in the analysis of both static and dynamic policy problems, whilst the second three are specific to the analysis of dynamic problems. Tinbergen's theorem, as generalised, provides a definitive statement on policy existence, uniqueness and design and has been appealed to constantly whenever one of these issues has appeared: it is unquestionably the fundamental tool of the linear theory of policy. The existence/uniqueness duality serves two functions: as an economising device for adducing uniqueness criteria from existence criteria (or conversely) and as a unifying device in proclaiming the duality of existence and uniqueness analyses.

Analysis of the interaction of a policy objective, specifying what is desired, with a policy model, stipulating what is feasible, is the nub of the theory of economic policy. This chapter illustrates the operation of this interaction and some of its more important features in the context of a review of the well-known static theory of policy. The theory itself stems principally from the work of Tinbergen, Hansen, Meade and Theil, and the chapter's task therefore is more expository than exploratory. By carefully reviewing familiar territory with an anticipatory eye on the sequel, some basic insights and issues can be identified which will guide the imposition of structure in the less familiar territory to follow.

Chapter 1 falls into two halves. The first half reviews the fixed objective theory of policy, the second half the flexible objective theory of policy. Whereas the focus of the former theory is the idea of hitting a target point exactly, the focus of the latter is the idea of approaching a target point closely. Because this is a difference of degree, and not of kind, it is not too surprising that major similarities will be found in the analyses of both problems.

Sections 1.1–1.5 refer to the fixed objective problem. Section 1.1 specifies the static policy model in its various guises, and Section 1.2 identifies the fixed policy objective as used by Tinbergen and Theil.

Stability issues, although a typical preoccupation of dynamic policy analysis, have so far not figured centrally in the current analysis of the dynamic theory of policy. Such issues were, it is true, but thinly disguised in the previous chapter as the correspondence of existence for the asymptotic flexible objective problem with the stability of the preference variables. Elsewhere Section 9.7 referred briefly to the property of instrument instability in the fixed target problem, promising to reconsider that phenomenon jointly with the flexible objective problem.

This final chapter not only redeems that promise but also explicitly considers a miscellany of stability questions pertinent to the asymptotic flexible objective problem when existence, pursued so vigorously in Chapter 12, is not the sole concern. The concentrated rigour of that chapter is deliberately counterbalanced here by a more discursive and leisurely approach: for relevant stability issues are too numerous to allow more than a broad-brush analysis. But the analytical policy framework evolved in the preceding chapters will, when required, readily permit sustained analysis of those issues only adumbrated here.

Verification of the existence of an optimal policy for a particular preference specification is typically only part of the process of designing an optimal policy; for it may also be necessary to alter the preference specification before the optimal policy is adjudged fully satisfactory. Indeed, optimal policies are no better than the policy model permits and the preference specification insists; and this tension between positive capability and normative compulsion is accordingly the theme of this final chapter.

Introduction

The most persuasive criticism levelled at the static theory of policy is that it is indeed static. Even if a major preoccupation of the theory of the trade cycle – namely, the inherent stability or otherwise of the economic system – is nowadays resolved in the affirmative, the work of Phillips (1954, 1957) and many others since points to other factors denying the adequacy of a purely static formulation of the theory of policy. For example, the speed of adjustment to the equilibria of the static formulation may be so sluggish as to render static policy implementation impracticable; and, apart from this, adjustment paths may possess undesirable characteristics such as excessive oscillations. If, in common with Phillips for example, it is inquired how the Tinbergen fixed-target problem might be directly embedded within a dynamic context, so as to answer its critics and yet keep faith with the basic Tinbergen conception, a major conceptualisation of the dynamic policy problem is obtained: the stationarity objective as defined in Chapter 4.

Given a dynamic policy model, this objective entails the hitting and holding of a desired stationary point of the system. The stationary points of the dynamic policy model are the target/instrument solution pairs of the steady-state model embedded in the dynamic model; thus the Tinbergen fixed objective remains the focus of the stationarity objective but with adjustment dynamics superimposed. So defined, many authors have implicitly utilised this objective.