General Concepts

Statement of the Problem

To predict an unobserved variable is to find an approximation of it that is a function of the observations (see Chapter 2). Prediction problems are frequent in economic applications. They arise naturally in dynamic contexts when, for instance, one desires to know the level of unemployment in the next years, or when one's goal is to complete a time series of which some intermediate values are missing. Prediction problems, however, are also important in numerous other situations. For instance, a researcher wants to determine the expected change in consumption of a household whose income changes by 5%. Alternatively, one wants to find the price that would have prevailed in equilibrium on a market in desequilibrium (see Example 1.20).

We shall see in the following sections, that estimation problems and prediction problems are often closely related. As an example, consider estimation by ordinary least squares. This estimation will naturally appear when looking for an optimal prediction of a nonobserved endogenous variable by a linear function of the observed endogenous variables.

Thereafter, we let Y1,…,Yn denote the observations and W the variable to predict. To simplify, it is assumed that W takes its values in ℝ. A prediction or predictor of W is a function Ŵ(Y1,…,Yn) of the observations. The prediction error is the discrepancy between the predicted variable and the prediction.

This conclusion successively considers four points: it first attempts to analyse the main shortcomings of the study while remaining within the justificatory framework of Chapter 1 (A); it then provides a critical assessment of the polar assumptions of chapter 1 (B); third, it shows that our 'monist' option stressing theoretical unity has led us to interpret the results in a much too restrictive way (C); finally it discusses some of the most promising directions of research in the theory of second-best taxation (D).

(A) The model under consideration in chapters 2,3 and 5 has features which fit the conclusions of chapter 1 except in one respect: there are a number of commodities for which taxation should be non-linear rather than linear. Whatever the merits of the linearity assumption, non-linear taxation of, for example, labour income is suggested both by theoretical analysis and real world observation. Even taking into account the detailed analysis of quotas policies - that introduce non-linearities in the tax system, albeit of a special kind - in the section 4.2 of chapter 4, and the discussion of the non-linear taxation model of section 4.4, the treatment of non-linearities in this book is incomplete. This is an obvious shortcoming.

The previous chapter has led us to investigate in depth, from the complementary viewpoints of optimization and reform, the normative properties of our model. At the present stage we should however consider two opposite drawbacks of our study: on the one hand, we have assumed too many degrees of freedom for the planner's action; on the other hand we have assumed too few such degrees of freedom for the same planner's action.

We have assumed too many degrees of freedom. This objection can be seen as an objection which is external to our approach: our assumptions on the degrees of freedom of the planner's action are grounded in more basic informational and observational assumptions and, if correct, the objection should concern these more basic assumptions. Indeed, the assumption that all transactions between the production sector and the consumption sector can be observed at no cost is an extreme idealization of the conditions under which the taxation power of the government can be exercised. In fact, the complete disconnection between consumption and production prices, permitted by the no-cost assumption, is not accessible to actual tax systems. Administrative costs - or what are usually considered as such - lead to a definition of a small number of categories of commodities, which are treated similarly - in the sense of being (usually) taxed at the same ad valorem rate. On the grounds of realism, a further step to our analysis would be to take into account these additional constraints. Note also that the above objection makes sense even as an objection internal to our approach.

Introduction

The first chapter is concerned with an analysis of the rationale of the tax structure which we are going to study later. It presents a reflection on the design of fiscal tools, that gives a central role to informational constraints. The analysis has to be contrasted with the analysis of following chapters, which focus attention on tax incidence or tax design within given tax institutions.

This chapter addresses one question which should be of central relevance for those who are concerned with the foundations of the field and particularly for theorists. Indeed, pure theorists and mathematical economists have often viewed real tax systems either as pragmatic constructs which are not rooted in satisfactory theory or even as rough devices which could be advantageously replaced by more efficient systems. The underlying objection that applies, in particular, to the direct and indirect taxes that we are going to consider here often echoes the second theorem of welfare economics by stressing that, in the absence of lump-sum taxes, individual choices are distorted in an inefficient way. Such a sceptical appraisal parallels a long tradition, associated in particular with the names of L. Walras, H. George, etc. Again the debate is not purely intellectual. Notwithstanding the ambitious reform proposals à la Walras that advocate the switch to taxation schemes more neutral than the existing ones, the distortion issue creeps into many recurrent policy discussions.

This introduction aims at presenting an overview of the content of the book. For each chapter it presents the problem, the model and the main results. Whenever possible, an analysis or even a sketch of the proof is given within an appropriately simplified version of the model. The information can also be read by section, each section, or overview, acting as an extended introduction to the corresponding chapter. Finally, it provides a summary to which the reader may want to refer after reading the whole book. Naturally, these are not mutually exclusive uses of the Introduction!

As stressed in the foreword, and as emphasized in the title, the monograph is a contribution to pure theory. As such, it aims at exploring the logic of an abstract model, the stylized features of which describe a (so-called) polar world. Within such a model - again ‘théorie pure’ a la Walras - derivations and also comments concern the polar world under consideration and not the specific issues, debates, and controversies that the analysis of the polar world aims at clarifying. Although the separation of theory and policy analysis is a standard modern procedure, it has the inconvenience of making the theorist’s logic and motivation more obscure to practitioners than it should be. As an attempt to overcome this difficulty, at the beginning of each overview a subsection, entitled ‘Motivation of the chapter: issues, studies and debates’, sketches a description of background materials, underlying issues, related applied studies and possible controversies that put each chapter in a better perspective. Throughout the book bibliographical references are kept to a minimum. In fact, they are gathered as much as possible in the bibliographical note appearing at the end of each chapter.

In the previous chapter, the taxation of transactions has been justified by incentive compatibility arguments. These arguments can be briefly recalled as follows: as the agents' characteristics upon which transfers should be based are private information, a central authority is bound to the use of mechanisms whose enforceable outcomes are verifiable variables. Assuming that the (observable) variables were the net trades between the production and the consumption sector, we have shown conditions under which the outcomes which can finally be achieved are the same as those which would prevail in the presence of tax systems mixing linear and nonlinear taxes.

As noted earlier, the basic argument of the preceding chapter holds true with a fixed bundle of public consumption. The model we consider in the following actually incorporates public goods, but their level is possibly variable. We leave it to the reader to see how and under which conditions - concerning the nature of conditional information on the taste parameters for public goods - the conclusions of the previous chapter extend to the case of this chapter.

In this chapter, we shall only consider linear taxation and shall rule out non-linear taxes associated in the previous chapter with commodities L2. This modelling option is first justified by its simplicity. Non-linear schedules are difficult to handle in a general context: understanding fully what is going on in the simple case of linear taxes is a natural first step towards an in-depth investigation (sketched in chapter 4) of the more difficult nonlinear problem. Linearization, here as in many other scientific fields, is a useful simplifying device. Also, simplicity is not here so much at odds with realism. Non-linearities in real tax systems are generally concentrated in income tax schedules that display a limited number of marginal tax rates.

Introduction

The previous chapters have successively considered institutional aspects (chapter 1), positive aspects (chapter 2), and normative aspects of taxation (chapters 3 and 4). Throughout the analysis, the task of implementing fiscal systems is supposed to be accomplished by some central body. The nature of this central body has remained deliberately vague, a fact that is reflected in a hesitant terminology: central fiscal authorities, state, government, planner. Indeed the role of this central body is not necessarily the same across the chapters. It is (implicitly) in charge of implementing the different tax systems which are compared in the positive analysis of taxation of chapter 2. In chapter 1, the central body faces informational constraints that are inherent to the problem and that impose restrictions on its taxation power. The situation is somewhat different in chapter 3, particularly when the emphasis is switched from the Pareto criterion to a social welfare criterion. Endowed with such a social welfare function, the central authorities are no longer a passive coordinating body, but an active institution selecting taxation schemes through discriminatory criteria. This latter assumption raises the question of the nature of the social welfare criteria chosen for the selection of taxation schemes or, equivalently, of the nature of the central body which uses such criteria.