Introduction

In this chapter we present results from a number of counter factual experiments designed to analyze structural features of the tax/subsidy system. We consider various partitions of the whole set of UK taxes and subsidies, and provide indications as to which components of the tax system are more important in terms of their distributive and allocative impacts. We examine the separate producer and consumer price distortions in the tax and subsidy system, together with some of the more narrowly defined ‘legal’ components of the tax system. A number of specific policy options are also considered: changes in personal tax progressivity, restructuring consumer prices to redistribute income, industry aid, and altering the balance between direct and indirect taxation. One application of this analysis is in indicating possible directions for future tax reform in the UK.

Model-Type Taxes and Legal Tax Instruments

In Table 8.1 we present the results of cases which separately analyze the producer and consumer price distortions implicit in the tax/subsidy system. We report our central case analysis from Table 7.1 and two further cases where we separately abolish the distortions of consumer and producer prices with the same tax replacement as in our central case. All other characteristics of our central case, including the elasticity values used and the terms of trade neutralization, are unchanged.

The main taxes included as producer price distorting are the corporate and property taxes, while those classified as consumer price distorting are excise taxes and local authority housing subsidies.

Introduction

In this chapter we have as initial data a production set Y, a collection of preference relations ≳1,…, ≳n and a commodity vector ω ᄐ Rℓ. After defining the notion of attainable allocation, we introduce the concepts of optimality in the sense of Pareto and of price equilibrium. This chapter is devoted to the study of the interrelationships of these two concepts. Under general continuity hypotheses, Section 4.2 gathers basic definitions and the well-known facts for the convex preferences case. Under smoothness, but not convexity, assumptions, Sections 4.3 and 4.4 deal, respectively, with local first- and second-order theory, and Section 4.5 contains some results on approximate global supportability of optima by prices. Finally, Section 4.6 investigates the simplest facts about the topological and manifold structure of the set of optimal allocations.

Introduction

In this and the next chapter an extensive account is given of the Walrasian price equilibrium theory of exchange and production. In comparison to the previous chapter, the key new element is that consumers are now characterized not only by their preferences but also by their initial endowments of commodities. To streamline the exposition, the present chapter concentrates on economies without production, whereas the next goes into any distinct issue raised by the latter.

Section 5.2 presents the basic definitions of consumers, exchange economies, and exchange price equilibrium. This is done both for the general and the smooth case. Different systems of equations whose zeroes describe the equilibrium state are presented. One, the excess uitility map, is in the spirit of the previous chapter. We shall nevertheless emphasize an approach via excess demand functions. This is done partly because situations with a continuum of consumers can be accommodated at almost no cost.

Placing ourselves in a smooth framework, Section 5.3 presents the central concept of regular economy. Roughly speaking, an economy is regular if the relevant systems of equations, for example, the excess demand function, is not (first-order) degenerate, that is, singular, at equilibrium. In Section 5.8 and more deeply in chapter 8, we shall argue that, in a precise sense, nonregular economies are pathological. At any rate, it is for regular economies that the full power of the differentiate approach can be displayed. Thus, for example, the equilibria of a regular economy are well determined in the sense of being locally unique and persistent under perturbations of the economy. This is shown in Section 5.4 by an implicit function theorem argument that amounts, in essence, to the verification that the number of equations and unknowns is the same. Regular economies turn out thus to be the proper setting for the rigorous application of this classical technique.

In this chapter we introduce the basic concept of the production set of an economy and discuss some of its properties (Section 3.2). We shall describe how the production set can be viewed as the aggregate of a population of firm-specific technologies (Section 3.3). Associated with each production set we will define a number of derived, but nevertheless important, concepts such as the distance function, the normal manifold, and the profit function (Section 3.4). Finally, smoothness concepts and hypotheses (Sections 3.5 and 3.6), examples (Section 3.7), and a notion of proximity for production sets (Section 3.8) are discussed.

Production sets and efficient productions

The technological possibilities open to an economy are represented by a set Y ⊂ Rℓ, called the production set, of feasible input-output, or production, vectors. If y ᄐY, then it is understood that it is technologically possible to produce the output vector defined by max {yi, 0} by using the input vector defined by max{–yi, 0).

Until the end of World War II mathematical economics was almost synonymous with the application of differential calculus to economics. It was on the strength of this technique that the mathematical approach to economics was initiated by Cournot (1838) and that the theory of general economic equilibrium was created by Walras (1874) and Pareto (1909). Hicks's Value and Capital (1939) and Samuelson's Foundations of Economic Analysis (1947) represent the culmination of this classical era.

After World War II general equilibrium theory advanced gradually toward the center of economics, but the process was accompanied by a dramatic change of techniques: an almost complete replacement of the calculus by convexity theory and topology. In the fundamental books of the modern tradition, such as Debreu's Theory of Value (1959), Arrow and Harm's General Competitive Analysis (1971), Scarf's Computation of Equilibrium Prices (1982), and Hildenbrand's Core and Equilibria of a Large Economy (1974), derivatives either are entirely absent or play, at most, a peripheral role.

Why did this change occur? Appealing to the combined impact of Leontief's input-output analysis, Dantzig and Koopmans's linearprogramming, and von Neumann and Morgenstern's theory of games, would be correct but begs the question. Schematizing somewhat (or perhaps a great deal), we could mention two internal weaknesses of the traditional calculus approach that detracted from its rigor and, more importantly, impeded progress.

This chapter has two purposes. The first is to make good on the repeated promise to justify our attention to regular objects by establishing their typicality or, in the terminology of this chapter, their genericity. Hence, we shall argue that whenever a property has been called regular, the term is deserved. The second purpose is to illustrate by reiterated application the uses of transversality theory, an important mathematical technique that can be informally described to economists as a sophisticated version of the old counting of equations and unknowns.

A general introduction to the generic point of view is presented in Section 8.2. Section 8.3 describes the formal setting underlying the mathematical transversality theorems. Those are then applied to an investigation of the generic structure of demand (Section 8.4), production (Section 8.5), optima (Section 8.6), equilibria (Section 8.7), and some aspects of the equilibrium correspondence (Section 8.8). The aim is not to be exhaustive but to discuss a repertoire of typical situations and useful tricks. It is part of the objective of the chapter to make clear that, although they may differ in degree of complexity, most genericity arguments are, at bottom, very similar.

We define and study in this chapter the characteristics of an individual consumer. In particular, we introduce the concepts of preference relation, utility function, and demand function.

For a considerable part of this book it would suffice to accept the notion of demand function as the definitional characteristic of consumers. It should be emphasized, however, that demand functions are a poor conceptual foundation for economic theory and that the grounding of the latter on preferences goes beyond an aesthetic convenience. Demand functions are good devices for the study of price equilibrium theory but are inadequate for the analysis of welfare issues (see Chapter 4) or theoretical problems that do not emphasize prices (see, for example, Chapter 7).

Our aim in this chapter is not merely to get the existence of demand functions. We want them to be smooth. Mathematically speaking, a demand function exhibits nothing but the parametric dependence of a maximizer element.

Introduction

In this chapter we study economies with nontrivial production possibilities. The notions of equilibrium and regular equilibrium will be defined and the corresponding index theorem established.

Most of the analysis and results of the previous chapter could be generalized without much difficulty to cover the production case. It would, however, be pointless, not to say dull, to devote this chapter to doing so in detail. Hence, we shall limit ourselves to present the basic concepts and dwell only on those aspects that are specific to the presence of production.

Definitions of production economy, equilibrium, regular equilibrium, and index are presented in Sections 6.2 and 6.3. Section 6.4 is devoted to the special important case of constant returns economies. The index relation for production economies is stated and proved in Section 6.5. Section 6.6. parallels Section 5.7 and spells out the implications of production for the uniqueness of equilibrium. The point will be made that to get the latter the presence of production is helpful and definitely not a complicating factor.

This chapter gathers a number of mathematical definitions and theorems that, to different extents, will be needed for the economic theory of the following seven chapters. It is not meant to be read systematically before the rest of the book. The chapter is divided into twelve sections. The ordering of the sections is one of convenience and not of intrinsic importance for later developments. This being a book on differentiable techniques, it stands to reason that the central sections are Sections B on linear algebra, C on differential calculus, D on optimization, H on differentiable manifolds, I on transversality theory, and J on degrees of functions and indices of zeros of vector fields.

The economic theory of this book is presented in Chapters 2 through 8 with a fairly strict adherence to the axiomatic method and without presupposing much. No similar claim is made, however, for this chapter. A quick reading of the headings of the different sections will convince the reader that their content cannot be a systematic, complete, or rigorous exposition. It serves to fix terminology and to facilitate reference, but if one wishes to go deeper into the purely mathematical aspects, this chapter is no substitute for the study of the pertinent mathematical sources cited at the end of every section. It should suffice to say that the chapter does not contain any proofs.

We saw in Chapter 5 that the Walrasian allocations of an economy are optima or, in more descriptive terms, that they exhaust the gains from trade. The converse is, of course, not true. This raises the following question: Among the optima, which further properties characterize Walrasian equilibria? This chapter is devoted to investigating this problem in the context of exchange economies. Under the hypothesis of a continuum of agents we shall obtain a number of important results. In Walrasian theory individual agents optimize with respect to price vectors given independently of their own actions. Hence, the proper reference framework of the theory is one where single agents lack any macroscopic significance. The continuum hypothesis embodies this requirement, and it is thus natural that it should be an essential ingredient of a characterization of equilibria.

In Section 7.2 we shall offer precise definitions for the following three properties of an allocation x:

1. (i) No group of traders can allocate their initial endowment vector among themselves in a manner unanimously preferred (core property);

2. (ii) no individual trader would be better off with the net trade of any other trader (anonymity property); and

3. (iii) individual agents appropriate all the gains from trade they contribute (no-surplus property).

Remarks

Since functions can be viewed as singleton-valued correspondences, Brouwer's fixed point theorem can be viewed as a fixed point theorem for continuous singleton-valued correspondences. The assumption of singleton values can be relaxed. A fixed point of a correspondence μ is a point x satisfying x ∈ μ(x).

Kakutani [1941] proved a fixed point theorem (Corollary 15.3) for closed correspondences with nonempty convex values mapping a compact convex set into itself. His theorem can be viewed as a useful special case of von Neumann's intersection lemma (16.4). (See 21.1.) A useful generalization of Kakutani's theorem is Theorem 15.1 below. Loosely speaking, the theorem says that if a correspondence mapping a compact convex set into itself is the continuous image of a closed correspondence with nonempty convex values into a compact convex set, then it has a fixed point. This theorem is a slight variant of a theorem of Cellina [1969] and the proof is based on von Neumann's approximation lemma (13.3) and the Brouwer fixed point theorem. Another generalization of Kakutani's theorem is due to Eilenberg and Montgomery [1946]. Their theorem is discussed in Section 15.8, and relies on algebraic topological notions beyond the scope of this text. While the Eilenberg-Montgomery theorem is occasionally quoted in the mathematical economics literature (e.g. Debreu [1952], Kuhn [1956], Mas-Colell [1974]), Theorem 15.1 seems general enough for many applications. (In particular see 21.5.)

The theorems above apply to closed correspondences into a compact set.