# (p.xi) Preface

# (p.xi) Preface

This book provides a panorama of developments in our understanding of some general equilibrium models in the past three decades. These properties go far beyond the existence of equilibrium or the welfare theorems to which many current textbooks seem to limit microeconomics. The equilibrium manifold and the way that set is projected into the parameter space play a fundamental role in these developments.

Now a few words to justify the term *postmodern general equilibrium theory*. There is widespread consensus to associate modern mathematics with van der Waerden’s *Moderne Algebra* (1931) and the Bourbaki series, *Elements of Mathematics*. Debreu’s pivotal *Theory of Value* (1959) is the Bourbaki-style formulation of modern general equilibrium theory. Jacques Dréze once described that book as the “last definitive posthumous edition” of Walras’s Elements of Political Economy (1874). This reminds us that, according to Weil (1991, 104), the initial goal of the Bourbaki group was to reformulate *Goursat’s Cours d’analyse mathématique (1902)* in the style of modern mathematics. The main feature of modern general equilibrium theory is, in addition to the abstract and axiomatic style, the focus placed on properties that are satisfied by all, or almost all, equilibria and economies, in other words, absolute properties.

But many problems have no answer within the setup of modern general equilibrium theory. An example is the stability of competitive equilibrium. Properties like stability are not satisfied by all equilibria. Only some equilibria can be stable, and only some economies have stable equilibria. More generally, only some equilibria and economies are going to satisfy an economically meaningful property. This is where the equilibrium manifold takes the central role because it enables us to characterize and study sets of equilibria that satisfy any given property (p.xi) as subsets of the equilibrium manifold. From absolute, properties become relative. This change in perspective is sufficiently important to differentiate the relativistic postmodern approach from Debreu’s absolute (modern) approach.

Postmodern general equilibrium theory has nothing to do with the particular tone and political agenda of the postmodernism of political science and philosophy. Nevertheless, if postmodern theories in science and art seem to have little in common, they all stress to varying extents the idea that properties are relative rather than absolute. Postmodern general equilibrium theory thus fits very well into this setup.

The first two chapters of this book are devoted to a short presentation of the evolution of the theory of general equilibrium from a theory of rational economics (Divisia 1928) into the style epitomized by Debreu’s *Theory of Value* (1959) and finally into a theory where the main goal is the characterization through the equilibrium manifold approach of sets of economies and equilibria that satisfy specific economic properties.

Chapters 3 and 4 deal with the core of the equilibrium manifold approach applied to the study of the equilibrium equation of the pure exchange Arrow-Debreu model.

Chapter 5 describes a dual formulation of the equilibrium manifold approach. That approach generalizes to the case of many consumers the indirect utility of Hotelling (1938) and Roy (1942) considered in many textbooks. The interest of this dual formulation comes from its superiority in dealing with rather complex properties of the equilibrium equation, properties that prove crucial in later chapters of the book.

Chapter 6 illustrates the versatility of the equilibrium manifold approach by applying it to a version of the Arrow-Debreu model where preferences are price-dependent. That model is shown to have the same properties as the standard Arrow-Debreu model with price-independent preferences, provided total resources are variable. That model plays an important role in the analysis of the temporary equilibrium model, which is taken up in chapter 9.

Chapter 7 addresses the definition of a realistic dynamic adjustment process for nonequilibrium prices. Stability is typical of those properties that cannot be satisfied by all equilibria. Samuelson’s (1941) definition of what he calls Walras tatonnement suffers from a speed of adjustment of the price dynamics that is totally arbitrary. The problem with such a definition is that the stability properties of equilibria depend, (p.xiii) among other things, on the speed of adjustment. In this chapter, the speed of adjustment is derived from the structure of the exchange process, out of and at equilibrium. Stability is then studied for this intrinsically defined speed of adjustment.

Chapter 8 deals with an extension of the Arrow-Debreu model, namely, the fully stationary intertemporal Arrow-Debreu model where some consumers face restrictions in their ability to transfer wealth between time periods. This extension represents one of the first steps into a genuinely general equilibrium analysis of economic fluctuations. The question here becomes whether a fully stationary model can feature equilibrium solutions that are not stationary. It turns out that if there are no restrictions on intertemporal transfers, then all equilibrium solutions are asymptotically stationary, with little room left for fluctuations. The picture becomes totally different, however, if some restrictions exist on individual intertemporal transfers. Nonstationary equilibrium allocations can then exist. The equilibrium manifold approach is used to gain insight on these economies and to assess the role of restrictions in intertemporal wealth transfers in creating or amplifying economic fluctuations.

Chapter 9 deals with the two-period temporary equilibrium model with financial assets and arbitrary forecast functions of future prices, an approach in the tradition of Lindahl (1939) and Hicks (1946). At variance with earlier treatments of the temporary equilibrium model, future prices are included in the states of nature. In other words, economic agents forecast probability distributions of future prices. This approach enables us to give to the temporary equilibrium model with assets a reduced form that is the Arrow-Debreu model with the price-dependent preferences of chapter 6.

An appendix is devoted to a detailed and rigorous analysis of the properties of the set of Pareto optima in the Arrow-Debreu model. Most of these properties are well-known though rarely treated with sufficient rigor. Since their role is crucial in several parts of the book, they deserve a presentation on a par with the rigor of the rest of the book.

The mathematical prerequisites for reading this book have been keptto a minimum. Readers are expected to have a basic knowledge of point-set topology. They are also expected to be comfortable with properties of smooth maps from open sets of Euclidean spaces into Euclidean space. These properties include the inverse function theorem and the implicit function theorem. A very neat and accessible presentation
(p.xiv)
of this material can be found in Spivak’s *Calculus on Manifolds* (1965). As a reminder, it is worth recalling how the implicit function theorem is a workhorse of Samuelson’s *Foundations of Economic Analysis* (1947).

The theory of smooth manifolds and smooth mappings provides the mathematical setup for the study of the properties of general equilibrium models through the equilibrium manifold and the natural projection approach. But only the most elementary properties of smooth manifolds are needed to go through chapters 2–5, and these properties are not even strictly necessary in the later chapters. By skipping the sections in chapters 2–4 that deal with the equilibrium manifold without using an explicit set of coordinates, no knowledge of the theory of smooth manifolds is even strictly necessary. These mathematics can be found in the first chapters of Milnor’s marvelous little book *Topology from the Differentiable Viewpoint* (1997). Guillemin and Pollack’s more lengthy treatment of the same subjects in *Differential Topology* (1974) nicely complements Milnor’s book with the bonus of pictures that help develop a strong geometric intuition. A mathematically more advanced presentation that goes far beyond the needs of the current book can be found in, for example, Hirsch’s *Differential Topology* (1976).

An introductory knowledge of microeconomics at the graduate level is also sufficient but not even strictly necessary provided readers work through some of the easy exercises that complement several sections of the book.