Equilibrium in Economics

Scope and Limits

Valeria Mosini

• Publisher: Taylor & Francis
• Year: 2008
• Print ISBN: 9780415391375
• eBook ISBN: 9781134165865
• Edition: 1
• Topic: Business
• Subtopic: Business General
• Index: Business

General Equilibrium Theory, which became the dominating paradigm after the Second World War, is founded on the postulated existence, uniqueness, and stability of equilibrium in economic processes. Since then, the concept has come under sustained attack from all points of the heterodox compass, from Austrian economists to Marxists.

Partly in response to these pressures, mainstream economics has changed and moved away from the rigid framework of GET. Nonetheless, economists are continually arguing in terms of equilibrium and the existence of a variety of equilibrium concepts continues to stir controversy.

The contributions in this book, which include articles from Tony Lawson, Ivor Grattan-Guinness and Roger Backhouse, highlight current notions of equilibrium in economics and provide a guide to understanding the links between economic theory and economic reality.

Part I
The interplay of equilibrium notions between the natural sciences and economics

1 Equilibrium in mechanics and then in economics, 1860–1920
A good source for analogies?

Ivor Grattan-Guinness

It is well known that mathematical economics took much positive influence from classical mechanics, especially from the 1860s until the 1920s, and that equilibrium was a prime target for imitation. After a review of the main traditions of mechanics, this chapter considers the history of its effect upon economics, first by noting some branches of economics so influenced where the effect was far from clear or rapid, and second by considering the place of mechanical principles, especially equilibrium, in the work of some major neoclassical economists. Finally, the merits of the analogies will be considered, and a rather ironic conclusion drawn.

The varieties of equilibrium in mechanics

Mechanics has a very long history; we are concerned here with the developments of it during the later eighteenth and especially the nineteenth centuries, when emerged the versions of which the effects on economics are most apparent. The first feature to emphasise is that Newtonian mechanics was a major version but far from the only one, especially on the Continent, where almost all the main advances were made from the 1740s onwards. For further details of this huge story, see Dühring (1873), Mach (1883) (with caution), Rühlmann (1881–85), Wolf (1889–91), Voss (1901), Duhem (1903), Stäckel (1905), Jouguet (1909), Dugas (1955), Grattan- Guinness (1990a, b, 2005a, 1994a, esp. pts 8 and 9), Roche (1998), Heilbron (2002) and Pulte (2005), and the many further original and historical references given in these sources; and the extensive bibliography to this chapter.

The range of mechanics

First we need briefly to consider the range of phenomena that had come to fall under mechanics. They may be conveniently divided into five branches: the adjectives below are mine.

Corporeal mechanics concerned ‘ordinary-sized’ objects as found and handled on the Earth: bodies, fluids and elastic surfaces. Sound was then often regarded as part of mechanics (Cannon and Dostrovsky 1981). In celestial mechanics the planets and satellites were taken to be point masses; prime concerns included the fine details of their orbits and rotations, the former considered also for comets. Planetary mechanics was concerned with the shape of the heavenly bodies: the Earth was the most important one, followed by the Moon. Important topics included precession and nutation, tides, and topography and cartography. Several aspects of engineering mechanics involved friction; for example, the stability of embankments and the construction and stability of buildings or structures of various kinds, such as towers, cupolas, arches and bridges. Of the many machines in this branch, several related to water, both in their design and (in)efficiency of their operation: waterwheels, turbines, pumps, valves and pistons. Connected topics included the building and steerage of boats and ships, and the use of sails. Finally, as the least developed branch, molecular mechanics treated the interaction of the supposed ‘molecules’ somehow comprising the intimate structure of matter (in, for example, elasticity and friction studies) and/or the even smaller molecules that (for some researchers) comprised the structure of the assumed aether (Körner 1904).

By 1800 three distinct traditions had emerged, in competition not only over the question of heuristics versus formalisation but also concerning the territory of legitimate application. Let us note them in turn.

Newtonian mechanics

Newton’s approach was prominent, especially in celestial and planetary mechanics (Gautier 1817; Todhunter 1873; Greenberg 1995). His laws were at once both mathematical and mechanical. The second one was often used in the form

force = mass×acceleration (1)

including by Newton himself; but he actually formulated it in terms of a relationship between increments of impulse and increments of momentum (with mass assumed constant) (Cohen 1971; Brackenridge 1995; Maltese 1992). The first law was well understood to apply both to static and to dynamic equilibrium. However, within dynamics the derivation of some results was problematic, until it was realised (by Leonhard Euler among others) that the principle of angular momentum had to be adopted as a fourth law (Truesdell 1968, ch. 5). The notion of central forces and actions between bodies (balanced by reaction according to Newton’s third law) was widely adopted; however, the inverse square law was taken up with more enthusiasm in Britain than on the Continent, where other laws were also mooted (Guicciardini 1999).

Energy–work mechanics

An alternative tradition, with quite a long history, drew upon the relationship between kinetic and potential energies. (I use the modern terms: vis viva and forces vives then were popular names for the former notion, while the latter, involving (force×distance) in some form or other, received various names.) G.W. Leibniz advocated it in the late seventeenth century as a means of mathe-maticising René Descartes’s vortex theory of celestial motion, which Newton had come to loathe (Bertoloni Meli 1993). This tradition gained its best credentials in engineering and technology; by the 1780s it was elevated into a general approach to dynamics, with special utility in cases of impact and percussion where disequilibrium occurred. A pioneer of this tradition was Charles Coulomb (Heyman 1972), and the main advocate of its generality was Lazare Carnot (Gillispie 1971; Gillmor 1971), both men with strong engineering backgrounds.

One of the strengths of this tradition is that it allowed for the role of friction in the losses of energy and/or work. However, studies of the properties of friction were limited, for it was and remains a major stumbling block in all traditions of mechanics, in all its manifestations; perhaps the greatest advances have come in fluid friction, with the use and adaptations of the Navier–Stokes equations.

Variational mechanics

Energy mechanics challenged not only the Newtonian tradition but especially the third tradition, which grew up from the mid-eighteenth century onwards in reaction against Newton’s. Puzzled by the notion of force, Jean d’Alembert proposed that (1) should be taken as its definition; but then some new law is needed to replace it, and he offered a rather incoherent statement, now known as ‘d’Alembert’s principle’, about the relationship between the motions of masses when left in their current state of equilibrium and when affected by imposed actions such as forces or impacts (Fraser 1985).

This tradition also adopted ‘the principle of least action’, an optimising law formulated during the 1740s with the help of the calculus of variations; ‘action’ was a technical term, denoting (force×velocity×distance) in a variety of contexts: for example, for infinitesimal displacements ds it required an integral in distance s. As in other contexts in mechanics, some tricky metaphysical issues arose, concerning the relationship between force and substance; here Euler also invoked religious grounds in order to guarantee its generality. However, he used it only fitfully; it was to be utilised comprehensively first by J.L. Lagrange, but in a secular spirit (Pulte 1989).

This new tradition was enhanced by ‘the principle of virtual velocities’ (not then ‘work’, a word that hinted at the disliked notion of force): a refinement of d’Alembert’s principle, it stated how masses move after disturbance from equilibrium. But it assumed that mechanical situations could always be reduced to equilibrate ones, and that dynamics could be reduced to statics. Various efforts were made to prove it from other statical and dynamical principles, such as that of the lever (Lindt 1904).

This tradition was formulated and developed in algebraic terms. Indeed, it is often called ‘analytical’, and Lagrange’s treatise Méchanique analitique (1788) was the definitive account of it for a long time. There are no diagrams in the book, the author tells us early on, and he meant it, seriously. His use of the calculus of variations inspired the alternative adjective ‘variational’ to characterise this tradition.

A highlight of this book is an analysis, already rehearsed in papers, of the stability of the planetary system, a major use of perturbation theory (Wilson 1980). Rejecting the religious explanation of stability put forward by predecessors such as Newton and Euler, Lagrange thought that it could be proved from Newton’s laws, together with the assumption that the planets move in the same direction round the Sun. At that time the problem was conceived as showing that the inclination to the ecliptic and the radius vector of every planet will always remain bounded (while not constant). Although not fully sound, his proof brilliantly launched the study of a problem of extraordinary difficulty.

Parallel progresses

All through the nineteenth century these traditions progressed, especially in dynamics. Every aspect was advanced, from theoretical principles through properties of solids and fluids to the precise definition and measurement of quantities (passim in Klein and Müller 1896–1935; Schwarzschild et al. 1904–34; Royal Society 1909).

Statics benefited greatly in 1803 from Louis Poinsot’s theory of the ‘couple’ (his word), denoting two forces equal in value, parallel but opposite in direction and not collinear (Poinsot 1803). It is strange that this major modification of statics should have taken so long to be recognised. It also bears upon the (partial) understanding of static equilibrium held by economists and many others.

Another oddity was the small response to the ‘paradox of statics’, as Euler and others called it. This is the fact that it was not possible to analyse the equilibrium of a loaded table with four or more legs; for there were only three basic equations of static equilibrium, so that a further condition of some kind would have to be assumed. A two-dimensional analogue of this paradox obtains for a bench supported on more than two legs. Given the link that will be made by economists between equilibrium and the numbers of unknowns and of equations required to state the theory, it is ironic that insufficient equations are available in this elementary mechanical situation itself.

Among the three traditions, the variational was substantially advanced by the contributions of W.R. Hamilton from the 1830s onwards, which greatly extended the range and techniques of Lagrange’s legacy (Prange 1933). All traditions were elaborated in new contexts: continuum mechanics proved some enthralling challenges, especially in fluid mechanics and elasticity theory (Truesdell 1954, 1955, 1960). However, progress was slow on the task of analysing, especially mathematically, the interactions of the molecules which many scientists held to compose the basic components of physical bodies and substances (Weiss 1988; Rowlinson 2002), and the mysteries of the behaviour of materials, especially the many ways in which they rupture and break (Gordon 1976).

Perhaps the most significant mathematical elaboration, which affected all traditions, was the gradual growth of potential theory, especially from the 1830s (Bacharach 1883). Many of its initial contexts came from mathematical physics, which had grown rapidly in the new century, especially in French hands – heat theory, physical optics, and electricity and magnetism, and from 1820 their interaction in electromagnetism and electrodynamics (Grattan-Guinness 1990a, chs 9, 12–14). The last two subjects were new; the others just named grew massively, and not only on the mathematical side but also in their theoretical and experimental aspects (Bogolyubov 1976, 1978; Harman 1982; Garber 1999). The place in them of mechanical principles was very strong, so much so that in the early nineteenth century P.S. Laplace (1749–1827) advocated that all physical as well as mechanical theories should be developed in terms of short-range central forces acting between the elementary ‘molecules’ of which they were presumed to be composed and the cumulative actions of a body or physical regimes be taken to be the appropriate integral of its basic inte...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Illustrations
• Contributors
• Foreword
• Preface
• Acknowledgements
• Introduction: Three Ways of Looking At Economic Equilibrium
• Part I: The Interplay of Equilibrium Notions Between the Natural Sciences and Economics
• Part II: Equilibrium In Pre-Neoclassical Economics
• Part III: Equilibrium In Present-Day Economic Theory and Practice