Werner Hildenbrand and the general equilibrium theory
Will we have to rewrite our textbooks?
K.Vela Velupillai

 Imagine, if you can, the Pope, invited to give a Lecture to the Clergy of Northeastern Italy, in an appropriately refurbished hall at the Buonconsiglio, say on the occasion of a suitable anniversary of the celebrated meeting of the Council of Trent. Suppose the Pope, on such an occasion, decides to talk on the topic of the Immaculate Conception and it dawns, gradually, on the learned clerical audience, that, in fact, he was questioning its veracity.
 Professor Hildenbrand’s eloquent and erudite Lectio Magistralis on a topic that almost characterises the citadel of economic theory, general equilibrium theory, was, in conception and execution of the nature of a Pope calling into question the Immaculate Conception. Of the five books he has authored or co-authored, three are on mathematical general equilibrium theory of one kind or the other. By any standard, he is not only one of the great modern authorities of this subject; but also an architect of many parts of it.
 That he, for most of his professional life, an inhabitant of the citadel of economic theory, questions the economic foundation of the building that has been erected with painstaking care, elegance and professionalism by a distinguished array of economic theorists, including himself, was akin to the Pope questioning the Immaculate Conception.
 Many of us in the audience - woefully lacking in the kind of numbers appropriate for such an exalted academic occasion - had been nurtured in the outstanding texts through which professor Hildenbrand, over a period stretching to almost 30 years, had been expounding the theory of general economic equilibrium. In essence, this is a theory that is based on simple - even simplistic - hypotheses about the activities of a suitably abstract economic system. Such a system, assumed to be populated by rational decision makers achieves an equilibrium of supply and demand, under certain explicit conditions about consumer behaviour, production possibilities activated by profit-seeking entrepreneurs encapsulated in institutions such as firms. The crowning achievement of modern general economic theory, a subject on which all students of economics are weaned, was encapsulated in what mathematicians call an existence theorem: a rigorous proof of the conditions under which a decentralized market economy, populated by the kind of agents mentioned above, could achieve an equilibrium allocation of scarce resources in a precisely specified optimal manner, first articulated by Vilfredo Pareto.
 Even although the basis for a formal statement about the achievement of a balance between the forces of supply and demand via the invisible hand was first articulated majestically by Adam Smith, it is only with the founders of the Lausanne School, Léon Walras and Vilfredo Pareto that the idea was made formal in a precise and mathematical sense. However, professor Hildenbrand did caution us to sympathise with all economists who attempt to formalize any aspect of economics mathematically. His eminently reasonable point of view was that there would always be an unbridgeable chasm between the economic vision of the theorist and the feasible mathematical formalism of the vision. As his story unfolded, we became aware that this was an important thread that held his own vision of the path towards modernity in economic theory: the chasm had widened massively in the three-quarters of a century from Walras to Arrow-Debreu.
Professor Hildenbrand’s excellent exposition of the story of this saga of the mathematical formalization of the fundamental problem of supply and demand was Whig History at its best. His starting point was the modern mathematical formalization of optimal market equilibrium in a decentralized economy made famous by Kenneth Arrow and Gerard Debreu in the very early 50s. His almost counterfactual question was: if this is where we are, where must we have started? Of course, such a question, by its very nature, can be answered in many different ways. The great merit of Hildenbrand’s expository skills and impressive erudition was that he was able to locate, in the works of Leon Walras, a precise starting point and a believable path that may have been traversed in the development of general equilibrium theory to arrive at the impressive culmination with the work of Arrow and Debreu. This path, as professor Hildenbrand quite convincingly demonstrated, was dotted with outstanding partial contributions by an array of impressive economic theorists and even ‘outsiders’ to the profession, including a banker, Schlesinger, and a mathematician, Wald, who, step by step, refined the original Walrasian framework, till it was amenable to mathematical manipulations without compromises with logic or economic intuition. The latter included, above all, the need to make sure that mathematical reasoning did not lead to economic absurdities, such as negative prices equating supply and demand.
Till this stage of the story, professor Hildenbrand was not the Pope who was questioning the immaculate conception. He was telling a good story, well substantiated with accurate textual details, about how we got to where we are. One had to agree to some of the ground rules of story telling: in this case, agree on the virtues of a Whig History. Having, however, described the path that may have brought us to the present state of general equilibrium theory, he did not pause to wonder whether the same or slightly different starting point in Walras may not have taken alternative routes towards another, more, interesting present. That was not his aim.
Instead, at this point, he switched gears, so to speak, and suggested, with impeccable logic that where we are now is, from an economic point of view, a pointless position. The exercise in proving the existence of equilibrium mathematically had become a self-indulgent pastime of scholars with inadequate economic intuition or education but possessed with a modicum of mathematical knowledge, which enabled them to explore alternative “economic” hypotheses, ungrounded in economic reality, which may produce economic equilibria of almost any variety and in any number, almost to order.
Those of us who, as students or teachers, had rationalised the learning and teaching of general equilibrium theory as a possible benchmark to anchor our explorations of economic realities, were grateful to professor Elisabetta De Antoni’s perceptive question on the relevance of such an attitude. Professor Hildenbrand’s unambiguous answer was that general equilibrium theory was not a benchmark for anything, let alone as an anchor for the pursuit of realistic studies of economies near or far from any kind of equilibrium.
He concluded with suggestion on how we may progress beyond the sterility of academic exercises of proving the existence of an economic equilibrium. His major suggestion was that economists must unshackle themselves of the dangerous habit of postulating metaphysical entities such as utility functions. Instead, he suggested that a fruitful line of research was to start from obvious economic constructs: demand and supply functions and locate them in the characteristics common to identifiable groups of economic actors, studied probabilistically. 
The Lectio Magistralis was delivered by professor Hildenbrand with authority, vision and panache on a topic of central importance to economic theorists and those who claim its relevance to an interpretation and a study of the vicissitudes of economic reality. I doubt that anyone in the audience expected the startling negative verdict at which he arrived on a topic to whose phenomenal development and formal elegance he had himself so famously contributed.
We will now have to rewrite our textbooks and unlearn some cherished theories!