# «3 – 2013/I It is not too late - Economics Nobel Prizes for a Trio of pioneers Uzawa, Scarf and Negishi K. Vela Velupillai March 2013 ...»

Algorithmic Social Sciences Research Unit

ASSRU

Department of Economics

University of Trento

Via Inama 5

381 22 Trento, Italy

Discussion Paper Series

3 – 2013/I

It is not too late - Economics Nobel Prizes for a

Trio of pioneers

Uzawa, Scarf and Negishi

K. Vela Velupillai

March 2013

**Tennyson’s noble lines in Ulysses (italics added) are the inspiration for this phrase:**

Death closes all; but something ere the end,

Some work of noble note, may yet be done, Not unbecoming men that strove with Gods.

The lights begin to twinkle from the rocks;

The long day wanes; the slow moon climbs; the deep Moans round with many voices. Come, my friends.

'T is not too late to seek a newer world.

Algorithmic Social Sciences Research Unit, Department of Economics, University of Trento & Department of Economics, The New School of Social Research, New York.

§ 1. A Brief Preamble “The emotion of art is impersonal. And the poet cannot reach this impersonality without surrendering himself wholly to the work to be done. And he is not likely to know what is to be done unless he lives in what is not merely the present, but the present moment of the past, unless he is conscious, not of what is dead, but of what is already living.” T.S. Eliot: Tradition and the Individual Talent As befits one who was born and spent his formative years in Serendip1, it was serendipitous that I began studying economics, in Sweden, inspired by Gunnar Myrdal’s Asian Drama, literally as the first Nobel Memorial Prizes in Economics were being awarded to Ragnar Frisch and Jan Tinbergen, then Paul Samuelson, Simon Kuznets, John Hicks and Kenneth Arrow, Wassily Leontief, Gunnar Myrdal and Friedrich von Hayek, in the first half a decade after its inception.

I am old enough to have been just adequately young to have befriended some of these giants who fashioned and moulded the analytical core of twentieth century economics – in particular, Tinbergen, Samuelson, Hicks, Arrow and Myrdal – but also had the privilege to corresponded with some of the others (Leontief and Hayek).

I doubt anyone of my generation, or any of my own teachers or their contemporaries, had too many doubts about the suitability and desirability of honouring these outstanding pioneers of analytical and applied economics, even if some – or even many – of them may have felt such a prize conferred an undeserved parity of status, by an uncomfortable adherence of the name of Nobel, with the more established sciences of physics, chemistry, medicine & physiology2.

Despite occasional grumbles, I think it is fairly safe to surmise that there was considerable agreement in the discipline at large on the award of the prize to the distinguished recipients till at least 1989 (Trygve Haavelmo). Most of the dissatisfaction in the first two decades was more on the omissions – Joan Robinson, Roy Harrod, Oskar Morgenstern, Piero Sraffa, Jacob 1 The name by which old Ceylon, now Sri Lanka, was once known, and the word, in the title of The Three Princes of Serendip, which inspired Horace Walpole to coin serendipitous.

2 The Nobel Prizes for Peace and Literature, although often controversial, did not claim an anchoring in ‘science’ – however that word may be defined; the economics prize, from the outset, was supposed to be an award for contributions to ‘economic science’ (’ekonomisk vetenskap’). Moreover, the Nobel family did not reconcile themselves to granting this new entrant parity of status with the original five.

2 Marschak3, Nicholas Georgescu-Roegen, etc., – than on the selections (with notable exceptions, especially on the timing, of the awards to Leonid Kantorovich and Milton Friedman).

Some of the perplexities were on the joint awards to Hicks & Arrow, Myrdal & Hayek, on the entirely justifiable grounds that each of these four maestros deserved not just single awards – perhaps, at least in the case of Arrow, even multiple prizes (for initiating the wholly new field of research, Mathematical Social Choice Theory and laying one of the significant foundations for what eventually became Endogenous Growth Theory – quite apart from General Economic Equilibrium Theory and Welfare Economics, for which he was actually awarded the prize, shared with John Hicks).

Then, there is the wholly different – more philosophically grounded – criticism, not voiced only by heterodox economists, but also by humanists and other social scientists, who deplore the possible unavoidability of ideological motives in the selection process, the tenuous ‘scientific’ underpinning of economic analysis, the consolidation of the increasing ‘imperialism’ of economists and economic theory vis a vis the other social sciences and, above all, the inevitable subjective biases of a selection committee unable to transcend visions – in the strict Schumpeterian senses – of an economic theory unable to encapsulate, and to be underpinned, by broader social and political concerns.

These are short-comings which we in economics share copiously with our Literary brethren, where – often – the award of the Nobel Prize for Literature highlights omissions, due, ostensibly, for cultural, historical and political reasons and for the understandable impossibility of creating a ‘level playing field’ for alternative literary traditions, has bedeviled the award, time and time again.

3 On reading my Obituary of Herbert Simon, in February, 2001, my old friend and former colleague, Michael Intrilligator wrote me that Mrs. Marschak had mentioned, in a casual conversation, that ‘they’ thought Jacob Marschak was to share the Nobel Memorial Prize in Economics with Herbert Simon, in 1978. Intrilligator added that it was the unforeseen death of Marschak that thwarted this shared honour. I am not entirely sure the dates ‘add up’, although it would certainly have called forth yet another usage of the word serendipitous, had it materialized (and given me even more professional pleasure as a current member of the economics department at NSSR, of which Marschak was a distinguished past member!).

3 Had Joyce written Ulysses or Finnegans Wake in Irish, could the Nobel Committee have had the competence to judge an award very easily? Why were Borges and Casares passed over for the award, in an era that saw the rise and celebration of Latin American literary magic realism? Why Kawabata and not Mishima? And so on, as we ask, in Economics too, why not John McCall rather than – or before, or jointly with4 - Diamond, Mortenson and Pissarides, why not Armen Alchian, at least jointly with Ronald Coase or, why not Masahiko Aoki, jointly with Elinor Ostrom and Oliver Wiiliamson, why not Oskar Morgenstern, long before Harsanyi, Nash5 and Selten or, indeed, why not Shapley for Cooperative Game Theory before celebrating non-cooperative game theory6.

§ 2. Three Non-Nobel Nobles "The most remarkable achievements of modern microeconomic theory are the proof of the existence of an equilibrium and the First and Second Theorems of Welfare Economics which establish the relation between equilibria and Pareto efficiency.

David G. Luenberger (1994), Optimality and the Theory of Value, Journal of Economic Theory, Vol. 63,# 2,, p. 147.

If, as Luenberger – and countless others – are substantially correct in their evaluation of the ‘remarkable achievements of modern microeconomic theory’, and if the profession has already paid formal homage to the acknowledged architects of these ‘remarkable achievements’ (Arrow and Debreu, above all), then, surely, it is also time we acknowledged those who have made these achievements computationally meaningful.

I aim, therefore, to make a case for awarding the Nobel Memorial Prize in Economic Sciences to two outstanding Japanese economists – Hirofumi Uzawa7 and Takashi Negishi –jointly 4 Perhaps there is a rule against four awardees for any one Prize, at any one point in time?

Many of us, of my generation of graduate studies in the early 1970s, felt – when Kantorovich and 5 Nash were recognized as having contributed sufficiently original work which had decisive importance in economic analysis – that Bruno De Finetti, too, should have, eventually, been awarded the Economics Nobel. Such thoughts would have also applied to Frank Ramsey, George Dantzig and Leonard Savage – and who, in economics, would have begrudged an Economics Nobel to von Neumann?

6 Shapley, serendipitously, is our most recent Economics Nobel, but awarded the Prize for his contributions to “the theory of stable allocations and the practice of market design" - not for his celebrated work on the ‘Shapley Value’, whose ramifications, even in ostensibly mundane applications like imputing joint costs by accountants, aircraft landing fees, production under conditions of alternative land ownership regimes, and so on remain almost inexhaustible.

Uzawa’s age cannot, of course, be held against the award of the Prize to him. After all, our current 7 Nobel Memorial Prize winner in economics, Lloyd Shapley, was born in 1923 – and Uzawa in 1928;

4 with Herbert Scarf. The case I try to make is for their pioneering contributions towards making it formally possible to underpin the purest of pure economic theory - general equilibrium theory in the Walras–Arrow–Debreu tradition, supplemented by the two fundamental theorems of welfare economics – computational8 in a rigorous, yet applicable sense.

It will not be considered an exaggeration to claim that one of the research frontiers in economics – both from theoretical and applied points of view, and from microeconomic, macroeconomic and game theoretic9 vantage points – is the search for computational foundations for the concepts, formal framework and entities of basic economic theory. The computational core of dynamic stochastic general equilibrium theory can be traced back to Scarf’s pioneering work on Computable General Equilibrium theory (Scarf, 1973), which, in turn, had as its own analytic foundations in Uzawa’s remarkably original equivalence theorem10 (Uzawa, 1962).

Almost simultaneously with Uzawa’s ‘rigorous’ derivation of his equivalence theorem, Negishi (1960) developed an alternative to the Scarf approach by devising a method to compute (approximately) the traditional Arrow-Debreu equilibrium, starting from the first fundamental theorem of welfare economics, via a concentration on a mapping of the set of weights of a social welfare function to itself.

A characterisation of the difference between the standard approach to proving the existence of an Arrow-Debreu equilibrium, and its computation by a tâtonnement procedure - i.e., algorithm - of a mapping from the price simplex to itself, and the alternative Negishi method Scarf and Negishi were born in 1930 and 1933, respectively. Compared to Shapley’s age, the latter three are mere younglings!

8 I would like to emphasise that it is their contributions towards making aspects of core areas of economic theory computational – not computable – that is of relevance here. The difference between the two terms is outlined in careful detail in many of my writings of the last decade, or so.

9 I have in mind, here, the recent burgeoning interest in, and contributions to, varieties of algorithmic game theory. The methodology here is exactly analogous to that followed by computable general equilibrium theorists – i.e., to devise procedures, implementable as meaningful algorithms on digital computers, to compute, even if only approximately, equilibria; in the one case, approximations to the traditional (Walras)-Arrow-Debreu equilibrium and the other case concentrates on Nash and related game theoretic equilibria.

10 The equivalence Uzawa demonstrated, formally, was that between the Brouwer fix-point theorem and his – i.e., Uzawa’s – formulation of a version of what he referred to as the Walrasian Equilibrium Existence Theorem (WEET).

5 of iterating the weights assigned to individual utility functions that go into the definition of a social welfare function which is maximised to determine - i.e., compute - the equilibrium, captures the key innovative aspect of the two approaches.

Essentially, therefore, the difference between the standard approach to the proof of existence of equilibrium Arrow-Debreu prices, and their computation, and the Negishi approach boils down to the following. The standard approach proves the existence of Arrow-Debreu equilibrium prices by an appeal to a fixed point theorem and computes them - the equilibrium prices - by invoking the Uzawa equivalence theorem (Uzawa, op.cit.,) and devising an algorithm for the excess demand functions that map a price simplex into itself to determine the fixed point (Scarf, 1973).

The Negishi approach proves, given initial endowments, the existence of individual welfare weights defining a social welfare function, whose maximization (subject to the usual constraints) determines the identical Arrow-Debreu equilibrium. The standard mapping of excess demand functions, mapping a price simplex into itself to determine a fixed point, is replaced by a mapping from the space of utility weights into itself, appealing to the same kind of fixed point theorem (in this case, the Kakutani fixed point theorem) to prove the existence of equilibrium prices.

In other words, the method of proof of existence of equilibrium prices in the one approach is replaced by the proof of existence of `equilibrium utility weights', both appealing to traditional fixed point theorems (Brouwer, 1910, von Neumann, 1937, and Kakutani, 1941).

In both cases, the computation of equilibrium prices on the one hand and, on the other, the computation of equilibrium weights, algorithms are devised that are claimed11 to determine (even if only approximately) the same fixed points.

I use the word ‘claimed’ guardedly and deliberately because none of the frameworks used by 11 Uzawa, Negishi or Scarf are based on either computability or constructivity theories (both interpreted in their strict mathematical senses). Even an ‘approximate computation’ of an uncomputable or nonconstructive configuration, to be meaningful, requires significant departures from the ‘paradise’ that is conventional real analysis.