1
Negishi's Contributions to the Development
of Economic Analysis: research programs and outcomes*
Warren Young
Bar Ilan University
* Forthcoming, International Journal of Economic Theory (special issue on Negishi)
Copyright , International Journal of Economic Theory (Blackwell), not to be cited
without the permission of the publisher and author.
The following persons assisted me greatly in understanding the importance of
Negishi's path-breaking contributions to economics: Kenneth Arrow, Jean-Pascal
Benassy, Jacques Drèze, Victor Ginsburgh, Peter Howitt, Tim Kehoe, Kazuo
Nishimura, Roy Radner, Herbert Scarf, Nancy Stokey, Steve Spear, Makoto Yano and
last, but not least, Takashi Negishi himself.
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Introduction
In 1992, Negishi described himself "as a mainstream economist who has made
some contributions to the areas of general equilibrium, international trade and neoKeynesian economics" (1992, 227). Indeed, few economists have written on such a
wide range of topics as Negishi. Among other things, Negishi attempted to extend the
multi-market Neo-Walrasian system in a number of directions so as to encompass
stability, imperfect competition, money, trade and unemployment – both demarcating
some of the boundaries of mainstream economic theory and setting research agenda,
as Samuelson in Foundations of Economic Analysis. Interestingly enough, it was
Arrow and Hahn (1971), who initially outlined some key elements in the Negishian
research program, and described the initial impact of his early work, while Negishi
himself (1972) completed the immediate picture, as will be seen below.
Over just a short five year period, 1958-62, Negishi made fundamental
contributions to General Equilibrium theory which changed the course of modern
economics. His 1958 paper article reduced the number of conditions for stability of
tatonnement to gross substitution.
In his pioneering 1960 paper, Negishi provided a completely new way of proving
the existence of equilibrium, via the Second Welfare Theorem. He established
equivalence between the equilibrium problem set out by Arrow-Debreu and what has
been called "mathematical programming", thereby developing a "method" which has
been utilized with much success by later economists working in both theoretical and
applied General Equilibrium modeling, as will be shown below.
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Negishi's 1961 paper "On the formation of prices" (1961a) was, according to
Arrow and Hahn (1971, 346) "the first study of a process without recontract" and,
according to them provided fertile ground for extensions by Uzawa (1962a) and Hahn
(1962).
Negishi's 1961 incorporation of imperfect competition into General Equilibrium
models(1961b), while problematic to some (see, e.g. Hart, 1985, 107) has been seen
by others as his most important contribution, as will be seen below. In this paper, he
initiated the study of imperfect competition in general equilibrium analysis. He
assumed consumers to be price takers and firms to be monopolistically competitive.
In his model, firms exhibited subjective functions, these being consistent with the
given information regarding the current state of the market. He assumed convexity of
possible production set of firms, and clearly indicated the problems associated with
non-convexities in a Walrasian system. Finally, he proved the existence of
equilibrium in an imperfect market setting.
His joint 1962 paper with Hahn (1962a), introduced the "Hahn-Negishi Process"
of non-tatonnement stability, while Negishi's survey article on stability, published in
1962, is still considered to be the most authoritative on the subject published over the
past four decades (1962b).
Any attempt to analyze all of Negishi's "multifarious contributions", as Drèèze
recently put it, would be a Herculean effort. The object of this paper, therefore, is to
assess the impact of what can be considered his "most significant" contributions
(Drèèze, 2006, personal communication), in terms of the research programs that
emanated from his early work (1958, 1960, 1961a, 1961b, 1962a, 1962b) and their
outcomes..
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Negishi on Gross Substitutes, Welfare, and Existence of Equilibrium
Gross Substitutes (1958)
Already in his Master's dissertation, entitled "Existence and stability of an
economic equilibrium" [in Japanese] (1957, cited in Negishi 1972, 12, note 4), as he
recalls, he had proven "the local stability of the gross substitute case" (2000 b, 326).
His first publication in English was in one of the leading journals, Econometrica, in
July 1958, and remains, along with papers by Arrow and Hurwicz (1958) and Hahn
(1958), one of the fundamental papers dealing with issues of existence of stability of
the tatonnement process. What is interesting to also recall here is the editor's note
regarding Negishi's first published paper (1958, 445): "These papers [Hahn; Arrow
and Hurwicz; Negishi] were written independently of one another and were submitted
for publication at about the same time". This illustrates not only the importance of
Negishi's contribution in the context of the "multiple discovery", but also highlights
the importance the editors of Econometrica placed upon it.
Now, according to Negishi, in his 1958 paper, he "proved the local stability of
the tatonnement process under the assumptions of gross substitutability and the
homogeneity of excess demand functions" (2000 b, 326). While Hahn's 1958 paper
did not cite Negishi, it did refer to the forthcoming work of Arrow and Hurwicz and
their proof of "a similar theorem" (Hahn, 1958,169, note 1). Arrow and Hurwicz, for
their part, referred to both Hahn's paper and that of Negishi and wrote (1958, 546,
note 44) "the general case [regarding stability of equilibrium under the assumption of
gross substitutability] has been demonstrated independently by Hahn... and Negishi".
And, almost 50 years later, the legacy of Negishi's 1958 paper is still "well alive"
(Hamada and Endo, 2004, 22).
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Welfare, Existence of Equilibrium and Competitive Economy (1960) and its
impact
Theoretical extensions and modifications
What has been called Negishi's " approach" or " method" (1960) has been
widely utilized on both theoretical and applied levels, the former in the context of
dynamic models and infinite dimensional general equilibria; the latter in the context
of their numerical computation, as will be seen below. Similar results to Negishi's
1960 proof of the existence of equilibria based on what can be termed mathematical
programming techniques, were obtained by Takayama and El Hodiri (1968) and
Diewert (1970).
Mantel (1971) was the first to consider the utilization of Negishi's1960 approach
for computing equilibria. This extension, or tatonnement algorithm, encompassed
what Mantel called "the welfare adjustment process" so as to enable "the computation
of an equilibrium solution, adjusting the weights assigned to each individual in the
social welfare function" (1971, 415).
Negishi's 1960 approach also showed that a vector of welfare weights existed
and that transfers necessary for decentralizing the Pareto efficient outcome were zero.
In seminal papers, Bewley (1982 [1980]) and Yano (1984a, b) applied this to dynamic
models in what Kehoe later termed characterizing "equilibria as solutions to social
planning problems" (1991, 2090). Lucas and Stokey (1984) also utilized Negishi's
approach and that of Bewley (1982 [1980]) in their treatment of optimal dynamic
growth, and while not directly citing Negishi (1960), they took his 1960 approach to
be part of the corpus of economic theory, and utilized it as such (Stokey, 2006,
personal communication).
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Kehoe and Levine (1985) modified the approaches of Negishi and Bewley to
deal with the properties of an infinite horizon economy. More specifically, they
developed an approach by which to deal with the problem of determinacy in settings
with infinite dimensional characteristics based upon Negishi's characterization of
equilibrium as zero on the map of excess spending (1960). Further modification
formed the basis for Kehoe, Levine and Romer (1990). This involved conversion of
the infinite dimensional problem "into a finite dimensional Negishi problem" (1990,
1). Following this, Kehoe, Levine and Romer (1992), utilized the the Negishi
approach to deal with equilibria of economies with externalities and taxes. The
Kehoe and Levine (1985) approach of utilizing the map of excess spending assuming
additively separable preferences was further extended by Belasko (1997) and
Chichilnisky and Zhou (1998).
Mas-Colell(1986) also followed Negishi's 1960 approach in his reconsideration
of the existence problem regarding price equilibrium and exchange economy
characterized by unbounded number of commodities with a finite number of
consumers, and developed a topological version of Negishi's approach, as manifest in
Mas-Colell and Zame (1991). Dana and Le Van (1991) also extended Negishi's
approach, albeit into a "dual version", by utilizing a Pareto optimal weighting system.
This Negishi-based "weight approach" has, in fact, come to be one of the main tools
in equilibrium asset pricing (Duffie, 1996; Becker and Boyd, 1997).
Following from the work of Dana and Le Van (1991), Duran and Le Van (2003)
applied Negishi's method to prove existence of equilibrium in a one- sector Ramsey
economy. The "Negishi-type" social planner also provided the basis for the analysis
by Jensen (2004) of unbounded growth with consumers who are heterogeneous.
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Another extension of Negishi's 1960 approach can be found in Ghiglino and
Olszak-Duquenne (2001) and Ghiglino and Sorger (2002). The former, by utilizing
Negishi's 1960 approach, showed that the dynamics in a two-sector neoclassical
general equilibrium model with exogenously determined labor supply-- with a single
consumption good-- is affected by the initial distribution of capital. The latter, using
the Kehoe-Levine-Romer (1991) modification of Negishi's 1960 approach, analyzed
how initial individual values of wealth, and the wealth distribution, impacted on the
dynamics of equilibrium. And, as Ghiglino put it (2002, 5) "the natural framework to
analyze the effects of income inequalities in a general equilibrium framework is
provided by a generalization of the Negishi [1960] approach".
Crockett, Spear and Sunder (2005) have further extended Negishi's 1960
approach to a learning rule that could ensure convergence to competitive equilibrium.
Indeed, on the theoretical level, then, it can be said that Negishi's 1960 approach has
become a "standard tool" (Ghiglino and Sorger, 2002, 122).
Applications and Computational Analysis
With regard to the influence of Negishi's 1960 paper upon the development of
"applied", that is "computable" general equilibrium models, in a private
communication to the present author, Ginsburgh gave a comprehensive albeit "brief
history" of the impact this paper made. He wrote (2006, personal communication):
Waelbroeck and I were among the first to use in an applied model Negishi's way
of proving the existence of a competitive equilibrium in Ginsburgh and… Waelbroeck
(1981)….The book came out in 1981, but there were papers published [by me] before
that, the oldest being "Computational experience with a large general equilibrium
model"(1976) and a 1975 Cowles Foundation Discussion Paper "A general
equilibrium model of world trade. Part I: Full format computation of economic
equilibria"… Dixon played with the same idea in his 1972 Harvard Ph.D. thesis which
was published in 1975 as The theory of joint maximization, but he only refers very
loosely to Negishi, though it is exactly the same model as Negishi's. He calls it "joint
maximization". Earlier than that, a paper in the spirit of Negishi's proof... is
Trzeciakowski, (1971).
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More recent applications of Negishi's 1960 approach to the computation of
equilibria can be found in Ginsburgh and Van der Heyden (1988), who apply it to the
case of the government price supports; Kehoe (1991), who dealt with multiplicity of
equilibria; Backus, Kehoe and Kydland (1992 a, b, c), who apply the approaches of
Negishi (1960) and Mantel (1971) to calculate equilibria in international real business
cycles and dynamic general equilibrium models of international trade; Nordhaus and
Yang (1996), who apply it to develop a computable general equilibrium model of
alternate climate-change strategies; Ginsburgh and Keyzer (1997), who deal with the
structure of AGE models in their book; Esterban-Bravo (2004), who surveys the
computation of equilibria in GE models via interior-point methods; Kehoe et. al
(2005), who present the "frontiers" of CGE models; and Judd (2005), who extends the
approach to dynamic stochastic GE models.
Monopolistic Competition: 1959, 1961, and 1972 vintages and their impact
Negishi first gave this paper at the December 1959 Washington, D.C. meeting of
the Econometric Society .According to the report of the Washington meeting, he
presented his paper on Monday afternoon, 28 December 1959, at the session
"Economic Theory I". The three papers presented at the session included one by Hahn
entitled "The existence of competitive equilibrium". Koopmans also presented a
paper, entitled "Stationary Ordinal Utility and Timing Preferences". Ando and
McKenzie were the discussants for all the papers. Negishi's paper, however, was the
only one that had an abstract published in the Washington meeting Report (1960, 67778).
A decade later, Arrow and Hahn (1971, 151-167) extended the existence theorem
and formal model contained in what they called Negishi's "brilliant paper" (1971,
167) via more general assumptions such as non-convex production possibility sets
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(1971, 152). Negishi's monopolistic competition paper (1961b) has been cited almost
as many times as his 1960 paper, and also originally stimulated a Negishian-based
research program in the 1970's as described by Roberts and Sonnenschein in their
Econometrica survey (1977). As they wrote (1977, 101) "Since the pioneering work
of Negishi [1961b], a number of studies have been directed towards incorporating
firms which recognize their ability to influence prices (but behave non-cooperatively
towards one another) into the Arrow-Debreu model of general equilibrium". They
listed the works of Arrow and Hahn (1971), Fitzroy (1974), Gabszewicz and Vial
(1972), Laffont and Laroque (1976), and Marschak and Selten (1974).
Drèèze (1975), Grandmont and Laroque (1975) and Benassy (1976) had extended,
implicitly, in the former cases, and explicitly, in the latter case, Negishi's "brilliant
paper" (1961b) in another direction, into the sphere of disequilibrium, that is to say
non-tatonnement situations, where, as Benassy put it, "transactions can actually occur
outside equilibrium" (1976, 69).
Silvestre (1977a, 1977b) , for his part, extended Negishi (1961b) to encompass
increasing returns to scale, including Negishian-style subjective demand functions. He
also developed (1978, 397) an alternative existence theorem to overcome what he
called the "shortcomings" in the Arrow-Hahn (1971) extension of Negishi (1961b).
In his 1980 Econometrica survey, Drazen noted that the "the basic work on
monopolistic competition in general equilibrium models is the pioneering contribution
of Negishi [1961b]..." (1980,290). Drazen went on to say that besides the work of
Benassy [1976] and Drèèze[1975] "a model in which price setting is integral to agent's
attempts to "break" the constraints is that of Hahn [1978] , also based on Negishi's
work [1961b], which uses the Drèèze [1975] basic framework" (1980,291). There is,
however, a problem in Drazen's linkage of Hahn [1978] to Negishi [1961b]. This is
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due to the fact that Hahn said (1978, 7) that his 1978 approach "is not the model
Negishi [1961b] considered. I have not found it in Benassy. The kink which my
formulation gives rise to has recently been used by Negishi ... in a different context"
referring the reader to an unpublished 1974 " Memo" by Negishi, which Hahn cited
as "Unemployment Equilibrium". In a recent communication to this author, Negishi
(2006d) cleared up the matter by saying that "Judging from the date, the paper seems
to be the one I read in a conference organized by the Institute for Advanced Studies,
Vienna, in 1974. The exact title of the paper is "Existence of an under-employment
equilibrium." This paper was published in G. Schwoediauer, ed., Equilibrium and
Disequilibrium in Economic Theory, D. Reidel Publishing, 1978, pp. 497-510"
[Negishi 1974,1978].
Hart (1985), in his overview of work on "imperfect competition in general
equilibrium" claimed that both Negishi's approach (1961b) and that of Arrow and
Hahn (1971) were not "very satisfactory" (1985, 107) and "suffered from…weakness"
as "the class of possible equilibria…is very large"; the outcome of being based upon
"subjective", rather than "objective" demand (1985, 139). Hart's points may be
relevant; however, as Gabszewicz (1985, 152) noted in his comment on Hart, the
"overview" did not deal with the issue of imperfect competition and product
differentiation, as set out by Negishi in the 1972 revision of his paper (1972, 104), an
approach that was implicitly taken up, for example, by Mas-Colell (1975) and Drèèze
and Hagen (1978). In contrast to Hart's critical view of Negishi (1961b), then, and the
fact that he did not deal with the work it stimulated, Gabszewicz (1985, 151) wrote
"The Negishi approach…received a considerable revival of interest".
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Very significant outcomes of Negishi (1961b) are found in the non-tatonnement
approach of Benassy (1976, 1978, 1982, 1991, 1995), and in the attempt to
reformulate Negishi's approach (1961b) by Gary-Bobo (1989). It should also be
recalled at this point that Negishi's volume Microeconomic Foundations of Keynesian
Macroeconomics appeared in 1979, but before this he had presented papers, and
published articles which formed the basis for this book (1974, 1978). Perhaps the
most influential of Negishi's articles in this field, that appeared prior to his 1979 book,
was entitled "Existence of an under-employment equilibrium". As mentioned above,
this paper was originally presented at a conference in Vienna in July 1974, and was
published in the volume Equilibrium and Disequilibrium in Economic Theory (1978,
497-510). In this volume, Negishi's paper was complemented by that of Benassy, "A
Neo-Keynesian model of price and quantity determination in disequilibrium" (1978,
511-544).
It is important to point out here that both papers, in addition to Benassy (1976),
preceeded the so-called "New Keynesian" works that appeared in the 1980s (e.g.
Rotemberg 1982; Mankiw, 1985). Indeed, in his 1976 paper, Benassy wrote (1976,
69) "monopolistic price setting was incorporated for the first time in a brilliant paper
by Negishi [1961b]". The impact of Negishi (1961b) on "New Keynesian economics"
has been recounted by Peter Howitt (2006, personal communication): "most of
modern New Keynesian Economics owes its intellectual origins to his [Negishi's]
work in imperfect competition in General Equilibrium, although it is, like Benassy's
work which was also very influential, hardly ever cited. I would say these works were
influential because so much subsequent work is a straightforward extension of the
concepts and techniques in them, and because although they are not much cited, they
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were available in prominent publication outlets when the work that is cited
(Blanchard-Kiyotaki, for example) was done".
More recently, Dehez, Drèèze and Suzuki, in their work on imperfect competition
and fixed costs have asserted (2003, 220) Negishi's 1961 "perceived demand
approach" to be, in their opinion, "more realistic, in many situations, than the
alternative, objective demand approach". Moreover, Negishi's approach (1961b) has
been even further extended into the area of international trade theory by Neary (2003),
in his attempt to analyze globalization and the structure of markets.
Econometrica survey article and "Hahn-Negishi process", 1962
Negishi's classic 1962 survey article "The stability of a competitive economy"
was originally presented, in part, at the September 1960 Naples meeting of the
Econometric Society (Negishi 1962b, 635, note 1). The abstract of this paper was
published in the Report of the Naples meeting in Econometrica (1962, 192). Among
those who took part in the discussion also published in the Report (1962, 192-194)
were Malinvaud and Wold, with a brief reply by Negishi. In his comments,
Malinvaud focused on the implications of Negishi's analysis of "tatonnement" and
"non-tatonnement" processes. Negishi's reply to the commentators was "I would say
that the tatonnement processes is a special case or limiting case of non-tatonnement
processes. It must be noted that both of them lead to a point which is optimal in the
sense of Pareto" (1962, 194). Writing with the perspective of over four decades,
Drèèze has recently commented on what Negishi said as follows (2006, personal
communication):
"I agree that limit points of both Walrasian tatonnement and Walrasian nontatonnement are Pareto-efficient (under standard assumptions). The point on which
there has been occasional misunderstanding is that limit points of Walrasian non-
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tatonnement are not competitive equilibria relative to initial endowments, so the
efficiency proof is a bit different."
His 1962 survey article contained important "suggestions for future studies"
(1962b, 665-666), some of which Negishi himself took up in his 1964 papers (1964a,
b) and in Chapter 13 of his 1972 book. For example, in the 1962 survey paper, he
proposed studying the "price formation process... over Hicksian weeks", with
reference to "the cobweb process" and "the process with interactions between
expectations and inventory fluctuations" (1962b, 666). In his 1964 paper (1964b,
649), Negishi linked Arrow-Debreu (1954), and Enthoven and Arrow (1956), with
Muth (1961); which he repeated in Chapter 13 of his 1972 book (1972, 201). The
importance of Negishi's survey article was recognized early on. For example,
Patinkin, in Money, Interest and Prices wrote (1965, 540): "The extensive literature
which has grown up in recent years on the stability question has been usefully and
critically surveyed by Takashi Negishi".
In 1962, Negishi also published his joint paper with Hahn. The first note in this
paper (1962a, 463, note 1) said that "this is the outcome of two papers by the authors,
each written independently. Hahn's paper [Hahn, 1960] formulated the process of
adjustment and proved some theorems which were then generalized by Negishi...";
"Negishi's contribution" was "a sequel" to his 1961 International Economic Review
paper "On the formation of prices", written while he was at Stanford (1961a, 26).
As Negishi recalled (2006, personal communication) "Frank and I got acquainted
…before the Washington meeting, in [the] Stanford-Berkeley regular joint seminars
in 1959-60. Frank was at Berkeley in 1959-60 and I was at Stanford, though, because
of [the] 1958 gross-substitute paper, we knew each other's names in 1958".
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In his paper "The stability of exchange and adaptive expectations" (1964a)-which also emanated from the research program he outlined in his 1962 survey
article--Negishi attempted to model the relationship between prices and expectations
in the non-tatonnement process he had originally developed independently of Hahn
(Negishi, 1961a), based upon the approach developed in Hahn and Negishi (1962a)
and Arrow and Nerlove (1958).
Perhaps one of the most important outcomes of the "suggestions for further
studies" in his 1962 survey article, is Negishi's utilization of rational expectations in
his 1964 "Note" in Econometrica entitled "Stability and rationality of extrapolative
expectations"(1964b). Negishi did not attend the sessions of Muth and Mills at the
1959 Washington meeting of the Econometric Society where he gave his own paper
("Monopolistic Competition"), but as he recalled in a letter to this author (9 Dec.
1991):
I recognized, however, the significance of their contributions soon after they
were published, at least in my own way of interpretations. I published a small note
“Stability and Rationality of Extrapolative Expectations” in Econometrica (1964), in
which I referred to Muth’s 1961 Econometrica paper. Also, in the Chapter on
Rational Expectations of my 1965 book Kakaku to Haibun no Riron (Theory of Price
and Allocation), I referred to Mills’ 1962 book as well as Muth’s 1961 paper.
In fact, Negishi’s 1964 “small note” is, in my view, quite significant, for in it he
used Muth’s approach to “give some rational basis to extrapolative expectations”
(1964b, 649). In other words, Negishi proposed rational expectations as the basis for
the endogenous expectational assumption underlying “the dynamic stability of
multiple markets” in the system originally proposed by Arrow and Debreu (1954) as
manifest in Enthoven and Arrow (1956). In fact, Negishi actually extended Muth’s
approach in this regard. As he put it (1964b, 649):
The rational expectation hypothesis advanced by Muth...is that expectations are
essentially the same as the predictions of the relevant economic theory; that the
economy generally does not waste information; and that expectations depend
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specifically on the structure of the entire system. However, since there is cost of
information and computation, expectations may also be called rational when they are
formed as the prediction based on a simplified and approximated version of the
economic theory, using only limited amounts of information on a part of the system.
Extrapolative expectations will be derived below as the prediction of the equilibrium
by the use of estimated excess demand functions, and it will be shown that the
coefficients of expectations thus derived are such that the system of multiple markets
is stable when gross substitutability and tatonnement are assumed.
By making rational expectations the expectational basis of the Arrow-Debreu
general equilibrium model, Negishi provided fertile ground for Radner to further
develop the Arrow-Debreu approach. For, as Roy Radner also wrote in a letter to this
author (1992, personal communication):
My own interest in the subject arose from my attempt to extend the ArrowDebreu model to the case of incomplete markets. The first results of this attempt were
published in 1967...This paper dealt simultaneously with two aspects of “rational
expectations”: consistency in the expectations of future prices, and making inferences
about other agents’ information from equilibrium prices. I like to think it had little
impact because it was published in French!" [on these and related issues, see Young
and Darity, 2001 and Young, Leeson and Darity, 2004].
Negishian Research Programs: variations on the themes of his contributions
Negishi 1960 and 1961
In order to gauge the impact of Negishi's contributions in the eyes of his peers,
this author contacted those leading economists who worked with, or were directly
influenced by Negishi, and asked them to assess the impact of his work on the
General Equilibrium and Non-Walrasian research programs. In his reply, Kenneth
Arrow wrote (2006, personal communication):
Negishi's work...pioneered in the study of "non-tatonnement" stability, in a paper
of his own and a subsequent one jointly with Frank Hahn... He also developed a
model of general imperfectly competitive equilibrium, based on subject[ive] demand
curves, which impressed me at the time, though it doesn't seem to have had a
permanent effect on the literature. Finally, and perhaps most importantly, he had a
proof of the existence of equilibrium based on the idea that a competitive equilibrium
maximizes a suitably weighted sum of individual utilities. The existence proof
depends on a fixed point in the space of weights. This proposition has been used in
solving applied general equilibrium models and was also extensively used by Lucas
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and Stokey in the analysis of intertemporal equilibrium as applied to the study of
economic fluctuations.
In his reply, Drèèze took the opposite position to that of Arrow regarding the
importance of Negishi's "extension of general equilibrium to monopolistic
competition". As Drèèze wrote (2006, personal communication)
Regarding Negishi, I have all along admired his work, mostly for the reason
which he himself lists as his main motivation in his entry for "Who's who in
Economics" (3rd ed. p. 819): "I have always tried, not so much to generalise theory
mathematically, as to enrich it with economic significance, so that it can be applied to
the problems of the real-world economy". Of course, this remark should be applied
primarily to General Equilibrium Theory, not to any brand of theory. The remarkable
feature is the combination of respect for abstract theory as evidenced by privileged
attention to GE, and primary interest in real-world problems [Drèèze's emphases].
Two obvious applications of the principle are: (i) the extension of GE to
monopolistic competition; and (ii) the attention to non-Walrasian equilibria. To my
own eyes, (i) stands as the most significant among Negishi's multifarious
contributions. There is really no palatable alternative to his approach (given the
existence problems and lack of realism of the so-called "objective demand"
alternative). I am confident that it will be retained again and again by future
researchers in the area. Also, his exposition in the 1961 article and 1972 book is very
good: rigorous, concise and understandable - what more could one ask?
In his reply, Benassy (2006, personal communication), strongly supported
Drèèze's postition when he wrote "To me the major contribution by Negishi has been
his short but incredibly influential 1961 article "Monopolistic competition and general
equilibrium". All authors writing on the subject have been directly influenced by this
highly elegant article, so I would place it at the center of any piece on Negishi's
contributions".
Scarf, for his part, noted in his reply (2006, personal communication) to this
author:
Takashi Negishi has made many significant contributions to economic theory,
primarily, though not exclusively, in the area of general equilibrium theory. One of
his most striking innovations is his proof of the existence of equilibrium prices based
on maximizing a weighted sum of individual utilities. Each such maximization will
produce a vector of prices and an allocation of society’s resources in which the value
of consumption will be different from income for the typical consumer. A fixed point
argument is then applied in the space of social welfare weights to find the appropriate
set of weights for which the value of consumption is equal to income for all
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consumers. Fixed point algorithms, or their equivalents, are used to find equilibrium
prices in applied general equilibrium analysis. Computational demands depend on the
dimension of the space in which the fixed point argument is applied - typically the
number of commodities in the model. The number of consumers is usually much
smaller than the number of commodities. Applied general equilibrium practitioners
have found Negishi’s approach extremely useful in reducing computational time and
allowing much larger problems to be solved.
Stokey, for her part, when asked by this author about the connection made by
Arrow in his reply to the present author as cited above, that is, between her joint work
with Lucas (1984), and Negishi's 1960 "method", replied (2006, personal
communication): "Ken has a good memory. Bob Lucas and I used Negishi's method
in our 1984 JET paper on optimal growth with Koopmans-Diamond-Williamson
preferences, which are recursive but not additively separable over time. In our
setting, the key idea was to define a mapping on the Pareto weights, which with those
preferences evolve over time"; and this, despite the fact that Negishi's 1960 paper is
not cited in Stokey and Lucas (1984).
Indeed, as Spear has recently observed in a communication to this author (2006):
…my exposure to this body of work has been most focused on the work on existence,
the work on tatonnement stability conditions, and the non-tatonnement, Hahn-Negishi
processes….I was probably first exposed to the existence paper in graduate school,
and came to particularly appreciate it as a way of showing existence of equilibrium in
the neo-classical capital model with finitely-many agents… I think some results can
become so well-known and so well discussed in surveys that people stop citing
them. This would certainly explain the absence of a citation by Lucas and Stokey.
This observation has been confirmed by Stokey, who, with Lucas (1984) "took
Negishi's method to be part of the corpus of modern economic theory" (2006,
personal communication).
Analytical Extension of Negishi (1960) and the "Negishi-Mantel algorithm"
But more is involved here than simple theoretical and computational application
of Negishi (1960), and its absorption into the "corpus" of modern theory. In his
seminal paper, Yano (1998) extended the analytics of Negishi's 1960 approach to
18
gauge the efficacy of "temporary fiscal policy". By extending Negishi's approach he
found that, among other things "the inefficacy of temporary fiscal policy… may be
thought of as one of the most fundamental properties of dynamic general equilibrium"
(1998, 440). The crucial implication of this result has yet to be digested by those who
deal with macroeconomic policy and analysis.
On another level, there has been increasing utilization of what has been called
the "Negishi-Mantel algorithm" (Backus et al. 1992b, 9). For example, Cunat and
Maffezzoli (2004), apply the "Negishi-Mantel algorithm" to Heckscher-Ohlin
business cycles, while Mendoza and Oviedo (2005), apply it to the analysis of fiscal
policy and macroeconomic uncertainty in emerging markets. Tim Kehoe has recently
provided this author with his important account of the development of what Scarf
originally termed the "Negishi-Mantel algorithm". According to Kehoe (2006,
personal communication):
I started in the Ph.D. program at Yale in the fall of 1975. I took Herb Scarf’s
general equilibrium course (mathematical economics as it was called then) that
fall. In the spring, Herb was a visitor at Harvard …Rolf Mantel, who was a visitor at
Yale, taught the second semester of Herb’s course. That is when I first heard about
what Rolf called the “Negishi approach” to proving the existence of equilibrium via
the second welfare theorem and a fixed point problem in welfare weights. Rolf
stressed that Negishi’s approach relied on an underlying fixed point problem. He
explained that, in his Ph.D. thesis, he (that is, Rolf) had tried to circumvent using
fixed point methods in developing a computational algorithm for general equilibrium
models. Of course, by the time the principal paper taken from Rolf’s thesis, Mantel
(1971), was published, he had realized that the stability properties of a tâtonnement
process on welfare weights depended on the properties of the underlying utility
functions and endowments of the economy. In fact, an argument due to Uzawa
(1962)[1962b] said that any proof of existence of equilibrium in an economy whose
excess demand function was arbitrary except for the assumptions of continuity,
homogeneity of degree zero, and Walras’s law could be trivially manipulated into a
proof of Brouwer’s fixed point theorem. Rolf was intrigued by the question of
whether excess demand functions were arbitrary except for these basic assumptions or
whether they satisfied some stronger properties that would make computation of
equilibria easier. He came up with the answer in Mantel (1974): Aggregate excess
demand functions are, in fact, arbitrary except for these assumptions unless we place
strong restrictions on utility functions, endowments, or numbers of consumers…
The next year, Herb Scarf was back at Yale, and he asked me to be the grader for
his course….Many of my friends and classmates, now in their second year, including
19
Dave Backus, took this course. In the second semester, Herb spent a couple of classes
going over the results of Negishi, Uzawa, and Mantel. If anyone was first to refer to a
Negishi-Mantel approach or algorithm, it was Herb. (Or it was Dave and me talking
about what we had learned in his course.) Herb stressed that any general algorithm
for computing equilibria that would be guaranteed to converge had to be a fixed point
algorithm. The dimensionality of the problem, however, was determined by the
minimum of the numbers of goods and consumers. For models with lots of
consumers and few goods, we should solve a problem in price space. For models
with lots of goods and few consumers, we should use the Negishi-Mantel approach
and solve a problem in the space of welfare weights. Herb gave the class a homework
problem with two infinitely lived consumers and asked the students to solve it using
the Negishi-Mantel approach. Versions of this problem still survive in the problems
that I give first year graduate students at Minnesota in my macro course …and in
exam questions. It is also the basis of the approach that David Levine and I took in
our papers on determinacy of equilibria…
To sum up, a number of major research programs can be identified, therefore, as
emanating from Negishi's now classic papers, that of 1960 and 1961 (1961b)
respectively. Negishi's 1960 paper forms the basis for both "theoretical" an "applied"
research programs in general equilibrium analysis, and his 1961 paper (1961b) has
been almost as influential in demarcating ongoing research up to the present in the
field of imperfect competition and non-tatonnement processes. These papers, as has
been shown above, attest to Negishi's considerable influence on the development of
modern economic theory and analysis.
20
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21
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