Léon Walras’s work holds an important role in general equilibrium economic theory. Written at the end of the 19th century, Elements of Pure Economics (1874) is part of a legacy that many economists (Hicks, Samuelson, Allais…) have commented on throughout the 20th century. More recently, the Walras Center at the University of Lyon 2 and the Walras Pareto Center at the University of Lausanne have collected, published, and translated numerous articles, lectures, and correspondence by Léon Walras. Among all the contributors on Walras’s work, Donald Walker takes an important place.

Donald Anthony Walker (born 1934, March 6th) died in January 2023 at Memorial Regional Hospital in Hollywood. His education, research and academic activities were extremely prolific in the field of history of economic thought and microeconomics. Donald Walker got his AB(Artium Baccalaureus) of Social Science(1952) andhisMaster of Arts (1956) from Texas State University, his PhD ofPhilosophy at Harvard University. He started his career as assistant professor in economics at Miami University (1961-1967), before joining Indiana University Pennsylvania as economics professor (1969-1988) and Chairman Department (1988-1998). In 1998, he received the Distinguished University Professor Awardand served as Professor Emeritus of Economics in 1999¹.

Donald Walker published a lot of books and over 100 peer-reviewed articles. His field of research is microeconomics, the history of thought and, more specifically, Walrasian economic theory ([163]). Indeed, Donald Walker is often associated with work relating to Walrasian thought ([76]). His desire to make Walras’s work accessible to an English-speaking audience led him to continue Jaffé’s work (publication of Léon Walras’s correspondence, 1965), to analyze the process of tâtonnement² and equilibrium in markets and to push back the frontiers of general equilibrium theory ([61]). However, such a panorama would lead us to forget that he tackled the foundations and origins of microeconomics at a very early stage. His early work (1970-1980) focused on trade theory, the static analysis of market equilibria and disequilibria, and the process of tâtonnement through the works of Marshall, Edgeworth, Veblen and Clarence Ayres. Donald Walker is also interested in issues of applied microeconomics, such as inflation, monopsony, short- and long-term labor supply, and the economic recession caused by the first oil shock (1973-1975) in the United States. It was not until the early 1980s, in particular the publication of the Jaffé Essays de Walras, that the focus really shifted to Walras’ work ([158]; [149]). But Donald Walker’s activities as a researcher are not limited to writing his own productions. Walker commented on and reviewed numerous books and articles ([74]). Some of these comments ([47]) underline his interest in economic issues and, more broadly, in the works of his contemporaries, as in his analysis of [8] book Remaking American Values, or [15] book Institutional Economics,the Changing System. One of his reviews played a key role in his work on Walrasian general equilibrium theory ([65]). His analysis of [23] book Walras’economics,A Pure Theory of Capital and Money, relayed by the work of [20], led him to analyze the question of tâtonnement in Walrasian theory ([10]).

Finally, the portrait of Donald Walker would not be complete without mentioning his institutional role in the Society for the History of Economic Thought in the United States and in the creation of theInternational Walras Association(AIW). Donald Walker was the president of the History of Economics Society (1988-1989). During his presidency, the History of Economics Societydecided that instead of publishing the conference programs and paper abstracts in the HES Bulletin as it used to do, a selection of papers presented at the annual meetings would come out in edited volumes. Thus, from 1989 to 2000, selected conference papers were published in Perspectives in the History of Economic Thought, a hardcover series published first by Edward Elgar (1989-1995) and then by Rout-ledge³ (1996-2000). From 1990 to 1998, Donald Walker was the Editor of the Journal of the History of Economic Thought (JHET). Donald Walker was also a key figure in the creation of the Association Internationale Walras (AIW)⁴. The AIW was founded on October 3, 1997 at the University of Lyon 2, but it was at the Association’s second meeting (Paris, September 16, 1999) that the statutes and rules were adopted. The creation of the AIW was published in the JournalOfficiel on February 26, 2000. Donald Walker was President of the Association from October 1997 to September 2000 ([137]). During his presidency, the AIW website was set up by [2], conference papers were published in the Cahiers of CERAS, and the AIW aroused the interest of many young researchers. The first AIW colloquium (1999) was a resounding success, with 15 high-quality papers and a keynote address by economist Gérard Debreu.

In this article, we propose to enlighten readers on Donald Walker’s contributions to microeconomics, economic thought and Walrasian economic theory ([93]). To do so, we begin by presenting Donald Walker’s contributions to microeconomics, the history of economic thought and applied economics. We will then look at his contributions to Walrasian theory. We will distinguish three interdependent phases: 1) an entry through Walras’ correspondence and in particular the archives held by William Jaffé (1970-1986), 2) a reflection on the process of tâtonnement and the question of equilibria-disequilibria in Walrasian theory (1987-1996), 3) a reflection on the new insights concerning the theory of general equilibrium and a desire to make Walras’ works accessible to Anglo-Saxon researchers (1997-2018).

Microeconomics, History of Economic Thought and Applied Economics are interdependent in Walker’s research ([40]). His first paper, untitled Marshall’s Theory of Competitive Exchange and published in the Canadian Journal of Economics(1969) is a good example of the scientific and analytical approach adopted by Walker ([84]). This paper was focused on Marshall’s assumption of the constancy of the marginal utility of money u’(m). This question has been studied by many economists, [26], [165], [29], [16]. The dilemma was double: 1) does u’(m) refer to the marginal utility of a flow of money income or a flow of the numéraire? 2) does u’(m) is a constant with respect to money, income, numeraire or prices. For Walker, the many reviewers of Marshall’s work were concentrated upon its possible meaning and implications in the context of the theory of behaviour of an individual ([1]). They never examined its character and function in Marshall’s theory of a competitive exchange market ([57]). For example, [4], [25] recognized Marshall’s use of the assumption in reference to a stock of money, but their treatment was confined to the consequences for the trader’s indifference map or Marshall’s theory of bilateral exchange. According to Walker, “there did not exist any adequate explanation of the properties of the model ofcompetitive exchange in which Marshall assumed that u’(m)was constant” ([33]). His article has been written to fill this gap in exchange literature. In particular, Walker intends to show that Marshal is pursuing two objectives: 1) the first one is to develop a theory of value in which disequilibrium transactions are part of the equilibrium process of a competitive market; 2) the second one is to predict the equilibrium price with the use of the initial market demand (D) and Supply (S) functions. According to Walker, the reconciliation of these two objectives requires that the last price at which trade occurs during a market day must be equal to the Initial Solution Price (ISP). The hypothesis that Marshall used to achieve that result was that u’(m) was some constant to each trader. This assumption enables him to assert that the equilibrium price would be equal to the ISP in a competitive disequilibrium transactions market ([42]).

According to Walker, Marshall’s disequilibrium transactions model does not really find the equilibrium price to be the outcome of the action of economic forces: “The solution price is not a variable of the model. It isa constant,and therefore it is unaffected by the sequence of trading prices and the redistribution of the assets. The value of the constant is determined independently of the market functions in any stage of the system” ([33]).

This article is quite important for identifying the areas of research that Donald Walker intends to pursue. Recognition of the existence of disequilibrium transactions and the analysis (static and dynamic) of disequilibrium transactions markets will provide the common thread running through all his work ([112]). Two years after this publication, [35] published in the India Economic Journal, a paper entitled The Static Analysis of Disequilibrium Transactions Markets. The fundamental concepts of this study were first presented in papers delivered at Northwestern University and the University of Michigan in 1964. This research has been financed by grants from the Miami University (Ohio) Faculty Research Committee (Donald Walker was Assistant Professor from 1961 to 1967). In this paper, Walker aims to answer a question that is the subject of much debate in the economics community (this question would occupy him for the rest of his life). He intends to develop some of the foundations of a static method of analysis of the equilibrium values of the variables in competitive disequilibrium transactions markets (see also, [36]). Walker remind us that most of the literature on the theory of exchange in competitive “assumes that disequilibrium transactions do not occur,and that the excess demand functions of the participants do not change,with the results that the vector of prices at which all excess demands are equal to zero remains unchanged” ([35]). When we have a look at the neoclassical process ([24]), dynamics are the changes in the set of quoted prices at which no final transactions are allowed to occur until the set of initial solutions prices is found. This adjustment of a system market has been described as a Tâtonnement process, and a system in which disequilibrium transactions are part of the equilibrating process has been called a non-tâtonnement system([18]). Walker will use the terms of neoclassical adjustment process and disequilibrium transactions adjustment process to study the determinateness of equilibrium in isolated competitive markets ([36]) and to describe the competitive tâtonnement exchange markets ([37]).

This analysis of the process of tâtonnement, and more broadly of price formation, led Walker to examine the Walrasian arguments as well as those of Edgeworth and Marshall, and to a lesser extent those of Veblen and Ayres ([102]). The Edgeworth theory of recontract, more precisely Edgeworth’s contribution to the theory of exchange, plays a key role⁵ ([12]). [38] considers that the current interpretation of competitive recontract is erroneous and he proposes to provide a better understanding of Edgeworth’s thought. According to Walker, the current interpretation of Edgeworth’s theory of competitive recontract has its origin in an article of Kaldor, A Classification Note on the Determinateness of Equilibrium, published in 1934 by the Review economic Studies. In Kaldor’s view, Edgeworth’s and Walras models should be considered as attempts to formulate the conditions under which buyer and seller are able to find out the true equilibrium price before they undertake any exchange activity. Kaldor concluded “While Edgeworth’s analysis may be slightly obscure and Walras’s assumption slightly ridiculous,the main idea stands clear: the formation of prices must precede the process of exchange and not be the result of it” ([21]). To provide better understanding of the Edgeworth Theory, Walker introduces a transitional phase of Edgeworth’s thought, around ten years after 1881 (publication of Mathematics Psychics). In an article entitled The Mathematical Theory of Political Economy (1889), Edgeworth reviewed the new edition of Elémentsd’EconomiePolitique Pure of [163] and has reservations about the analysis of the Tâtonnement process. Edgeworth explained that Walras described a hypothetical equilibrating process. The equations of exchange define a position of equilibrium, but they afford no information as to the path by which that point is reached: “Prof Walras’slabouredlessons indicate a way,not the way,of descent to equilibrium” ([13]; [38]). Two years later, Edgeworth proposes his theory of recontact to analysis of a market in which there are flows of buyers and sellers: “Nouspouvonseneffetconstruired’autrestypes quipourrontprétendreaussià bon droitreprésenterl’étatd’unmarché.Considérez par exemple,le cas qui est le plus fréquemment indiqué dans l’hypothèse où le nombre des A qui sont vendeurs de la marchandise a est plus petit que le nombre des B qui sont vendeurs de la marchandise b. En ce cas,chaque A virtuellement est en marché avec trois ouquatre B et peut obtenir des prix plus avantageux pour lui à la façon d’unmono-poleur.Voilà le point de départ. Le second degré,ce sera si un des A essaie de dérober la clientèle des autres A,en abaissant son prix au degré voulu pour les attirer à lui et en bénéficier. Les A qui se trouvent ainsi dépouillés de leur clientèle,rétorquent en abaissant encore plus leurs prix. Et ainsi de suite,jusqu’à ce qu’on soit descendu à un point tel qu’aucun des vendeurs A ne trouverait plus de profit à attirer à lui les clients des autres A,parce que l’accroissement des frais résultant de l’accroissement de production contrebalancerait juste le profit qu’on pourrait tirer d’un nouvel acheteur” ([14]).

According to Donald Walker, the theory that Edgeworth developed in 1891 and more precisely in 1902, differs from the one currently described, and this for two major reasons: 1) Edgeworth analyzes a type of normal price, not market price. 2) His model of recontract is concerned with markets that are characterized by irrevocable disequilibrium prices and transactions, and that may be in a state of disequilibrium for an extended period of time. In his model, the formation of equilibrium prices does not precede the actual exchange of commodities. Exchange at disequilibrium prices is an essential part of the equilibrating process: “But it is supposed thatagreements are renewed or varied many times. A final settlement is not reached until the market has hit upon a set of agreements which cannot be varied with advantagetoall the re-contracting parties. The re-contract mostfavourableto the disturbance of a temporary equilibrium is one in which an A deals with a great many B’s” ([14]).

But the study of tâtonnement’s process and more broader the history of economic thought are not the only contributions of Walker, his comments about applied economics and economic facts were also interesting contributions. We may introduce here his study of microeconomics foundations of inflation ([39]), his review of The 1973-1975 Recession in the United States of America ([43]) or his analysis of Giffen’s Paradox ([156]). This interest in economics in general should not leave the reader indifferent to the scope of Donald Walker’s work, even if Walrasian theory was to occupy a large part of his life.

In 1999, Donald Walker published a french book entitled LaThéoriedel’équilibregénéral,de nouveauxéclairages ([109]).This book is based on the Hennipman Lecture given by Donald Walker to the Royal Netherlands Academy of Sciences on 12 June 1995. Walker presents a study that goes beyond the simple Walrasian framework and seeks to shed new light on the question of general equilibrium. New directions for future research are presented: “My study of these topics draws on several fundamental concepts that will be developed throughout this book. These are the concepts of models that are complete enough to be functional systems,of an understanding of the meaning of perfect competition in different contexts and of the aspects of markets that create it,of the differences between virtual models and models in which disequilibrium occur at the transaction and production levels,of the difference between models and equations,and of the implications for modeling the specific processes that occur in the real economy” ([133]). According to Walker, the primary focus of the book is on general equilibrium models, but most of the main developments can be used to explain other types of economic models, such as consumer, firm and specific market models.

The reformulation and perspectives of the general equilibrium model take the following forms:

1) The construction of functional models: Walker argues that one of the fundamental aspects of economic modelling is the construction of functional models. Such models are hypothetical economic systems, containing a multitude of essential characteristics, with institutions, technologies, laws and regulations, companies, households, other economic sectors of various horizons, financial institutions, and different ways of doing business. In short, all the entities that make up markets and their functional relationships. According to Walker, “the general equilibrium models found in economic literature are not functional systems. Some of their less detailed postulates have no basis in a model,while others arebased on incomplete models” ([133]).

2) Building realistic models: The other fundamental aspect of economic modeling is the attractiveness of building models linked to reality. Models in the literature are not. They use unrealistic or absurd assumptions, and therefore do not simulate the real economy. The conclusions drawn from them therefore have no relation to the real economy: “As the precision and density of descriptive detail of the structural and behavioral characteristics of markets,of different ways of doing business and of the general functioning of the economy increase,the more the new general equilibrium models will correspond to the real economy,and the stronger their power to explain and predict the economy will be,once they are converted into empirical models” ([133]).

3) The new general equilibrium model: Walker considers that we need to introduce into a new model, a number of elements which in the past have not all been presented in general equilibrium models. Preferences, production functions and quantities of economic resources are important, but represent only a fraction of the complex of phenomena that determine the model’s behavior: “From the beginning It will incorporate functional sub-models of particular markets with realistic aspects,such as instruments,procedures,regulations,laws,physical structures,technologies,information and communication dissemination systems,and other important features. These sub-models must be correlated as real markets are,to form a functional and realistic system of multiple markets” ([133]).

The new model must involve the government, not only because laws affect markets, but also because it participates in the production and sale of goods and services. The model can also be studied in terms of the paths taken by its variables, the results of its disequilibrium change processes, the existence of equilibrium, and the values of the variables in equilibrium. According to Walker, elements can be taken into account, such as net saving and investment, usually positive but sometimes negative. Finally, business fluctuations in the model are one aspect of its disequilibrium change processes. The growth path of the model is another aspect of its behavior. The model may reach the full-employment growth path or experience interruptions in its convergence process and fluctuate without ever reaching its path.

Moving somewhat away from Walrasian general equilibrium theory, Walker already anticipates what a general equilibrium model might be. This model, based on correlations between markets, “will be useful because of its effectiveness in explaining,predicting and finding solutions to problems in the real economy” ([133]).

All these suggestions and proposals highlight a certain open-mindedness but also a broad understanding of the problems of economic modeling. Like Leon Walras, Walker is aware that models can be abstract and simplified representations, however they cannot turn away from the real economy.

Nevertheless, it is Walrasian theory that interests Donald Walker most, and he will devote much of his work to making the Lausanne economist’s research accessible.

To account for Donald Walker’s many contributions, we will distinguish three interdependent phases.: 1) an entry through Walras’ correspondence and in particular the archives held by William Jaffé (1970-1987), 2) a reflection on the process of tâtonnement and the question of equilibria-disequilibria in Walrasian theory (1987-1996), 3) a reflection on the new insights concerning the theory of general equilibrium and a desire to make Walras’ works ([159]; [153]) accessible to Anglo-Saxon researchers (1997-2018).

1) “Léon Walras in the Light of His Correspondence and Related Papers”, published in the Journal of Political Economy(1970), is the first paper written by Donald Walker ([34]). This article provides an insight into the origins of Donald Walker’s interest in Walrasian theory, and in particular his connection with William Jaffe⁶, the “leading authority on the work of Léon Walras” ([56]). As Walker points out in his preface to the William Jaffé Essays on Walras, his links with William Jaffe go back to the early 1970s: “in 1970 I wrote to Jaffé about his view that Walras has assumed disequilibrium transactions occur in his example of a real exchange market but was unaware of the effects of such transactions. In opposition to that view,I argued that Walras knew that disequilibrium transactions would make it impossible in most cases to predict equilibrium prices with the use of the excess demand functions based on initial endowment of assets…” ([60]). In July 1979, Walker wrote to Jaffee suggesting that they should bring together all the resources they had in order to publish a collection of his Essays on Walras ([160]; [28]).

“My reasoning was based on several considerations. There is,and will continue to be,adeep interest in Walras’ work;Jaffé was the foremost authority⁷on that work;and there consequently has been,and will continue to be,a deep interest in Jaffé Essays. The essays are individually of great value,displaying soundness of scholarship,perceptiveness of interpretation,wisdom in evaluation,and scrupulousness in bibliographical accuracy: and,taken as a collection,they form a well-rounded body ofWalrasiana. Some of them were published in sources that are not easily accessible,three of them… were written in French,and one of them consistslargely of quotations in French” ([60]).

Although Jaffé responded favorably to Walker’s request and personally asked him to edit the correspondence, it was not until 1983 that William Jaffé’s Essays on Walraswere published. Jaffé’s death in 1980 (August, 17th) led Walker to interrupt his work and publish a few articles on Jaffé’s work ([54]).

2) The study of Walras’s Tâtonnement process is the main research of Donald Walker. The articles of [18], Walras’s theory of Tâtonnement: a critique of recent interpretationsand Another look at Léon Walras’s theory ofTâtonnement, were great materials for more investigation. In 1987, Walker published Walras’s theory of tâtonnement in the Journal of Political Economy ([62]). This paper introduced what Walker called the conflicting interpretations of Léon Walras on Tâtonnement. To evaluate these interpretations, Walker showed that Walras developed two separate theories of tâtonnement. The one is a disequilibrium production model. The other is a pledges model⁸ in which transactions and production occur only at the equilibrium set of prices. Walker concluded that “the disequilibrium production model is more representative of Walras’s work on economic process than the pledges model” ([83]).

Subsequently, [87] inquiries led him to tackle the question of Léon Walras’ theory of general equilibrium (1988a) and, more generally, the mathematical issue of tâtonnement(1988b). Once again, Walker used some arguments developed by [20], he proposed an examination of Walras’s investigations of mathematical iteration⁹, its connection with the behavior of real markets, the properties of Walras’s economic theory in regard to iterative pricing, including the means by which prices are changed in his theory and the rapidity of their adjustment. According to Walker, these issues have to be examined in relation to the disequilibrium production theory that Walras has presented in the first editions of the Eléments d’économie politique pure: “In that theory,he recognized that there are disequilibrium rates of production and use of consumer commodities and capital goods,and that the related trading of productive services,consumer commodities,and capital goods occurs at disequilibrium prices” ([88]). Walker considered that there are two reasons for dealing with Walras’s work on iterative processes. Firstly, the disequilibrium production system and its tâtonnement process have a dominant place in Walras’s writings. Secondly, when he applied mathematical iteration to the output, pricing and exchange of productive services, consumer commodities and capital goods, he did so in all cases in reference to a disequilibrium production model. Walker concluded that Walras tried to develop a technique of mathematical iteration to solve the equations of a general equilibrium model and to construct a theory of economic tâtonnement: “Walras constructed the foundations of a theory of economic tatonnement that describes its character and outcome in general terms. He did this by identifying the economic agents that change prices,by summarizing their iterative behavior in the theory ofWalrasianpricing,by describing the ways that prices are adjusted in different markets,by identifying the role of the entrepreneur in the tatonnement process,and by asserting that a system of freely competitive markets converges to equilibrium through iterative pricing and the related iterative adjustments of the allocation of resources”([88]).

Of course, laying the foundations for an economic theory of tâtonnement comes up against a number of dilemmas, not least that of aggregation in Walrasian general equilibrium theory ([157]), institutions and structure of the model ([98]), disequilibrium and equilibrium interactions ([132]), equilibrating process ([100]) or monetary model ([104]).

- [157] analyzed and evaluated the procedure of aggregating economic quantities and relations that Léon Walras introduced into economic theory. They explained (i) the procedure that Walras adopted, (ii) presented a brief summary of the conditions required for consistent aggregation and their implications for the form of demand and supply functions, and finally showed “that Walras’s procedure can create inconsistency between the microeconomic and macroeconomic relations that he constructed,because he did not make the assumptions requisite for consistency” ([157]: p. 489).

- [98] argued that the current view of Walras’s work on the theory of exchange ([20])—considering that he did not pay attention to market institutions—is erroneous. Walras dealt with several kinds of markets and he expended much effort on specifying their institutional features and the characteristics and behavioral patterns of the participants ([22]). Walker took a special look at Walras’s model of markets in which the participants make oral pledges to buy and sell commodities. By an oral pledges market, Walker meant the freely competitive market that Walras developed. Walker examined the institutions, participants, kinds of commodities and sorts of trades that are made in the model. According to Walker, there are three reasons for studying Walras’s model of oral pledges markets: “First,they are important in the modern economy,so it is interesting to learn what Walras contributed to their understanding. Second,the oral pledges model is an important part of his theoretical work. Third,the markets with which he dealt in his oral pledges model have been described as auctions,thereby greatly affecting perceptions of how organized competitive markets behave,and it is therefore interesting to determine whether the model is accurately described in this way” ([98]: p. 652). Walker used the organization of the market operations on the Bourse described by [164] to conclude that an oral pledges market is not a place where atomistic unregulated competitive forces act in the absence of institutional constraints. Walras explored the significance of many conventions and rules ([27]) that were adopted by the traders (for example, the securities exchange). Walras provided many details about the structure of his oral pledges model such as the types of participants and their functions ([7]; [5]), the institutional provisions for changing and publicizing prices, the types of contract, the settlement procedures, the kinds of assets that are traded.

Five other essays ([99])—concerned the examination of its equilibrating conditions—give an overall view of Léon Walras’s work on the topic of trading with the use of pledges. [132] argued that Walras provided an extensive account of how those market features interact in disequilibrium to result in the determination of prices and quantities traded. To build a complete model of exchange, Walras needed to specify whether or not disequilibrium transactions are allowed, the disequilibrium conditions in response to which the price is changed and that determine its direction of change, how the sign of the market excess demand quantity is known, how it is signaled that trade is allowed to occur… Walker concluded that “it must be recognized that Walras’s model was a significant achievement. It was a brilliant expression of his recognition that traders’behavior is conditioned by and takes place within a framework of institutions and rules. Walras related his statements of theoretical principles to the functioning of real markets and provided in many respects a realistic theoretical treatment of the behavior of the traders in disequilibrium” ([132]: p. 976). [100] also examined the structural aspects of Walras’s model that influence its equilibrating process. He was concerned with presenting and interpreting Walras’s treatments of the parameters of the model, of written pledges, of the numeraire, and of the participants, institutions, and trading conventions in the markets for consumer commodities, capital goods, and productive services ([130]). Walker argued “that the written pledges markets are purely competitive organized exchanges;that the structure of the model ensures that the supply functions for goods and services do not change in disequilibrium;that the use of pledges eliminates the endogenous processes in disequilibrium that would invalidate Walras’s equations of general equilibrium,namely transactions and all the disequilibrium processes;that they make possible;that written pledges are made only by suppliers;that they transmit all-or-none written pledges to agents;that the agents are the direct participants in the markets;that producers and service suppliers are supposed to exchange goods and services;and that trade in the model is a barter process” ([133]).

In his essay entitled Theequilibrating process in Walras’s barter model of written pledges markets(1990f), Walker went back over the behavior of the market participants in relation to the use of written pledges, analyzing the pricing of commodities, and evaluating the structure and functioning of the model. Walker argued that (1) the model was incomplete because the participants did not express their demands for consumer commodities, capital goods, or productive services; (2) there were serious equilibrating problems in the model because of the difficulties of distributing the pledges to consumers and producers and of arranging fulfillment of the pledges: “The logical structure of the [Walras’s] model is deficient in four major ways. First,Walras’s decision not to introduce pledges to buy deprives his model of a way of enabling demands to be expressed,and that in turn deprives it of the action of the forces of supply and demand that would be necessary to achieve the equilibrium set of prices. Walras’s equations and references to demand and excess demand functions are not descriptions of the behavior of the participants in the written pledges model,do not provide for the behavior to which they should refer,and are inconsistent with its properties. Second,themodel lacks systems of information collection and dissemination. Third,even if the reader assumes that the equilibrium prices are found,the model could not reach equilibrium because it does not solve the problems of the distribution and fulfillment of pledges nor therefore of the allocation of goods and services. The agents cannot distribute the pledges so as to make the model mimic a real market system. Without pledges to buy and money it is not possible to devise a market clearing mechanism that overcomes the problems presented by the model. Fourth,Walras represented the written pledges model as a general equilibrium model of a hypothetical economy[Walras,1900,§ 274,p. 302],but in actuality it is a partial equilibrium model,albeit one of wide scope,because he did not include markets for primary materials”([101]).

The last contribution of [104] is concerned by the markets for circulating capital and money in Walras’s last monetary model. Walker evaluated Léon Walras’s modeling of written pledges markets in his last monetary model, concentrating upon their characteristics in regard to the pricing and determination of the rates of output of goods and services. Walker argued that the markets do not have a demand side, that the model has no means to enable the distribution and fulfillment of pledges, and that it is lacking markets for a number of important commodities and money. He concluded that Walras’s monetary model of written pledges markets does not have a pricing process and is not a functioning system (See also, [6]): “This essay has examined the written pledges markets in Walras’s monetary model with the objective of discovering their structural and behavioral features. The conclusion is that Walras did not endow them with enough of the features that would be necessary to make them viable. The model is fatally flawed by the lack of any expressions ofdemands,and it lacks written pledges markets for money and fora number ofimportant commodities. Without those markets the consumer and capital goods industries in the model cannot function” ([104]). [105] completed this analysis by assessing Léon Walras’s modeling of markets for circulating physical capital and money and his last monetary model as a whole. Walker argues that Léon Walras’ reasoning concerning the markets mentioned in his model is based on two contradictory postulates. The first of these postulates indicates that written pledges are used in the markets for certain goods in order to eliminate disequilibrium in transactions and production, while the second indicates precisely the presence of such phenomena in these same markets. Furthermore, by assuming the existence not only of a market for non-durable goods, but also of a market for services for these same goods, Walras is using a redundant and illogical hypothesis. According to Walker, these observations confirm the idea that the resulting model cannot function as a system: “The weakening of Walras’s intellectual powers in the revision of theElémentsthat he undertook in 1900 was manifested in many ways. His barter model of written pledges is deficient in majorrespectsand his last monetary model is seriously logically defective. Lessons 29 and 30 convey a strong impression of disorganization and analytical confusion. They are full of poor definitions,illogical constructions,badly arranged statements,conflations of separate topics,and incomplete treatments” ([105]).

3) In 1996, Donald Walker published Walras’s Market Models ([122]). The book differs from previous articles, Walker supports the view that we should dissociate two phases in the works of Walras (see also, [27], [11]). During his mature phase of theoretical activity (phase 1), Walras was concerned with a competitive economy that passes through a stage of disequilibrium in the production and sales of commodities (see also, [32]). In his last phase of theoretical activity (phase 2), he tried to construct a model in which hiring, production, sales, consumption, and saving would be suspended until an equilibrium of prices was found. According to Walker, the defective structure of that model and its fragmentary nature prevented it from becoming a functioning system. Until now, the models of the two phases of Walras’s theoretical work have not been subjected to an accurate analysis and evaluation. Walras’s Market Models was written to meet this challenge. In the first part of the book, Walker presented the general characteristics of economic behavior that Léon Walras identified (Walras’ conception of a competitive market economy, mature model of the barter of stocks of commodities, institutions and participants in the model of monetary oral pledges markets, disequilibrium and equilibrium in the model of monetary oral pledges markets, structure of the mature non durable consumer commodities model, equilibrating processes, critics on the maximum utility of new capital goods, model of the money markets, iteration in the model of tâtonnement, the mature model of the behavior of the entrepreneur, the differences between Walras and Edgeworth on tâtonnement processes). He reminded us that his objective in the mature phase was to understand and model the behavior of a freely competitive capitalist market economy. He defined that as one in which there is private ownership of most economic resources and other commodities, and in which commodities of all types and money are exchanged in freely competitive markets. In his Eléments and other writings he elaborated on the properties of such an economy, setting forth his message about them in a number of propositions, models and analyses. Walker insisted on three important general features of a Walrasian market economy: “First,he conceived of a market economy as a system in which price competition is strong and widespread but not universal. He recognized that it is absent or very limited in somemarkets,anddeveloped a theory of monopoly to take account of them. He believed,however,that “free competition is the principal mode of exchange in the real economy,practiced on all markets with more or less precision and therefore with less or more efficiency” ([122]: p. 19). In the second part of the book, Walker introduced the idea of a marked decline in the quality of Walras’s work during his final phase of theoretical activity. During that phase he tried to revise the Eléments into a written pledges system of markets. Walker attributed the inconsistencies in Léon Walras’s writing to his state of health (see L. Walras’s correspondence, [17]).

This thesis was taken up and developed in his book WalrasianEconomics (2006) but the challenges are not the same ([145]). We are in what we called the third phase of the Walker works. Donald Walker engaged in a reflection on the new insights concerning the theory of general equilibrium (see also, [9]) and a desire to make Walras’ works accessible to Anglo-Saxon researchers ([160]): “this book isaboutWalrasianEconomics,meaning the economics of Léon Walras and the economics in which other scholars have used his idea” ([145]: p. 1). According to Walker, this book had two general objectives: (1) present an accurate account of the foundations and basic components of Walras’s models of general economic equilibration and equilibrium; (2) trace the influence of that legacy on his contemporaries and successors. To treat the subject, Walker made an effort to reach an audience wider than general equilibrium specialists. He explained some theoretical propositions that seemed elementary knowledge to general equilibrium specialists but not to some students or economists who concentrated on other fields of research. The book has been divided into 2 parts. The first part presented Walras’s ideas, including the foundations of Walras’s mature comprehensive model, the written pledges sketch and some of Walras’s writings. The second part dealt with Walras’s influence ([9]), such as the models constructed by Walras’ contemporaries and immediate successors (Pareto, School of Lausanne, Wicksell, Schumpeter, Schultz, Moore) and the models drawing upon the heritage of the written pledges sketch, from 1930 to 1971 (Cassel, Schlesinger, Neisser, Wald, Von Neumann, Hicks). According to Walker, “Walras’legacy in special respects and in general has therefore been to contribute many important strands of the mainstream of modern economic research and knowledge. Indeed,his legacy is inseparable from that mainstream” ([122]: p. 315).

This desire to disseminate the Walrasian heritage as widely as possible was naturally reflected in the statutes of the International Walras Association (IWA), of which Donald Walker was the first president (1997-2000). The IWA’s objectives were clearly stated: “1)To promote research on the works of Léon Walras and the intellectual and historical issues related to them; 2)To facilitate communication between people working in the field ofWalrasianresearch; 3) To help disseminateWalrasianresearch”. Donald Walker has never ceased to contribute to the many conferences, publications “Les Cahiers du CERAS” ([134]; [155]) and books of the Walras International Association (Baranzini, Diemer, & Mouchot, 2004, [3], [10]).

During his career, Donald Walker made interesting contributions to Microeconomics, Economic Thought, Walrasian Economic Theory and History of Science ([152]). Drawing on Jaffé’s numerous essays, he helped popularize Walrasian theory in the Anglo-Saxon scientific community through 1) his study of Walras’ correspondence and discussions on the Elements (1970-1986), 2) a reflection on the trial and error process and the question of equilibria-disequilibria in Walrasian theory (1987-1996), 3) a reflection on new perspectives on general equilibrium theory and a desire to make Walras’s work accessible to Anglo-Saxon researchers (1997-2018).

While [136] left behind many areas of Walrasian theory that are still debated in the scientific community (the principle of competition, the analysis of the tâsonnement process, the question of money), he also brought international recognition to the work carried out over many years by researchers in Lausanne (Pascal Bridel, Roberto Baranzi, etc.) and Lyon (Pierre Docks, Jean-Pierre Potier, etc.). As the first president of the International Walras Association, he succeeded in placing the work of economic historians at the heart of modern economic theory and in giving young economists specializing in general equilibrium theory a taste for reading Walras.

¹In 2023, Indiana University of Pennsylvania (IUP) created the Donald A. Walker Department of Economics Scholarship Fund to offer scholarship awards to IUP students majoring in economics, in economics pre-law, in mathematics/economics, or in social science education with a concentration in economics.

²Walrasian tâtonnement is based on the metaphor of an auction. The price is set by an auctioneer, who determines the price based on the supply of goods and the price that consumers are willing to pay. The price is not determined in a single step, but through a process of trial and error: the auctioneer announces a price, then gathers information on supply and demand intentions. If these intentions do not coincide, he or she announces a new price. The auctioneer announces prices based on previous prices until a price is found that matches supply and demand.

³In 1998, the HES President David Colander proposed to the Executive Committee that the book series to be terminated and that selected conference papers be then published in one dedicated issue of the Journal of the History of Economic Thought(the journal had recently moved from publishing two issues per year to four).

⁴As a member and treasurer of AIW during his presidency, I appreciated his ability and willingness to bring people together around a shared project.

⁵It should be remembered that Edgeworth’s theory of exchange had its origins in debates in the early 1870s, which highlighted the inability of the theory of supply and demand to account for the functioning of markets with a small number of exchangers. These debates led Edgeworth to propose a representation of economic behaviour that could replace the notions of supply and demand at given prices and apply to situations of both imperfect and perfect competition.

⁶In his article in honour of Jaffé (who died on 17 August 1980), [54] recalls Jaffé’s desire to make Walras’ work known to the international scientific community: “Jaffé returned to the U.S in 1928 to take a position at Northwestern university as a teacher of Marshall and the continental economists. At thattimehe became interested in the work of Walras, and began to translate theElémentsd’économiepolitique pure… For many years Jaffé concentrated obsessively in that effort, expending enormous amounts of time on trying to achieve the most faithful possible rendition of Walras’s meaning, and on preparing a detailed annotation of the volume” ([54]). Jaffé struck up a friendship with Henri Schultz, economist at the University of Chicago. Schultz was familiar with Walras theories and he published a number of articles about Walras ([30]). It was Schultz who gave enthusiasm to Jaffé for reading TheEléments in 1929 ([56]).

⁷In 1954, his translated edition of Walras’s book Elements of Pure Economics was printed. In 1965, Jaffé published the correspondence of Léon Walras.

⁸Donald Walker uses the term of (written) pledges model to refer to the monetary submodel of markets of that type that Walras constructed.

⁹In his investigations of mathematical iteration, Walras described how the theorist, who is sitting in his study pondering a model, must hold certain prices constant by assumption and change its prices in certain directions in order to find the solutions of its system of equations of general equilibrium. The theorist must transform the equations relating economic variables repeatedly to diminish the difference between the demand and supply quantities of each commodity.