# Is Walras’s Theory so Different from Marshall’s?

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## Abstract

This paper shows that Marshall’s theory is generally equivalent to Walras’s one.

It shows that Walras used two types of demand functions: (1) the original (ordinary) demand curve (function); and (2) the derived (general) demand function.

Marshall also used both types of demand curves (function); however he did so in a very simplified and vague manner.

Therefore, Friedman’s statement that they are alternative theories is mistaken.

This paper strives to demonstrate that Marshallian and Walrasian demand (supply) curves are indeed very similar, if not equivalent for several theoretical purposes.

In my view a discussion on similarities and differences in the demand functions of Walras and Marshall makes sense in the context of examining which is the most productive analytical tool. In his original article on the Marshallian demand curve Friedman (1949), endorses this view and argues that his interpretation is more useful for the analysis of practical problems. Sixty years after Friedman’s paper I would add that a discussion on the demand (supply) functions in Walras and Marshall should address the following critical question: how does a revised analysis help solve the problems related to stability analysis in a general equilibrium context?

The first point of similarity considered in the paper pertains to the fact that for both Marshall and Walras demand functions have a negative slope. This is clear from their work and I have no problem with that first result. What is less clear to me is why the paper does not mention immediately that this idea has been now proven erroneous by the work of Sonnenschein (1973), Mantel (1974) and Debreu (1974). Market demand functions have a negative slope only under very special conditions (i.e., for example when preferences are homothetic or if Engel curves are parallel). These results have been around for almost four decades and have earned their place in any analysis in the history of economic thought. A discussion without this reference can be somewhat misleading, giving the impression that we should still be working out the details of a theory or a research program that is essentially valid.

On page 2 of his article, professor Davar continues his claim that the demand functions of both authors have more similarities than differences. One of his arguments is that they both share a common method for the “establishment” and “re-establishment” of general equilibrium. This method, according to the paper, is that “they assumed that the given basic data does not change during the process of equilibrium establishment and considered the problem of equilibrium re-establishment as a result of changes in the given basic data (…)”. For Walras, professor Davar refers us to the famous paragraph at the end of Lesson 20.

This is a strange proposition that needs to be clarified. The terminology leaves much to be desired. The word “re-establishment” of equilibrium suggests a situation in which an original equilibrium has been disturbed and a process to re-establish that initial position has been set in motion. This of course is the typical scenario for the analysis of local stability of equilibrium. But is this what professor Davar means when he uses the word “re-establishment”? I guess the answer is in the negative.

Professor Davar states that conditions 2.5 and 2.6 in his paper are identical to the ones used by Walras to determine equilibrium prices. He also argues that his conditions 2.8 and 2.9 are different because they include the quantities of commodities and the utility functions of commodities as arguments of the demand (supply) functions. Now, the functions 2.8 and 2.9 are determined by the mathematical model described in 2.1 – 2.4. Because in linear programming, unknowns depend upon all the parameters in the model, professor Davar asks if Mr. Walras ignored the fact that his demand functions were incomplete? He thinks Walras knew this. And to prove it, he argues that Walras divided the process of “equilibrium establishment” into two stages: “The first stage of the process is the establishment of equilibrium prices (external parameters) for the given available quantities and utility function (internal parameters). The second stage of the process is the analysis of the variation of prices (equilibrium re-establishment) when initial quantities and utility functions are changed. (…) Walras used his famous tâtonnement for equilibrium establishment (vide infra).”

So what is then the distinction between the first and second stage?

Can it be that the first stage concerns a pure exchange economy (i.e., quantities do not change). The second stage would then appear to be the economy with production, i.e. an economy in which quantities are allowed to change. This is suggested by the fact that professor Davar makes a reference (on page 2) to the paragraph in which Walras introduces the need for his system of tickets (“bons”) as a way to solve the problem of production at disequilibrium prices (Walras 1954:242).

This interpretation poses a problem because the tâtonnement is needed and used in both stages of the process, and not only in the first (pure exchange) economy. In fact, the tâtonnement process is used in all economies, including production and capitalization. Thus, the introduction of tâtonnement is not what differentiates the first from the second stage.

As noted above, the text states that in the second stage quantities change. Walras notes that in the economy with production there is a complication not present in pure exchange: production at disequilibrium prices needs to be avoided. This is done through the introduction of ‘tickets’ (“bons”). But, in fact, this system of tickets is also required in the pure exchange economy because effective trading at ‘false’ (disequilibrium) prices is as problematic in pure exchange as it is in a production economy. This is the direction of the critique by Bertrand: normally the equilibrium prices arrived at when disequilibrium trading takes place will not be the solution of the original equations in the pure exchange economy. There will be as many equilibrium positions as trading paths in the process of price formation. This path-dependency of trading processes is confirmed by all non-tâtonnement models (see Negishi 1962). This strengthens the interpretation that tâtonnement does not distinguish the first (pure exchange where quantities do not vary) from the second stage (production where quantities are supposed to vary). We must discard this interpretation, but then the author needs to clarify just exactly what are these two stages in Walras’ theory.

Another problem with this text is that it gives the impression that equilibrium is established (or re-established) in the model used by Walras without problems. This is of course quite inexact. Walras’ account of the dynamic process for the formation of equilibrium prices (for example, in the case of the pure exchange economy, paragraphs 125-129) did not prove that the competitive forces in the market lead to general equilibrium. Today this is non controversial. Again, I’m sure professor Davar agrees with me on this point. But his text suggests repeatedly the establishment of equilibrium is not a problem.

It is important to recognize the fact that in the field of price formation processes (or stability) general equilibrium theory has failed to deliver satisfactory results. The Arrow-Block-Hurwicz line of work showed that convergence was guaranteed through the assumption of gross substitutability or with the weak axiom of revealed preferences at the market level. Scarf’s counterexample destroyed hope for the famous conjecture about the validity of these limited results. The Sonnenschein-Mantel-Debreu theorems have shown that excess demand functions do not have any structure, thus proving it is not possible to ascertain convergence to equilibrium prices in general equilibrium theory. These results have been around for many decades. At the very least, they deserve a footnote.

One final remark on Walras: the prices that appear as arguments of the demand (supply) functions examined by professor Davar’s paper make up a system of prices. This means there is only one price for each commodity regardless of the path followed by bilateral exchanges. This is a feature of the model when prices are announced in terms of a numéraire. But this is a very strong assumption. The existence of a price system is something that cannot be assumed ex ante, it is a characteristic of equilibrium allocations and should not be assumed for disequilibrium situations. Walras’ claim that arbitrage operations lead to price systems is false.

This commentary is already too long, but one last word on Marshall’s functions is perhaps warranted. The paper mentions the way in which Marshall constructs his notion of representative firm. But Marshall’s construct was the object of a very serious critique by Sraffa: a critique that basically showed these functions to be inexistent. This is something the paper should mention.

To summarize, the point is not whether the demand functions in Marshall and Walras are similar or not, but whether they can play a constructive role in analytical terms. Professor Davar’s paper is filled with references that seem to indicate the formation of equilibrium prices in Walrasian theory is non-problematic. The very serious problems afflicting price formation theory in the context of general equilibrium models deserve to be mentioned in any analysis of demand functions and price formation processes.

References

• Debreu, G. (1974). “Excess-demand functions”. Journal of Mathematical Economics 1: 15–21.

• Friedman, Milton (1949). “The Marshallian Demand Curve”. The Journal of Political Economy. Vol. LVII (6).

• Mantel, R. (1974). “On the characterization of aggregate excess-demand”. Journal of Economic Theory 7: 348–353.

• Negishi, Takashi (1962) “The Stability of a Competitive Economy: A Survey Article”, Econometrica Vol. 30, No. 4, Oct., , pp. 635-669

• Sonnenschein, H. (1973). “Do Walras’ identity and continuity characterize the class of community excess-demand functions?”. Journal of Economic Theory 6: 345–354.

• Walras, L. (1969). Elements of Pure Economics. New York: Augustus Kelley.

Replay to Prof. Alejandro Nadal

I would like to thank Prof. Alejandro Nadal for reading and commenting on my paper, especially since he is a well-known expert on general equilibrium theory. Unfortunately, however, his lengthy comment is useless, and shows his misreading, misunderstanding and misinterpretation, not only of my paper, but also of Walras’s and Marshall’s Theories. The crucial reason for that is the fact that Prof. Alejandro Nadal, as well as the majority of economists, have been identifying their theories with the Neo Classical (Walras, Marshall) theory, but this is erroneous.

First, Prof. Alejandro Nadal states that “This paper strives to demonstrate that Marshallian and Walrasian demand (supply) curves are indeed very similar,” while in my paper Walras’s and Marshall’s whole theory is discussed, where demand (supply) curves are one of components of the whole theory. This is an essential difference, because in the following Prof. Alejandro Nadal did not remind at all (Davar, 1994 and 2012).

Second, Prof. Alejandro Nadal does not relate in any way to the fact that, in my paper, two types of demand (supply) curve are considered: original and derived, and to the connection between them, which is the key tool in the process of equilibrium establishment and re-establishment.

Third, the reviewer absolutely ignores the fact that in my paper it is clearly demonstrated that Marshall’s approach includes not only partial equilibrium, as it is generally argued in the modern economic literature, but also general equilibrium, and this is equivalent to Walras’s approach. Yet, it seems as if Walras discussed only quantitative adjustment and Marshall discussed only prices adjustment; actually, both of them discussed both adjustments simultaneously.

Fourth, the reviewer also ignores the special attributes of the mathematical model for individual economy, formulated and presented for the first time. This model allows us to directly obtain the demand and the supply of each commodity (service), which might be less or equal to the available (initial) quantity, and to formulate the excess demand (supply) between them; opposed to the modern approach, where excess demand (supply) is determined between the available (initial) and final quantities; which prevents the discussion of the crucial economic problem of employment (unemployment) in the modern general equilibrium theory.

Fifth, I agree with Prof. Alejandro Nadal that equilibrium establishment is a very serious problem, and Walras, unfortunately, did not prove its existence strongly enough, though his approach guarantees the approximate establishment of equilibrium. But, for the main task of the paper, the identity of Walras’s and Marshall’s theory, equilibrium establishment is secondary and does not influence the result.

Sixth and finally, the reviewer totally misses one of the central attributes of Walras’s method revealed in my paper, namely: the two-stage process of equilibrium establishment-reestablishment. He writes: “This is a strange proposition that needs to be clarified. The terminology leaves much to be desired. The word “re-establishment” of equilibrium suggests a situation in which an original equilibrium has been disturbed and a process to re-establish that initial position has been set in motion. This of course is the typical scenario for the analysis of local stability of equilibrium. But is this what professor Davar means when he uses the word “re-establishment”? I guess the answer is in the negative.” Of course it is negative, because local stability of equilibrium must be solved in the first stage. Walras assumed the changing of the initial data so that new equilibrium establishment is required; there might be situations where that it is not required; which would depend on assumptions of the model in question. The reviewer continues: “Can it be that the first stage concerns a pure exchange economy (i.e., quantities do not change). The second stage would then appear to be the economy with production, i.e. an economy in which quantities are allowed to change.” What about Capital and Credit Economy? Or, Circulation and Money Economy?; What stages are they?

Despite the above reflections, I am grateful to Prof. Alejandro Nadal for his comments and to the editors for giving me chance to reply.

References

Davar, E. (1994), The Renewal of Classical General Equilibrium Theory and Complete Input-Output System Models, Avebury, Aldershot, Brookfield USA, Hong Kong, Singapore, Sydney

Davar, E. (2012) Is ‘Walras’ Law’ Really Walras’s Original Law?, World Review of Political Economy, V.3, N.4, 478-500