When Arrow and Debreu published their famous proof of the existence of competitive equilibrium in 1954 their work was met with extraordinary praise. In fact, approbation was so intense that there was hardly any substantive criticism of the paper. Not a good omen. But then again, who needs to cast doubts when you want to have faith?
The existence question is not only a technical question (i.e., finding out if a system of equations has a solution). In the grand narrative of market theory, the issue of existence of equilibrium is relevant because it concerns the reference point towards which disequilibrium prices (and allocations) are supposed to converge. The idea of market forces leading an economy to a point of equilibrium would be meaningless without certitude about the existence of the promised land.[i] In macroeconomic theory, this is so important that in some extreme cases (i.e., dynamic general equilibrium models), it is assumed that the economy is always in an equilibrium position.[ii]
The existence of a general competitive equilibrium haunted neoclassical economics since Walras. He knew that equality in the number of equations and unknowns was not enough to guarantee the existence of a solution.[iii] The problem was assumed away and the question of existence remained without an adequate answer for decades. [iv]
The Arrow-Debreu model changed the name of the game. As Koopmans (1957) noted, “in this model the problem is no longer conceived as that of proving that a certain set of equations has a solution.” The idea was now to show there was a specific state of the economic system where at some prices of goods the manifold individual price-taking maximization behaviors are mutually consistent.
The Arrow-Debreu (A-D) model was acclaimed as the first rigorous solution to this problem.[v] The essential idea of A-D is to build a model of an economy and to utilize tools developed in game theory by Nash (1950). The center of attention here is a fixed-point theorem, considered today as the most important tool in mathematical economics (Geanakoplos 1989).
Arrow and Debreu’s intuition was that because every equilibrium is a rest point, the fixed-point theorem lent itself beautifully for the proof of existence. The trick was to be able to interpret the fixed point as an (general) economic equilibrium.
In its simplest version, the procedure using this tool can be schematized as follows.[vi] Imagine you have a set P of price vectors each with the property Σpi = 1 (i.e., for each price vector in P the sum of its components is equal to 1). Now suppose you build a mapping f that transforms price vectors into new price vectors. This mapping transforms each price vector p in the set P to a point p’, that is, another price vector in P. The fixed point theorem tells us that under certain conditions, the mapping has a fixed point, i.e., there is a p* in P such that f(p*) = p*. The validity of the theorem depends on the properties of both the mapping and the set on which it is defined.[vii]
Suppose you build this mapping so that it represents the law of supply and demand, i.e., if demand is greater than supply, the mapping ensures that the price of that commodity increases. If it is less, the price will decrease. If the excess demand equals zero (i.e., supply equals demand) the price will remain unchanged. Now, under certain mathematical conditions, imagine you found that your mapping transformed one price vector into precisely the same vector. This is a fixed point at which the prices would remain unchanged. Suppose you also demonstrated that at this fixed point for every commodity with a strictly positive price, supply would be equal to demand and where a commodity’s price was zero the excess demand would be negative (i.e., positive excess supply). Your proof of existence would be complete.
But this is where one needs to look at the proof in more detail, especially at the structure of the mapping. In both Brouwer’s and Kakutani’s fixed point theorems, the mapping must be defined on a convex, compact set (i.e., a set that is closed and bounded). The unit price simplex P has this property. It is composed of all price vectors such that the addition of its components equals unity, Σpi = 1. This is strictly a mathematical necessity in order to be able to use the fixed-point theorem.
There are various mappings (transformational rules) that change price vectors into new price vectors and represent the law of supply and demand.[i] But we need to guarantee that the new price vectors are elements of P, the simplex, for one simple reason: the fixed-point theorem requires that the mapping be defined on a compact set. Thus, the mapping must also involve a normalization procedure in order to ensure that all of the new price vectors belong to the price simplex (i.e., they must have the desired property: Σpi = 1). And here is where things go awry: this normalization destroys the interpretation of the mapping as a representation of the law of supply and demand.
In our paper “The law of supply and demand in the proof of existence of general competitive equilibrium” (by Carlo Benetti, Alejandro Nadal and Carlos Salas) we demonstrate that none of the mappings used in the various proofs of existence of a general equilibrium involving a fixed point theorem is a good representation of the law of supply and demand.[ii] In some cases, commodities where demand is greater (less) than supply may have their price reduced (increased). The reason is straightforward: the normalization process needed to ensure that the new price vectors belong to the price simplex destroys the possibility of interpreting the mapping as consistent with the law of supply and demand. With the normalization procedure, price changes for one commodity not only depend on the sign of the excess demand of that specific commodity, but also on the sign of the excess demands of the other n – 1 commodities. This is not what the law of supply and demand indicates.[iii]
Our conclusion: the proof of existence of a general competitive equilibrium using a fixed-point theorem fails to possess an economic meaning.
In 1994 I had a brief meeting with professor Kenneth Arrow. I showed him the abstract of our paper. As he read it, he smiled and said “You and your colleagues are mistaken, you are confusing stability with existence and this is of course the source of your error”.
We had anticipated this type of reaction, so I replied: “No, we are clear on that. This has nothing to do with stability. We know the existence proof does not replicate a dynamic process. But you and professor Debreu claim that the mapping used in the proof of existence represents the law of supply and demand, and what we are saying is that the normalization procedure you have to use to ensure that the new price vector is in the unit simplex contradicts your claim”.
“Oh, it’s about the simplex”, exclaimed Professor Arrow, “Then you’re right”.
That simple comment confirmed our results: the mappings do not represent the law of supply and demand. As a consequence, the proof of existence of equilibrium using a fixed-point theorem is devoid of economic sense.[iv]
Arrow, K. and F. Hahn (1971) General Competitive Analysis. San Francisco: Holden Day.
Geanakoplos, John (1989) “Arrow-Debreu Model of General Equilibrium”, General Equilibrium. The New Palgrave. (Eatwell,J., M. Milgate and P. Newman, eds.). New York: Norton.
Koopmans, T. (1957), Three Essays on the State of Economic Theory.
Nikaido, H. (1989) “Fixed-Point Theorems”, General Equilibrium. The New Palgrave. (Eatwell,J., M. Milgate and P. Newman, eds.). New York: Norton.
Smale, Steve (1982) “Global Analysis and Economics”, Handbook of Mathematical Economics, (Arrow, K. and M. Intrilligator, eds.). North-Holland.
Walras, L. (1952) Éléments d’Économie Politique Pure. Paris: Librairie Générale de Droit et de Jurisprudence.
Wickens, M. (2008) Macroeconomic Theory: A Dynamic General Equilibrium Approach. Princeton: Princeton University Press.
[i] See the exposition of Arrow and Hahn (1971: 25-7).
[ii] We analyzed four different mappings: Arrow-Debreu, Nikaido, Debreu, Arrow-Hahn. In Debreu’s version, commodities with a positive excess demand may see their price reduced to zero just because there are other commodities with greater positive excess demands!
[iii] In the mapping used by Debreu, the price of commodities with positive excess may be reduced all the way to zero just because there are other excess demands that are greater!
[iv] We leave out of this analysis the existence results of global analysis à la Smale (1982) because they rely on stronger assumptions. It is true that this line of analysis “is closer to the older traditions” as Smale claims but this is precisely what Arrow and Debreu wanted to leave behind by relying on a fixed point theorem.
[i] In Nikaido’s terms (Nikaido 1989: 139) existence of an equilibrium “is a primary premise of the theory, on which all its developments are built (…). Without this consistency, the theory is void.”
[ii] See Wickens (2008).
[iii] In his Éléments d’économie politique pure he showed that multiple equilibria were possible. He also considered the case where supply and demand curves lacked an intersection point. See Walras (1952: 66-68).
[vi] For a detailed description of the entire procedure, see chapter Nikaido (1968), Chapter 5.
[vii] Brouwer’s fixed point theorem for single valued functions requires that P be a non-empty, convex, closed and bounded set, and that f be continuous. Kakutani’s theorem requires that f be an upper-semicontinuous correspondence for which the image set f(p) is a non-empty, convex subset of P.