Market Reality vs. Economic Theory
The Impossibility of Neoclassical Aggregate Demand
The Law of Demand has been a cornerstone of economics since the times of Adam Smith. Yet, modern economic theory systematically failed to derive this Law within its highly sophisticated, axiomatic models of the economy. These models eventually contributed to the precipitation of the Global Financial Crisis.
An Intellectual Shock
The story starts with the groundbreaking work of Nobel laureates Kenneth Arrow and Gerald Debreu in the 1950s. Arrow and Debreu successfully proved that there exists a set of non-negative prices that can simultaneously clear all markets in the economy. The proof, which relies on advanced mathematical techniques, was a major achievement in the profession. But then, economists noticed something strange: The behavior of the aggregate economy in this model was arbitrary and did not follow the established economic wisdom. Despite economists’ best efforts, it was impossible for the market demand in a general equilibrium model to abide by the Law of Demand.
The Law of Demand states that if the price of a commodity rises, holding other things equal, then the demand for that commodity should fall. The Law is a bedrock against which Adam Smith and subsequent classical economists founded the then-emerging economic science. So, when 20th-century economics concluded that the Law does not hold at the aggregate market level, only at the individual level, it was a shock to the majority of economists.
The triumph of modern theory in proving the existence of general equilibrium was overshadowed by the failure to formalize the most cherished law of economics since the field was established 250 years ago.
The Big Four
The roots of this failure go back to 1951. That year, Kenneth Arrow published his famous impossibility theorem of social choice. As explained in a previous article, the theorem establishes that there is no formal procedure to translate individual rationality to collective or aggregate rationality. Arrow made it clear that the impossibility result includes not only voting schemes but the market mechanism as well.
In 1974, Gerard Debreu and Rolf Mantel independently proved a conjecture made a year earlier by Hugo Sonnenschein that aggregate excess demand function needs only to satisfy three conditions: (1) continuity, (2) homogeneity of degree zero with respect to prices, and (3) Walras’ law. Subsequent works by the three authors and others confirmed these results (Rizvi, 2006).
Since consumers are rational, their individual behavior follows a negatively sloped demand curve. However, the theorems of Sonnenschein, Mantel, and Debreu (commonly referred to as SMD) conclude that the market demand function (which is the sum of the individual demand functions) can be arbitrary as long as it satisfies the three above conditions.
In the words of Shafer and Sonnenschein (1982, p. 672), generally, “market demand functions need not satisfy in any way the classical restrictions which characterize consumer demand functions.” Under the title “Anything Goes,” Mas-Colell, Whinston, and Green (1995, p. 598) write: “Anything satisfying the few properties that we have already shown must hold, can actually occur.” Even with boundary conditions, notes Alan Kirman (1989, p. 132), “we have no restrictions which will help to obtain global uniqueness or stability.” According to Donald Saari (1995, p. 224), no mechanism can always promise convergence to a price equilibrium.
“Market demand functions need not satisfy in any way the classical restrictions that characterize consumer demand functions” — W. Shafer and H. Sonnenschein
The implications of these findings are profound. It is not possible to treat the aggregate economy as an idealized consumer. The rationality of the individual does not carry over to the market as a whole. There is “a fundamental separation between collective rationality and individual rationality” (Hong, 2008, p. 896). This is a very awkward result whereby each agent is fully rational, yet their aggregate behavior can be completely irrational.
Markets Succeeded Where Theory Failed
The SMD theorems are in direct conflict with real-world experience. Since Adam Smith, economics was established to explain economic phenomena, the most important of which is the Law of Demand.
Not only do these theorems contradict the experience accumulated over two centuries, but they also contradict well-designed and controlled lab experiments. Nobel laureate Vernon Smith and his coworkers have conducted a large number of experiments over more than three decades, and they found how markets, especially markets of goods and services rather than speculative asset markets, systematically satisfy the Law of Demand. V. Smith contrasts the neoclassical theory with experiments, stating: “Markets succeeded where theory failed” (Inoua & Smith, 2022, pp. 1, 10).
An Emptiness Theorem
The pioneers of the field, mainly Kenneth Arrow and Gerard Debreu, were long skeptical of the ability of general equilibrium theory to provide insights into aggregate market behavior.
For Arrow, as pointed out earlier, it was obvious to him since 1951 that there is no formal mechanism to translate individual rationality to collective rationality, the market mechanism included. He made this even clearer decades later when he wrote: “In the aggregate, the hypothesis of rational behavior has in general no implications” (Arrow, 1986, p. 204).
Gerard Debreu held similar views. According to his pupil, Weiner Hildenbrand, “Debreu was always convinced that the microeconomic foundations do not imply sufficient structure for total excess demand to allow a satisfactory treatment of these problems” (Ingrao and Israel, 1990, p. 272).
“The near emptiness of general equilibrium theory is a theorem of the theory” — C. Bliss
Oxford economist Christopher Bliss made a profound remark about the pattern emerging from these and similar results: “The near emptiness of general equilibrium theory is a theorem of the theory” (Rizvi, 2006, p. 232).
The Emperor Has No Clothes
Many brilliant economists were deeply disappointed by the results of SMD. Werner Hildenbrand, a pupil of Gerard Debreu and a leading theoretical economist, wrote in the preface of his 1994 book Market Demand (p. ix):
“When I read in the seventies the publications of Sonnenschein, Mantel, and Debreu on the structure of the excess demand function of an exchange economy, I was deeply consternated. Up to that time I had the naive illusion that the microeconomic foundation of the general equilibrium model, which I admired so much, does not only allow us to prove that the model and the concept of equilibrium are logically consistent (existence of equilibria), but also allows us to show that the equilibrium is well determined. This illusion, or should I say rather, this hope, was destroyed, once and for all, at least for the traditional model of exchange economies.”
The majority of practicing economists, however, learned to stop worrying and love general equilibrium models. While the “emperor” was visibly naked (Kirman, 1989), many pretended he was not. They simply assumed that the aggregate demand has the same restrictions as the individual agents and thus followed the “representative agent” modeling approach. The SMD theorems were therefore largely dismissed as a theoretical curiosity with little practical significance.
Despite the profound consequences of the SMD theorems, there is no clear explanation of why these results are obtained. They seem to emanate from a “black box” of the formal axiomatic theory of general equilibrium that defeats reasonable justification.
The failure of the theory of aggregate demand is not an accident; it is a logical result of the way the models are designed
The failure of the theory of aggregate demand is not an accident; it is a logical result of the way the models are designed. As a contribution to the large literature criticizing general equilibrium theory (e.g., Ackerman and Nadal, 2004), we argue that Gödel’s incompleteness theorems can offer precious insights into this puzzling phenomenon.
From Hilbert to Debreu
The prominent mathematician of the early 20th century, David Hilbert, was the father of the axiomatization movement, not only in mathematics but in all fields of mature knowledge, including economics (Weintraub, 2002, p. 88).
Despite the devastating effect of Gödel’s incompleteness theorems on the ambitious goals of axiomatization, the movement continued, albeit probably at a slower pace. The first major axiomatized work in economics was done by John von Neumann, the star pupil of Hilbert. His Games and Economic Behavior, in collaboration with economist Oskar Morgenstern, was published first in 1945. Kenneth Arrow followed with his axiomatized impossibility theorem published in 1951. In 1954, Gerard Debreu, a leading mathematician of the Bourbaki tradition, joined forces with Arrow to prove the existence of general equilibrium theory.
Axiomatization reached a landmark when, in 1959, Debreu published his Theory of Value, a completely axiomatized system of economic theory. In his introduction, Debreu argued that the theory’s formal structure must be constructed axiomatically and with no reference to the interpretative values of the concepts. He wrote (p. 3):
“Allegiance of rigor dictates the axiomatic form of the analysis where the theory, in its strict sense, is logically entirely disconnected from its interpretation.”
Axiomatization, according to Debreu, has the merits of clarity and rigor on one hand and the freedom of interpreting formal symbols on the other (Ingrao and Israel, 1990, p. 250).
The Price of Axiomatization
There are certainly benefits of axiomatization. But there are costs as well. The most critical price is the incompleteness of formal systems. Kurt Gödel proved in 1931 that a formal axiomatic system, rich in arithmetic (with addition and multiplication), is prone to two crucial limitations (Smullyan, 1990):
- The system cannot prove all valid sentences formulated in the language of the system. Truth extends beyond provability. This is, roughly, the First Incompleteness Theorem.
- The system, if consistent, cannot assert its own consistency. If it does, it ceases to be consistent. Consistency must be “invisible” within a consistent system. This is, roughly, the Second Incompleteness Theorem.
The formal systems for which Gödel’s theorems apply can be in any field of knowledge. Such systems, by design, disconnect the formal structure from its interpretation, as Debreu pointed out. Accordingly, Gödel’s theorems apply to any such formal axiomatic system, rich in arithmetic, regardless of the domain in which the system is applied.
For various reasons, many mathematicians (and, by implication, many mathematical economists) considered the incompleteness phenomenon as an intellectual curiosity hovering at the edge of mathematics. It is the same weird position that economists took with respect to the SMD theorems.
Gödel’s Insight
The Second Theorem shows that, for global properties like consistency, there is a critical difference between the view from the inside and the view from the outside of a sufficiently complex system.
In his Gibbs Lecture of 1951, Gödel (1995, p. 309) elaborates on this point as follows:
“For it makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion: All of these axioms and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics. If someone makes such a statement he contradicts himself. For if he perceives the axioms under consideration to be correct, he also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable from his axioms.” (Bold added.)
We argue that this critical insight has been largely ignored in mainstream economic modeling. The price of this ignorance was enormous.
Invisible Equilibrium
The Arrow-Debreu axiomatic model of general equilibrium is built on the assumption of equilibrium of markets. The model assumes that agents trade at prevailing prices; they are price-takers. Agents trade at prevailing prices only if they believe these prices are unique, stable equilibrium prices; otherwise, they would have negotiated them.
Since an individual agent need not trade all commodities, the agent may not assume that all prices are equilibrium prices; only those prices of the commodities of his interest are assumed to be so.
So far, so good.
But what happens when we aggregate across all agents? Although for any individual agent, no assumption is made about all prevailing prices, aggregation necessarily implies that all prevailing prices are equilibrium prices. This implies that the global market system asserts its own equilibrium.
Equilibrium, like consistency, must be invisible within the system, just as Adam Smith proclaimed two centuries ago
Equilibrium is a special form of consistency: it implies not only logical consistency but also the consistency of supply and demand across all markets. Since the economy is modeled as a formal axiomatic system liable to incompleteness, by asserting global equilibrium within, the system becomes prone to the inconsistency implied by the Second Theorem. This type of inconsistency is not trivial; otherwise, it would not have taken someone with the caliber of Gödel to discover it.
Equilibrium, like consistency, must be invisible within the system, just as Adam Smith proclaimed two centuries ago about the “invisible hand.”
Rationality
Another way to think about the above result is through the flip side of equilibrium: rationality. In economics, rationality means consistency; a rational agent has complete and consistent preferences (Jehle & Reny, 2011, pp. 5–6).
If we model the individual agent as a formal system, then rationality implies the consistency of such a system. It is not necessary, however, to assume that an agent asserts his own consistency, but it is quite reasonable that each agent assumes that other agents are consistent. These assumptions seem harmless at the individual level. The problem arises, once again, with aggregation.
The market system aggregates the agents by summing their individual systems. If each agent assumes the others to be consistent, it follows that, within the aggregate system, all agents are deemed to be consistent. If the market is simply the sum of the agents, this amounts to the market system asserting its own consistency. This might render the aggregate system inconsistent, as per the Second Theorem.
Rational expectations modeling is in direct opposition to the Second Theorem
But there is more. Rationality is commonly modeled based on the Rational Expectations Hypothesis. According to Thomas Sargent (2008, p. 878), a rational expectations equilibrium asserts that the same model is shared by (1) all of the agents within the model, (2) the econometrician estimating the model, and (3) nature, or the data generating mechanism.
This clearly shows how economic models fail to differentiate the view from the outside and the view from the inside as implied by the Second Theorem. The result is that the model asserts its own consistency, exposing it to logical inconsistency, as the Theorem dictates.
Inconsistency and Instability
General equilibrium and rational expectations are at the core of state-of-the-art Dynamic Stochastic General Equilibrium (DSGE) models, the workhorse models used by major central banks and private institutions (e.g., Athreya, 2013). As Nobel laureate Joseph Stiglitz (2018) notes, these models failed to predict the Global Financial Crisis (GFC) or to provide policy guidance on how to deal with the consequences.
In light of the above analysis, one important reason for the failure of these models is their self-assertion of equilibrium and rationality. This makes these models structurally immune to economic cycles, which itself contributed to instability. The very use of crash-free models, as discussed in a previous article, was a major factor in triggering the crash of the GFC.
Smart Modeling
A proper economic theory is supposed to explain how prices are formed in a competitive market. Instead, the Arrow-Debreu theory bypasses the process of price discovery and assumes equilibrium upfront.
Vernon Smith offers a visionary roadmap of how economic theory should progress. The theory should explain how the market mechanism can successfully coordinate the actions of millions of agents, each with only local information and limited computing power, so that equilibrium emerges globally. Equilibrium is an emergent phenomenon that economics should seek to explain rather than assume upfront. Inoua and Smith (2022, p. 14) remark:
“The classical economists were certainly aware that almost all markets yielded stable prices, but that was the mystery to be understood, not taken as given to define quantities demanded or supplied.”
This approach is supported by vast experimental evidence that shows how ordinary people, with limited information and computing machinery, can quickly reach equilibrium in trading normal goods (in contrast to speculative assets).
It is also supported by the experiments conducted by the ABC Group, led by Gerd Gigerenzer. In their Simple Heuristics that Make Us Smart (1999), they show how ordinary people follow simple and intuitive rules (heuristics) that lead to outcomes that are optimal with respect to the complex environments of the tasks at hand. Bounded rationality serves to achieve “ecological rationality” (Gigerenzer, 2008).
Conclusion
To sum up, the general equilibrium theory is built from the ground up on the assumption of equilibrium. Since equilibrium is a strong form of consistency, the theory, as a formal axiomatic system rich in arithmetic, becomes prone to Gödel’s Second Theorem. This may provide valuable insights into the puzzling results of the divergence between individual and collective rationality implied by the SMD theorems.
Gödel’s theorems not only pinpoint the potential flaws of mathematical modeling, they also point to the way forward. Equilibrium is an emergent property that cannot be asserted within the model. A good theory shows how local behavioral rules can unexpectedly lead to the wonders of a stable and efficient market.
This article is part of a series on Economic Impossibility Theorems.
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