Adam Smith Meets Kurt Gödel: The Incompleteness of the Invisible Hand
The deep insights of Adam Smith and Kurt Gödel are highly pertinent to modern economic modeling. The invisibility of efficiency within the market, as implied by the “invisible hand,” and the invisibility of the consistency of a formal system within the system, as implied by the Incompleteness Theorems, have striking similarities. For a system to remain consistent, its consistency must be invisible. Likewise, for the market to remain efficient, its efficiency must be invisible. Contradicting these insights may contribute to financial instability, as has been witnessed in the Global Financial Crisis.¹
Modern economics started with Adam Smith (1723–1790). He was one of the most prominent philosophers of the Enlightenment Age. His remark on the “invisible hand” is probably the most remembered of his works. The term appeared only once in each of his two major works: Theory of Moral Sentiments and The Wealth of Nations. But the basic idea of how self-regarding can coordinate productive exchange is clearly explained in both works.
A famous quote from The Wealth of Nations is the following:
“It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard to their own interest. We address ourselves, not to their humanity but to their self-love, and never talk to them of our own necessities but of their advantages.”
In another paragraph, Smith writes:
“Every individual is continually exerting himself to find out the most advantageous employment for whatever capital he can command. … He generally, indeed, neither intends to promote the public interest, nor knows how much he is promoting it. … he intends only his own gain, and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention.”
See Muller (2002, pp. 62, 65).
As we shall see, Smith had deep insights into how an enlightened self-regard would ultimately make everyone better off.
Enlightened Self-interest
Smith thought the ability to trade was unique to humans, making the human species uniquely capable of progress. Like most Enlightenment intellectuals, Smith believed commerce promoted civilized and virtuous behavior.
The self-regard that Smith was concerned with was enlightened self-interest that, combined with empathy, allows market participants to achieve mutual benefit. He clearly disdained greed, selfishness, and shortsightedness (Muller, 2002, pp. 62, 72; Skousem, 2016, p. 32).
Because of the centrality of exchange in Smith’s analysis, he considered the average person to be “in some measure a merchant” (cited in Muller, 2002, pp. 61, 72).
Smith saw trade not as a zero-sum game but as a transaction that promotes coordination to achieve mutual gain. His writings brilliantly weave moral virtues into economic reasoning. “The road to virtue and that to fortune … are, happily in most cases, very nearly the same,” Smith wrote in Theory of Moral Sentiments. His Wealth of Nations was intended to make men better, not just better off (Muller, 2002, pp. 52, 69–73; Skousem, 2016, pp. 19, 33).
“The road to virtue and that to fortune are, happily in most cases, very nearly the same.” — Adam Smith
Upon his death in 1790, reports Jerry Muller (2002, p. 54), Smith’s estate was found to be minimal, as he used to privately donate most of his fairly sizable income during his life.
Smith’s Vision
Smith had a deep insight into why the market hand is “invisible”: each participant has the incentive and the ability to gather information related only to the local market conditions, but they don’t have neither the ability nor the incentive to collect information for the entire economy (Muller, 2002, p. 67).
Accordingly, each participant focuses on reaching a mutually-beneficial trade with his direct partners. When all participants follow the same strategy, it is not difficult to conclude that resources will tend to be efficiently allocated even though each participant is concerned primarily with his local or direct counterparts.
The strategy coordinates the actions of millions of participants to produce system-wide efficient outcomes. Local mutual gains pave the way for global harmony and efficiency.
Local mutual gains pave the way for global efficiency.
In modern terms, the key to the invisible hand is that information is scarce and dispersed. The market, therefore, is a decentralized system to integrate scattered information to achieve a globally efficient allocation of resources. Smith’s vision is not only appealing but also quite advanced by modern standards.
The Rise of Neoclassical Economics
During the first half of the 20th century, economics underwent major changes (Ingrao & Israel, 1990). Regarding Smith’s vision, the change involved establishing two critical maxims:
- The moral dimension is no longer integral to economic analysis. Economics is supposed to be “value-free.”
- The average person is no more assumed to be a “merchant” but instead a homo economicus: a cold, selfish calculator with infinite information-storage capacity and infinite information-processing powers.
Economics, from Adam Smith to J.M. Keynes, was understood to be a moral science (Staveren & Peil, 2009). But few economists today would complain about the absence of morality from economic analysis.
Economics, from Adam Smith to J.M. Keynes, was understood to be a moral science.
Frederich Hayek (1945) was a leading critic of the assumption of perfect information. He argues that knowledge used by economic agents never exists in concentrated or integrated form but as dispersed pieces of incomplete knowledge owned by different agents. He writes (1945, p. 530):
“The problem is thus in no way solved if we can show that all of the facts, if they were known to a single mind … would uniquely determine the solution; instead, we must show how a solution is produced by the interactions of people, each of whom possesses only partial knowledge.”
With all these information powers of economic agents, “the hand running the economy is very visible indeed,” notes Kenneth Arrow (1986, p. 208).
Despite the frequent reference to Adam Smith, mainstream economics departed significantly from his core vision of how the invisible hand is supposed to work.
Enter Kurt Gödel
The first few decades of the 20th century also witnessed a transformation of mathematics, although few economists noticed. The most significant development in this regard was the work of Kurt Gödel (1906–1978), published in 1931 at the age of 25. His “Incompleteness Theorems” established beyond doubt the limits of formal axiomatic systems for mathematics and, by implication, to any field that applies axiomatic modeling.
Polymath John von Neumann was among the first prominent mathematicians to recognize the significance of Gödel’s works. Later, on the occasion of the presentation of the Albert Einstein Award to Gödel in 1951, von Neumann remarked:
“Kurt Gödel’s achievement in modern logic is singular and monumental — indeed, it is more than a monument; it is a landmark which will remain visible far in space and time.”
von Neumann then gave a brief description of the incompleteness results:
“Gödel was the first man to demonstrate that certain mathematical theorems can neither be proved nor disproved with the accepted, rigorous methods of mathematics. …
He proved furthermore that a very important specific proposition belonged to this class of undecidable problems: The question as to whether mathematics is free of inner contradictions. The result is remarkable in its quasi-paradoxical ‘self-denial’: It will never be possible to acquire with mathematical means the certainty that mathematics does not contain contradictions.”
“It must be emphasized,” von Neumann pointed out, that “this is not a philosophical principle … but the result of a rigorous mathematical proof of an extremely sophisticated kind.”
von Neumann elaborated that Gödel’s results are not specific to the field of mathematics but are general to any field that applies formal methods:
“Gödel actually proved this theorem, not with respect to mathematics only, but for all systems which permit a formalization, that is a rigorous and exhaustive description in terms of modern logic: For no such system can its freedom from inner contradiction be demonstrated with the means of the system itself.”
Axiomatics and Economics
The renowned mathematician David Hilbert championed the axiomatization program during the late 19th and early 20th centuries. He advocated for the axiomatization of not only mathematics but also other mature fields of knowledge, including economics. Hilbert’s vision was driven by his firm belief in the power of formal mathematics to construct complete and consistent systems, ultimately capturing knowledge in the most reliable way possible (Weintraub, 2002, pp. 87–90).
This approach influenced his disciple John von Neumann, who initiated the axiomatization of various fields, including economic game theory (von Neumann and Morgenstern, 1953). The trend peaked during the 1950s, on the hands of Kenneth Arrow and Gerard Debreu, among others, in what is known as the “Formalist Revolution” (Blaug, 2003). Today, the axiomatic approach is part of graduate and advanced textbooks on microeconomic theory and general equilibrium models.
Hilbert’s program, however, was greatly derailed by the Incompleteness Theorems of Gödel, as he proved that a formal axiomatic system, rich in arithmetics, can be either complete or consistent but not both. With Gödel’s results, the dream of achieving certainty through axiomatization evaporated (Kline, 1980). However, axiomatization might still be useful as a means to improve and enhance the progress of the respective field.
The Invisibility of Consistency
Let us give a brief overview of Gödel’s two Incompleteness Theorems (see Hofstadter, 1999, and Smith, 2013):
- The First Theorem states that a formal axiomatic system, rich in arithmetics, can be complete or consistent, but not both. In economics terminology, an axiomatic formal system is a mathematical model that starts with certain postulates or axioms, with a set of variables and rules of inference, and accordingly derives the key results of the model as theorems.
The model or the system is complete if it is possible to prove or disprove all sentences formulated within the system. To be consistent means, it should be free from contradiction, as von Neumann pointed out. Gödel demonstrated that, in a consistent formal system, there will be (many) sentences that are true but cannot be proved within the system. Truth extends beyond proof.
Truth extends beyond proof.
- The Second Theorem states that a consistent system cannot prove its own consistency. The system, in other words, cannot derive a sentence or even include an axiom that says it is consistent. If it does, the system becomes inconsistent! The consistency of the system is said to be undecidable within. Put differently, if the system is consistent, then consistency will be invisible within the system. It can be verified or “observed” only from the outside (Davis, 2000, p. 117).
In the following sections, we draw parallels between the invisibility of consistency established by The Second Theorem and the invisibility of market efficiency envisioned by Smith. The implications are striking.
The Invisibility of Market Efficiency
If the market mechanism is invisible, as Smith points out, then market efficiency, as a property of the entire market, shall also be ex-ante invisible. That is, agents cannot assert upfront the efficiency of the market.
If we model the market as a formal axiomatic system, then the consistency of the model implies the harmony of all individual transactions across the entire market. As discussed earlier, this harmony implies the efficient allocation of resources. Hence, there is an intimate relationship between model consistency and market efficiency. Model consistency shall be invisible within the model if the model is to remain consistent. Likewise, market efficiency shall be invisible within the market if the market is to remain efficient. Market efficiency depends on remaining invisible, just as the consistency of a formal model does.
The invisibility of consistency and efficiency shows that the deep insight of Adam Smith is very relevant to modern mathematical modeling. This is not the end of the story, however. We shall see below how contradicting this insight may result in very costly consequences.
Inconsistent Modeling
A crucial insight from Smith and Gödel is that market efficiency cannot be assumed a priori by market players. If they do, they will undermine the efficiency. This seemingly paradoxical result has been observed and discussed by economists on several occasions.
One way to see how belief in market efficiency becomes self-defeating was pointed out by Sanford Grossman and Joseph Stiglitz (1980, p. 404). The argument is that information is essential for achieving efficiency. However, since information is scarce, collecting and processing information is costly. If agents believe the market to be efficient, then prevailing prices will reflect (roughly) all relevant information. Thus, there is no need to invest in information. But if market participants stop investing in information, the market becomes inefficient!
Modern finance theory is formulated on the assumption of no arbitrage. However, assuming no arbitrage, like assuming market efficiency, invites inefficiency, thus bypassing arbitrage opportunities. Paul Wilmott and David Orrell (2017, p. 143) note: “Ironically, the assumption of no arbitrage creates another opportunity for arbitrage.”
“Ironically, the assumption of no arbitrage creates another opportunity for arbitrage” — P. Wilmott & D. Orrell
Market efficiency is a result of the invisible hand of the market mechanism. It is an emergent or equilibrium property that cannot be assumed or asserted at the individual player’s level. From the perspective of Gödel’s theorem, the hand is invisible in the sense that it cannot be explicitly asserted within the system, despite being “visible” from the outside, as explained earlier. Thus, to assume market efficiency within the model means that the market’s hand is not invisible anymore.
Moreover, imposing an ex-post equilibrium property onto an ex-ante choice model induces the system to change its behavior in response. This is the basic idea behind the Lucas Critique (Savin & Whitman, 1992), which argues that an aggregate regularity cannot be exploited without inducing rational agents to adjust their optimal behavior accordingly. A model may not be stable if it is used to recommend actions or behaviors that are not accounted for in the model itself, as we shall see shortly.
Failure of Models that Predict Failure
This has been empirically documented concerning the Global Financial Crisis of 2008 by Rajan, Seru, and Vig (2015). They analyze how the securitization of subprime mortgages changed the underlying relations in a manner that caused the failure of the models estimating the probability of default. The authors describe the result as “Failure of Models that Predict Failure.”
As Joseph Stiglitz (2010, p. 95) points out, models based on data from the pre-securitization era were used to create complex financial instruments like CDOs and CDSs. These instruments alter the data-generating processes, which makes these models inconsistent. The models assumed “crash-free” markets, which itself contributed to the crash, as Bouchaud (2008) points out.
“Ironically, it was the very use of a crash-free model that helped to trigger a crash.” — J.P. Bouchaud
These events were presciently perceived by Hyman Minsky decades earlier when he formulated his Financial Instability Hypothesis. The Hypothesis argues that if agents perceive the economy as stable, they are inclined to take on additional risks, eventually destabilizing the economy — “stability is destabilizing” (Minsky, 1982, ch. 5; Wray, 2016). Since the period preceding the Global Financial Crisis was considered to be a low-risk environment, the Crisis was a manifestation of Minsky’s Hypothesis.
The consistency of economic models in the light of the Incompleteness Theorems isn’t just a matter of rigor; it has direct implications for the behavior of market participants and financial instability.
Conclusion: A Scientific Revolution?
There has been much debate on the role of mathematics in economics (Weintraub, 2002). The problem is not about the use of mathematics; it is more likely about the misuse of it. Mathematics is a means to sharpen ideas, identify patterns, and discover the truth. However, when mathematical modeling becomes an end, or worse, a means to seek financial gains at the expense of the public, the price that society will pay might be enormous. Jean-Philippe Bouchaud (2008) wrote in Nature:
“Economics needs a scientific revolution.”
The revolution would likely start with internalizing the core insights of Gödel into mainstream economics curricula. Combined with the priceless insights of Adam Smith and other great economists, economics will be able to successfully balance economic wisdom with intelligent modeling, thus providing invaluable guidance for market participants and policymakers.
[1] This article draws on Al-Suwailem (2020).
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