The Mirage of Complete Markets
A complete world, if ever achievable, would be dull and stagnant. To envisage a vibrant and creative realm, we need to embrace incompleteness and undecidability.
Commodities Everywhere
Our hero this time is, once again, Kenneth Arrow. In 1953, while actively working with Gerard Debreu on the existence of general economic equilibrium, Arrow introduced an ingenious concept of goods: Rather than think of 1 kg of rice today as 1 kg of rice tomorrow, he considered these to be two different commodities. Moreover, tomorrow, it might or might not be raining in Chicago. So there will be now four different commodities, and so on.
A commodity, therefore, is defined not only by its physical characteristics but also by its date, location, and the environment’s specifications. These are called “state-contingent” commodities. The number of state-contingent commodities in this imaginary world would be unbelievably gigantic compared to the real world, but that is only a minor issue for these great theorists.
The price of a state-contingent commodity is paid upfront for an uncertain delivery, depending on the specifications of the contract. To obtain a commodity for certain, the buyer must pay the prices of the commodity under all states. A state-contingent market provides a way to transfer resources from one state of the world to another, allowing agents, in theory, to insure against risks.
Although it is unknown a priori whether it will be raining tomorrow or not, agents are assumed to know upfront all possible states of the world and all possible dates and places of delivery of each commodity for the entire time horizon of the market. In such a world, uncertainty only relates to which particular state will be realized at what period. (See: Arrow, 1964; Debreu, 1959, chapter 7; Flood, 1991.)
With state-contingent commodities, Arrow and Debreu were able, in one stroke, to extend the theory of general equilibrium from the case of certainty to that of uncertainty.
Arrow didn’t think of his state-contingent world as a mere intellectual exercise. He and many leading theorists believed that “complete markets” are the ideal benchmark towards which real-world markets should be progressing (Arrow, 1972, p. 127).
Market incompleteness might endogenously emanate from logical undecidabilities, independent of the agents’ cognitive capacities
A common critique of the complete markets theory is that it is unrealistic (e.g., Radner, 1982, pp. 930–931). In most studies, market incompleteness is exogenously given due to imperfections or bounded rationality (see Magill and Quinzii, 2008). We argue that market incompleteness might endogenously emanate from logical undecidabilities independent of the agents’ cognitive capacities. This might have substantial consequences, as we shall see.
The Description Problem
To trade state-contingent commodities, agents need a priori a complete world description, as pointed out above. Such a description is obviously the most important “commodity” in this world. Yet, the Arrow-Debreu theory offers little, if any, on how such a commodity is produced or exchanged. We will not get into these details but instead focus on a fundamental logical problem with the concept.
Consider a state-contingent economy W for which a complete description D is available. We ask a simple question: Is D part of W?
Suppose it is.
Then, a full description of the economy W with D shall include a second layer description, Dʹ. If Dʹ belongs to the economy Wʹ, there will be a third layer description … ad infinitum. This infinite regress problem means the description can never be complete.
If D is not part of the economy, then by construction, the economy cannot be complete. So, in either case, it is logically impossible to have a complete state-contingent economy. This purely logical problem has nothing to do with the frictions or transaction costs of our messy real world.
The Paradox of Tristram Shandy
According to Bertrand Russell (1937, p. 358), Tristram Shandy wanted to write a full story of his life, spending more time describing every event than the time it took him to live through it. Instead of being reasonably up to date, his autobiography must become hopelessly out of date the longer he works on it.
This paradox mirrors the challenge we face when attempting to write a complete description D of all possible states of the world for a state-contingent economy. In such a world, time is extremely important since, together with other properties of the environment, it defines the nature of the commodity. So, D must be accurately up to date.
However, the process of writing D unfolds in time, just like any other process in the economy. There is no reason why this process should have a different speed than other processes of W. So, when the writing is concluded, time will have necessarily progressed from, say, t₀ to t₁. By then, the description would become outdated.
Is it possible to write D at an infinite speed to avoid the time lag? With an infinite speed, all events will take place at once. In such a world, there will be no time nor uncertainty to worry about.
No matter how hard we try to keep the description of the world up to date, it will necessarily be behind regardless of how time is measured
No matter how hard we try to keep the description D up to date, it will necessarily be behind regardless of how time is measured. This means there will never be a current a priori full description of the state-contingent economy. The economy must necessarily be incomplete.
Self-Prediction
The complete markets theory assumes that agents’ knowledge will not grow over time. All necessary knowledge is embodied in the upfront description D. In such a world, there are no discoveries, innovations, or breakthroughs that open up new possibilities for progress. Is this the ideal world the theory aspires for?
Aside from this negative attitude towards knowledge and creativity, how do the agents know their knowledge will never grow over time? To make such a statement, they are making a self-prediction. Is such a prediction logically feasible?
In The Open Universe, the prominent philosopher of science, Karl Popper, develops an extensive argument for “The Impossibility of Self-Prediction” (1956, pp. 67–77). Building on the work of Kurt Gödel and Alan Turing, Popper contends that, even with perfect information on present or past initial conditions, it is impossible for us, using deductive methods, to predict our future state. The argument ultimately hinges on self-reference, which creates a “blind spot” whereby the prediction itself can influence the outcome, even for inanimate systems (see also: Ismael, 2016, chapter 7).
To see the connection with Gödel’s Incompleteness Theorem, we follow the mathematician Raymond Smullyan (1987) in his book Forever Undecided, where he poses a thought-provoking question (p. xi):
“Is it possible for a rational human being to be in a position in which he cannot believe that he is consistent without losing his consistency in the process? That is one of the main themes of this book. It is modelled on the famous discovery of Kurt Gödel (the so-called Second Incompleteness Theorem) that any consistent mathematical system with enough power to do what is known as elementary arithmetic must suffer from the surprising limitation that it can never prove its own consistency!”
This conclusion points to an essential insight: Incompleteness presupposes an open universe with unbounded possibilities.
Enter Emmy Noether
Defining goods by time and space means such commodities cease to be invariant with respect to space or time. But we learn from the theorem of the German mathematician Emmy Noether (1882–1935) that there is a deep link between symmetry or invariance in physical systems and conservation laws (Stewart, 2013). Noether’s theorem, in essence, asserts that every symmetry (invariance) corresponds to a specific conservation law.
The symmetry of physical laws with respect to time corresponds to the conservation of energy. This implies the laws of nature are as valid today as they were 1000 years ago and 1000 years from now. Similarly, symmetry with respect to space corresponds to the conservation of momentum, ensuring the laws of nature are as valid on Earth as on the Moon, Mars, or in the deepest reaches of outer space.
Noether’s theorem establishes that for every symmetry in the laws of nature, there is a corresponding conservation law
The production of goods and services is ultimately governed by the laws of nature. Production commodities, therefore, must be state-invariant and cannot generally be state-contingent. If consumption commodities are predominantly state-contingent, the discrepancy between production and consumption jeopardizes the possibility of equilibrium in such a world.
Moreover, the stability of a dynamic system is closely related to its set of invariant properties (Rosen, 2008, pp. 279–281). A system lacking sufficient invariances, as seems to be the case with a complete state-contingent market, would be prone to instability. This contrasts with the theory’s claim of providing a perfect framework for risk distribution.
The lack of space and time invariance in a state-contingent market shows how economic theory might be disconnected from natural sciences, especially physics. This seems ironic, considering the economics profession is often accused of “physics envy.”
The Invariance Problem
Clearly, there must be parts of a state-contingent economy that are state-invariant. In addition to the above discussion, consumers’ preferences must be stable over time (in the absence of new knowledge). If preferences, in principle, were not stable, the claims consumers make would not be credible. With the absence of trustworthy commitments, the market would collapse.
Let us be more specific and focus only on commodities. By construction, consumption goods are supposed to be state-contingent. However, some commodities are not. For example, the full description of the states of the world, D, is supposed to be state-invariant. Similarly, capital commodities required to set up the trading platform should be state-invariant. Moreover, laws and regulations (and the associated services) are required in all states. The question, therefore, becomes:
Is there a rule to decide which commodities are state-contingent and which are state-invariant?
To answer this question, suppose we encode the description of each commodity in the state-contingent economy into a specific computer program. The program, when executed, will print out the features of that commodity except whether it is state-contingent or not.
Now, we want to see if there is a master computer program that can go through all these commodity-specific programs and, based on the code, decide if the commodity should be state-contingent or not. We are looking for a general algorithmic rule that, for an arbitrary state-contingent economy, can decide which commodity should be state-contingent and which should not.
Based on the work of Alan Turing on the unsolvability of the Halting Problem, the answer to the above question is in the negative. To see this, suppose that, for an arbitrary state-contingent economy, we have such a rule, R, which will classify computer-encoded descriptions of all commodities in the economy into two sets:
- A = {x : R(x) = 1}, the set of encoded state-invariant commodities.
- B = {x : R(x) = 0}, the set of encoded state-contingent commodities.
By construction, sets A and B should be mutually exclusive.
The rule R should obviously be state-invariant, and thus, sets A and B should be state-invariant descriptions as well. This implies that R(A) = 1, and so, A belongs to itself. This should not be problematic. However, if B is also state-invariant, then R(B) = 1, which implies that B belongs to A. This contradicts the assumption that the two sets are mutually exclusive.
Without a universal rule to identify state-contingent from state-invariant commodities, agents cannot agree on terms of trade, and the market may fall apart
It follows that the rule R cannot exist: No general rule can decide which commodities should be state-contingent and which should be state-invariant. Without such a universal rule, agents cannot agree on terms of trade, and the market may fall apart.
Arrow’s Securities
The creativity of Arrow (1964) did not stop at state-contingent commodities. He quickly realized that it might be more efficient, financially, to split the state-contingent claim of a commodity into two components:
- A state-contingent security exchanging money in one state for money in another, and
- A spot trade of the underlying commodity at the maturity date.
The equivalence of the two formulations assumes a linear world in which the whole is equal to the sum. State-contingent securities became known as “Arrow’s Securities,” which formed the theoretical basis for financial derivatives.
Arrow was careful in his formulation to assert the resource constraint of all claims on any commodity. However, financial market applications tended to ignore these constraints to concentrate on trading risks (Flood, 1991, p. 42). Moreover, the belief that the market was, in principle, completable, fostered the view that more derivatives would be better.
This view persisted despite the conclusive work of Oliver Hart (1975) showing that if the market is incomplete, then removing securities and hence decreasing the possibilities for trade would result in a Pareto superior allocation (Magill and Quinzii, 2008, p. xv). Nonetheless, the financial market ignored these considerations and continued its exponential growth out of sync with the real economy.
Financial Weapons of Mass Destruction
Derivatives grew from almost nothing in the 1970s to a $670+ trillion in notional value by the end of 2007, shortly before the onset of the Global Financial Crisis. This amount was more than ten times the global GDP at the time.
Attempting to complete an incompletable system is not only futile but potentially destabilizing
Derivatives were a crucial factor contributing to the Global Financial Crisis, as Arrow (2012, p. 967) himself pointed out; see also FSA (2009) and Stiglitz (2010). The Crisis demonstrated, among other things, that attempting to complete an incompletable system is not only futile but potentially destabilizing.
In 2002, Warren Buffet presciently warned of derivatives as “time bombs, both for the parties that deal in them and the economic system” (p. 13).
He argued that derivatives can exacerbate the risks at the levels of the firm and the market and that the “macro picture is dangerous and getting more so” (p. 14). He concluded his letter by saying: “In our view, however, derivatives are financial weapons of mass destruction, carrying dangers that, while now latent, are potentially lethal” (p. 15).
Confessions
Years before the Global Financial Crisis, Arrow expressed his dissatisfaction with the developments of the derivatives markets. In a 1995 interview, when asked if the massive expansion in financial markets has significantly enhanced our ability to allocate risks efficiently, he replied:
“I would like to think so, but I’m not so sure it’s true. The trouble is that the risks that are being hedged very well by new financial securities are financial risks. And it appears to me that the real things you want to hedge are real risks, for example, risks in innovation.” (Emphasis added.)
In another interview in 2003 with (the subsequently discontinued) Financial Engineering News, he noted that “derivatives and securities that offer methods of reducing risks are not necessarily used for that purpose. They are neutral and can be used to reduce risks, but people gamble on them.” Further:
“With derivatives, whether the risk-reducing aspects are predominant or the risk-enhancing aspects are predominant, they can be used for gambling. That means speculators are adding to the swings rather than reducing them.”
Conclusion
The theory of complete markets was an ingenious invention by Kenneth Arrow in collaboration with Debreu and other leading theorists. Despite the clever formulation, it suffers several logical flaws that undermine its validity. Markets are likely to be inherently incomplete regardless of imperfections or cognitive limitations.
Attempting to complete an incompletable system is not only futile but potentially destabilizing. A world with stagnant knowledge is far from the ideal world that we aspire for. To envision a stable, vibrant, and creative economy, we need to embrace incompleteness and undecidability.
This article is part of a series on Economic Impossibility Theorems.
References
- Arrow, K. (1964). The role of securities in the optimal allocation of risk-bearing. The Review of Economic Studies, 31, 91–96. (Original work published in French in 1953).
- Arrow, K. (1973). General economic equilibrium: Purpose, analytic techniques, collective choice. Nobel Memorial Lecture. Retrieved from https://www.nobelprize.org/
- Arrow, K. (2012). Economic theory and the financial crisis. Information Systems Frontiers. 14, 967–970.
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- Flood, M. (1991). An introduction to complete markets. Federal Reserve Bank of St. Louis, March/April.
- Financial Services Authority. (2009). The Turner Review: A regulatory response to the global banking crisis. UK Financial Services Authority.
- Hart, O. (1975). On the optimality of equilibrium when market structure is incomplete. Journal of Economic Theory, 11, 418–443.
- Ismael, J. (2016). How physics makes us free. Oxford University Press.
- Magill, M., & Quinzii, M. (Eds.). (2008). Incomplete markets (Vol. 1). Edward Edgar.
- Popper, K. (1956). The open universe: An argument for indeterminism (W.W. Bartley, III, Ed.). Routledge.
- Radner, R. (1982). Equilibrium under uncertainty. In K. J. Arrow & M. D. Intriligator (Eds.), Handbook of mathematical economics (Vol. II, pp. 923–1006). North-Holland.
- Rosen, J. (2008). Symmetry rules: How science and nature are founded on symmetry. Springer.
- Russell, B. (1937). The principles of mathematics (2nd ed.). George Allen & Unwin.
- Smullyan, R. (1987). Forever Undecided: A Puzzle Guide to Gödel. Alfred A. Knopf.
- Stewart, I. (2013). Symmetry: A very short introduction. Oxford University Press.
- Stiglitz, J. (2010). Free fall. W.W. Norton.
