A complete market is “a world where every imaginable good can, in principle, be traded in every possible situation at competitive prices.” In this setup, “all trades in this imaginary world occur at the initial time (time 0). Agents trade state-contingent claims for consumption at all future dates and states.” Once these trades are made, “no further trading occurs; the contracts are honored as states unfold over time.”
“In such an imaginary world, every risk can be hedged, and every opportunity can be seized.” But as both economists and lawyers recognize, “almost all contracts are incomplete (Halonen-Akatwijuka and Hart, 2024).”
Jean Tirole (1999) remarks:
“Incomplete contracting arguably underlies some of the most important questions in economics and some other social sciences … For all its importance, there is unfortunately no clear definition of ‘incomplete contracting’ in the literature.”
John Moore (2016) adds:
“We’ve had 25 years to come up with a watertight theory of contractual incompleteness and we haven’t succeeded yet.”
Contracts as Algorithms
“A contract, in essence, is a set of ‘if–then’ statements. This means we can view contracts as algorithms or computer programs.” But the assumption that such programs can make a market perfect is flawed.
“It was proved to be flawed not by economists but by first-class mathematicians almost a century ago.” Alonzo Church and Alan Turing (1936) showed that the question of whether a program will halt cannot, in general, be answered. “This means that the accuracy of program Q is algorithmically unverifiable.”
Newton da Costa and Francisco Doria (1991) extended these insights, demonstrating that “certain ‘interesting’ or ‘useful’ properties in these domains are generally undecidable, just as Rice’s Theorem shows for non-trivial semantic properties of programs.”
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Algorithms and Flash Crashes
“Even if the computer program can execute normally, it will not be possible to tell in advance how it may respond to different stimuli. Programs may behave unexpectedly in some circumstances, even if the code was technically bug-free.”
This is exemplified by financial “Flash Crashes.” The most famous occurred on May 6, 2010, in the U.S., initially wiping out about $1 trillion before partially recovering. According to data from Nanex, “mini flash crashes have exceeded 20,000 since 2006.”
“The moral of these examples is that in complex environments, deterministic algorithms can result in unpredictable, emergent behavior. Being algorithmic does not guarantee being predictable — it might be exactly the opposite.”
Market Boundaries and Undecidability
A key question follows: Can the market determine its own boundaries? “Suppose we can design an algorithm that systematically identifies all contractable objects.”
But as the article explains, “the function k cannot exist. No algorithm can universally identify which objects are contractable and which are not.” This undecidability reinforces the conclusion that markets cannot be fully complete.
Why Incompleteness Matters
Oliver Hart (1975) argued that “when markets are incomplete, competitive equilibrium may not exist in general, and even if it does, it may not be Pareto optimal.” Samuel Bowles (2016) warns that “trying to complete the market when it is inherently incompletable might make things even worse.”
Ken Arrow (1969) pointed out that “social norms, including moral and ethical codes, serve to compensate for market incompleteness.” Amartya Sen (2017) remarked that “the ultimate guarantee of individual liberty may rest not on rules, but on individual values.”
Free Market, Ownership, and Completeness
Oliver Hart (2016) explained that ownership becomes valuable because contracts are incomplete. “If a contract is incomplete, then the right to decide about the missing parts of the contract is the residual control or decision right.”
“In a complete market with complete contracts and zero transaction costs, there is no need for firm ownership.” But in reality, ownership fills the gap left by incompleteness.
This leads to the central conclusion:
“A market system can be either free or complete, but not both.”
Reason, Free Will, and Incompleteness
Gödel’s First Incompleteness Theorem shows that “a consistent system will have many true statements that cannot be proved within the system.” The only way forward is “going outside” the system — the essence of free will.
“Incompleteness is a feature, not a bug.” It explains why free choice, ownership, and creativity matter in markets.
Conclusion
“While economic theory celebrates completeness as the crown jewel of free market efficiency, a complete market, technically speaking, is neither free nor efficient.”
“The incompleteness of mathematics is one of the greatest discoveries of the 20th century. The implications of such a discovery for social sciences are yet to be fully internalized.”
Source: This article is part of a series on Economic Impossibility Theorems.
