The first welfare axiom of mathecon
Lesson 3
The First Welfare Axiom vs. the First Welfare Theorem
(or Lesson 3)
by Joseph Mark Haykov
August 1, 2024
The first welfare axiom of mathematical economics states that symmetrically informed and unfettered trade increases the welfare of all parties directly involved, except in cases of unforeseen events, such as dropping purchased eggs on the way home from the grocery store. The reason is self-evident to any rational person; under whatever definition of rationality you choose to apply, no one would ever voluntarily enter into an arm's-length commercial transaction unless they expected to benefit in some way from doing so.
That’s what the word "commercial" means—trade motivated purely by profit measured by money. The reason the first welfare axiom holds true in mathematical economics universally is precisely because it is only true conditionally: when describing commercial trade, and is not applicable to non-arm's-length transactions. For example, leaving money to your children would not classify as an arm's-length commercial transaction. Here, the word "commercial" refers to the fact that in any such “arm's-length and commercial transaction,” the parties involved are indifferent as to the counterparty from whom the goods or services are purchased. For example, as long as your pool is clean, it doesn’t matter to you who your condo association hires to clean it.
In any such “arm's-length, commercial transaction,” the only reason anyone would willingly, of their own free will— which is how we define “unfettered trade,” entirely voluntary, without any coercion whatsoever—enter into any such transaction is that they expect to benefit from it in some way. This means that ex-ante, or before entering into any such transaction, no rational individual would do so unless they expected to benefit—increase their well-being, happiness, welfare, or equivalently, utility—in some way. Being symmetrically informed about the goods and services being purchased guarantees that the expected benefit ex-ante, or before the purchase, is realized ex-post, or after the fact, barring “acts of God” as explained above.
The first welfare theorem of mathematical economics, in the context of the Arrow-Debreu framework, states that competitive markets yield Pareto-efficient outcomes. What Pareto-efficient means in this context is that there is no more mutually beneficial trade to be had—no one can be made better off without making someone else worse off. If we are maximizing the overall welfare or utility of everyone in the economy collectively as a group—akin to a real-world example of gradient descent optimization by Pareto-improving trade—then at the maximum, when the gradient is zero, the condition that no one can be made better off without making someone else worse off, or equivalently that no more mutually beneficial trade is possible, is what is referred to as Pareto-efficiency: an optimal, welfare-maximizing outcome.
Now, in the context of the first welfare theorem, “competitive markets” assumes not only symmetrically informed and unfettered exchange but also multiple additional conditions necessary to guarantee that a global, as opposed to merely a local, maximum is reached. This is a very nuanced and important point that is often overlooked in such discussions. Merely requiring that trade is unfettered and symmetrically informed guarantees that it is mutually beneficial—or equivalently, Pareto-improving—and results in a local maximum. However, real-world constraints, as exemplified by externalities, monopolies facilitated by barriers to entry into the market, and several other violations of perfect market conditions, preclude the achievement of global maximum welfare. This is why only a “perfectly competitive market” guarantees a theoretical Pareto-efficient equilibrium.
In theory, all violations of perfect market conditions reduce not only overall welfare but also economic efficiency—as often measured by real per capita GDP or its growth rate. However, in reality, while violations of perfect market conditions, such as no externalities—as exemplified by environmental pollution—do not reduce efficiency as measured by per capita GDP, they do in fact reduce overall quality of life in other ways, such as by having poisonous water in Flint, Michigan, next to Detroit. In reality, reduced efficiency as measured by low real per capita GDP inevitably results from violations of two key competitive market assumptions: symmetrically informed and unfettered exchange. Any real-world violations of these two conditions (or axiomatic assumptions) are absolutely guaranteed to inevitably result in huge real-world inefficiencies each and every time they are violated. The reason is obvious: while violations of other additional assumptions—beyond symmetrically informed and unfettered trade—preclude the achievement of a global optimum welfare, any trade that is either involuntary or asymmetrically informed is not guaranteed to be even mutually beneficial or Pareto-improving, thus failing to achieve even a local, let alone a global, maximum.
This is precisely what George Akerlof explained in "The Market for Lemons," and this is also exactly the reason why Haiti’s per capita GDP is five times lower than that of the neighboring Dominican Republic. The prevalence of lawlessness in Haiti results in involuntary exchanges, such as robbery, theft, extortion, kidnapping, and other acts that violate the unfettered trade assumption necessary for achieving real-world Pareto-efficiency.
In this sense, the difference between the first welfare axiom and the first welfare theorem is that while the first welfare axiom holds true universally in reality—though only when applied to commercial trade—the first welfare theorem never holds true in reality, on account of the fact that in reality, multiple assumptions, including the assumption of unfettered and symmetrically informed exchange, are routinely violated in all real-world economies. In this sense, while the first welfare theorem proves the existence of a theoretical Pareto-efficient outcome, the first welfare axiom tells us what in reality precludes the achievement of this outcome by real-world economies: real-world exchange that is either involuntary or asymmetrically informed.
Rent-Seeking Axiom vs. Public-Choice Theory
The rent-seeking axiom is rooted in the evidence-based claim that the theft of grain stored in a granary by vermin, such as rats, or in the field by locusts, directly reduces the efficiency of any economy. This is because the grain eaten by vermin represents wealth lost to parasites who did not contribute to the production of the grain they consumed. The rent-seeking "axiom" classifies economic rents as wealth obtained by "economic parasites" who, like rats stealing grain from a warehouse, consume a portion of the real GDP produced by others without contributing to the production of that GDP. Therefore, all economic rents, like grain stolen by rats in a warehouse, represent a market failure and reduce not only labor productivity but also Pareto efficiency.
Formally, the rent-seeking axiom states that the unearned extraction of wealth by unproductive "economic parasites" universally reduces economic efficiency. It is an empirical fact—independently verifiable for its real-world accuracy and evidence-based—that robbery, theft, and fraud are universally detrimental to economic growth and therefore punishable by imprisonment in all legal systems, no matter how corrupt. Unfortunately, the extraction of unearned wealth is accomplished daily by perpetrators of various crimes, including theft, robbery, extortion, and ransom collection—whether involving people or computer hard drives. These crimes allow the perpetrators to consume goods and services without contributing to their production, consistent with the definition of rent-seeking in public choice theory.
Public choice theory, deduced from the rent-seeking axiom, was developed by Gordon Tullock and James Buchanan (1986 Nobel Prize) and applies economic principles to political processes, analyzing how self-interested behavior influences decision-making in government. It explores how individuals in the public sector (politicians, bureaucrats) pursue their interests, sometimes leading to outcomes that do not maximize social welfare. Unearned wealth extraction occurs when these individuals use their positions to benefit themselves or specific groups at the expense of the general public.
Rent-seeking, according to public choice theory, involves efforts by individuals or firms to gain economic benefits through manipulation or exploitation of the political and economic environment rather than through productive economic activities. This includes lobbying for favorable regulations, subsidies, or tariffs that provide them with unearned wealth. Rent-seeking is inefficient because it diverts resources away from productive activities and creates economic inefficiencies, much like other forms of theft.
However, the distinction between public choice theory and the rent-seeking axiom is that, unlike public choice theory, the rent-seeking axiom cannot turn out to be false, as it is fully supported by real-world facts and not contradicted by any. The ability of rent-seekers to obtain unearned wealth represents a market failure, no different in theory or reality from robbery, theft, ransom, extortion, or blackmail—all illegal crimes. This holds true universally, given the underlying definition of economic rents as goods and services pilfered by parasites but produced by others, enabling parasites to obtain wealth that they did nothing to earn, thus facilitating parasitic consumption of real GDP produced by others—resulting in a Pareto-inefficient outcome.
Public choice theory—which is actually a theorem, since it follows logically from the rent-seeking axiom—argues that the existence of rent-seeking opportunities to effectively rob the public without risking going to jail will inevitably lead rational, utility-maximizing politicians and others to systematically engage in rent-seeking in reality. In fact, while it is theoretically possible to mitigate rent-seeking, in reality, it is not possible to do so—under the rational utility maximizer axiom—without the ability to identify and punish rent-seekers. Similarly, the axiom of rational utility maximization guarantees with absolute certainty that the prevalence of theft and robbery will always increase unless swiftly and reliably punished. This is evidenced by recent events in San Francisco, where the decision not to prosecute perpetrators of thefts under $950 directly led to such a crime wave that several retailers were forced to close their stores—an evidence-based claim, independently verifiable for accuracy, that cannot in reality prove false, barring mass hallucinations or similar extraordinary circumstances.
The purpose of this discussion is to reiterate the fact that just as Peano's axioms of arithmetic define the precise conditions under which any resulting theorems, such as "2+2=4," are guaranteed to be universally true, the following three axioms define key conditions under which any resulting theorems, such as the first welfare theorem, are true, but only as long as the underlying axioms remain inviolate in reality; otherwise, all bets are off.
Representative Agent: Rational subjective utility maximizer.
First Welfare Axiom: Unfettered and symmetrically informed trade is guaranteed to be mutually beneficial ex-post, barring acts of God.
Rent-Seeking Axiom: Unearned extraction of wealth by "economic parasites" universally reduces Pareto efficiency in any economy.
Indeed, just as the first welfare theorem—competitive markets result in Pareto-efficiency—is logically deduced from the first welfare axiom, public choice theory that rational utility-maximizing representative agents will seek to obtain economic rents is logically deduced from the rent-seeking axiom.
However, just as the Pythagorean Theorem is false in reality if the underlying Euclidean axiom that the shortest distance between two points is a straight line does not hold (e.g., GPS triangulation), the First Welfare Theorem will not hold true in reality unless all the underlying axioms—from which it is derived—hold true in reality and remain unviolated. Barring a few rare exceptions, most trade that occurs in any real-world economy is either asymmetrically informed or involuntary, violating the conditions necessary for the first welfare theorem to hold true in reality, as explained above.
In the next lesson, we will learn that cognitive biases do not violate the Rational Utility Maximizer axiom at all but instead can be precisely modeled in game theory and mathematical economics as individuals who are asymmetrically informed about the rules of the game—using a different set of axioms that lead to different conclusions—while employing the same deductive logic on which all mathematical proofs are based.
However, the key point of this third lesson is to reiterate that the only assumptions, or axiomatic logical claims, in whose accuracy we can be absolutely certain, are those assertions that cannot turn out to be false in reality, namely: evidence-based claims, not contradicted by any facts. This is very different from many theoretical claims such as 2+2=4 that can always turn out to be false, as they are not evidence-based, but conditional upon assumption-dependent axioms, such as Peano’s fifth axiom, as explained.