Kenneth Arrow’s (Ignored) Impossibility Theorem

By William K. Black
February 22, 2017     Bloomington, MN

Kenneth Arrow, one of the giants of economics, has died at the age of 95.  He became a Nobel Laureate in 1972.  As a young lawyer in 1977, I saw him in action as an expert witness on the subject of risk.  The context was setting the rates for shipping oil through the Trans-Alaska Pipeline System (TAPs).  Arrow testified about the risks of oil prices falling.  The FERC administrative law judge thought such a scenario was ridiculous.  Within four years, oil prices fell sharply.  Arrow’s experience was a common one for economists dealing with lawyers – the ALJ ignored him.

The New York Times obituary for Arrow is revealing about how the conventional wisdom distorts economic theory in a predictably skewed fashion.  It begins by discussing Arrow’s “impossibility theorem,” which states that where there are more than two choices it is impossible to construct perfect majority choice systems.

The author of the obit stressed the impossibility of such systems being optimal.  Contrast that emphasis with the author’s treatment of Arrow’s work on “general equilibrium.”

Professor Arrow proved that their system of equations mathematically cohere: Prices exist that bring all markets into simultaneous equilibrium (whereby every item produced at the equilibrium price would be voluntarily purchased). And market competition puts society’s resources to good use: Competitive markets are efficient, in the language of economists.

Professor Arrow’s theorems set out the precise conditions under which Adam Smith’s famous conjecture in “The Wealth of Nations” holds true: that the “invisible hand” of market competition among self-serving individuals serves society well.

That is one way to phrase it, but a more accurate, parallel way to phrase Arrow’s work on general equilibrium would be as an “impossibility theorem.”  Arrow actually proved that it was impossible for general equilibrium to occur.  The “precise conditions” in which economists can guarantee that a market transaction “serves society well” is the null set.  There is no market that meets those “precise conditions” because they are impossible to meet.

Market competition does not inherently “put society’s resources to good use” and “competitive markets” can be enormously inefficient.  In a Gresham’s dynamic, for example, the more competitive the market the more CEOs put society’s resources to bad uses and the more inefficient the results.  When hit men compete to murder spouses, the price of hiring hit men declines, but this does not serve society well and it is not efficient.  (The same is true for competition by cigarette company CEOs.  When companies prosper by increasing greenhouse gas emissions it, eventually, does not serve society well and it is not efficient.

The author of the obit accurately reflects the conventional economic wisdom in the two paragraphs that I quoted.  The conventional economic wisdom is flat out wrong.  As I have emphasized in prior posts, a prominent economist (who loves general equilibrium) admits that the conventional wisdom would only be true under the “silent” “assumption” that “God” mystically ensures that buyers and sellers do not act in a “self-serving” manner that harms society and reduces efficiency (Athreya 2013: 104).  We should call it the Arrow-Debreu-McKenzie (ADM) “impossibility theorem,” but instead orthodox economists make the hilarious (implicit) assumption that God loves laissez faire so much that he prevents all predation.  As Dr. Athreya phrases it, the “ADM God” prevents CEOs from even considering the possibility of predating on customers.

The author of the obit understands most of these points.

[Arrow] made clear, his powerful conclusions about the workings of competitive markets held true only under ideal — that is to say, unrealistic — assumptions.

His assumptions, for example, ruled out the existence of third-party effects: The sale of a product by Harry to Joe was assumed not to affect the well-being of Sally — an assumption routinely violated in the real world by, for example, the sale of products that harm the environment.

Note, however, that the author does not go back and correct his earlier errors and he never states that the ADM general equilibrium model shows that it is impossible for laissez faire to produce general equilibrium.

He also fails to inform readers that orthodox economists’ twin “dystopian” assumptions (Athreya 2013) make it impossible for laissez faire to guarantee either efficiency or socially desirable outcomes.  The twin dystopian assumptions are self-interested behavior and rationality.  Orthodox economists such as N. Gregory Mankiw define actions by CEO, such as the refusal to “loot” the firm, as “irrational” rather than “moral” because it would harm the CEO’s “self-interest” (Akerlof & Romer 1993: 65).

The punch line to the defining economist joke is “assume a can opener.”  To “silently” assume an ADM God,” however, takes that joke to an unprecedented level worthy of a superlative humorist.

“Modern macro economists” have surpassed micro economists’ supreme act of humor.  Modern macro silently assumes an ADM God – and invariably describes its use of general equilibrium models as “rigorous.”  One imagines the rigor of “modern macro” proponents as they begin their incantations, using an analog to a pilot’s pre-flight checklist.  Step one:  silently assume an “ADM God.”  Step two: silently assume that the “ADM God” is the patron of laissez faire and acts rigorously to prevent any predation by CEOs (even though the express dystopian assumptions would produce widespread CEO predation).  Step three: Define steps 1 and 2 as the “rigorous” treatment of “micro foundations.”  Step four: chant the mantra endlessly and with a straight face.  Step five: do not fly on the ADM plane – and blame the crashes on “governmental interference” with the otherwise inerrant ADM checklist.

A last nerdy note.  The author of the obit stresses repeatedly how impressed he is by Arrow’s general equilibrium math.  The math only general equilibrium, however, because the model assumes away reality.  If the model attempted to deal with reality, without the “ADM God’s” aid, the math would produce indeterminacy or spiral away from equilibrium into bubbles and market breakdowns.  The math would also show that laissez faire is frequently criminogenic and would produce epidemics of elite fraud and other predatory abuse.  Arrow made his absurd assumptions in his model not because they reflected reality, or proved reliable in prediction, but to make the “system of equations mathematically cohere.”  When the math fails to explain reality and predict events it is a grave error (rather than a cause for celebration) when economists assume out of existence reality and torture the model until the math “coheres.”

The ultimate failure of economics as a field is to:

  1. worship an economic model that is criminogenic,
  2. hide that disaster from the public by assuming “silently” an “ADM God” that contradicts the model’s express assumption,
  3. continue to worship and proselytize that model when its silent assumption of an “ADM God” repeatedly produces criminogenic policies and epic predictive failures, and
  4. praise your models as “rigorous,” “scientific,” and “transparent,” and
  5. define critics as anti-scientific and demand that their critiques be excluded as heresy.

Arrow was brilliant and well meaning.  We celebrate his life and mourn his passing.  The opportunity cost to our field is how much he could have accomplished had his research not been so distorted by neoclassical dogma.

6 Responses to Kenneth Arrow’s (Ignored) Impossibility Theorem

  1. Any function appears linear over a short-enough interval, right? And given the “right” initial conditions, can produce a rational result. So much for the real world…

  2. David Harold Chester

    If the interval is sufficiently short what Richard writes is true, with the exception of very sudden changes. A simulation with all factors included is preferable and if some are not included the results are likely to be wrong.

    • Consider the function: let f(x) = q for all x such that x = p/q where p and q are integers, expressed with lowest common denominator. (In other words, for all x that are rational numbers. => Function is undefined for irrational numbers.)

      Show that this function has linearity on any scale. There are an infinite number of such functions.

      • That should be ‘p and q are integers, expressed so that p/q are reduced to lowest terms’.

        The function has no linearity on any interval. And no linear approximation.

  3. The error here has to do with ‘any function’. Not all functions are polynomial. Not all functions can be approximated by a polynomial. Some functions have no continuity on any scale, and no linearity on any scale.