Kenneth Arrow made his PhD thesis into a book in 1951 (Social Choice and Individual Values). It remains one of the most remarkable works in the strange field of economics in the twentieth century. This is the book that launched a thousand (or more than fifty, anyway) attempts at refutation or compartmentalization by troubled neo-classical economists in the years following its publication.
The book starts off with the voting paradox, which Arrow attributes to E.J. Nansen (1896). This goes as follows: if A, B, and C are three alternatives and 1, 2, and 3 are three individuals and supposing that 1 prefers A to B and B to C (and therefore A to C), 2 prefers B to C and C to A (and therefore B to A), and 3 prefers C to A and A to B (and therefore C to B), then a majority (of two) prefer A to B but also B to C. A “rational” conclusion would be that the community prefers A to C; but, in fact, a majority prefers C to A. So the voting procedure has led to a result that is not “rational”. Unfortunately, this paradox is not resolved, but rather deepened as Arrow proceeds.
Starting from Bentham’s hedonistic calculus (refuted forcefully by Plato 2500 years ago in the Philebus) and the voting paradox (attributed to Condorcet by many), Arrow proves in math / logical notation that, given five conditions that most neo-classical economists would think should be honored in any theory of social choice, any solution to the social welfare question in which there are at least three alternative choices must either be imposed or dictatorial, thus contradicting two of the assumed conditions. His “possibility” theorem (in the original; it has been referred to as his “impossibility theorem” by many) “shows that, if no prior assumptions are made about the nature of individual orderings, there is no method of voting which will remove the paradox of voting . . . neither plurality voting nor any scheme of proportional representation, no mater how complicated. Similarly, the market mechanism does not create a rational social choice.” (Chapter V, Section 4).
In his last chapter Arrow shows that for “single-peaked preferences” majority decision does lead to an outcome satisfying five revised conditions, including the revision to Condition 1 to the effect that “the tastes of individuals fall within certain prescribed realms of similarity” and “provided the number of individuals is odd.”(!) Arrow then asks “Do these or possibly other mathematical restrictions have any social significance?” His answer is “I do not pretend to have any definite answer” but that uniformity of social attitudes actually is a condition of one prominent school of political philosophy, namely the “idealist school” of Rousseau, Kant, and T.H.Green. In a footnote he quotes Rousseau to the effect that the law of plurality is established by agreement, which must have been unanimous, at least in the beginning. Rousseau’s unanimous consent would, of course, satisfy Arrow’s revised Condition 1.
Arrow’s conclusion, however, is that even if some degree of consensus is taken as true, “Any view which depends on consensus as the basis for social action certainly implies that the market mechanism cannot be taken as the social welfare function since that mechanism cannot take account of the altruistic motives which must be present to secure that consensus.”
So there you have it; the smartest (oddest?) economist of his generation has proven that his “science” cannot be placed on a rational basis given the assumptions that are required for a market economy; and that a market economy cannot produce social benefit! My guess is that the only neo-classical economists that are not worried about this are the ones that remain blind to the issues that Arrow has raised or those who don’t care that their “science” can’t even describe a logically possible world, much less our actual one.