Perfect competition, methodologically contemplated

Abstract:

The concept of perfect competition embodies a formal contradiction, precisely as would that of “the largest integer.” The ascription of legitimate meaning to the concept, as in George J. Stigler’s well-known 1957 essay, “Perfect Competition, Historically Contemplated,” is demonstrably circular, hence methodologically unacceptable. Mathematical problems arising from the concept are explored and it is these that give rise to the contradiction, as is demonstrated. Paradoxically, the elimination of perfect competition and its supporting canon comes at no cost to a scientific, that is, empirically oriented economics.

Key words::

• mathematical methods
• microeconomics
• theory

JEL classifications::

• B41
• C00
• D00

Notes

¹In the interests of both focus and brevity, I have excised part of the Knight quotation as given by Stigler and also bold-faced the concepts in it that are of particularly methodological interest here.

²Robinson (1965, p. 51) treats the concept as a template that guides theorizing rather than a constitutive element within the resulting theory. Chamberlin (1933 [1948], pp. 6, 25) distinguishes “pure” from “perfect” competition but characterizes the latter only negatively and rather loosely.

³This line of argument was greatly stimulated by Chomsky (1965, 1975 [1957]).

⁴The argument here rests on the work of the German mathematician, Georg Cantor who first systematically distinguished the existence of and the properties of different kinds of infinites (Cantor, 1952 [1915], pp. 103–110). For an equivalently authoritative but more reader-friendly account, one may consult the still classic Courant and Robbins (1969 [1941], pp. 77–86).

⁵From the very first days of my very first economics course I was troubled by the insouciance with which the professor drew mighty consequences from the supply/demand diagrams he kept sketching on his blackboard. But how could one belie the evidence of one’s eyes? For the exceptionally formidable assumptions demanded by preference orders, thus by extension even the most conventional supply/demand curves, see, for example, McDermott (2011, pp. 176–177).

⁶Walras’s (1977 [1874]) device of the auctioneer who takes in and sorts out all the possible bids and offers in the market, is an effective narrative device but it also obscures the very considerable logical-mathematical leap from an iterative procedure of bids and offers being received and processed one by one to the continuum of all possible bids and offers being processed simultaneously. It is the latter that is needed.

⁷The construction to follow has some elements of “realism” in it but, following Friedman (1951), they are unnecessary though they may, as here, serve heuristic purpose.

⁸We assume here that all transactions are money denominated; the assumption is harmless in the present context, merely allowing for simpler exposition.

⁹This assumption underlines that one of the normal, even customary formulations offered for a perfect market is methodologically unsatisfactory. By postulating an “indefinite number” of transactors, we intend that each of them will have but a “negligible effect” on prices/quantities. But we need for each of them to have a “mathematically negligible effect” on prices/quantities; thus we need an infinite number. Stigler’s appeal to “intuitive induction” (1957, p. 7, para. 3) appears to recognize this. On the other hand, his repeated use of the prefix indefinite throughout the article introduces some unnecessary ambiguity.

¹⁰Because of the function f(x, y), greater than or equal to, two of its markets may be different (i.e., have at least one different transactor) but share the same location in the series.

¹¹This step rests on Cantor (1952 [1915], p. 108, theorem D).

¹²According to Arrow and Hahn (1971, p. 1), “The notion that a social system moved by independent actions in pursuit of different values is consistent with a final coherent state of balance, and one in which the outcomes may be quite different from those intended by the agents, is surely the most important intellectual contribution that economic thought has made to the general understanding of social processes.”

Additional information

Notes on contributors

John F.M. McDermott

John F.M. McDermott is professor emeritus of social sciences at the State University of New York, College at Old Westbury.