Two results showing the limitations of the “as if” methodology are proved under relatively mild assumptions. In an interpretation of the results, a competitive market cannot simulate the outcome of a market M in which the single price assumption does not hold. In a second interpretation, the market M resulting from the aggregation of a finite number of competitive markets is not competitive. In both cases there is no ground to sustain the fiction that M operates as if it were competitive.“As if” methodology; Competitive market; Market aggregation; Market simulation; Price aggregation.

Antonio Quesada

English

Research Papers in Economics

-1-
Competitive markets and “as if” methodology
Antonio Quesada
†
Departament d’Economia, Universitat Rovira i Virgili, Avinguda de la Universitat 1, 43204 Reus, Spain
1st April 2005
28.10
Abstract
Two results showing the limitations of the “as if” methodology are proved under relatively mild
assumptions. In an interpretation of the results, a competitive market cannot simulate the outcome of a
market M in which the single price assumption does not hold. In a second interpretation, the market M
resulting from the aggregation of a finite number of competitive markets is not competitive. In both cases
there is no ground to sustain the fiction that M operates as if it were competitive.
JEL Classification: D40, E10
Keywords: “As if” methodology; Competitive market; Market aggregation; Market simulation; Price
aggregation.
                                                
† E-mail address: aqa@urv.net. Financial support from the Spanish Ministerio de Educación y Ciencia
under research project SEJ2004-07477 is gratefully acknowledged.-2-
1. Introduction
The term “‘as if’ methodology” (AIM) has come to define the methodological position
expressed in Friedman (1953). For a critical review of Friedman’s position, see
Caldwell (1980; 1982, Ch. 8); for a defence, see Boland (1979; 1982, Ch. 9) and
http://www.sfu.ca/~boland/summation.PDF.
In essence, the basic principle of AIM is that everything should be subordinated to the
goal of obtaining a successful prediction; for a recent justification of AIM, see Nordberg
and Røgeberg (2003). Friedman (1953) himself admits restrictions to the application of
AIM: while considering the problem of determining the effect on retail prices of
cigarettes of an increase in the federal cigarette tax, he ventures to predict that “broadly
correct results will be obtained by treating cigarette firms as if they were producing an
identical product and were in perfect competition”, whereas he contends this
presumption to be a false guide under the existence of price controls.
Motivated by this example, the aim of this note is to make apparent stronger limitations
of AIM when attempting the “simulation” or “replication” of the outcomes of a non-
competitive market through a competitive market. In fact, as in Friedman’s example,
textbooks at an elementary and even intermediate levels typically apply the competitive
market model to the analysis of real markets that are far from being competitive. In
addition, the outcome of a competitive market (market equilibrium) is also typically
justified, even in more advanced textbooks, by resorting to mechanisms inconsistent
with the competitive market structure.
For instance, Mas-Colell, Whinston and Green (1995 p. 315) justify equilibrium in a
competitive market using arguments that violate the price-taking assumption. Similarly,
Kreps (1990, p. 264) suggests four conditions describing how a market operates in order
to define a competitive market. Implicit in those conditions is the presumption that each
seller is able to set the price at which he is willing to exchange. This presumption is in
principle at odds with the fact that a single price regulates all the exchanges in a
competitive market. It is nonetheless claimed that the conditions ensure the existence of
a single price in the market. In both cases, AIM seems to be implicitly in use.
Those considerations, and the nature itself of AIM, point to the problem of ascertaining
the extent to which links between competitive and non-competitive markets can be
consistently established. This note deals with what appears to be a favourable case for
the application of AIM: a real market for a certain commodity is assumed to be-3-
accurately represented by a non-competitive market M which differs from a competitive
market essentially only in that the single price assumption fails. This appears to be the
situation Friedman considers in his example, as he invokes a competitive market to
explain what happens to retail prices in an actual market. The two results in this note
show that, under relatively mild conditions, it is not possible for a competitive market m
to consistently replicate the outcomes of M because there is no way to translate the
information these outcomes express in terms that are meaningful in m. Section 2 defines
the framework in which the results are obtained; these are stated and proved in Section
3; and briefly discussed in Section 4.
2. Framework
Let W be a non-empty set of states and N = {1, ... , n}, with n ³ 2, a finite set whose
members designate producers-sellers of a given commodity C in a certain non-
competitive market M (true representation of a real market). For each i Î N, there are
functions pi : W ® Â+ and qi : W ® Â+, with Â+ being the set of non-negative real
numbers. The interpretation is that, in state w Î W, i Î N produces qi(w) units of C and
sets price pi(w) for each one of those qi(w) units of C. In trying to conceptualize M as
close as possible to a competitive market, it is presumed that producers believe M to be
competitive but, having incomplete information, do not know with full accuracy which
is the market price. Thus, in state w, producer i resorts to available information to
compute an estimate of what he believes to be the market price; once he estimates price
pi(w), he merely ascertains what is the amount of production qi(w) = si(pi(w)) that his
(standard) supply function si indicates that must be produced at price pi(w).
For the purposes at hand, this framework can be simplified to a structure (N, D, (si)iÎN),
where D Í Â+ is a domain of possible prices and, for i Î N, si : D ® Â+ is a standard
individual supply function. The structure (N, D, (si)iÎN) is supposed to provide true data
from the working of M. Specifically, at a given state of the world, M generates two
observable n-tuples, a vector price (p1, … , pn) and a supply vector (s1(p1), … , sn(pn)).
AIM is then invoked to associate with M a competitive market m in such a way that a
correspondence can be established between the outcomes of M and those of m. Since M
and m can be viewed as generating information in different languages, the
correspondence defines the translation rules between languages. Without such a
correspondence, it is not possible to determine whether treating M as if it were m makes
m yield “broadly correct” predictions in M: if the increase of a tax in M raises some
prices but lowers some others, it is necessary some rule to clarify whether this means an-4-
increase or a reduction in “the” price in order to compare the result with what occurs in
m, where there is a unique price.
The first problem is then to define a merging function P : Dn ® Â+ that constructs a
price level P(p1, … , pn) for M from the corresponding vector price (p1, … , pn) in M. If
m is assumed to simulate or replicate M (or, simply, be the “as if” counterpart of M)
then this P would actually embody the way to associate price vector (p1, … , pn) from M
with price p = P(p1, … , pn) in m. But once prices from M and m have been linked, a
second problem emerges: everything determined by prices (quantities supplied) should
also be linked in a way consistent with the price link. In particular, it is necessary to
define a second merging function S : rang(P) ® Â+ such that S(P(p1, ... , pn)) = åiÎN
si(pi), where rang(P) = {x Î Â+: for some x Î Dn, P(x) = x}. Just as P defines a single
price for M (the one supposed to be replicated by m), S defines the (hypothetical) market
supply function that associates with price level P(p1, ... , pn) the total amount of
production generated under price vector (p1, ... , pn), that is, the amount åiÎN si(pi).
Accordingly, the links between M and m are at least two. On the one hand, when M
generates the price vector (p1, … , pn), m must generate the price P(p1, … , pn). On the
other, when M generates the output vector (s1(p1), … , sn(pn)), m must generate the
same total output åiÎN si(pi) and make it equal to the corresponding total output S(P(p1,
... , pn)).
In this interpretation, it is possible to treat M as if it were m provided there are such
functions P : Dn ® Â+ and S : rang(P) ® Â+. It is through P and S that the “as if” fiction
operates, acting as a procedure to translate M into m or, more precisely, to make m
capable of simulating or replicating outcomes of M. Hence, the way M forms prices and
determines the total supply could be simulated by a competitive market, so that, under P
and S, it could be argued that M operates as if there were a single price.
In this respect, it could be claimed that there exists a competitive “rationalization” of the
non-competitiveness of M: the departure from m consisting in the existence of several
prices for the commodity can be accommodated through AIM by defining P and S. The
existence of such functions would make m pass a “robustness” test: the “perturbed”
market M obtained from the competitive market m by dropping the single price
assumption remains, in a way, competitive. This robustness interpretation would be
related to results like the one showing that a competitive market can be thought of as the
limit of certain oligopolistic markets; see Mas-Colell, Whinston and Green (1995, pp.
411-412).-5-
3. Results
There are plausible requirements to be imposed on P, S and the functions si. Proposition
3.1 suggests four such conditions and shows that no P and S can be found that satisfy all
of them, together with the primary “as if” condition S(P(x)) = åiÎN si(xi) for all x Î Dn.
Condition (i) is a unanimity property: for all p Î D, P(p, ... , p) = p. This is reasonable
under the presumption that P selects a representative price for every price vector, for in
the case in which all producers set the same price p, p itself appears to be the best
candidate to represent the price vector (p, ... , p).
Condition (ii) means that it is not always worth while to produce: for every producer
there is some price inducing the producer to leave the market (he does not produce).
Condition (iii) is in part symmetric with respect to (ii) and expresses the existence of
competitors: there is some price inducing at least two producers to be in the market.
Finally, condition (iv) holds that if, for some price vector x Î Dn, only one producer is
in the market then this producer’s price should count as “the” market price. The
justification is that when there is only one producer serving the market, the prices the
rest of potential producers would have liked to receive is irrelevant information to
define a representative price for the commodity. For i Î N, x Î D and x of Dn, define
(xi, x–i) to be the member of z Î Dn such that zi = x and, for all j Î N\{i}, zj = xj.
Proposition 3.1. Let N = {1, ... , n}, with n ³ 2, and D Í Â+ be non-empty sets. For all i
Î N, suppose there are functions si : D ® Â+, P : Dn ® Â+ and S : rang(P) ® Â+. If
(i) for all p Î D, P(p, ... , p) = p
(ii) for every i Î N there is p Î D such that si(p) = 0
(iii) there are p Î D, i Î N and j Î N\{i} such that si(p) ¹ 0 and sj(p) ¹ 0 and
(iv) for all i Î N and x Î Dn, if sj(xj) = 0 for all j Î N\{i} and si(xi) ¹ 0 then P(x) = xi
then it is not the case that
(v) for all x Î Dn, S(P(x)) = åiÎN si(xi).
Proof. Assume (i)-(v). By (iii), let p Î D, i Î N and j Î N\{i} satisfy si(p) ¹ 0 ¹ sj(p).
By (ii), there is z Î Dn such that, for all k Î N, sk(zk) = 0. By (v), S(P(pi, z–i)) = åkÎN\{i}
sk(zk) + si(p). By (iv), P(pi, z–i) = p. Hence, by (ii), (iii) and (iv), S(p) = S(P(pi, z–i)) =-6-
åkÎN\{i} sk(zk) + si(p) = si(p) ¹ 0. By (i), S(p) = S(P(p, ... , p)). As S(p) = si(p), by (v),
S(P(p, ... , p)) = åkÎN sk(p). Thus, si(p) = åkÎN\{i} sk(p) + si(p), so åkÎN\{i, j} sk(p) + sj(p)
= 0. Since, for all k Î N and q Î D, sk(q) ³ 0, it follows that sj(p) = 0, contradicting sj(p)
¹ 0.v
Proposition 3.2 states another impossibility result when the following three conditions
are added to “S(P(x)) = åiÎN si(xi) for all x Î Dn”. Condition (i) is, as in Proposition 3.1,
the unanimity principle. Condition (ii) requires from P to restrict its values to the set D
of prices that producers can set. The converse, D Í rang(P), follows from the unanimity
principle. By condition (ii), every price that could count as a price set by some producer
can also count as a market price and vice versa. Finally, condition (iii) asserts that there
is some price in D encouraging some producer to produce and, moreover, that either (a)
there is a second producer producing for some price or (b) that the total supply function
does not coincide with the individual supply of some producer, so that no producer can
always be identified with the total supply of the market. Condition (iii) again expresses
the existence in some case of a minimal amount of competition in the market.
Proposition 3.2. Let N = {1, ... , n}, with n ³ 2, and D Í Â+ be non-empty sets. For all i
Î N, suppose there are functions si : D ® Â+, P : Dn ® Â+ and S : rang(P) ® Â+. If
(i) for all p Î D, P(p, ... , p) = p
(ii) rang(P) = D and
(iii) there are q Î N and x Î Dn such that sq(xq) ¹ 0 and
(a) either there are k Î N\{q} and z Î Dn with sk(zk) ¹ 0
(b) or there is no k Î N such that sk = S
then it is not the case that
(iv) for all x Î Dn, S(P(x)) = åiÎN si(xi).
Proof. Assume (i)-(iv). By (iii), let q Î N and x Î Dn satisfy
sq(xq) ¹ 0. (1)
Choose j Î N\{q} and z Î Dn such that zq = xq. With z := zj, let f : D ® rang(P) be the
function such that f(x) = P(xj, z–j). Clearly, P(z) = f(z). By (i), S(f(z)) = S(P(f(z), ... ,
f(z))). By (iv), S(P(f(z), ... , f(z))) = åiÎN si(f(z)). By (iv), S(f(z)) = S(P(z)) = åiÎN\{j} si(zi)
+ sj(z). Hence, åiÎN si(f(z)) = åiÎN\{j} si(zi) + sj(z). Set a := åiÎN\{j} si(zi), so-7-
sj(z) = åiÎN\{j} si(f(z)) + sj(f(z)) – a. (2)
By (ii), f(z) Î D. When f(z) replaces z, the preceding reasoning leads to conclude that
sj(f(z)) = åiÎN\{j} si(f2(z)) + sj(f2(z)) – a    (3)
where f2(z) = f(f(z)). Inserting (3) into (2) yields sj(z) = åiÎN\{j} [si(f(z)) + si(f2(z))] +
sj(f2(z)) – 2a. By (ii), f2(z) Î D and the same reasoning yields sj(z) = åiÎN\{j} [si(f(z)) +
si(f2(z)) + si(f3(z))] + sj(f3(z)) – 3a. Since, for all r ³ 1, fr(z) Î rang(P), it follows that,
for all r ³ 1, sj(z) = g(z, r) – ra, where g(z, r) := å1
￿
t
￿
r åiÎN\{j} si(ft(z)) + sj(fr(z)). Given
that sj(z) is a number independent of r, the limit of g(z, r) – ra as r goes to infinity is the
constant value sj(z). If a = 0 then 0 = a = åiÎN\{j} si(zi); in particular, since the values of
each si are non-negative, sk(zk) = sk(xk) = 0, contradicting (1). If a ¹ 0 then, in view of
(1), the sequence {ra}r³1 approaches infinity, so the limit of g(z, r) – ra as r tends to
infinity must be 0. Since both j Î N\{k} and zj = z were arbitrary, for all j Î N\{q} and x
Î D, sj(z) = 0. This contradicts (a). If it is (b) that holds then, by the preceding result, (i)
and (iv), for all x Î D, S(x) = S(P(x, ... , x)) = åiÎN si(x) = sq(x), contradicting (b).v
4. Comments
Propositions 3.1 and 3.2 identify two situations in which the way a non-competitive
market M operates cannot be replicated or simulated by a competitive market m and,
therefore, one cannot have recourse to the fiction that it is possible to deal with markets
like M as if it were competitive. In addition, the non-existence of P and S seems to
remove the possibility of a non-competitive interpretation of the way a competitive
market is supposed to work: without P and S, there does not appear to be room to
contend that M operates as if there were a single price determining total supply.
The same formal results can be interpreted in a different context and for a different
problem. Suppose the aim is to ascertain whether n competitive markets for the same
commodity can, from a theoretical point of view, be treated as if they constituted a
unique competitive market. In this case, members of the set N could designate
geographically scattered competitive markets for the same commodity. For i Î N, pi
would be the equilibrium price in competitive market i and si(pi) would be the total
supply in market i at market price pi. The question is then whether those n markets can
be merged to form one competitive market that reproduces the result of aggregating the
outcomes of the n markets. This problem is formally identical to the one described in-8-
Section 2: the construction of the summarizing market would require a single price
P(p1, ... , pn) and a competitive supply function S such that S(P(p1, ... , pn)) = åiÎN si(pi).
Propositions 3.1 and 3.2 establish conditions under which an all-embracing competitive
market cannot represent independent competitive markets. Under the assumed
conditions, these results make it difficult to sustain the fiction that the market resulting
from the aggregation of competitive markets operates as if it were competitive.
References
Boland, L. (1979): “A Critique of Friedman’s Critics”, Journal of Economic Literature
17, 503–522.
Boland, L. (1982): The Foundations of Economic Method, George Allen & Unwin,
London.
Caldwell, B. (1980): “A Critique of Friedman’s Methodological Instrumentalism”,
Southern Economic Journal 47, 366–374.
Caldwell, B. (1982): Beyond Positivism: Economic Methodology in the Twentieth
Century, George Allen & Unwin, London.
Friedman, M. (1953): “The Methodology of Positive Economics”, in Essays in Positive
Economics, Chicago University Press, Chicago, 1-43.
Kreps, D. (1990): A Course in Microeconomic Theory, Harvester Wheatsheaf, New
York.
Mas-Colell, A., Whinston, M. D., and Green, J. R. (1995): Microeconomic Theory,
Oxford University Press, Oxford.
Nordberg, M. and O. Røgeberg (2003): “Defence of Absurd Theories in Economics”,
Working Paper 2003:18, University of Oslo.

A Course in Microeconomic Theory, Harvester Wheatsheaf,

A Critique of Friedman’s Critics”,

Beyond Positivism: Economic Methodology in the Twentieth Century,

Defence of Absurd Theories in Economics”, Working Paper 2003:18,

Microeconomic Theory,

The Methodology of Positive Economics”,

Competitive markets and “as if” methodology

http://128.118.178.162/eps/mic/papers/0504/0504003.pdf

Competitive markets and “as if” methodology

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics