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promoting in this book says the only way we learn is through criticism; and
or activity analysis and these do not require such restrictive assumptions.
of course, testing is one form of criticism. Incomplete theories are very
Thus it would seem that if we are able to show that any one of Wald™s
difficult to criticize because they leave so much room for conceivable
conditions is not satisfied (in the ˜real world™) we do not necessarily refute
contradictions. Because I want to learn, I want to be able to criticize any
the original incomplete theory. From a methodological position, this state
theory, and attempting to complete a theory is an important means of
of affairs is rather perplexing. We may wish to complete an axiomatic
exposing a theory to decisive criticism. The unintended outcome of this
version of neoclassical price theory and then criticize it. But, if our
view of knowledge is that when we attempt to explain an economic
criticism deals only with those conditions which we add (for completion
equilibrium (such as Walras™) it is necessary to explain why all other
purposes), then we are not really criticizing the original price theory. Some
possible equilibrium positions are not obtained. In effect, this says we must
think this can be overcome by attempting to deduce testable statements
be concerned with uniqueness, since to be complete (and thus testable) our
from the incomplete theory and submitting these to tests. No matter how
explanation of any alleged equilibrium must not allow for other contrary
the theory is eventually completed, should any one of them be shown to be
situations such as ˜multiple equilibria™. 11 This view is contrary to the
false, the theory as a whole will be false “ otherwise, the apparent
popular myth (all too often promoted by those economists who ˜picked up
falsifying fact must be explained away! Either way, this is a very difficult
mathematics on the side™) that satisfying the calculus conditions of a ˜stable
task and not much has been attempted or accomplished so far. 10
equilibrium™ is sufficient to explain the equilibrium in question. A stable
The question of testability (or criticizability in general) is above all a
equilibrium structure (such as a negatively sloped demand curve and
logical problem. And since axiomatic analysis is concerned with the logical
positively sloped supply curve) is necessary, of course, but without
properties of a theory, it can have something to say about empirical
behavioural assumptions concerning price adjustment dynamics, we still
testability as well as being able to offer a means of theoretically testing a
have not explained why the system is in ˜equilibrium™ where it is. All that
theory. For example, we should probably view most of the theoretical
the calculus stability conditions accomplish is the avoidance of confusing a
analysis of neoclassical textbooks as failures of indirect attempts to test the
possibly unstable ˜balance™ situation with a stable equilibrium situation. I
completeness of the neoclassical theory (i.e. failures to show the
will return to the matter of the importance of stability conditions in Chapter
neoclassical theory to be incomplete). Actually, what we read in the
14.
textbooks should be viewed as the only aspects of the theory which are
considered complete (often only on the basis of apparent, but untested,
consistency). A THEORY OF COMPLETENESS
This disagreement in viewpoints is not just apparent. It would seem that
In spite of what economists think they are doing, they can be seen to have
few economists are directly concerned with completeness because most of
been indirectly concerned with completeness, and the evidence is the
them (implicitly or explicitly) view economic knowledge as a logical
development of neoclassical economic theory. One way to understand this
system which is supported by positive evidence. ˜Supported™ usually means
development on the basis of a theory of the development of theories is to
that at least some predictions (or propositions) that logically follow from
characterize all theories as systems of assumptions where each assumption
their theories have been verified or confirmed. An unintended outcome of
is in the logical form of an ˜all-and-some™ statement. As I briefly discussed
this view of knowledge is that most economists are satisfied with an
in Chapter 1, an ˜all-and-some™ statement is one of the form ˜for all x there
argument whenever it allows for the possibility of the truth of their theory
is some y such that ...™. The ˜such that ...™ clause may or may not be
even though the theory at the same time may imply propositions which are
completely specified depending on whether or not, and to what extent, the
false. For example, a model may have several solutions, one of which is
theory has been completed. Thus an attempt, such as Wald™s, to complete a
© LAWRENCE A. BOLAND
62 Principles of economics Axiomatic analysis of equilibrium states 63
since it is limited to the comparison of two points. A strong version would refer
model of a Walrasian theory is in effect an attempt to specify the ˜such that
to a chain of comparisons of many points [see Houthakker 1950, 1961]. None
...™ clauses of the theory. Whether an ˜all-and-some™ statement is
of the discussion in this book will require us to be concerned with this
empirically testable is a question of how the ˜such that ...™ clause has been distinction so I will not be emphasizing the ˜weakness™ of this axiom.
completed. It is always possible to complete a theory without making it 7 The arrowhead on the price“consumption curve indicates the direction along
testable; for example, by making it circular. 12 which the changing price increases for the given income and price of the other
good.
The specification of the ˜such that ...™ clauses is almost always ad hoc,
8 This interpretation of the Axiom of Revealed Preference will be the subject of
and so is the completion of an axiomatic system. The history of formal
Chapter 13.
model-building in neoclassical economics is one of a sequence of efforts to 9 Note Figure 4.5 can be used to represent two kinds of appropriate Z-lines
complete systems of ideas which rationalize certain enduring propositions. simply by swapping the X1 and X2 labels (and the P1 and P2 labels).
The specification of the nature of indifference curves by Hicks and Allen 10 Paul Samuelson has in effect attempted to deal with this in Chapter 5 of his
published PhD thesis [1947/65]. I have discussed his attempt in Boland [1989,
[1934], the specification of imperfect competition by Robinson [1933/69],
Chapter 1].
the specification of the idea of a market equilibrium by Samuelson
11 Whether multiple equilibria represent contrary situations depends on what we
[1947/65], and the attempts of Franco Modigliani [1944] and Donald are trying to explain. For example, if we were trying to explain the
Patinkin [1956] to explain Keynes, are all examples of developments in the price“quantity in market A and we found that it was compatible with various
neoclassical theory which amount to completions of ˜such that ...™ clauses. equilibria in market B, there would be no problem. But, if there are various
possible equilibria in market A allowed by our explanation of market A, then we
These are also examples of placing requirements on theories which are
have an incomplete explanation.
similar to requirements of typical axiomatic analyses.
12 To the statement ˜for every rationalizable choice there is a maximizing choice
If an axiomatic analysis of a theory manages to posit requirements ...™ we might add ˜such that if it is not a maximizing choice it is not
which are necessary for the sufficiency of any given model of that theory, it rationalizable™.
is an important achievement which should not be left only to mathematical 13 The axiom just happens to be the one used in Wald™s and others™ attempts to
formally analyze their invented models of neoclassical equilibrium. The role of
economists to pursue. Wald™s Axiom of Revealed Preference, for example,
this axiom in the formalization of neoclassical economics will be further
is such a requirement. Any requirement (or ˜condition™) that is necessary
explored in Chapter 13.
for the completion of a theory may offer an important opportunity for
critically testing that theory. However, the Axiom of Revealed Preference
by itself is not an essential element in economic analysis. 13 What is
essential in neoclassical economics is the notion of a state of equilibrium.
In the next chapter I examine other ways to view equilibrium analysis.

NOTES
1 Of course, there are some neoclassical economists who even put the existence
of a state of equilibrium beyond question.
2 This type of analysis began in the nineteenth century with studies of the
axiomatic structure of Euclid™s geometry [see Blanch© 1965].
3 Many other axiomatic studies have been published since Wald™s, for example
Arrow and Debreu [1954], Arrow and Hahn [1971], Debreu [1959, 1962], Gale
[1955], McKenzie [1954, 1959].
4 Note well that he does not say only if.
5 Specifically, by replacing it with a duality assumption [see Kuhn 1956]. It
should be noted that Wald recognized the possibilities of using other
mathematical techniques which did not require such a condition. See Quirk and
Saposnik [1968] for a survey of the other well-known axiomatic studies of
Walrasian economics.
6 Today, Wald™s condition is called the Weak Axiom of Revealed Preference
© LAWRENCE A. BOLAND Axiomatic analysis of disequilibrium states 65
analyzing the logic of the situation facing the producers. 2 In this chapter, I
5 Axiomatic analysis of will follow this tradition by focusing on the theory of the individual
disequilibrium states producer to determine how the logic of the situation facing the firm may be
used to account for any state of disequilibrium.

COMPETITION BETWEEN THE SHORT AND LONG RUNS
In regard to the theory of the firm facing a general equilibrium situation, I
want to examine the role played by two particular assumptions. One is the
assumption that prices are fixed givens which in turn is based on an
assumption that the firm is a ˜perfect competitor™ (perhaps because it is too
small to be able to affect its price by altering the supply). I wish to show
The theory of stable equilibrium of normal demand and supply helps why dropping the fixed-price assumption would severely restrict our choice
indeed to give definiteness to our ideas; and in its elementary stages it of assumptions regarding other aspects of the firm. The other assumption to
does not diverge from the actual facts of life, so far as to prevent its
be examined is one concerning the applicability of the assumption of profit
giving a fairly trustworthy picture of the chief methods of action of the
maximization. In Chapter 3 I noted that Marshall defined a short run where
strongest and most persistent group of economic forces. But when
everything but the input of labour and the level of resulting output are
pushed to its more remote and intricate logical consequences, it slips
away from the conditions of real life. fixed. At the other extreme is his long run where everything but technology
Frank Hahn [1973, p. 1] is variable (and thus subject to his Principle of Substitution). Here I will
examine what might transpire in the shadowy area between Marshall™s
short and long runs, that is, in what I will call the intermediate run. The
While the axiomatic analysis of equilibrium models can determine whether
distinction between the Marshallian runs is solely a matter of the time
a given model is consistent and complete, little analysis has been done
available in the period under consideration and a recognition that some
concerning consistency and completeness of models of disequilibrium
inputs are easier to change than others (i.e. change takes less time). In
states.1 Obviously, we cannot expect to be able to assess solvability as a
Marshallian terms (i.e. assuming just two inputs, labour and capital 3) the
means of assuring consistency since, as discussed in Chapter 4, the
question is the speed by which capital can be physically changed. While it
solutions of the equilibrium models were sets of equilibrium prices that
is commonplace to define the short run as a period of time so short that
could be used possibly to explain existing prices. In this chapter I will offer
there is not enough time to change capital, the long run presumes that both
a few elementary axiomatic analyses of models of ˜disequilibrium™ states.
inputs are unrestrictedly variable. Now, the purpose of recognizing an
Eventually, we will need to consider how they may be used to critically
intermediate run is to recognize that there are two ways of changing
assess any axiomatic analysis of disequilibrium models.
capital, internally and externally. The period of time corresponding to the
There are two ways to use disequilibrium models. One is to explain why
intermediate run is defined to be too short to allow wholesale changes in
disequilibrium phenomena occur and the other is to explain away
the physical type of the capital used in the firm but long enough to allow
disequilibrium phenomena as mere appearances. Both utilize underlying
the firm to vary internally the quantity of the existing type of capital used.
equilibrium models in which it is assumed that all consumers are
In the intermediate run the firm must decide upon the optimum quantity of
maximizing utility (either directly or indirectly by maximizing personal
capital. In the long run, however, there is sufficient time to change to a
wealth) subject to given equilibrium prices and all producers are
different type of capital as is usually the case when a firm switches from
maximizing their profit subject to given technology and given market
one industry to another. Thus, in the long run the firm must decide upon the
equilibrium prices.
optimum type of real capital.
Since virtually all neoclassical equilibrium models take for granted that
One reason why many theorists wish to drop either the perfect-
there are no barriers to any consumer quickly responding to changing
competitor assumption or the profit-maximizer assumption is simply that
prices, if there is a state of disequilibrium, such a state will be found by
these assumptions in many cases are ˜unrealistic™ in disequilibrium models.
© LAWRENCE A. BOLAND
66 Principles of economics Axiomatic analysis of disequilibrium states 67
Some just complain that these assumptions are plainly ˜unrealistic™ in the reached once the optimum amount of labour has been hired. The necessary
sense that it would be realistic to assume that the firm is a perfect condition for this is that the price of the good being produced equals its
competitor only when there are an extremely large number of firms, each marginal cost (MC ) or, in terms of the decision concerning labour, that the
of which is relatively small “ for example, an economy of ˜yeoman marginal physical product of labour (MPP L ) equals the real cost of one unit
farmers™ or perhaps an economy consisting of only small businesses. A of labour. Specifically, the existence of a short-run equilibrium assures us
small firm has to take its product™s price as given only because it will go that MC = Px or MPPL = W/Px (where the good produced is X and the
out of business if a higher price is charged since its customers can go to prices of X and labour are, respectively, P x and W). Given a price of capital
any of the large number of competing firms. Similarly, if it charges less (Pk ), an intermediate-run equilibrium assures that the optimum quantity of
than the given price when the given price is the ˜long-run equilibrium capital has been utilized such that the marginal product of capital (MPP K)
price™ (which equates with average cost) then it will be losing money and equals the real cost of capital (P k /P x ). And since the intermediate run is
will still eventually go out of business. It is thus said that with a large longer than the short run (i.e. there is sufficient time to satisfy both sets of
number of small firms competition can be ˜perfect™. conditions), we can also be assured that the marginal rate of technical
Would-be ˜realists™ argue that the modern economy consists of rela- substitution (MRTS) between labour and capital equals the relative costs of
tively large firms or few firms in each industry (or both) and thus, they say, those inputs (W/Pk ). Except when we limit the notion of a production func-
in the real world there is ˜imperfect™ competition. Imperfect competition tion to the special case of linear-homogeneous production functions, we
allows for two possible circumstances. First, it is possible for the firm to be will see that the attainment of an intermediate-run equilibrium does not
a price-taking ˜competitor™ and also be one of a few producers such that assure a long-run equilibrium. Specifically, an intermediate-run equilib-
changes in its output do affect the market-determined equilibrium price. rium will not assure us that total profit is zero. The absence of zero total
The second is to assume that the firm is a price setter such as the usual profit means that there may be an incentive for new entries or exits and
textbook™s monopolist. The first approach will be the one adopted here thereby means that there may be incentives which deny an equilibrium
since it does not require the producer to know the full nature of the demand state (since there is sufficient time for such reactions).
curve facing the firm. The second approach can be considered a special Most textbooks go straight to the long-run equilibrium from the short-
case of the first “ namely where the firm™s demand curve is the market™s run equilibrium. That is, they go from where, while MPP L = W/P x , it is
possible that MRTS ≠ W/Pk (since not all short-run equilibria are long-run
demand curve and the firm has full knowledge of the market.
equilibria) to a long-run equilibrium where MRTS = W/P k and TP = 0. It is
interesting to note that the long-run equilibrium is the starting point for an
THE ˜PERFECT-COMPETITOR™ FIRM IN THE LONG RUN: A
Adam Smith type of philosophical discussion of the virtues of competition
REVIEW
and self-interest. That is, if every firm is making ˜zero profits™ with the
In order to examine the axiomatic role of the assumptions of the given production functions (i.e. given technology) the only way a firm can
Marshallian theory of the firm, we need to discuss the effect that dropping obtain positive ˜excess™ profits is to develop new cost-reducing technolo-
the perfect-competitor assumption would have on equilibrium models and gies. In the absence of competition such ˜greed™ (in this case, the pursuit of
in particular on the assumptions concerning the production function. Before extra profits) would mean that one firm might gain at the expense of others,
we drop this assumption, however, let us review the basic logic of the but if we also have ˜free enterprise competition™ any improvements in
perfect-competitor firm with respect to its production function. productive efficiency which reduce costs will eventually be shared by all
Since by definition the intermediate run involves less time than the long the firms and thus benefit everyone through lowered prices.
run, it can be argued that a long-run equilibrium must also be an intermedi- All this seems to be taken for granted or ignored in most textbooks.
ate-run equilibrium and similarly it must also be a short-run equilibrium. Everyone seems to be satisfied with discussing only the necessary
Most important in the recognition of the intermediate run is the separation properties of the long-run equilibrium “ as if there were virtue in zero
of the zero total profit idea (TP = 0) from the idea of complete profit profit itself! There is some virtue to having the lowest possible price for a
maximization (i.e. with respect to all inputs). To do this we need to recog- given technology but it leaves open the question from a broader perspective
nize the explicit conditions necessary for each of the three types of equilib- of the choice of optimal production or the optimal ˜quality™ of capital and
ria. In the short run, since only labour can be varied, an equilibrium is its associated technology.
© LAWRENCE A. BOLAND
68 Principles of economics Axiomatic analysis of disequilibrium states 69
What the recognition of an intermediate-run equilibrium allows is the assures that, given Px (as well as W and Pk ), whenever the firm is internally
discussion of situations where profit is maximized with respect to all inputs maximizing profit with respect to both labour and capital, the following
but TP ≠ 0. The basis for this discussion is that while zero profit is due to two equations are true:
decisions which are external to the firm, the efficiency of production MPPL = W/ Px [5.2a]
(MRTS = W/Pk ) is due to an internal decision whereby profit is maximized MPPK = Pk / Px . [5.2b]
with respect to all inputs. The intermediate run is often ignored because the
Now, the combination of [5.1], [5.2a] and [5.2b] leads to the following:
properties of the long-run equilibrium are considered more interesting “
X = (W/ Px )…L + (Pk / Px )…K [5.3]
usually, this is because they are mathematically determinant and thus
available for applications. Unfortunately, the long-run equilibrium or rearranged by multiplying both sides by P x :
Px…X = W…L + Pk…K.
conditions are considered so interesting that models of the firm are [5.3′]
designed to guarantee that it is logically impossible to have an
The left side of [5.3′] is total revenue (TR) and the right side is total cost
intermediate-run equilibrium which is not a long-run equilibrium. I shall
(TC), hence it implies TP = 0. This means that in the usual long-run model,
now show how this is done and as well show how such models are also
with its typical everywhere-linear-homogeneous production function,
incompatible with imperfect competition.
intermediate-run equilibrium implies all necessary conditions of long-run
equilibrium. That is to say, one cannot obtain an intermediate-run
PROFIT MAXIMIZATION WITH CONSTANT RETURNS TO equilibrium without obtaining the necessary conditions for a long-run
SCALE equilibrium of the firm.
Given that we try to explain to students the importance of competition
The basic ingredient of long-run models of the firm is the assumption that

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