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particular, they depend on the assumptions made about the relevant eighteenth century problem was the elimination of both the monarchy and

exogenous variables. Changes are explained only as the effects of changed the Church. This revolutionary social programme collapsed in Europe with

conditions. the failures of the French Revolution. Although in many ways this

Again, unless the changeability (or fixity) of the â€˜conditionsâ€™ is programme lived on in the economic principles of the Classical School as

explained, the Marshallian method of explanation runs the risk of profound well as in the Americansâ€™ Declaration of Independence, those intellectuals

circularity. Circularity might be avoided by adopting the Walrasian disappointed with the failures of classical Rationalism hastily retreated

approach, but doing so would only risk an infinite regress. 5 Moreover, the from the objective world of â€˜reasonable menâ€™ to the Romantic worlds of

completion of the Walrasian programme of representing the economy with subjective psychology, poetry and introspection.

a set of simultaneous equations turns out to depend intimately on the math- In this sense it is easy to see how many intellectuals identified the

ematical form of those equations. Thus, where Marshallâ€™s programme runs classical school of economics with the failure of classical rationalism and

the risk of circularity, Walrasâ€™ programme runs the more obvious risk of thus economic analysis was considered suspect in many circles. The

arbitrariness if one does not attempt to explain oneâ€™s choice of hypoth- shortcomings of the subsequent Romantic view were not so apparent

esized mathematical forms. during most of the nineteenth century. Yet Marshall rejected Jevonsâ€™

Romantic theory of value (which was based on demand rather than supply)

because in Marshallâ€™s eyes this was probably seen as a retreat from one

MARSHALLâ€™S REJECTION OF MECHANICS AND

extreme (namely, exclusive mechanics of supply) to another extreme

PSYCHOLOGY

(namely, exclusive mechanics of demand). Later, Keynes, dissatisfied with

Summarized this way, Marshallâ€™s research programme sounds rather Marshallâ€™s neoclassical economics, was to go all the way. In order to reject

mechanical. Marshall states that he wishes to avoid identifying economics the mechanics of classical economics, Keynes endorsed a psychological

basis for all businessmenâ€™s decision-making. 6 But the methodological

with the immutable laws of physics [p. 37]. Yet he thinks economics can be

more rigorous and less subjective than the â€˜scientificâ€™ study of history. In question here is whether the rejection of mechanics necessarily entails the

effect, he sees biology as an intermediate stage on a continuum between espousal of subjective psychology. Clearly, Marshall opted for a more

inexact, subjective historical studies at the one extreme and precise, liberal compromise.

Â© LAWRENCE A. BOLAND

42 Principles of economics Marshallâ€™s â€˜Principle of Continuityâ€™ 43

A psychological basis for decision-making would seem too much like assume the existence of a market equilibrium whenever we are also

the â€˜immoral hedonismâ€™ often identified with the Benthamite programme assuming universal maximization, we need only consider the contrary

of explanation where all human behaviour is considered to be the conse- implications of the non-existence of a market equilibrium. Whenever there

quence of utility maximization. The major problem with psychologistic is excess demand, some of the demanders are unable to maximize due to an

explanations is that they presume an immutable â€˜human natureâ€™ â€“ for insufficiency of supply at the going price. Such a disequilibrium in the

example, permanently given tastes. John Stuart Millâ€™s Principles came very market would thus deny universal maximization. And thus, when it is

close to being such a theory of human behaviour. As Marshall saw this, the assumed that everyone is a maximizer, disequilibria are logically

precluded.7

difficulty was not maximization, but rather the view that human nature is

immutable. If human nature were immutable there would be little reason The more general assumption of the existence of a competitive

for social or economic change. To a Victorian scientist, the immutability of equilibrium meets a similar fate simply because the assumption of a

the human character was unthinkable. In summary, Marshall saw additional competitive equilibrium implies the absence of excess profits; that is, it

significance in the support his biological analogies gave to his discussion implies the absence of any reason to exit one industry and enter another. It

of continuity. He embraced biology because evolutionary biological is easy to show that whenever Marshallâ€™s Principle of Continuity is

analogies were the obvious and most palatable alternative to mechanical or applicable (such that all relevant factors of production are variable), total

hedonistic theories of economics and society. revenue must equal total costs if it is also assumed that all the factors are

paid their marginal product. First, whenever a price-taking firm is

maximizing its profit with respect to every factor, it must be paying each

COMPREHENSIVE MAXIMIZATION MODELS

factor its marginal product. Second, whenever all factors are variable,

Keynes identified Marshall with the mechanistic Classical School. Eulerâ€™s theorem is applicable: output equals the weighted sum of all the

Disagreement would be difficult on the sole basis of Book V of the input factors, each weighted by its respective marginal product. Putting

Principles. But Marshall insisted that mathematical models of dynamics these two considerations together, we see that whenever all factors are

(and hence mechanics) would be inappropriate [pp. 382 and 637]. variable there must be constant returns to scale and thus paying factors

Nevertheless, Marshallâ€™s protestations notwithstanding, it is easy to see that their marginal product in order to maximize profit will exhaust the output.

all economic behavioural assumptions can be reduced to maximization (or In other words, whenever the Principle of Continuity applies, universal

minimization). profit maximization precludes excess profit. Thus we can see that there is

To see how the idea of equilibrium can be reduced to one of universal no need to add an assumption which asserts the existence of a competitive

maximization alone, consider the two most common assumptions regarding equilibrium if we are already assuming universal maximization as well as

equilibrium: (1) the assumption of the existence of a specific market equi- assuming that all factors are variable!

librium and (2) the assumption of the existence of a general competitive These considerations would seem to lend considerable support to those

equilibrium. It is easy to see that both can be shown to follow from the neoclassical economists who, by accepting that everything reduces to the

assumption of successful maximization alone. mathematics of maximization, wish to consider other territories to conquer

First, let us consider the elementary idea of a market equilibrium, that is, with their maximization hypothesis [e.g. Becker 1976; Stigler and Becker

of the existence of a price at which demand equals supply. There are two 1977]. Their research programme is rather straightforward. Every decision-

structural elements in any market: the demand curve and the supply curve. maker faces constraints and possesses an objective (utility) function and

In neoclassical economics, the demand curve is the dominant logical thus every equilibrium in society or an economy can be seen to follow from

consequence of utility maximization in the sense that the curve is the locus universal maximization. The theoristâ€™s task is only to describe the

of price and quantity combinations for which at any given price the constraints and the objective function which is consistent with the absence

indicated quantity is the total demand which results when every consumer of any incentive for change â€“ that is, for example, with zero excess profit

is maximizing utility while facing that price. Likewise, the supply curve and zero marginal profit. Thus, the appearance of imperfections in

indicates the consequence of profit maximization where for any given price competition can easily be explained away as the misperception of some

the curve indicates the total supply which is achieved when every firm is economic theorists who incorrectly calculate the transaction costs of

facing that price and is maximizing its profit. To see what it means to encouraging additional competition. That is, even the constraints facing all

Â© LAWRENCE A. BOLAND

44 Principles of economics Marshallâ€™s â€˜Principle of Continuityâ€™ 45

short-run maximizers can supposedly be explained as the consequences of Continuity that neoclassical models are relevant only if the Principle of

all individualsâ€™ maximization efforts by realistically assessing the cost of Continuity can be shown to apply.

further substitutions in the constraints.

Such a programme has been applied to unusual questions such as those

NOTES

concerning an optimal amount of charity, an optimal marriage contract, an

1 The systematic research programme based on the universal application of

optimal capital punishment or deterrent, an optimal institutional environ-

maximization is the explicit methodological agenda of neoclassical economics

ment, the optimality of being altruistic or even of voting, and so on. Of

which I discussed in Chapter 1.

course, one is free to do or assume anything one likes, even to attempt to 2 Note that this says that it is necessary for the sufficiency of any argument

explain everything as an effect of maximization. Intellectual honesty, how- employing the maximization assumption.

ever, seems to require that all the necessary conditions of maximization 3 The remainder of this chapter is based on an invited paper which appeared as

Boland [1990]. The copyrighted parts are reprinted here with the permission of

must be fulfilled. One of them is the requirement of a continuity of options.

lâ€™Institut de Sciences MathÃ©matiques et Ã‰conomiques AppliquÃ©es and Les

By giving prominence to the Principle of Continuity (and the related

Presses Universitaires de Grenoble.

â€˜element of Timeâ€™) Marshall, to his great credit, recognized the limitations 4 I have discussed these notions of continuity and discreteness in more detail in

of applying the Principle of Substitution. In the absence of universal conti- Boland [1986a, Chapter 5].

nuity and variability, Marshall implies that the assumption of maximization 5 For example, to the extent that Walrasian economics is about the allocation of

given resources, the question can always be begged as to where they come

is not an appropriate method of analysis for all situations.

from.

The major methodological question for proponents of neoclassical

6 I will discuss this in more detail in Chapter 9.

economics is â€˜Can maximization be the sole basis for the neoclassical 7 It might be argued that the stability of the equilibrium is a separate assumption,

research programme?â€™. I have argued above that the assumption of but Samuelson [1947/65, p. 5] argues that even stability conditions are formally

maximization alone is not sufficient; one must also assume or establish a equivalent to maximization conditions.

8 For more on this methodological strategy, see Boland [1986a, pp. 75â€“8].

minimum degree of continuity. For those who wish to extend the

maximization hypothesis as a method of analysis, it is a moot point to show

that the variables in question are in fact variable in both directions over a

continuous range. It is all too easy to just assume that the decision-maker

faces a continuum even when the choice to be made involves integer values

such as when one cannot choose a half of an automobile tire or half of a

radio. There are two ways to avoid this possible impasse. One could change

the choice question to one involving rental time or sharing such that the

choice variable more easily fits the notion of an equilibrium. Unfortunately,

this type of shift in perspective usually is merely an attempt to hide the

original question.8

Given the futility of direct criticism of the assumption of maximization

behaviour, as I argued in Chapter 1, critics of the neoclassical research

programme would be advised to shift their attention to the methods used

(implicitly or explicitly) by neoclassical economists to establish the

applicability of the maximization hypothesis. Surely, questions such as

whether to execute a murderer or whether to vote or whether to make any

irreversible decision must be a dubious territory for the method of

maximization analysis. Marshall explicitly limited his analysis to those

territories amenable to the Principle of Continuity. Perhaps modern

â€˜imperialistsâ€™ such as the followers of Stigler and Becker ought to learn

from Marshallâ€™s avowed appreciation of the necessity of the Principle of

Â© LAWRENCE A. BOLAND Axiomatic analysis of equilibrium states 49

maximization hypothesis which is deliberately put beyond question in

4 Axiomatic analysis of every neoclassical model, the assumption of equilibrium is usually open to

equilibrium states question.1 Some models are designed to explain phenomena as equilibrium

phenomena (such as prices or resource allocations). Models which offer

equilibrium explanations must at least provide logically possible

equilibrium states. Clearly, such equilibrium models are open to question

and thus can be critically examined to determine whether a state of

equilibrium is consistent with the other behavioural assumptions made.

There are some equilibrium models which are not easily criticized such as

those which put the existence of equilibria beyond question (e.g. those

which involve the Coase theorem or unobserved transaction costs). These

necessary-equilibrium models are most often used to explain away alleged

Often mathematical formulas are used to describe certain events disequilibrium phenomena (e.g. involuntary unemployment or socially

without awareness of the assumptions on which the applicability of the unacceptable levels of pollution).

formulas depends. Even less is there thought of an investigation to

In this chapter I will be concerned only with models that explicitly claim

determine whether the requisite assumptions are fulfilled in the real

to offer explanations in which it is asserted that the phenomena in question

world. Therefore it is not surprising that the results are often quite

are equilibrium phenomena. In the next chapter the focus will be models

unsatisfactory.

On the other hand, conclusions have often been drawn from which by claiming that the phenomena are disequilibrium phenomena posit

mathematical formulas, which, strictly speaking, are not conclusions at the equilibrium state as an unattainable ideal.

all and which at best are valid only under restrictive assumptions. The

Equilibrium models which explain why the phenomena occur usually do

latter may not have been formulated, not to mention efforts to discover

so by stating a series of explicit assumptions which together logically entail

to what extent these further assumptions are fulfilled in the real world.

statements representing the phenomena in question. Now, the most

Thus, for a fruitful application of mathematics in economics it is

essential, first, that all the assumptions on which the given common models are ones which represent each assumption with an

mathematical representation of economic phenomena depends be equation and thus show that the solution of the system of equations is a

enumerated completely and precisely; second, that only those

statement representing the phenomena. Where there is a solution there must

conclusions be drawn which are valid in the strictest sense, i.e., that if

be a problem (except perhaps in chemistry). In this case the problem is to

they are valid only under further assumptions, these also be formulated

find values for the endogenous variables which (given the values of the

explicitly and precisely.

If these directions are strictly adhered to, then the only objection exogenous variables) allow all the assumptions to be simultaneously true.

which can be raised against a theory is that it includes assumptions There may be many sets of such values. When there is just one, we call it a

which are foreign to the real world and that, as a result, the theory

unique solution. If none is possible we say the model is unsolvable. If one

lacks applicability.

could never solve the system of equations, then the model cannot explain

Abraham Wald [1936/51, pp. 368â€“9]

the phenomena as equilibrium phenomena.

When do we know that we are successful in explaining something?

Whenever economics is used or thought about, equilibrium is a central

organising idea. Chancellors devise budgets to establish some There are two necessary conditions. The first is the easiest. Most

desirable equilibrium and alter exchange rates to correct â€˜fundamental economists seem to agree that we are successful when the theory we

disequilibriaâ€™. Sometimes they allow rates to â€˜find their equilibrium

construct is shown to be internally consistent and is shown to allow for the

levelâ€™. For theorists the pervasiveness of the equilibrium notion hardly

possibility of the phenomena, that is, when the theory does not contradict

needs documenting.

the phenomena to be explained. If we look closer at the notion of

Frank Hahn [1973, p. 1]

explanation we will find that this consistency criterion for success is

insufficient. The condition that causes difficulty is the second one.

One common avenue for criticism of neoclassical economics is to analyze

Specifically, if one is to explain why prices are what they are then for a

the assumptions required for a state of equilibrium. Unlike the neoclassical

complete explanation (i.e. beyond just possibilities) one must also explain

Â© LAWRENCE A. BOLAND

50 Principles of economics Axiomatic analysis of equilibrium states 51

why prices are not what they are not. In this chapter I shall examine these any other allowed point on the flat spot. Nevertheless, consistency is obvi-

two necessary conditions of a successful explanation. Namely, I shall ously important since we cannot tolerate contradictions or inconsistencies.

examine why we are successful in explaining any particular phenomena

Completeness is the requirement that an explanation does not allow for

only when our theory is not only consistent but is also â€˜completeâ€™ with

the possibility of competing or contrary situations. Completeness rules

respect to those phenomena.

out the possibility of a false explanation accidentally appearing to be

true. That is, if our explanation is complete and happens to be false, we

ANALYZING THE LOGICAL STRUCTURES IN ECONOMICS shall be able to show it to be false directly. For example, if we assume

that the production possibilities curve has no flat spot and is concave (to

Analyzing the success or failure of logical structures such as equilibrium

the origin) then our explanation would be logically complete since each

models is not a new enterprise. Indeed, for a long time it has been an

point on the curve is compatible (tangent) with only one price ratio and

interest of pure mathematicians and some mathematical economists who

each price ratio is compatible with only one point on the curve. In other

engage in what they call axiomatic analysis or axiomatics. 2 Their efforts

words, our equilibrium point is unique given any particular price ratio.

have been directed only at the formalistic aspects of logical structures and

Should any other equilibrium point be possible for the same price ratio,

thus they have too often been more concerned with axioms of language

then we would also have to explain why we observe the one point rather

models where the form of the axiomatic structure remains the same and the

than the other possible points. That is, our model must explain why we

interpretations of the axioms differ to produce various languages [e.g. see

do not observe what we do not observe. The logical possibility of other

Koopmans 1957]. I think axiomatics can also be of considerable

compatible points would mean that our model is not complete.

importance for our critical understanding of economic phenomena. The

primary importance of axiomatics is that it can offer a means of The standard method of demonstrating the consistency of a theory is to

systematically criticizing a given theory (i.e. a given set of assumptions). construct a mathematical model of that theory and prove that it necessarily

For the purpose of critical understanding, the two primary tools of possesses a sensible solution â€“ that is, demonstrate the existence of a

axiomatics are the two necessary conditions of successful explanations. sensible solution. The standard method of demonstrating the completeness

They are the inquiry into the consistency of a theory, and the inquiry into of a theory is to show that the equilibrium solution of the model is unique.

the completeness of a theory. Since these tools are the basis of any criticism Although there is some danger of confusion, these two attributes of theories

of an equilibrium explanation, I briefly explain how they are used in are usually analyzed separately. There are other, secondary, aspects of

economics. axiomatics such as inquiries into the independence and â€˜weaknessâ€™ of the

various assumptions that make up a theory. I will not discuss these topics

Consistency requires that the set of assumptions (which form any par-

here since they are questions of aesthetics rather than of the explanatory

ticular theory) does not lead to inconsistencies such as would be the case

power of any equilibrium model.

if both a given statement and its denial were logically allowed by our

Usually the question of consistency can be dealt with in a rather direct

theory. For example, the statement â€˜the economy at time t is on its

way: try to solve the system of equations constituting the model of the

production possibilities curveâ€™ and its denial â€˜the economy is not on that

theory. If a sensible solution cannot always be obtained, it may be possible

curveâ€™ could not both follow from a consistent theory. This requirement,

to specify additional assumptions to guarantee such a solution. Eliminating

however, does not rule out the possibility of a theory allowing for

non-sensible solutions is a low-order completeness criterion â€“ that is, the

competing or contrary situations such as multiple equilibria. For

model must be complete enough to exclude them but it may not be

example, all points on a production possibilities curve are potential

complete enough to allow only one sensible solution.

equilibria that differ only with regard to the given price ratio. If there is

The conditions which assure consistency are usually much less

a flat spot on the curve, there is a set of points (along the flat spot) all of

restrictive than those which assure completeness. For this reason the

which are potential equilibria for the same price ratio.

question of completeness can be a serious source of important fundamental

Thus, if our explanation of why the economy is at one particular point criticism. One of the pioneers of axiomatic analysis in economics,

along the flat spot is that it is faced with the corresponding price ratio, then Abraham Wald, offered such a criticism of Walrasian economics. A well

consistency alone will not enable us to explain why the economy is not at known but minor aspect of his analysis was a simple proof that the popular

Â© LAWRENCE A. BOLAND

52 Principles of economics Axiomatic analysis of equilibrium states 53

condition that â€˜the number of equations be equal to the number of consequence of the assertion that Walrasâ€™ system does explain all the

unknownsâ€™ was neither necessary nor sufficient to guarantee a solution, let (endogenous) variables? In particular, what are the logical conditions

alone a unique solution. Waldâ€™s 1936 axiomatic study of Walrasian general placed on the system for it to be truly â€˜in equilibriumâ€™? When we say the

competitive equilibrium, which now may be merely of interest to historians system explains all the prices and quantities, we are saying that all the

of mathematical economics, can serve as an interesting case study to explicit and implicit (i.e. unstated) assumptions necessary for the

demonstrate the importance of completeness. Subsequently, I will present sufficiency of the explanation are satisfied. In other words, we are claiming

my theory of completeness which I think is relevant for general economists that the system of assumptions is complete. We know what the explicit

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