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think may be the only effective means of criticizing equilibrium models. implicit assumptions? To conjecture what the implicit assumptions are is

the task of an axiomatic analysis of the completeness of a general

equilibrium system such as Walrasâ€™. However, before the search for

WALDâ€™S AXIOMATIC WALRASIAN MODEL: A CASE STUDY

implicit assumptions can begin, we must first show that the explicit

Rarely will we find axiomatic studies of Marshallian economics. The assumptions form an incomplete system, that is, an incomplete system with

reason is simple but misleading. The reason is that Marshallâ€™s statical respect to the task of explaining all prices and quantities of traded goods.

method focuses primarily on the necessary equilibrium requirements for Wald, in his famous 1936 paper, attempted to do both of these tasks,

just one market at a time. The key notion is a partial equilibrium which is namely, to demonstrate the incompleteness of the Walrasian system (which

partial because all other markets are impounded in the ceteris paribus supposedly Walras at first thought was complete merely because the

condition invoked in the determination of each individualâ€™s demand (or number of equations equalled the number of unknowns) and to posit some

supply). But each individual still needs to know the prices of other goods. possible implicit assumptions. His paper represents one of the first rigorous

In other words, the individual makes substitution choices on the basis of a (axiomatic) studies of the mathematical implications of a Walrasian

economic system (in general equilibrium). 3 His version of a Walrasian

knowledge of relative prices. Thus, in effect, the partial equilibrium

method is actually predicated on all other markets providing equilibrium system is the following:

ï£¼

prices â€“ otherwise, the equilibrium of the market in question will not

ri = ai1 X1 + ai2X2 + ... + ain Xn + Ui (i = 1, 2, ..., m)

ï£´

persist. The absence of such a general market equilibrium will usually lead

ï£´

UiVi = 0 (i = 1, 2, ..., m)

ï£´

to price changes in the other markets followed by appropriate substitution

m

Î£a V ï£½

responses in the demand and supply curves of the market in question. So [4.1]

Pj = ( j = 1, 2, ..., n)

ï£´

ultimately a complete Marshallian explanation of an equilibrium price

ï£´

ij i

i=1

involves a form of general equilibrium since only when there is a general

ï£´

market equilibrium can we be sure there is a partial equilibrium in the Pj = fj (X1, X2, ..., Xn ) ( j = 1, 2, ..., n)

ï£¾

market in question. Thus Marshall and Walras differ only in their

methodological procedures. Since the ultimate equilibrium state in one where the exogenous variables are as follows:

market depends on all other markets being in equilibrium, the most direct ri is the quantity available of the ith resource

way to analyze the requirements of a general market equilibrium would be

aij is the quantity of the ith resource needed per unit of the jth good

to consider all individuals simultaneously and try to determine a set of

and the endogenous variables are as follows:

prices that would allow all individuals to be maximizing. This latter

procedure is the Walrasian approach to equilibrium explanations. Although Ui is the unused portion of the available ith resource

Marshallâ€™s procedure may appear to differ, any analysis of a Walrasian Pj is the price of the jth good

equilibrium state will have implications for any successful application of

Vi is the value of the ith resource

the statical method even when focused on just one market.

Xj is the output quantity of the jth good

The Walrasian system of general equilibrium thus purports to explain

simultaneously all (relative) prices and all (absolute) quantities of traded

goods (in the system). The question of interest here is: What is the logical

Â© LAWRENCE A. BOLAND

54 Principles of economics Axiomatic analysis of equilibrium states 55

n

âˆ‘ Pjâ€² âˆ†Xj < 0,

This system of equations is the beginning of an axiomatic version of a

Walrasian economic system. The first class of equations (r i = ...) represents

where Pjâ€² = fj (X1 + âˆ†X1, ..., Xn + âˆ†Xn) ( j = 1, 2, ..., n).

the production or resource allocation relations. The second class is a special

consideration which says that if a resource is not scarce then some of it will

Furthermore, he noted that if the rank of the matrix [a ij] is equal to m, then

be unused (Ui > 0), and thus the resource price (V i) must be zero (i.e. it is a

the solution is also unique for the variables V 1, ..., V m.

free good). Walras was claimed to have ignored this consideration (perhaps

Now let us try to see what Wald has imposed on the well known

because he thought it would be obvious which resources are scarce). The

Walrasian economic explanation of prices and outputs. The first three

third class of equations is the typical long-run competitive equilibrium

conditions are the usual economic considerations. Condition (1) says that

condition where price equals unit cost. Now the fourth class is actually a

the resources must exist in positive amounts in order to be used. Condition

set of Marshallian market demand curves. Waldâ€™s axiomatic version of the

(2) says that input requirements are not negative (i.e. they are not outputs).

Walrasian system then differs slightly from the textbook version of

And condition (3) says the output of any good must require a positive

Walrasian neoclassical economics. In particular, his version makes no

amount of at least one input.

attempt to explain the market demand curves by explaining individual

Conditions (4) and (5) are required for the method of proving his

consumer behaviour.

existence and uniqueness theorem. That is, in order to use calculus-based

Waldâ€™s study involved the question â€˜Does the system of equations [4.1]

mathematics, he must simplify the mathematical aspects of the system. But,

have a unique non-negative system of solutions where r i and a ij are given

whereas condition (4) involves only the usual assumption of continuity,

numbers, fi (X1, ..., Xn) are given functions, and the U i, X i, V i and P i are

condition (5) is a more serious simplification. Condition (5) says that for

unknowns?â€™ On the basis of his method of rationalizing his affirmative

the quantity demanded of a good to be zero, the price must be infinitely

answer to this question, he formulated the following theorem which he said

large. He says that this condition is not necessary for an existence proof but

he proved elsewhere [Wald 1933/34, 1934/35].

it does help by making the mathematics simple (this condition was the first

Theorem. The system of equations [4.1] possesses a set of non-negative to be discarded by subsequent developments in mathematical economics

twenty years later).5

solutions for the 2m + 2n unknowns and a unique solution for the

unknowns X1, ..., Xm, P1, ..., Pn, U1, ..., Um, if the following six conditions Now we reach (6), the most important condition. It is so important that it

are fulfilled:4 has been given a special name: the Axiom of Revealed Preference. 6 It says

that the demand functions must be such that if combination A of goods is

(1) ri > 0 (i = 1, 2, ..., m).

purchased rather than any other combination B that cost no more than A at

aij â‰¥ 0

(2) (i = 1, 2, ..., m; j = 1, 2, ..., n). the given prices then, for combination B ever to be purchased, the prices

(3) For each j there is at least one i such that a ij > 0. must change such that combination B costs less than combination A at the

new prices. A rather reasonable assumption if we were speaking of

(4) The function fj (X1, X2, ..., Xn) is non-negative and continuous for

individual consumers, but these are market demand curves! Unfortunately,

all n-tuples of non-negative numbers X 1, X2, ..., Xn for which

Xj â‰ 0 ( j = 1, 2, ..., n). it does not follow that if the axiom holds for each individual consumerâ€™s

demand function, then it necessarily will hold for the market function.

(5) If the n-tuple of non-negative numbers X 1k, ..., X nk (k = 1, 2, ... âˆž)

Similarly, when it holds for the market, it does not necessarily hold for all

in which Xjk > 0 for each k, converge to an n-tuple X1, ..., Xn in

the individuals. One behavioural interpretation of condition (6) is that all

which Xj = 0, then

consumers act alike and thus are effectively one. Thus condition (6)

lim fj (X1k, X2k, ..., Xnk) = âˆž ( j = 1, 2, ..., n). imposes constraints on the â€˜community indifference mapâ€™ which may be

kâ†’âˆž

difficult or impossible to satisfy.

(6) If âˆ†X1, âˆ†X2, ..., âˆ†Xn are any n numbers in which at least one < 0, We should thus ask (as did Wald): Do we need the axiom of revealed

and if preference (in order to assure completion)? His answer was â€˜yesâ€™, and he

n

âˆ‘ Pj âˆ†Xj â‰¤ 0, demonstrated it with a simple model of system [4.1]. Note that if it is

necessary for system [4.1] it is necessary for every model of the system;

then

Â© LAWRENCE A. BOLAND

56 Principles of economics Axiomatic analysis of equilibrium states 57

thus if we could show that it is unnecessary for any one model, we could We know that whenever we base consumer theory on indifference

refute its alleged necessity for the systems. analysis we can derive the demand curve for a good by considering what is

Conditions (1) to (5) are necessary for Waldâ€™s proof of the consistency usually called the â€˜priceâ€“consumption curveâ€™. To illustrate, consider the

of his version of the Walrasian system. Condition (6) is necessary to two goods, X1 and X2. Specifically, all the possible non-negative

complete the system. To show this we shall specify a model which satisfies combinations of them, and let us assume that income is given. Note that in

conditions (1) to (5), and then we show the necessity of condition (6) by Figure 4.1, for a particular combination of goods, say point Z, there is only

describing a case in which condition (6) is not fulfilled and for which a one set of prices which will be compatible with a choice of combination Z,

in particular P11 and P21. If we were to change P11 to P12 without changing

unique solution does not exist. Consider Waldâ€™s special case of system

[4.1] involving the unknowns X1, X2, P1, P2 and V1: P2, we should find that point Zâ€² is the combination which is compatible

with the new price(s).7

ï£¼

r1 = a1X1 + a2X2

ï£´ 2

P1 = a1V1 3 1

P1 P

4 P1

X2 1

P

ï£½

P2 = a2V1 [4.1â€²] 1

PCC 1

ï£´ 1

P1 = f1(X1, X2 ) P2

PCC 2 Z4

ï£¾

P2 = f2(X1, X2 ) 2

P2

And to satisfy conditions (1), (2) and (3), we can simply let a 1 = a 2 = a 3

P2

where a > 0 and let r1 > 0. To satisfy (4) we assume f j (X 1, X 2) to be Z3

continuous and positive. To satisfy (5) we assume that as X j approaches 4

P2

zero, Pj â†’ âˆž. The heart of the matter is the inverse demand functions, Z2

Z1

fj (X1, X2 ).

X2

X1

1

2

B/P2 = B/P2

Figure 4.2 The Z-line (incomeâ€“consumption curve)

PCC

In this manner we can trace all the combinations which are compatible

Zâ€²

with a particular P2 (i.e. where P2 is constant). The curve traced is simply

the priceâ€“consumption curve for X1 from which we derive the demand

curve for X1 or, in terms of model [4.1â€²], it is all the combinations of X 1

Z

and X2 such that f2(X1, X2 ) = constant. Now, instead of drawing an

indifference map, we could simply draw a representative set of the possible

priceâ€“consumption curves (assuming income given) and get something like

Figure 4.2. In this figure each curve is labelled with the appropriate fixed

1 2 X1 level representing the fixed price of the other good. On this diagram we can

B/P1 B/P1

see that point Z1 is compatible only with given prices P 14 and P 24. If we

Figure 4.1 Priceâ€“consumption curve (PCC)

hold P2 constant and move outward from point Z1, in neoclassical

consumer theory we should find that P1 falls along the priceâ€“consumption

Let us therefore look more closely at them by first reviewing textbook

curve labelled with the fixed price P24 (see also Figure 4.1). Similarly, if

indifference analysis, and in particular, we want to look at the nature of the

we hold P1 constant and move outward along the other priceâ€“consumption

set of combinations of X1 and X2 which give the same demand price (i.e.

curve from Z1, then P2 falls. Thus note in Figure 4.2 that the superscripts

for Pj constant).

Â© LAWRENCE A. BOLAND

58 Principles of economics Axiomatic analysis of equilibrium states 59

indicate an ordering on prices. Also we note that conditions (4) and (5) can X2 which satisfy the first equation as a line (resembling a budget line)

be satisfied; for example, as we move horizontally toward the vertical axis which satisfies conditions (1), (2) and (3). Condition (4) says that through

(i.e. X1 goes to zero) the price of X1 rises. If we let P 11 = P 21, P 12 = P 22, ..., each and every point in Figure 4.3 there is exactly one priceâ€“consumption

P1k = P2k, we can trace all the combinations for which P 1 = P 2, viz. Z 1, Z 2, curve for good X1 and exactly one for good X2. Condition (5) says that as

Z3, etc. The line connecting these Zs is what is usually called the we trace out any priceâ€“consumption curve for good X 1 in the direction

â€˜incomeâ€“consumption curveâ€™ but since the definition of priceâ€“consumption indicated by the arrowhead (i.e. for a rising P 1) the priceâ€“consumption

curves is based on a fixed budget or income, I will call this the Z-line. curve will never touch the X2 axis. Condition (6) is less obvious. It says

that no priceâ€“consumption curve for good X1 will have a shape illustrated

f(X1, X2 ) = P1 = constant in Figure 4.4.8 The reason for excluding such a shape is that the inverse

X2

demand function implied by such a shape might not be sufficiently well

Zâ€² P /P2 = constant defined. Condition (6) also assures a sufficient degree of convexity of the

1

V r /P = r1/a2 P

2

11 2 underlying preference map (which would have to be a communityâ€™s map in

P

1 Waldâ€™s model). In my diagrams, this means that if you face in the direction

indicated by the arrowhead on any particular Z-line, then to your left the

W

PCC 1 ratio of P1/P2 will always be higher than the one corresponding to this

f(X1, X2 ) = P2 = constant Z-line.

What Waldâ€™s proof establishes is that there is at least one stable

equilibrium point on the quasi-budget line through which passes the correct

Z

PCC 2 Z-line. The correct Z-line will be the one drawn for a P 1/P 2 ratio that equals

the slope of the quasi-budget line. That is, he proves that there is at least

X1

V r /P = r /a1 one point like either the one on the positively sloped Z-line illustrated in

11 1

1

Figure 4.3 or like the one on a negatively sloped Z-line which has its

Figure 4.3 Priceâ€“consumption curves and Waldâ€™s special case

arrowhead outside of the feasible production points limited by quasi-budget

line as illustrated in Figure 4.5.9

X2

P X2

2

Zâ€² P2

P1

PCC 1

Z

X1

X1

Figure 4.4 A denial of condition (6)

Figure 4.5 A possible negatively sloped Z-line

Returning to system [4.1â€²], we see that the first equation can be

represented on the commodityâ€“space diagram as shown in Figure 4.3.

Since r1, a1 and a2 are given we describe the set of combinations of X 1 and

Â© LAWRENCE A. BOLAND

60 Principles of economics Axiomatic analysis of equilibrium states 61

COMPLETENESS AND THEORETICAL CRITICISM true (i.e. agrees with the observed facts) but the others are false. A

completed model, however, leaves no room for errors (viz. for

Although the inclusion of Waldâ€™s six conditions in the axiomatic structure

disagreement with facts). Unfortunately, most economists would be

of the Walrasian system fulfills the task of completing an explanation of

satisfied with the incomplete model because at least one of its many

prices and outputs, it does not follow that they are necessary for the

solutions is true.

original theory. As it was later shown, the existence and uniqueness of the

There are different theories of knowledge. Obviously, the one I am

entire Walrasian system can be proved by using either linear programming

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