ñòð. 21 |

information than is conceivably possible. 26 At one time many thought that

rules out too much if we only want to rule out Giffen goods.

there might be a short-cut to actually constructing the map; we could

observe the personâ€™s choices and ex post deduce from the actual

Interdependence of elasticities

observations what the personâ€™s preference ordering was [e.g. Little 1949].

On the assumption that the consumerâ€™s income is entirely spent, the Without a known ordinal preference map it would seem to be quite

following simple situation is always maintained: arbitrary whether we specify ex ante certain properties of the map that are

assumed to exist, or deduce that map ex post on the basis of a simple notion

PxÂ·X + PyÂ·Y = B [13.1]

of consistent choice.27 The question then is, when does a particular differ-

And, using a little calculus, for any PCC one can generate the following

ence in priceâ€“incomeâ€“choice combinations imply different preferences?

relationship involving the elasticity of demand, Îµ i, for good i, and the slope

Samuelsonâ€™s answer [1948, pp. 243-4] was in effect that any time two

of the PCC, (âˆ‚Y/âˆ‚X)i, for PCCi :

different priceâ€“incomeâ€“choice combinations satisfy the Axiom of Revealed

for good X, Preference, we can utilize the neoclassical theory of the consumer (i.e.

[1 + (1/Îµx)] + [(âˆ‚Y / âˆ‚X)x / (Px / Py)] = 0 [13.2a] utility maximization or optimum choice) to infer the preference map that

for good Y, this individual consumer was assumed to be using. As it turns out, satisfy-

[1 + (1/Îµy)] + [(Px / Py) / (âˆ‚Y / âˆ‚X)y] = 0 [13.2b] ing the Axiom of Revealed Preference is like satisfying the usual

which taken together gives the following relationship between elasticities conditions of Ordinal Demand Theory. These two approaches are suffi-

at one chosen point in Xâ€“Y space since at any one point these two ciently alike that they have important consequences in common which have

relationships must have the same (P x /P y): led Houthakker [1961] and others to consider them equivalent.

What I am going to do here is a little different. Since it has been shown

[1 + (1/Îµx)]Â·[1 + (1/Îµy)] = (âˆ‚Y/âˆ‚X) x / (âˆ‚Y/âˆ‚X) y [13.3]

that certain versions of the axioms of Revealed Preference Analysis imply

That is to say, the ratio of the slopes of the two PCCs indicates directly the

the existence of a preference ordering [Houthakker 1950, 1961; Arrow

product involving the two demand elasticities. This result only conflicts

1959a], I want to apply one of the axioms, the Axiom of Revealed

with the conceivable situation Ee represented by the dotted lines and the

Preference, to specific situations which were derived from preferences.

solid case Ae. The ratio of the slopes of PCCx to PCCy must be less than

There should be no danger of contradiction here even though I may be

one, by definition of demand elasticities, but in the Ee dotted case and the

violating the intentions for inventing the Axiom of Revealed Preference. In

Ae solid case that ratio would be greater than one.

particular, I am going to apply the Axiom of Revealed Preference to two

points on any given PCC. There is no way two points on the same PCC can

CHOICE THEORY FROM REVEALED PREFERENCE ANALYSIS directly violate the Axiom of Revealed Preference if we always assume

â€˜greedâ€™ (lowering one price alone always means that the consumerâ€™s real

Referring back to the schemata [A], one can see the logic of options avail-

income has increased). The question here is, what are the implications of

able to the ordinary neoclassical demand theorist. Neoclassical demand

the Axiom of Revealed Preference for the shape of the PCC?

theorists up to the time of the acceptance of Samuelsonâ€™s Revealed Prefer-

To answer this, a way must be found to express that axiom in terms of

ence Analysis would have us assume a given and known ordinal preference

PCCs and budget lines rather than in terms of quantities of goods and/or

map [e.g. Hicks and Allen 1934, pp. 55, 198]. With a known map and any

indifference curves. It will be recalled from Chapter 4 that the Axiom of

Â© LAWRENCE A. BOLAND

188 Principles of economics Revealed Preference vs Ordinal Demand 189

Y

Revealed Preference says that point A (in Xâ€“Y space) is â€˜revealed Budget line

preferredâ€™ to point B when A is bought at prices P xA and P yA and B is

B/Py

bought at prices PxB and PyB such that

PCC x

PxAÂ·XA + PyAÂ·YA â‰¥ PxAÂ·XB + PyAÂ·YB

if [13.4a]

PxBÂ·XA + PyBÂ·YA > PxBÂ·XB + PyBÂ·YB

then [13.4b]

âˆ†Y

Of course, this must be true for any two points on any PCCx where by A

âˆ†X

definition PyA = PyB (= Py ). Hence the Axiom of Revealed Preference can

âˆ‚X

be stated in this particular case as: âˆ‚Y P

y

PxAÂ·(XA â€“ XB) â‰¥ PyÂ·(YB â€“ YA)

if [13.5a]

P

B

PxBÂ·(XA â€“ XB) > PyÂ·(YB â€“ YA) x

then [13.5b]

Parenthetically, at this point it becomes possible to point out a potential

error in Houthakkerâ€™s [1961] famous survey of consumer theory. He says

X

B/Px

that the Axiom of Revealed Preference

Figure 13.4 Comparing slopes of PCC and budget line at a point

is nothing but a generalization of the Law of Demand to arbitrary price

changes. To see how it relates to the ordinary Law of Demand we need If, as is usual, the consumer is assumed to be maximizing his or her

only put Î£iPAÂ·QB equal to Î£iPAÂ·QA and assume vectors PA and PB are satisfaction, then the slope of the budget line, (âˆ†Y/âˆ†X), equals the negative

identical except for one (say [good X]) price. After some subtractions of the going price ratio, that is, â€“ (Px /P y) = (âˆ†Y/âˆ†X), (see Figure 13.4) then

we then get that one gets the following:

if Î£iPAÂ·QB = Î£iPAÂ·QA then Î£i(PxA â€“ PxB)Â·(XA â€“ XB) < 0 if the slope of PCCx â‰¥ the slope of the budget line of the preferred point

or in words: if a price changes in such a way that in the new situation then the slope of PCCx > the slope of the budget line of the inferior point

the consumer can buy what he bought in the old, then the price change

that is,

and the quantity change are necessarily of opposite signs. [1961, p. 707]

(âˆ‚Y/âˆ‚X) â‰¥ (âˆ†Y/âˆ†X) at A

if [13.8a]

Unfortunately for Houthakkerâ€™s attempt to apply the Axiom of Revealed then (âˆ‚Y/âˆ‚X) > (âˆ†Y/âˆ†X) at B [13.8b]

Preference to demand theory, his â€˜if-clauseâ€™ can never be satisfied on any

When the slope of the PCCx along that curve between the two points is

one PCC curve (and hence on a demand curve). It must always be an

positive (i.e. demand is relatively inelastic) this hypothetical condition is

inequality if only one price is varied and all the income is spent because all

easily satisfied. When the slope of PCCx is negative, the situation gets

the points on any PCC are optimum (â€˜equilibriumâ€™) points. In neoclassical

problematic again. In this case, the Axiom of Revealed Preference says that

textbook terms, no two different points on one budget line can be on the

the slope of the budget line must be steeper than the slope of the PCC x at

same PCC as PCCs and budget lines necessarily cross at only one point.

point B if the slope of the PCCx is not steeper than the budget lineâ€™s slope

Perhaps I am misinterpreting Houthakker, so I will push on. If one

at A. To see what this says, consider the two cases shown in Figures 13.5a

defines âˆ‚X = (XA â€“ XB) and âˆ‚Y = (YA â€“ XB) then the Axiom of Revealed

and 13.5b which represent columns a and e, respectively, of Figure 13.2.

Preference in this particular case says that:

Since the slopes can be compared directly by comparing âˆ‚Y with âˆ†Y for a

PxAÂ·âˆ‚X â‰¥ â€“ PyÂ·âˆ‚Y âˆ‚X = âˆ†X > 0, the first clause of the Axiom of Revealed Preference requires

if [13.6a]

PxBÂ·âˆ‚X > â€“ PyÂ·âˆ‚Y

then [13.6b] that

âˆ‚Y â‰¥ âˆ†Y at A

By specifying merely that âˆ‚X > 0 and Py > 0, one can say that [13.9]

and this is true in Figure 13.5b and is false in Figure 13.5a since both âˆ‚Y

(PxA/Py) â‰¥ â€“ (âˆ‚Y/âˆ‚X)

if [13.7a]

and âˆ†Y are negative. Now the Axiom of Revealed Preference can be

(PxB/Py) > â€“ (âˆ‚Y/âˆ‚X)

then [13.7b]

restated as follows:

Â© LAWRENCE A. BOLAND

190 Principles of economics Revealed Preference vs Ordinal Demand 191

if at point A the demand curve is not positively sloped [13.10a] is that demand curves as shown in Figure 13.6(a) are made impossible by

the Axiom of Revealed Preference (although those as in Figure 13.6(b) are

then at any point B (corresponding to a higher Px ) that

still possible).

demand curve is definitely negatively sloped. [13.10b]

P P

x x

Y

B

B/Py A

âˆ‚X

B

âˆ†X

A

âˆ†Y

A

âˆ‚Y Budget line

X X

(a) (b)

B

Figure 13.6 Possible Giffen demand curves

PCC x

While this interpretation and use of the Axiom of Revealed Preference

B/Px may not seem surprising on its own, it is still interesting to note that Hicks

X

Figure 13.5a Giffen PCC gives precisely the demand curve of Figure 13.6(a) as the plausible

description of the case of a Giffen good [see Hicks and Allen 1934, Figure

6, p. 68]. If my interpretation of the Axiom of Revealed Preference is

Y correct, then one can see that the axiom does say something more than the

Budget line Ordinal Demand Theory (of Hicks and Allen) which alone will not exclude

Giffen goods except by excluding â€˜inferior goodsâ€™. By adding the Axiom of

B/Py

Revealed Preference to Ordinal Demand Theory, however, we can get

slightly closer to the Law of Demand.

METHODOLOGICAL EPILOGUE

B

âˆ†Y Clearly, writing about a subject that has received so much attention in the

âˆ‚Y past is difficult to justify. Some would accept this reconsideration if it had

âˆ‚X PCC x

pedagogical utility â€“ that is, on the presupposition that we all know all

âˆ†X A there is to know about neoclassical demand theory but we always can use

some clever device with which to help teach undergraduates. I think that if

there is a use for better pedagogical devices, such a potentiality reflects a

poor understanding of the matter at hand. Of course, others would accept

B/Px X

this reconsideration merely if it involves the demonstration of some new

Figure 13.5b Non-Giffen PCC mathematical devices or techniques. Although most seem unwilling to

admit it, the application of a complicated mathematical technique to a

The direct implication of this reformulation (at least in the case used here) simple concept always â€˜costsâ€™ more than the resulting â€˜benefitsâ€™ warrant.

Â© LAWRENCE A. BOLAND

192 Principles of economics Revealed Preference vs Ordinal Demand 193

of the demand curve for good X), buying more than 10 percent less of good X

The years of clothing demand theory in a mathematized fabric has left us

whose price has risen by 10 percent means that the consumer is spending less

where we began â€“ Hicksâ€™ half of the 1934 Hicks and Allen article. All that

on good X. This leaves more money to be spent on good Y with its price fixed.

we have to show for our heroic efforts are a few vacuous generalities such To keep the budget fixed, the consumer must buy more of the good with the

as â€˜the generalized law of demandâ€™. Our explanation of consumer fixed price. Thus we see that at point e an increase in the price of X means that

behaviour has not changed, nor has our understanding of our explanation the consumer buys more Y, which fulfills the definition of a point of elastic

demand.

changed. The Emperor has no more clothes on today than he had prior to

11 Actually there are thirty cases since five of the cells represent two cases. I have

1934. Above all, our task of establishing the Law of Demand has neither

represented the two alternative cases by representing one of them with dotted

been assisted nor corrected by our sophistication. rather than solid lines.

Now, rather than dismissing the Law of Demand, as many would seem 12 As always, multiplying all prices and income by the same scalar does not

willing to do [Samuelson 1953, p. 106; Lipsey and Rosenbluth 1971], we constitute a changed situation.

13 Of course, L-orderings are excluded, too.

must attempt to deal with it, one way or another. First, because, as claimed

14 Once you know the family of PCCs for good X, you have enough information

here, Revealed Preference Analysis and Ordinal Demand Theory are not

to determine the family of PCCs for good Y as well as the implicit family of

equivalent with respect to the Law of Demand. 28 And second, but more ICCs. In other words, there is sufficient information in any one set of PCCs to

important, because its significance is intimately involved with our theory of deduce the other PCCs and thus the ICCs.

prices, as I will explain in the next chapter, to dismiss ad hoc the necessity 15 Specifically, in Figure 13.3 every intersection point can be represented by the

solid lines of cell Cc of Figure 13.2.

of the Law of Demand without examining its broader significance cannot

16 See above, pp. 52-60.

help us understand economic behaviour, nor can it foster the development

17 They may be in some sense â€˜inverse demand functionsâ€™ but they contain more

of â€˜testableâ€™ implications of neoclassical theories. information than a single inverted demand function.

18 The arrowhead of the ICC will always be in the shaded area.

19 Consider the location of the ICCâ€™s arrowhead in the dual-purpose cells. When

NOTES considering the dotted-line PPCy, higher income is represented by the white

area demarcated by the extensions of the tail ends of the two PCC arrows.

1 For the derivation of the Slutsky equation from Revealed Preference Analysis,

While there are parts of this area that are not â€˜south-westâ€™ of the intersection, if

see McKenzie [1957] and Samuelson [1947/65, Chapter 5].

we wish to preclude the possibility of violation of the assumption of â€˜greedâ€™ it is

2 Samuelson calls this the â€˜Fundamental Theorem of Consumption Theoryâ€™.

the possibility of any higher-income points being â€˜south-westâ€™ of the intersec-

3 See further Lipsey and Rosenbluth [1971, p. 132] and Samuelson [1947/65, p.

tion which necessitates the exclusion of cells Aa and Ee.

115, footnote 17].

20 That is, the DY/DX needed to remain on the same indifference curve.

4 Such a task is impossible, quite apart from the â€˜integrability problemâ€™, since it

21 That is, DY/DX, which is the measure of the slope of the indifference curve,

requires an impossibly faultless inductive logic [see further, Wong 1978].

must be more negative.

5 For example, one person may be allowed to have a positively sloping demand

22 That is, we are not comparing â€˜goodsâ€™ with â€˜badsâ€™.

curve as long as no other person does.

23 Which means that the indifference curves cannot pass through the intersection

6 I discussed the methodology of individualism in Chapter 2, note 8 and Chapter

point in question and be found â€˜south-westâ€™.

8, note 14. For more detail see Boland [1982a, Chapter 2].

24 This occurs in both Hicks [1956] and [1939], which has been copied by virtu-

7 Except we do exclude a change in response to any homogeneous change where

ally everyone who has wanted to assume the possibility of inferior goods.

all prices and income are multiplied by the same scalar.

25 Quite apart from the problem of induction if we know the consumerâ€™s prefer-

8 Not a very â€˜riskyâ€™ prediction, however.

ences, they are no longer subjective.

9 Note that I have not included a point representing where the slope would be

26 This is the problem of induction â€“ more information is required than is conceiv-

positive and the arrowhead would indicate a rising price. The reason is simple.

ably possible.

Since the income and the price of Y are assumed fixed, whenever only the price

27 This, too, is probably arbitrary without a known map (ex post or ex ante).

of X increases, the purchasing power of the income must fall, yet the excluded

28 The late Cliff Lloyd suggested to me that I have said the following. Since the

point would imply the opposite, which is impossible (viz. more of both goods is

Axiom of Revealed Preference implies more than the Slutsky relations (S+) and

bought as the price of X rises).

the Axiom of Revealed Preference can be deduced from Ordinal Demand

10 The relationship between the elasticity of the implied demand curve and the

Theory (ODT), then, it must be true that ODT implies S+, which is contrary to

slope of the PCC is entirely mechanical. Recall that the definition of demand

what seems to be the consensus concerning ODT. If Cliff was correct then we

elasticity of good X says that if the price of good X rises by 10 percent, an

should be able, by means of the PCC analysis of this chapter, to show that ODT

elastic demand means that the consumer buys more than 10 percent less of good

does imply S+.

X. Since the budget (or income) and the price of good Y are fixed (by definition

Â© LAWRENCE A. BOLAND Giffen goods vs market-determined prices 197

Tradition, casual knowledge or perhaps theoretical imperatives have ruled

14 Giffen goods vs market- out these two approaches to demand theory. George Stigler, decades ago,

determined prices noted that although the dictates of casual knowledge were strong enough to

reject Casselâ€™s notions on the utility of utility analysis, â€˜it could not reject

ñòð. 21 |