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elastic demand for any good!). Obviously this conjunction of assumptions 1934, Part II; Allen 1950], even though a complete map still requires more
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 person™s choices and ex post deduce from the actual
Interdependence of elasticities
observations what the person™s preference ordering was [e.g. Little 1949].
On the assumption that the consumer™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
Px·X + Py·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 price“income“choice combinations imply different preferences?
relationship involving the elasticity of demand, µ i, for good i, and the slope
Samuelson™s answer [1948, pp. 243-4] was in effect that any time two
of the PCC, (‚Y/‚X)i, for PCCi :
different price“income“choice combinations satisfy the Axiom of Revealed
for good X, Preference, we can utilize the neoclassical theory of the consumer (i.e.
[1 + (1/µx)] + [(‚Y / ‚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/µy)] + [(Px / Py) / (‚Y / ‚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 X“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/µx)]·[1 + (1/µy)] = (‚Y/‚X) x / (‚Y/‚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
˜greed™ (lowering one price alone always means that the consumer™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 Samuelson™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
© LAWRENCE A. BOLAND
188 Principles of economics Revealed Preference vs Ordinal Demand 189
Y
Revealed Preference says that point A (in X“Y space) is ˜revealed Budget line
preferred™ 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
PxA·XA + PyA·YA ≥ PxA·XB + PyA·YB
if [13.4a]
PxB·XA + PyB·YA > PxB·XB + PyB·YB
then [13.4b]
∆Y
Of course, this must be true for any two points on any PCCx where by A
∆X
definition PyA = PyB (= Py ). Hence the Axiom of Revealed Preference can
‚X
be stated in this particular case as: ‚Y P
y
PxA·(XA “ XB) ≥ Py·(YB “ YA)
if [13.5a]
P
B
PxB·(XA “ XB) > Py·(YB “ YA) x
then [13.5b]
Parenthetically, at this point it becomes possible to point out a potential
error in Houthakker™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 ΣiPA·QB equal to ΣiPA·QA and assume vectors PA and PB are satisfaction, then the slope of the budget line, (∆Y/∆X), equals the negative
identical except for one (say [good X]) price. After some subtractions of the going price ratio, that is, “ (Px /P y) = (∆Y/∆X), (see Figure 13.4) then
we then get that one gets the following:
if ΣiPA·QB = ΣiPA·QA then Σi(PxA “ PxB)·(XA “ XB) < 0 if the slope of PCCx ≥ 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]
(‚Y/‚X) ≥ (∆Y/∆X) at A
if [13.8a]
Unfortunately for Houthakker™s attempt to apply the Axiom of Revealed then (‚Y/‚X) > (∆Y/∆X) at B [13.8b]
Preference to demand theory, his ˜if-clause™ 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 (˜equilibrium™) 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 line™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 ‚X = (XA “ XB) and ‚Y = (YA “ 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 ‚Y with ∆Y for a
PxA·‚X ≥ “ Py·‚Y ‚X = ∆X > 0, the first clause of the Axiom of Revealed Preference requires
if [13.6a]
PxB·‚X > “ Py·‚Y
then [13.6b] that
‚Y ≥ ∆Y at A
By specifying merely that ‚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 ‚Y
(PxA/Py) ≥ “ (‚Y/‚X)
if [13.7a]
and ∆Y are negative. Now the Axiom of Revealed Preference can be
(PxB/Py) > “ (‚Y/‚X)
then [13.7b]
restated as follows:
© 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
‚X
B
∆X
A
∆Y
A
‚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 ˜inferior goods™. 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
∆Y Clearly, writing about a subject that has received so much attention in the
‚Y past is difficult to justify. Some would accept this reconsideration if it had
‚X PCC x
pedagogical utility “ that is, on the presupposition that we all know all
∆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 ˜costs™ more than the resulting ˜benefits™ warrant.
© 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 “ Hicks™ 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 ˜the generalized law of demand™. 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 ˜inverse demand functions™ but they contain more
of ˜testable™ 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 ICC™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 ˜south-west™ 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 ˜greed™ it is
2 Samuelson calls this the ˜Fundamental Theorem of Consumption Theory™.
the possibility of any higher-income points being ˜south-west™ 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 ˜integrability problem™, 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 ˜goods™ with ˜bads™.
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 ˜south-west™.
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 consumer™s prefer-
8 Not a very ˜risky™ 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 “ 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
© 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 Cassel™s notions on the utility of utility analysis, ˜it could not reject

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