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or on Samuelson™s [1938, 1948] Revealed Preference Analysis, is a major
part of the neoclassical theory of prices and, as such, has as its purpose the As one half of a neoclassical theory of prices, consumer theory is a particu-
explanation of demand in general and the Law of Demand in particular. lar conjunction of ideas that is intended to explain why the quantity
The Law of Demand is the commonly accepted notion that the demand demanded is what it is at the going market price. In neoclassical economics
curve for any commodity is downward sloping. This ˜basic proposition™, it is usually taken for granted that no individual in society should have the
says Hicks [1956, p. 59], ˜remains ... the centre of the whole matter™. power to influence (substantially) the going prices directly, yet together (in
Unlike ˜the generalized law of demand™, the Law of Demand is not conjunction with supply) large groups of individuals do determine the
restricted only to ˜normal goods™. prices of all the goods that they buy. Although the necessity that this
The essentialness of the Law of Demand will not be put at stake in this determination involves only downward sloping market demand curves will
chapter. I will take up that matter in Chapter 14. Here I want to critically not be examined until Chapter 14, in this chapter that requirement will be
examine the alleged equivalence of Ordinal Demand Theory and Revealed assumed to hold. Moreover, it is a sufficient argument that if all individuals
Preference Analysis with regard to the Law of Demand. have downward sloping demand curves for any particular good, then the
It is well known that necessary and sufficient reasons for the Law of market demand curve will be downward sloping for that good. The
Demand have yet to be established using Ordinal Demand Theory with a neoclassical notion of demand curves is always in terms of partial
set of conditions or specification that are placed on preference orderings equilibrium “ that is, nothing is required regarding other individuals, other
(except, of course, by ruling out inferior goods). 3 Contrary to the popular markets, etc. Particularly, we do not require that other markets be in
opinions concerning equivalence [e.g. Samuelson 1950; Houthakker 1961], equilibrium. This is the basic feature of both Marshall™s and Pareto™s
in this chapter I will attempt to provide necessary and sufficient reasons for approaches to economics [Pareto 1916/35, footnote to Section 1978;
the Law of Demand by showing how the Axiom of Revealed Preference Marshall 1920/49, Book V]. This approach reveals their view of what is
can be interpreted as saying more than Ordinal Demand Theory about the ˜scientific™: one must begin with the smallest element and work up to broad
Law of Demand. The approach taken here is to examine consumer generalities [Pareto 1916/35, Section 2078; Marshall 1920/49, footnote p.
behaviour without first specifying an ex ante preference ordering (such a 315; see also Schumpeter 1909, pp. 214“17]. If in our theory we allow any
specification would not be directly testable anyway) and I will not be individual to have an upward sloping demand curve, we must then explain
requiring that we must have observed all possible points in goods-space so why the net outcome for the whole market will still be a downward sloping
as to construct ex post a preference ordering. 4 I shall develop the primary demand curve as required. This would in turn require some theoretical
entailment of consumer behaviour that is directly relevant for the Law of statement about consumers as a group (perhaps, about the distribution of
© LAWRENCE A. BOLAND
178 Principles of economics Revealed Preference vs Ordinal Demand 179
people with negative sloping demand curves5). That everyone has a does limit the shape of the consumer™s preferences is the requirement that
the individual be responsive to changes in his or her constraints 7 “ without
downward sloping demand curve is not merely sufficient but also desirable
for the maintenance of strict methodological individualism. 6 For the responsiveness we could not say that prices in any way influence the
purpose of this chapter, the necessity of downward sloping demand curves consumer™s choice. The only place where all of the consumer™s constraints
will continue to be accepted without question as well as the necessity of are influential is on the boundary of his or her ˜attainable set™, that is,
maintaining the strict methodological individualistic view of economics. So where the consumer is spending all of his or her income (or budget) on the
I will thus assume that we must explain why every individual™s demand goods in question.
curve is negatively sloped. This, I think, is the meaning of Hick™s statement The question of determinateness of the choice situation facing an opti-
that the Law of Demand is ˜the centre of the whole matter™ and that ˜centre™ mizing consumer leaves open several different ways for the theorist to
is focused on the individual™s demand curve as the outcome of the approach the explanation of the consumer™s behaviour. We could, on the
individual™s behavioural response both to his or her economic constraints one hand, begin with a consumer™s fully specified ordering (i.e. with ex
(going prices and income) and to his or her disposition regarding the goods ante specified properties) and then examine the expected logical conse-
bought (i.e. tastes). quences of that consumer facing different price and income situations. On
the other hand, we might avoid the requirement that the consumer in ques-
tion be able to specify ex ante his or her preferences, and instead attempt to
THE LOGIC OF EXPLANATION
deduce the nature of those preferences from observed coincidence of
Let us then begin with a general look at the two-step logic of explaining all different price“income situations and actual choices made on the basis of a
individuals™ behaviour regarding their choice of the quantities of goods that static preference ordering. We could then use the deduced preference
ordering as a basis for our ˜prediction™ of the consumer™s behaviour. 8
they buy. We say that consumers are maximizing when they buy the best
quantity combination subject to their economic constraints and subject to On the basis of our theory that the consumer wishes to maximize his or
their criterion as to what is ˜best™. We say, in effect, that their choice of her utility (or, equivalently, pick the ˜best™ point), certain logical require-
quantities is optimally determined conjointly by the state of those ments must be satisfied either by the unobservable ex ante preference
constraints and the nature (shape) of their criterion. We say that their ordering or by the revealed ex post behaviour in observable price“
choice is optimal, hence it can be rationalized by anyone whenever the income“choice situations. Because of the determinateness of our explana-
consumers can clearly state the nature of their tastes. That is, given a tions and the responsiveness of the consumer™s behaviour to all aspects of
specified preference ordering, if the choice is optimal, then we can the price“income situation, satisfaction by one implies satisfaction by the
independently determine what that choice would be. In that manner we say other. That is to say, the theoretical and philosophical necessity of
that we can explain the consumer™s choice. The determinateness is the determinateness and responsiveness is what gives rise to the apparent
crucial element in this theory of explanation. To summarize schematically, equivalence between Ordinal Demand Theory and Revealed Preference
we have the following elements in our explanation of the consumer™s Analysis. I say ˜apparent™ because it is only true in the case where the
choice of quantities of n goods: Marshallian requirement of being able to rationalize the Law of Demand is
Not not imposed upon the optimal choice determination. As yet, the recognized
observable
Directly observable
conditions for an optimal choice determination that is placed on ordinal
preferences are either insufficient or unnecessary for the exclusion of
Optimal
˜Giffen goods™. I will try to show here that the Axiom of Revealed Prefer-
[ X1, X2 , ... Xn ] [ P , P2 , ... Pn , Income, Tastes ]
[ A] 1
determination ence can be interpreted consistently with the above dual approach to
˜Constraints™/Situation
˜Choice™ bundle Criterion
consumer behaviour to show that it does seem to say something more than
of quantities (objective) (subjective)
(objective) the assumptions of Ordinal Demand Theory can and also try to show that
where Xi is the quantity of good i purchased at price P i. some well-known interpretations of demand theory are contradicted by this
That the consumer optimally picks the best point (or bundle) in no way interpretation of the Axiom of Revealed Preference. More specifically, I
requires that the tastes as represented by a preference ordering be of any will attempt to use the determinateness to specify indirectly the nature of
particular shape whatsoever, except that the ˜best™ be well defined. What the preferences which allow inferior goods while still excluding Giffen
© LAWRENCE A. BOLAND
180 Principles of economics Revealed Preference vs Ordinal Demand 181
goods. This indirect specification will be based on the properties of price“ sloped since as the price of X rises the consumer will move from point a
consumption curves. toward b, and thus we note that the consumer must buy more of good X (a
Giffen good situation). At point b the demand would be perfectly inelastic;
relatively inelastic at point c; at point d it would have ˜unit™ elasticity; and
PRICE“CONSUMPTION CURVES (PCCS)
at point e it would be relatively elastic. 10 Similarly, in Figure 13.1(b), I
To compare the assumptions of Ordinal Demand Theory with those of have represented the possible cases for good Y. This means that (ignoring
Revealed Preference Analysis I need to identify something that they have collinear configurations) there are five cases for each good which in turn
in common. The one thing that they do have in common is the behavioural can be combined in twenty-five different ways, see Figure 13.2. Since
consequences entailed in the assumptions. The Slutsky equation, for passing through every point on a consumer™s indifference map there will be
example, is entailed in both sets of assumptions, and both seem to be one PCCx and one PCCy (each with its own slope), with Figure 13.2 I have
catalogued each point as being one of the twenty-five cases. 11
insufficient to deal with the Law of Demand “ they only describe the
behaviour at one point, and do not help us to explain it in relation to other
points. My approach here will be to examine the behaviour by first
examining the families of the PCCs which can be considered either the
logical consequences of using any preference ordering (map) or the
implications of any set of observed choices. The properties of these PCCs A
are the central concern of the theory of consumer behaviour. To examine
the properties of a PCC family or grid, I will lay out all the conceivable
options which must be dealt with and then try to explain the significance of
the various options with respect to either Ordinal Demand Theory or B
Revealed Preference Analysis. To keep this task manageable in two-
dimensional diagrams, I will deal only with two-good cases. And to assist
in the task, I am again going to enhance the usual representation of a PCC
by adding an arrowhead to indicate in which direction (along the PCC) the C
price rises for the good in question.
Y Y Budget line
Budget line
Indifference curve D
Indifference curve
BA
C
a
B/Py
b E
D
PCC x c
d e
E d c b a
PCC y
e
X
X B/Px higher
Key: N
higher
P /Py income
(a) (b) x W E
Figure 13.1 Possible slopes of price“consumption curves
PCC x PCC y S
In Figure 13.1(a), I have drawn a PCC for good X representing all five
possible slopes.9 At point a the implicit demand curve would be positively Figure 13.2 Possible relative slopes of PCCs
© LAWRENCE A. BOLAND
182 Principles of economics Revealed Preference vs Ordinal Demand 183
Now, before examining the logic of the situation, we need to get a clear As the major concern of this chapter is the ability to derive the Law of
idea of what is meant by ˜responsiveness™ and ˜determinateness™ in our Demand and hence explain price behaviour, I will always assume that
explanation of behaviour. Responsiveness means simply that whenever the income or budget, B, is fixed and thus the same for all PCCs considered.
consumer faces significantly different price“income situations, 12 the This is only a minor concern since all income changes can be represented
consumer will choose to buy different combinations of quantities of goods. (inversely) by proportional changes in all prices simultaneously. This
That is, no two different price“income situations determine the same leaves only Py, the fixed price of Y, to be the identifying feature of any
goods“quantity combination. In other words, the mapping from goods- particular PCC for good X. If we change the fixed and given Py we will get
space into situation-space is unique. Determinateness means that for any a different PCCx. Similarly Px, the price of X, is the identifying feature of
particular price“income situation there is only one particular any particular PCCy . If we assume the budget or income is fixed, then to be
goods“quantity combination that will be chosen “ the mapping from on any particular PCCx, the consumer is faced with an implied P y and thus
situation-space into goods-space is unique (i.e. ˜well defined™). To keep the PCCx is labelled with the given Py . Figure 13.3 thereby represents a
within the Hicksian tradition, my concern here will be only with preference grid of PCCs or more important a grid of iso-price lines where at each line
orderings which are representable by indifference maps. 13 A particular there is a PCCx and a PCCy for the given prices, Px and Py . Armed with
indifference map may (when used with neoclassical behavioural assump- such a grid we can say what the prices must be for the consumer to choose
tions) allow for more than one of the choice situations of Figure 13.2, but at any point in the goods-space (given a fixed income, of course). The usual
any given point in goods-space, only one choice situation. By considering income“consumption curve will be generated as the locus of intersecting
all possible price situations, a particular indifference map will give rise to a PCCs with the labels in a constant ratio, Px /Py. In Figure 13.3 the line
family of PCCs, that is, one sub-family for all PCC xs and implicitly one for representing the ICC has an arrowhead indicating the direction of increas-
all PCCys in the two-goods case plus a set of income“consumption curves ing income. The map formed for all the implicit PCCs for any particular
(ICCs) which are merely generated from the PCCs. 14 This relationship indifference map will be called the ˜PCC grid™ for that indifference map.
between the curves is illustrated in Figure 13.3 where representative curves On the basis of either responsiveness or determinateness there is a one-to-
are drawn in the form of a grid.15 Any particular PCC for good X (PCC x) is one correspondence between PCC grids and indifference maps.
drawn by definition only on the basis that the income (or budget, expend- Assuming income constant, note that if we consider a particular PCC as
a vector function on goods-space into price-space, P = φ(X), or an inverse
iture, etc.) is held constant and the price of good Y is held constant.
function on price-space into goods-space, X = φ“1(P), then its projection,
Y Xi = φi“1(Pi), is the individual™s demand function but the projection need
P=8
x
not be ˜well defined™ even though the PCC itself is. The PCCs can be
P=6
PCC x x interpreted in other ways: as a mapping from all-goods-space into one-
P=4
PCC y price-space, X’Pi, such as Wald™s demand function 16 or a mapping from
x
ICC ( P /P = 1) all-price-space into one-good-space, P’Xi such as Cassel™s demand
xy
functions [Cassel 1918, p. 80]. But the entire PCC grid is not in any way an
Py = 4 indirect utility function.17
The importance of PCC grids here is that the PCC grid is the one thing
Py = 6 that Ordinal Demand Theory and Revealed Preference Analysis necessarily
have in common. Conditions placed on preference orderings of demand
Py = 8
theory ultimately must be reflected in the nature of the consequential PCC
grid. Likewise, ˜axioms™ of Revealed Preference Analysis are direct state-
ments about the nature of the implicit PCC grid.

X
Figure 13.3 An implicit ICC
© LAWRENCE A. BOLAND
184 Principles of economics Revealed Preference vs Ordinal Demand 185
CHOICE ANALYSIS WITH PREFERENCE THEORY an assumption about the ˜marginal rate of substitution™ (MRS) “ that is,
ASSUMPTIONS about the slope of an indifference curve. Since the PCC grid is based on
maximization, we can easily determine the value of the marginal rate of
It would seem that the alleged equivalence between the Revealed Prefer-
substitution. Specifically, a necessary condition for maximization is that
ence Analysis and Ordinal Demand Theory should be apparent in the cat-
MRS equals the given Px /Py . To see what the Hicksian assumption of a
egorical logic of the consumer™s situation that has been outlined in Figure
diminishing MRS means, consider any indifference curve drawn through
13.2. In particular, the logical significance of the assumptions concerning
the point of intersection of two PCCs. If the MRS is diminishing then (1) at
preferences or choices is always in terms of which PCC situations (of Fig-
all points along the curve that are ˜north-west™ of the intersection point, the
ure 13.2) are ruled out as impossible by those behavioural assumptions.
slope of the indifference curve20 must be higher (i.e. steeper) than the slope
The primary tool in this section will be the array of cells illustrated in
at the intersection point and (2) at all points along the curve that are ˜south-
Figure 13.2. Let me be clear about what that figure shows. Each cell is
east™ of the intersection, the slope must not be higher. In Figure 13.2 there
drawn for one point with the two PCCs intersecting as shown. Note that at
are a few cells which would contradict this requirement. This is most
each point of an indifference map there is a PCC x for a given price of Y and
clearly seen in situation Bb where below and to the right all points (whether
a PCCy for a given price of X and implicitly a specific P x /P y. Relative to
or not they are on the one indifference curve in question) must necessarily
this Px /Py I have identified the cross-hatched areas (points) where P x /P y
have a steeper MRS21 which contradicts directly the assumption of
would definitely have to be higher for those points to be chosen. Similarly,
diminishing MRS. Note that so long as indifference curves are negatively
there are shaded areas (points) where income would definitely have to be
sloped (which is the only way we would ever use them) 22 and they are not
greater for those points to be chosen. With Figure 13.2 in mind, I will now
straight lines, indifference curves must be drawn in a direction which lies
examine the consequences of some of the usual assumptions concerning the
in the angle formed by the arrowhead of one PCC and the tail end of the
shape of preferences. Throughout this examination I will be referring to the
other PCC. That is, as one moves along a curved indifference locus, MRS
various cells in Figure 13.2 by identifying the row with a capital letter and
is changing (diminishing or increasing). Now, in the context of diminishing
a column with a lower-case letter, (e.g. the lower left cell is Ee). Each cell
MRS along a negatively sloped indifference curve, one can see that more
in Figure 13.2 represents the possible relative slopes of the two PCCs at the
situations are ruled out. In addition to Bb, situations Ab, Aa and Ba are
point of intersection and corresponding to the points labelled in Figures
clearly seen as logically impossible. Recognizing that ˜greed™ implies
13.1(a) and 13.1(b).
negatively sloped indifference curves, 23 situations Ad, Ac, Bc, Cb, Ca and
Da are also impossible. Situations Bd and Db, and the conceivable cases
Greed (dominance, non-satiation) represented by the dotted lines in cell Cc, are also impossible. The
situations Ae and Ea are problematic under the assumption of diminishing
The most common neoclassical assumption is to rule out ˜bliss points™. The
MRS since some of the cases allowed contradict diminishing MRS. But
effect of ruling out bliss points is that people will always prefer more of
since they are extreme cases, Hicks argued, they are unlikely. 24 If his
any good if none of any other has to be given up. The major implication of
argument were considered sufficient we could see how Hicks™ assumption
this assumption is that indifference curves are always negatively sloped.
might eliminate Giffen goods since they are to be found anywhere in rows
For my purposes here, this assumption rules out those situations in Figure
A and B or columns b and a.
13.2 where the ICC is positively sloped but has the arrowhead (increasing
income) pointing ˜south-west™.18 Such cases as those represented by the
dotted lines in Aa and Ee become problematic here if we require that the Assumption of ˜normal goods™
slope of the PCCs guarantee that this assumption is fulfilled. 19
There is one assumption which is more than sufficient for ruling out Giffen
goods. If one assumes that all goods are such that any increase in income
Hicksian assumption of diminishing MRS (convexity) (or lowering of all prices) would mean that more of all goods would be
bought, then the ICC would always be positively sloped with its arrowhead
Next let us consider the effect of utilizing the Hicksian assumption that
pointing ˜north-east™ (i.e. the shaded area would be restricted to appear only
indifference curves should be convex to the origin. Convexity is assured by
˜north-east™ of the intersection point). This assumption can be seen in
© LAWRENCE A. BOLAND
186 Principles of economics Revealed Preference vs Ordinal Demand 187
Figure 13.2 to rule out the situations Ae, Ad, Be, Bd, Db, Da, Eb and Ea as given price“income situation, we can deduce the optimal choice in goods-
being impossible (since they contradict this assumption). Also, the space. Unfortunately, this approach is based on a presumption of the avail-
additional situations Ac, Ab, Aa, Ba, Ca, Ce, De, Ee, Ed and Ec would be ability of the consumers™ subjective knowledge. Without such knowledge,
problematic as above. If the assumption is that the goods must necessarily it would be impossible to apply this version of demand theory directly to
any person without some heroic philosophical jumps. 25 Many thought that
be ˜normal™ (never inferior) then only the remaining situations would be
possible. If one uses both the diminishing MRS and the ˜normality™ this was merely a minor difficulty since we could ask the individual what
assumptions then only situations Cd, Cc, Dd and Dc are possible (i.e. no his or her relative preferences were at any given point [e.g. Hicks and Allen

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