MICROECONOMICS 1 – REVEALED PREFERENCE
Our earlier discussions on the behaviour of the consumer
relied on fundamental assumptions about preferences and
the budget constraint to derive demand functions for
individual consumers.
MICROECONOMICS 1 – REVEALED PREFERENCE
In the case of revealed preference, we want to go about
things the other way.
That is, we want to use information about the consumer’s
demand to discover information about his/her preferences.
MICROECONOMICS 1 – REVEALED PREFERENCE
Why this the case? In other words why the reliance on
revealed preference?
In real life, preferences are not directly observable. We
discover them by observing their behaviour.
MICROECONOMICS 1 – REVEALED PREFERENCE
The notion of revealed preference was introduced into
economics by Paul Samuelson (1938; 1947) in his
investigation of the empirical content of the theory that
consumers maximize their utility.
MICROECONOMICS 1 – REVEALED PREFERENCE
His analysis was based on observable data, and thus attempted to
characterize the data sets that are consistent with the existence of
some utility function.
The main criticism of the ordinal approach was from a
methodological point of view, in that it used non-observable
concepts and propositions.
MICROECONOMICS 1 – REVEALED PREFERENCE
As Samuelson (1938) argued, one ought to analyse the
consumer’s behaviour without having recourse to the
concept of utility at all, since this did not correspond to
directly observable phenomena.
MICROECONOMICS 1 – REVEALED PREFERENCE
Basically the theory of revealed preference makes a virtue of
assuming nothing whatsoever about the psychological causes of
our choice behaviour.
Instead, it pays attention only to what people do. It assumes that
we already know what people choose in some situations, and uses
this data to deduce what they will choose in other situations.
MICROECONOMICS 1 – REVEALED PREFERENCE
A few Assumptions are in order:
Preferences are stable: they remain unchanged whilst we observe
consumers’ behaviour.
Preferences also indicate rational choices by the consumer
Rational choices indicate optimization-based approach to decision
making
MICROECONOMICS 1 – REVEALED PREFERENCE
A few Assumptions are in order:
Consistency, that is, the consumer’s choice behaviour must be
consistent. .
Strict Preferences: It is thus usual to write A≺ B to mean that the
consumer likes B strictly more than A. Such a strict preference
relation is said to be consistent if it is both asymmetric and transitive.
MICROECONOMICS 1 – REVEALED PREFERENCE
A few Assumptions are in order:
A preference relation ≺ is transitive if a≺ b and b≺ c implies a≺ c.
It is only when transitivity holds that we can describe the consumer’s
preferences by simply writing a≺ b≺ c. Without transitivity, this
information wouldn’t imply that a≺ c.
MICROECONOMICS 1 – REVEALED PREFERENCE
A few Assumptions are in order:
A preference relation ≺ is asymmetric if we don’t allow both A ≺ B
and B≺ A.
It represents a full set of strict preferences on a set X if we insist that
either A≺ B or B≺ A must always hold for any A and B in X that
aren’t equal (complete or total).
MICROECONOMICS 1 – REVEALED PREFERENCE
The revealed preference axiom: The consumer, by choosing a collection of
goods in any one budget situation, reveals his preference for that particular
collection. The chosen bundle is revealed to be preferred among all other
alternative bundles available under the budget constraint.
The chosen ‘basket of goods’ maximises the utility of the consumer. The
revealed preference for a particular collection of goods implies the
maximisation of the utility of the consumer.
MICROECONOMICS 1 – REVEALED PREFERENCE
Figure 1: Revealed Preference: the bundle (x1,y1) that the consumer chooses is
revealed preferred to the bundle (x2,y2) that he could have chosen
Y
•(x1, y1)
• (x2,y2)
0
X
MICROECONOMICS 1 – REVEALED PREFERENCE
In Figure 1, the consumer is faced with two bundles (x1,y1) and (x2,y2).
Both bundles are clearly affordable to the consumer, as is any bundle on
or beneath the budget line.
However, bundle (x1,y1) is the optimal bundle and thus a unique
demanded bundle (for reasons that are familiar to us).
MICROECONOMICS 1 – REVEALED PREFERENCE
Thus, from Figure 1 we can conclude that all other bundles on or
beneath the budget line are revealed worse than the chosen bundle
(x1,y1). This is because those other bundles are affordable and could
have therefore been chosen, but were rejected in favour of bundle
(x1,y1).
MICROECONOMICS 1 – REVEALED PREFERENCE
The algebra of revealed preference
With quantities (xi, yi) and prices (px, py), and a given income, m the
two bundles can be expressed algebraically as follows:
For the chosen bundle (x1,y1) this condition must be satisfied 𝑝𝑥 𝑥1 +
𝑝𝑦 𝑦1 = 𝑚, whilst for the other bundle this condition must be satisfied
𝑝𝑥 𝑥2 + 𝑝𝑦 𝑦2 ≤ 𝑚.
MICROECONOMICS 1 – REVEALED PREFERENCE
The algebra of revealed preference
Thus, putting these two together, the fact that (x2,y2) is affordable
at the budget (px, py, m) means that 𝑝𝑥 𝑥1 + 𝑝𝑦 𝑦1 ≥ 𝑝𝑥 𝑥2 + 𝑝𝑦 𝑦2 .
MICROECONOMICS 1 – REVEALED PREFERENCE
The algebra of revealed preference
If the above inequality is satisfied and (x2,y2) is actually different from
(x1,y1), we say that (x1,y1) is directly revealed preferred to (x2,y2). Thus
revealed preference is a relation that holds between the bundle that is
actually demanded at some budget and the bundles that could have been
demanded at that budget.
MICROECONOMICS 1 – REVEALED PREFERENCE
The principle of revealed preference
Let (x1,y1) be the chosen bundle when prices are (px, py), and let
(x2,y2) be some other bundle such that 𝑝𝑥 𝑥1 + 𝑝𝑦 𝑦1 ≥ 𝑝𝑥 𝑥2 +
𝑝𝑦 𝑦2 . Then if the consumer is choosing the most preferred bundle
he/she can afford, we must have (𝑥1 , 𝑦1 ) ≻ (𝑥2 , 𝑦2 ).
MICROECONOMICS 1 – REVEALED PREFERENCE
The principle of revealed preference
A point worth noting! If oat porridge is revealed preferred to maize
porridge, it doesn’t automatically mean that oat porridge is preferred to
maize porridge. This is because ‘revealed preferred’ just means that oat
porridge was chosen when maize porridge was affordable (and
available).
MICROECONOMICS 1 – REVEALED PREFERENCE
Indirect revealed preference
In our earlier discussion, we noted that (𝑥1 , 𝑦1 ) ≻ (𝑥2 , 𝑦2 ). Now
suppose we know that (x2,y2) at prices (p1, p2) and that (x2,y2) is
itself revealed preferred to some other bundle (x3,y3).
MICROECONOMICS 1 – REVEALED PREFERENCE
Indirect revealed preference
That is, 𝑝1 𝑥2 + 𝑝2 𝑦2 ≥ 𝑝1 𝑥3 + 𝑝2 𝑦3 . Then we know that
(𝑥1 , 𝑦1 ) ≻ (𝑥2 , 𝑦2 ) and that (𝑥2 , 𝑦2 ) ≻ (𝑥3 , 𝑦3 ).
Thus, from the transitivity assumption we can conclude that
(𝑥1 , 𝑦1 ) ≻ (𝑥3 , 𝑦3 ).
MICROECONOMICS 1 – REVEALED PREFERENCE
Indirect revealed preference
Hence, from revealed preference and transitivity, we can conclude
that (x1, y1) is indirectly revealed preferred to (x3, y3). In Figure 2
we depict the idea of indirect revealed preference. The bundle (x1,
y1) is indirectly revealed preferred to (x3, y3).
MICROECONOMICS 1 – REVEALED PREFERENCE
Figure 2: Indirect Revealed Preference
F
Y
• (x1, y1)
•
(x2, y2)
• (x3, y3)
0
X
MICROECONOMICS 1 – REVEALED PREFERENCE
Derivation of the Demand Curve
We have discussed the concept of revealed preference. Now we can
use this concept to derive the demand curve.
As usual we make use of the budget line and the well-known
concept of compensated budget line.
MICROECONOMICS 1 – REVEALED PREFERENCE
Derivation of the Demand Curve
Suppose the consumer is faced with the budget constraint AB in
Figure 3 and chooses bundle Z, thus revealing his preference. Of
course everything else we learnt previously is obvious; within the
class of bundles affordable, Z is revealed preferred to all.
MICROECONOMICS 1 – REVEALED PREFERENCE
Figure 3: Derivation of the Demand Curve
y
A
Z
A’
N
•
•
W
X1
X2 B
•
x3 B’
C
x
MICROECONOMICS 1 – REVEALED PREFERENCE
Derivation of the Demand Curve
Now suppose the price of commodity X falls, such that the new
budget line rotates outwards, to become AC.
But first, suppose we make a ‘compensating variation of income so
that the consumer is left with just enough money to continue
purchasing bundle Z if he/she so wishes.
MICROECONOMICS 1 – REVEALED PREFERENCE
Derivation of the Demand Curve
The compensated budget line is shown by A’B’, which passes
through bundle Z to illustrate the idea of income compensation.
Because bundle Z is still available to the consumer, he/she will not
choose any bundle to the left of Z. Why?
MICROECONOMICS 1 – REVEALED PREFERENCE
Derivation of the Demand Curve
Thus, the consumer will continue to consume bundle Z, in which
case the substitution effect of the price fall is zero, or choose a batch
on the segment ZB’, such as bundle W (which includes larger
quantities of commodity X, for which the substitution effect is
negative).
MICROECONOMICS 1 – REVEALED PREFERENCE
Derivation of the Demand Curve
Now if we allow the consumer to move back to the new budget
line, AC, the consumer may choose a bundle to the right of W, say
N (if commodity X is normal with a positive income effect).
MICROECONOMICS 1 – REVEALED PREFERENCE
Derivation of the Demand Curve
The new revealed equilibrium position (N) includes a larger
quantity of commodity X (x3) resulting from the fall in its price.
Thus, the revealed preference axiom and the implied consistency of
choice lead us to a direct derivation of the demand curve: as price
falls of a commodity, more of it is purchased.
MICROECONOMICS 1 – REVEALED PREFERENCE
The Weak Axiom of Revealed Preference
How do we know the consumer is following the maximising
model?
What kind of observation would lead to us to conclude that the
consumer was not maximising?
MICROECONOMICS 1 – REVEALED PREFERENCE
The Weak Axiom of Revealed Preference
The weak axiom of revealed preference can be stated as follows: if
(x1, y1) is directly revealed preferred to (x2, y2), and the bundles are
not the same, then it cannot happen that (x2, y2) is directly revealed
preferred to (x1, y1). The weak axiom can be explained using
Figures 4 and 5.
MICROECONOMICS 1 – REVEALED PREFERENCE
Figure 4: Violation of the Weak Axiom of Revealed Preference
Y
•
(x1, y1)
• (x2, y2)
X
MICROECONOMICS 1 – REVEALED PREFERENCE
The Weak Axiom of Revealed Preference
In Figure 4, we observe the consumer making choices that do not
follow from the logic of revealed preference.
This is because two conclusions can be arrived at: 1) we observe that is
one case, (x1, y1) is revealed preferred to (x2, y2); and 2) in another
instance, (x2, y2) is revealed preferred to (x1, y1).
MICROECONOMICS 1 – REVEALED PREFERENCE
Figure 5: Satisfying the Weak Axiom of Revealed Preference
Y
A
●
(x1, y1)
B
●
(x2, y2)
X
MICROECONOMICS 1 – REVEALED PREFERENCE
The Weak Axiom of Revealed Preference
The Weak Axiom of Revealed Preference is however satisfied in
Figure 5. Here we observe that his/her choices are consistent with
the logic of revealed preference. That is, when either bundle of
goods is chosen, the other is not affordable to the consumer.
MICROECONOMICS 1 – REVEALED PREFERENCE
The Weak Axiom of Revealed Preference
This leads us to restate the weak axiom of revealed preference
algebraically as follows:
MICROECONOMICS 1 – REVEALED PREFERENCE
The Weak Axiom of Revealed Preference
If a bundle of goods (x1, y1) is purchased at prices (p1, p2) and
different bundle (x2, y2) is purchased at prices (p3, p4), then if
𝑝1 𝑥1 + 𝑝2 𝑦1 ≥ 𝑝1 𝑥2 + 𝑝2 𝑦2 ,
then it must not be the case that
◦
𝑝3 𝑥2 + 𝑝4 𝑦2 ≥ 𝑝3 𝑥1 + 𝑝4 𝑦1
MICROECONOMICS 1 – REVEALED PREFERENCE
The Weak Axiom of Revealed Preference
In simple words, if bundle B is not affordable when bundle A is
purchased, then when bundle B is purchased, bundle A must not be
affordable.
MICROECONOMICS 1 – REVEALED PREFERENCE
The Strong Axiom of Revealed Preference
The weak axiom requires that if A is directly revealed preferred to
B, then we should never observe B being directly revealed preferred
to A.
The Strong Axiom of Revealed Preference requires that the same
sort of condition hold for indirect revealed preference.
MICROECONOMICS 1 – REVEALED PREFERENCE
The Strong Axiom of Revealed Preference
More formally, the Strong Axiom of Revealed Preference can be
stated as follows: if (x1, y1) is revealed preferred to (x2, y2) (either
directly or indirectly),and (x2, y2) is different from (x1, y1), then
(x2, y2) cannot be directly or indirectly revealed preferred to (x1, y1).
MICROECONOMICS 1 – REVEALED PREFERENCE
The Strong Axiom of Revealed Preference
Thus, the Strong Axiom of Revealed Preference is a necessary
implication of optimizing behaviour: if a consumer is always
choosing the best things that he/she can afford, this his/her
observed behaviour must satisfy the strong axiom of revealed
preference.
MICROECONOMICS 1 – REVEALED PREFERENCE
The Strong Axiom of Revealed Preference
Further, we can also conclude that the Strong Axiom of Revealed
Preference is a necessary condition for optimizing behaviour: if the
observed choices satisfy the strong axiom of revealed preference,
then it is always possible to find preferences for which the observed
behaviour is optimizing behaviour.