Chapter 18: Revealed Preference and Revealed Technology
18.2: Introduction
This chapter is in two parts – the first relating to preferences and the second to technology.
Intellectually the material in the two parts is effectively the same. In each part we do the same: first
we discuss how we might use empirical observations to test the assumptions we have been making;
then we discuss how we might use empirical observations to tell us something about preferences or
about technology. In a sense we have already done some of the second already – as we know, for
example, how to distinguish between perfect substitutes, perfect complements or Cobb-Douglas
preferences (or technology) on the basis of observations1. But it is important to be aware of the
methods that we can use to test directly the assumptions that we have been making. Some
economists would argue that these assumptions are definitions of ‘rational behaviour’ and thus must
be true – but empirical evidence would suggest that we should be careful in jumping to such a
conclusion.
The bulk of the material relates to testing our assumptions - using observations on behaviour. We
should be a little careful in using this material for two reasons: (1) we know that any theory we use
is empirically false (since any useful theory must be an approximation to reality); and (2) we know
from countless experiments that human beings make mistakes when taking decisions. In a sense
these reasons are inter-related: given the fact that human beings make mistakes, any theory must
only be approximately true (unless we have some theory about how people make mistakes – which
seems to be almost definitionally impossible).
Be all that as it may, the material in this chapter is useful as it tells us a little about what we might
infer from observations. We begin with direct inferences and then move on to indirect inferences.
18.3: Direct Inferences about Preferences
In this section we ask what we might infer about preferences on the basis of observations of
demand. Let us start with a simple example in which we are given some information about an
individual’s demands in various scenarios. Let us suppose that we have two observations on the
individual’s behaviour. The first observation is when the individual has a money income of 80 and
where the prices of good 1 and good 2 are 2 and 1 respectively. The second observation is when the
individual has a money income of 80 and where the prices of good 1 and good 2 are 1 and 2
respectively. Suppose we observe the individual demanding 10 units of good 1 and 60 units of
good 2 in the first situation and 60 units of good 1 and 10 units of good 2 in the second. We show
all this in the figure 18.1.
1
Recall that: (1) with perfect substitutes the demand for one of the two goods is usually zero; (2) with perfect
complements the ratio of the quantities purchased is constant; (3) with Cobb-Douglas the ratio of the amounts spent is
constant.
The first observation gives us a budget constraint going from (40, 0) to (0, 80): with an income of
80 and with prices 2 and 1 for goods 1 and 2, the individual can either buy 40 of good 1 and 0 of
good 2 or 0 units of good 1 and 80 of good 2 (or any combination in between). This budget line is
shown in the figure as is the observation that the individual, faced with this budget constraint,
bought 10 units of good 1 and 60 of good 2 – this is indicated by the point labelled ‘1’. The second
observation gives us a budget constraint going from (80, 0) to (0, 40): with an income of 80 and
with prices 1 and 2 for goods 1 and 2, the individual can either buy 80 of good 1 and 0 of good 2 or
0 units of good 1 and 40 of good 2 (or any combination in between). This budget line is shown in
the figure as is the observation that the individual, faced with this budget constraint, bought 60 units
of good 1 and 10 of good 2 – this is indicated by the point labelled ‘2’.
Let us test the assumption that the preferences underlying these demands are represented by strictly
convex indifference curves. If this assumption is true, what can we infer? More importantly, are the
inferences that we can make consistent with each other? Or are they inconsistent with each other? If
the latter then we must conclude that our assumption about convex preferences must be false.
What can we infer from the above observations given our assumption? From observation 1 we can
infer, given that the individual chose the point 1 when in fact he or she could have chosen any point
within the triangle defined by the vertices (0, 0), (40, 0) and (0, 80), that he or she strictly prefers
point 1 to any other point in this triangle. Actually when you think about it – this is quite a strong
inference. It tells us that there must be a (smoothly convex) indifference curve tangential to the
budget constraint at the point 1. From observation 2 we can infer, given that the individual chose
the point 2 when in fact he or she could have chosen any point within the triangle defined by the
vertices (0, 0), (80, 0) and (0, 40), that he or she strictly prefers point 2 to any other point in this
triangle. Again this is a strong inference – once again it tells is that there must be a (smoothly
convex) indifference curve tangential to the budget constraint at the point 2.
So far so good. Are these inferences inconsistent with each other? Well no – there is no
contradiction between the two sets of inferences. One way to see this is to ask whether it is possible
to draw a family of indifference curves – that do not intersect – that are consistent with these
observations. The answer is obviously ‘yes’. Moreover we can do this in various ways – so that
point 1 is preferred to point 2 and vice-versa. Note that the observations do not tell us anything
about the individual’s preference between point 1 and point 2. This is because when the individual
chose 1 the bundle 2 was not available (it was not within the budget constraint); and when the
individual chose 2 the bundle 1 was not available (it was not within the budget constraint).
Let us consider now a second individual. In the first observation he or she has an income of 80 and
the prices of the two goods are 2 and 1 respectively. In the second observation the income is once
again 80 and the prices are 1 and 2 respectively. Let us suppose that this individual is observed to
buy 35 units of good 1 and 10 of good 2 in the first situation and is observed to buy 10 units of good
1 and 35 units of good 2 in the second. These observations are shown in figure 18.4.
What can we infer? From the first observation – that with the budget constraint going from (40, 0)
to (0, 80) - that he or she strictly prefers point 1 to all points within the triangle [(0, 0), (40, 0) (0,
80)]. Why? Because he or she chose point 1 when all the points within the triangle were available in
the sense that they were all purchasable. But note that point 2 is within this triangle. So we can infer
from this first observation that point 1 is strictly preferred to point 2. Point 1 was chosen when point
2 was available (purchasable).
The problem is that, from the second observation, we can infer exactly the opposite. In this second
observation the individual chose point 2 when all the points within the triangle [(0, 0), (80, 0), (0,
40)] were available. But note that point 1 is within this triangle. So we can infer from this second
observation that point 2 is strictly preferred to point 1. Point 2 was chosen when point 1 was
available (purchasable).
So we have two inconsistent observations:
From the first observation that point 1 is strictly preferred to point 2
From the second observation that point 2 is strictly preferred to point 1
What do we conclude? That this individual’s behaviour is inconsistent with our assumption (of
smoothly convex indifference curves). Indeed we might be tempted to conclude that this individual
is crazy – one moment preferring point 1 to point 2 and the next moment preferring point 2 to point
1. You might note that it is impossible to draw smoothly convex indifference curves – that do not
cross – which leads to the observed choices2.
This observed behaviour is a violation of the assumption that we have been using. It suggests that
the individual is not predictable. Our assumption is that the individual is predictable in that if he or
she has a preference then he or she has that preference. It seems to me that if we want to make
economics a science which can predict behaviour we need to make some such assumption.
Incidentally and mainly for the record, I should note that the behaviour of our second individual is a
violation of an assumption that economists call the Weak Axiom of Revealed Preference. This says
2
If you try and draw one indifference curve through point 1 which is tangential to the first budget constraint and a
second indifference curve through point 2 which is tangential to the second budget constraint – and which are convex
everywhere - they are doomed to cross, which we know is crazy. If we try to rationalise the observations with concave
indifference curves, this fails because the individual would choose (almost) always an extreme (where either none of
good 1 or none of good 2 is bought).
that if a bundle X is revealed directly3 preferred to another bundle Y then it cannot be the case that Y
is revealed directly preferred to X.
18.4: Indirect Inferences about Preferences
In the section above we discussed how we might make direct inferences about preferences. For
example, from the first observation shown in figure 18.4 above we could directly infer that point 1
was preferred to point 2 (because 2 was available but 1 was chosen). If we make the assumption of
transitivity we can use a series of observations to make indirect inferences about preferences.
Transitivity is the following assumption: if 1 is preferred to 2 and 2 is preferred to 3 then it must be
true that 1 is preferred to 3. Some people regard this assumption as almost tautological but it is an
assumption. Consider how it might be used.
Suppose we have the following three observations on an individual’s behaviour:
observation 1: the individual has income 120 and faces prices 3 and 1 for goods 1 and 2, and
demands 10 units of good 1and 90 of good 2;
observation 2: the individual has income 80 and faces prices 1 and 1 for goods 1 and 2, and
demands 20 units of good 1 and 60 units of good 2;
observation 3: the individual has income 120 and faces prices 1 and 3 for goods 1 and 2, and
demands 48 units of good 1 and 24 units of good 2.
These are pictured in figure 18.7.
What can we infer?
From observation 1 we can infer that point 1 (which was chosen) is preferred to all the points within
the triangle [(0, 0), (40, 0), (0, 120)] including point 2 (which is within that triangle and hence
available). From observation 2 we can infer that point 2 (which was chosen) is preferred to all the
points within the triangle [(0, 0), (80, 0), (0, 80)] including point 3 (which is within that triangle and
hence available). From observation 3 we can infer that point 3 (which was chosen) is preferred to all
the points within the triangle [(0, 0), (120, 0), (0, 40)]. We have no direct evidence of the
individual’s preferences between points 1 and point 3 (because when 1 was chosen 3 was not
available and when 3 was chosen 1 was not available) but using observations 1 and 2, we know that
3
Through the kinds of observations that we have been discussing above.
1 is preferred to 2 and that 2 is preferred to 3, from which it follows (using transitivity) that 1 must
be preferred to 3. This is an example of an indirect inference.
Once again for the record we should note that implicit in what we have assumed (if we include
transitivity as one of our assumptions) is a second assumption that economists call the Strong Axiom
of Revealed Preference which says that if a bundle X is revealed directly or indirectly preferred to
another bundle Y then it cannot be the case that Y is revealed directly or indirectly preferred to X. Of
course if this axiom is violated it simply means that the individual cannot have preferences of the
type that we have assumed – in essence the individual seems to be unaware of whether he or she
prefers X to Y or vice versa.
18.5: Inferring Preferences
If the individual’s behaviour does not violate either the Weak or the Strong Axiom of Revealed
Preference then it must be the case that we can draw in indifference curves that are consistent with
that behaviour. Obviously with just a few observations we cannot infer very much but as the
number of observations increases we can build up a clearer picture. This is one way to use
observations to infer preferences.
However most economists prefer an alternative strategy. That is to assume that preferences are one
of a smallish set of possible preferences (such as perfect 1:a substitutes, perfect 1-with-a
complements, Cobb-Douglas with parameter a) and then use the observations to try and distinguish
between these possibilities and to estimate the parameter a. Of course there is almost certainly some
degree of approximation involved – as it is almost impossible that any one preferences fits the data
exactly. A decision must be then taken to see if the approximation is ‘good enough’. If it is, then
fine; if not then the set of possible preferences must be widened in the search for a better
approximation. We have said a little about this in chapter 16.
18.6: Direct Inferences about Technology
The material in this section is effectively the same as in section 18.2. The only difference is the
context. In section 18.2 we discussed how observations on demand for goods by a consumer could
be used to make inferences about his or her underlying preferences. We used this material to see
how we might use such observations to test whether these underlying preferences could be
represented by smoothly convex indifference curves – as we have been assuming.
In this section we change context. The context is now that of a firm buying inputs for its production
process. Observations relate to the prices of the two inputs and the incurred costs of the firm and the
actual demand by the firm for the two inputs. We want to infer from these observations whether the
production process of the firm is consistent with the assumption that smoothly convex isoquants
exist.
Consider the following two observations. The first observation is when the firm reports input costs
of 80 and where the prices of input 1 and input 2 are 2 and 1 respectively. The second observation
is when the firm reports input costs of 80 and where the prices of input 1 and input 2 are 1 and 2
respectively. Suppose we observe the firm demanding 10 units of input 1 and 60 units of input 2 in
the first situation and 60 units of input 1 and 10 units of input 2 in the second. We show all this in a
figure.
What can we infer from these observations? First that (input) bundle 1 produces a greater level of
output than all the bundles inside the triangle [(0, 0), (40, 0), (0, 80)] - because bundle 1 was chosen
while all the points inside the triangle were possible purchases. Second, that bundle 2 produces a
greater level of output than all the bundles inside the triangle [(0, 0), (80, 0), (0, 40)] - because
bundle 2 was chosen while all the points inside the triangle were possible purchases. We might also
ask what can we infer about the relative outputs of bundles 1 and 2. On this we can infer very little.
Consider the first situation. From figure 18.11 we see that that when the firm chose bundle 1, it
could not have bought bundle 2. So we can not infer whether bundle 1 or bundle 2 produces a
higher output from this observation. Furthermore, from the figure we see that that when the firm
chose bundle 2, it could not have bought bundle 1. So we can not infer whether the output of the
firm is greater with bundle 1 or bundle 2 from this observation.
However consider the following set of observations. Suppose we observe a firm facing the same set
of prices and reporting the same costs and we observe that the firm buys 35 units of input 1 and 10
of input 2 in the first situation and buys 10 of input 1 and 35 of input 2 in the second. What can we
infer? Consider figure 18.14.
From the first observation, that bundle 1 leads to a higher output than all the points in the triangle
[(0, 0), (40, 0), (0, 80)] - including bundle 2. And from the second observation that bundle 2 leads
to a higher level of output than all the points inside the triangle [(0, 0), (80, 0), (0, 40)] - including
bundle 1. This is manifestly inconsistent with our assumptions about technology - the observations
are telling us that simultaneously bundle 1 produces a higher output than bundle 2 and bundle 2
produces a higher output than bundle 1 It is a violation of our assumptions. We might even
conclude that the firm is crazy. We might call this a violation of the Weak Axiom of Revealed
Technology which asserts that this can not happen. Another way of seeing this is to see that it is
impossible to draw isoquants which do not cross which would imply the choices made by the firm
in the two situations. Try it!4
18.6: Indirect Inferences about Technology
It should now be clear that we may be able to infer technologies from observations (and to check
whether these technologies are consistent with our assumptions). Such inferences may be direct or
indirect. Consider the following three observations
observation 1: costs 120 and input prices 3 and 1 – the firms demands 10 and 90 of inputs 1 and 2;
observation 2: costs 80 and input prices 1 and 1 – the firm demands 20 and 60 of inputs 1 and 2;
observation 3: costs 120 and input prices 1 and 3 – the firm demands 48 and 24 of inputs 1 and 2;
We show these in figure 18.17.
Now let us consider them one by one, particularly looking at the firm's revealed technology for the
three bundles chosen With the first observation we can infer that bundle 1 produces a bigger output
than bundle 2 - because bundle 1 was chosen when bundle 2 was available. But we can say nothing
about the relative outputs of bundles 1 and 3 or of 2 and 3. The second observation allows us to
infer that bundle 2 produces a bigger output than bundle 3 - because 2 was chosen when 3 was
available. But it tells us nothing about the relative outputs of 1 and 2 nor of 1 and 3. The third
observation does not tell us anything about the relative outputs of the three bundles.
However, if we combine the inferences from the first two observations we can infer that bundle 1
must produce a bigger output than bundle 3 - because 1 produce a bigger output than bundle 2 and
bundle 2 produces a bigger output than bundle 3.
The Strong Axiom of Revealed Technology says that we should not be able to find any
inconsistency in inferred output whether the inferences are direct or indirect.
4
Note - concave technologies do not work as it is clear that with such technologies the firm would
always choose a point on one of the two axes.
18.7: Inferring Technology
If the firm’s behaviour does not violate either the Weak or the Strong Axiom of Revealed
Technology then it must be the case that we can draw in isoquants that are consistent with that
behaviour. Obviously with just a few observations we cannot infer very much but as the number of
observations increases we can build up a clearer picture. This is one way to use observations to infer
technology.
Most economists prefer an alternative strategy. That is to assume that the technology is one of a
smallish set of possible technologies (such as perfect 1:a substitutes, perfect 1-with-a complements,
Cobb-Douglas with parameter a) and then use the observations to try and distinguish between these
possibilities and to estimate the parameter a. Of course there is almost certainly some degree of
approximation involved – as it is almost impossible that any one technology fits the data exactly. A
decision must be then taken to see if the approximation is ‘good enough’. If it is, then fine; if not
then the set of possible technologies must be widened in the search for a better approximation. We
have said a little about this in chapter 16.
18.8: Summary
This chapter has been concerned with what we can infer from observations – particularly
concerning whether the assumptions we have been using are consistent with the observations. We
saw how we might make direct or indirect inferences. One important assumption is that known as
the Weak Axiom of Revealed Preference. This states that:
If a bundle X is revealed directly preferred to another bundle Y then it cannot be the case that Y is
revealed directly preferred to X.
A second important axiom is the Strong Axiom of Revealed Preference which states:
If a bundle X is revealed directly or indirectly preferred to another bundle Y then it cannot be the
case that Y is revealed directly or indirectly preferred to X.
There are similar axioms for revealed technology.