[Note for bibliographic reference: Melberg, Hans O. (1997), Words, Figures or
Mathematics - A review of Rødseth's Consumer Theory , http://www.oocities.org/hmelberg/papers/970512.htm]
Words, Figures or Mathematics?
A review of Rødseth's Consumer Theory
by Hans O. Melberg
Asbjørn Rødseth
Konsumentteori (Consumer Theory)
Oslo, Universitetsforlaget, 1992
ISBN: 82-00-21568-7
184 pages
Introduction
The purpose of Konsumentteori (Consumer Theory) by Asbjørn Rødseth, is to give
graduate students an introduction to standard consumer theory. Judged by this, the book is
a great success. It is concise, pedagogic and formal - in short, a great book if you are
preparing for an exam in the subject.
This review can be divided into three parts. First, I shall explain in more detail why
I think this is a good book. Second, I will try to criticise the book without questioning
standard consumer theory. Finally, in the third part I shall criticise standard consumer
theory. The distinction between the second and the third part is important. It is
perfectly possible to criticise an approach, while admitting that a particular book is
very good at explaining the basics of this approach. I shall use the label
"endogenous" for the criticism of Rødseth's explanation of the approach, and
"exogenous" for criticism of the approach itself.
Why is this a good book?
A good textbook presents the reader with a map of the topic before the student is thrown
into the finer details. The value of this map is to give the student a feeling for how
everything fits together - to see how the various topics relate to each other. I have seen
few do this better than Rødseth. In less than five pages he explains the main content of
the book. The first topic, of course, is how maximization of utility (for a given level of
income) leads to downward sloping demand curves. He then shows how mimimization of
expenses (given a certain fixed level of utility) leads to the same result as maximization
of utility. The reader is then given a guided tour of the various concepts derived from
the solutions to the maximization and the minimization problem: Compensated and
uncompensated demand curves, the cost function, the indirect utility function. These
concepts are defined, the properties of the functions are examined, and Rødseth shows how
they are all related (see especially p. 99). In short, all the major concepts in standard
consumer theory is covered, explained formally as well as verbally.
Although the discussion often relies on mathematics, Rødseth often hints at
non-standard topics in his verbal interpretations. For example, chapter three is devoted
to a closer examination of the concept of utility. Is it an ordinal or a cardinal concept?
What is the relationship between the satisfaction of preferences and welfare? (It seems
wrong to count the satisfaction of a desire for drug as an increase in welfare) What if
preferences are lexicographic? (That I always prefer A instead of B regardless of the size
of B.) What about interpersonal comparisons of utility? (Necessary in order to have a
meaningful discussion of aggregate consumer surplus in welfare economics) What if we
demand a good because it is expensive (in order to show-off)? I found many of these
questions to be fascinating, and Rødseth deserve credit for at least mentioning these
more philosophical topics.
Altogether, to the question 'why this is a good book', the short answer is that it
succeeds well in giving the reader an understanding of standard consumer theory. Moreover,
it also makes the reader aware of some of the problems with the standard theory. Finally,
it does so in a clear and concise fashion.
"Endogenous" problems
No book is absolutely perfect. First of all the book does not cover uncertainty or risk.
This topic is omitted on purpose, but I think it is so central that it deserves to be
included. Second, the book could have profited from a worked example using Cobb-Douglas
preferences (for example, in chapter 6.2). Combined with some real numbers this would
increase the students understanding of the various concepts. Moreover, a concrete example
of how one may go from compensated demand curves to the cost function, would help students
realize why the dual problem is worth their attention. (Click to see a
worked example of Cobb-Douglas preferences.)
Third, I have some small quibbles. In chapter 7.3 Production in the household
(p. 110-110) I think Rødseth should have distinguished between Becker's theory of
production in the household and Lancaster's characteristics' approach (which, in fact, was
first suggested by Gorman in 1955 - but not published until 1980). In Rødseth's book the
title of the chapter suggests a presentation of Becker's views, while the discussion
mainly presents Lancaster's theory. (The standard reference to Becker is Economic
Journal, 1965. Both theories are discussed in Deaton and Muellbauer, 1980)
Another small quibble, is the failure to mention that uncompensated demand curves are
often called Marshall demand, while compensated demand curves are called Hicks demand
(which he noes on p. 54). For readers who later go on to read English textbooks and
journals this is useful information. Similarly, he could have mentioned the term
"revealed preferences" on in his discussion of the problematic relationship
between preferences and welfare (p. 48-49). He could also have offered a more formal
definition of a good (which includes its physical properties, the time and place of
delivery, and the state of nature obtaining).
A final, more serious, problem was the somewhat boring emphasis on technical problems
as opposed to substantive issues. However, this problem really belongs to the third
category of exogenous criticism, since it is more a problem of the standard approach to
consumer theory, than this specific book.
"Exogenous" criticism
What is the main content of books on consumer theory? A quick look at Rødseth's book, and
others, reveal that much space is devoted to the discussion of the mathematical properties
of the various functions given certain assumptions. The outline of a standard book would
be something like this. First, the author introduces the standard assumptions (a complete
preference ordering, reflexive preferences, transitive preferences, continuity,
rationality, perfectly divisible commodities, and given unit prices i.e. no bulk
discounts). Next, the Marshallian (uncompensated) demand curves are derived. A large
section is then devoted to discussing the properties (homogeneity, convexity, increasing
or decreasing in various variables, continuity) of the various functions (the utility
function - direct and indirect, the cost function, the demand curves - compensated and
uncompensated) as derived from the assumptions regarding preferences, and as they follow
from each other (e.g. prove that the cost of living function is concave if the
indifference curve is covex). This section often include a discussion of concepts such as
the Engel curve, elasticity, complements, substitutes, normal goods, inferior goods etc.
Lastly, the author may give some applied examples dealing with the choice between leisure
and labour, or the choice between consumption and saving.
The question is how useful it is to know all this. According to Rødseth, one aim with
the exercise above, is to find restrictions on the demand functions (p. 92). Thus, one
point of spending a lot of time deriving the mathematical implications of a theory, is
that it enables us to test the theory. If real-world data are consistent with our
implications, we are more certain that the standard theory is correct. The problem with
this, as Rødseth points out, is that consumer theory is unfalsifiable unless we assume
that the utility function of different people have at least some common characteristics
(p. 93). As Rødseth briefly notes "Economists often make the assumption that these
common characteristics exist, but there is as yet no coherent and common theory as to what
these traits are" (p. 93, my translation). This is interesting. Why bother spending a
lot of time on deriving implications of something that is close to unfalsifiable?
A second critique of standard consumer theory, is that it pays a low marginal return in
terms of increased understanding of consumption - the question of why we buy the
quantities we do. I think this is related to the insistence on presenting the theory in a
mathematical language. This, in turn, leads us to ignore some potentially important
variables (either because of lack of space and time, or because the arguments cannot be
treated in a simple mathematical framework). For example, Rødseth mentions the snowball
effect and Veblen-goods. In the first, my demand for a good depends on other peoples
consumption (I will not by a BETA video if most other people buy VHS). In the second,
people buy goods because they are expensive (to show off). In this case, reducing
the price might reduce even uncompensated demand. These are only two examples of the
complexities of consumption. I think there are many more, and I believe it is much more
interesting and useful (in terms of understanding) to hunt these complexities (For an
attempt to do so, see a sub-chapter of my observation Russians
about Russians.)
Maybe one could justify the standard theory by arguing that one has to learn the basics
before one is allowed to explore the complexities - one has to learn how to walk before
one can run. I would agree with this. Nevertheless, one might still argue that the balance
is skewed too far toward learning the mathematical formalities of the basics, and not
enough toward the broader theory. There is also an additional argument: What if the basic
frame actually hinders the exploration of the broader theory? For example, within the
formal language of mathematics it is easy to focus on technical problems: Is the income
effect larger than the substitution effect (Giffen goods)? (p. 85) What if we use
Laspeyres instead of Paasche index? (p. 66) Maybe I am wrong, but these are more technical
issues than substantive. It is very difficult to find a real-life example of a Giffen good
(but it is easy to find examples of Veblen goods); The difference between Laspeyres and
Paasche is usually not of great practical significance, but it is easily discussed within
a mathematical frame - in contrast to a verbal discussion about the problems of
interpersonal comparison of utility. In sum, when we use a mathematical model, there is a
tendency to focus on these technical questions instead of the more substantive and
practically important questions. This arguments makes me more inclined to reduce (not
eliminate) the importance of learning the mathematical formalities of the basic theory.
(See Boettke's article "What is wrong
with neo-classical economics" for an attempt to discuss how economists are
distracted from the main questions by their excessive use of mathematics).
Conclusion
There is no doubt that this is a good book. It is relatively short, yet surprisingly
rigorous and complete. All the standard topics - with the exception of expected utility
theory - are covered, and the author gives many verbal hints at more non-standard topics.
He is, moreover, very good at explaining (see, for example the discussion of the concavity
of the utility function on p. 18). Still, there is room for improvement. My main
suggestion would be a a slight reduction of the emphasis on technical details, and more
verbal discussions of non-standard topics. This would make the book less boring, and
increase understanding of the basic question: Why do we consume the quantities we do of
various goods.
A note on sources
The review relies heavily on lecture notes from P.J.N. Sinclair and M. E. Williams,
University of Oxford (1992). Another reference is the chapter on Consumer Theory in the
book The Theory of Choice: A Critical Guide.
A worked example: Cobb-Douglas Preferences
(from a lecture by P. J. N. Sinclair at the University of Oxford)
The direct utility function expresses utility as a function of the quantities consumed
of two goods, q1 and q2:
(1) U = q1a q21-a
We assume the consumer maximize this utility function, subject to the budget
constraint:
(2) m = p1q1 + p2q2
In order to solve this problem, we form the Lagrangean function:
(3) L = q1a q21-a - s (p1q1
+ p2q2 - m)
Setting the partial derivatives of (3) equal to zero, gives us the following first
order conditions:
(4) dL/dq1 = a q1a-1 q21-a - s p1
= 0
(5) dL/dq2 = (1-a) q1a q2a1-a-1
- s p2 = 0
(6) dL/ds = m - p1q1 - p2q2 = 0
We can simplify (4) and (5) using (1):
(4') a U/q1 - s p1 = 0
(5') (1-a) U/q2 - s p2 = 0
From (4') and (5') we get:
(7) p2q2 = (1-a)/a q1p1
Or,
(7') p1q1 = a/(1-a) p2q2
By substituting (7) into (6), and solve, we get find the following solution:
(8) q1 = a m/p1
To find the solution for q2, we substitute (7') into (6):
(9) q2 = (1-a) m/p2
We have now completed the first part of our problem: To find an expression for the
demand curves starting from constrained optimization of a utility function. The demand
curves found in this way are called Marshallian demand, and express the demand for a good
as a function of prices and income.
The next step is to find the cost function and the compensated demand curves. I'll
start with the cost function i.e. the level of income needed to achieve a certain level of
utility given a set of prices. To find this expression, we by substitute the demand
functions (8) and (9) into (1). This gives us the indirect utility function
(utility expressed as a function of prices and income, not goods):
(10) U = (a m/p1) a [(1-a) m/p2]1-a
Or, simplified:
(10') U = m (a/p1)a [(1-a)/p2] 1-a
Inverting (10') gives the cost function:
(11) m = (p1/a)a [p2/(1-a)]1-a U
The compensated demand curves (also called Hicksian demand curves) are found by
minimizing m for a given U. Taking the partial derivatives of (11) we have the following
first order conditions (after some simplification):
(12) dm/dp1 = q1 = (p2/p1)1-a
U/aa(1-a)1-a
(13) dm/dp2 = q2 = (p1/p2)a U/aa(1-a)1-a
(12) and (13) are the compensated demand curves, i.e. they show how the demand for a
good changes when its relative price change. In other words, it shows the pure
substitution effect - that you would demand less of a good when its price increases even
when you are given extra income to compensate for the increase in price.
To sum up. We started with the problem of maximizing utility subject to a budget
constraint. The solution to this problem gave us the uncompensated (Marshallian) demand
curves i.e. demand expressed as a function of income and prices. By substituting these
demand curves into the original utility function, we found the indirect utility function -
utility expressed as a function of prices and income. Inverting this we found the cost
function - an expression of how much income we would need to reach a certain level of
utility given a set of prices. Mimimization of the cost function gave us the compensated
demand curves i.e. a function showing how demand changes in response to relative price
changes when you are given the extra income need to maintain the previous level of
utility.
One might ask - I certainly did - why we would want to learn this. I am still uncertain
about the answer, but here are som suggestions (The book, by the way, could have
benefitted from a some concrete numerical examples discussing the use of learning duality
theory):
1. In empirical research some kind of information is sometimes easy to find, while some
is close to impossible to get. So, while we may want to estimate the uncompensated demand
function, we only have information on the compensated demand function. The set of
relationships outlined shows us how we should derive uncompensated demand from compensated
demant data.
2. In welfare economics it is very useful to know the cost function and uncompensated
demand curvces (when we are estimating the benefits of a public project and subsidies to
compensate those who loose from the project).
3. Røedseth gives an example of the use of duality theory: he proves how
cash-subsidies is never worse than subsidies when we try to lift people to a certain level
of utility (p. 72). Thus, consumer theory is useful when we design the tax-system, and
more generally when we decide on public policies.
These were some suggestions as to the use of duality in consumer theory. More are
welcome.
[Note for bibliographic reference: Melberg, Hans O. (1997), Words, Figures or
Mathematics - A review of Rødseth's Consumer Theory , http://www.oocities.org/hmelberg/papers/970512.htm]