NAME: OLUCHI HAPPINESS ORJI
PROGRAMME: MSC 2022/2023
MATRIC NUMBER: 2022116006FA
FACULTY: ECONOMICS
COURSE: ADVANCED MICRO ECONOMICS
LECTURER: DR BEN UZOECHINA
TOPIC: CONSUMER PREFERENCE AND UTILITY
AUGUST 2023
ABSTRACT
Today’s consumers are too smart to buy their needs through various
means. But before buying their needs, they go through various online sites
and social media review about product performances and price. While
surfing this information they can able to evaluate its real value and price
advantages, since online establishment need not spend cost for showroom
with staff. Consumers need not roam here and there to various shops to
evaluate the product performance and its cost. Moving from one place to
other is tedious journey and time consuming part. It is also difficult to
ensure their required models are available or not. Moreover, consumers
can view forthcoming new models in the manufacture’s site whereas these
details may not be shared in showrooms. Earlier accessing internet is
complicated and a system to view. Now this can be accessed through
smart phone. The prices of smart phone were also drastically lowered.
This became an affordable price for the common people.
Understanding consumers' decision-making process is one of the most
important goal for any producer. However, the traditional tools (e,g,
surveys, personal interviews and observations) used in research are often
inadequate to analyse and study consumer behaviour. Since people's
decisions are influenced by several unconscious mental processes, the
consumers very often do not want to, or do not know how to, explain their
choices.
The relationship between Consumer preference and Utility theory is
discussed in this paper. We describe why Producers and consumers should
take into account this Consumer preference and Utility to maximize thier
resources.
Keywords; Utility, Preferences,Consumers, producers, behaviour, theory,
function,Axioms, assumptions, curve, graphically, substitute, choice,
preference, demand, supply
Reference to this research work should be made to:Intermediate micro
Economics by Harl .R. Varian, Google search. How to win customers and
keep them for life by Micheal Labeouf. The undercover Economist by Tim
harford.
TABLE OF CONTENTS
Introduction ………………………………………………………………………………
Consumer preference…………………………………………………………………..
Assumptions of consumer preference …………………………………………..
Completeness axioms of consumer theory…………………………………….
Transitivity axioms of consumer theory…………………………………………
Non-satiation axioms of consumer theory …………………………………….
Strict convexity axioms of consumer theory …………………………………
Relative axioms of consumer theory ……………………………………………
Continuity axioms of consumer theory …………………………………………
Indifference axioms of consumer theory ……………………………………..
Cob douglas Preference …………………………………………………………….
Indifference curve …………………………………………………………………….
Terms of preference relatives …………………………………………………….
Utility ………………………………………………………………………………………
Cardinalist Utility ……………………………………………………………………….
Marginalist Utility ………………………………………………………………………
Law of diminishing marginal utility ………………………………………………
Utility for commuting …………………………………………………………………
Forms of utility …………………………………………………………………………
Characteristics of Utility ………………………………………………………………
Utility Function ………………………………………………………………………….
Utility monotonic transformation …………………………………………………
Consumer choice rule ………………………………………………………………..
Lexicography Ordering ………………………………………………………………
Demand function …………………………………………………………………….…
Revealed preference …………………………………………………………………
Weak axiom of revealed preference ……………………………………………
Expenditure function ………………………………………………………………..
Hicksian demand function …………………………………………………………
The von neumann - Morgenstern utility function …………………………
Conclusion ………………………………………………………………………………
Questions and Answers ………………………………………………………….
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CONSUMER PREFERENCE AND UTILITY
Introduction
Consumer preference theory is a theory that explains how consumers make
decisions. It is based on the idea that consumers are rational and will
choose the product or service they believe will satisfy their needs While
Utility is the total satisfaction derived by a consumer from the consumption
of a particular good or service, Utility was thought of as a numeric measure
of a person’s happiness.
Consumer preferences allow a consumer to rank different bundles of goods
according to levels of utility, or the total satisfaction of consuming a good
or service. It is important to understand that consumer preferences are not
dependent upon consumer income or prices.19 Jan 2022. The price a
consumer is willing to pay for a good depends on its marginal utility, which
declines with each additional unit of consumption, according to the law of
diminishing marginal utility. Therefore, the price decreases for a normal
good when consumption increases.
Preferences can be represented by a utility function when they are able to
be quantified and assigned numerical values. When the ordering of
alternatives is complete. In other words, for all pairs of alternatives (a, b)
you can say of your utilities, a>b, a<b, or a=b, and if a>b and b>c then
a>c (transitivity). Utility is dependent on the bundles of goods consumed
by an individual, with greater quantities of goods representing greater
levels of happiness or utility. For consumers, their decisions are driven,
quite simply, by what they want! All consumers make decisions to
maximize their utility.
Consumer preference is the subjective taste of customers gauged by their
satisfaction. Utility is the key element to understanding the preferences of
the consumer.
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CONSUMER PREFERENCE
Consumer Preference is simply the choice of a consumer. Consumer
preferences allow a consumer to rank different bundles of goods according
to levels of utility, or the total satisfaction of consuming a good or service.
Consumer preference is a significant part of microeconomics. Customer
preferences include the concepts of the budget line, utility, indifference
map, and indifference curve which are very closely associated with
customer satisfaction.
It is important to understand that consumer preferences are not dependent
upon consumer income or prices. We call the objects of consumer choice
consumption bundles. This is a complete list of the goods and services that
are involved in the choice problem that we are investigating. The word
“complete” deserves emphasis: when you analyze a consumer’s choice
problem, make sure that you include all of the appropriate goods in the
definition of the consumption bundle.
If we are analyzing consumer choice at the broadest level, we would want
not only a complete list of the goods that a consumer might consume, but
also a description of when, where, and under what circumstances they
would become available. After all, people care about how much food they
will have tomorrow as well as how much food they have today. A raft in
the middle of the Atlantic Ocean is very different from a raft in the middle
of the Sahara Desert. And an umbrella when it is raining is quite different
good from an umbrella on a sunny day. It is often useful to think of the
“same” good available in different locations or circumstances as a different
Good, since the consumer may value the good differently in those
situations. However, when we limit our attention to a simple choice
problem, relevant goods are usually pretty obvious. We’ll often adopt the
idea described earlier of using just two goods and calling one of them “all
other goods” so that we can focus on the trade-off between one good and
everything else. In this way, we can consider consumption choices
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involving many goods and still use two-dimensional diagrams. So let us
take our consumption bundle to consist of two goods,
let x1 denote the amount of one good
x2 the amount of the other.
The complete consumption bundle is therefore denoted by
(x1, x2).
As noted before, we will occasionally abbreviate this consumption bundle
by X.
We will suppose that given any two consumption bundles, (x1, x2) and (y1,
y2), the consumer can rank them as to their desirability. That is, the
consumer can determine that one of the consumption bundles is strictly
better than the other, or decide that she is indifferent between the two
bundles.
We will use the symbol to mean that one bundle is strictly preferred over
another so that
(x1, x2) (y1, y2)
Should be interpreted as saying that the consumer strictly prefers
(x1, x2) to (y1, y2),
in the sense that she definitely wants the X-bundle rather than the Ybundle. This preference relation is meant to be an operational notion. If
the consumer prefers one bundle to another, it means that he or she would
choose one over the other, given the opportunity. Thus the idea of
preference is based on the consumer’s behavior. In order to tell whether
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one bundle is preferred to another, we see how the consumer behaves in
choice situations involving the two bundles. If she always chooses (x1, x2)
when (y1, y2) is available, then it is natural to say that this consumer
prefers (x1, x2) to (y1, y2).
If the consumer is indifferent between two bundles of goods, We use the
symbol ∼ and write
(x1, x2) ∼ (y1, y2).
Indifference means that the consumer would be just as satisfied, according
to her own preferences, consuming the bundle (x1, x2) as she would be
consuming the other bundle,(y1, y2).
If the consumer prefers or is indifferent between the two bundles we say
that she weakly prefers (x1, x2) to (y1, y2) and write (x1, x2) (y1, y2).
These relations of strict preference, weak preference, and indifference are
not independent concepts; the relations are themselves related!
For example,
If (x1, x2) (y1, y2) and (y1, y2) (x1, x2)
We can conclude that (x1, x2) ∼ (y1, y2).
That is, if the consumer thinks that (x1, x2) is at least as good as (y1, y2)
and that (y1, y2) is at least as good as (x1, x2), then the consumer must
be indifferent between the two bundles of goods.
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ASSUMPTIONS ABOUT PREFERENCES
The theory has been used in marketing for decades to help companies
understand what products and services consumers prefer. It can also be
used to determine whether a product or service is worth an investment.
There are many different ways to determine consumer preferences, such
as surveys, interviews, focus groups, and ethnographic research.
There are three types of assumptions: completeness, transitivity, and nonsatiation, we assume that more is better. The first two assumptions reflect
a broader belief that consumers are rational that they make logically
consistent decisions.
Economists usually make some assumptions about the “consistency” of
consumers’ preferences. For example, it seems unreasonable—not to say
contradictory—to have a situation where (x1, x2) (y1, y2) and, at the
same time, (y1, y2) (x1, x2). For this would mean that the consumer
strictly prefers the x-bundle to the y-bundle ... and vice versa. So we
usually make some assumptions about how the preference relations work.
Some of the assumptions about preferences are so fundamental that we
can refer to them as “axioms” of consumer theory. Here are some of such
axioms about consumer preference.
COMPLETENESS AXIOMS OF CONSUMER THEORY
Consumers assume that consumers have full knowledge of the
commodity they prefer and its substitutes. They cannot say they don’t
know which one they prefer.
It is also assumed that individuals must have a preference relationship
between any two sets of goods; either we must be able to say that they
weakly prefer A to B, or that they weakly prefer B to A, or both
(indifference). If we are told that Dave strictly prefers larger chocolate bars
to smaller ones, this gives us enough information to completely define
Dave’s preferences over the entire space of chocolate bars. If Susie says
that she always prefers the bigger and darker chocolate bar, we do not
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have enough information to define a preference relationship across the
entire space of chocolate bars – if one bar is darker and smaller than
another, but lighter and bigger than a third, which is preferred to which?.
The completeness assumption also states that consumers are rational and
make decisions based on all the information they have. This assumption is
made because consumers control their own preferences and are not
influenced by external factors.
There is a lot of evidence that this assumption does not hold true.
Consumers often make decisions based on incomplete information, which
means that they may be making decisions for irrational reasons.
One important factor to always remember is that the consumer is not
indifferent, which means that consumers want to buy what they want
when they want it and from the place where they want it. The consumer
wants to feel confident that everything is in their control. They want to
know that the product will be delivered on time and that the quality is good
enough for them. However, the consumer may not know what product or
service will make them feel or experience what they want. We assume that
any two bundles can be compared. That is, given any x-bundle and any ybundle, we assume that (x1, x2) (y1, y2),
Or (y1, y2) (x1, x2), or both, in which case the consumer is indifferent
Between the two bundles.
TRANSITIVITY AXIOMS OF CONSUMER THEORY
Transitivity is a relation between three elements such that if it holds
between the first and second and it also holds between the second and
third it must necessarily hold between the first and third.
This assumption implies that if at first an individual chooses good A over
good B, and if a second time chooses good B over good C, with B being the
same in both cases, then it is logical that the consumer will select good A
over good C
Transitivity assumptions state that the choice of goods by consumers is
rational. The assumption implies that indifference curves must not cross
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each other as it relates to consumer behavior as well as individual
preferences hence transitive.
Another Example is this;
If (x1, x2) (y1, y2) and (y1, y2) (z1, z2),
Then we assume
That (x1, x2) (z1, z2).
In other words, if the consumer thinks that X is at least as good as Y and
that Y is at least as good as Z, then the consumer thinks that X is at least
as good as Z.
It isn’t clear that the transitivity of preferences is necessarily a property
that preferences would have to have. The assumption that preferences are
transitive doesn’t seem. Compelling on the grounds of pure logic alone. In
fact, it’s not. Transitivity is a hypothesis about people’s choice behavior,
not a statement of pure logic. Whether it is a basic fact of logic or not isn’t
the point: it is whether or not it is a reasonably accurate description of how
people behave that matters.
On a price-income line, the consumer selects only one combination.
Consumers prefer larger sets of goods to smaller ones. Consumers have
complete and transitive preferences, meaning that if A is preferred over B,
and B is preferred over C, then A is preferred over C. Consumer behavior
exhibits consistency over time.
The six types of Transitivity processes are Material Process, Mental Process,
Relational Process, Behavioral Process, Verbal Process, and Existential
Process. The concept of transitivity in Halliday's theory is a grammatical
system that is a powerful tool for analyzing the meanings expressed in
clauses.
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Transitivity refers to the property of preference relationships that if one
bundle (bundle A) is preferred to another (bundle B), and that bundle is
preferred to a third (bundle C), then the first bundle must be preferred to
the third.
The relationship between the first and the third bundles will be governed
by the strongest preference relationship in the set; if A is strictly preferred
to B and B is weakly preferred to C (at least as preferred as C), A is strictly
preferred to C. If A is weakly preferred to B and B is strictly preferred to C,
A is strictly preferred to C.
Given that the measurement of cardinal utility is beyond the realms of
practicality (its measurement is theoretically possible by making use of
nuclear imaging such as a Functional Magnetic Resonance Imager), we are
left able only to perform ordinal analysis. If all we are able to do is rank
preferences, it is important that we are able to compare these ranks.
NON-SATIATION AXIOMS OF CONSUMER THEORY
Non-satiation is the state of never being satisfied. Consumers will
always prefer more to less. Economic theory expresses non-satiation
through a mathematical property of the utility function called local nonsatiation, which, simply put, states that for every bundle of goods, there
always exists a better bundle of goods - a bundle that gives higher utility.
Decreasing marginal utility. The assumption is that a consumer will always
benefit from additional consumption. The demand for some goods may
have a finite limit, but it is likely that there is some good or service a
consumer would benefit from having more of. Consumers lose satisfaction
with a product the more they consume it.
Non-satiation assumes that if one person has X amount of something, it
does not mean they will not want more options. People are seldom
satisfied with one trip to the shops and always want to consume more.
The indifference curve has a negative slope. This is caused because of
nonsatiation. The indifference curve cannot slope upward because the
consumer cannot be indifferent between two commodity bundles if
contains more of both goods.
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The assumption that a consumer will always benefit from additional
consumption. The demand for some goods may have a finite limit, but it is
likely that there is some good or service a consumer would benefit from
having more of.
STRICT CONVEXITY AXIOMS OF CONSUMER THEORY
This is a situation where consumers prefer average to extra e.g
B= burgers, C= Ice cream
X1= [10B, 1C]
X2= [1B, 10C]
X3= [5B, 5C]
The consumer under strict convexity will go for X3.
So, in two dimensions, with strictly monotonic preferences,
strict convexity says that if two consumption bundles are each on the same
indifference curve as x, then any point on a line connecting these two
points (except for the points themselves) will be on a higher indifference
curve than x.
Strict convexity comes from the fact that a convex combination of any pair
of bundles is strictly preferred to the pair. Strict convexity implies that the
second derivative or F double prime of X is greater than zero if it's equal to
zero. and that implies that the line can be straight or linear. there's no
slope associated with it.
What is an example of a strictly convex function?
If Hf(x) ≻ 0 for all x ∈ C, then f is strictly convex. But the implication
doesn't go both ways. For example, f(x) = x4 has f//(0) = 0, but is still
strictly convex. The simplest and most important operations that preserve
convexity are addition and multiplication by a positive scalar.
A function f : Rn → R is convex if for any x, y ∈ Rn and any θ ∈ (0,1), θf
( x) + (1 − θ)f ( y) ≥ f (θ x + (1 − θ) y). (1) The function is strictly convex
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if the inequality is always strict, i.e. if x = y implies that θf ( x) + (1 − θ)f
( y) > f (θ x + (1 − θ) y).
A function f is strictly convex if whenever x = y, and 0 <θ< 1, strict
inequality holds, that is, we have f(θx + (1 − θ)y) < θf(x) + (1 − θ)f(y).
REFLEXIVE AXIOMS OF CONSUMER THEORY
Reflexivity in simple terms means that an identical product to X is just
as preferred as X to the customer e.g
X= X i.e. coke is as preferred to Pepsi, just like Pepsi is as preferred to
Coke because they are identical and close substitutes.
Reflexivity is the fact of someone being able to examine their own
feelings, reactions, and motives (=reasons for acting) and how these
influence what they do or think in a situation.
Relationships are reflexive if they can be applied when both sides of the
relationship are the same – i.e. I am at least as old as myself (I am in fact
exactly as old as myself, but the statement is not incorrect, merely
imprecise). Weak preference relationships are reflexive; a bundle of goods
can be said to be weakly preferred to itself, but not strictly preferred to
itself (in fact, it can be more accurately said to be exactly as preferred as
itself).
Reflexivity: this identity condition says that the consumer is indifferent
when comparing a bundle to itself. We assume that any bundle is at least
as good as itself: (x1, x2) (x1, x2).
Reflexivity in simple terms means that an identical product to X is just as
preferred as X to the customer e.g. X= X i.e. coke is as preferred to Pepsi,
just like Pepsi is as preferred to coke because they are identical and close
substitutes.
Reflexivity is the fact of someone being able to examine their own feelings,
reactions, and motives (=reasons for acting) and how these influence what
they do or think in a situation.
Relationships are reflexive if they can be applied when both sides of the
relationship are the same – i.e. I am at least as old as myself (I am in fact
exactly as old as myself, but the statement is not incorrect, merely
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imprecise). Weak preference relationships are reflexive; a bundle of goods
can be said to be weakly preferred to itself, but not strictly preferred to
itself (in fact, it can be more accurately said to be exactly as preferred as
itself).
Reflexivity: this identity condition says that the consumer is indifferent
when comparing a bundle to itself.
We assume that any bundle is at least as good as itself: (x1, x2) (x1, x2).
CONTINUITY AXIOMS OF CONSUMER THEORY
Continuity is a property of a consumer's preference relation that captures
the idea that if a bundle x is preferred to a bundle y then bundles close to
x are preferred to bundles close to y.
A preference relation R (≽) is continuous if, given a sequence. {xn }∞ n=1
with xn → x and xn ≽ y ∀n, then x ≽ y. In words, this means that if you
weakly prefer points that are very close to x to y, then you should also
weakly prefer x to y. Continuity is probably the least intuitive property of
preferences, yet it is not implausible. P.5 The "Continuity"
Property. Preferences are continuous if the set of all choices that are at
least as good as a choice x' and the set of all choices that are no better
than x' are both closed sets.
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INDIFFERENCE CURVE
An indifference curve shows a combination of two goods in various
quantities that provides equal satisfaction (utility) to an individual. It is
used in economics to describe the point where individuals have no
particular preference for either one good or another based on their relative
quantities.
An indifference curve is downward sloping, usually with substitute goods
being straight lines and complementary goods being right angles. The two
types of indifference curves are perfect complements indifference curves
and perfect substitutes indifference curves.
It is used in economics to describe the point where individuals have no
particular preference for either one good or another based on their relative
quantities. It turns out that the whole theory of consumer choice can be
formulated in terms of preferences that satisfy the three axioms described
above, plus a few more technical assumptions. However, we will find it
convenient to describe preferences graphically by using a construction
known as indifference curves. An indifference curve is downward sloping,
usually with substitute goods being straight lines and complementary
goods being right angles. The two types of indifference curves are perfect
complements indifference curves and perfect substitute indifference curves.
One problem with using indifference curves to describe preferences is
that they only show you the bundles that the consumer perceives as being
indifferent to each other—they don’t show you which bundles are better
and which bundles are worse. It is sometimes useful to draw small arrows
on the indifference curves to indicate the direction of the preferred bundles.
We won’t do this in every case, but we will do it in a few of the examples
where confusion might arise.
Properties of Indifference Curve:
1. IC slopes downward:
2. IC is convex to the origin:
3. IC curves never cut each other:
4. Higher indifference curve represents more satisfaction:
5. IC neither touches X-axis or Y-axis:
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6. IC need not be parallel to each other:
A budget line shows combinations of two goods a consumer is able to
consume, given a budget constraint.
The budget line shows all the different combinations of the two
commodities that a consumer can purchase, given his money income and
the price of two commodities. The equation of a budget line is given
by: M=PX. QX+PY.
Formula for the slope of the indifference curve; The slope (d x2 / d
x1) of the tangent at any point on an indifference curve is the rate at
which x1 must be substituted for x2 or vice versa. The negative of the
slope (− d x2 / d x1) is the marginal rate of substitution of x1 for x2.
An indifference curve shows the different consumption points
between two goods that give the same utility. On indifference curves,
substitute goods tend towards straight lines, while complementary goods
tend towards right angles. An indifference curve is downward sloping
because it is representing quantity demanded, which has an inverse
relationship with price. In other words, when price increases, demand
decreases, and vice versa.
The image provided shows what an indifference curve looks like:
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The Graph above represents the number of commodities Y and X. The
straight line that forms a right-angle triangle ‘M’ is the marginal rate of
substitution and the straight line forming the triangle is the Income of the
Consumer. The movement of the line outwards shows that there has been
an increase in the income of the consumer. Assuming that the commodity
X and Y are the only things he spends his money on. The graph shows that
an increase in income will lead to an increase in the quantity demanded of
X and Y In the preference of the consumer. The graph also shows that the
consumer will prefer to buy more units of X than Y. The marginal utility
between both products is really small and we can assume that between the
two commodities. The consumer is indifferent.
Indifference curves are negatively sloped and there is a theory that says
more is better i. e F (x, y)
Cobb-Douglas Preferences
Suppose that the utility function is of the Cobb-Douglas form, u(x1, x2) =
xc
1xd
2. In the Appendix to this chapter we use
ESTIMATING UTILITY FUNCTIONS
calculus can be used to derive the optimal choices for this utility function.
They turn out to be
x1 = c
c+d
m
p1
x2 = d
c+d
m
p2.
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These demand functions are often useful in algebraic examples, so you
should probably memorize them.
The Cobb-Douglas preferences have a convenient property. Consider the
fraction of his income that a Cobb-Douglas consumer spends on good 1. If
he consumes x1 units of good 1, this costs him p1x1, so this represents a
fraction p1x1/m of total income. Substituting the demand function for x1
we have
p1x1
m = p1
m
c
c+d
m
p1
=c
c+d
Similarly the fraction of his income that the consumer spends on good 2 is
d/(c + d).
Thus the Cobb-Douglas consumer always spends a fixed fraction of his
income on each good. The size of the fraction is determined by the
exponent in the Cobb-Douglas function.
This is why it is often convenient to choose a representation of the CobbDouglas utility function in which the exponents sum to 1. If u(x1, x2) = xa
1x1−a 2 , then we can immediately interpret a as the fraction of income
spent on good 1. For this reason we will usually write Cobb-Douglas
preferences in this form.
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SOME TERMS OF PREFERENCE RELATIONS
 Marginal rate of substitution
Change in y
Change in X
 Budget Line
Using the graph above as an assumption.
M= Px X+ Py Y
Where Px= Price of X
X=Quantity Of X
And
Py= Price of Y and
Y= Quantity Of Y
M = Marginal rate of substitution
M= Px X + Py Y
M- Px X = Py Y
Py
Py
Py Y
Py
= M- Px X
Py
Y = M - Px X
Py
Py
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 Budget
Assuming that the consumer’s income is 1000 naira and he wants to buy
10 of X and 5 of Y, the function will look like this
10x+ 5y= 1000
If he decided to spend it all on Y, this will be
10y=1000/10
Y=100
If he decides to spend on commodity x only, this will be
5x=1000
X=200
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UTILITY
In economics, Daniel Bernoulli was a prominent 18th-century
Swiss mathematician. He defined the term "utility".
As a topic of economics, utility is used to model worth or value. Its usage
has evolved significantly over time. The term was introduced initially as a
measure of pleasure or happiness as part of the theory of utilitarianism by
moral philosophers such as Jeremy Bentham and John Stuart Mill.
It was Alfred Marshall who first discussed the role played by the theory of
utility in the theory of value. In Marshall's theory, the concept of utility is
cardinal. The price that a consumer is willing to pay for a good is an
indication of the utility of that good to the consumer.
Utility refers to the comprehensive benefits obtained from consuming
an item or service. Is an economic term referring to the satisfaction
received from consuming a good or service? Given this idea, it was natural
to think of consumers making choices so as to maximize their utility, that is,
to make themselves as happy as possible. Because of these conceptual
problems, economists have abandoned the old-fashioned view of utility as
being a measure of happiness. Instead, the theory of consumer behavior
has been reformulated entirely in terms of consumer preferences, and
utility is seen only as a way to describe preferences.
The consumer preference relation can be represented by a utility
function only if it is natural.
Economists gradually came to recognize that all that mattered about utility
as far as choice behavior was concerned was whether one bundle had a
higher utility than another—how much higher didn’t really matter.
What are the methods of calculating utility?
Utility comes in two types: cardinal and marginal. Cardinal utility assigns a
number to the utility, such as a basket of grapes gives a utility of 15 and a
bushel of corn is 30. Marginal utility is based on the idea utility functions
diminish as quantities of products are increased."Ordinal" utility refers
to the concept of one good being more useful or desirable than
another. "Cardinal" utility is the idea of measuring economic value
through imaginary units, known as "utils." Marginal utility is the utility
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gained by consuming an additional unit of a service or good. Cardinal
utility analysis of consumer's behaviour is based on which
combination of the following assumptions: (i) Utility is measurable in
terms of cardinal number. (ii) Constancy of the marginal utility of
money. (iii) Utilities of different goods are interdependent.
Cardinal numbers tell 'how many' of something, they show
quantity. Ordinal numbers tell the order of how things are set, they
show the position or the rank of something.
CARDINALIST UTILITY
The cardinal utility states that the level of satisfaction a consumer acquires
after consuming any goods and services can be measurable and expressed
in quantitative numbers. The price that a consumer is willing to pay for a
good is an indication of the utility of that good to the consumer.
Utility is Additive: The cardinalists believe that not only the utility is
measurable but also the utility derived from the consumption of different
commodities are added up to realize the total utility.
There are some theories of utility that attach a significance to the
magnitude of utility. These are known as cardinal utility theories. In a
theory of cardinal utility, the size of the utility difference between two
bundles of goods is supposed to have some sort of significance.
We know how to tell whether a given person prefers one bundle of goods
to another: we simply offer him or her a choice between the two bundles
and see which one is chosen. Thus we know how to assign an ordinal
utility to the two bundles of goods: we just assign a higher utility to the
chosen bundle than to the rejected bundle. Any assignment that does this
will be a utility function. Thus we have an operational criterion for
determining whether one bundle has a higher utility than another bundle
for some individual.
But how do we tell if a person likes one bundle twice as much as another?
How could you even tell if you like one bundle twice as much as another?
One could propose various definitions for this kind of assignment: I like one
bundle twice as much as another if I am willing to pay twice as much for it.
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Or, I like one bundle twice as much as another if I am willing to run twice
as far to get it, or to wait twice as long, or to gamble for it at twice the
odds.
There is nothing wrong with any of these definitions; each one would
give rise to a way of assigning utility levels in which the magnitude of the
numbers assigned had some operational significance. But there isn’t much
right about them either. Although each of them is a possible interpretation
of what it means to want one thing twice as much as another, none of
them appears to be an especially compelling interpretation of that
statement.
Even if we did find a way of assigning utility magnitudes that seemed
to be especially compelling, what good would it do us in describing choice
behavior? To tell whether one bundle or another will be chosen, we only
have to know which is preferred—which has the larger utility. Knowing how
much larger doesn’t add anything to our description of choice. Since
cardinal utility isn’t needed to describe choice behavior and there is no
compelling way to assign cardinal utilities anyway, we will stick with a
purely ordinal utility framework.
MARGINAL UTILITY
Marginal utility is the amount of additional satisfaction that a consumer
gets from having one more unit of a good or service. Marginal utility is the
added satisfaction a consumer gets from having one more unit of a good
or service. The concept of marginal utility is used by economists to
determine how much of an item consumers are willing to purchase.
Marginal Utility = Change in total utility/Change in number of units
consumed.
What is the law of diminishing utility?
The law of diminishing marginal utility holds that as we consume
more of an item, the amount of satisfaction produced by each additional
unit of that good declines. The change in utility gained from utilizing an
additional unit of a product is known as marginal utility. Even though the
two ideas are related, marginal utility focuses on how much of a product a
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customer will use, while the law of diminishing marginal returns focuses on
how much of a certain factor of production should be used to make the
product.It is important to note that marginal cost is derived solely from
variable costs, and not fixed costs. The marginal cost curve falls briefly at
first, then rises.
Marginal utility, in economics, the additional satisfaction or benefit (utility)
that a consumer derives from buying an additional unit of a commodity or
service. In economics, the standard rule is that marginal utility is equal to
the total utility change divided by the change in the amount of goods. The
formula appears as follows: Marginal utility = total utility
difference/quantity of goods difference. Find the total utility of the first
event. Marginal utility is the extra benefit derived from consuming one
more unit of a specific good or service. The main types of marginal utility
include positive marginal utility, zero marginal utility, and negative marginal
utility.
Consider a consumer who is consuming some bundle of goods, (x1, x2).
How does this consumer’s utility change as we give him or her a little more
of good 1? This rate of change is called the marginal utility with respect to
good 1. We write it as MU1 and think of it as being a ratio,
MU1 = ΔU = u(x1 + Δx1, x2) − u(x1, x2)
Δx1
Δx1
Consider a consumer who is consuming some bundle of goods, (x1, x2).
How does this consumer’s utility change as we give him or her a little more
of good 1? This rate of change is called the marginal utility with respect
to good 1. We write it as MU1 and think of it as being a ratio,
MU1 = ΔU
Δx1
= u(x1 + Δx1, x2) − u(x1, x2)
Δx1
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that measures the rate of change in utility (ΔU) associated with a small
change in the amount of good 1 (Δx1). Note that the amount of good 2 is
held fixed in this calculation.3
Utility definition implies that to calculate the change in utility associated
with a small change in consumption of good 1, we can just multiply the
change in consumption by the marginal utility of the good:
ΔU = MU1Δx1.
The marginal utility with respect to good 2 is defined in a similar manner:
MU2 = ΔU
Δx2
= u(x1, x2 + Δx2) − u(x1, x2)
Δx2
Note that when we compute the marginal utility with respect to good 2 we
keep the amount of good 1 constant. We can calculate the change in utility
associated with a change in the consumption of good 2 by the formula
ΔU = MU2Δx2.
It is important to realize that the magnitude of marginal utility depends
on the magnitude of utility. Thus it depends on the particular way that we
choose to measure utility. If we multiplied utility by 2, then marginal utility
would also be multiplied by 2. We would still have a perfectly valid utility
function in that it would represent the same preferences, but it would just
be scaled differently.
This means that marginal utility itself has no behavioral content. How can
we calculate marginal utility from a consumer’s choice behavior? We can’t.
Choice behavior only reveals information about the way a consumer ranks
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different bundles of goods. Marginal utility depends on the particular utility
function that we use to reflect the preference ordering and its magnitude
has no particular significance. However, it turns out that marginal utility
can be used to calculate something that does have behavioral content.
That measures the rate of change in utility (ΔU) associated with a small
Change in the amount of good 1 (Δx1). Note that the amount of good 2 is
held fixed in this calculation.
Marginal Utility and MRS
A utility function u(x1, x2) can be used to measure the marginal rate of
substitution (MRS) defined in Chapter 3. Recall that the MRS measures
the slope of the indifference curve at a given bundle of goods; it can be
interpreted as the rate at which a consumer is just willing to substitute a
small amount of good 2 for good 1.
This interpretation gives us a simple way to calculate the MRS. Consider a change in the consumption of each good, (Δx1, Δx2), that keeps
utility constant—that is, a change in consumption that moves us along the
indifference curve. Then we must have
MU1Δx1 + MU2Δx2 = ΔU = 0.
Solving for the slope of the indifference curve we have
MRS = Δx2
Δx1
= −MU1
MU2
. (4.1)
(Note that we have 2 over 1 on the left-hand side of the equation and 1
over 2 on the right-hand side. Don’t get confused!)
The algebraic sign of the MRS is negative: if you get more of good 1 you
have to get less of good 2 in order to keep the same level of utility.
However, it gets very tedious to keep track of that pesky minus sign, so
economists often refer to the MRS by its absolute value—that is, as a
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positive number. We’ll follow this convention as long as no confusion will
result.
Now here is the interesting thing about the MRS calculation: the MRS can
be measured by observing a person’s actual behavior—we find that
rate of exchange where he or she is just willing to stay put.
The utility function, and therefore the marginal utility function, is not
uniquely determined. Any monotonic transformation of a utility function
leaves you with another equally valid utility function. Thus, if we multiply
utility by 2, for example, the marginal utility is multiplied by 2. Thus the
magnitude of the marginal utility function depends on the choice of utility
function, which is arbitrary. It doesn’t depend on behavior alone; instead it
depends on the utility function that we use to describe behavior. But the
ratio of marginal utilities gives us an observable magnitude namely the
marginal rate of substitution. The ratio of marginal utilities is independent
of the particular transformation of the utility function you choose to use.
Look at what happens if you multiply utility by 2. The
MRS becomes
MRS = −2MU1
2MU2
The 2s just cancel out, so the MRS remains the same.
The same sort of thing occurs when we take any monotonic transformation of a utility function. Taking a monotonic transformation is just relabeling the indifference curves, and the calculation for the MRS described
above is concerned with moving along a given indifference curve. Even
though the marginal utilities are changed by monotonic transformations,
the ratio of marginal utilities is independent of the particular way chosen
to represent the preferences.
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First, let us clarify what is meant by “marginal utility.” As elsewhere in economics, “marginal” just means a derivative. So the marginal utility of good
1 is just
MU1 = lim Δx1→0
u(x1 + Δx1, x2) − u(x1, x2)
Δx1
= ∂u(x1, x2)
∂x1
.
Note that we have used the partial derivative here, since the marginal
utility of good 1 is computed holding good 2 fixed. Now we can rephrase
the derivation of the MRS given in the text using calculus. We’ll do it two
ways: first by using differentials, and second by using implicit
functions.
For the first method, we consider making a change (dx1, dx2) that keeps
utility constant. So we want
du = ∂u(x1, x2)
∂x1
dx1 +
∂u(x1, x2)
∂x2
dx2 = 0.
The first term measures the increase in utility from the small change dx1,
and the second term measures the increase in utility from the small change
dx2. We want to pick these changes so that the total change in utility, du,
is zero. Solving for dx2/dx1 gives us
dx2
dx1
= −∂u(x1, x2)/∂x1
25
∂u(x1, x2)/∂x2,
which is just the calculus analog of equation in the text.
As for the second method, we now think of the indifference curve as
being described by a function x2(x1). That is, for each value of x1, the
function x2(x1) tells us how much x2 we need to get on that specific
indifference curve. Thus the function x2(x1) has to satisfy the identity
u(x1, x2(x1)) ≡ k,
where k is the utility label of the indifference curve in question.
We can differentiate both sides of this identity with respect to x1 to get
∂u(x1, x2)
∂x1
+
∂u(x1, x2)
∂x2
∂x2(x1)
∂x1
= 0.
Notice that x1 occurs in two places in this identity, so changing x1 will
change the function in two ways, and we have to take the derivative at
each place that x1 appears.
We then solve this equation for ∂x2(x1)/∂x1 to find
∂x2(x1)
∂x1
= −∂u(x1, x2)/∂x1
∂u(x1, x2)/∂x2,
just as we had before.
The implicit function method is a little more rigorous, but the differential
method is more direct, as long as you don’t do something silly.
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Suppose that we take a monotonic transformation of a utility function, say,
v(x1, x2) = f(u(x1, x2)). Let’s calculate the MRS for this utility function.
Using
the chain rule
MRS = −∂v/∂x1
∂v/∂x2
= −∂f /∂u
∂f /∂u
∂u/∂x1
∂u/∂x2
= −∂u/∂x1
∂u/∂x2
since the ∂f /∂u term cancels out from both the numerator and
denominator.This shows that the MRS is independent of the utility
representation. This gives a useful way to recognize preferences that are
represented by different utility functions: given two utility functions, just
compute the marginal rates of substitution and see if they are the same. If
they are, then the two utility functions have the same indifference curves.
If the direction of increasing preference is the same for each utility function,
then the underlying preferences must be the same.
EXAMPLE: Cobb-Douglas Preferences
The MRS for Cobb-Douglas preferences is easy to calculate by using the
formula
derived above.
If we choose the log representation where
u(x1, x2) = c ln x1 + d ln x2,
then we have
MRS = −∂u(x1, x2)/∂x1
∂u(x1, x2)/∂x2
= − c/x1
d/x2
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=−c
d
x2
x1.
Note that the MRS only depends on the ratio of the two parameters and
the quantity of the two goods in this case.
What if we choose the exponent representation where
u(x1, x2) = xc
1xd
2?
Then we have
MRS = −∂u(x1, x2)/∂x1
∂u(x1, x2)/∂x2
= − cxc−1
1 xd
2
dxc
1xd−1
2
= − cx2
dx1,
which is the same as we had before. Of course you knew all along that a
monotonic transformation couldn’t change the marginal rate of substitution!
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UTILITY FOR COMMUTING
Utility functions are basically ways of describing choice behavior: if a bundle of goods X is chosen when a bundle of goods Y is available, then X
must have a higher utility than Y . By examining choices consumers make
we can estimate a utility function to describe their behavior.
This idea has been widely applied in the field of transportation economics
to study consumers’ commuting behavior. In most large cities commuters
have a choice between taking public transit or driving to work. Each of
these alternatives can be thought of as representing a bundle of different
characteristics: travel time, waiting time, out-of-pocket costs, comfort,
convenience, and so on. We could let x1 be the amount of travel time
involved in each kind of transportation, x2 the amount of waiting time for
each kind,and so on.
If (x1, x2,...,xn) represents the values of n different characteristics of
driving, say, and (y1, y2,...,yn) represents the values of taking the bus, we
can consider a model where the consumer decides to drive or take the bus
depending on whether he prefers one bundle of characteristics to the other.
More specifically, let us suppose that the average consumer’s preferences
for characteristics can be represented by a utility function of the form
U(x1, x2,...,xn) = β1x1 + β2x2 + ··· + βnxn, where the coefficients β1, β2,
and so on are unknown parameters. Any monotonic transformation of this
utility function would describe the choice behavior equally well, of course,
but the linear form is especially easy to work with from a statistical point of
view.
Suppose now that we observe a number of similar consumers making
choices between driving and taking the bus based on the particular pattern
of commute times, costs, and so on that they face. There are statistical
techniques that can be used to find the values of the coefficients βi for i =
1,...,n that best fit the observed pattern of choices by a set of consumers.
These statistical techniques give a way to estimate the utility function for
different transportation modes.
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One study reports a utility function that had the form4
U(TW, T T, C) = −0.147TW − 0.0411T T − 2.24C, (4.2)
where
TW = total walking time to and from bus or car
T T = total time of trip in minutes
C = total cost of trip in dollars
The estimated utility function in the Domenich-McFadden book correctly
described the choice between auto and bus transport for 93 percent of the
households in their sample.
1. A utility function is simply a way to represent or summarize a preference ordering. The numerical magnitudes of utility levels have no intrinsic
meaning.
2. Thus, given any one utility function, any monotonic transformation of
it will represent the same preferences.
3. The marginal rate of substitution, MRS, can be calculated from.
Utility may take any of the following forms:
(1) Form Utility
(2) Place Utility
(3) Time Utility
(4) Service Utility
(5) Possession Utility
(6) Knowledge Utility
(7) Natural Utility
Characteristics of Utility:
 The utility has no Ethical or Moral Significance
 The Utility is Psychological
 The utility is always Individual and Relative
 Utility is not Necessarily Equated with Usefulness
 The utility cannot be Measured Objectively
 The Utility Depends on the Intensity of Want
 Utility is Different from Pleasure
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UTILITY FUNCTION
What Is Utility Function? Utility describes the benefits gained or satisfaction
experienced with the consumption of goods or services. Utility function
measures the preferences consumers apply to their consumption of goods
and services. The main functional forms of the utility functions: CobbDouglas, CES, and quasi-linear. One of the most common is the CobbDouglas utility function, which has the form u(x, y) = x a y 1 - a. Another
common form for utility is the Constant Elasticity of Substitution (CES)
utility function. This function has the form u(x, y) = (a x r + b y r) 1/r.
A utility function is a way of assigning a number to every possible
Consumption bundles such that more-preferred bundles get assigned larger
numbers than less-preferred bundles. That is, a bundle (x1, x2) is
preferred to a bundle (y1, y2) if and only if the utility of (x1, x2) is larger
than the utility of (y1, y2): in symbols, (x1, x2) (y1, y2) if and only if u(x1,
x2) > u(y1, y2).
The only property of a utility assignment that is important is how it
orders the bundles of goods. The magnitude of the utility function is only
important in so far as it ranks the different consumption bundles; the size
of the utility difference between any two consumption bundles doesn’t
matter. Because of this emphasis on ordering bundles of goods, this kind of
utility is referred to as ordinal utility.
Consider for example below, where we have illustrated several different
ways of assigning utilities to three bundles of goods, all of which order the
bundles in the same way. In this example, the consumer prefers A to B and
B to C. All of the ways indicated are valid utility functions that describe the
same preferences because they all have the property that
A is assigned a higher number than B, which in turn is assigned a higher
number than C.
Bundle
U1
U2
U3
A
B
C
3
2
1
17
10
002
-1
-2
-3
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Since only the ranking of the bundles matters, there can be no unique
Way to assign utilities to bundles of goods. If we can find one way to
assign utility numbers to bundles of goods, we can find an infinite number
of ways to do it. If u(x1, x2) represents a way to assign utility numbers to
the bundles (x1, x2), then multiplying u(x1, x2) by 2 (or any other
positive number) is just as good a way to assign utilities.
UTILITY monotonic transformation is a way of transforming one set of
numbers into another set of numbers in a way that preserves the order of
the numbers. We typically represent a monotonic transformation by a
function f(u) that transforms each number u into some other number f(u),
in a way that preserves the order of the numbers in the sense that u1 > u2
implies f(u1) > f(u2). A monotonic transformation and a monotonic
function are essentially the same thing.
Examples of monotonic transformations are multiplication by a positive
Number (e.g., f(u)=3u), adding any number (e.g., f(u) = u + 17), raising u
to an odd power (e.g., f(u) = u3), and so on.1
The rate of change of f (u) as u changes can be measured by looking at
The change in f between two values of u, divided by the change in u:
Δf
Δu = f (u2) − f(u1)
u2 − u1
For a monotonic transformation, f (u2) – f (u1) always has the same sign
as u2 − u1. Thus a monotonic function always has a positive rate of
change. This means that the graph of a monotonic function will always
have a positive slope.
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CONSUMER CHOICE RULE
The theory of consumer choice is the branch of microeconomics that
relates preferences to consumption expenditures and to consumer demand
curves. The range of competing products and services from which a
consumer can choose. It is based on the assumption that individuals
maximize their utility and are willing to pay a specific price for a product or
service if they perceive it as better than an alternative.
Three types of consumer choice processes:
Affective Choice.
Attitude-Based Choice.
Attribute-Based Choice.
Consumer choice theory helps us understand a consumer's behavior that
results from the combination of their income and preferences. Having a
clear understanding of this behavior is necessary to be able to construct
the demand curve and set the price of a good appropriately. Other
consumer rationality assumptions include: consumers' choices are
independent, consumers have fixed preferences, consumers can gather all
the information and review all available alternatives, and consumers always
make optimal choices regarding their preferences.
The consumer decision-making process involves five basic steps. This is the
process by which consumers evaluate making a purchasing decision. The 5
steps are problem recognition, information search, alternatives evaluation,
purchase decision, and post-purchase evaluation. For example, when you
decided to keep the ice cream bar and return the cookies, you, consciously
or not, applied the marginal decision rule to the problem of maximizing
your utility: You bought the ice cream because you expect that eating it
will give you greater satisfaction than would consuming the box of cookies.
LEXICOGRAPHY ORDERING
What do lexicographic preferences mean?
In economics, lexicographic preferences or lexicographic orderings
describe comparative preferences where an agent prefers any amount of
one good (X) to any amount of another (Y). A model used in the study of
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consumer decision processes to evaluate alternatives; the idea that if two
products are equal on the most important attribute, the consumer moves
to the next most important, and, if still equal, to the next most important,
etc. With the lexicographic method, preferences are imposed by ordering
the objective functions according to their importance or significance, rather
than by assigning weights.
Completeness means that for all x and y, either x ≽ y or y ≽ x. Suppose
that without loss of generality only two goods, x1 ≥ y1, then x ≻ y. If x1 =
y1 and x2 > y2, then x ≻ y; if x1 = y1 and x2 = y2, then x ∼ y.
Then lexicographic preferences are complete. Explaining the lexicography
in simple terms
(a,b) is preferred to (c,d)
a>c or a>d
Example 2
There is a group of words that are to be arranged in Lexicographical order,
they include:
Apple, Car, Beans, Band, Bus,
1. Apple
2. Band
3. Beans
4. Bus
5. Car
This is because Apples start with the letter A and the next is B, but there
are Three Bs so we check for the second letter of the Bs, they have a, e
and u respectively
Example 3
We have two commodities, Ice cream, and coke. In this example, the Ice
cream is lexically preferred to coke based on my taste.
If these two sets are given, this is the most preferred bundle
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A
1. [4I, 10C]
B
[3I, 20C]
2. [4I, 10C]
[3I, 10C]
3. [4I, 10C]
[4I, 20C]
Consumer will prefer set A because Ice
The cream is lexically preferred
Consumer will prefer set A because Ice
The cream is lexically preferred
Consumers will prefer set B because
Ice cream is lexically preferred and it's the
satisfaction with A and there is more coke
with the same quantity of Ice cream in B.
More is better.
CRITICISM
1. It is criticized because it does not satisfy the continuity assumption
2. It assumes that two commodities are indifferent.
3. It is not possible to represent the use of utility function through
lexicographic ordering.
DEMAND FUNCTIONS
A demand function associates the price of a good, the consumer's income,
and his preferences to the quantity of the good he consumes. The shape of
the demand curve depends on the utility function. The price elasticity of
demand measures the responsiveness of quantity demanded to a change
in the good's relative price.
From the lexicographic Ordering, we assume that the drinking man has M
income and X1 represents the demand. Suppose He faces a price P1 for
one bottle of wine and a price P2 per loaf of bread then he is free to spend
his entire income on wine. The demand function can be written as
35
X1= M/P1, X2=0
Here X2 = 0 because the drinking man does not spend their income on
bread.
We already know that lexicographic ordering satisfies the completeness,
reflexivity, transitivity, and non-satiable assumptions. Although it does not
satisfy the demand for goods. It only gives the preference of two
goods/commodities. The continuity assumption guarantees that a
continuously increasing utility can be found to represent the preference
order.
REVEALED PREFERENCE
Revealed preference is an economic theory regarding individual
consumption patterns, which asserts that the best way to measure
consumer preference is to observe their purchasing behavior. It assumes
that consumer spends all their income on the commodity and that the
consumer’s choice is consistent. When we talk of determining people’s
preferences from observing their behavior, we have to assume that the
preferences will remain unchanged while we observe the behavior. Over
very long time spans, this is not very reasonable. But for the monthly or
quarterly time spans that economists usually deal with, it seems unlikely
that a particular consumer’s tastes would change radically. Thus we will
adopt a maintained hypothesis that the consumer’s preferences are stable
over the time period for which we observe his or her choice behavior.
Before we begin this investigation, let’s adopt the convention that in this
chapter, the underlying preferences—whatever they may be—are known
to be strictly convex. Thus there will be a unique demanded bundle at
each budget. This assumption is not necessary for the theory of revealed
preference, but the exposition will be simpler with it.
Consider Figure below, where we have depicted a consumer’s demanded
bundle, (x1, x2), and another arbitrary bundle, (y1, y2), that is beneath
the consumer’s budget line. Suppose that we are willing to postulate that
36
this consumer is an optimizing consumer of the sort we have been studying. What can we say about the consumer’s preferences between these
two
bundles of goods?
Well, the bundle (y1, y2) is certainly an affordable purchase at the given
budget—the consumer could have bought it if he or she wanted to, and
would even have had money left over. Since (x1, x2) is the optimal bundle,
it must be better than anything else that the consumer could afford. Hence,
in particular it must be better than (y1, y2).
The same argument holds for any bundle on or underneath the budget
37
line other than the demanded bundle. Since it could have been bought at
the given budget but wasn’t, then what was bought must be better. Here
is where we use the assumption that there is a unique demanded bundle
for each budget. If preferences are not strictly convex, so that indifference
curves have flat spots, it may be that some bundles that are on the budget
line might be just as good as the demanded bundle. This complication can
be handled without too much difficulty, but it is easier to just assume it
away.
In Figure above all of the bundles in the shaded area underneath the
budget
line are revealed worse than the demanded bundle (x1, x2). This is
because
they could have been chosen, but were rejected in favor of (x1, x2). We
will
now translate this geometric discussion of revealed preference into algebra.
Let (x1, x2) be the bundle purchased at prices (p1, p2) when the consumer
has income m. What does it mean to say that (y1, y2) is affordable at
Those prices and income? It simply means that (y1, y2) satisfies the
budget
constraint
p1y1 + p2y2 ≤ m.
Since (x1, x2) is actually bought at the given budget, it must satisfy the
the budget constraint with equality.
p1x1 + p2x2 = m.
Putting these two equations together, the fact that (y1, y2) is affordable at
the budget (p1, p2, m) means that
p1x1 + p2x2 ≥ p1y1 + p2y2.
38
If the above inequality is satisfied and (y1, y2) is actually a different
bundle from (x1, x2), we say that (x1, x2) is directly revealed preferred
to (y1, y2).
Note that the left-hand side of this inequality is the expenditure on the
bundle that is actually chosen at prices (p1, p2). 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.
The term “revealed preference” is actually a bit misleading. It does not
inherently have anything to do with preferences, although we’ve seen
above
that if the consumer is making optimal choices, the two ideas are closely
related. Instead of saying “X is revealed preferred to Y ,” it would be better
to say “X is chosen over Y .” When we say that X is revealed preferred to
Y , all we are claiming is that X is chosen when Y could have been chosen;
that is, that p1x1 + p2x2 ≥ p1y1 + p2y2.
THE WEAK AXIOM OF REVEALED PREFERENCE
When a consumer prefers to buy product A instead of product B, It simply
means that their choice is product A. If you see them using product B, then
it means that A is not available to them. Consumers will purchase what
they want and make constant choices. WARP is a criterion that needs to be
studied to know the needs of the consumers that need to be satisfied in
other to make sure that the consumer is consistent with preference.
Clearly, this consumer cannot be a maximizing consumer. Either the
The consumer is not choosing the best bundle she can afford, or there is
some. The aspect of the choice problem that has changed that we have
not observed. Perhaps the consumer’s tastes or some other aspect of her
economic environment have changed. In any event, a violation of this sort
is not Consistent with the model of consumer choice in an unchanged
environment.
39
The theory of consumer choice implies that such observations will not
Occur. If the consumers are choosing the best things they can afford, then
Things that are affordable, but not chosen, must be worse than what is
Chosen. Economists have formulated this simple point in the following
Basic axiom of consumer theory.
EXPENDITURE FUNCTION
It assumes that the prices of commodities are fixed. In order to achieve
certain expenditures at a given set of prices. At any given moment,
consumers come across a variety of commodities available at a variety of
prices. Most consumers have fixed incomes. Therefore each consumer
decides how much to spend on different commodities to achieve a
particular level of utility.
To find his expenditure function we set u = w min {px, py} and solve for w.
We have e (px, py, u) ≡ w = u min {px, py} . Expenditure to get u = 100
when px = 5 and py = 7.
If u(·) is continuous and strictly increasing, then e(p,u) is: Zero when u
takes on the lowest level of utility in U. ++ × u. For all p ≫ 0, strictly
increasing and unbounded above in u.
The expenditure minimization problem (EMP) looks at the
reverse side of the utility maximization problem (UMP). The UMP
considers an agent who wishes to attain the maximum utility from a
limited income. The EMP considers an agent who wishes to find the
cheapest way to attain a target utility. There is a related problem
which takes the opposite view: given a level of utility and market
prices, what is the least amount of wealth necessary to achieve this
level of utility? This problem is referred to as the expenditure
minimization problem. the expenditure function gives the minimum
amount of money an individual needs to spend to achieve some level
of utility, given a utility function and the prices of the available goods.
Cost is minimized at the levels of capital and labor such that the
40
marginal product of labor divided by the wage (w) is equal to the
marginal product of capital divided by the rental price of capital (r).
THE HICKSIAN DEMAND FUNCTION
In microeconomics, a consumer's Hicksian demand function or
compensated demand function for a good is his quantity demanded as
part of the solution to minimizing his expenditure on all goods while
delivering a fixed level of utility. In Hicks's approach, the consumer
does not move to higher IC which means the same satisfaction. The
Hicksian compensated demand curve is where agents are given
sufficient income to maintain them on their original utility curve. Using
Hicks' method, the income effect is removed by returning the
consumer to the same level of utility as before the price change.
Hicksian (or Compensated or Utility constant demand functions) yield
the amount of good x1 purchased at prices p1 and p2 when income is
just high enough to get utility level u0. Hicksian Demand Curves must
slope down.
Hicksian demand is the derivative of the expenditure
function. ∇p e(p, v) = h∗(p, v) − 0 since F does not depend on p.
THE VON NEUMANN- MORGENSTERN UTILITY FUNCTION
The von Neumann–Morgenstern utility function adds the
dimension of risk assessment to the valuation of goods, services, and
outcomes. As such, utility maximization is necessarily more subjective
than when choices are subject to certainty. Von Neumann and
Morgenstern proposed an index for measuring utility in situations
involving risk for the decision-maker. Utility index is designed for
predictive purposes. Allows to predict which of several choices a
person would prefer and thus enables him to take decisions.
41
Von Neumann and Morgenstern were the first to construct a cooperative
theory of n-person games. They assumed that various groups of players
might join together to form coalitions, each of which has an associated
value defined as the minimum amount that the coalition can ensure by its
own efforts. Theorem (von Neumann 1932). The general form of the
function which fulfills these requirements is given by, E(|ψ>, O) = Tr(ρψO)
(4) where ρψ is a positive operator with the property Tr(ρψ) = 1, (5)
otherwise known as the density operator for the state |ψ>. VNM The
expected utility or von Neumann-Morgenstern (VNM) utility of a lottery is
given by the utility of each outcome multiplied by its probability.
Again, note that the expected utility function is not unique, but
several functions can model the preferences of the same individual
over a given set of uncertain choices or games. What matters is that
such a function (which reflects an individual's preferences over
uncertain games) exists. The expected utility theory takes into
account that individuals may be risk-averse, meaning that the
individual would refuse a fair gamble (a fair gamble has an expected
value of zero). Risk aversion implies that their utility functions are
concave and show diminishing marginal wealth utility.
Expected utility theory provides a way of they are: the higher the
expected utility, the better it is to choose the act. (It is, therefore,
best to choose the act with the highest expected utility—or one of
them, in the event that several acts are tied.).
42
CONCLUSION
There is a strong relationship between the preference of consumers
and the utility. Utility is the satisfaction that consumers derive in the
consumption a good or service. Consumer preference is something
that the producer needs to have full knowledge of, in other for them
to know what to produce, for whom to produce and also the market
target that they would focus on. On the other hand consumer will
always go for the good or service that gives them the most Utility.
Consumers are rational thinkers and sometimes the reason they make
a particular preference may not be conscious and deliberate,
sometimes they don’t know the reason why they prefer a particular
product to another, although price,Income and taste is a major factor
that influences the preference of the consumer and the quantity they
would be willing to get, just like the examples given in this book.
However, the research conducted shows that Economists gradually came to
recognize that all that mattered about utility as far as choice behavior was
concerned was whether one bundle had a higher utility than another—how
much higher didn’t really matter.
We can conclude that Consumer preferences can directly impact an
economy too. When demand for one product rises and decreases for
another, the economy changes from one stage to another. Depending on
this change, economies must adapt themselves. Consumers have some
degree of control over the type of goods they buy, but they cannot always
choose what they want.
Consumer research is conducted to boost sales. The objective of consumer
research is to look into various territories of consumer psychology and
understand their buying pattern, what kind of packaging they like and
other similar attributes that help brands to sell their products and services
better.
43
QUESTIONS
1. Explain the consumer preference relation in detail.
Consumer preference theory is a theory that explains how consumers make
decisions. It is based on the idea that consumers are rational and will
choose the product or service they believe will satisfy their needs While
Utility is the total satisfaction derived by a consumer from the consumption
of a particular good or service, Utility was thought of as a numeric measure
of a person’s happiness.
Consumer preferences allow a consumer to rank different bundles of goods
according to levels of utility, or the total satisfaction of consuming a good
or service. It is important to understand that consumer preferences are not
dependent upon consumer income or prices.19 Jan 2022. The price a
consumer is willing to pay for a good depends on its marginal utility, which
declines with each additional unit of consumption, according to the law of
diminishing marginal utility. Therefore, the price decreases for a normal
good when consumption increases.
Preferences can be represented by a utility function when they are able to
be quantified and assigned numerical values. When the ordering of
alternatives is complete. In other words, for all pairs of alternatives (a, b)
you can say of your utilities, a>b, a<b, or a=b, and if a>b and b>c then
a>c (transitivity). Utility is dependent on the bundles of goods consumed
by an individual, with greater quantities of goods representing greater
levels of happiness or utility.
The crucial point of consumer preference theory is this law. It states
that as more and more of a commodity is consumed, consumers receive
less and less satisfaction from its consumption. More formally, it means
that the Marginal utility of a commodity declines as successive units of it
are consumed. Basic preference relations can be grouped into
more comprehensive preference relations. A well-known example of a
44
comprehensive preference relation is the Outranking relation S = P ∪ Q ∪
I , where for any pair of decision alternatives ( a , b ) ∈ A × A , aSb means
“a is at least good as b”.
2. Define the following terms
a)Completeness
This also assumed that individuals must have a preference
relationship between any two sets of goods; either we must be able to say
that they weakly prefer A to B, or that they weakly prefer B to A, or both
(indifference). If we are told that Dave strictly prefers larger chocolate bars
to smaller ones, this gives us enough information to completely define
Dave’s preferences over the entire space of chocolate bars. If Susie says
that she always prefers the bigger and darker chocolate bar, we do not
have enough information to define a preference relationship across the
entire space of chocolate bars – if one bar is darker and smaller than
another, but lighter and bigger than a third, which is preferred to which?.
The completeness assumption also states that consumers are rational and
make decisions based on all the information they have. This assumption is
made because consumers control their own preferences and are not
influenced by external factors.
b) Transitivity
Transitivity is a relation between three elements such that if it holds
between the first and second and it also holds between the second and
third it must necessarily hold between the first and third.
This assumption implies that if at first an individual chooses good A over
good B, and if a second time chooses good B over good C, with B being the
45
same in both cases, then it is logical that the consumer will select good A
over good C
Transitivity assumptions state that the choice of goods by consumers is
rational. The assumption implies that indifference curves must not cross
each other as it relates to consumer behavior as well as individual
preferences hence transitive.
c) Reflexivity
Reflexivity in simple terms means that an identical product to X is just
as preferred as X to the customer e.g
X= X i.e. coke is as preferred to Pepsi, just like Pepsi is as preferred to
Coke because they are identical and close substitutes.
Reflexivity is the fact of someone being able to examine their own feelings,
reactions, and motives (=reasons for acting) and how these influence what
they do or think in a situation.
Relationships are reflexive if they can be applied when both sides of the
relationship are the same – i.e. I am at least as old as myself (I am in fact
exactly as old as myself, but the statement is not incorrect, merely
imprecise). Weak preference relationships are reflexive; a bundle of goods
can be said to be weakly preferred to itself, but not strictly preferred to
itself (in fact, it can be more accurately said to be exactly as preferred as
itself).
Reflexivity: this identity condition says that the consumer is indifferent
when comparing a bundle to itself.
We assume that any bundle is at least as good as itself: (x1, x2) (x1, x2).
Reflexivity in simple terms means that an identical product to X is just as
preferred as X to the customer e.g
X= X i.e. coke is as preferred to Pepsi, just like Pepsi is as preferred to
coke because they are identical and close substitutes.
46
d) Nonsatiation
Non-satiation is the state of never being satisfied. Consumers will
always prefer more to less. Economic theory expresses non-satiation
through a mathematical property of the utility function called local nonsatiation, which, simply put, states that for every bundle of goods, there
always exists a better bundle of goods - a bundle that gives higher utility.
Decreasing marginal utility. The assumption is that a consumer will always
benefit from additional consumption. The demand for some goods may
have a finite limit, but it is likely that there is some good or service a
consumer would benefit from having more of. Consumers lose satisfaction
with a product the more they consume it.
Non-satiation assumes that if one person has X amount of something, it
does not mean they will not want more options. People are seldom
satisfied with one trip to the shops and always want to consume more.
e) Strict convexity
Strict convexity says that if two consumption bundles are each on the same
indifference curve as x, then any point on a line connecting these two
points (except for the points themselves) will be on a higher indifference
curve than x.
Strict convexity comes from the fact that a convex combination of any pair
of bundles is strictly preferred to the pair. Strict convexity implies that the
second derivative or F double prime of X is greater than zero if it's equal to
zero. and that implies that the line can be straight or linear. there's no
slope associated with it.
What is an example of a strictly convex function?
If Hf(x) ≻ 0 for all x ∈ C, then f is strictly convex. But the implication
doesn't go both ways. For example, f(x) = x4 has f//(0) = 0, but is still
strictly convex. The simplest and most important operations that preserve
convexity are addition and multiplication by a positive scalar.
47
3. Define the utility function of consumer preference
The utility function describes consumer preference. Consumers maximize
their well-being or pleasure from consumption, subject to the
constraints they face. The utility function measures consumers'
preferences for a set of goods and services. Utility is measured in
units called utils the Spanish word for useful— but calculating the
benefit or satisfaction that consumers receive is abstract and difficult
to pinpoint.
4. How is consumer preference difference from consumer choice?
In the preference relation approach, we assume that individuals have
preferences over goods. However, in the choice rule approach, at least
under WARP, we must observe a choice being made between two goods in
order to determine which is preferred.
5. What is lexicography ordering and why is it criticised in
Economics
In economics, lexicographic preferences or lexicographic orderings
describe comparative preferences where an agent prefers any amount of
one good (X) to any amount of another (Y). A model used in the study of
consumer decision processes to evaluate alternatives; the idea that if two
products are equal on the most important attribute, the consumer moves
to the next most important, and, if still equal, to the next most important,
etc. With the lexicographic method, preferences are imposed by ordering
the objective functions according to their importance or significance, rather
than by assigning weights.
Example 1
There is a group of words that are to be arranged in Lexicographical order,
they include:
48
Apple, Car, Beans, Band, Bus,
1. Apple
2. Band
3. Beans
4. Bus
5. Car
This is because Apples start with the letter A and the next is B, but there
are Three Bs so we check for the second letter of the Bs, they have a, e
and u respectively.
Example 2
We have two commodities, Ice cream, and coke. In this example, the Ice
cream is lexically preferred to coke based on my taste.
If these two sets are given, this is the most preferred bundle
A
4. [4I, 10C]
B
[3I, 20C]
5. [4I, 10C]
[3I, 10C]
6. [4I, 10C]
[4I, 20C]
Consumer will prefer set A because Ice
The cream is lexically preferred
Consumer will prefer set A because Ice
The cream is lexically preferred
Consumers will prefer set B because
Ice cream is lexically preferred and it's the
satisfaction with A and there is more coke
with the same quantity of Ice cream in B.
More is better.
CRITICISM
49
a) It is criticized because it does not satisfy the continuity assumption
b) It assumes that two commodities are indifferent.
c) It is not possible to represent the use of utility function through
lexicographic ordering.
6. Explain briefly the revealed preference theory?
Revealed preference is an economic theory regarding individual
consumption patterns, which asserts that the best way to measure
consumer preference is to observe their purchasing behavior. It assumes
that consumer spends all their income on the commodity and that the
consumer’s choice is consistent. When we talk of determining people’s
preferences from observing their behavior, we have to assume that the
preferences will remain unchanged while we observe the behavior. Over
very long time spans, this is not very reasonable. But for the monthly or
quarterly time spans that economists usually deal with, it seems unlikely
that a particular consumer’s tastes would change radically. Thus we will
adopt a maintained hypothesis that the consumer’s preferences are stable
over the time period for which we observe his or her choice behavior.
7. Explain the weak axiom of revealed preference with reference
to the substitution and income effects
Weak Axiom of Revealed Preference (WARP): This axiom states
that given incomes and prices, if one product or service is purchased
instead of another, then, as consumers, we will always make the
same choice. Consumers will purchase what they want and make constant
choices. WARP is a criterion that needs to be studied to know the needs of
the consumers that need to be satisfied in other to make sure that the
consumer is consistent with preference.
Generally, consumers are expected to spend more when their income rises
and less when their income falls. Income and spending correlations can
also trend with economic cycles which are known to heavily affect the
50
consumer discretionary and consumer staples sectors. How changes in
income and prices affect consumer choices?
The income effect is that a higher price means, in effect, the buying power
of income has been reduced (even though actual income has not changed),
which leads to buying less of the good (when the good is normal). The
income effect causes indifference curves to move up or down. If the price
of the good decreases, our real income increases, and the indifference
curve will move upwards and vice versa. The substitution effect occurs due
to a decrease in the price of one good while the other good's price remains
the same.
According to the principle of the substitution effect, if the price of the
first item (the one the consumer normally buys) goes up, but the price
of the second item remains the same, the consumer will be more
likely to substitute the second item for the first. The decrease in sales
for a product can be attributed to consumers switching to cheaper
alternatives when its price rises. When the price of a product or
service increases but the buyer's income stays the same, the
substitution effect generally kicks in.
8. Critically examine the indirect utility function along with it
various properties
A consumer's indirect utility function is a function of prices of goods and
the consumer's income or budget. The function is typically denoted as v(p,
m) where p is a vector of prices for goods, and m is a budget presented in
the same units as the prices. Therefore, the first restriction placed on a
utility function is that it has a positive first derivative. The second principle
of a utility function is an assumption of an investor's taste for risk. Three
assumptions are possible: the investor is either averse to risk, neutral
towards risk, or seeks risk.
Utility is given
U=XY
51
Budget constraint M=Px X + Py Y
Where
Px=price of X
Py= price of Y
X=quantity of X
Y=quantity of Y
We will use the marginal utility to get the consumer's indirect utility
MUx = Δu = Y
Δx
Muy = Δu =X/Δy
Equilibrium condition for consumers
Mux = Muy
Px
Px
= Muy = Δu =X
Δx
Y/Px = X/Py
Y/Px =Y/Py
Y = Px/Py = X
Solving for Y and X separately; solving for X
M=PxX +Py( Px/Py X)
M=PxX + PxX
M=2PxX
M= M/2Px
52
Solving for Y
Y/Px = Y/Py
X = Py/Px .Y
M= Px (Py/Px .Y) +PyY
M = 2PyY
Y= M/2Py
Consumer indirect Utility = U= XY
U=(M/2Py) (M/2Px)
U = =(MU/4P*Py)
It is also assumed that all income is spent and the function adheres to
the law of demand, which is reflected in increasing income m and
decreasing price p. Last, but not least, the indirect utility function is
also quasi-convex in price.
9. Examine the expenditure function along its properties
The expenditure function yields the minimum expenditure required to
reach utility u at prices p. The expenditure function is an essential tool
for making consumer theory operational for public policy analysis.
Using the expenditure function, we can monetize otherwise in
commensurate tradeoffs to evaluate costs and benefits
53
It assumes that the prices of commodities are fixed. In order to achieve
certain expenditures at a given set of prices. At any given moment,
consumers come across a variety of commodities available at a variety of
prices. Most consumers have fixed incomes. Therefore each consumer
decides how much to spend on different commodities to achieve a
particular level of utility.
To find his expenditure function we set u = w min {px, py} and solve for w.
We have e (px, py, u) ≡ w = u min {px, py} . Expenditure to get u = 100
when px = 5 and py = 7.
If u(·) is continuous and strictly increasing, then e(p,u) is: Zero when u
takes on the lowest level of utility in U. ++ × u. For all p ≫ 0, strictly
increasing and unbounded above in u.
10) Write short notes on the following
a) Hicksian demand function;
In microeconomics, a consumer's Hicksian demand function or
compensated demand function for a good is his quantity demanded as
part of the solution to minimizing his expenditure on all goods while
delivering a fixed level of utility. In Hicks's approach, consumer does
not move to higher IC which means the same satisfaction. The
Hicksian compensated demand curve is where agents are given
sufficient income to maintain them on their original utility curve. Using
Hicks' method, the income effect is removed by returning the
consumer to the same level of utility as before the price change.
Hicksian (or Compensated or Utility constant demand functions) yield
the amount of good x1 purchased at prices p1 and p2 when income is
just high enough to get utility level u0. Hicksian Demand Curves must
slope down.
54
Hicksian demand is the derivative of the expenditure
function. ∇p e(p, v) = h∗(p, v) − 0 since F does not depend on p.
b) Expenditure minimization problem
The EMP considers an agent who wishes to find the cheapest
way to attain a target utility. There is a related problem that takes the
opposite view: given a level of utility and market prices, what is the
least amount of wealth necessary to achieve this level of utility? This
problem is referred to as the expenditure minimization problem, the
expenditure function gives the minimum amount of money an
individual needs to spend to achieve some level of utility, given a
utility function and the prices of the available goods. Cost is minimized
at the levels of capital and labor such that the marginal product of
labor divided by the wage (w) is equal to the marginal product of
capital divided by the rental price of capital (r).
11. Critically examine the von Neumann- Morgenstern utility
function
The von Neumann–Morgenstern utility function adds the
dimension of risk assessment to the valuation of goods, services, and
outcomes. As such, utility maximization is necessarily more subjective
than when choices are subject to certainty. Von Neumann and
Morgenstern proposed an index for measuring utility in situations
involving risk for the decision-maker. The utility index is designed for
predictive purposes. Allows to predict which of several choices a
person would prefer and thus enables him to take decisions.
Von Neumann and Morgenstern were the first to construct a cooperative
theory of n-person games. They assumed that various groups of players
might join together to form coalitions, each of which has an associated
value defined as the minimum amount that the coalition can ensure by its
own efforts. Theorem (von Neumann 1932). The general form of the
55
function which fulfills these requirements is given by, E(|ψ>, O) = Tr(ρψO)
(4) where ρψ is a positive operator with the property Tr (ρψ) = 1, (5)
otherwise known as the density operator for the state |ψ>. VNM The
expected utility or von Neumann-Morgenstern (VNM) utility of a lottery is
given by the utility of each outcome multiplied by its probability.
Again, note that the expected utility function is not unique, but
several functions can model the preferences of the same individual
over a given set of uncertain choices or games. What matters is that
such a function (which reflects an individual's preferences over
uncertain games) exists. The expected utility theory takes into
account that individuals may be risk-averse, meaning that the
individual would refuse a fair gamble (a fair gamble has an expected
value of zero). Risk aversion implies that their utility functions are
concave and show diminishing marginal wealth utility.
56