The Theory of Consumer

Behaviour
 CONSUMER BEHAVIOUR
Theory of consumer behavior Description of how consumers
allocate incomes among different goods and services to
maximize their well-being.

Consumer behavior is best understood in three distinct steps:

1.Consumer preferences
2.Budget constraints
3.Consumer choices
 CONSUMER PREFERENCES
Market Baskets
● Market basket (or bundle) List with specific quantities
of one or more goods.

TABLE Alternative Market Baskets

Market Basket Units of Food Units of Clothing

A 20 30
B 10 50
D 40 20
E 30 40
G 10 20
H 10 40

To explain the theory of consumer behavior, we will ask

whether consumers prefer one market basket to another.
Market baskets…
Some Basic Assumptions (Axioms) about Preferences
1. Completeness: Preferences are assumed to be complete.
In other words, consumers can compare and rank all possible baskets.
Thus, for any two market baskets A and B, a consumer will prefer A to
B, will prefer B to A, or will be indifferent between the two.
By indifferent we mean that a person will be equally satisfied with
either basket.
Note that these preferences ignore costs.
A consumer might prefer fresh juice to manufactured juice but buy
manufactured juice because it is cheaper.
Preferences by a binary relation: For all A and B baskets of
goods, either 𝐴 ≿ 𝐵 or 𝐵 ≿ 𝐴
It says that, the consumer can examine any two distinct consumption
plans A and B and decide whether “A is at least as good as B” or “B is
at least as good as A”.
Market baskets…
2. Transitivity: Preferences are transitive.
Transitivity means that if a consumer prefers basket A to basket B and
basket B to basket C, then the consumer also prefers A to C.
Transitivity is normally regarded as necessary for consumer
consistency.
Preferences by a binary relation: For any three basket of goods A,B, and C,
if 𝐴≿𝐵 and 𝐵≿C, then 𝐴≿C.
The assumptions (axioms) of completeness and transitivity describe a
consumer who can make consistent comparisons among alternatives.
These two assumptions together imply that the consumer can
completely rank any finite number of elements in the consumption set,
X, from best to worst, possibly with some ties.
Market baskets…
3. More is better than less (Local Non-satiation): Goods are
assumed to be desirable—i.e., to be good.
Consequently, consumers always prefer more of any good to less.
In addition, consumers are never satisfied or satiated; more is always
better, even if just a little better.
This assumption is made for pedagogic reasons; namely, it simplifies
the graphical analysis.
Of course, some goods, such as air pollution, may be undesirable, and
consumers will always prefer less.
Some Basic Assumptions about Preferences…
4. Continuity: For all basket goods which belongs to positive real
numbers, the at least as good as’ set, ≿C and the ‘no
better than’ set, (C)
5. Strict Monotonicity.
 Indifference Curves
Indifference curve: Curve representing all combinations of
market baskets that provide a consumer with the same level
of satisfaction. An Indifference Curve
The indifference curve U1 that
passes through market basket
A shows all baskets that give
the consumer the same level
of satisfaction as does market
basket A; these include
baskets B and D.

Consumer prefers
basket E, which lies
above U1, to A, but
prefers A to H or G,
which lie below U1.
Indifference Maps
Indifference map: Graph containing a set of indifference curves
showing the market baskets among which a consumer is indifferent.
An indifference map is a set An Indifference Map

of indifference curves that

describes a person's
preferences.

Any market basket on

indifference curve U3,
such as basket A, is
preferred to any basket
on curve U2 (e.g.,
basket B), which in turn
is preferred to any
basket on U1, such as
D.
Indifference Maps…

If indifference curves U1 Indifference Curves Cannot Intersect

and U2 intersect, one of

the assumptions of
consumer theory is
violated.

According to this
diagram, the consumer
should be indifferent
among market baskets
A, B, and D. Yet B
should be preferred to D
because B has more of
both goods.
 The Marginal Rate of Substitution
Marginal rate of substitution (MRS): Maximum amount of a good that a consumer is
willing to give up in order to obtain one additional unit of another good.

The magnitude of the slope of The Marginal Rate of Substitution

an indifference curve
measures the consumer’s
marginal rate of substitution
(MRS) between two goods.

In this figure, the MRS between clothing

(C) and food (F) falls from 6 (between A
and B) to 4 (between B and D) to 2
(between D and E) to 1 (between E and
G).
Convexity The decline in the MRS
reflects a diminishing marginal rate of
substitution. When the MRS diminishes
along an indifference curve, the curve is
convex.
 Perfect Substitutes and Perfect Complements

Perfect substitutes: Two goods for which the marginal rate of substitution of
one for the other is a constant.

Perfect complements: Two goods for which the MRS is zero or infinite; the indifference
curves are shaped as right angles.

Bads

Bad: Good for which less is preferred rather than more.

Perfect Substitutes and Perfect Complements…

Perfect Substitutes and Perfect Complements

In (a), Bob views orange juice and In (b), Jane views left shoes and
apple juice as perfect substitutes: right shoes as perfect complements:
He is always indifferent between a An additional left shoe gives her no
glass of one and a glass of the extra satisfaction unless she also
other. obtains the matching right shoe.
 Utility and Utility Functions
 Utility: Is a numerical score representing the satisfaction that a consumer gets from a given
market basket.
 A utility function is a way of assigning a number to every possible consumption bundle 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 ).

A utility function i.e. 𝑢(𝐹, 𝐶) = 𝐹𝐶

can be represented by a set of
indifference curves, each with a
numerical indicator.
This figure shows three indifference
curves (with utility levels of 25, 50,
and 100, respectively) associated
with the utility function FC.
Utility and Utility Functions…
Some examples of utility functions
Perfect Substitutes
Here a and b are some positive numbers that measure
Perfect Complements the “value” of goods 1 and 2 to the consumer.

Perfect Complements
a and b are positive numbers that indicate the
proportions in which the goods are consumed.

Cobb-Douglas
Where c and d are positive numbers that describe the preferences
of the consumer
 Constant Elasticity of Substitution (CES)
Elasticity of substitution for a utility function is defined as the elasticity
of the ratio of consumption of two goods to the MRS.
Therefore it is a measure of how easily the two goods are substitutable
along an indifference curve.
For a class of utility functions this value (elasticity of substitution) is
constant for all (𝑥1 , 𝑥2 ).
These utility functions are called Constant Elasticity of Substitution
(CES) utility functions.
The following utility functions are special cases of the general CES
utility function:
Linear Utility: Linear Utility is of the form of Perfect Substitutes
Leontief Utility: Leontief utility is of the form of Perfect Complements
 BUDGET CONSTRAINTS
Budget constraints: Constraints that consumers face as a result of limited incomes or is
the set of all the bundles a consumer can afford given that consumer's income.
𝑃𝐹 𝐹 + 𝑃𝐶 𝐶 ≤ 𝐼
Budget line: All combinations (set) of goods for which the total amount of money
spent is equal to income.
PF F  PC C  I

TABLE Market Baskets and the Budget Line

Market Basket Food (F) Clothing (C) Total Spending

A 0 40 $80
B 20 30 $80
D 40 20 $80
E 60 10 $80
G 80 0 $80
The table shows market baskets associated with the budget line
F + 2C = $80
BUDGET CONSTRAINTS…

• The Budget Line A Budget Line

A budget line describes the

combinations of goods that
can be purchased given the
consumer’s income and the
prices of the goods.
Line AG (which passes
through points B, D, and E)
shows the budget
associated with an income
of $80, a price of food of PF
= $1 per unit, and a price of
clothing of PC = $2 per unit.
The slope of the budget line
(measured between points
B and D) is −PF/PC = −10/20 C  ( I / PC )  ( PF / PC ) F
= −1/2.
BUDGET CONSTRAINTS…
The Effects of Changes in Income and Prices Effects of a Change in Income on the
Budget Line

Income Changes A
change in income (with
prices unchanged) causes
the budget line to shift
parallel to the original line
(L1).
When the income of $80
(on L1) is increased to
$160, the budget line shifts
outward to L2.
If the income falls to $40,
the line shifts inward to L3.
 BUDGET CONSTRAINTS…
The Effects of Changes in Income and Prices Effects of a Change in Price on the
Budget Line

Price Changes A
change in the price of
one good (with income
unchanged) causes the
budget line to rotate
about one intercept.
When the price of food
falls from $1.00 to $0.50,
the budget line rotates
outward from L1 to L2.
However, when the price
increases from $1.00 to
$2.00, the line rotates
inward from L1 to L3.
 CONSUMER CHOICE
The maximizing market basket must satisfy two conditions:
1. It must be located on the budget line.
2. It must give the consumer the most preferred combination
of goods and services.
Maximizing Consumer Satisfaction

A consumer maximizes
satisfaction by choosing market
basket A. At this point, the
budget line and indifference
curve U2 are tangent.
No higher level of satisfaction
(e.g., market basket D) can be
attained.
At A, the point of maximization,
the MRS between the two
goods equals the price ratio. At
B, however, because the MRS
[− (−10/10) = 1] is greater than
the price ratio (1/2), satisfaction
is not maximized.
 CONSUMER CHOICE

Satisfaction is maximized (given the budget constraint) at the

point where

MRS  PF / PC (3)

● marginal benefit Benefit from the consumption of one

additional unit of a good.
● marginal cost Cost of one additional unit of a good.

The condition given in equation (3) illustrates the kind of

optimization conditions that arise in economics. In this instance,
satisfaction is maximized when the marginal benefit—the
benefit associated with the consumption of one additional unit of
food—is equal to the marginal cost—the cost of the additional
unit of food. The marginal benefit is measured by the MRS.
 CONSUMER CHOICE
 corner solution Situation in which the marginal rate of substitution of one good for another in a
chosen market basket is not equal to the slope of the budget line. A corner solution arises when a
consumer's optimal consumption bundle contains zero quantity of one or more goods. This typically
happens when a consumer gets no utility from consuming a specific good or when the marginal utility
per unit of price for a good is lower than that for other goods available
Figure

A Corner Solution

When the consumer’s marginal rate

of substitution is not equal to the price
ratio for all levels of consumption, a
corner solution arises. The consumer
maximizes satisfaction by consuming
only one of the two goods.
Given budget line AB, the highest
level of satisfaction is achieved at B
on indifference curve U1, where the
MRS (of ice cream for frozen yogurt)
is greater than the ratio of the price of
ice cream to the price of frozen
yogurt.
 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,

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…
This 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:

The marginal utility with respect to good 2 is defined in a similar

manner:

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
 Marginal Utility and MRS
A utility function u(x1, x2) can be used to measure the marginal rate of
substitution (MRS).
MRS measure 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.

Solving for the slope of the indifference curve we have

MARGINAL UTILITY AND CONSUMER CHOICE
Example
Suppose that the utility function from consuming goods 1 and 2 is

a) What type of function is the above

b) Find the marginal utilities for xs
c) Find the MRS
 Consumer Demand
The consumer problem is to maximize utility. The consumers choose
the best combination of goods to maximize their utility, given their
budget and the prices of goods
The optimal choice of goods 1 and 2 at some set of prices and income is
called the consumer’s demanded bundle.
In general when prices and income change, the consumer’s optimal
choice will change.
The demand function is the function that relates the optimal choice—
the quantities demanded—to the different values of prices and incomes
Consumer Demand…
We will write the demand functions as depending on both prices and
income: and
For each different set of prices and income, there will be a different
combination of goods that is the optimal choice of the consumer.
Different preferences will lead to different demand functions
Consumer Demand…
Perfect Substitutes: We have three possible cases.
If 𝑝2 > 𝑝1 , then the slope of the budget line is flatter than the slope of
the indifference curves.
The optimal bundle is where the consumer spends all of his or her
money on good 1.
If 𝑝1 > 𝑝2 , then the consumer purchases only good 2.
Finally, if 𝑝1 = 𝑝2 , there is a whole range of optimal choices—any
amount of goods 1 and 2 that satisfies the budget constraint is optimal
in this case. Thus the demand function for good 1 will be
Optimal choice with perfect substitutes
If the goods are perfect substitutes, the optimal choice will usually be
on the boundary.
Consumer Demand…
Perfect Complements
Note that the optimal choice must always lie on the diagonal, where the
consumer is purchasing equal amounts of both goods, no matter what
the prices are.
We know that the consumer is purchasing the same amount of good 1
and good 2, no matter what the prices.
If this amount is denoted by 𝑥, then we have to satisfy the budget
constraint

Solving for 𝑥 gives us the optimal choices of goods 1 and 2:

Optimal choice with perfect complements
If the goods are perfect complements, the quantities demanded
will always lie on the diagonal since the optimal choice occurs
where 𝑥1 equals 𝑥2 .
Consumer Demand…
Cobb-Douglas Preferences
Suppose that the utility function is of the Cobb-Douglas form
Its demand functions will look like:

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.
Consumer Demand…
uncompensated and compensated demand
What we have been calling demand up to now is Marshallian (uncompensated) or
ordinary demand which maximizes utility u given prices p1 and p2 and income m, so
is a function of p1, p2, m.
 Notation 𝑥1 (𝑝1 , 𝑝2 , 𝑚) and 𝑥2 (𝑝1 , 𝑝2 , 𝑚).
Compensated (Hicksian) demand minimizes the expenditure (cost) of obtaining
utility u at prices p1 and p2 and is a function of utility u, p1 , p2.
 Notation 𝑥1ℎ (𝑝1 , 𝑝2 , 𝑢) and 𝑥1ℎ (𝑝1 , 𝑝2 , 𝑢).
The Hicksian demand curve—the one with utility held constant—is sometimes called
the compensated demand curve.
This terminology arises naturally if you think of constructing the Hicksian demand
curve by adjusting income as the price changes so as to keep the consumer’s utility
constant.
Hence the consumer is “compensated” for the price changes, and his utility is the
same at every point on the Hicksian demand curve.
This is in contrast to the situation with an ordinary demand curve.
In this case the consumer is worse off facing higher prices than lower prices since his
income is constant.
 Optimization of Utility Functions
Optimization is a procedure which aims to find the optimal solutions to the
objective function or functions under constraints.
There are two approaches for optimizing a utility function
While substitution approach is simple enough, there are situations where it will be
difficult to apply.
The procedure requires that, as we know, before the calculation, the budget
constraint actually binds.
In many situations there may be other constraints and we may not know whether
they bind before demands are calculated.
 Consider the (two good) problem of
 Examples
Examples 1: A consumer has a utility function
𝑢(𝑥1 , 𝑥2 ) = min(𝑥1 , 𝑥2 )
a) What type of utility function is that?
b) Find the consumer demand functions
Examples 2: A consumer has a utility function
𝑢(𝑥1 , 𝑥2 ) = min(𝛼𝑥1 , 𝛽𝑥2 )
a) What type of utility function is that?
b) Find the consumer demand functions
Example 2: You are given the following utility function
𝑢 𝑥1 , 𝑥2 = 𝑥1 + 𝑥2
a) State what type of utility function is that?
b) Find consumer demand functions
Example 3: The consumer want to maximize the following utility
𝑢(𝑥1 , 𝑥2 ) = 𝑥1 𝑥2
Find consumer demand functions
Example 4: Suppose that the utility function is of the form

a) State what type of utility function is that?

b) Find consumer demand functions
Example 5: Leah spends all of her food budget on chips and eggs. Leah’s
utility function is:
𝑈(𝑥1 , 𝑥2 ) = 𝑥1 𝑥2
Where x1 is the quantity of chips and x2 is the quantity of eggs. Her
monthly food budget is TSh 120,000, the price of chips (p1) is 3,000, and
the price of egg (p2) is 2,000. Find Leah’s utility-maximizing choice of chips
and eggs.
Example 6: Mansoor has a utility function 3750 = 𝑥1 𝑥2 . Where x1 is the
quantity of beef and x2 is the quantity of rice. His monthly food budget is
TSh 120,000, the price of beef (p1) is 3,000, and the price of rice (p2) is
2,000. He aims to find the minimum amount of money that he needs to
spend to reach that utility level of 3750.
a) Find Mansoor’s demand functions
b) Find the minimum amount of money that he needs to spend to reach a
utility level of 3750
c) Find Mansoor’s expenditure function.
 The Slutsky equation
Generally, if the price of something goes down, we buy more of it.
This is down to two effects:
Income effect: because it’s less expensive, we have more purchasing power
because it is a smaller drain on our personal finances.
Substitution effect: because it offers more utility per unit of money, other
alternatives become less attractive.
Eugen Slutsky managed to find an equation that decomposes this
effect based on Hicksian and Marshallian demand curves.
The Slutsky equation…
Mathematically, it is based on the derivatives of Marshallian and Hickisan
demands:

The left hand side of the equation is the total effect- that is, the derivative of x
(quantity) respect p (price).
It shows how much the total quantity of x that we consume varies when we change price.
The next part is the substitution effect- how much the variation is due to us
finding similar options.
It is obtained from the derivative of the Hicksian demand with regards price.
The right hand side is the income effect, how much changes in our purchasing
power affect the amount we consume of a certain good.
It is the derivative of the Marshallian demand with regards wealth (multiplied by the
quantity).
 Indirect utility and expenditure
The ordinary utility function, u(x), is defined over the consumption set X
and represents the consumer’s preferences directly, as we have seen.
It is referred to as the direct utility function.
Given prices p and income y, the consumer chooses a utility maximizing
bundle x(p, y).
The level of utility achieved when x(p, y) is chosen thus will be the
highest level permitted by the consumer’s budget constraint facing prices
p and income y.
The relationship among prices, income, and the maximised value of
utility can be summarised by

The function v(p, y) is called the indirect utility function.