CHAPTER ONE: THE THEORY OF CONSUMER BEHAVIOR.

1.1 Consumer Preferences

Everyday observation tells us that consumers differ widely in their preferences: some like liver,
others dislike it; some smoke cigarettes, others avoid cigarette smoke etc. Economists make three
assumptions about the typical consumer’s preferences.
First, we assume that preferences are complete in the sense that a consumer can rank (in order of
preference) all market baskets. For any two commodity bundles X and Y,a consumer will prefer
X to Y,Y to X or will be indifferent between the two. We say a consumer is indifferent between
two options when both are equally satisfactory. Importantly, this preference ranking reflects the
relative desirability of the options themselves and ignores their cost. Preferences and budgets
both influence consumer choice, but for the moment we will focus only on preferences.
Second, we assume that preferences are transitive. Transitivity means that if a consumer prefers
market basket A to B, and B to C, then the consumer prefers A to C. In a sense this condition
simply requires that people have rational or consistent preferences.
Third, a consumer is presumed to prefer more of any good to less. This characteristic is termed
non satiation and is expressed as “more is preferred to less.”
The theory of consumer choice lies on the assumption of the consumer being rational to
maximize level of satisfaction. The consumer makes choices by comparing bundle of goods.
There are two approaches to analyze consumer‘s decision making process. These are, the
cardinal and ordinal utility approaches.

2.2 Theories of Utility

Before discussing the concept of utility, let us first point out some of the assumptions that
economists make about the average consumer.
1. An average consumer is rational. That means:
a) A consumer has a clear-cut preference,i.e, the consumer is able to compare any
two bundles X and Y and decide which one he/she prefers.
b) A consumer has a persistent (transitive) preference. For example, given three
consumption bundles X, Y and Z, if X> Y, and Y>Z, then he/she prefers X to Z
(X>Z).
2. The consumer is not free in his/her choice. This means the consumer’s choice is constrained
by his/her income level and the prices of goods and services.

Definitions:
 Utility is defined as the power of a product to satisfy human wants. In other words
“Utility is the quality of good to satisfy a want.”

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  According to Mrs. Robinson, “Utility is the quality in commodities that makes
individuals wants to buy them”
 Utility is the word used to describe the pleasure or satisfaction or benefit derived by
a person from consuming goods.
Important Points to note About Property of Utility:

1. Utility is Subjective: as it deals with the mental satisfaction of a man. A thing may have
different utility to different persons.
E.g. Liquor has utility for drunkard but for person who is teetotaler, it has no utility.
2. Utility is Relative: As a utility of a commodity never remains the same. It varies with time
and place.
E.g. Cooler has utility in summer not during winter season.
3. Utility is not essentially useful: A commodity having utility need not be useful.
E.g. Liquor and cigarette are not useful, but if these things satisfy the want of addict then they
have utility for him.
4. Utility is independent of Morality: It has nothing to do with morality. Use of opium liquor
may not be proper from moral point of view, but as these intoxicants satisfy wants of the opium
– eaters, drunkards, they have utility.

2.2.1 Approaches to measure Utility:- There are two approaches to measure or compare
consumer’s utility derived from consumption of goods and services.

3.2.1.1 The Cardinal Utility Theory

According to the cardinalist approach, utility is measurable using arbitrary unit of measurement
called ‘util’ in the form of 1 util, 2 utils, 3 utils,etc. For example, consumption of an orange gives
Almaz 10 utils, and a banana gives her 7 utils. From this we can say that an orange gives Almaz
more utility than that of a banana. Hence, utility can be quantitatively measured.

Assumptions of Cardinal Utility theory

1. Rationality of Consumers. The main objective of the consumer is to maximize his/her
satisfaction given his/her limited budget or income. Thus, in order to maximize his/her satisfaction,
the consumer has to be rational.
2. Utility is cardinally Measurable. According to this approach, the utility or satisfaction of each
commodity is measurable. Money is the most convenient measurement of utility. In other words, the
monetary unit that the consumer is prepared to pay for another unit of commodity measures utility or
satisfaction.
3. Constant Marginal Utility of Money. According to assumption number two, money is the most
convenient measurement of utility. However, if the marginal utility of money changes with the level
of income (wealth) of the consumer, then money can not be considered as a measurement of utility.
4. Limited Money Income. The consumer has limited money income to spend on the goods and
services he/she chooses to consume.
5. Diminishing Marginal Utility (DMU).The utility derived from each from each successive units
of a commodity diminishes. In other words, the marginal utility of a commodity diminishes as the
consumer acquires larger quantities of it.

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6. Utility is additive: The total utility of a basket of goods depends on the quantities of the individual
commodities.
If there are n commodities in the bundle with quantities, X1, X2, …, Xn, the total utility is given by:
TU=f (X1, X2, …, Xn)
Total Utility (TU): It refers to the total amount of satisfaction a consumer gets from consuming or
possessing some specific quantities of a commodity at a particular time. As the consumer consumes
more of a good per time period, his/her total utility increases. However, there is a saturation point for
that commodity in which the consumer will not be capable of enjoying any greater satisfaction from
it.
Marginal Utility (MU): It refers to the additional utility obtained from consuming an additional
unit of a commodity. In other words, marginal utility is the change in total utility resulting from
the consumption of one or more unit of a product per unit of time. Graphically, it is the slope of
total utility.
Mathematically:
MU= ΔTU Where, ΔTU = change in total utility
ΔQ ΔQ = change in the amount of the product consumed
dTU
MU 
dQ
Marginal utility can be:
a) Positive Marginal Utility: If by consuming additional units of commodity, total utility goes
on increasing, and then marginal utility of these units will be positive.
b) Zero Marginal Utility: If the consumption of additional unit of commodity causes no change
in the total utility, it means the marginal utility of additional unit is zero.
c) Negative Marginal Utility: If the consumption of an additional unit of a commodity causes
fall in total utility, it means the marginal utility is negative.
For N commodities case
nth
MU = TUn – TUn-1; Where
MUnth = Marginal utility of nth unit.
TUn = Total utility of n units.
TUn-1 = Total utility of n-1 units

Is the utility you get from consumption of the first orange is the same as the second orange?
The utility that a consumer gets by consuming a commodity for the first time is not the same as the
consumption of the good for the second, third, fourth, etc.
Example : Table 3.1 , Hypothetical table showing TU and MU of consuming Oranges (X)
Units of quantity consumed (x) Total utility in utils Marginal utility
0 0 -
1 10 10
2 16 6
3 20 4
4 22 2
5 22 0
6 20 -2

Graphically, the above data can be represented as follows.

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Figure Total and marginal utility

As indicated in the above figures, as the consumer consumes more of a good per time period, the
total utility increases, at an increasing rate when the marginal utility is increasing and then increases
at a decreasing rate when the marginal utility starts to decrease and reaches maximum when the
marginal utility is Zero.

The law of diminishing marginal utility

This law states that other things remaining constant, consuming successive units of a product
gives a consumer less and less extra satisfaction (utility). This law is based on the assumptions
that:
 The consumer is rational
 The consumer consumes identical or homogenous product.
 There is no time gap in the consumption of the good.
 The consumer taste/preference remains unchanged.

Consumer Equilibrium: - Cardinal utility approach

Consumer Equilibrium: is the point where rational consumers maximize his/her total utility
given his/her income and prices of the commodities. As mentioned earlier, a consumer is
assumed to be a utility maximizer. Analyzing consumer’s equilibrium requires answering the
question as to how a consumer allocates his money income among the various goods and
services he/she consumes.

1. Consumer equilibrium: - One commodity case

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We begin with a simple model of one commodity case (say X). The consumer can either buys X
or retain his money income M. Under these conditions the consumer is in equilibrium when the
marginal utility of X is equated to its market price (PX). Symbolically we have: MUX  PX
Proof
Given the utility function
U  f ( X ) If the consumer buys commodity X, then his expenditure will be Qx.Px. The
consumer maximizes the difference between his utility and expenditure.

Max(U  QxPX )
The necessary condition for maximization is equating the derivative of a function to zero. Thus,
dU  d (Q P )  0
X X
dQ X dQ X
dU  P 0MU P
X X X
dQ X

If, MUX >PX, the consumer can increase his welfare by purchasing more units of X. (Which
result in decrease in MU of X)

Similarly if MUX < PX, the consumer can increase his total satisfaction by cutting down the
quantity of X and keeping more of his income unspent.

Therefore, he/she attains maximum utility when:

MUX= PX
Table 2 Utility schedule for a single commodity

quantity orange Total utility in MU MUper Birr Marginal utility

utils (price=2) of money
0 0 - - 1
1 6 6 3 1
2 10 4 2 1
3 12 2 1 1
4 13 1 0.5 1
5 13 0 0 1
6 11 -2 -1 1

For consumption level lower than three quantities of oranges, since the marginal utility of orange is
higher than the price, the consumer can increase his/her utility by consuming more quantities of
oranges. On the other hand, for quantities higher than three, since the marginal utility of orange is
lower than the price, the consumer can increase his/her utility by reducing its consumption of
oranges.

2. A case of more than one commodity: If there are more commodities, the condition for the
equilibrium of the consumer is the equality of the ratio of the marginal utilities of the individual
commodities to their prices. i.e.
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 MU X MU Y MU N
i) = …………… , and
PX PY PN

ii) PXX + PYY +-----------------+ PNN =M , where M is money income.

If income of a given consumer is 20 Birr and he wants to buy, Orange and Banana,
Orange, price=2 Banana, price=4
quantity TU MU MU/P quantity TU MU MU/P

0 0 - - 0 0 - -
1 6 6 3 1 6 6 1.5
2 10 4 2 2 22 16 4
3 12 2 1 3 32 12 3
4 13 1 0.5 4 40 8 2
5 13 0 0 5 45 5 1.85
6 11 -2 -1 6 48 3 0.75

Utility is maximized when the condition of marginal utility of one commodity divided by its market
price is equal to the marginal utility of the other commodity divided by its market price. Thus, the
consumer will be at equilibrium when he consumes 2 quantities of Orange and 4 quantities of
banana, because

How much is total utility at equilibrium? TU=10 + 40=54

Note: In determining the optimal level, at the equilibrium point should fulfill the basic budget line
equation. i.e PXX + PYY +-----------------+ PNN =M , where M is money income.

Example: Given a utility function of the form:

U(x,y) = 4x2 + 3xy +6y2: maximize utility subject to the budget constraint:

x + y = 56

Solution: The equilibrium condition is given by:

MU X MU Y
=
PX PY

Thus, MUX = 8x + 3y, and px = 1

MUY = 12y + 3x, and py = 1

Applying the equilibrium condition , we have:

X = 36, and Y = 20. Therefore, the consumer purchases 20 units of good Y and 36 units of good X.

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 Derivation of the Demand Curve of the Consumer
The derivation of demand curve is based on the axiom of diminishing marginal utility. The MU
of the commodity X may be depicted by a line with a negative slope. Geometrically, MU is the
slope of the TU function U=f (QX). MUX declines continuously, and become negative beyond
quantity X*. If MU is measured in monetary units, the demand curve for X is identical to the
positive segment of the marginal utility curve.

MUX PX

MU1 P1 Demand curve

MU2 P2

MU3 P3

QX QX
X1 X2 X3 X* MU X1 X2 X3 X*
X

At X1 the marginal utility is MU1, which at equilibrium is equal to P1 and at X2 marginal utility is
MU2, which in turn is equal to P2 at equilibrium. The negative section of the MU curve does not
form part of the demand curve, since negative quantities and price do not make sense in
economics.

Weaknesses of cardinal utility approach:

The assumption of cardinal (measurable) utility is doubtful. The satisfaction derived from
the consumption of various commodities cannot be objectively measured.
The assumption of constant marginal utility of money is also unrealistic. The utility
derived from a unit of money varies with the level of income of the consumer. MU of a
unit of money for a poor person is by far higher than the MU of a rich person. Thus,
money cannot be used as a measuring rod since its own utility changes.

2.2.1.2 The Ordinal Theory of Utility (The Indifference Curve Approach)

Unlike the cardinal utility theory, the ordinal utility theory says that utility is not measurable,
rather the consumer can rank or order the utility he/she derives from consuming different goods
& services.
For example, if a consumer is subjected to three consumption bundles X, Y and Z, he/she can
rank or order his/her preference as :

1st preference = Y 1st = X

2nd preference =Z or 2nd = Z

3rd preference =X 3rd = Y, or any other ordering.

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The ordinal utility theory provides another method of studying the consumer behavior. Since it
uses indifference curves to study consumer’s behavior, this theory is also known as the
indifference curve approach. A consumer’s preferences across various market baskets or
combinations of goods can be shown in a diagram with indifference curves. An indifference
curve plots all the market baskets that a consumer views as being equally satisfactory. In other
words, it identifies the various combinations of goods among which the consumer is indifferent.

Indifference Set, Curve and Map

The indifference curve approach is based on the following assumptions;

1. The consumer is rational (utility maximizer )

2. The consumer can simply order /rank his/her performances.

3. There is diminishing marginal rate of substitution.

4. The consumer’s preference is transitive.

A) Indifference set: An indifference set is a combination of goods for which the consumer is
indifferent.

Eg. Combination Qx Qy Utility

A 10 2 U

B 6 4 U

C 3 6 U

D 2 8 U

Note: - Each combination gives the consumer equal level of total utility.

B). Indifference Curve: - An indifference curve is a line (curve) that connects

all possible combinations of goods and services which gives the consumer equal
total utility.

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 U
Qx
Fig 3.1 an indifference curve
Indifference Map: - So far we have examined only one indifference curve. To show a
consumer’s entire preference ranking, we need a set of indifference curves, or an indifference
map.
Qy

U3

U2

U1

Qx

Fig: 3.2 an indifference curve and map.

Properties of an indifference curve:

I. An indifference curve is downward slopping. First, an indifference curve must slope
downward if the consumer views the goods as desirable.
II. A consumer prefers a market basket lying above (to the northeast of) a given
indifference curve to every basket on the indifference curve.
III. It is convex to the origin (reflects diminishing marginal rate of substitution).
IV. Indifference curves never cross each other. If they do, the consumer is not consistent in
his/her choice. The assumption of transitivity will be violated.

Consider the graph below

Y

BA

I2
I1
X

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From the above graph, A  B ^ A ~C, thus by the axiom of transitivity B  C, which is
wrong. Thus, since both bundles lie on the same indifference curve I1 , they should be

indifferent .that is, why ICs can never intersect each other.

Marginal Rate of Substitution (MRS)

The MRS is the rate by which the consumer is willing to give up (scarify) a good so as to obtain
more of another good holding total utility constant. Consider the following figure. The marginal
rate of substitution is related to the slope of the consumer’s indifference curves.
Y
a

In the above figure, in moving from point ‘a’ to ‘b’ the consumer is willing to scarify good Y to
get more of good X. goods X&Y, the MRS of good X for Y shows the amount of good Y the
consumer is willing to give up so as to get more units of good X, holding total utility constant.
Mathematically:
MRS xy = Δy = The amount of good Y sacrificed = The slope of the
Δ x The amount of good X gained Indifference curve

i) Conceptual derivation of MRS

Given U= f(x,y)
Lets suppose x is substituted for y. when the consumption of y decreases, the stock of
y declines by y
i.e level of marginal sacrifice will be -MUy. y
When the consumption of x increases, the stock of x will increase by x
i.e level of marginal gain will be MUx . x
thus, for the total utility to remain constant, the level of marginal gain should be equal
with the level of marginal sacrifice .
 MUx. x =-MUy. y
MUx  y
= =MRSxy
MUy x

ii) mathematical derivation of MRS

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 Given U=f(x,y)
Total differential of the utility function will give us
U U
dU  . dX  . dY =0
X Y
MUx . dx + MUy. dy  0
MUx. dx= - MUy. dy
MUx  dy
   MRSxy
MUy dx The law of Diminishing MRS
ICs are usually convex or bowed inward .the term convex
means the slope of the IC increases (becomes less negative ) as we move down along the
curve. In other words, an IC is convex if MRS diminishes along the curve
Exceptional Indifference Curves
In a standard case indifference curves are convex to the origin and downward sloping as we have
seen earlier and this shape of indifference curve is true for most goods. In this situation, we
assume that the two commodities such as X and Y can substitute one another to a certain extent
but are not perfect substitutes. However, the shape of the indifference curve will be different if
commodities have some other unique relationship such as perfect substitution or complementary.
Here, are some of the ways in which indifference curves/maps might be used to reflect
preferences for three special cases.
1. Perfect substitutes: perfect substitutes are goods which can be replaced for one another at a
constant rate. If two commodities are perfect substitutes (if they are essentially the same), the
indifference curve becomes a straight line with a negative slope. MRS for perfect substitutes is
constant. (Panel a)
2. Perfect complements: perfect complements are goods which are to be consumed jointly at a
constant rate. If two commodities are perfect complements the indifference curve takes the
shape of a right angle. Suppose that an individual prefers to consume left shoes (on the
horizontal axis) and right shoes on the vertical axis in pairs. If an individual has two pairs of
shoes, additional right or left shoes provide no more utility for him/her. MRS for perfect
complements is zero i.e. there is no substitution between the two goods.
3. A useless good: Panel C in the above figure shows an individual’s indifference curve for
food (on the horizontal axis) and an out-dated book, a useless good, (on the vertical axis).
Since they are totally useless, increasing purchases of out-dated books does not increase
utility. This person enjoys a higher level of utility only by getting additional food
consumption. For example, the vertical indifference curve shows that utility will be the same
as long as this person has same units of food no matter how many out dated books he/she
has.

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 Panel a Panel b Panel c

Fig.2.6 Special cases of indifference curves

Budget
udget constraint of the consumer
The consumer has a given income, which set limits to his maximizing behavior. Income acts as a
constraint in the attempt for maximizing utility. The income constraint, in the case of two
commodities, may be written as:
M= PXQX+PYQY
Where, M= money income
PX = price of good X
PY = Price of good Y
We may represent the income constraint graphically using the budget line; whose
equation
tion is derived from the budget equation by solving for QY.
M
QY=  PXQX ,if QX=0, i.e ,if the
PY
Consumer spends all his income on Y, he/she can buy
QY=M/PY units of Y. similarly
M  PYQY
QX= , If the consumer spends all his income on X) i.e at QY =0
PX
QX= M
PX
This assumption shows that the commodities can substitute one another,

M
=A
Px

Slope=  PX
PY

M
0 B=
Px
Mathematically, the slope of the budget line is the derivative

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QY M / PY PX
 PX  
Qx PY M / PX PY

2.12 Factors Affecting the Budget Line

Budget line depends on the price of the two goods and the income of the consumer. Any change in the
income of the goods or the income of the consumer results in a shift in the budget line. Let us examine the
impact of these changes one by one.
a. Effects of changes in income
If the income of the consumer changes (keeping the prices of the commodities unchanged) the budget line
also shifts (changes). Increase in income causes an upward shift of the budget line that allows the
consumer to buy more goods and services and decreases in income causes a downward shift of the budget
line that leads the consumer to buy less quantity of the two goods. It is important to note that the slope of
the budget line (the ratio of the two prices) does not change when income rises or falls. The budget line
shifts from Bo to B1 when income decreases and to B2 when income rises.
M2/Py
Where
M2>Mo>M1
Mo/Py
Bo B2
M1/Py
B1

M1/PX Mo/PX M2/PX

Fig.2.8 Effects of change in income

b. Effects of Changes in Price of the commodities

Changes in the prices of X and Y is reflected in the shift of the budget lines. In the figures below
(fig.a), a price decline of good X results in the shift from B to B1.A fall in the price of good Y in
figure (b) is reflected by the shift of the budget line from B to B1.We can notice that changes in
the prices of the commodities change the position and the slope of the budget line. But,
proportional increases or decreases in the price of the two commodities (keeping income
unchanged) do not change the slope of the budget line if it is in the same direction.

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 Y Y

B1 B1
B
B
X X

Fig. a Fig.b
Fig.2.9 Effects of change in price

Let us now consider the effects of each price changes on the budget line
 What would happen if price of x falls, while the price of good Y and money incme
remaining constant?

Here PX1, < Pxo, hence M/Pxo <M/Px1

B B’

M/PX0 M/Px1
Fig. 2.10 Effect of a decrease in price of x on the budget line
Since the Y-intercept (M/Py) is constant, the consumer can purchase the same amount of
Y by spending the entire money income on Y regardless of the price of X. We can see from the
above figure that a decrease in the price of X, money income and price of Y held constant,
pivots the budget line out-ward, as from AB to AB’.
 What would happen if price of X rises, while the price of good Y and money incme remaining constant?
Since the Y-intercept (M/Py) is constant, the consumer can purchase the same amount of
Y by spending the entire money income on Y regardless of the price of X. We can see from the

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figure below that an increase in the price of X, money income and price of Y held constant,
pivots the budget line in-ward, as from AB to AB’.

A
M/Py

B
B’

M/Px1 M/Px2

Fig. 2.11 Effect of an increase in price of x on the budget line

 What would happen if price of Y rises, while the price of good X and money incme remaining
constant?
Y
A
M/py

A’
M/py'

B
M/Px X
Fig.2.12 Effect of a raise in price of Y on the budget line

Since the X-intercept (M/Py) is constant, the consumer can purchase the same amount of
X by spending the entire money income on X regardless of the price of Y. We can see from the
above figure that an increase in the price of Y, money income and price of X held constant,
pivots the budget line in-ward, as from AB to A’B.

 What would happen if price of Y falls, while the price of good X and money incme remaining
constant?

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 Y
M/py' A’

M/py A

B M/Px X

Fig.2.13 Effect of a fall in price of Y on the budget line

The above figure shows what happens to the budget line when the price of Y increases while the
price of good X and money income held constant. Since Py decreases, M/Py increases thereby the budget
line shifts outward.
 Dear colleague some times the price of the two commodities may change proportionally in the same
direction and this will have a shifting impact on the budget line.
Proportional increase in the prices of both goods (X and Y), income remaining the same, reduces
the total quantity of the two goods that the consumer can buy with the given income. For example if the
price of the two goods doubles, the quantity of the two goods bought decreases by half. Since the slope
of the budget line which is the ratio of the prices of the two goods remains the same, there will be a
parallel inward shift of the budget line.
When there is proportional (equal) decrease in price of the two goods income remaining the
same, the quantity bought of the two goods increases which leads o a parallel out ward shift in the
budget line

M/Py2
Where PX1>PXo > PX2 and

M/Pyo
Where PY1>PYo > PT2

M/Py1 Bo B2

B1

M/PX1 M/PXo M/PX2

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 Example
A person has Birr 60 to spend on two goods(X, Y) whose respective prices are Birr 3 and
Birr 6.
a) Draw the budget line.
b) What happens to the original budget line if the budget decreases by 50%?
c) What happens to the original budget line if the price of Y doubles?
d) What happens to the original budget line if the price of X falls to Birr 2?
e) What happens to the original budget line if price of both X and Y is doubled?

From our previous discussion, the budget line for two commodities is expressed as:
PX X  PY Y  M
3 X  6Y  60
6Y  60  3 X
60 3
Y  X
6 6
1
Y  10  X
2
Y  10  0.5 X
When the person spends all of his income only on the consumption of good Y, we can get
the Y intercept that is(0,10).However, when the consumer spends all of his income on the
consumption of only good X, then we get the X intercept that is (20,0). Using these two points
we can draw the budget line. Thus, the budget line will be:

10 A

A’

B’ B
20 X

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 If the budget decreases by 50%, then the budget will be reduced to 30.As a result, the
budget line will be shifted in-ward as indicated by (A’B’).This forces the person to buy less
quantity of the two goods. The equation for the new budget line can be solved as follows:
3 X  6Y  30
6Y  30  3 X
30 3
Y   X
6 6
1
Y 5 X
2
Y  5  0 .5 X
Therefore, the Y-intercept decreases to 5 units while the X-intercept is only 10 units.
However, since the ratio of the prices does not change the slope of the budget line remains
constant.
If the price of good Y doubles the equation of the budget line will be 3 X  12Y  60 and
if the price of good X falls to Birr 2, the equation for the new budget line will be 2 X  6Y  60 .
If price of both X and Y is doubled, the new budget line equation will be
6X+12Y=60.The X-intercept and Y-intercept decreases to 10 units and 5 unity respectively. The
slope remaining the same (-0.5), the budget line shifts inward in a parallel way.
Consumer equilibrium: ordinal utility approach
A consumer attains his equilibrium position when he maximizes his total utility given is
income and price of the commodities.
Technically, the conditions that make the consumer in equilibrium are:
i) first order condition (Necessary condition)
Px
MRSxy should be equal with price ratio. i.e. MRSxy=
Py

ii) second order condition ( Sufficient condition)

Px
MRSxy= at higher IGraphically,
Py

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 Com y

I3
e.  equilibrium po int
I2

I1
 budgetline
Com x
At point of equilibrium e,
Slope of the budget line = slope of I2
Px
  MRSxy
Py
Mathematical derivation of equilibrium point
To determine the consumer’s optimum point, we will maximize
U=f(Qx,Qy) subject to the budget constraint Qx.Px +Qy.Py=Y
To do so, we will follow the following steps:
i) Rewrite the budget equation as Qx.Px+Qy.Py-Y=0
ii) then multiply left hand side of the above equation by the
lagrangian multiplier (  ), in doing so, we get  (Qx.Px  Qy.Py  Y )
iii) subtract the above rewritten constraint from the utility function and
construct the composite function as
  U   (Qx.Px  Qy.Py  Y )
iv)Then maximize the composite function and find the optimal values of Qx,
Qy and 
 U  (Qx.Px  Qy.Py  Y )
    0...............................................................(1)
Qx Qx Qx
 U  (Qx.Px  Qy.Py  Y )
    0...............................................................(2)
Qy Qy Qy
 U  (Qx.Px  Qy.Py  Y )
    0...................................................................(3)
  
Then,
 U
  Px  0............................ from(1)
Qx Qx

U MUx
  Px  MUx   
Qx Px

19
  U
  Py  0............................ from(2)
Qy Qy
U MUy
  Py  MUy   
Qy Py

 (Qx.Px  Qy.Py  Y )  0............................ from (3)

 Qx.Px  Qy.Py  Y  0
Solving the above simultaneous equations we will get
 
MUx MUx
 
Px Py
rearranging the above expression ,we will arrive at the consumer’s
MUx Px
equilibrium point   MRSxy
MUy Py
Interpretation of 
 is interpreted as the marginal utility of income(MUI).
Thus, MUI is interpreted as the extra satisfaction derived from having one
more birr and it is amounted to  .
Example
Suppose a consumer having a disposable income of 600 birr consumes only two
commodities X and Y and his utility function is given as U(X,Y)=100X-X2+50Y
Given further that prices of X and Y are 2 birr and 5 birr respectively, determine:
a) the consumer’s optimal bundles
b) his Marginal utility of income and its interpretation

Solution
Technique 1 (using the lagrangian)
a) Construct the budget equation, i.e. X.Px+Y.Py=M
2X+5Y=600
i) 2X+5Y-600=0
ii)  (2 X  5Y  600)
iii)  =100X –X2 +50Y-  (2 X  5Y  600)

 100  2 X  2  0.....................................................(1)
X

 50  5  0.....................................................................( 2)
Y

 ( 2 X  5Y  600)  0.....................................................(3)

then solve these three equations simultaneously, in doing so,
50=5     50  10  MUI …………………….from (2)
5
100-2X-2  =0………………………..from (1)

20
  100  2 X  2(10)  0  80  2 X  0  X  80  40
2
Optimal X=40 units
2X+5Y=600 ………………………….from (3), then substitute the optimal X
2(40) +5Y=600  5Y  600  80  Y  520  104
5
Optimal Y=104 units
 the optimal bundles of the consumer are 40 units of X and 104 units of Y
b) marginal utility of income(MUI)=   10 ,thus, the extra satisfaction that the consumer
will derive from one more birr is 10.
Technique 2: the short cut (the marginal analysis)
i) determine the marginal utilities
U  (100 X  X 2  50Y )
MUx=   100  2 X
X X
U  (100 X  X 2  50Y )
MUy=   50
Y Y
MUx Px
ii) from the equilibrium point we have  ,thus we can determine the
MUy Py
equilibrium proportions of X and Y
100  2 X 2
   100  500  10 X  10 X  400  X  40units
50 5
ii) then substitute this proportion in to the third equation ( the budget equation ) and
solve fro the optimal values ,
2(40)+5Y=600  Y  520  104units
5
 the consumer’s optimal bundles are 40 units of X and 104 units of Y.
Then after, similar procedures to the 1st technique will be employed to determine the
rest requirements.

Example 1
A consumer consuming two commodities X and Y has the following utility function U  X 1.5Y
.If the price of the two commodities are 3 and 4 respectively and his/her budget is birr 100.
a) Find the quantities of good X and Y which will maximize utility.
b) Total utility at equilibrium.
c) Find marginal utility of income and provide its interpretation
d) Find the MRS X ,Y at optimum point

e) Show optimum point graphically

Example 2
A consumer consuming two commodities X and Y has the following utility function U  X 2Y 2
.If the price of the two commodities are Birr 1 and 4 respectively and his/her budget is birr 10.

21
 A) Find the quantities of good X and Y which will maximize utility.
B) Find the MRS X ,Y at optimum.

C) Total utility at optimum point

D) Find marginal utility of income and provide its interpretation
E) Find the MRS X ,Y at optimum point

F) Show optimum point graphically

Exercise 3
Assume the consumer having a given income of 200 birr consumes only two
1
commodities Qx and Qy and his utility function is given as U(Qx,Qy)= Qx.Qy
4
Assuming further that prices of Qx and Qy are 5 birr and 10 birr respectively.
Determine:
A. Find the quantities of good X and Y which will maximize utility.
B. Find the MRS X ,Y at optimum.

C. Total utility at optimum point

D. Find marginal utility of income and provide its interpretation
E. Find the MRS X ,Y at optimum point

F. Show optimum point graphically

Effects of change in Income on Consumption
a) Income effect on Normal good (eI>0)
Good Y

ICC- Income Consumption Curve (Income Offer Curve)

Or Income Expansion Path

I4
I3
I2

I1
Good X
ICC- is a curve which is a locus of various consumer equilibrium points resulting from changes
in income, cetris paribus.

b) Income effect on Inferior good (eI<0)

22
 i) Let commodity x be inferior good
Com y
ICC

I3

I2

I1
Com x

ii) Let commodity y be inferior good

Com y

I2 I3
I1

ICC

Com x

Engle curve
It is a curve depicting the relationship between equilibrium quantity purchased of a
commodity and the level of income.

23
 Graphical derivation of an Engle curve for a Normal good X
Com y

ICC

M3

M2
M1
Com x

Money
Engle curve

M3

M2

M1

X1 X2 X3 Com x

For a general case, the Engle curve is

Money

Inferior good

Normal good

Com x

24
 Distinction
ICC - traces the utility-maximizing combinations of two goods as a consumer’s income
Changes.
Engle curve - depicts the relationship between consumer’s income & equilibrium
Quantity consumed of a commodity.

Effects of change in price on Consumption

a) On Normal good
Com y

PPC – Price Consumption Curve (Price offer curve)

C Or price expansion path
A B
I3
I1 I2
Com x

Price

P3 A
P2 B

P1 C

Demand curve

X1 X2 X3 com x
PPC - is a locus of equilibrium points on Indifference curves resulting from changes in the
price of a commodity.

25
b) On Giffen good
 Giffen good is a type of good whose demand decrease as its price decreases.
 Giffen good is an inferior good but not all inferior goods are giffen

Com y price

ICC Demand curve

Com x quantity

Substitution and Income Effects of Price change

A fall in price of a good has two effects:
1) Consumers will tend to buy more of the good that has become cheaper and less of
those goods that are now relatively more expensive. This response to a change in the
relative prices of goods is called the substitution effect.
2) Because one of the goods is now cheaper, consumers enjoy an increase in real purchasing
power. They are better off because they can buy the same amount of the good for
less money, and thus have money left over for additional purchases . The change in
demand resulting from this change in real purchasing power is called the income
effect.
OR in general;
Substitution effect : change in consumption ( demand )due to a change in its
relative price, with the level of utility held constant.
Income effect : change in consumption ( demand ) resulting from a change in
real purchasing power , with relative prices held constant.
N.B:
 The fundamental reasons for the existence of the law of Demand are the
substitution & income effects
- Total Effect(TE)=Price Effect (PE)= Substitution Effect (SE) + Income Effect(IE)

Normally, these two effects occur simultaneously , but it will be useful to

distinguish between them for purposes of analysis . To illustrate this ,there are two
decomposition techniques: Slutsky & Hicksian approaches , however , due to its
popularity, we will apply the Hicksian decomposition technique

.
a) Substitution& Income Effects of price decrease on Normal good

26
 Com y

Y1 P
Y2 R
Y3 Q I2
I1

X1 X3 F X2 H Com x
Illustration:
Substitution Effect (SE)
The first step is to eliminate the income effect : to do this , we assume that
accompanying to fall in the price of X, there is a compensating variation in income which
leaves the consumer at the same level of utility ( real income) as before the price decrease.
The original budget line is labeled AF while the new budget line after the price decrease is
is labeled until it becomes tangential to the original indifference curve I1. The imaginary
budget line ( compensating variation income ) is GH and the movement of consumer
equilibrium from P to R is the Substitution Effect
-the consumer is no better off but has substituted X1X3 of X for Y1Y3 of Y because of the
change in relative prices, thus the rest of the price change effect will be the income effect.
⟹ = 1 3
Income Effect (IE)
To illustrate the income effect, we will eliminate the substitution effect by holding the relative
prices constant. Accordingly, this will be reflected by movement from one indifference, I1 to
the other indifference curve ,I2 OR movement between the new budget line , A and the
imaginary budget line, GH .thus , movement from R to Q is due to income effect – the
consumer buys X3X2 of X and Y3Y2 of Y because of increase in his real income.
⟹ = 3 2 b/c EI >0 for Normal goods
TE = PE =SE + IE= X1X3 + X2X3 =X1X2 ( reflected by movement fom P to R)
N.B: Graphically in general:
Substitution Effect (SE): is reflected by movement from the original equilibrium
To the point of between the original indifference curve
And the imaginary budget line, along the original
Indifference curve.
Income Effect (IE) : is reflected by movement between the two parallel new and
Imaginary budget lines or movement from the point of
Tangency between the original IC and the imaginary budget

27
 Line to the new equilibrium, between the two ICs.

b) Substitution & Income effects of price decrease on Inferior good

com y
A

Y2 Q
G I2
Y1 P

Y3 R

I1
Com x
X1 X2 X3 F H
Applying the same decomposition procedures:
SE- is reflected by movement from P to R
SE= X1X3 and positive
IE- is reflected by movement from R to Q
IE = X3X2 and negative b/c EI<0 for inferior goods
TE- is reflected by movement from P to Q
⟹ =PE=SE+IE= X1X3+(-X3X2)=X1X3-X3X2=X1X2
∴ The total effect of a decrease in the price of an inferior good positive because the
positive substitution effect outweighs ( exceeds ) the negative income effect. Thus , the
decrease in the price of an inferior good will totally lead to increase in its demand and
consumption.
c) Substitution and Income effects of price decrease on Giffen goods

Com y
A
Q
Y2

G I2
Y1 P

Y3 R

I1
Com x
X2 X1 X3 F H
Applying the same decomposition procedures:

28
SE- is reflected by movement from P to R
SE= X1X3 and positive
IE- is reflected by movement from R to Q
IE = X3X2 and negative b/c EI <0
TE – is reflected by movement from P to Q
TE=PE= SE + IE = X1X3 + (-X3X2)
the total effect a decrease in the price of a Giffen good is negative because the negative
income effect outweighs (exceeds ) the positive substitution effect. Thus , a decrease in the
price a Giffen good leads to a decrease in its demand ( consumption ). We can say that
Giffen goods are Inferior good but inferior goods are not always Giffen good.
Exercise
1) Graphically illustrate the substitution ,income & total effects of a price increase on:
a) Normal good
b) Inferior good
c) Giffen good
2) Graphically illustrate the substitution, income & total effects of a price decrease
on:
a) Apple & orange juice
b) Left & right shoes

The Slutsky Equation

As we have discussed earlier,when the price of a good decreases,there are two effects in
consumption.The change in relative prices makes the consumer to consume more of the cheaper
good(substitution effect) and the increase in purchasing power or real income due to the lower
price may increase or decrease consumption of the good (income effect).
Generally,the Slutsky equation says that the total change in demand is the sum of the
substitution effect and the income effect.
Example
Suppose that the consumer has a demand function for good X is given by
2
X  20  MPX
Originally his income is $ 200 per month and the price of the good is 5 per killogram.
200
Therfore,his demand for good X will be 20   28 per month.
52
Suppose that the price of the good falls to 4 per kilogram.Therfore,the new demand at the
200
new price will be: 20   32.5 per month.
42
Thus,the total change in demand is 4.5 that is 32.5-28.

29
 When the price falls the purchasing power of the consumer changes.Hence,in order to
make the origiinal consumption of good X,the consumer adjusts his income.This can be
calculated as follows:

M 1  P1' X  P2Y
M  P1 X  P2Y
Subtracting the second equation from the first gives:
M 1  M  X [ P1'  P1 ]
M  XP1
Therfore,new income to make the original consumption affordable when price falls to 4
is:
M  XP1
M  28 * [4  5]  28
Hence,the level of income necessary to keep purchasing power constant is
M 1  M  M  200  28  172
The consumers new demand at the new price and income will be :
172
X (4,172)  20   30.75
42
Therfore,the substitution efffect will be:
X  X (4,172)  X (5,200)  30.75  28  2.75
The income effect will be:
X (4,200)  X (4,172)  32.5  30.75  1.75
Since the result We obtained is positive we can conclude that the good is a normal good.
Example
Suppose the consumer has a demand function for milk of the form
= 10 +
Originally his income is 120 birr per week and the price of milk is 3 birr per milk.
Now suppose that the price of milk falls to 2 birr per cup.
Given the above information, find
a. change in level of income required to make original consuption affordable
b. level of income required to make original purchasing power constant
c. find substitution effect of change in price
d. find income effect of change in price
e. find total effect of change in price

30
 CONSUMER SURPLUS (CS)
CS is the difference between the willingness to pay and the actual payment. It is
area below demand curve and above price line.
Producer surplus (PS): is the difference between the actual receipts and the
willingness to sell. It is area above supply curve and below
price line.
i) Geometrical approach

a) Price

A CS

e Px
P*
Demand curve
O q* Quantity x

CS = Willingness to pay (demand) actual payment (price)

= a(OAeq*) a( OP*eq*) = a(P0Ae)
⟹ = (P*A)(q*) unit2 and PS= 0
b)

Supply curve

e
P*

Demand curve

O quantity x
q*
First determine the equilibrium price and quantity by setting DD=SS
CS= willingness to pay (demand) – actual payment (equilibrium price* equilibrium qty)
= OAeq* OP*eq* = P*Ae
⟹ = (P*A)(q*) unit2
PS= Actual receipts (equilibrium price *equilibrium qty) – willingness to sell (supply)
= OP*eq*− Oeq*=Oeq*
⟹ = (q*)(q*e) unit 2
ii) Integration approach
Case (a)
Given the inverse demand function: P= − , to determine CS &PS at q =q*

31
 ∗
CS =∫ ( − ) – [( ∗)( ∗)] unit2
PS= 0
Case (b)
Given the inverse demand & supply functions as:
DD: P= −
SS: = +
To determine CS&PS, the First step is find the equilibrium price & quantity by setting
DD=SS , thus you will get equilibrium price P* & equilibrium quantity q*,then:
∗
CS=∫ ( − ) − [( ∗)( ∗)]
∗
PS= [( ∗)( ∗)] − ∫ ( + )
N.B: CS+PS= Welfare
Example
Given inverse demand & supply functions:
DD: P=200−2
SS: = 100 + 3
Then determine the consumer surplus & producer surplus
Solution
i) Geometrical approach
Determine the equilibrium Price & quantity
DD=SS …………………………..at equilibrium
200−2 = 100 + 3 ⟹ 5 = 100 ⟹ = 20
= 200 − 20(20) ⟹ = 200 − 400 ⟹ = 160
∴ = 160 & = 20

Then depict these values on the graph

Price

200 CS
= 100 + 3

160

100 = 200 − 2
PS
Quantity
20
CS = (200 − 160)(20)= (40)(20) = 400 2

PS = (160 − 100)(20) = (60)(20) = 600 2

ii) Integration approach

CS = ∫ (200 − 2 ) − [(160)(20)] = 200 − − 3200

= [200(20) − (20 )] − 0 − 3200 = 4000 − 400 − 3200
= 400 2

32
 PS = [(160)(20)] − ∫ (100 + 3 ) = 3200 − 100 +

= 3200 − 100(20) − 3 − 0 = 3200 − 2000 − 600

= 600 2

Example
Given inverse demand & supply functions:
DD: P=180−2
SS: P=100 + 2 Q
Then determine the consumer surplus & producer surplus
i. Geometrical approach
ii. Integration approach.

33