CONSUMER THEORY Ogada 0721101472

THE BUDGET CONSTRAINT

Resources are scarce relative to human wants. Because of scarcity, we are forced to make a choice for the needs to be
fulfilled first using available resources. When making the choice, it is assumed that we choose the best option/bundles that
we can afford with our limited income.

Let us assume two goods X and Y

Unit price of X → P x , unit price of Y → P y. Let the income or consumer budget be represented by M.

P X X represents the amount of money the consumer is spending on good X and P y Y is amount of money spent on good Y

The budget constraint of the consumer requires that the amount of money spent on the two goods be no more than the
total amount the consumer has to spend. The consumer affordable consumption bundles are those that don’t cost any more
than M

Px X + P y Y ≤ M

This Set of affordable consumption bundles at prices P x ∧P y and income M is called the budget set of the consumer. The
budget line is the set of bundles that cost exactly M

P x X + P y Y =M

These are the bundles of goods that just exhaust the consumer income

Qty of Y
M
Vertical Intercept = P y

A
G
D

H
C

B Qty of X
M
Horizontal Intercept = P x

M
The maximum amount of X that the consumer can buy (point B).
Px

M
The maximum amount of Y that he can buy is (point A).
Py
Between points A and B there are other possible combinations (consumption bundles) of X and Y that the consumer can
afford and this is achieved by reducing amounts of X and increasing Y or vice versa.

Beyond line AB we also have some combinations of X and Y but unfortunately consumer cannot afford them because his
budget/income cannot allow him, e.g. G

i.e. P x X + P y Y > M The consumer is therefore constrained to choose the bundles on the line AB. If he chooses at
point H

Px X + P y Y < M This is less than the budget. A rational consumer/ consumer out to optimize his income or choosing
the best bundle.

Changes in the budget line

M Px
From the budget line equation Y = − X the budget line can only change if P y or P x or M changes.
Py P y

An increase in income will increase the vertical intercept and not affect the slope of the line. Thus, an increase/ decrease
in income will result in a parallel shift outward/inward of budget line.
Qty of Y

A
} } over {{P } rsub {y } ¿
M ¿

M
Py
'
M
Py B Qty of X
'
M M
} } over {{P } rsub {x } ¿
Px Px M ¿

If price of one of the commodities changes holding the other price and M constant, the slope of the budget line will also
change.

If price of X increases (X is a normal good) the consumer will reduce quantity demanded of X. the vertical intercept will

Qty of Ybut the budget line will be steeper since P x /P y will become larger
not change

A
} } over {{P } rsub {y } ¿
M ¿ Px
Slope=−
Py
Px
Slope=−
Py
Simultaneous change in both prices ( P x ∧P y ) by the same factor, will affect income by the inverse of that factor.

Suppose, for example, that prices of both goods X and Y are doubled, both the horizontal and vertical intercepts will shift
inward by a factor of one half and therefore the budget the shifts inward by one half as well. Multiplying both prices by
two is just like dividing income by 2

P x X + P y Y =M

Multiplying both prices by t will yield

tP x X +tP y Y =M

t (P ¿ ¿ x X+ P y Y )=M ¿

M
Px X + P y Y =
t

M M
If M decreases and both P x ∧P y increase the intercepts and must both decrease. This means that the budget line
Px Py
will shift inward. In a situation where both prices and income changes with the same margin, then the budget line will not
change.

Taxes, subsidies and rationing

These are economic policy tools used to affect consumer budget constraint

Taxes
There are 2 types of taxes that can be imposed on commodities
1. Quantity tax
2. Ad valorem (value) tax

Quantity tax is levied per unit of commodity consumed or purchased from the consumer’s viewpoint; the tax is just like a
higher price. Thus, a quantity tax of t shillings per unit of goods x simply changes the price of good X from P x to P x +t .
this will make the budget line to be steeper.
Qty of Y

A
} } over {{P } rsub {y } ¿
M ¿ Px
Slope=−
Py
Px
Slope=−
Py

B Qty of X
Ad-Valorem tax is levied on the value (price) of the commodity, rather than on the quantity of goods purchased. Value tax
is usually expressed in percentage terms (a certain% of the price)

P x +r P x ∨Px (1+r )

Where r represents the tax %

The consumer will pay P x to the producer and rP x to the government for each unit of the good. Both taxes have the
effects of rotating the budget line inwards. They make X more expensive.

Subsidies

A subsidy is the opposite of tax. In the case of a quantity subsidy, the government gives an amount to the consumer that
depends on the amount of the good purchased.

P y −t

It makes the budget line flatter.

Ad valorem subsidy is based on the prices of the goods being subsidized

P y +r P y → P y (1 −r )

Lump-sum tax / Subsidy

In the case of a tax, the government takes away some fixed amount of money, regardless of the individual consumption
behavior. Doesn’t affect price of X or Y but the consumer money income has been reduced. Lumpsum subsidy will make
the budget line to shift outward.
Qty of Y

Qty of X

Rationing
This means that the government fixes the level of consumption of some good to be no larger than some amount. It is
restriction on consumption

Qty of Y Qty of Y

A A

Qty of X Qty of X
0 X
0
B
X B

The shaded area shows the new budget Set.

Sometimes the government may combine rationing with taxes or subsidies. For example, a consumer could consume good
X at a price of P X up to some level X and then pay a tax t on all consumption in excess of X then the budget Set will be
depicted as

Qty of Y

Qty of X
0
X B

PREFERENCES

The budget line deals with what the consumer can afford. Preferences deals with the concept of “best things”. People
choose the best things that they can afford.

The objects of consumer choice is called consumption bundles. This is a complete list of goods and Services that are
involved in the choice problem being investigated. Continuing with the assumption of only two goods X & Y,
Bundles A ( X a , Y a )

B ( X b , Y b)

We would expect that the consumer will rank the 2 bundles according to his taste such that he can prefer A ( X a , Y a ) to B (
X b , Y b)

i.e. A> B

or B is preferred to A

i.e. B> A

The consumer can also be indifferent between the two bundles of goods i.e. A B . Indifference means that the consumer
would be just as satisfied, according to his own preferences, by consuming the bundle A as she would be by consuming
the other bundle B.

To be able to make the above 3 statements there are assumptions about preferences which have to be made. The are also
called ‘axioms’ of consumer theory.

Axiom 1: The axiom of completeness.

It assumes that any two bundles can be compared. i.e. given any 2 bundles, we assume that A>B or B>A or both,
in which case the consumer is indifferent between the two bundles

Axiom 2: Reflexive

This is a truism. We assume that any consumption bundle is at least as good as itself or is as good as an
identical bundle ( X a , Y a) > (Y a , X a)

Axiom 3: Transitivity

This assumption is made for the purpose of ensuring consistency on the part of consumer’s preference ordering. If
the consumer thinks that A is at least as good as B and that B is at least as good as C, then the consumer thinks
that A is at least as good as C

Alternatively, if A>B & B>C then A>C

The preference order is conveniently described by using indifference curve.

Indifference curve
Qty of Y

This is a locus of Series of points (consumption bundles) yielding equal satisfaction(utility) to the consumer

Ya
The curve may take different shapes depending on the relationship between good X and good Y. All the bundles above the
curve will be weakly preferred. All bundles below the curve are less preferred to bundle A or B. they don’t yield much
utility as compared to A and B. The consumer will prefer bundles that are at the highest indifferences curve for it contain
more of each of the goods.

Examples of preferences/shapes of IC

1. Perfect substitutes

Two goods are perfect substitutes if the consumer is willing to substitutes one good for the other at a constant rate. In this
case, the consumer does not care of how much of each good he gets but only cares about the total amount of the 2 goods

Qty of Y

0
U1 U2 Qty of X
U0

In this kind of perfect substitution, if it is assumed that one unit of x is substituted for 1 unit of y then the slope this will be
-1. The important fact about perfect substitutes is that the indifference curve has a constant slope. But the slope is
determined by the rate of substitution of the 2 goods i.e. slope is not always -1.

Perfect compliments

Are goods that are always consumed together in fixed proportion e.g. right shoes and left shoes. Having only one doesn’t
Qty of Y
do the consumer any good.

A
The consumer always wants to consume the goods in fixed proportion to each other thus the indifference curve is L
shaped with consumption occurring at the vertices e.g. A, B and C

2. Bads

A bad is a commodity that the consumer doesn’t like. The goods don’t yield any utility to the consumer. The consumer
wouldQty of to
like Y consume the least of the good.

U1

F U2
E
C
D
U3
B
U4
A

0 Qty of X

In this case Y is a ‘’bad’’ and X is a “good” for the consumer thus, the indifference curve has a +vet slope.

In the example D, E, F yield lower utility than A, B, C U1<U2. The consumer would wish to use the least of the bad good
Y and thus move to U2, U3, U4

3. Neutrals

A good is a neutral good if the consumer doesn’t care about it in one way or the other

Qty of Y

U1 U2 U3

0 Qty of X
The consumer likes good X but is neutral about good Y, so the indifference curve are vertical lines. The more of good X,
the better but adding more of Y doesn’t affect him in one way or the other

4. Well behaved preference

They are convex to the origin. This implies that x and y are related but not as perfect substitutes or complements and the
rate of substitution is not 1 or 2 but changes at every point. i.e. the slopes are not the same since there is different
substitution of the two goods at all points.

Qty of Y

Ya

Yb
Qty of X
0 Xa Xb B

Characteristics

1. Well behaved preferences don’t cross each other (intersect)

2. They have a negative slope i.e. in order to gain a unit of one of the commodities, the consumer must lose some
units of the other commodity. The negative slope also means that the well-behaved preferences are monotonic
(more is preferred to less) so the tendency of the consumer is to move to the highest level of satisfaction
(indifference) curve in search of more.
3. Averages are preferred to extremes. This is because they contain more of each good.

Slope of well-behaved preference

Marginal rate of substation (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 an amount of good Y for good X

∆U
MU x =
∆X

This measures the rate of change in utility (∆ U ) associated with a small change

in the amount of goods X (∆ X )

∆ U =MU x ∆ X
Similarly;

∆U
MU Y =
∆Y

∆ U =MU Y ∆ Y

Consider a change in the consumption of each good ¿) that keeps utility constant i.e a change in consumption that moves
us along the IC, Then

MU X ∆ X + MU Y ∆Y =¿ ∆ U =0

Solving for the slope of the indifference curve

MU X ∆ X =MU Y ∆ Y

∆ Y MU X
− = = MRS
∆ X MU Y

The slope is negative since to get more of one good, lesser unit of the other good must be consumed in order to keep the
same level of utility

Consumer equilibrium

This is attained by combining the budget line and the indifference curve in order to determine the optimal choice of the
consumer. As noted from economic model, consumer chooses the best bundle they can afford.

Qty of Y

0
X1 Qty of X

The budget Set and the well-behaved consumer preference are drawn on the same diagram. The bundle of goods that is
association with the highest indifference curve and is affordable is (X1, Y1) - point F.

At this point 2 things must agree

 1. Market rate of substitution between X and Y
2. Consumer rate of substitution of X and Y

The two are equal where the budget line is tangential to the highest indifference curve, so that the slope of the indifference
curve is equal to the slope of the budget line at the equilibrium point (optimal choice)

PX
MRS=−
PY

MU X PX
− =−
MU Y PY

Or simply

MU X P X
=
MU Y P Y

This is the consumer equilibrium condition

CONSTRAINED UTILITY MAXIMIZATION

The consumer demand function

The consumer demand function gives the optimal amount of each of the goods as a function of the price and income faced
by the consumer.
The demand functions are within as;
X =f ¿ ¿

Y =f ¿ ¿

The optimal choice (bundle) is a function of its price, price of related good and income.

Using A Mathematical Approach,

We can determine the demand function for the optimal choice bundle of the consumer. The goal of the consumer is to
maximize utility subject to the budget constraints. The consumer problem is stated as

Max U (X , Y )……………………………………….…(i)

s.t P X X + P Y Y =M …………………………. ….…(ii)

To solve the problem, we introduce the Lagrangian Multiplier.

This leads to the converting of equation (i) and (ii) into an auxiliary function known as the Lagrangian.

L=U ( X , Y ) − λ ( P ¿ ¿ X X + PY Y − M )¿
The new variables λ is called a Lagrange multiplier since it is multiplied by the constraint. The lagrangian theorem says
that an optimal choice ( X ∗ ,Y ∗) must satisfy the three first order conditions:

δL δU ( X ,Y )
= − λ P X =0 ………………………………… (iii)
δX δX

δL δU ( X , Y )
= − λ PY =0 ………………………………… (iv)
δY δY

δL
=P X X + PY Y − M =0 ………………………………… (v)
δλ

Re writing equation iii and iv and dividing them yields

δU ( X , Y )
δX PX
=
δU ( X , Y ) PY
δY

This is a similar concept to the consumer equilibrium condition. It corresponds to the point of tangency between an
indifference curve and the budget line from the graphical approach.

i.e. MRS = Price ratio

Solving equations (iii), (iv)and (v) above simultaneously we obtain the demand function for the optimal choice bundle.

This is best demonstrated using a specific consumer preference and in this case a Cobb-Douglas preference

Suppose a utility function is specified as

a b
U ( X 1 , X 2)= X 1 X 2

And that P1 is price of good X1, P2 is price of good X2and M is the consumer’s income

The consumer’s problem is therefore stated as:

a b
Max U= X 1 X 2

s.t P1 X 1+ P2 X 2=M

L=( X a1 X b2 ) − λ(P1 X 1 + P2 X 2=M )

δL a −1 b
=a X 1 X 2 − λ P1=0 ………………………………… (i)
δ X1

δL a b −1
=b X 1 X 2 − λ P2=0 ………………………………… (ii)
δ X2
δL
=− P 1 X 1 − P2 X 2 + M =0 …………………………… (iii)
δλ

Rewriting and dividing equations (i) & (ii);

a−1 b
a X1 X2 λ P1
a b− 1
=
bX X 1 2
λ P2

a − 1− a
a X1 λ P1
b − 1− b
=
bX 2
λ P2

−1
a X1 P1
−1
=
bX 2
P2

a X2 P1
=
b X1 P2

bP1
X2= X ……………………………. (iv)
a P2 1

bP2
X1= X …………….……………. (v)
a P1 2

Equations (iv) and(v) represent the income expansion path or the income offer curve

Substituting equation (iv) and (v) into equation (iii) one at a time, we obtain:

P1 X 1+ P2 X 2=M

bP1
P 1 X 1+ P2 ( X )=M
a P2 1

b
P1 X 1+ P1 X 1=M
a

M
X1=
b
(1+ )P 1
a

∗ a M
X1 =
a+ b P1

⟹ Demand funcion for X 1

 a P2
P1 ( X )+ P2 X 2=M
b P1 2

a
P X + P X =M
b 2 2 2 2

M M
X2= =
a b +a
(1+ )P 2 ( )P2
b b

∗ b M
X2 = ⟹ Demand funcion for X 2
a+ b P2

∗ ∗ ∗ ∗
X 1 ∧ X 2 referred to as the demand functions that represent the optimal choice bundle X 1 X 2 . They provide the solution
to the consumer utility maximizing problem.
Given the market price and the income level for a consumer, then the functions describe the exact amount of both goods
that the consumer would have to consume so as to maximize utility.

Comparative static analysis

Here we look at change in the optimal bundle as price and income change. The optimal choice X ∗ is a function of its,
price of related good and income.
∗
X =f (P X , PY , M )

∗
Y =f ( P X , PY , M )

Qty of
Y

∗ E
Y

0
X
∗ Qty of X

E will change if P X , PY ∨M changes. i.e. The consumer will move to a new equilibrium condition. From the analysis we
deduce:

a) Price consumption curve

b) Income consumption curve
Price Consumption Curve
Qty of Y

Price Consumption Curve /

Price offer curve

E2
Y2 E1
Y1
Y0 E0

X2
0 X0 X1 Qty of X

Let P X change i.e. decrease and PY and M remain constant. Them the budget line will rotate outwards. The horizontal
intercept will increase and more of X can be bought than before. The consumer will also move to a new equilibrium
position where the amount of X will be X 1 greater than X 0 .
If the price of X continues to decline the budget line will continue to rotate outwards. Joining the equilibrium points
E0 , E1∧E 2 we can trace the optimal consumption points of X as price of X changes. This is called Price Consumption
Curve (PCC). From PCC of X, we can trace the price demand curve. This strictly means relationship between price and
quantity demanded.

Qty of Y

P0

P1

P2
B Qty of X
X0
Income Consumption Curve X1 X2

Qty of Y
Income Consumption Curve /
Income offer curve
 M

M2

M1

M0
B Qty of X
X0 X1 X2

Assuming that the consumer income is changing while holding P X and P y constant. Further assume that X and Y are
normal goods, consumption will increase as income increases thus leading to the income consumption curve. From the
ICC we can also trace out the relationship between income and one of the commodities e.g. X

This leads to the Engel curve. For a normal good it is expected that an angle curve will have a positive slope. However,
there are some exceptional cases where will have negatively sloping engel curve i.e as M increases, quantity consumed of
X decreases. Such are called inferior goods.

INCOME AND SUBSTITUTION EFFECTS

When there is a change in price of one of the goods holding constant the other price and income, there will be two types of
effects

i. Substitution effects
ii. Income effects

PX M
When the price changes, the rate of exchange ( ) and also the consumers purchasing power also changes ( ). These
PY PX
two will result to change in the consumer demand for a good.
Substitution Effect (SE)

PX
If the price of X changes, holding constant PY & M , then the slope of the budget line will also change. The change
PY
can be defined as substitution effect because the change in price is affecting the rate at which the market allows the
consumer to substitute good Y for X.

Substitution effect therefore is a change in an item’s consumption associated with a change in its price holding constant the
utility level. Substitution effects is always negative, meaning that change in price will always lead to change in quantity
demanded in the opposite direction.

Income Effect (IE)

The 2nd change is referred to as Income Effect because, even though income (M) remains constant, but for example, as
M
price of X falls will increase i.e the consumers purchasing power will increase and he will therefore buy more of X.
Px

Income Effect therefore is the change in an item’s consumption brought about by the change in consumer’s purchasing
power with the price of the item being held constant.

When a person’s purchasing power increases, the quantity demanded for the item may increase or decrease depending on
the nature of the good and Income Effect can therefore be positive or negative

S.E & I.E for a Fall in the Price of a Normal Good

Qty of Y

A
C
B
IC 2
IC 1
S . E + I . E=T . E
(X ¿ ¿1 − X 2)¿ (X ¿ ¿ 2− X 3)¿ (X ¿ ¿1 − X 3)¿
PX ↓ QX ↑ PX ↓ QX ↑ PX ↓ QX ↑
−Ve −Ve −Ve

In the diagram DD is the initial budget line and A is the optimal choice (tangency between indifference curve & budget
line). The diagram illustrates a fall in the price of X while PY & M remain constant. The reduction in the price of X makes
the consumer to substitute Y for more of the now cheaper X

Since the consumer considers bundles on the indifference curve IC 1 to be desirable, he would still prefer to remain on the
same curve even with the change in price. Therefore, while still remaining on the same indifference curve, we introduce a
new (pivoted) budget line which is also just tangent to the original indifference curve IC 1 and this leads to a new
equilibrium B with a new amount of X at X 2 . The movement from the original equilibrium at A to the pivoted equilibrium
B is the substitution effect. i.e movement from X 1 to X 2

However, the reduction in the price of X has an equivalent effect as an increase in income which shifts the budget line
outwards. The consumer will therefore shift his consumption from pivoted budget line to a new budget line with a new
equilibrium C. The movement from the pivoted equilibrium to the final equilibrium C is called the Income Effect (from
X 2 to X 3 ).

The purpose of the pivoted budget line is to enable us to put the consumer at the same level of utility even with the new
price ratio. For this to happen, there must be some compensation to the consumer. In the case of an increase in price, the
compensation will be in the form of the consumer being given some money in order to still operate along the same
indifference curve. In case of a fall in price, the compensation will be in form of some money being taken from the
consumer in order for him to still operate on the same indifference curve.

S.E & I.E for a Rise in Price of a Normal Good

Qty of Y
S . E + I . E=T . E
(X ¿ ¿1 − X 2)¿ (X ¿ ¿ 2− X 3)¿ (X ¿ ¿1 − X 3)¿
PX ↑ QX ↓ PX ↑ QX ↓ PX ↑ QX ↓
−Ve −Ve −Ve

S.E & I.E for a Fall in the Price of Inferior Non-Giffen Good
Qty

C
A
B IC 2
IC 1

S . E + I . E=T .X
0 E1 X 3 X 2
Qty of X
(X ¿ ¿1 − X 2)¿ (X ¿ ¿ 2− X 3)¿ (X ¿ ¿1 − X 3)¿
PX ↓ QX ↑ PX ↓ QX ↓ PX ↓ QX ↑
−Ve +Ve −Ve

S.E & I.E for a Rise in the Price of Inferior Non-Giffen Good
Qty of Y

B
S . E + I . E=T . E
(X ¿ ¿1 − X 2)¿ (X ¿ ¿ 2− X 3)¿ (X ¿ ¿1 − X 3)¿
PX ↑ QX ↓ PX ↑ QX ↑ PX ↑ QX ↓
−Ve +Ve −Ve

For an inferior non giffen good, the S.E is still negative while I.E is positive. However, the negative S.E is stronger than
the positive I.E thus making the Total Effect to be negative.

S.E & I.E for an Inferior Giffen Good

Fall in price of X

Qty of Y

C
IC 2
A
B
IC 1
 S . E + I . E=T . E
(X ¿ ¿1 − X 2)¿ (X ¿ ¿ 2− X 3)¿ (X ¿ ¿1 − X 3)¿
PX ↓ QX ↑ PX ↓ QX ↓ PX ↓ QX ↓
−Ve +Ve+Ve

Rise in price of X
Qty of Y

0
X2 X1 X3 Qty of X

S . E + I . E=T . E
(X ¿ ¿1 − X 2)¿ (X ¿ ¿ 2− X 3)¿ (X ¿ ¿1 − X 3)¿
PX ↑ QX ↓ PX ↑ QX ↑ PX ↑ QX ↑
−Ve +Ve+Ve

For an inferior giffen good S.E is still negative and I.E is positive. However, the positive I.E is stronger than the negative
S.E thus making the Total Effect to be positive.

SUMMARY
 Normal Good Inferior Non-Giffen Giffen Good
S . E + I . E=T . E S . E + I . E=T . E S . E + I . E=T . E
−Ve −Ve −Ve −Ve +Ve −Ve −Ve +Ve+Ve

Demand Curve

P P P

D D D
X X X
T.E: -Ve T.E: +Ve
T.E: -Ve
ENGEL CURVE

M M M

D D D

X X X
I.E: +Ve I.E: -Ve I.E: -Ve

The Slutskys Equation/ Identity

We have been looking at the S.E where the consumer’s utility level is held constant. We therefore have only one
indifference curve being tangent to both the original BL and the pivoted BL as a result of the price change. Such S.E
where we hold constant the consumer utility is called Hicksian’ Substitution Effect

Slutsky’s Substitution Effect is the name given to change in demand when price changes but the consumer’s purchasing
power is held constant so that the original bundle remains affordable even after the price change. The budget line is
therefore pivoted around the original consumption bundle.

Hicksian S.E therefore has 2 indifference curves while Slutsky has 1

Qty of
Qty
The equation:

S . E + I . E=T . E

or

s n r
∆ X +∆ X =∆ X

Is called the Slutsk y ' s Equation∨Identity

Calculation

Using two goods X 1 and X 2 , price of X 1 is P1, price of X 2 is P2 and the consumers income M. Supposing there is a
change in price of X 1 so that

Original price is P1

1
New price is P1

1
M will therefore be the amount of money income that will just make the original consumption bundle affordable. This
will be the amount of money associated with the piroted budget line.

Since( X 1 , X 2 ) is affordable at both ( P1 , P2 , M ) and ( P11 , P2 , M ) we therefore have

1 1
M =P1 X 1 + P2 X 2 ……………………………………… 1
M =P1 X 1 + P2 X 2 ……………………………………… 2
Substracting ( 2 ) ¿ ( 1 ) ,
1 1
M − M =P1 X 1+ P 1 X 1
Simplifying;
1 1
M − M =X 1 (P1 + P1 )
This is the change in money income necessary to make the old bundle affordable at the new prices

Let;

1
∆ P1=P 1 − P1 : Change in Price
1
∆ M ❑=M − M ❑ : Is change in Income necessary to make the old bundle just affordable, then
∆ M =∆ P1 X 1

From the above, we can precisely define the S.E as the change in the demand for good X 1 when the price of good X1
1
changes from P1 to P1 and at the same time money income changes from M to M 1

∆ X s1=X 1 ( P11 , M 1 ) − X 1 ( P , M )

S.E is therefore sometimes called the change in compensated demand. The consumer is being compensated for the price
change by having his income adjusted accordingly. If the price rises, the compensation will be in the form of the consumer
being given some money income to make the same amount of good affordable at a higher price & vice versa.

Example

Given a consumers demand function for good X as

M
X =10+
10 P

Let the original income be Ksh. 120 per day and the price of good X be Kshs 3 per unit. Then the demand for good X per
day is

120
X (P , M )=X (3,120)=10+
10 (3)

= 14 units per day.

Suppose the price of X falls to Ksh 2 per unit, his new demand at the new price would be

120
X ( P❑ , M )=X ( 2,120 ) =10+
1

10 (2)

=16 Units per day

In order to calculate the S.E we must first calculate by how much income would have to change in order to make the
original demand of 14 units just affordable when the price is Kshs 2 per unit

Recall that:

∆ M =∆ P1 X 1
 =14(2-3) = -14

This is the amount of compensation as a result of the price reduction so as to make the original bundle just affordable. The
level of income necessary to keep purchasing power constant is therefore:

From ∆ M ❑=M 1 − M ❑

1
M =∆ M ❑ + M ❑

= 120 – 14 = 106 Shillings

The consumer demand at the new price of Kshs 2 and the new income level of Kshs 106 is

106
( X ( P❑1 , M 1 ) )=X ( 2,106 )=10+ = 15.3 Units per day
10 ( 2 )

∆ X s= X ( P❑1 , M 1 ) − X (P , M )

= 15.3 – 14

= 1.3

Income effects

We have so far considered the pivot on the budget line. The second stage of the price adjustment is the shift movement. A
parallel shift of the budget line is the movement that occurs when income changes while relative prices remain constant. It
is thus called Income Effect which is the change of the consumer’s income from M 1back to M while holding price of X
fixed at P1❑

∆ X n=X ( P1❑ , M ) − X ( P1❑ , M 1 )

Using the previous example;

120
X ( P❑ , M )=X ( 2,120 ) =10+
1
=16 Units
10 ( 2 )

106
X ( P❑ , M ) =x ( 2,106 )=10+
1 1
=15.3Units
10 ( 2 )

∆ X n=X ( P1❑ , M ) − X ( P1❑ , M 1 )

¿ 16 −15.3=0.7

Total Effect

This is the total change in demand due to the change in price holding income constant
∆ X T = X ( P❑1 , M ) − X (P , M )

T
∆ X =16 − 14=2

Alternatively,

T s n
∆ X =∆ X + ∆ X

= 1.3 + 0.7 = 2

Reading assignment

 Revealed preference
o Weak axiom of RP
o Strong axiom of RP