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Chapter 5
Price, Income, and Substitution Effect
Macroeconomics For Financial Decisions
MS Finance, Spring 2008
By Kornkarun Kungpanidchakul, Ph.D.
When the price of a good changes, there is a change in the quantity demanded of
good 1 due to a change in its price is referred to as the ‘price effect’. The price effect is
decomposed into two sort of effects : the substitution effect and the income effect.
I Slusky Equation
The substitution effect
When the price of good 1 decreases, you have to give up less of good 2 to
purchase good 1. Therefore, the rate of exchange between the two goods changes. This is
what we call the substitution effect.
Slutsky isolated the change in demand due only to the change in relative prices
(substitution effect) by asking “What is the change in demand when the consumer’s
income is adjusted so that, at the new prices, she can only just buy the original bundle?”
In other word, if the ‘real income’ or ‘purchasing power’ remains unchanged, how will
the consumer adjust his consumption at the new price?
In order to find the substitution effect, we will draw the pivoted budget line where
the slope of the budget line changes while its purchasing power constant (the original
bundle of goods lies on the pivoted budget line).
Let m’ be the amount of money income that will just make the original consumption
bundle affordable, and from (x1, x2) is affordable at both (p1,p2,m) and (p’1,p2,m’), we
have
m'  p1' x1  p2 x2
(1)
m  p1 x1  p2 x2
(2)
(1)-(2), we have
m  x1p1
(3)
We can now find the pivoted budget line. It is the budget line at the new price
with income changed by m .
Although (x1, x2) is affordable, the optimal consumption bundle at the pivoted
budget line is not (x1, x2). The optimal consumption bundle at the pivoted budget line is
(x’1, x’2). The movement from (x1, x2) to (x’1, x’2) is the substitution effect.
s
Precisely, the substitution effect x1 is the change in the demand for good 1
when price of good 1 changes to p’1 and the income changes to m’ simultaneously (the
income changes to m’ in order to make the purchasing power constant). Therefore,
x1s  x1 ( p1' , m' )  x1 ( p1 , m)
(4)
If the price of a good goes down, he substitution effect is always nonnegative and
vice versa. The explanation is that the original consumption bundle (x1, x2) is always
affordable on the pivoted budget line. However, the bundle (x1, x2) is not purchased on
the pivoted budget line. Instead, the optimal consumption bundle of the pivoted line is
(x’1, x’2). Therefore, the optimal bundle on the pivoted line must not lie underneath the
s
original budget line which means that x1  0 when price of good 1 goes down. In
x1s
 0.
conclusion, the substitution effect must always be negative ( p1
)
The income effect
The income effect is the change in demand due to having more purchasing power
or more real income. To find the income effect, we change the consumer’s income from
m’ to m, keeping the price constant at the new price (p’1,p2).
From the picture below, the income effect is the change from the point (x’1, x’2)
n
to (x’’1, x’’2). Precisely, the income effect x1 is the change in the demand for good 1
when income changes from m’ to m, keeping price constant at (p’1,p2).
x1n  x1 ( p1' , m )  x1 ( p1' , m' )
(5)
The total change in demand
Slutsky identity:
x1  x1s  x1n
(6)
x1 ( p , m)  x1 ( p1 , m)  [ x1 ( p , m )  x1 ( p1 , m)]  [ x1 ( p , m)  x1{( p , m )] (7)
'
1
'
1
'
'
1
'
1
'
1. Normal goods
Higher income increases demand, so the income and substitution effects reinforce each
other. So the total effect is negative for normal goods. The Law of Downward-Sloping
Demand therefore always applies to normal goods.
x1s
x1n
x1
0
0
0

p

p

p
1
1
1
Therefore, we have
,
, and
.
2. Inferior goods
For inferior goods, demand is reduced by higher income. Therefore, The substitution and
income effects oppose each other when an income-inferior good’s own price changes.
x1s
x1n
x1
0
0
Therefore, we have p1
, p1
, and p1 can be either positive or negative.
In rare cases of extreme income-inferiority, the income effect may be larger in
size than the substitution effect, causing quantity demanded to fall as own-price rises.
Such goods are Giffen goods.
Example 1 : Suppose The consumer has an income of $200 and his utility function is in
the form of U(x1, x2) = x1x2. The price of good 1 is $2 while for good 2 is $4. If the
price of good 1 decreases to $1, find the (Slutsky) substitution and income effects.
II Hicks Substitution effect
The difference between Slusky and Hicks substitution effect is how we
‘compensate’ the consumer. We have learnt that slusky substitution effect compensates
consumers so that the real income or purchasing power is constant. For Hicks substitution
effect, it will compensate consumers so that their utilities are constant. In other words,
Hicks substitution effect implies that if the level of utility remains unchanged, how will
the consumer adjust his consumption at the new price?
Expenditure Minimization Problem
So far we know that the consumer will maximize his utility subject to the budget
constraint. What if the problem for the consumer when the level of the utility is given?
This is what we call the dual problem. Instead, the consumer will minimize his
expenditure subject to a certain level of utility :
Min E = P1x1+P2x2
s.t. U(x1,x2) = U0
Under expenditure minimization problem, the demand function we get is Hicksian
Demand: xh(P,U0) while the demand function from utility maximization problem is what
we call Marshallian Demand: x(P,m).
2
1
3 3
Given the utility function u ( x1 , x 2 )  x1 x 2 and the budget constraint
Example 2 :
m  p1 x1  p2 x2
a) find marshallian and hicksian demands of each good.
b) Given that the original income is $100 and the original price of good x1 and x2 is
$3 and $2 respectively. Now suppose that the price of good x 1 decreases to $2,
find the hicksian substitution effect and income effect.
III Rate of Change
Suppose that from the price and income set (p1,p2,m), the consumer gets the utility
at the level of U0. Therefore we have:
x1h ( p1 , p2,U 0 )  x1 ( p1 , p2 m)
Differentiate wrt to p1, we have:
x1h ( p1 , p2,U 0 ) x1 ( p1 , p2 m) x1 ( p1 , p2 m) m


.
p1
p1
m
p1
(8)
m
 x1
From (3), p1
, so we have:
h
x1 ( p1 , p 2 m) x1 ( p1 , p2,U 0 ) x1 ( p1 , p 2 m)


.x1
p1
p1
m
(9)
Slusky equation
If we evaluate at the optimal consumption bundle with the set of price and income
at (p1,p2,m) and the consumer gets the utility at the level of U0 as the optimal level, then
(9) can be rewritten as:
h
x1 ( p1 , p2 m) x1 ( p1 , p2,U ( p1 , p2 , m)) x1 ( p1 , p2 m)


.x1
p1
p1
m
(10)
(10)
(10) is what we call Slusky Equation.
IV The Price Effect when there is endowment
In this chapter, we suppose that, instead of the fixed income, consumers have
endowment. The list of resource units with which a consumer starts is his endowment.
We will denote the endowment with  .
i) Budget constraint and budget line
Denote the endowment of good 1 and good 2 by ( 1 ,  2 ). Then the budget
constraint becomes:
p1 x1  p2 x2  p11  p22
The above equation is represented in terms of gross demands (the amount of
goods that consumers end up consuming). If we express the budget in terms of net
demands (the difference between the amount of good that consumers end up consuming
and the initial endowment), we have:
p1 ( x1  1 )  p2 ( x2  2 )  0
If net demand is positive ( xi  i ) , the consumer is a net buyer of that good. If the
net demand is negative, the consumer is a net seller or net supplier.
We still have the budget line similar to the case that the income is fixed. The
endowment bundle is always on the budget line as the initial endowment is always
affordable from the consumer’s point of view.
The objective of a consumer is still to maximize his utility, given the level of
endowment and prices. Consumers can trade their endowments with others so that they
can reach a higher level of utility
ii) The movement of budget line
Changing in the endowment
Price ratio (slope)is constant. The direction of the movement depends on the
change in value of the total endowment.
Changing in prices
The above picture is the example when price of good 1 changes. The price ratio will
changes and the budget line will be pivoted around the endowment point as this point is
always on the budget line.
iii) Consumption with Endowment
Consumer’s objective function is still to maximize is utility given the budget
constraint.
Max U(x1,x2)
s.t. p1 x1  p2 x2  p11  p22
Also, we still have the dual problem, the expenditure minimization problem.
Min E = p1 ( x1  1 )  p2 ( x2  2 )
s.t. U ( x1 , x 2 )  U
Example 1: Suppose that the utility function is U ( x1 , x2 )  x1 x2 . Find the marshallian
and hicksian demands for both goods.
iv)Change in consumption when price changes
Example 2 : Suppose that the consumer is a seller of good 1
and price of good 1 decreases.
If he remains a supplier, then his new consumption bundle must be on the blue
part of the new budget line. So the consumer is worse off (by revealed preference).
However, if the consumer decides to switch to be a buyer of good 1, it is unsure whether
he will be worse of or better off.
Example 3: Suppose that the consumer is a buyer of good 1 and price of good 1
decreases.
If the consumer is originally a net buyer of good 1, when the price of good 1
decreases, he will always continue to be a net buyer of good 1. If he switches to be a
supplier, his budget line is on the blue part, which lies underneath the original budget
line. However, we knows that originally, ( x1* , x2* ) is preferred to any point from yintercept to the endowment point. Also, under new budget line, ( x1* , x2* ) is feasible
(inside the budget set). Therefore, consumers will not consume on the blue part which is
less favourable than ( x1* , x2* ) . We can draw a conclusion that the consumer always
remains a net buyer of good 1.
v) Offer curves and Demand curves
Offer curve contains all the utility-maximizing gross demands or the
combinations of both goods demanded by a consumer at difference levels of prices.
If we depict the curve representing the relationship between the price and the
quantity demanded of some good, we get the demand curve. The gross demand curve
represents the total amount of good a consumer chooses to consume. The net demand
curve represents the difference between the gross demand and the endowment of good 1
if this is positive, and zero otherwise. Finally, the net supply curve is the difference
between the initial endowment and the gross demand if this term is positive, and zero
otherwise.
vi) The slutsky equation revisited
Total change = substitution effect + ordinary income effect + endowment income effect
If we consider the case that endowment is involved, when the price of some good
changes, there is a change in money income as well.
How to find slutsky equation?
I)
Fixed money income and consider only the effect of a change in prices.
II)
Find the pivoted budget line that has the same price ratio as the new budget
line but passes through the original consumption bundle.
III)
The movement from the original consumption bundle to the optimal
consumption bundle on the pivoted budget line is substitution effect (the
movement from point 1 to point 2).
IV)
The movement from the optimal consumption bundle on the pivoted budget
line to the optimal consumption bundle on the new budget line that has fixed
income (from point 2 to point 3) is the ordinary income effect.
V)
The new budget line that has a change in money income is the budget line that
has the new price ration and passes through the initial endowment.
VI)
The movement from the optimal consumption bundle on the new budget line
that has fixed income to the new optimal consumption bundle (point 3 to point
4) is the endowment income effect.
x2
1  2 Substitution Effect
2  3 Ordinary Income Effect
3  4 Endowment Income Effect
1
3
2

4
2

x1 ( p1 , p 2 m) x1s x1 ( p1 , p 2 m)
 1 
.x1 + endowment income effect
p1
p1
m
x1 ( p1 , p2 m) m
Where endowment income effect =
.
m
p1
m
 1 . Therefore,
With m  p11  p22 , we have
p1
x1 ( p1 , p 2 m)
endowment income effect =
.1 .
m
Substitute (3) into (1), we have
x1 ( p1 , p 2 m) x1s x1 ( p1 , p 2 m)


.(1  x1 )
p1
p1
m
(4) is the slutsky equation when a consumer has endowment.
x1
(1)
(2)
(3)
(4)
Example 4 : From example 1, given the initial endowment is 1  100 and 2  100 .
Suppose further that p1  $2 and p2  $4 . Then price of good 1 changes to $1, find the
substitution effect, income effect and the endowment income effect.
V Intertemporal Choice
i) The Budget Constraint
Let consider the case that a consumer lives for 2 periods. The amount of money
the consumer will have in each period is (m1,m2) and the amount of consumption in each
period is (c1,c2). Suppose further that the prices of consumption in each period are
constant at 1. First consider when there is no borrowing. Then the only way the consumer
has in order to transfer money from period 1 to period 2 is by saving it without earning
interest. Then we can draw the budget constraint as follows:
Now, let suppose that the consumer can borrow and lend money at the same
interest rate, r. Then we can write the budget constraint as:
(1)
(1  r )c1  c2  (1  r )m1  m2
Or
1
1
c1 
c 2  m1 
m2
(2)
1 r
1 r
(1) is the budget constraint in terms of future value and (2) is the budget
constraint in terms of present value.
ii) Preferences for consumption
The optimal consumption bundle is still a point at which the indifference curve is
tangent to the budget line. If the consumer chooses the point where c1 < m1, then he is a
lender while if he chooses the point where c1 > m1, then he is a borrower.
iii) Comparative Static
Suppose that the rate of interest increases. Then the budget line goes to a steeper
position. Suppose that he is a lender before, then using the revealed preference, he must
remain a lender and he will be better off.
Consider the slutsky equation to decide the change in today’s consumption.
Total effect = income effect + substitution effect
 Income effect: the consumer is getting richer in the future, he will increase his
consumption, both today and in the future. (IE > 0)
 Substitution effect: the price of consumption is relatively higher. So he will
substitute the consumption in the future for tomorrow. (SE < 0).
Therefore, it is uncertain whether the consumer will increase his consumption or
not. If IE > SE, the consumer will have higher today consumption. If IE < SE, the
consumer will reduce his consumption today.
However, if the consumer is initially borrower, revealed preference can also be
used to make judgment that he must be worse off at the new interest rate. However, a
borrower may remain to be a borrower or switch to being a lender after an increase in the
rate of interest.
Consider the slutsky equation in this case to decide the change in today’s
consumption.
 Substitution effect : the consumer will consume less today and consume more in
the future.
 Income effect: the consumer is poorer as he has to pay higher interest for today’s
loan.
 For the net borrower, the income effect and the substitution effect works in the
same direction. So the total effect is that the today’s consumption decreases.