# Notes 8 -Substitution effect and Income effects

Document Sample

```					 Notes 8 - Substitution eﬀect and Income eﬀects
When some price changes our demand changes for two reason:

• Change in the ratio between prices (slope of the Budget Line= px ); one
py
good is relatively cheaper, the other is relatively more expensive. Suppose,
for example, one of the prices increases. So we’ll tend to buy more of
the good whose price is unchanged and less of the good whose price has
increased (the opposite if one price decreases).
We call this SUBSTITUTION EFFECT
• Change of the real income. If one price increases I.m relatively more poor,
since I can aﬀord a smaller quantity of goods. We call this INCOME EFFECT.

Our goal is to be able to discriminate these 2 eﬀects.
For example, suppose, after an increase of a price, my demand of good
decreases by 20. How much of this drop is due to Substitution eﬀect? How
much is due to Income eﬀects?
Let’s see how to do it with two examples:

Example 1 Suppose u(xy) = xy, m = 80, px = 1; py = 1
Solving the optimization problem I ﬁnd
m             m
x=         = 40; y =     = 40                       (1)
2px           2py

Now suppose the price of x increases up to 2, (the rest is the same). So now
m               m
x0 =     0
= 20; y 0 =     = 40                     (2)
2px             2py

So after an increase of px the demand of x dropped by 40 − 20 = 20 units.
How much of this change is due to income eﬀect? How much is due to
substitution eﬀect?
Let’s try to eliminate the income eﬀect. How?
We compensate the lost of purchase power that the consumer suﬀered because
of the increase in price, giving him more income.
Since before he was able to aﬀord 40 units of x and 40 units, now we’ll give
him money enough to aﬀord the same bundle (40,40) at the new prices.
Since now px = 2, py = 1 , now the bundle (40,40) costs 40 ∗ 2 + 40 ∗ 1 =
80 + 40 = 120.
I will give the consumer exactly 120, so that he can exactly aﬀord the old
bundle (40,40).
Having now an higher income that compensate the increase in price, we have
eliminated the income eﬀect!!!
With m = 120, px = 2, py = 1 what will be the demand of the consumer?

1
m    120              120
x00 =       =     = 30; y 00 =     = 60                (3)
2p0
x    4                2

As you can see since now x is relatively more expansive the consumption of
x has decreased, and consumption of y has increased.
But as you can see, since we have eliminated the income eﬀect the demand
of x has decreased only by 10 (from 40 to 30). So the substitution eﬀect is 10.
How much is the income eﬀect?
Since the total eﬀect was 20 (from 40 to 20) and the substitution eﬀect is
10, it means that the remaining 10 is to be attributed to income eﬀect.

Example 2 Suppose u(x, y) = x + ln y, m = 100, px = 20, py = 1.
The optimization problem yields:
m − px   80          px
x=            =    = 4, y =    = 20                   (4)
px     20          py

Now suppose the price of x increase up to 50 (the rest stays the same). Now

m − p0
x   50         p0
x0 =          =    = 1, y = x = 50                    (5)
p0
x     50         py

So due to the change in px , the demand of x has decreased by 4 − 1 = 3
units.
To ﬁnd substitutuion eﬀects, we’ll increase the income suche that the old
bundle (4,20) is aﬀordable at new prices.
The new income must be (4 ∗ 50 + 20 ∗ 1) = 220
With m = 220, px= = 50, py = 1 the optimal bundle is

m − p0
x   220 − 50   170             p0
x00 =          =          =     = 3.4; , y = x = 50             (6)
p0
x        50       5              py

So due the substitution eﬀect the demand of x dropped only by 4 − 3.4 = 0.6.
The total change was 3, so the substitution eﬀect is 3 − 0.6 = 2.4.

So in general how can following this procedure if we want to study the
consequence of a change in px.

1. Solve the optimization problem for the initial m, px, py . Call the solution
x and y
2. Solve the optimization problem with the same m, py but the new p0 . Call
x
this solution x0 and y 0 .
3. The total eﬀect is ∆x = x0 − x
4. Find the income that compensate the increase in prices (m0 = p0 x + py y)
x

2
5. Solve the optimization problem for the new income m0 , and the new price
p0 ( the price py is always unchanged).
x
Call this solution x” and y”.
6. The substitution eﬀect is SE = x00 − x
7. The income eﬀect is IE=∆x − SE

Exercise 3 In example 2. Suppose the price of y changes, and increases up to
2.
Find the income and substitution eﬀect.

Exercise 4 Suppose u = min{x, y}.m = 10, px = 2, py = 2.(This is the case of
complementary goods, i.e. good that are not substitute)
Find the income and substitution eﬀect if the price of x increases to 2
Fine the income and substitution eﬀect if the price of y decreases to 1
Give an economic explanation of this particular result.

Exercise 5 Suppose u = x + y; m = 10, px = 4, py = 2.(this is the case of
perfect substitute good).
Find the income and substitution eﬀect when the price of x decreases up to
1.

3

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 198 posted: 9/2/2010 language: English pages: 3