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Proposed solutions for tutorial 5
Intermediate Microeconomics (UTS 23567)*
Preliminary and incomplete
Available at https://backwardinduction.blog/tutoring/
Office hours on Mondays from 9 am till 10 am in building 8 on level 9
Please whatsapp me on 0457871540 so I could meet you at the door, I donāt have an internal phone
Also please whatsapp if you have questions, I wonāt be able to answer through whatsapp but I will give answer during office hours
or, since you are very unlikely to have ask a unique question, in the beginning of next tutorial
Sergey V. Alexeev
2018
NavigationPage numbers are clickable
Tutorial 5 (17/04/2018) 6
Question 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Answer to 1.a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Answer to 1.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Question 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Answer to 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Question 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Answer to 3.a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Answer to 3.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Answer to 3.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Answer to 3.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Answer to 3.e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Answer to 3.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Answer to 3.g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Answer to 3.h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
*Questions for the tutorial were provided by Massimo Scotti, slide and textbook are by Nechyba (2016). Solutions and commentary are by SergeyAlexeev (e-mail: [email protected]). (Btw this is a much better book Varian (1987))
1
A price change in š„1 creates two effects
Say a guy has
š¢ = š„1š„2
then
if
2š„1 + š„2 = 10
the optimal bundle is
(2.5, 5) ā” arg maxš„1,š„2
{š„1š„2
2š„1 + š„2 = 10
letās call it (2.5, 5)š“
if1
2š„1 + š„2 = 10
the optimal bundle is
(10, 5) ā” arg maxš„1,š„2
{š„1š„2
12š„1 + š„2 = 10
letās call it (10, 5)šµ
2 4 6 8 10 12 14 16 18 20 22
1
2
3
4
5
6
7
8
9
10
11
12
(2.5, 5)š“ (10, 5)šµ
š„1
š„2š„2 = 10 ā 2š„1
š„2 = 10 ā š„1/2
š„2 = 12.5/š„1
š„2 = 50/š„1
2
The change in demand due to the change in the rate of exchange between the two goods ā the substitution
effect
2 4 6 8 10 12 14 16 18 20 22
1
2
3
4
5
6
7
8
9
10
11
12
(2.5, 5)š“
(10, 5)šµ
(6.25, 3.125)š“ā²
š„1
š„2š„2 = 10 ā 2š„1
š„2 = 10 ā š„1/2
š„2 = 19.53/š„
š„2 = 6.25 ā š„1/2
If prices are1
2š„1 + š„2
and the guy is taken away
āš¼ = (2 ā 0.5)2.5 = 3.75
then
(2.5, 5)
is still affordable, but not optimal because
(6.25, 3.125) ā” arg maxš„1,š„2
{š„1š„2
12š„1 + š„2 = 6.25
is optimal. Letās call it š“ā²
Then
6.25 ā 2.5 = 3.75
is a substitution effect
It indicates how the consumer āsubstitutesā one good for the other when a price changes but purchasing power remains
constant.
The substitution effect always moves opposite to the price movement. Substitution effect is negative
3
The change in demand due to having more purchasing power ā the income effect
2 4 6 8 10 12 14 16 18 20 22
1
2
3
4
5
6
7
8
9
10
11
12
(2.5, 5)š“
(10, 5)šµ
(6.25, 3.125)š“ā²
š„1
š„2š„2 = 10 ā 2š„1
š„2 = 10 ā š„1/2
š„2 = 19.53/š„
š„2 = 6.25 ā š„1/2
If income is changed from1
2š„1 + š„2 = 6.25
to1
2š„1 + š„2 = 10
keeping the prices constant we have an income effect
10 ā 6.25 = 3.75
Income effect increases if good is a normal or decreases if good is inferior
Total change it demand is a sum of income and substitution effects
3.75 + 3.75 = 7.5
4
We go through these troubles only to formulate Law of Demand
Microeconomics says that demand can go in any direction as circumstances change, which makes it a terrible science.
Theory has to impose limitations (e.g. Law of Gravity), otherwise all of these math is for nothing. But now we know this:
If
š„1 is a normal good
and
š1 ā
then
āš„1(ā)
= āš„š 1
(ā)
+ āš„š1
(ā)
substitution effect āš„š 1 and income effect āš„š
1 reinforce each other
If
š„1 is an inferior good
and
š1 ā
then
āš„1(?)
= āš„š 1
(ā)
+ āš„š1
(+)
substitution effect āš„š 1 and income effect āš„š
1 opposite each other
If the income effect āš„š1 is larger āš„š
1, the total change in demand is positive.
Which is weird.
Such a good called Giffen.
A Giffens good could be understood as a very very inferior good.
Note that if we already know that the demand increases when income increases
āš„š1
(ā)
i.e. the good is normal
then the substitution effect and the income effect reinforce each other, and an increase in price reduces demand.
This logic is known as Law of Demand
If the demand for a good increases when income increases, then the demand for that good must decrease when its price increases.
It follows directly from the Slutsky equation.
However, you are told that a good satisfies Law of Demand if
š1 ā ā š„1 ā
and
š1 ā ā š„1 ā
which is a good enough explanation
5
Tutorial 5 (17/04/2018)
Question 1
ā Suppose you find that income and substitution effects for Good 1 go in opposite direction when the own price
of Good 1, š1 changes.
ā Suppose you also find that the income effect is smaller than the substitution effect.
1.a.
Does Good 1 satisfy the law of demand?
Answer to 1.a
Say
š1 ā
We know
āš„1(?)
= āš„š 1
(ā)
+ āš„š1
(?)
and we are also told that income effect and substitution effect have opposite signs
āš„1(?)
= āš„š 1
(ā)
+ āš„š1
(+)
and that
āš„š1
(+)
< āš„š 1
(ā)
clearly
āš„1(ā)
= āš„š 1
(ā)
+ āš„š1
(+)
In short we have
š1 ā ā š„1 ā
by symmetry
š1 ā ā š„1 ā
which indicates that law of demand holds
5 10 15 20
2
4
6
8
10
12
š„1
š„2
6
Question 1
ā Suppose you find that income and substitution effects for Good 1 go in opposite direction when the own price
of Good 1, š1 changes.
ā Suppose you also find that the income effect is smaller than the substitution effect.
1.b
What is good 1 called?
Answer to 1.b
In situation like this
āš„1(ā)
= āš„š 1
(ā)
+ āš„š1
(+)
the good is inferior, but not too inferior to make it a Giffen good.
7
Question 1 Checklist
ā Suppose you find that income and substitution effects for Good 1 go in opposite direction when the own price
of Good 1, š1 changes.
ā Suppose you also find that the income effect is smaller than the substitution effect.
1.a.
Does Good 1 satisfy the law of demand?
1.b
What is good 1 called?
8
Question 2
ā Suppose there are only two goods, Good 1 and Good 2.
ā Further, suppose that Good 2 is an inferior good.
ā Then, it must be that an increase in the price of Good 1, š1 leads on an increase in the consumption of Good
2.
ā True or False? Explain.
Answer to 2
Take
āš„2(?)
= āš„š 2
(?)
+ āš„š2
(?)
We know
š1 ā ā š¼ ā
because š„2 is inferior
š¼ ā ā āš„š2
(+)
and consumer also starts substituting with š„2
š1 ā ā āš„š 2
(+)
Taken together
āš„2(+)
= āš„š 2
(+)
+ āš„š2
(+)
9
Question 2 Checklist
ā Suppose there are only two goods, Good 1 and Good 2.
ā Further, suppose that Good 2 is an inferior good.
ā Then, it must be that an increase in the price of Good 1, š1 leads on an increase in the consumption of Good
2.
True or False? Explain.
10
Question 3
ā Suppose that a consumer has preferences described by the following utility function:
š¢(š„1, š„2) = š„1š„2
ā Assume that we put š„1 on the horizontal axis and š„2 on the vertical axis.
ā Also, let š1, š2 and š¼ respectively denote the price of š„1, the price of š„2 and the income of the consumer.
3.a
Find the demand functions for good 1 and good 2.
Answer to 3.a {šš š(š„1, š„2) = āš1/š2
š1š„1 + š2š„2 = š¼ā
{āš„2/š„1 = āš1/š2
š1š„1 + š2š„2 = š¼
ā
{š„1š1 = š„2š2
š1š„1 + š2š„2 = š¼
ā
{š„1š1 = š„2š2
š1š„1 + š1š„1 = š¼
ā
{š„1š1 = š„2š2
2š1š„1 = š¼
ā
{š„1š1 = š„2š2
š„1 = š¼/2š1
ā
{š„1 = š¼/2š1
š„2 = š¼/2š2
11
Question 3
ā Suppose that a consumer has preferences described by the following utility function:
š¢(š„1, š„2) = š„1š„2
ā Assume that we put š„1 on the horizontal axis and š„2 on the vertical axis.
ā Also, let š1, š2 and š¼ respectively denote the price of š„1, the price of š„2 and the income of the consumer.
3.b
Suppose
š1 = 1
š2 = 3
and
š¼ = 90
Find the optimal bundle chosen by the consumer and provide a graphical representation.
Answer to 3.b
A particular solution with given parameters {š„1 = š¼/2š1
š„2 = š¼/2š2
ā
{š„1 = 45
š„2 = 15
To plot the budget line
š„1 + 3š„2 = 90
3š„2 = 90 ā š„1
š„2 = 30 ā š„1/3
To plot the indifference curve
š„1š„2
(45,15)
= 675
š„2 = 675/š„1
10 20 30 40 50 60 70 80 90
5
10
15
20
25
30
(45, 15)š“
š„1
š„2š„2 = 30 ā š„1/3
š„2 = 675/š„1
12
Question 3
ā Suppose that a consumer has preferences described by the following utility function:
š¢(š„1, š„2) = š„1š„2
ā Assume that we put š„1 on the horizontal axis and š„2 on the vertical axis.
ā Also, let š1, š2 and š¼ respectively denote the price of š„1, the price of š„2 and the income of the consumer.
3.c
Suppose š1 increases to 9. How does the optimal bundle change?
In the same graph you have used for part b, provide a graphical representation of how the optimal bundle changes.
Answer to 3.c
A particular solution with given parameters {š„1 = š¼/2š1
š„2 = š¼/2š2
ā
{š„1 = 5
š„2 = 15
To plot the budget line
9š„1 + 3š„2 = 90
3š„2 = 90 ā 9š„1
š„2 = 30 ā 3š„1
To plot the indifference curve
š„1š„2
(5,15)
= 75
š„2 = 75/š„1
10 20 30 40 50 60 70 80 90
5
10
15
20
25
30
(45, 15)š“(5, 15)š¶
š„1
š„2š„2 = 30 ā š„1/3
š„2 = 675/š„1
š„2 = 30 ā 3š„1
š„2 = 75/š„1
13
Question 3
ā Suppose that a consumer has preferences described by the following utility function:
š¢(š„1, š„2) = š„1š„2
ā Assume that we put š„1 on the horizontal axis and š„2 on the vertical axis.
ā Also, let š1, š2 and š¼ respectively denote the price of š„1, the price of š„2 and the income of the consumer.
3.d
Find the size and direction of the income and substitution effects for good 1 and for good 2?
[Note: You are require to provide a numerical answer]
Answer to 3.d {š¢(š„1, š„2)|šµ = š¢(š„1, š„2)|š“
šš š(š„1, š„2)|šµ = āšššš¤1 /šššš¤
2
ā
{š„1š„2 = 675
āš„2/š„1 = ā9/3
ā
{š„1š„2 = 675
9š„1 = 3š„2
ā
{š„1š„2 = 675
3š„1 = š„2
ā
{3š„1š„1 = 675
š„2 = 3š„1
ā
{š„21 = 225
3š„1 = š„2
ā
{š„1 = 15
š„2 = 45
10 20 30 40 50 60 70 80 90
10
20
30
40
50
(45, 15)š“(5, 15)š¶
(15, 45)šµ
š„1
š„2š„2 = 30 ā š„1/3
š„2 = 675/š„1
š„2 = 30 ā 3š„1
š„2 = 75/š„1
š„2 = 90 ā 3š„1
14
š1 ā
(45, 15)š“ ā (15, 45)šµ ā (5, 15)š¶
āš„1(ā40)
= āš„š 1
(ā30)
+ āš„š1
(ā10)
š2 ā¼
(45, 15)š“ ā (15, 45)šµ ā (5, 15)š¶
āš„2(0)
= āš„š 2
(+30)
+ āš„š2
(ā30)
15
Question 3
ā Suppose that a consumer has preferences described by the following utility function:
š¢(š„1, š„2) = š„1š„2
ā Assume that we put š„1 on the horizontal axis and š„2 on the vertical axis.
ā Also, let š1, š2 and š¼ respectively denote the price of š„1, the price of š„2 and the income of the consumer.
3.e
Is good 1 normal, inferior or quasi-linear?
Answer to 3.e
š1 ā
(45, 15)š“ ā (15, 45)šµ ā (5, 15)š¶
āš„1(ā40)
= āš„š 1
(ā30)
+ āš„š1
(ā10)
š„1=š¼
2š1
16
Question 3
ā Suppose that a consumer has preferences described by the following utility function:
š¢(š„1, š„2) = š„1š„2
ā Assume that we put š„1 on the horizontal axis and š„2 on the vertical axis.
ā Also, let š1 , š2 and š¼ respectively denote the price of š„1, the price of š„2 and the income of the consumer.
3.f
Is good 2 normal, inferior or quasi-linear?
Answer to 3.f
š2 ā¼
(45, 15)š“ ā (15, 45)šµ ā (5, 15)š¶
āš„2(0)
= āš„š 2
(+30)
+ āš„š2
(ā30)
š„2=š¼
2š2
17
Question 3
ā Suppose that a consumer has preferences described by the following utility function:
š¢(š„1, š„2) = š„1š„2
ā Assume that we put š„1 on the horizontal axis and š„2 on the vertical axis.
ā Also, let š1, š2 and š¼ respectively denote the price of š„1, the price of š„2 and the income of the consumer.
3.g
In the same graph you have used for parts b and c:
ā plot the compensated budget line
ā graphically identify the income and the substitution effects for good 1 and for good 2
ā use an arrow to show the direction of these effects
Answer to 3.g
20 40 60 80
10
20
30
40
50
(45, 15)š“(5, 15)š¶
(15, 45)šµ
š„1
š„2š„2 = 30 ā š„1/3
š„2 = 675/š„1
š„2 = 30 ā 3š„1
š„2 = 75/š„1
š„2 = 90 ā 3š„1
20 40 60 80
10
20
30
40
50
(45, 15)š“(5, 15)š¶
(15, 45)šµ
š„1
š„2š„2 = 30 ā š„1/3š„2 = 30 ā 3š„1
š„2 = 90 ā 3š„1
10 20 30 40 50
10
20
30
40
50
(45, 15)š“(5, 15)š¶
(15, 45)šµ
š„1
š„2
10 20 30 40 50
10
20
30
40
50
(45, 15)š“(5, 15)š¶
(15, 45)šµ
š„1
š„2
āš„1(ā40)
āš„š1
(ā10)
āš„š 1
(ā30)
18
Question 3
ā Suppose that a consumer has preferences described by the following utility function:
š¢(š„1, š„2) = š„1š„2
ā Assume that we put š„1 on the horizontal axis and š„2 on the vertical axis.
ā Also, let š1 , š2 and I respectively denote the price of š„1, the price of š„2 and the income of the consumer.
3.h
Explain what happens to the consumption of good 2 as š1 increases to 9.
Answer to 3.h
5 10 15 20 25 30 35 40 45 50 55
10
20
30
40
50
(45, 15)š“(5, 15)š¶
(15, 45)šµ
āš„š 2
(+30)
= āš„š2
(ā30)
š„1
š„2
19
References
Nechyba, Thomas (2016). Microeconomics: an intuitive approach with calculus. Nelson Education.
url: https://sergeyvalexeev.files.wordpress.com/2018/03/1nechyba_t_microeconomics_an_intuitive_
approach_with_calculus.pdf.
Varian, Hal R (1987). Intermediate Microeconomics; a modern approach.
url: https://sergeyvalexeev.files.wordpress.com/2018/04/varian-intermediate-microeconomics.pdf.
20
