Microeconomics Mid-Term Mock
Paolo Nicola Barbieri
October 23, 2014
Ex. 1
1
Let p = 30 − x be the inverse demand function, where p is price and q is total output.
4
1. Find the price which corresponds to = 3;
2. Would the seller find profitable to increase the price when price equals 3? Why?
Solution
1. The direct demand function is x = 120 − 4p, using the formula for the elasticity we find
that
p
∂x p =2
=−4·
|| =
∂p x
120 − 4p (1)
so
−4p = 240 − 8p
p∗ = 60
(2)
2. With p = 3 we have that
3 1
|| = − 4 ·
=
108 9
(3)
Yes! Very inelastic demand, so the consumers will not be able to react so quickly to the price change.
Ex. 2
Quasimodo consumes good x and y his utility function is the following
U (x, y) = 100x −
x2
+y
2
(4)
1. What kind of utility functions is?
2. What is the inverse demand function for x? (HINT : p = U 0 (x))
3. What is his total utility foe earplugs and other things if the px is 50 and 80? What is the change in
(net) consumer’s surplus when the pX goes from 50 to 80?
1
Solution
1. Quasi-linear
2. p = 100 − x
3. 5,250-4,200
4. 1,050
Ex. 3
Suppose preferences are
U (x, y) = ln(x + 1) + y
(5)
suppose px and py are the prices of each goods and m is income.
1. Write the equilibrium condition.
2. Find the demand functions
Solution
1. 1/(x + 1) = px /py
2. x = 1/px − 1 and y = m − 1 + p
Ex. 4
For the Hi-Fi market we have a demand equals to
q D = 300 − 5p
(6)
q S = 20p − 400
(7)
and a supply of
1. Find the equilibrium
2. Find the price elasticities of demand and supply.
3. The new supply is qS = 20p − 600, find the new equilibrium and the new elasticities.
4. Find the elasticity in during the change of supply.
5. Which equilibrium ensures a demand elasticity equals to = 1
Solution
1. In order to find the market equilibrium let’s solve the following system
q D = 300 − 5p
q S = 20p − 400
by equating the two quantities
300 − 5p = 20p − 400
from which p∗ = 28 and q ∗ = 160
2
(8)
2. The demand elasticity is
D ∂q p 28 7
D = = −5·
=
∂p q 160 8
(9)
The supply elasticity is
S
∂q
S = ∂p
p 28 7
= 20 ·
=
q 160 2
(10)
3.
q D = 300 − 5p
q S = 20p − 600
by equating the two quantities
300 − 5p = 20p − 600
(11)
from which p∗∗ = 36 and q ∗∗ = 120. The demand elasticity is
D 9
∂q p 36
=−5·
=
D = ∂p q
120 5
(12)
The supply elasticity is
S
∂q
S = ∂p
p 36 = 20 ·
=6
q 120 (13)
4. We can find the elasticity over the two cases as
D
∂q
ˆD = ∂p
5.
p∗ +p∗∗
2
q ∗ +q ∗∗
2
8
=
7
(14)
D ∂q p =1
D = ∂p q (15)
5p
=1
300 − 5p
(16)
This means that
D =
Leading to p = 30 and q = 150. For every p > 30 and q < 150 > 1, while for every p < 30 and q > 150
<1
[Figure 1 about here.]
Ex. 6
The Italian CD market is characterised by the following demand and supply
q D = 100 − 2p
p
qS = − 5
2
3
1. Find the equilibrium.
2. Find the elasticity of demand and supply.
3. A big foreign company imports CD in Italy at the p̄ = 30 regardless of what the other are doing. What
happens to the market? What is the new equilibrium?
4. P2P and the demand becomes q D = 80 − 2p. What is the effect? What happens if the foreign company
stops its imports?
Solution
1. In order to find the market equilibrium let’s solve the following system
q D = 100 − 2p
p
qS = − 5
2
by equating the two quantities
100 − 2p =
p
−5
2
(17)
from which p∗ = 42 and q ∗ = 16.
2. The demand elasticity is
D ∂q p 42 21
D = = 2· =
∂p q 16 4
(18)
S
∂q
S = ∂p
(19)
The supply elasticity is
p 1 42 21
=
=
·
q 2 16 16
3. If there are no commercial barrios is impossible for the Italian distributors to set the equilibrium price.
It is an example of a price cap that the foreign distributors imposes on the Italian market. Which will also
create a over-demand since
q S (30) = 10 < 40 = q D (30)
(20)
Since the price cap is too low than the one ensuring a market clearing in Italy.
4. Given the new demand and the price cap we will have a new equilibrium quantity equals to qD (30) = 20.
If the foreign importer exit the market
q D = 80 − 2p
p
qS = − 5
2
by equating the two quantities
80 − 2p =
from which p∗ = 34 and q ∗ = 12.
4
p
−5
2
(21)
[Figure 2 about here.]
Ex. 7
Given the preferences
U (x1 , x2 ) = x1 x2
(22)
and p1 = 2, p2 = 10. Find the Engel’s curve for this consumer.
Solution
To compute and Engel curve means to see how the equilibrium changes in terms of income
given the prices. Let’s impose as our main variable income R. Given our preferences let’s compute the MRS
M RS1.2 =
x2
x1
(23)
The solution for the consumer is found by solving the following system
1
x2
=
x1
5
R = 2x1 + 10x2
from which we have our Engel curves
R
4
R
x2 =
20
x1 =
Ex. 8 Assume that and individual in endowed with income R = 100, he has the following utility function
U (x1 , x2 ) = 3x21 2x32
(24)
with p1 = 5 and p2 = 20.
1. Write the budget constant and illustrate it graphically.
2. What is the amount of x1 and x2 demanded?
3. Find and comment the demand functions for the two goods.
4. How much does the demand changes if the income becomes R = 200?
Solution
1. The budget constraint is
5x1 + 20x2 = 100
[Figure 3 about here.]
5
(25)
2. The solution for the consumer is found by solving the following system
M RS1,2 =
5
2x2
=
3x1
20
100 = 5x1 + 20x2
from which x1 = 8 and x2 = 3
3. In order to find the demand functions solve the same system as before but keeping the prices as
unknown variables
2x2
p1
=
3x1
p2
100 = 5x1 + 20x2
from which x1 = 60/p1 and x2 = 40/p2 . The particular feature of these demand functions is that each
demand does not depend on the price of the other good.
4. If R double the purchasing power of the individual doubles thus x1 = 16 and x2 = 6.
Ex. 9
A consumer has the following preferences
√
U (x1 , x2 ) = (x1 − 5) x2
(26)
with p1 = 1 and p2 = 3 and total income R = 185.
1. Find the optimal choice of the consumer.
2. Find the demand functions of the two goods.
3. Write and graphically illustrate the Engel curves. What kind of goods we these?
Solution
1. The solution for the consumer is found by solving the following system
√
M RS1,2 =
x2
2x2
x1 −5
=
1
2x2
=
x1 − 5
3
185 = x1 + 3x2
from which x1 = 125 and x2 = 20
3. In order to find the demand functions solve the same system as before but keeping the prices as
unknown variables
2x2
p1
=
x1 − 5
p2
185 = p1 x1 + p2 x2
6
from which x1 = (370 + 5pi )/3p1 and x2 = (185 − 5p1 )/3p2 .
4. To compute and Engel curve mean to see how the equilibrium changes in terms of income given the
prices. Let’s impose as our main variable income R. Given our preferences let’s compute the MRS
2x2
1
=
x1 − 5
3
R = x1 + 3x2
from which we have our Engel curves
2R + 5
3
R+5
x2 =
9
x1 =
Ex. 10
Assume a market with the following demand and supply functions
q D = 100 − 4p
q S = 60 + p
1. Find the market equilibrium.
2. Find the price elasticity of demand and supply.
3. Find the new equilibrium with demand q D = 160 − 4p4
Solution
1. In order to find the market equilibrium let’s solve the following system
q D = 100 − 4p
q S = 60 + p
by equating the two quantities
100 − 4p = 60 + p
(27)
D ∂q p 8 8
D = = −4· =
∂p q 68
17
(28)
from which p∗ = 8 and q ∗ = 68
2. The demand elasticity is
The supply elasticity is
S
∂q
S = ∂p
p 8 8
= 1 · =
q
68
68
7
(29)
3.
q D = 160 − 4p
q S = 60 + p
by equating the two quantities
160 − 4p = 60 + p
(30)
from which p∗∗ = 20 and q ∗∗ = 80. The demand elasticity is
D ∂q p 20 D = = −4· =1
∂p q 80
(31)
The supply elasticity is
S
∂q
S = ∂p
p 20 1
= 1·
=
q 80 4
(32)
[Figure 4 about here.]
Ex. 11
Assume a market characterised by the following demand and supply functions
qD =
160
p
q S = 10 −
20
p
1. Sketch the graph. Does the supple functions yields for p ≤ 2?
2. Find the market equilibrium and the elasticities of demand and supply, in the equilibrium?
3. What happens if there is a price cap of p = 5?
Solution
1.
[Figure 5 about here.]
2. In order to find the market equilibrium let’s solve the following system
qD =
160
p
q S = 10 −
20
p
by equating the two quantities
160
20
= 10 −
p
p
8
(33)
from which p∗ = 18 and q ∗ =
160
18 .
The demand elasticity is
D ∂q p 160 18 =
= 0, 49
D = ·
∂p q 324 160
18
(34)
S
∂q
S = ∂p
(35)
The supply elasticity is
p 20 20 =
·
= 0, 06
q 324 160
18
3. With a price cap of p = 5 we will have
q D (5) = 32
q S (5) = 6
(36)
and thus there is an excess demand.
Ex. 12
Assume a market characterized as follows
q D = 120 − 4p
q S = 2p − 30
1. Find the equilibrium
2. Find the demand elasticities
3. Assume that the supply shift to q D = 2p − 60 find the new equilibrium
4. The government decides to levy a tax on consumers of 5 dollars, what is the new equilibrium?
Solution
1. In order to find the market equilibrium let’s solve the following system
q D = 120 − 4p
q S = 2p − 30
by equating the two quantities
120 − 4p = 2p − 30
(37)
from which p∗ = 25 and q ∗ = 20. The demand elasticity is
D ∂q p 25 D = = − 4 · = 5, 5
∂p q 18
(38)
The supply elasticity is
S
∂q
S = ∂p
p 25 = 2·
= 2, 7
q 18 9
(39)
2.
q D = 120 − 4p
q S = 2p − 60
by equating the two quantities
120 − 4p = 2p − 60
(40)
from which p∗ = 30 and q ∗ = 0.
3. With the introduction of this tax the price paid by the consumer and received by the supplier is pD + 5
This means that we have the following new system of equations to solve
q D = 120 − 4p
q S = 2p − 50
by equating the two quantities
120 − 4p = 2p − 50
(41)
from which p∗ = 170/6 and q ∗ = 20/3.
[Figure 6 about here.]
Ex. 13
Demand and supply for ski-lessons are the following
q D = 100 − 2p
q S = 3p
1. Find the equilibrium
2. Find the elasticities.
3. A tax of 10 per ski-lesson is imposed to the consumers, find the new equilibrium
1. In order to find the market equilibrium let’s solve the following system
q D = 120 − 2p
q S = 3p
by equating the two quantities
120 − 2p = 3p
from which p∗ = 20 and q ∗ = 60.
10
(42)
2. The demand elasticity is
D ∂q p 20 2
D = = −2· =
∂p q 60
3
(43)
The supply elasticity is
S
∂q
S = ∂p
p 20 = 3·
=1
q 60 (44)
3. The new price for the consumers will be pD = p + 10 Thus we have the following new equilibrium
conditions
q D = 120 − 2(p + 10) = 100 − 2p
q S = 3p
by equating the two quantities
100 − 2p = 3p
(45)
from which p∗ = 26 and q ∗ = 48. The demand elasticity is
D ∂q p 26 13
= −2· =
D = ∂p q 48
12
(46)
The supply elasticity is
S
∂q
S = ∂p
Ex. 14
p 26 13
= 3·
=
q 48 8
(47)
The demand and supply functions for silk are
qD =
100
p
qS = p
1. Find the equilibrium and elasticities
2. An ad valorem tax of 300% is imposed on merino ewes so that the price paid by demanders is four
times the price received by suppliers.
Solution
qD =
100
p
qS = p
by equating the two quantities
100
=p
p
11
(48)
from which p∗ = 10 and q ∗ = 10. The demand elasticity is
D ∂q p 100 = 0, 01
D = =
∂p q 10000 (49)
The supply elasticity is
S
∂q
S = ∂p
p =1
q
(50)
2.
100
4p
qD =
qS = p
by equating the two quantities
100
=p
4p
(51)
D ∂q p 100 = 0, 0025
D = =
∂p q 40000 (52)
from which p∗ 5 and q ∗ 5.The demand elasticity is
The supply elasticity is
S
∂q
S = ∂p
Ex. 15
p =1
q
(53)
The price elasticity of demand for oatmeal is constant and equal to = −1. When the price of
oatmeal is 10 per unit, the total amount demanded is 6,000 units.
1. What is the demand function?
2. If the supply is perfectly inelastic at q = 5, 000, what is the equilibrium price?
Solution
1. If the demand curve has a constant price elasticity equals to , then q D = αp for some
constant α. Thus we can write
6000 =
From which
qD =
α
10
60000
p
2. if q = 5000 we have that p = 12.
12
(54)
(55)
Figure 1: Market Equilibrium for Ex. 5
5
4
3
2
1
5
10
15
20
13
Figure 2: Market Equilibrium for Ex. 6
80
70
60
50
40
30
20
10
20
30
40
50
14
Figure 3: Budget Constraint for Ex. 8
5
4
3
2
1
5
10
15
20
15
Figure 4: Budget Constraint for Ex. 10
40
20
20
40
60
80
100
-20
-40
-60
16
Figure 5: Solution for Ex. 11
10
8
6
4
2
20
40
60
80
100
120
140
-2
-4
17
Figure 6: Market Equilibrium Ex. 12
18