MICROECONOMICS 1
- Mock exam 2 – solution -
Tutor: NGUYEN Thu Tra
A.Y. 2023-2024
EXERCISE 1
Two consumers A and B have an income 𝑴 = 𝟏𝟎𝟎 euros. Their
utility functions are, respectively, 𝑼𝑨 (𝒙, 𝒚) = 𝒎𝒊𝒏{𝒙, 𝒚} and
𝑼𝑩 (𝒙, 𝒚) = 𝒙𝟐 𝒚 . The price of good 𝒙 is 𝑷𝒙 = 𝟐, the price of good
𝒚 is 𝑷𝒚 = 𝟐.
a. Find the optimal consumption choice of A and represent it in the
plane (𝒙, 𝒚).
Consumer A has a perfect complement utility function with Lshape indifference curves; therefore, his optimal consumption
bundle must be on the line connecting all kinked points of the
indifference curves and satisfies: 𝒙 = 𝒚 (1)
Budget line: 𝟐𝒙 + 𝟐𝒚 = 𝟏𝟎𝟎 ⇔ 𝑥 + 𝑦 = 50 ⇔ 𝒚 = 𝟓𝟎 − 𝒙
Substitute (1) into the budget line, we have:
𝑦 ∗ = 50 − 𝑦 ∗ ⇔ 2𝑦 ∗ = 50 ⇔ 𝒚∗ = 𝟐𝟓
⇒ 𝒙∗ = 𝑦 ∗ = 𝟐𝟓
➔The best bundle is: (𝟐𝟓, 𝟐𝟓)
Two consumers A and B have an income 𝑴 = 𝟏𝟎𝟎 euros. Their
utility functions are, respectively, 𝑼𝑨 (𝒙, 𝒚) = 𝒎𝒊𝒏{𝒙, 𝒚} and
𝑼𝑩 (𝒙, 𝒚) = 𝒙𝟐 𝒚 . The price of good 𝒙 is 𝑷𝒙 = 𝟐, the price of good
𝒚 is 𝑷𝒚 = 𝟐.
a. Find the optimal consumption choice of A and represent it in the
plane (𝒙, 𝒚).
EXERCISE 1
Budget line (B1): 𝟐𝒙 + 𝟐𝒚 = 𝟏𝟎𝟎
x
0
𝑀
= 50
𝑃𝑥
y
𝑀
= 50
𝑃𝑦
0
Best bundle: 𝟐𝟓, 𝟐𝟓
y
⇔ 𝑦 = 50 − 𝑥
IC
50
A’s Best choice
25
Perfect-complement Utility
function => L-shape
indifference curve
B1
25
50
x
EXERCISE 1
Two consumers A and B have an income 𝑴 = 𝟏𝟎𝟎 euros. Their
utility functions are, respectively, 𝑼𝑨 (𝒙, 𝒚) = 𝒎𝒊𝒏{𝒙, 𝒚} and
𝑼𝑩 (𝒙, 𝒚) = 𝒙𝟐 𝒚 . The price of good 𝒙 is 𝑷𝒙 = 𝟐, the price of good
𝒚 is 𝑷𝒚 = 𝟐.
b. Find the optimal consumption choice of B and represent it in the
plane (𝒙, 𝒚) using a new graph. Comment on the difference
between A and B.
Consumer B has a Cobb-Douglas Utility function:
𝜕𝑈𝐵 (𝑥, 𝑦)
2𝑥𝑦
2𝑦
𝟐𝒚
𝜕𝑥
𝑀𝑅𝑆𝑋𝑌 =
=
=
⇔ 𝑴𝑹𝑺𝑿𝒀 =
𝜕𝑈𝐵 (𝑥, 𝑦) 𝑥 2 × 1
𝑥
𝒙
𝜕𝑦
𝑷𝒙
2𝑦 ∗ 2
𝑴𝑹𝑺𝑿𝒀 =
⇔ ∗ =
⇔ 2𝑥 ∗ = 4𝑦 ∗ ⇔ 𝒙∗ = 𝟐𝒚∗
𝑷𝒚
𝑥
2
2
Budget line: 𝟐𝒙 + 𝟐𝒚 = 𝟏𝟎𝟎 ⇔ 𝑥 + 𝑦 = 50 ⇔ 𝒚 = 𝟓𝟎 − 𝒙
Substitute (2) into the budget line, we have:
𝑦 ∗ = 50 − 2𝑦 ∗ ⇔ 3𝑦 ∗ = 50 ⇔ 𝒚∗ = 𝟏𝟔. 𝟔𝟕
⇒ 𝒙∗ = 2𝑦 ∗ = 2 × 16.67 = 𝟑𝟑. 𝟑𝟒
Two consumers A and B have an income 𝑴 = 𝟏𝟎𝟎 euros. Their
utility functions are, respectively, 𝑼𝑨 (𝒙, 𝒚) = 𝒎𝒊𝒏{𝒙, 𝒚} and
𝑼𝑩 (𝒙, 𝒚) = 𝒙𝟐 𝒚 . The price of good 𝒙 is 𝑷𝒙 = 𝟐, the price of good
𝒚 is 𝑷𝒚 = 𝟐.
b. Find the optimal consumption choice of B and represent it in the
plane (𝒙, 𝒚) using a new graph. Comment on the difference
between A and B.
EXERCISE 1
Budget line (B1): 𝟐𝒙 + 𝟐𝒚 = 𝟏𝟎𝟎
x
0
y
𝑀
= 50
𝑃𝑦
𝑀
= 50
𝑃𝑥
y
⇔ 𝑦 = 50 − 𝑥
IC
50
0
Best bundle: 𝟑𝟑. 𝟑𝟒, 𝟏𝟔. 𝟔𝟕
Cobb-Douglas Utility function
16.67
=> diminishing MRS (convex
indifference curves) => first
order condition is sufficient
B’s Best
choice
Umax
B1
33.34
50
x
EXERCISE 1
Two consumers A and B have an income 𝑴 = 𝟏𝟎𝟎 euros. Their
utility functions are, respectively, 𝑼𝑨 (𝒙, 𝒚) = 𝒎𝒊𝒏{𝒙, 𝒚} and
𝑼𝑩 (𝒙, 𝒚) = 𝒙𝟐 𝒚 . The price of good 𝒙 is 𝑷𝒙 = 𝟐, the price of good
𝒚 is 𝑷𝒚 = 𝟐.
b. Find the optimal consumption choice of B and represent it in the
plane (𝒙, 𝒚) using a new graph. Comment on the difference
between A and B.
Comment:
• A has a perfect complement utility function, he/she can only
enjoy good x when it is consumed together with good y in a
specific ratio.
• B has a Cobb-Douglas utility function with diminishing
marginal rate of substitution, he/she is adapting his/her
choice to the price ratio.
Two consumers A and B have an income 𝑴 = 𝟏𝟎𝟎 euros. Their
utility functions are, respectively, 𝑼𝑨 (𝒙, 𝒚) = 𝒎𝒊𝒏{𝒙, 𝒚} and
𝑼𝑩 (𝒙, 𝒚) = 𝒙𝟐 𝒚 . The price of good 𝒙 is 𝑷𝒙 = 𝟐, the price of good
𝒚 is 𝑷𝒚 = 𝟐.
c. Consider only B and assume that the price of x becomes 𝑷′𝒙 = 𝟒.
Find B’s new optimal bundle and decompose the variation of 𝒙 in
substitution and income effect. Illustrate with a graph.
EXERCISE 1
New budget line:
𝟒𝒙 + 𝟐𝒚 = 𝟏𝟎𝟎 ⇔ 2𝑥 + 𝑦 = 50 ⇔ 𝒚 = 𝟓𝟎 − 𝟐𝒙
𝑃𝑥′
2𝑦 ∗∗ 4
𝑀𝑅𝑆𝑋𝑌 =
⇔ ∗∗ =
⇔ 4𝑥 ∗∗ = 4𝑦 ∗∗ ⇔ 𝒙∗∗ = 𝒚∗∗ 𝟑
𝑃𝑦
𝑥
2
Substitute (3) into the new budget line, we have:
𝑦 ∗∗ = 50 − 2𝑦 ∗∗ ⇔ 3𝑦 ∗∗ = 50 ⇔ 𝒚∗∗ = 𝟏𝟔. 𝟔𝟕
⇒ 𝒙∗∗ = 𝑦 ∗∗ = 𝟏𝟔. 𝟔𝟕
➔The new best bundle is: (𝟏𝟔. 𝟔𝟕, 𝟏𝟔. 𝟔𝟕)
Two consumers A and B have an income 𝑴 = 𝟏𝟎𝟎 euros. Their
utility functions are, respectively, 𝑼𝑨 (𝒙, 𝒚) = 𝒎𝒊𝒏{𝒙, 𝒚} and
𝑼𝑩 (𝒙, 𝒚) = 𝒙𝟐 𝒚 . The price of good 𝒙 is 𝑷𝒙 = 𝟐, the price of good
𝒚 is 𝑷𝒚 = 𝟐.
c. Consider only B and assume that the price of x becomes 𝑷′𝒙 = 𝟒.
Find B’s new optimal bundle and decompose the variation of 𝒙 in
substitution and income effect. Illustrate with a graph.
EXERCISE 1
New budget line (B2): 𝟒𝒙 + 𝟐𝒚 = 𝟏𝟎𝟎
x
0
y
𝑀
= 50
𝑃𝑦
𝑀
′ = 25
𝑃𝑥
y
⇔ 𝑦 = 50 − 2𝑥
IC
50
0
New best choice
B’s Best
choice
New best bundle:
(𝟏𝟔. 𝟔𝟕, 𝟏𝟔. 𝟔𝟕)
16.67
Umax
B2
16.67 25 33.34
B1
50
x
EXERCISE 1
Two consumers A and B have an income 𝑴 = 𝟏𝟎𝟎 euros. Their
utility functions are, respectively, 𝑼𝑨 (𝒙, 𝒚) = 𝒎𝒊𝒏{𝒙, 𝒚} and
𝑼𝑩 (𝒙, 𝒚) = 𝒙𝟐 𝒚 . The price of good 𝒙 is 𝑷𝒙 = 𝟐, the price of good
𝒚 is 𝑷𝒚 = 𝟐.
c. Consider only B and assume that the price of x becomes 𝑷′𝒙 = 𝟒.
Find B’s new optimal bundle and decompose the variation of 𝒙 in
substitution and income effect. Illustrate with a graph.
When 𝑃𝑥 = 𝑃𝑦 = 2, best bundle is A 𝑥 ∗ , 𝑦 ∗ = (33.34, 16.67)
and 𝑈𝑚𝑎𝑥 = 33.342 × 16.67
y
′
IC
When 𝑃𝑥 rises to 𝑃𝑥 = 4,
B3
new best bundle is B
50
𝑥 ∗∗ , 𝑦 ∗∗ = (16.67, 16.67)
To compute substitution and
income effect, we need to
y’
find point 𝑪 𝒙’, 𝒚’ where
𝑃𝑥′ = 4; 𝑃𝑦 = 2 and
16.67
𝑈 𝑥′, 𝑦′ = 𝑈𝑚𝑎𝑥 =
33.342 × 16.67
Income
effect
C
Substitution
effect
A
Umax
B
B2
16.67 25 x’ 33.34
B1
50
x
Two consumers A and B have an income 𝑴 = 𝟏𝟎𝟎 euros. Their
utility functions are, respectively, 𝑼𝑨 (𝒙, 𝒚) = 𝒎𝒊𝒏{𝒙, 𝒚} and
𝑼𝑩 (𝒙, 𝒚) = 𝒙𝟐 𝒚 . The price of good 𝒙 is 𝑷𝒙 = 𝟐, the price of good
𝒚 is 𝑷𝒚 = 𝟐.
c. Consider only B and assume that the price of x becomes 𝑷′𝒙 = 𝟒.
Find B’s new optimal bundle and decompose the variation of 𝒙 in
substitution and income effect. Illustrate with a graph.
EXERCISE 1
𝑀𝑅𝑆𝑋𝑌 =
𝑃𝑥′
𝑃𝑦
(tangency condition)
At point 𝑪 𝒙’, 𝒚’ : ቐ
𝑈 𝑥′, 𝑦′ = 𝑈𝑚𝑎𝑥 (utility condition)
⇔ ቐ
⇔ ൝
2𝑦′
𝑥′
𝑥′2 𝑦 ′
=
4
2
33.342 ×
=
16.67
𝑦 ′ = 𝑥′
2
𝑥 ′ (𝑥 ′ )
= 18,529.6
𝒚′ = 𝑥 ′ = 𝟐𝟔. 𝟒𝟔
⇔ቊ
𝒙′ = 𝟐𝟔. 𝟒𝟔
⇔൝
⇔ ൝
4𝑦′ = 4𝑥′
2
𝑥 ′ 𝑦′ = 18,529.6
𝑦 ′ = 𝑥′
3
𝑥 ′ = 18,529.6
Two consumers A and B have an income 𝑴 = 𝟏𝟎𝟎 euros. Their
utility functions are, respectively, 𝑼𝑨 (𝒙, 𝒚) = 𝒎𝒊𝒏{𝒙, 𝒚} and
𝑼𝑩 (𝒙, 𝒚) = 𝒙𝟐 𝒚 . The price of good 𝒙 is 𝑷𝒙 = 𝟐, the price of good
𝒚 is 𝑷𝒚 = 𝟐.
c. Consider only B and assume that the price of x becomes 𝑷′𝒙 = 𝟒.
Find B’s new optimal bundle and decompose the variation of 𝒙 in
substitution and income effect. Illustrate with a graph.
EXERCISE 1
Point C is 𝒙′ , 𝒚′ = (𝟐𝟔. 𝟒𝟔, 𝟐𝟔. 𝟒𝟔)
Substitution effect on good 𝑥:
𝒙′ − 𝒙∗ = 26.46 − 33.34 = −𝟔. 𝟖𝟖
Income effect on good 𝑥:
𝒙∗∗ − 𝒙′ = 16.67 − 26.46 = −𝟗. 𝟕𝟗
Substitution effect on good 𝑦:
𝒚′ − 𝒚∗ = 26.46 − 16.67 = 𝟗. 𝟕𝟗
Income effect on good 𝑦:
𝒚∗∗ − 𝒚′ = 16.67 − 26.46 = −𝟗. 𝟕𝟗
A firm produces good 𝒒 with the following total cost function
𝟑
𝟐
𝑪(𝑸) = 𝟏𝟎𝟎𝟎 + 𝟏𝟎𝑸 + 𝟐𝑸 , where 𝟏𝟎𝟎𝟎 is an avoidable fixed
cost
a. Find marginal cost function, average cost function, and average
variable cost function. Represent all curves, and their minima, in
the same graph. Has the firm economies or diseconomies of scale?
𝟏
3 3−1
𝑴𝑪 𝑸 = 𝐶 𝑄 = 0 + 10 × 1 + 2 × 𝑄2 = 𝟏𝟎 + 𝟑𝑸𝟐
2
EXERCISE 2
′
𝐶 𝑄
1000 + 10𝑄 +
𝑨𝑪 𝑸 =
=
𝑄
𝑄
3
2𝑄2
𝑉𝐶 𝑄 = 𝐶 𝑄 − 𝐹𝐶 = 1000 + 10𝑄
= 10𝑄 + 2𝑄
𝟏
𝟏𝟎𝟎𝟎
=
+ 𝟏𝟎 + 𝟐𝑸𝟐
𝑸
3
+ 2𝑄 2
− 1000
3
2
𝑉𝐶 𝑄
10𝑄 +
⇒ 𝑨𝑽𝑪 𝑸 =
=
𝑄
𝑄
3
2𝑄 2
= 𝟏𝟎 +
𝟏
𝟐𝑸𝟐
A firm produces good 𝒒 with the following total cost function
𝟑
𝟐
𝑪(𝑸) = 𝟏𝟎𝟎𝟎 + 𝟏𝟎𝑸 + 𝟐𝑸 , where 𝟏𝟎𝟎𝟎 is an avoidable fixed
cost
a. Find marginal cost function, average cost function, and average
variable cost function. Represent all curves, and their minima, in
the same graph. Has the firm economies or diseconomies of scale?
EXERCISE 2
𝑨𝑪 𝑸 reaches its minimum when: 𝑴𝑪 𝑸 = 𝑨𝑪 𝑸
1
2
⇔ 10 + 3𝑄 =
1000
+
𝑄
10 + 2𝑄
1
2
1
2
⇔𝑄 =
1000
𝑄
3
2
⇔ 𝑄 = 1000
⇔ 𝑸 = 𝟏𝟎𝟎
𝑨𝑪𝒎𝒊𝒏 = 𝐴𝐶 100 =
1000
+
100
1
2
10 + 2 × 100 = 𝟒𝟎
𝑨𝑽𝑪 𝑸 reaches its minimum when: 𝑴𝑪 𝑸 = 𝑨𝑽𝑪 𝑸
⇔ 10 +
1
3𝑄2
= 10 +
1
2𝑄2
⇔
1
𝑄2
1
2
=0 ⇔𝑸=𝟎
𝑨𝑽𝑪𝒎𝒊𝒏 = 𝐴𝑉𝐶 0 = 10 + 2 × 0 = 𝟏𝟎
A firm produces good 𝒒 with the following total cost function
𝟑
𝟐
𝑪(𝑸) = 𝟏𝟎𝟎𝟎 + 𝟏𝟎𝑸 + 𝟐𝑸 , where 𝟏𝟎𝟎𝟎 is an avoidable fixed
cost
a. Find marginal cost function, average cost function, and average
variable cost function. Represent all curves, and their minima, in
the same graph. Has the firm economies or diseconomies of scale?
EXERCISE 2
1
3𝑄2
𝑀𝐶 𝑄 = 10 +
1
1000
𝐴𝐶 𝑄 =
+ 10 + 2𝑄2
𝑄
𝐴𝑉𝐶 𝑄 = 10 +
1
2𝑄2
The firm has economies
of scale when 𝑸 ≤ 𝟏𝟎𝟎
(AC falls as firm produces
more) and diseconomies
of scale when 𝑸 > 𝟏𝟎𝟎
(AC rises as firm
produces more)
A firm produces good 𝒒 with the following total cost function
𝟑
𝟐
𝑪(𝑸) = 𝟏𝟎𝟎𝟎 + 𝟏𝟎𝑸 + 𝟐𝑸 , where 𝟏𝟎𝟎𝟎 is an avoidable fixed
cost
b. Assuming that the firm is a price-taker and that good q’s price is
𝑷 = 𝟑𝟕, find the profit maximizing level of output and the
corresponding profit.
Quantity rule: Price-taking firm ➔ At the profit-maximizing
EXERCISE 2
1
output: 𝑷 = 𝑴𝑪 𝑸
⇔
1
∗
𝑄 2
=9
⇔ 37 = 10
+ 3𝑄∗ 2
1
⇔
3𝑄∗ 2
= 27
⇔ 𝑸∗ = 𝟖𝟏
Shut-down rule: 𝐹𝐶 is avoidable, 𝑷 = 𝟑𝟕 < 𝑨𝑪𝒎𝒊𝒏 = 𝟒𝟎
⇒ the firm should shut down ⇒ 𝑸𝒔𝒉𝒖𝒕𝒅𝒐𝒘𝒏 = 𝟎
𝜋𝑠ℎ𝑢𝑡𝑑𝑜𝑤𝑛 = 0 (as FC is avoidable)
Explain how firms use their inputs to minimize costs. Provide an
example and consider both the long-run and the short-run scenario
EXERCISE 3
Firms use their inputs to minimize costs by choosing the combination
of inputs that produces a given level of output at the lowest possible
cost. This involves considering the prices of the inputs and their
productivity in the production process.
In the short run, firms have fixed inputs that cannot be changed, such
as the size of the factory or the number of machines. Therefore, firms
must choose the quantity of variable inputs, such as labor and raw
materials, to produce the desired output level. When only one input
is variable, the firm minimizes costs by using it in the most efficient
ways as given by the production function.
In the long run, all inputs are variable, and the firm can adjust its
production process by changing the size of the factory, the number of
machines, and the number of workers. Assuming that there are only
two inputs, a necessary condition for an optimal use of inputs is that
their marginal benefits, each given by the ratio of an input’s marginal
product and its price, are equal. The condition is also sufficient when
the marginal rate of technical substitution is decreasing.