Intermediate Microeconomics: Information Economics
Group Assignment 2 – Fall 2021
For instructions: Please check canvas.
Calculating Expectations / Conditional Expectations
Question 1
The variable 𝑥 is randomly distributed on the interval [1,7] with the cumulative distribution function 𝐹(𝑥).
0 𝑥<1
𝐹(𝑥) = {1 − (7 − 𝑥)2 /36 𝑥 ∈ [1,7]
0 𝑥>7
Which of the following expressions is correct:
A. 𝐸(𝑥) ≤ 2.5 and 𝑉(𝑥) > 2.5
B. 𝐸(𝑥) > 2.5 and 𝑉(𝑥) > 2.5
C. 𝐸(𝑥) ≤ 2.5 and 𝑉(𝑥) ≤ 2.5
D. 𝐸(𝑥) > 2.5 and 𝑉(𝑥) ≤ 2.5
Question 2
Assume that 𝐹(𝑥) in Question 1 represents the distribution of quality of cars in the market for second hand
cars. The sellers of cars value their car at 𝑉𝑠𝑒𝑙𝑙𝑒𝑟 = 3 ⋅ 𝑥. As a buyer of the car you do not observe the quality
of the car. Suppose that you pay a price 𝑝 = 15 for a second hand car. What is the expected quality of this car
(assuming you did not coerce anybody to sell their car)?
A. The expected quality cannot be calculated with the information provided.
8
B. The expected quality is equal to 3.
C. Only cars with a quality 𝑥 ∈ [1,5] are sold. The expected quality is therefore equal to 3
64
D. The expected quality is equal to 27
Lemon’s problem
Question 3
Consider the market for second-hand cars. Cars can be of different qualities 𝑞. The quality 𝑞 of a car is known
by the current owner but is not observable to prospective buyers. Buyers only know that in the population of
all available cars, 𝑞 is uniformly distributed between 1 and 4. There are many more sellers than buyers for
each type. The reservation value of sellers for a car of quality 𝑞 is given by 𝑉𝑆 = 1.5𝑞, while the reservation
value of potential buyers is given by 𝑉𝐵 = 1.5 + 𝑞. All agents try to maximize expected surplus.
Which of the following statements is true
A. All cars are sold and the equilibrium price is 3
B. No car is sold except the ones for which 𝑞 = 0
C. Only cars with quality 𝑞 ≤ 2 are sold
D. All cars are sold and the equilibrium price is 5.5
, Intermediate Microeconomics: Information Economics
Signaling
Question 4
Consider the following environment. There are two workers, H and L. The value to employers of the product
of H is 40, while the value of the product of L is 20. Employers cannot directly distinguish which worker is
which. Workers choose their level of education, e. This level is perfectly observed by employers. The cost to H
of acquiring education e is equal to e. The cost to L of acquiring education e is also equal to e. The utility
function of workers is given by the difference between the wage they earn and the cost of education.
Employers try to maximize expected profit, the difference between the expected value of output and the wage
they pay.
The timing of the game is as follows. First, both workers choose their level of education, which becomes known
to all players. Then, (many) employers compete à la Bertrand for hiring workers by posting wage schedules,
w(e), associating a given wage to each possible level of education. Finally, both workers choose for which
employer to work. We look for equilibria in which (i) given their beliefs about other players’ actions, players
maximize their payoff, and (ii) players hold beliefs that are consistent with other players’ equilibrium actions.
Which of the following set of actions is a pooling equilibrium?
A. H and L choose education level e=0 and all employers offer a wage schedule such that w=20 if e<5 and
w=30 if e≥5.
B. H and L choose education level e=7 and all employers offer a wage schedule such that w=30 for
education level just equal to 7, and w=20 otherwise.
C. H and L choose education level e=14 and all employers offer a wage schedule such that w=20 if e<14
and w=35 if e≥14.
D. H and L choose education level e=21 and all employers offer a wage schedule such that w=30 if
education level just equal to 21, and w=20 otherwise.
Question 5
In the same environment as in Question 4, which of the following set of actions is a separating equilibrium?
A. H chooses education level 2, L chooses education level 0, and all employers offer a wage schedule such
that w=20 if e<2 and w=40 if e≥2.
B. H chooses education level 8, L chooses education level 0, and all employers offer a wage schedule such
that w=20 if e<8 and w=40 if e≥8.
C. H chooses education level 12, L chooses education level 0, and all employers offer a wage schedule
such that w=20 if e<12 and w=40 if e≥12.
D. None of the above.
Adverse Selection / Price discrimination
Question 6
A monopolist faces two customers, one of which is of type 𝜃1 = 1 and the other of type 𝜃2 = 2. Customers’
𝑥2
utility function are 𝑢𝑖 = 𝑥 − 2𝜃 − 𝑇𝑖 where high 𝜃 indicates that the customer has stronger preferences for
𝑖
the good. 𝑇𝑖 is the total amount an individual pays to the monopolist in exchange for goods. The monopolist
maximizes profits, her unit costs are constant and equalized to 0.
2
Group Assignment 2 – Fall 2021
For instructions: Please check canvas.
Calculating Expectations / Conditional Expectations
Question 1
The variable 𝑥 is randomly distributed on the interval [1,7] with the cumulative distribution function 𝐹(𝑥).
0 𝑥<1
𝐹(𝑥) = {1 − (7 − 𝑥)2 /36 𝑥 ∈ [1,7]
0 𝑥>7
Which of the following expressions is correct:
A. 𝐸(𝑥) ≤ 2.5 and 𝑉(𝑥) > 2.5
B. 𝐸(𝑥) > 2.5 and 𝑉(𝑥) > 2.5
C. 𝐸(𝑥) ≤ 2.5 and 𝑉(𝑥) ≤ 2.5
D. 𝐸(𝑥) > 2.5 and 𝑉(𝑥) ≤ 2.5
Question 2
Assume that 𝐹(𝑥) in Question 1 represents the distribution of quality of cars in the market for second hand
cars. The sellers of cars value their car at 𝑉𝑠𝑒𝑙𝑙𝑒𝑟 = 3 ⋅ 𝑥. As a buyer of the car you do not observe the quality
of the car. Suppose that you pay a price 𝑝 = 15 for a second hand car. What is the expected quality of this car
(assuming you did not coerce anybody to sell their car)?
A. The expected quality cannot be calculated with the information provided.
8
B. The expected quality is equal to 3.
C. Only cars with a quality 𝑥 ∈ [1,5] are sold. The expected quality is therefore equal to 3
64
D. The expected quality is equal to 27
Lemon’s problem
Question 3
Consider the market for second-hand cars. Cars can be of different qualities 𝑞. The quality 𝑞 of a car is known
by the current owner but is not observable to prospective buyers. Buyers only know that in the population of
all available cars, 𝑞 is uniformly distributed between 1 and 4. There are many more sellers than buyers for
each type. The reservation value of sellers for a car of quality 𝑞 is given by 𝑉𝑆 = 1.5𝑞, while the reservation
value of potential buyers is given by 𝑉𝐵 = 1.5 + 𝑞. All agents try to maximize expected surplus.
Which of the following statements is true
A. All cars are sold and the equilibrium price is 3
B. No car is sold except the ones for which 𝑞 = 0
C. Only cars with quality 𝑞 ≤ 2 are sold
D. All cars are sold and the equilibrium price is 5.5
, Intermediate Microeconomics: Information Economics
Signaling
Question 4
Consider the following environment. There are two workers, H and L. The value to employers of the product
of H is 40, while the value of the product of L is 20. Employers cannot directly distinguish which worker is
which. Workers choose their level of education, e. This level is perfectly observed by employers. The cost to H
of acquiring education e is equal to e. The cost to L of acquiring education e is also equal to e. The utility
function of workers is given by the difference between the wage they earn and the cost of education.
Employers try to maximize expected profit, the difference between the expected value of output and the wage
they pay.
The timing of the game is as follows. First, both workers choose their level of education, which becomes known
to all players. Then, (many) employers compete à la Bertrand for hiring workers by posting wage schedules,
w(e), associating a given wage to each possible level of education. Finally, both workers choose for which
employer to work. We look for equilibria in which (i) given their beliefs about other players’ actions, players
maximize their payoff, and (ii) players hold beliefs that are consistent with other players’ equilibrium actions.
Which of the following set of actions is a pooling equilibrium?
A. H and L choose education level e=0 and all employers offer a wage schedule such that w=20 if e<5 and
w=30 if e≥5.
B. H and L choose education level e=7 and all employers offer a wage schedule such that w=30 for
education level just equal to 7, and w=20 otherwise.
C. H and L choose education level e=14 and all employers offer a wage schedule such that w=20 if e<14
and w=35 if e≥14.
D. H and L choose education level e=21 and all employers offer a wage schedule such that w=30 if
education level just equal to 21, and w=20 otherwise.
Question 5
In the same environment as in Question 4, which of the following set of actions is a separating equilibrium?
A. H chooses education level 2, L chooses education level 0, and all employers offer a wage schedule such
that w=20 if e<2 and w=40 if e≥2.
B. H chooses education level 8, L chooses education level 0, and all employers offer a wage schedule such
that w=20 if e<8 and w=40 if e≥8.
C. H chooses education level 12, L chooses education level 0, and all employers offer a wage schedule
such that w=20 if e<12 and w=40 if e≥12.
D. None of the above.
Adverse Selection / Price discrimination
Question 6
A monopolist faces two customers, one of which is of type 𝜃1 = 1 and the other of type 𝜃2 = 2. Customers’
𝑥2
utility function are 𝑢𝑖 = 𝑥 − 2𝜃 − 𝑇𝑖 where high 𝜃 indicates that the customer has stronger preferences for
𝑖
the good. 𝑇𝑖 is the total amount an individual pays to the monopolist in exchange for goods. The monopolist
maximizes profits, her unit costs are constant and equalized to 0.
2
