Problem set ECOS3003
Due date 27 Oct 2022 9am
Please keep your answers brief and concise. Excessively long and irrelevant answers will be
penalised. You can either type or neatly handwrite your answers.
Summit your answers in ONE file under PDF or Word format through the submission portal
on Canvas. Any late submission will incur a 5% penalty per day.
There are 5 short-answer problems. Total mark is 48. The problem set is worth 15% of
your final grade.
Question 1 (8 marks)
Consider the following simultaneous game (A) in normal form:
a) Calculate all the Nash equilibria of A. (4 marks)
Now consider the following game (B):
b) Do games A and B have the same Nash equilibrium/equilibria? Why or why not? (4
marks)
Question 2 (7 marks)
Tanya is an owner of Invest Co. Her utility of wealth is given by U(W) = W2 where W is her
wealth level. Tanya has been sponsored $M to invest in two potential investment projects (M
is an amount of money and simply disappears if it is not invested in a project). Investment A
pays $16 per dollar of investment or $0 with a 50% chance each. Investment B pays $9 per
dollar of investment with probability 2/3 or $0 with probability of 1/3. Which proportion of M
does Tanya want to invest in A? Explain your answer in reference to risk aversity discussed in
class. (7 marks)
Question 3 (10 marks)
DrugCo buys an input that costs $p from the market for its drug production. DrugCo knows
that 4/5 of suppliers will provide a good input that the firm can use to produce $100 value of
output and 1/5 of suppliers will provide a low quantity input that produces $60 value for the
firm. The firm is matched with a potential input supplier in the market, but cannot tell whether
it will supply a good or low quality product.
a. If DrugCo does not buy the input from the market, no production will occur. What is the
maximum price p that DrugCo is willing to pay for the input? Show your working. (2 marks)
b. Assume now that DrugCo can also buy the same input from its long-term trading partner -
Compound Corp at a price c. Compound Corp is supplying a good input for sure. What is the
most the firm is willing to pay to Compound Corp for the input if the input costs $20 from the
market? Show your working. (2 marks)
Now assume that Compound Corp can choose to supply a good input or a low-quality input.
The cost of producing a good input is $10 and that of producing a low-quality input is $5.
DrugCo will pay $25 to buy the input from Compound Corp.
c. If there are two years in which trade can occur, will Compound Corp supply a good input or
a low-quality input in the first period? In the second period? Explain your answer. (2 marks)
d. Will DrugCo sign a 2-year contract with Compound Corp or buy from the market? Explain
your answer. (2 marks)
e. Now trade can occur potentially an infinite number of times. DrugCo adopts a trigger-
strategy if Compound Corp ever supplies low-quality inputs. Both parties have a discount
factor of δ. For what value of δ will Compound Corp always supply good inputs? (2 marks)
Question 4 (8 marks)
A principal has to implement a decision that has to be a number between 0 and 1; that is, a
decision d needs to be implemented where 0 1d? ? . The difficulty for the principal is that she
does not know what decision is appropriate given the current state of the economy, but she
would like to implement a decision that exactly equals what is required given the state of the
economy. In other words, if the economy is in state s (where 0 1s? ? ) the principal would
like to implement a decision d = s as the principal’s utility Up (or loss from the maximum
possible profit) is given by = ?( ? )
2. With such a utility function, maximising utility
really means making the loss as small as possible. For simplicity, the two possible levels of s
are 0.5 and 0.8 which occur with probability 0.4 and 0.6 respectively.
There are two division managers A and B who each have their own biases. Manager A always
wants a decision of 0.3 to be implemented and incurs a disutility = (0.3 )
2. Similarly,
Manager B always wants a decision of 0.8 to be implemented and incurs a disutility =
(0.8 )2. Each manager is completely informed, so that each of them knows exactly what
the state of the economy s is.
a) The principal can opt to centralise the decision but before making her decision – given
she does not know what the state of the economy is – she asks for recommendations
from her two division managers. Centralisation means that the principal commits to
implement a decision that is the average of the two recommendations she received from
her managers. The recommendations are sent simultaneously and cannot be less than 0
or greater than 1.
Assume that the state of the economy s = 0.8. What is the report (or recommendation)
that Manager A will send if Manager B always truthfully reports s? Explain your
answer. (4 marks)
b) What if the principal instead delegates decision-making entirely to manager A (that is,
A can decide on her own d without any consultation). Does this make the principal
better or worse off than with centralisation as in part a? Provide some intuition for your
answer. (4 marks)
Question 5 (8 marks)
Consider the following workplace situation. A boss can implement an incentive scheme (I) or
not (NI). Following observing this a worker can work hard (H) or slack (S). The payoffs are as
follows. If I and H are chosen by the boss and worker, respectively, the payoffs are 25 to the
boss and 9 to the worker. If I is chosen by the boss, then L by the worker the payoffs are 12 to
the boss and 5 to the worker. If the choices are NI then H, the returns are 30 and 7 to the boss
and worker respectively. Finally following NI and L the payoffs are 17 (boss) and 11 (worker)
a) What is the subgame equilibrium of the game? Explain your answer in the context of
the principal-agent incentive contracts. (3 marks)
b) Now assume that the worker, through their union workplace agreement has a veto right
on any incentive scheme, and there is no incentive scheme currently in place. The boss
P a g e 5 | 5
and the worker can negotiate to allow for incentive scheme to be legally implemented
(and compensation/side payments can be made as part of this agreement). If the worker
has all of the bargaining power during these negotiations, will the incentive scheme be
implemented? Is the outcome surplus-maximising? Explain your answer in the context
of the value-maximisation principle? (5 marks)
Question 6 (7 marks)
Consider a version of the property‐right model studied in class involving a buyer (Simon) and
a seller (Jonny). The timing of the game is as follows. Initially the parties cannot write a
contract on investment or on sharing surplus. At this point, Jonny can make an investment that
costs either $50 (low investment) or $300 (high investment). This investment is sunk and
specific to trading with Simon. After the investment has been sunk, and contracting becomes
possible, so the parties negotiate. The ex post surplus generated is either $200 with low
investment or $500 if Jonny made a high investment. Trade occurs (or does not) between the
two parties and the game ends.
a) Draw a timeline of the game. What is the ex‐ante period and the ex‐post period in this
model? (1 mark)
b) What is the first‐best level of investment? Explain. (2 marks)
c) Now assume that property rights can be assigned to either party of an asset critical to
this production process. If Simon owns the asset he will receive 50% of the ex post
surplus. If Jonny owns the asset he will receive 90% of the ex post surplus. In this
incomplete contracting environment, who should own the asset? Explain your answer
in the context of the key predictions of the property‐rights model. (4 marks)