game theory, undergraduate economics

pdf file attached. due in 4 hours. 5 questions with 3-4 short parts. 

ECON 421 – 002
Jonathan L. Graves Assignment 4 March 24, 2021

Instructions: This is the Module 3 assignment. This assignment is individual, but you
should feel free to discuss with other students as your work on the assignment. Do not copy
another person’s analysis or answers; if you use an outside source, cite it in your solutions.
Try to clearly and logically explain each part of your analysis.

Structure: This assignment has 5 questions, for a total of 105 points and 5 bonus points.

Hand-in: This assignment is due the last class prior the date listed on Canvas or online
by the deadline. Late assignments will be penalized at a rate of 2% per hour.

1. Consider the “cheap talk” model we saw in class. Suppose everything else is the same
as in class, but now we have two agents who announce their reports simultaneously.
Suppose both agents have the same bias b > 0.

(a) (5 points) Is there still an uninformative equilibrium? Explain why or why not?

(b) (10 points) Suppose the government tells the experts “I will always set the policy
equal to the lower of your two signals”. Is truth telling now an equilibrium? Explain
why or why not.

(c) (10 points) Suppose the government just gives up, and says “I’ll implement what-
ever policy you tell me if you agree, otherwise I’ll set it randomly.” Is this better
or worse than the two-signal equilibrium we saw in class? Explain.

Total for Question 1: 25

2. In the herding model we saw in class, suppose there were two types of experts: skilled
and unskilled. Skilled experts have qs = 0.80 > qu = 0.55 for the unskilled experts; i.e.
the skilled experts are more accurate.

(a) (10 points) How many skilled experts would need to report G in a row in order to
start a herd? Unskilled?

(b) (5 points) If you, as a government, know the skill of each expert would you prefer
to hear from the skilled experts or unskilled experts first? Explain.

Total for Question 2: 15

3. Consider the game in Figure 1, where x ∈ [0, 1]
(a) (5 points) Is there a PBNE where S sets x = 0? Explain.

(b) (5 points) Is there a PBNE where both players set the same x? Explain.

(c) (5 points) Is there a PBNE where S sets x = 1? Explain.

(d) (5 points (bonus)) Can you find a better PBNE than in (c)? Explain why or why
not.

Total for Question 3: 15

ECON 421 – 002 Assignment 4, Page 2 of 4 March 24, 2021

P2

Nature

S(0.5) W(0.5)

S
x

2 −x,−1

A

0, 0

D

W
x

2 − 3x, 1

A
0, 0
D

Figure 1: A Game Tree

4. Consider the game in Figure 2.

(a) (5 points) Find all three of the NE in this game, especially including the MSNE and
calculate the expected equilibrium payoffs for both players. Suppose we change the

game slightly. Before the simultaneous game (below) occurs, a third party (nature)
observes an event which occur with equal probability and indicates one of the three
non-zero cells. Nature then communicates to each player which action they play in
this cell, but not what cell it it.

(b) (5 points) Draw this game tree, being careful to label information sets.

(c) (10 points) Suppose each player follows the strategy “Do what nature tells me to
do”, believing their opponent will do the same thing. Is this a PBNE? Explain.

(d) (5 points) What is the ex ante payoff in the PBNE and how does it compare to
your answers to (a)?

Total for Question 4: 25

Player 1

Player 2
S NS

S 0,0 7,2
NS 2,7 6,6

Figure 2: A Matrix

5. (F) Here is a very famous model of credit rationing. In this model there are two
players, a bank and a large number of firms (i). Each firm has a project which costs i
to undertake. The project has a potential return Ri which occurs with probability pi.
Both the bank and the firms are risk-neutral profit maximizers.

ECON 421 – 002 Assignment 4, Page 3 of 4 March 24, 2021

• Firms are either risky (r) or safe (s), which refers to their project

• Both firms have the same expected return which is profitable: piRi = m > 1

• However, ps > pr (and therefore Rs < Rr)

• Firms are safe with probability β

We will represent the large number of firms as a unit mass; i.e. you can imagine there
are an infinite number of firms who “add up” to 1 firm in aggregate. In particular, this
means that (a) exactly β of the firms are safe, and (b) if you finance a fraction of the
firms f, it costs you f in total.

• This is a common technique in many economic models. Essentially, it lets you
imagine the “fraction” of the populations the “number” of firms.

• For example, since each firm demands 1 unit of financing, this means that the whole
market, in aggregate demands 1 unit of financing, since each firm is an infinitestimal
element of that.

• Formally, the totals in this model are given by:

f =

∫
f(j)dj

where j is the index for each “individual” firm.

• This also implies, by the LLN, that any probabilistic elements are exact fractions
in the model; hence why there are exactly β safe firms.

On the bank side of things, the bank has a total amount to lend α < 1, so they can’t finance everyone. However, α > max{β, 1−β} to they have enough funds to (potentially)
lend to both the safe and risky firms if they want to.

• The bank provides lending using a contract. The contract says “If I finance your
project, you must pay me D in exchange.”

• If firm cannot repay their loan, (i.e. their project doesn’t succeed) they go bankrupt,
and the bank can’t recover any of their expenditure

Answer the following questions, using the above set-up. If necessary, you can assume we
are looking for a symmetric PBNE here (i.e. where all firms of a given type behave in
the same way).

(a) (5 points) Suppose that D > Rs. Will a safe firm borrow from the bank? What
about risky firms? Who will accept this contract? Would the bank set D > Rr if
it wants to attract borrowers? Why or why not?

(b) (3 points) Continuing from (a), in equilibrium what will the bank believe about the
type of firm who accepts a loan? What is the bank’s expected profit from lending
to these types?

ECON 421 – 002 Assignment 4, Page 4 of 4 March 24, 2021

(c) (5 points) What should the bank set D to in order to maximize profits (under the
condition that D > Rs)? What is the bank’s optimal profit? (Remember: there
are 1 −β risky firms in total)

(d) (5 points) Okay, now suppose that the bank decides to set D ≤ Rs. Who would
borrow from the bank in this case? What is the bank’s optimal value of D? What
is their profit? (Remember, again, that there are β safe firms and 1−β risky firms,
and the bank only has α dollars total to lend out.)

(e) (2 points) Compare the profits in (c) and (d) when β is very close to zero. What
would the bank do?

(f) (5 points) Can all of the risky firms find financing? Why is this efficient or prob-
lematic? Explain and discuss your answer

Total for Question 5: 25

End of Assignment