BPUB 250
Final Examination Spring 2006
Name:
Please write your name at the top of the exam and hand this sheet in with your blue book(s). You
must show your work to receive full or partial credit. Partial credit is available, so you should
attempt to answer all of the questions, even if you are not sure of the answer. This exam is worth
a total of 100 points. You have two hours to complete the exam. Good luck!
1) (20 points) Two candidates, Ed and Lynn, are running against one another in the
Pennsylvania Gubernatorial race. Initially, each expects to get half the votes. Ed has settled
on using one of two campaign strategies: run positive ads or run negative ads. Lynn has three
possible strategies: run positive ads, run negative ads, or run no ads at all. Because there are
only two candidates, any increase in the percentage of voters choosing Ed equals a loss in the
percentage of voters choosing Lynn. If Ed runs negative ads, he will gain 8 percentage points
if Lynn runs positive ads, 1 percentage point if Lynn runs negative ads, and 7 points if Lynn
runs no ads. If Ed runs positive ads, he will lose 5 percentage points if Lynn runs positive ads,
he will lose 2 points if Lynn runs negative ads and he will gain 10 points if Lynn runs no ads
at all.
a. (4 points) Show the payoff matrix for this game.
b. (4 points) Does either candidate have a dominant strategy?
c. (5 points) Define Nash equilibrium. Which if any of the pairs of pure strategies (i.e. one
for each candidate) constitute a Nash equilibrium?
d. (7 points) Lynn decides not to run negative ads and to randomize between positive and no
advertising: on some days Lynn will run positive ads, on some days he will not run any
ads at all. What is Lynn’s equilibrium randomizing strategy?
2) (20 points) Mark owns a small biotechnology company. His utility function is given by
U = 0.5,
where  is the company’s annual profit. Mark is considering whether his company should
invest in bringing a new drug to market. If the drug is a huge success in the market, the
annual profit will be $2.5 billion. If the drug is less successful the profit will be $36 million.
Mark assesses the probability that the drug is a huge success to be 0.25.
a. (6 points) Find both expected profits and expected utility from developing the drug.
b. (8 points) Pfizer comes along and offers to buy the rights to market the drug and earn
whatever profits the drug yields. What is the minimum amount of payment that Mark
would accept to sell the right to market the drug?
c. (6 points) Assume Pfizer offered the minimum amount required for Mark to sell the
rights as identified in the previous problem. However, before they start the transaction, a
market research company approaches Mark and tells him that they can predict whether
the profits will be $2.5 billion or $36 million with certainty. The market research
company is willing to sign a confidentiality agreement so that Pfizer will never learn of
this service. How much is Mark willing to pay the company for its service?
(Page 1 of 2)
3) (30 points) The production of paper produces a negative externality (water pollution). The
marginal cost to society of this externality is MCS = Q. The paper supply curve based on the
private marginal cost is MCP = 3Q. The inverse demand curve faced by the paper producer is
P(Q) = 30 – Q.
a. (5 points) How much output would the company produce in an (unregulated) perfectly
competitive equilibrium?
b. (7 points) What level of production corresponds to the socially optimal output?
c. (6 points) Make a picture illustrating this market, labeling all important features.
d. (6 points) How much dead weight loss does the company’s output choice in part (a)
generate?
e. (6 points) What specific tax could the government impose on the competitive equilibrium
to restrict output to the socially optimal level?
4) (30 points) The Flourtown Golf Club, the only golf club in the area, serves two towns,
Flourtown (F) and Wyndmoor (W). The club charges a price of P per round of golf.
Each of the 100 residents of Flourtown has demand for rounds of golf of: Q F = 100 – P,
Each of the 50 residents of Wyndmoor has demand of: Q W = 80 – 2P.
The club incurs total costs of TC (Q) 
1
10
Q 2  Q , where Q is total rounds across all
2200
11
residents of Flourtown and Wyndmoor.
Assume initially that the club CANNOT price discriminate between the residents of each
town.
a. (13 points) What price does it charge per round of golf? How many rounds do the
residents from the two towns purchase together? What is the club’s marginal cost for the
number of rounds of golf that it sells?
Now assume that the club CAN price discriminate between the residents of each town.
b. (7 points) What price does the club charge Flourtown residents? What price does it
charge Wyndmoor residents? What is the club’s total profit?
c. (10 points) Instead of the simple per-round fee, the club decides to charge a membership
fee and a per-round fee. It decides to set the per-round fee based on marginal cost pricing
of the total number of rounds sold. What per-round fee does the club charge? What
membership fee does the club charge to each resident of the two towns if it discriminates
between the residents of each town (i.e. the membership fees can potentially be
different)?
Have a great summer!
(Page 2 of 2)
BPUB 250 Final Exam Answers
Spring 2006
Question 1
a)
The payoff matrix is given by:
Ed
Negative
Positive
Lynn
Negative
(51,49)
(48,52)
Positive
(58,42)
(45,55)
None
(57,43)
(60,40)
b)
The players do not have dominant strategies. For Ed, running negative ads is best
when Lynn is running positive or negative ads, but running positive ads is better when
Lynn is running no ads at all. Lynn’s best strategy is to run negative ads when Ed runs
negative ads, but positive ads when Ed is running positive ads.
c)
The only pure strategy equilibrium is (Negative, Negative), as indicated in the
matrix. Neither candidate has an incentive to deviate given what the other is doing.
d)
The key for finding mixed strategies is to randomize so that the opponent has
equal payoffs across his strategies.
The new payoff matrix is:
Lynn
Ed
Negative
Positive
Positive
(58,42)
(45,55)
None
(57,43)
(60,40)
Assume p 1 is the probability that Lynn will run positive ads and 1-p 1 is the probability
that he will run no ads.
Ed's expected payoff from negative ads: E ( E )  .58 p1  .57(1  p1 )
Ed's expected payoff from positive ads: E ( E )  .45 p1  .60(1  p1 )
E ( E )  E ( E )
p1  3 / 16
(We can similarly solve for Ed’s mixed strategy. The solution is p 1 = 15/16.)
Question 2
a)
Expected profits:
E ( )  (0.25)(2500)  (0.75)(36)  652 M
Expected utility:
E (u )  (0.25)( 2500 )  (0.75)( 36 )  17
b)
This question asks about the certainty equivalent of the gamble. From a) we
know the expected utility is 17. Then
  17   289
He will be indifferent between a (certain) offer of 289 million and taking the gamble.
c)
Before purchasing the information, the value of marketing the drug is now $2.5
billion with probability 0.25 (Pfizer’s offer is rejected) or $289 M (drug turns out to be a
bust and Pfizer’s offer is accepted).
E ( )  (0.25)(2500)  (0.75)(289)  841.75M
E (u )  (0.25)( 2500 )  (0.75)( 289 )  25.25
The certainty equivalent for this second gamble with information is
25.252 = 637.56
This means that knowing the information will increase the certainty equivalent from
289M to 637.56M. Therefore he is willing to pay up to the difference between the two,
or 348.56M.
Question 3
a)
Without regulation, the firm will just consider its own cost. For profit
maximization,
MCp = P
3Q = 30 – Q
Q = 7.5
P = 22.5
b)
From society’s point of view, total marginal cost should be considered.
MCTotal = MCp + MCs = 3Q + Q
=>3Q + Q = 30 – Q
Q=6
P = 24
c)
P
DWL
MCtotal
MCp(tax)
tax
MCp
24
22.5
MCs
6
d)
Q
7.5
DWL = (7.5 – 6) (30 – 22.5) / 2 = 5.625
e)
A constant per-unit tax is easiest to implement. We want the tax to shift the
producer’s marginal cost curve up to entail the socially optimal equilibrium quantity.
(MCp + tax) = 30-Q where Q = 6
tax = $6 per unit
(Other, more complicated tax schemes could also work.)
Question 4
a)
We first derive the total demand curve.
Q F = 100(100-P) = 10000 – 100P
Q W = 50(80-2P) = 4000 – 100P
Q T = 10000 – 100P if P>=40
Q T = 14000 – 200P if P<=40
(when W’s demand = 0)
(when W’s demand > 0)
P = 100 – (1/100)Q
P = 70 – (1/200)Q
if Q<=6000
if Q>=6000
TC = (1/2200) Q2 – (10/11)Q
MC = (1/1100)Q – (10/11)
MC = P => (1/1100)Q – (10/11) = 70 – (1/200)Q
Q = 6500
MC = P = 37.50
(When Q>=6000)
b)
We want to find price/quantity combination for each town so that the marginal
revenue from each town is exactly the marginal cost at Q = 6500.
MC = (1/1100)(6500) – 10/11 = 5
MR F = 5 => 100 – (2/100) Q F =5
Q F = 4750
P F = 52.5
MR W = 5 => 40 – (2/100) Q W =5
Q W = 1750
P F = 22.5
In fact we can verify that Q = Q F + Q W = 6500
Profit is just the total revenue from the two towns minus the total cost of producing 6500:
  TR  TC  (4750)(52.5)  (1750)(22.5)  (1 / 2200)(6500) 2  (10 / 11)(6500)  275454.5
c)
Q T = 10000 – 100P if P>=40
Q T = 14000 – 200P if P<=40
(when W’s demand = 0)
(when W’s demand >0)
The per-round fee price is the marginal cost of selling Q T . Using Q T when W’s demand
is positive:
(1/200)(14000 – Q) = (1/1100)Q – (10/11)
Q = 12000
MC = P (1/1100)12000 – (10/11) = 10
The membership fee is the consumer surplus under each demand curve, respectively: For
For W:
P = 40 – (½)Q =10 => Q=60
Fee = 900
For F:
P = 100 – Q =10 => Q=90
Fee = 4050
BPUB 250
Managerial Economics
Spring 2007
Final Exam
Instructions
1. Read over the whole exam, and do the easier questions first.
2. Show your work.
3. No credit will be given for work shown on even-numbered pages. Use
labeled “overflow space” for work that does not fit between questions.
4. Write legibly!
5. Initial the proctor’s class list when you hand in your exam.
6. Write your name on every page of this exam.
7. Do not disassemble the exam.
Your
Name:_____________________________________________________
Section number:_____________________________________________
Cohort name:_______________________________________________
TA name:__________________________________________________
Professor’s name:___________________________________________
1.
DO NOT START UNTIL 6PM.
(30 points) Elmo and Cookie Monster are the only two consumers of political
magazines. Elmo’s demand for magazines is P=100-q, and Cookie Monster’s is
P=100-2q. Magazines are competitively produced according to the supply
function P=20+2Q.
a. (5 points) What is the aggregate private market demand for magazines?
b. (5 points) What are the price and quantity of magazines consumed?
c.
(5 points) Elmo reads magazines in the morning and Monster in the
afternoon, so now they decide to purchase the magazines together and
share them. What is the marginal social benefit function of the new public
good “magazines”?
d. (5 points) What is the optimal level of provision of the public good?
e.
(10 points) Monster moves away, and Elmo is the only remaining
consumer. However, after Elmo reads and discards his magazines, Oscar
uses them to play. Oscar places a value of $20 on each magazine he finds
in the trash can. What is Elmo’s consumption of magazines? What is the
socially optimal consumption of magazines? Illustrate graphically,
labeling all relevant axes, quantities, and prices.
2.
a. (10 points) Firm 1 is the monopolist in the market of chandeliers. Its
production function is f ( K , L) = K ⋅ L . The cost of capital and labor are
r=4 and w=1. Demand is characterized by P=20-Q. What are firm 1’s
profits, based on its long-run total costs?
b. (15 points) Firm 1 faces the potential entry into the market of a new
competitor, firm 2. If firm 2 enters the market, Cournot competition will
ensue. Assume that firm 1’s cost structure now is:
1
TC1 = Q12 + 5
2
If firm 2 enters, its costs will be:
1
TC2 = Q22 + 40 .
2
Will firm 2 enter this market, assuming that it has not yet incurred any costs?
c. (10 points) Assume that firm 2 decides not to enter the market. Instead, a
new competitor threatens to enter. Should that firm decide to enter, firm 1
can respond in one of two ways: it can engage in a price war or it can
choose not to compete on price, but instead to sue the entrant for
infringement of copyrights. Firm 1 is certain to win the case, but litigating
is costly. Consider the sequential game where the potential entrant moves
first and the payoff matrix is given by:
2
POTENTIAL
ENTRANT
FIRM 1
PRICE
WAR
SUE
ENTER
STAY OUT
40 , 40
100 , 0
60-X , 60-4X
120-X , 0
Provide a range for X for firm 1 to successfully prevent entry.
3. Demand for tickets for a Penn football game are given by
q
PR = 20 − R for the general public, and
4,000
q
PS = 8 − S for students.
2,000
Penn requires student ID to qualify for a reduced student price. Suppose that both
marginal and fixed costs are zero.
a. (10 points) If the stadium has an unlimited number of seats, then what prices
( PS , PR ) maximize total profit? How many seats are sold to each group? What is
the deadweight loss?
b. (15 points) Suppose that one end of the stadium is closed for repairs so only
36,000 seats are available. What are profit-maximizing prices and quantities now?
c. (10 points) Penn faces some uncertainty over the demand for football tickets.
Student demand is very stable and, as above, is
q
PS = 8 − S .
2,000
However, the demand from the general public fluctuates and is given by
q
PR = 20 − R with probability 0.5 and
4,000
q
PR = 12 − R otherwise.
4,000
Penn decides to sell tickets to the general public at $10 and tickets to students at $4.
For logistical reasons, Penn dislikes fluctuation in attendance: its utility from the
profits of a football game is given by U= Π .
Compute Penn’s expected profit from a football game. How much expected profit would
Penn be willing to forego to remove the uncertainty in the demand for a game?
3
Question 1
a) We have to add the two demand curves horizontally (i.e. add the quantities):
Elmo:
q = 100-P
1
Monster: q = 50- P
2
3
2
Market: Q = 150- P Î P= 100- Q
2
3
2
b) Market equilibrium D = S: 100- Q = 20 + 2Q Î Q* = 30 Î P* = 80
3
(note that both consumers demand a positive quantity: Elmo 20, Monster 10)
c) To obtain the MSB function, we add the demand functions vertically (i.e. add the prices):
Elmo:
P = MB = 100 − q
P = MB = 100 − 2q
Monster:
MSB = 200 − 3Q
Sesame Street
For Q<50. If Q>50 only Elmo derives positive MB from more units and MBS=100-Q
d) We want to make MSB = S: 200-3Q = 20 + 2Q Î Q* = 36 .
e) Now Elmo’s consumption generates positive externalities for Oscar. The marginal social
value of consumption is Elmo’s valuation of the good, plus Oscar’s:
Elmo:
P = MB = 100 − q
MB = 20
Oscar:
Sesame Street:
MSB = 120 − Q
The market will set Elmo’s private MB=S: 100-Q = 20 + 2Q Î Q* = 26.66 Î P* = 73.33
The social optimum is at: MSB=S: 120-Q = 20 + 2Q Î Q* = 33.33
Positive externality
P
MSB
S
MPB=marginal private benefit
MSB= marginal social benefit
MPB
73.33
DWL
26.66 Q*=33.33
Q
Question 2
a) First, we need to find the cost function of the monopolist. To do that, remember the solution
to the cost minimization problem of the firm (to obtain demand for inputs):
1 12 − 12
K L
MPL w
w
w
K w
2
MRTS = or
= , and
= . Thus
= . Recall f ( K , L) = K ⋅ L , so
1 1
r
MPK r
L r
1 −2 2 r
K L
2
K 1
therefore
=
L 4
Substituting L = 4 K into the production function yields Q= f ( K , L) = 4 K 2 = 2 K . Thus
Q
and L = 2Q .
K=
2
The cost function is TC = rK + wL = 4 K + L . Using the input demand functions:
1
TC = 4 ⋅ Q + 2Q = 4Q . Marginal cost is therefore MC=4.
2
Next, using the demand function P=20-Q, we obtain total revenue:
Re v = PQ = (20 − Q) ⋅ Q = 20Q − Q 2 , and thus marginal revenue: MR = 20 − 2Q .
Profit maximization entails MR=MC, or 20 − 2Q = 4 , so Q*=8.
At Q=8:
Re v = 20 ⋅ 8 − 82 = 96
TC = 4 ⋅ 8 = 32
∏ = Re v − TC = 96 − 32 = 64
b) If firm 2 enters, Cournot competition ensues with Q = Q1 + Q2 .
For firm 1 Re v1 = PQ1 = (20 − Q) ⋅ Q1 = (20 − Q1 − Q2 ) ⋅ Q1 = 20Q1 − Q12 − Q2Q1 , and
MR1 = 20 − 2Q1 − Q2
From the cost function, we obtain MC1 = Q1
For firm 1’s reaction function, we set MR=MC: 20 − 2Q1 − Q2 = Q1 , so Q1 =
20 1
− Q2
3 3
Notice that both MC and MR are symmetric for firm 2 and therefore the reaction function of
firm 2 is:
Q2 =
20 1
− Q1
3 3
The Cournot equilibrium is the intersection of the reaction curves. Solving the system of
reaction functions yields Q1 = Q2 = 5 .
Firm 2 will enter the market if profits are positive upon entry:
1
∏ 2 = Re v2 − TC2 = (20 − Q1 − Q2 ) ⋅ Q2 − Q22 − 40
2
1
= (20 − 10) ⋅ 5 − ⋅ 25 − 40 = −2.5
2
Profits are negative. Firm 2 will not enter!
c) In order to deter entry, firm 1’s management has to convince the potential entrant that it will
sue. If it did not sue, the entrant would choose to enter and obtain profit of 40.
For the threat of litigation to be credible, the profits accruing to firm 1 from suing have to
exceed the profits from not suing. In terms of the payoff this implies that: 60 − X > 40 , or
X < 20 .
However, even if firm 1 sues, the potential entrant will enter if the payoff from entry is larger
than the payoff from not entering of 0. In order to deter entry, therefore, the payoff of the
entrant needs to be negative whenever firm 1 sues: 60 − 4 X < 0 , implying X > 15 .
The acceptable range for X is therefore 15 < X < 20
Question 3
a) Since seats are unlimited, Penn optimally prices as a monopolist over the two segments of
demand:
q ⎞
⎛
∏ S = Re vS = ⎜ 8 − S ⎟ qS
2, 000 ⎠
⎝
2 qS
MRS = 8 −
= MC = 0
2, 000
qS * = 8, 000
PS * = 8 −
8, 000
=4
2, 000
q ⎞
⎛
∏ R = Re vR = ⎜ 20 − R ⎟ qR
4, 000 ⎠
⎝
2qR
= MC = 0
MRR = 20 −
4, 000
qR* = 40, 000
PR* = 20 −
40, 000
= 10
4, 000
The associated deadweight loss is the deadweight loss due to monopoly pricing. The social
surplus is maximized under perfectly competitive prices, which in this case are such that PS =
PR = MC = 0. Total surplus equals:
(20)(80, 000)
= 800, 000
2
(8)(16, 000)
SURPLUS s = Π s + CS s = 0 +
= 64, 000
2
TOTAL = 864, 000
SURPLUS R = Π R + CS R = 0 +
Under the monopoly prices, surplus equals only:
(20 − 10)(40, 000)
SURPLUS R = Π R + CS R = (10)(40, 000) +
= 600, 000
2
(8 − 4)(8, 000)
SURPLUS s = Π s + CS s = (4)(8, 000) +
= 48, 000
2
TOTAL = 648, 000
The deadweight loss is the difference between the two surpluses, or 216,000.
b) Now Penn chooses quantities to maximize total profit, subject to the constraint that no more
than 36,000 seats are sold, or that qS * + qR* ≤ 36, 000 . The Lagrangian is given by:
q ⎞
q ⎞
⎛
⎛
L = ⎜ 8 − S ⎟ qS + ⎜ 20 − R ⎟ qR + λ ( 36, 000 − qR − qS )
2, 000 ⎠
4, 000 ⎠
⎝
⎝
The first-order conditions are”
(1)
(2)
(3)
2 qS ⎞
∂L ⎛
= ⎜8 −
−λ = 0
∂qS ⎝
2, 000 ⎟⎠
2 qR ⎞
∂L ⎛
= ⎜ 20 −
−λ = 0
∂qS ⎝
4, 000 ⎟⎠
∂L
= 36, 000 − qR − qS = 0
∂λ
2 qS ⎞ ⎛
2qR ⎞
⎛
= ⎜ 20 −
.
Equations (1) and (2) imply that MRS=MRR, or that ⎜ 8 −
⎟
2, 000 ⎠ ⎝
4, 000 ⎟⎠
⎝
Solving for qS as a function of qR:
2 qS
2qR
= 8 − 20 +
2, 000
4, 000
2qS = (−12)(2, 000) + qR
1
qS = −12, 000 + qR
2
Substituting this expression for qS into the constraint (equation (3)) yields qR:
1 ⎞
⎛
qR + ⎜ −12, 000 + qR ⎟ = 36, 000
2 ⎠
⎝
3
qR = 48, 000
2
qR* = 32, 000
1
⎛
⎞
qS * = ⎜ −12, 000 + qR* ⎟ = −12, 000 + 16, 000 = 4, 000
2
⎝
⎠
The associated prices are:
PR* = 20 −
32, 000
= 12
4, 000
PS * = 8 −
4, 000
=6
2, 000
c) Penn’s expected profit is given by Π = PS qS + EΠ R . Given the chosen prices of PS= 4 and
PR=10, the number of tickets sold equals:
qS
⇒ qS = (8 − PS )2, 000 = (8 − 4)2, 000 = 8, 000
2, 000
q
PR1 = 20 − R ⇒ q1R = (20 − PR1 )4, 000 = (20 − 10)4, 000 = 40, 000
4, 000
q
PR2 = 12 − R ⇒ q1R = (12 − PR1 )4, 000 = (12 − 10)4, 000 = 8, 000
4, 000
PS = 8 −
Expected profit thus equals:
E Π = PS qS + 0.5( PR q1R ) + 0.5( PR qR2 )
= (4)(8, 000) + (0.5)(10)(40, 000) + (0.5)(10)(8, 000)
= $272, 000
Expected utility equals:
EU = 0.5 PS qS + PR q1R + 0.5 PS qS + PR qR2
= 0.5 32, 000 + 400, 000 + 0.5 32, 000 + 80, 000
= 495.97
The certainty equivalent amount of profit is:
2
Π cert = ( 495.97 ) = $245,981.82
Penn would be willing to forego up to ($272,000 – $245,981.82), or $26,018.18, to avoid the
demand uncertainty.
BPub 250. SECOND MID TERM EXAMINATION
Name ……………………………… ID…………………………….. Question 1 . John and Jane are married. They like spending time together and they each like playing
bridge and playing tennis. The problem is that, although John likes tennis, he prefers bridge. Similarly,
although Jane likes bridge, she prefers tennis. Each Friday morning, each wakes up and decides what they
will do that evening, bridge or tennis. These are independent decisions, made without consulting each
other.
The following are the payoffs in units of pleasure




If both play bridge, John gets 20 units of pleasure and Jane gets 10
If both play tennis, John gets 10 units of pleasure and Jane gets 20
If John plays tennis and Jane plays Bridge, both get 5 units of pleasure
If John plays bridge and Jane plays tennis, both get 11 units of pleasure
a.
b.
c.
d.
Draw the matrix form for this simultaneous game.
On the typical week, what do you think they would each play?
What kind of equilibrium have you described?
If they continue deciding in this way for a whole year, what is the average weekly level of
pleasure for each spouse?
Name ………………………………
ID……………………………..
Question 2. The Tropic country of Believze, has wonderful barrier reef and is perfect for Scuba Diving.
However, there are currently no hotels. The demand for hotel rooms is as follows
P = 140 – 0.1Q
where Q is the number of rooms demanded and the price per night is P.
Two hotel chains wishing to enter are, Diving Digs and Underwater Adventures. Diving Digs estimates
that its total cost of operation would be
TC D = 100 + 5Q D + 0.5Q D 2,
Underwater Adventures estimates that its total cost market would be
TC U = 200 + 35.6Q U
Note that Q D + Q U = Q is the total number of rooms supplied by both hotels. These firms are old rivals in
other markets and know each others’ cost functions. The firms are profits maximizers and they enter the
market simultaneously. Collusion is banned in Beleivze.
a. What the REACTION FUNCTIONS of the two firms?
b. What are the outputs of the two firms and the price?
Name ………………………………
ID……………………………..
Question 3: You own a small business and hire a manager to run it on your behalf. While you are risk
neutral (thus seek to maximize expected profit), the manager is risk averse and her utility function is:
U =
U =
OF EFFORT)
– 8
if she provides low effort in her work
if she provides high effort in her work (THE “-2” IS THE DISUTILITY
Your profits (BEFORE DEDUCTING THE MANAGER’S COMPENSATION) are as follows
Poor economy
Good Economy
Low Effort from Manager
2,000
5,600
High Effort from Manager
3,125
5,600
Notice that profits depend on both the manager’s effort and the state of the economy. There is a 50% chance of a good economy and a 50% chance of a poor economy. You are thinking about two compensation plans for the manager PLAN A 50% of profits (as shown in the table above) PLAN B 80% of profits (as shown in the table above) in excess of 2,000 (e.g. 0 if profit < 2000; 0.8(500) if profit = 2500 ; 0.8(1000) if profit = 3000) a. Which plan would you offer the manager? b. Given the plan you choose and the level of effort chosen by the manager under that plan, what is the CERTAINTY EQUIVALENT level of compensation (assuming effort is as you specified in part a) and what is the RISK PREMIUM built into the compensation? Name ………………………………
ID……………………………..
Question 4. Firms A and B both produce chemicals but emit toxic gasses into the atmosphere.
Assume that the optimal level of chemical production produces 20 units of emissions. Now firms A and
B can both reduce emissions and the following functions describe their Marginal Costs of Abatement
MCA A = 20 - 1E A
MCA B = 20 - 2E B
The government decides to grant 10 tradable pollution permits to each firm (making a total of 20 permits)
each permit grants the holder the right to discharge one unit of emissions E.
a. After trading pollution permits between each other, how many permits will firms A and
B each end up holding?
b. What will be the price of these permits?
c. What will the total abatement costs of each firm be?
The marginal benefit to firm A from producing chemicals is
MB A = 100 – 2Q A
Where Q A is the quantity of CHEMICALS produced by firm A.
Note Emissions, E, are measured in quite different units to the Q’s above so don’t get them confused.
And the marginal benefit to Firm B from producing chemicals is
MB B = 100 – 4Q B
Where Q B is the quantity of CHEMICALS produced by firm A. The marginal social cost of emissions
from these chemicals is given by
TC = 10 + 3⅔ Q
where Q = Q A + Q B
d. What is the socially optimal quantity of production if chemicals?
Name ………………………………
ID……………………………..
Question 5. It is unfortunate that Violent Video Arcade (VVA) and Totally Zen (TZ) are located across
the street from each other. The profit for VVA is 1000 and that for TZ would also be 1000 if it were not
for the awful noise from VVA. With the noise, the customers of TZ are freaked out and this reduces TZ
demand and therefore profit from 1000 to 200.
This problem could be mostly solved if VVA were to soundproof its premises but this would cost VVA
400. However, the noise reduction would increase TZ profits from 200 to 900 (almost as good as if VVA
were not there).
Alternatively, TZ could employ BOSEO noise cancelling technology in its premises which would almost
restore the tranquility required by TZ. clients. This technology would cost 500 and would also increase
TZ profits from 200 to 900 (less of course the 500 paid to Boseo)
If VVA introduced the soundproofing and TZ introduced noise cancelling technology, then TZ profits
would be restored to 1000 (minus any expense they incurred to reduce noise).

The following scenarios can address the problems and are summarized by the following table
which yield the following profits:
VVA profit
TZ profit
Scenarios
Business as usual
10000
200
VVA can close down
0
1000
TZ can close down
1000
0
VVA can soundproof
1000 - 400
900
TZ can introduce noise
reduction technology
1000
900 - 500
Soundproof & Noise
cancelling
1000 - 400
900 - 500
a. Which is the most efficient solution?
b. If VVA is required to compensate TZ for any losses incurred through noise, which
outcome will occur (explain and show how much profit each firm will make)
c. If VVA is NOT required to compensate TZ for any losses incurred through noise, which
outcome will occur (explain and show how much profit each firm will make)
Answer question 1 a. Jane plays Bridge Jane plays Tennis John plays Bridge 10 20 11 11 b. John plays bridge and Jane plays Tennis c. Nash d. 11 units each John plays Tennis 5 5 20 10 Answer to Problem 2: Rewrite the market demand curve as P = 140 – 0.1(Q D + Q U ) Diving Dig’s total revenue will be TR D = PQ D = (140 – 0.1Q D – 0.1Q U )Q D Or = 140Q D – 0.1Q D 2– 0.1Q U Q D . MR D = ∂TR D /∂Q D = 140 – 0.2Q D – 0.1Q U . MC D = dTC D /dQ D = 5 + Q D . To maximize profits, DD will set MR D = MC D or MR D = 140 – 0.2Q D – 0.1Q U = 5 + Q D = MC D Or 1.2Q D = 135 – 0.1Q U Or Q D = 112.5 – (1/12)Q U This is DD’s reaction function. UA’s total revenue will be TR U = PQ U = (140 – 0.1Q D – 0.1Q U )Q U Or = 140Q U – 0.1Q D Q U – 0.1Q U 2 MR U = ∂TR U /∂Q U = 140 – 0.1Q D – 0.2Q U . MC U = dTC U /dQ U = 35.6 To maximize profits, UA will set MR U = MC U or MR U = 140 – 0.1Q D – 0.2Q U = 35.6 = MC D Or 0.2Q U = 104.4 – 0.1Q D Or Q U = 522 – (1/2)Q D This is UA’s reaction function Substituting UA’ reaction function into DD’s reaction function gives: Q D = 112.5 – (1/12)( 522 – (1/2)Q D ) = 112.5 – 43.5 + (1/24)Q D Or (23/24)Q D = 69 Or Q D = 72 Substituting Q D = 72 into UA’ reaction function gives Q U = 552 – (1/2)(72) = 552 – 36 = 486 Thus, Q = Q D + Q U = 72 + 486 = 558. Substituting Q = 558 into the demand function yields P = 140 – 0.1(558) = 140 – 55.8 = 84.2 Answer 3a. PLAN A 0.5(PROFIT) Lo effort EU = 0.5(0.5[2,000])0.5 + 0.5(0.5[5,600])0.5 = 42.2689 Hi effort EU = 0.5(0.5[3,125])0.5 + 0.5(0.5[5,600])0.5 ‐ 8 = 38.2217 Therefore, MANAGER CHOOSES HIGH EFFORT since 42.2689 > 38.2217 PROFIT NET OF COMPENSATION 0.5 (2000 – 1000) + 0.5(5,600– 2,800) = 1900 _____________________________________________________________________________ PLAN B 0.8(PROFIT IN EXCESS OF 1,000) Lo effort EU = 0.5(0)0.5 + 0.5(0.8[5,600– 2,000])0.5 = 26.833 Hi effort EU = 0.5(0.8[3,125‐2,000])0.5 + 0.5(0.8[5,600– 2,000])0.5 ‐ 8 = 33.8328 Thus, the MANAGER CHOOSES HI EFFORT because 33.8328 > 26.833 PROFIT NET OF COMPENSATION IS = 0.5(3,125 – 0.8(1,125)) + 0.5(5,600– 0.8(3,600)) = = 0.5(2,225) + 0.5(2,720) = 2,472.5 The owner is better off with PLAN B because their expected profit is 2,472.5 instead of their expected profit of 1900 with PLAN A. Answer b. With PLAN B the expected utility is 33.8328 and the manager works hard. Thus we can estimate the CE as follows ‐ 8 33.8328 = Therefore CE = 1749.983 The expected compensation is 0.5(0.8[1125]) + 0.5(0.8[3,600]) = 1890 The risk premium is the difference between the CE and the expected compensation RP = 1890– 1749.983= 140.017 Question 4. Answer a‐c. Note each unit of E requires 1 permit. The firms will trade permits until their marginal costs of abatement are equal MCA A = 20 ‐ 1E A = 20 ‐ 2E B = MCA B However E A + E B = 20 Solving the simultaneous equations 20 ‐ 1E A = 20 ‐ 2(20 ‐ E A ) E A = 40/3; E B = 20/3; Note MCA A = 20 – 1(40/3)= 20/3; Note MCA B = 20 – 2(20/3) = 20/3; Answer b. The price of permits will equal MCA = 20/3. Answer c. Total abatement costs for A are the area underneath the curve from 40/3 to 20 which is a triangle of height (20/3) and length (20/3) which has an area of (200/9). Total abatement costs for B are the area underneath the curve from 20/3 to 10, which is a triangle of height (20/3) and length (10/3), and therefore has an area of (100/9). Total abatement costs are (300/9)=33.3333. Answer d. Rewrite the two MB curves as follows Q A = 50 – 0.5 MB Q B = 25 – 0.25 MB Now add to get the TOTAL private MB Q A = 50 – 0.5 MB Q B = 25 – 0.25 MB ∑Q = 75 – 0.75 MB Now social optimum where MB = MC 100 – 4/3 Q = 10 +11/3 Q Or Q = 18 MB = 100 – 4/3 Q Question 5 Answer Business as usual
VVA profit
10000
TZ profit
200
Total
1200
VVA can close down
0
1000
1000
TZ can close down
1000
0
1000
VVA can soundproof
1000 - 400
900
1500
TZ can introduce noise
reduction technology
1000
900 - 500
1400
Soundproof & Noise
cancelling
1000 - 400
900 - 500
1000
a. VVA soundproof b. Iv VVA is liable to compensate TZ, then VVA can either  pay 800 in compensation or  it can spend 400 in soundproofing and pay 100 in compensation  it will prefer the latter Business as usual
VVA profit
10000 - 800
TZ profit
200 + 800
Total
1200
VVA can close down
0
1000
1000
TZ can close down
1000
0
1000
VVA can soundproof
1000 – 400 - 100
900 + 100
1500
TZ can introduce noise
reduction technology
1000
900 - 500
1400
Soundproof & Noise
cancelling
1000 - 400
900 - 500
1000
c. d. VVA Not liable. TZ pays an amount “P” somewhere between 400 and 500 to VVA to persude the latter to soundproof Business as usual
VVA profit
10000
TZ profit
200
Total
1200
VVA can close down
0
1000
1000
TZ can close down
1000
0
1000
VVA can soundproof
1000 – 400+ P
900 - P
1500
TZ can introduce noise
reduction technology
1000
900 - 500
1400
Soundproof & Noise
cancelling
1000 - 400
900 - 500
1000
BPUB 250: Final Spring 2008
Question 1: Consumers’ demand for air travel between St. Cloud and Duluth is governed
by the following function: P = 100-Q. Two carriers, North Central and Republic, serve
the route. They each have the following total cost function: TC = 800 + 10*q, where q
represents each firm’s output (the number of passengers flown).
Part a (5 points): The production function for air services is Q = K*F, where Q is units
of output, K is units of capital, and F is units of fuel. This production function holds for
both North Central and Republic. They both have 10 units of capital and the price of a
unit of capital is 80. The price of a unit of fuel is 100. Show that each carrier’s total
cost function is TC = 800 + 10*q.
Part b (10 points). Find the industry equilibrium total output, price, and each firm’s
profit supposing that just these two firms serve the route and that they make their
output decisions simultaneously and independently. Illustrate the quantity choices
in a diagram with North Central’s output on the vertical axis and Republic’s output
on the horizontal axis and label relevant values on each axis and curves.
Part c (5 points). Fuel costs rise substantially. North Central has an older fleet that is
less fuel-efficient. With the new fuel prices North Central’s total cost function is now
TC = 800 + 20q, while Republic’s is unchanged. Find the new equilibrium quantity,
prices, and profit in the short run.
Part d (5 points). If fuel prices remain at their new levels – and assuming that no firms
with lower costs than North Central stand ready to enter this market – what do you
expect to happen to prices, quantities, and profits in this market in the longer run if
participation requires 10 units of capital? (Give quantitative answers).
Part e (10 points). Suppose we are back to the initial situation with both firms with the
same cost function, i.e., TC = 800 + 10*q, but that North Central moves first in
announcing how many passengers it will serve on the St. Cloud-Duluth route. (Note that
Minnesota state law requires that once output is determined on a route, it cannot be
changed). Find the equilibrium total output, price, and profit for each firm.
Answer 1a: Note that
Or
q = K*F
F = q/K
The total cost of a carrier is
TC = P K *K + P F *F
Substituting in P K = 80, P F = 100, K = 10, and F = q/K = q/10
TC = 80*10 + 100*(q/10) = 800 + 10q
Answer 1b. Write the demand as P = 100 – q N – q R .
North Central’s total revenue is
TR N = P*q N = (100 – q N – q R )q N = 100q N – q N 2 – q R q N
North Central’s marginal revenue is
MR N = ∂TR N /∂q N = 100 – 2q N – q R
North Central’s marginal cost is
MC N = dTC N /dq N = 10
To maximize profit, North Central will set MR N = MC N or
Or
Or
MR N = 100 – 2q N – q R = 10 = MC N
2q N = 90 – q R
q N = 45 – 0.5q R
This is North Central’s reaction function.
Republic’s total revenue is
TR R = P*q R = (100 – q N – q R )q R = 100q R – q N q R - q R 2
Republic’s marginal revenue is
MR R = ∂TR R /∂q R = 100 – q N – 2q R
Republic’s marginal cost is
MC R = dTC R /dq R = 10
To maximize profit, Republic will set MR R = MC R or
MR R = 100 – q N – 2q R = 10 = MC R
2q R = 90 – q N
q R = 45 – 0.qQ N
Or
Or
This is Republic’s reaction function.
Substituting Republic’s reaction function into North Central’s reaction function gives
q N = 45 – 0.5(45 – 0.5q N ) = 45 – 22.5 + 0.25q N = 22.5 + 0.25q N
0.75q N = 22.5
q N = 30
Or
Or
Substituting q N = 30 into Republic’s reaction function gives
q R = 45 – 0.5*30 = 45 – 15 = 30
Thus, Q = q N + q N = 30 + 30 = 60
Substituting Q = 60 into the demand curve gives
P = 100 – 60 = 40
North Central’s total revenue = TR N = P*q N = 40*30 = 1,200
North Central’s total cost = TC N = 800 + 10q N = 800 + 10*30 = 800 + 300 = 1,100
North Central’s profit is = TR N – TC N = Π N = 1,200 – 1,000 = 100
Republic’s total revenue = TR R = P*q R = 40*30 = 1,200
Republic’s total cost = TC R = 800 + 10q R = 800 + 10*30 = 800 + 300 = 1,100
Republic’s profit is = TR R – TC R = Π R = 1,200 – 1,000 = 100
qN
90
Republic’s reaction function
45
30
North Central’s reaction function
30
45
90
qR
Answer 1c. With the higher fuel cost, North Central’s marginal cost is
dTC N /dq N = 20 = MC N
To maximize profit, North Central will set MR N = MC N or
Or
Or
MR N = 100 – 2q N – q R = 20 = MC N
2q N = 80 – q R
q N = 40 – 0.5q R
This is North Central’s reaction function.
Substituting Republic’s reaction function into North Central’s reaction function gives
Or
Or
q N = 40 – 0.5(45 – 0.5q N ) = 40 – 22.5 + 0.25q N = 17.5 + 0.25q N
0.75q N = 17.5
q N = 23.33
Substituting q N = 23.33 into Republic’s reaction function gives
q R = 45 – 0.5*23.33 = 45 – 11.67 = 33.33
Thus, Q = q N + q N = 23.33 + 33.33 = 56.67
Substituting Q = 56.67 into the demand curve gives
P = 100 – 56.67 = 43.33
North Central’s total revenue = TR N = P*q N = 43.33*23.33 = 1,066.67
North Central’s total cost = TC N = 800 + 10q N = 800 + 20*23.33
= 800 + 466.67 = 1,266.67
North Central’s profit is = TR N – TC N = Π N = 1,066.67 – 1,266.67 = -255.56
Republic’s total revenue = TR R = P*q R = 43.33*33.33 = 1,444.44
Republic’s total cost = TC R = 800 + 10q R = 800 + 10*33.33 = 800 + 333.33 = 1,133.33
Republic’s profit is = TR R – TC R = Π R = 1,444.44 – 1,133.33 = 311.11
Answer d. Republic would act as a monopolist. Their total revenue would be
TR = P*Q = (100 – Q)Q = 100Q – Q2
Their marginal revenue would be
MR = dTR/dQ = 100 – 2Q
To maximize profit, they would set MR = MC or
Or
Or
MR = 100 – 2Q = 10 = MC
2Q = 90
Q = 45
Substituting Q = 45 into the demand curve gives
P = 100 – 45 = 55
Republic’s total revenue = P*Q = 55*45 = 2,475
Republic’s total cost = 800 + 10*45 = 800 + 450 = 1,250
Republic’s profit = TR R – TC R = Π R = 2,475 – 1,250 = 1,225
Suppose a new entrant decided to come into the market. Since their marginal cost of
operating is 20, it would appear that they would do well with the monopoly price of 55.
But their reaction function would be
q NE = 40 – 0.5q R
Substituting q R = 45 into the entrant’s reaction function gives
q NE = 40 – 0.5*45 = 40 – 22.5 = 17.5
Then Q = q R + q NE = 45 + 17.5 = 62.5
Substituting Q = 62.5 into the market demand function gives
P = 100 – 62.5 = 37.5
The potential entrant’s total revenue = TR NE = P*q NE = 37.5*17.5 = 656.25
The potential entrant’s total cost = TC NE = 800 + 20q NE = 800 + 20*17.5
= 800 + 350 = 1,150
The potential entrant’s profit is = TR NE – TC NE = Π NE = 656.25 – 1,150 = -493.75
Thus, the potential new entrant stays out, and Republic has P = 55, Q = 45, and Π =
1,255. So price increases, quantity decreases, and profit increases.
Answer 1e: North Central would substitute Republic’s reaction function into the market
demand curve, i.e.,
P = 100 – q N – q R = 100 – q N – (45 – 0.5q N ) = 55 – 0.5q N
North Central’s total revenue is
TR N = P*q N = (55 – 0.5q N )q N = 55q N – 0.5q N 2
North Central’s marginal revenue is
dTR N /dq N = MR N = 55 – q N
To maximize profit, North Central will set MR N = MC N or
Or
MR N = 55 – q N = 10 = MC N
q N = 45
Substituting q N = 45 in Republic’s reaction function gives
q R = 45 – 0.5*45 = 45 – 22.5 = 22.5
Then Q = q N + q R = 45 + 22.5 = 67.5
Substituting Q = 67.5 into the market demand curve gives
P = 100 – 67.5 = 32.5
North Central’s total revenue = TR N = P*q N = 32.5*45 = 1,462.5
North Central’s total cost = TC N = 800 + 10q N = 800 + 10*45
= 800 + 450 = 1,250
North Central’s profit is = TR N – TC N = Π N = 1,462.5 – 1,250 = 212.5
Republic’s total revenue = TR R = P*q R = 32.5*22.5 = 731.25
Republic’s total cost = TC R = 800 + 10q R = 800 + 10*22.5 = 800 + 225 = 1,025
Republic’s profit is = TR R – TC R = Π R = 731.25 – 1,025 = -293.75
Thus, a first mover advantage would doom the follower carrier to a loss position.
Question 2: Ace Industries produces a machine that is a critical input on the assembly
line for the production of widgets and is used by the many identical producers in the
widget industry. If Ace’s machine suffers no failures, each producer will earn $640,000
in profit. However, if Ace’s machine fails, each producer will only make $360,000 in
profit, i.e., the widget market can’t produce as much because of down time entailed in
repairing Ace’s machine. Ace’s machine has a 20% chance of failure.
Each widget producer is risk averse with a utility function (U) of
U = Π0.5
where Π is the profit of the widget producer.
Part a (2 points). What is a typical widget firm’s expected profit?
Part b (8 points). What is a typical widget firm’s expected utility? Illustrate with a
diagram with utility on the vertical axis and profit on the horizontal axis. Label
appropriate values on each axis.
Part c (8 points). What is the certainty equivalent of being uninsured? What is the
maximum amount that a typical widget firm would be willing to pay to eliminate the
risk of faulty input from Ace?
Suppose now that Ace offers to warranty its machine. That is, Ace will pay a widget
company all of the widget company’s lost profit if Ace’s machine fails. Ace will sell this
warranty at the time they sell their machine. Suppose that Ace is risk neutral.
Part d (2 points). What is Ace’s expected payout to a typical widget firm if Ace
offers this warranty?
Part e (2 points). If Ace is the only producer of this machine, what is Ace’s expected
profit from issuing such a warranty?
Part f (8 points). Instead of fully covering the widget producers from loss of profit, Ace
considers an alternative: a warranty with a $10,000 deductible (a $10,000 deductible
means that Ace pays for all the damages due to the failure of its machine except for
$10,000). How would you determine the maximum price a widget producer would
pay for a warranty with a $10,000 deductible assuming that the alternative is being
uninsured? You don’t have to solve for the price P, just set the problem up.
Answer 2a: A typical widget firm’s expected profit is
E(Π) = 0.8*640,000 + 0.2*360,000 = 512,000 + 72,000 = 584,000
Answer 2b: A typical widget firm’s expected utility is
E(U) = 0.8*(640,000)0.5 + 0.2(360,000)0.5 = 0.8*800 + 0.2*600 = 640 + 120 = 760
Utility
800
760
Certainty
Equivalent
600
360,000
577,600 584,000 640,000
Profit
Answer 2c: The certainty equivalent of a typical widget firm is 7602 =577,600. Thus, the
typical widget firm would have expected utility of 760 if their profit were 577,600 for
certain. Therefore, the typical widget firm would pay up to 640,000 – 577,600 =
62,400 for such a warranty. Notice this payment would guarantee that they make
577,600 with no risk, i.e.,
E(Π) = 0.8(640,000 – 62,400) + 0.2(640,000 – 62,400 – 280,000 + 280,000)
= 0.8(577,600) + 0.2(577,600) = 577,600
and would guarantee that they get 760 in utility with no risk, i.e.,
E(U) = 0.8(640,000 – 62,400)0.5 + 0.2(640,000 – 62,400 – 280,000 + 280,000)0.5
= 0.8(577,600)0.5 + 0.2(577,600)0.5 = 0.8*760 + 0.1*760 = 760
Answer 2d: If Ace’s machine fails, they owe the typical widget firm
640,000 - 360,000 = 280,000
Ace’s expected payout to a typical widget firm is
E(Payout) = 0.8(0) + 0.2(280,000) = 56,000
Answer 2e: Ace will charge each widget company 62,400 (minus ε) for the warranty and
they expect to pay each widget company 56,000, so Ace’s expected profit from each
warranty is
E(Π Ace ) = 62,400 – 56,000 = 6,400
Answer 2f: As shown above, without any warranty, the widget firm has expected utility
of 760 (but with lots of risk). If Ace can come up with a warranty that takes the widget
firm to an expected utility of 760 with perfect certainty, the widget firm will buy it. Thus
760 = 0.8(640,000 – P)0.5 + 0.2(640,000 – P – 280,000 + 270,000)0.5
= 0.8(640,000 – P)0.5 + 0.2(630,000 – P)0.5
where P is the price of the warranty with a deductible of $10,000. Solve for P
(and take ε off)
Question 3: Suppose the demand for used cars at Honest Sam’s Used Car Emporium can
be expressed as
P = 10 – Q for 8 < P < 10
P = 9 – 0.5Q for P < 8
where P is the price of a car (in thousands) and Q is the number of cars demanded.
Assume that cars can be sold in non-integers and all are the same (in terms of make,
vintage, and quality). Honest Sam has monopoly power because he’s located in the
middle of North Dakota and it’s a long way to the next used car dealer.
Sam is contemplating how to sell his cars. Should he negotiate with each potential
customer, attempting to estimate and then charge the customer his/her reservation price
for the car or should he just post a price of P* on each car and tell the customers to “take
it or leave it” at that price. It costs Sam a constant $2 (two thousand) to sell each car, i.e.,
his marginal cost is a constant $2/car, under the “take it or leave it” pricing policy.
Alternatively, it costs Sam a constant $4 (four thousand) to sell each car under the
“negotiate with each buyer” pricing policy (because it now takes more time to deal with
each customer and Sam has to employ more salespeople). For simplicity, Sam has no
fixed cost.
Part a (8 points). If he is going to post the same price on each car in the “take it or
leave it” context, what price should he post, how many cars will he sell, and what
will his profit be? Illustrate the demand curve and label appropriate values on each
axis.
Part b (5 points). Suppose his daughter returns home after her freshman year at
Wharton, investigates Sam’s market, and finds that the market demand curve can be
divided into two separate segments: segment one has demand expressed by P = 10 – Q 1
and segment two has demand of P = 8 – Q 2 , where Q 1 and Q 2 are the number of cars
demanded by members of each segment when the price of a car is P. Show how these
demand curves relate to the demand information given in the problem above.
Part c (8 points). It still costs Sam a constant $2/car to sell a car to a customer in either
segment. If Sam uses his daughter’s analysis to charge a separate “take it or leave it”
price in each segment, what price should he charge in each segment, how many cars
will he sell in each segment, and what will his profit be? Assume that he can keep the
two segments separate.
Part d (8 points). Suppose that Sam decides to go with the “negotiate with each buyer”
model of selling cars and suppose he is very good and can determine each demander’s
reservation price perfectly. However, his marginal cost of selling a car now increases to a
constant $4/car because of the additional time/expense involved in closing a sale. How
many cars will he sell and what will his profit be?
Part e (2 points). What pricing method (of the three above) will Sam choose? Why?
Part f (4 points). What method of selling cars (of the three above) would consumers
as a whole prefer? Explain.
Answer 3a: Sam’s total revenue is either
Or
TR = P*Q = (10 – Q)Q = 10Q – Q2
TR = P*Q = (9 – 0.5Q)Q = 9Q – 0.5Q2
Yielding marginal revenue of either
Or
MR = dTR/dQ = 10 – 2Q
MR = dTR/dQ = 9 – Q
Setting MR = MC for the first MR gives
Or
Or
MR = 10 – 2Q = 2 = MC
2Q = 8
Q=4
Substituting Q = 4 into the demand curve gives
P = 10 – 4 = 6
This cannot be the correct marginal revenue curve because it comes from a demand curve
that’s only relevant when 8 < P < 10 and P = 6 is not in this range.
Setting MR = MC for the second MR gives
Or
MR = 9 – Q = 2 = MC
Q=7
Substituting Q = 7 into the demand curve gives
P = 9 – 0.5*7 = 9 – 3.5 = 5.5
This is the correct marginal revenue curve because it comes from a demand curve that’s
relevant when P < 8 and P = 5.5 is in this range.
Sam’s total revenue is TR = P*Q = 5.5*7 = 38.5
Since Sam’s MC = 2, his TC must be TC = 2Q since dTC/dQ = 2 so Sam’s ATC = TC/Q
= 2Q/Q = 2
Sam’s total cost is
TC = ATC*Q = 2*Q = 2*7 = 14
So Sam’s profit is Π = TR – TC = 38.5 – 14 = 24.5
$
10
P = 10 – Q
Kink in demand curve
9
8
P = 9 – 0.5Q
5.5
MR = 9 - Q
2
MC
2
7
9
18
Q
Answer 3b. Rewrite the demand in segment one as
Q 1 = 10 – P 1
and rewrite the demand in segment two as
Q2 = 8 – P2
Then the market demand is Q = Q 1 + Q 2 or
Q = 10 – P + 8 – P = 18 – 2P
Rewrite this as
Or
2P = 18 – Q
P = 9 – 0.5Q
The latter only holds when we can add Q 1 and Q 2 , i.e., for prices less than or equal to 8
(there is no Q 2 for prices greater than 8). For prices between 8 and 10, only segment one
demands cars and hence the market demand is P = 10 – Q for 10 > P > 8.
Answer 3c. Sam will be practicing third degree price discrimination in this case. The
total revenue from the first segment will be
TR 1 = P 1 *Q 1 = (10 – Q 1 )Q 1 = 10Q 1 – Q 1 2
The marginal revenue in the first segment will be
MR 1 = dTR 1 /dQ 1 = 10 – 2Q 1
Setting this equal to marginal cost gives
Or
Or
MR 1 = 10 – 2Q 1 = 2 = MC
2Q 1 = 8
Q1 = 4
Substituting Q 1 = 4 into segment 1’s demand curve gives
P 1 = 10 – 4 = 6
Sam’s total revenue from segment 1 is
TR 1 = P 1 *Q 1 = 6*4 = 24
Sam’s total cost from segment 1 is
TC 1 = AVC*Q 1 = 2*4 = 8
Sam’s profit from segment 1 is
Π 1 = TR 1 – TC 1 = 24 – 8 = 16
The total revenue from the second segment will be
TR 2 = P 2 *Q 2 = (8 – Q 2 )Q 2 = 8Q 2 – Q 2 2
The marginal revenue in the second segment will be
MR 2 = 8 – 2Q 2
Setting this equal to marginal cost gives
Or
Or
MR 2 = 8 – 2Q 2 = 2 = MC
2Q 2 = 6
Q2 = 3
Substituting Q 2 = 3 into segment 2’s demand curve gives
P2 = 8 – 3 = 5
Sam’s total revenue from segment 2 is
TR 2 = P 2 *Q 2 = 5*3 = 15
Sam’s total cost from segment 2 is
TC 2 = AVC*Q 2 = 2*3 = 6
Sam’s profit from segment 2 is
Π 2 = TR 2 – TC 2 = 15 – 6 = 9
Thus, Sam’s profit from third degree price discriminating is
Π = Π 1 + Π 2 = 16 + 9 = 25
Answer 3d. Sam will be first degree price discriminating under this negotiation policy.
The demand curve becomes his marginal revenue curve and he will produce where MR =
MC. Thus,
MR = 9 – 0.5Q = 4 = MC
0.5Q = 5
Q = 10
Or
Or
Sam will collect revenues equal to the area under the demand curve out to Q = 10 and
incur costs equal to
TC = ATC*Q = 4*10 = 40
$
10
A
8
B
C
4
D
E
2
10
The revenues are A + B + C + D + E
A = 0.5(10 – 8)2 = 2
18
Q
B + D = 8*2 = 16
C + E = 0.5(8 + 4)8 = 48
So A + B + C + D + E = 2 + 16 + 48 = 66
Profit under perfect discrimination is Π = TR – TC = 66 – 40 = 26
Answer 3e. Since profit under first degree (26) exceeds profit under third degree
(25) exceeds profit under simple monopoly pricing (24.5), Sam will choose the
negotiating (first degree strategy)
Answer 3f. Sam extracts all consumer surplus under the negotiating (first degree
strategy) so consumers will not like that.
Under third degree price discrimination, the consumer surplus in segment 1 is
CS 1 = 0.5(10 – 6)4 = 8
and the consumer surplus in segment 2 is
CS 2 = 0.5(8 – 5)3 = 4.5
for a total consumer surplus of CS = CS 1 + CS 2 = 8 + 4.5 = 12.5
Under simple monopoly pricing, we can see the consumer surplus in the figure below
$
10
A
8
B
C
5.5
D
E
2
CS = A + B + C
where
7
18
Q
A = 0.5(10 – 8)2 = 2
B = (8 – 5.5)2 = 5
C = 0.5(8 – 5.5)(7 – 2) = 6.25
So consumer surplus under simple monopoly pricing is CS = A + B + C = 2 + 5 + 6.25 =
13.25
Since 13.25 > 12.5 > 0, the consumers as a whole would prefer simple monopoly
pricing.
Note that at the simple monopoly price of 5.5, the segment 1 consumers would demand
Or
Thus, they would obtain a CS of
5.5 = 10 – Q 1
Q 1 = 4.5
CS 1 = 0.5(10 – 5.5)4.5 = 10.125
Note that at the simple monopoly price of 5.5, the segment 2 consumers would demand
5.5 = 8 – Q 2
Q 2 = 2.5
Or
Thus, they would obtain a CS of
CS 2 = 0.5(8 – 5.5)2.5 = 3.125
This gives, as above, total consumer surplus of CS = CS 1 + CS 2 = 10.125 + 3.125 =
13.25
But under third degree price discrimination, the segment 1 consumers received consumer
surplus of 8 and segment 2 consumers received consumer surplus of 4.5. Thus, while
consumers as a whole prefer simple monopoly pricing to third degree pricing, segment 1
consumers prefer third degree (consumer surplus bigger by 10.125 – 8 = 2.125) and
segment 2 consumers prefer third degree to simple monopoly (consumer surplus bigger
by 4.5 – 3.125 = 1.375).
BPub 250. SECOND MID TERM EXAM Name___________________________________________________________ ID ___________________________________________________________ Instructions: 1. WRITE YOUR NAME and STUDENT NUMBER ON BOTH PAGES.
2. Read over the whole quiz first.
3. Answer ALL questions
4. We will only grade answers on the fronts of the sheets (not the backs). Do not unstaple the two pages.
5 Show your work and write legibly. You have a maximum of 15 minutes for this quiz. QUESTION 1. Two firms located in the scenic Valley of Ex Tranquility have factories that produce sulfurous produce emissions. The Marginal Costs of abating these emissions are given by the functions. MCA 1 = 100 – 2(50‐A 1 ) or MCA 1 = 100 – 2(E 1 ) MCA 2 = 100 – (100‐A 2 ) or MCA 1 = 100 – (E 2 ) Notice that abatement “A” is simply the reduction in the level of emissions, “E”. The government has decided that it is socially optimal to have just 100 units of emissions (this is the total for both firms). a. Suppose that the government decides to achieve this level of emissions by imposing a tax of “T” per unit of emissions. What is the appropriate tax rate “T”? b. What is the total cost of abatement? c. Now suppose that the government decides to control pollution by granting tradable emission permits. Each firm is offered 50 permits and each such permit allows the firm to produce one unit of emissions. What trade will tack place between the two firms? 
which firm will sell? 
which firm will buy? 
how many permits will be traded? 
What is the price of each permit? d. What is the total cost of abatement? ANSWER a. If a tax is imposed each firm abates until MCA = T. Thus MCA 1 = MCA 2 = T 100 – 2E 1 = 100 – E 2 2E 1 = E 2 But E 1 +E 2 = 100 So E 1 = 33.3 and E 2 = 66.7 TAX = MCA 1 = 100 – 2(33) = MCA2 = 100 – 67 = 33.3 $
100 Firm 2 Firm 1 33.3 33.3 50 66.7
100
Emmissions
b. TOTAL COST of ABATEMENT for firm 1 (abatement is 50 – E = 50 – 33.3 = 16.7 = ½ (MCA times LEVEL OF ABATEMENT) = ½ (33.3 times 16.7) = 278 TOTAL COST of ABATEMENT for firm 2 (abatement is 100 – E = 100 – 66.7 = 33.3 = ½ (MCA times LEVEL OF ABATEMENT) = ½ (33.3 times 33.3) = 554 TOTAL COST = 554 + 278 = 832 c.
Firms will trade permits until MCA 1 = MCA 2 Since there are only a total of 100 permits, then E 1 +E 2 = 100 Thus the equilibrium emissions must satisfy the same simultaneous equations as answer (A). So E 1 = 33.3 and E 2 = 66.7 Thus FIRM 1 will sell 16.7 permits (reduce E from 50 to 33.3) Thus FIRM 2 will buy 16.7 permits (increase E from 50 to 66.7) Permits will sell for MCA = 33.3 d. TOTAL COST OF ABATEMENT is identical to part (B) since the emissions are identical. QUESTION 2. The success of your new business depends on the economic outlook and on the effort provided by the manager you employ. The following table shows profits in dollars (before you deduct the manager’s compensation). Low Effort High Effort Poor Economy 4 million 5 million Good economy 6 million 9 million You have the opportunity to decide on the compensation design you offer the manager but the manager will decide on what level of effort she provides. You cannot monitor the manager’s effort and the manager will decide according to the following utility function U = U = if Low effort if High effort (the ‐50 is the disutility of effort) In thinking about compensation, you must be competitive which means offering at least an expected utility to the manager of 500. You come up with three plans that you think might meet this competitive standard. PLAN A. Flat salary of $250,000 PLAN B. Flat salary of $200,000 plus a bonus equal to 2% of profit as shown in the table above PLAN C. Flat salary of $200,000 plus a option‐like bonus equal to 16% of profits in excess of $7.5m (i.e., 16% of the difference between actual profit and $7.5m – but the bonus is only paid if the profit is above $7.5m.) Which compensation plan would you choose? a. ANSWER Flat salary EU (Lo) = (250,000)0.5 = 500 EU (Hi) = (250,000)0.5 ‐ 50 = 450 Net profit for owner 0.5(4m) + 0.5(6m) – 250K = $4,250,000 Flat salary of 200,000 + 3% bonus EU (Lo) = 0.5(200,000 + 4m times 0.02)0.5 + 0.5(200,000 + 6m times 0.02)0.5 = 547 EU (Hi) = 0.5(200,000 + 5m times 0.02)0.5 + 0.5(200,000 + 9m times 0.02)0.5 ‐ 50 = 532.1 {
}{1‐0.02} – 200K = $4,700,000 Net profit for owner 0.5(4m) + 0.5(6m)
Flat salary of 200,000 + 16% option bonus for profit in excess of $3.0m EU (Lo) = 0.5(200,000)0.5 + 0.5(200,000)0.5 = 447 EU (Hi) = 0.5(200,000)0.5 + 0.5(200,000 + 1.5m times 0.16)0.5 ‐ 50 = 505.27 {
} ‐ {0.5(.16)(1.5m)} – 200K = $6,680,000 Net profit for owner 0.5(5m) + 0.5(9m)
QUESTION 3. The government of the developing country of B’leaves has decided to license two companies to make solar panels. The companies will enter the market simultaneously and will share the market for which the demand is P = 100 – Q 1 – Q 2 The costs of the two firms are TC 1 = 100 + 10Q 1 TC 2 = Q 2 2 Assume that each firm knows the cost curves of its rival. To help you in your answer, the two reaction functions are : Q 1 = 45 – 0.5Q 2 Q 2 = 25 – 0.25Q 1 You need not verify that these are correct. a. What price will solar panels sell for in B’leaves? b. Suppose now that the minister of industry in B’leaves makes a secret offer to firm 1, which would allow firm 1 to set its capacity before the license was granted to firm 2. Thus, firm 2 would enter the market after firm 1 has chosen its output. How much would solar panels sell for in B’leaves? c. What is the maximum amount of money that the minister could charge to firm 1 for the right to enter the market first? ANSWER Part a is COURNOT Firm 1’s TOTAL REVENUE is TR = (100 – Q 1 – Q 2 )Q 1 = 100Q 1 – Q 1 2 – Q 1 Q 2 MR = 100 – 2Q 1 – Q 2 SET MR = MC 100 – 2Q 1 – Q 2 = 10 Q 1 = 45 – 0.5Q 2 Q 2 = 25 – 0.25Q 1 Firm 2’s TOTAL REVENUE is TR = (100 – Q 2 – Q 1 )Q 2 = 100Q 2 – Q 2 2 – Q 2 Q 1 MR = 100 – 2Q 2 – Q 1 SET MR = MC 100 – 2Q 2 – Q 1 = 2Q 2 Substituting Q 1 = 45 – 0.5(25 – 0.25Q 1 ) Q 1 = 37.14 Q 2 = 25 – 0.25(37.14) Q 2 = 15.7 So price is 100 – 37.14 – 15.7 = 47.16 PART b. The second part is STACKLEBERG or leader – follower Firm 1’s TOTAL REVENUE is TR = (100 – Q 1 – Q 2 )Q 1 TR = (100 – Q 1 – (25 – 0.25Q 1 ) )Q 1 TR = 75Q 1 – 0.75Q 1 2 MR = 75 – 1.5Q 1 Setting MR = MC 75 – 1.5Q 1 = 10 So Q 1 = 43.3 So Q 2 = 14.17 Substituting to get Q 2 Q 2 = (25 – 0.25Q 1 ) = 25 – 0.25(43.3) So price is 100 – 43.3 – 14.17 = 42.53 PART (c) Profit for firm 1 when leader is TR – TC = 43.3(42.53) – (100 + 10(43.3)) = 1308.3 Profit for firm 1 with Cournot is TR – TC = 37.14(47.16) – (100 + 10(37.14)) = 1278.6 So firm 1 should be willing to pay up to 1308.3 – 1278.6 = 29.7 for the right to enter the market first QUESTION 4 Business is not too good at the New Age For Profit Mystic Society. They used be located next to open fields and the quiet environment encouraged contemplation which was what their customers were paying for; profit was then 10,000. However, the Hard Punk Café has located just next door, with its noisy punk bands, and contemplation is strained. As a result, profit has fallen to just 1,000. On the other hand, Hard Punk Café makes a profit of 20,000. New Age is thinking of bringing a lawsuit to recover compensation for its losses. The law says that anyone causing harm will be responsible for the economic loss. However there is some uncertainty whether the law will apply. There are things that can be done 
Hard Punk Café can change its name to Soft Folk Café and employ folk singers. If so, its profits will fall to 14,000, but there will be no disturbance of New Age’s clientele. Thus New Age’s profit would once again be 10,000. 
New Age can soundproof its premises. This would cost 5,000 and it would cut out most of the noise, so New Age’s profit would be 7,500 minus the cost of soundproofing; i.e., 7500 ‐ 5000 
Hard Punk could change its name and New Age could insert soundproofing, in which case New Age’s profit would be 10,000 – 5,000 = 5,000 and Hard Punk (Soft Folk) profit would be 14,000. 
Finally, neither could attempt to mitigate in which case Hard Punk makes 20,000 and New Age makes 1,000 a. What result will occur if Hard Punk IS responsible to compensate New Age for its loss – show what mitigation, if any, is undertaken and by whom, and how much net profit will each party make? b. What result will occur if Hard Punk IS NOT responsible to compensate New Age for its loss – show what mitigation, if any, is undertaken and by whom, and how much net profit will each party make? Neither firm mitigates Café changes to Folk New Age soundproofs Folk + Soundproof CAFÉ PROFIT 20,000 14,000 20,000 14,000 NEW AGE PROFIT 1,000 10,000 7500 – 5000 = 2500 10,000 – 5000 = 2500 TOTAL 21,000 24,000 22,500 19,000 NEW AGE PROFIT 1,000 + 9,000 = 10,000 10,000 7,500 + 2,500 – 5,000 + 5,000 = 10,000 10,000 – 5,000 + 5,000 = 10,000 TOTAL 21,000 24,000 22,500 Now Café is liable to Compensate Neither firm mitigates Café changes to Folk New Age soundproofs Folk + Soundproof CAFÉ PROFIT 20,000 ‐ 9,000 = 11,000 14,000 20,000 – 5,000 ‐ 2500 = 12,500 14,000 – 5,000 = 9,000 19,000 Café can either 
pay compensation of 9,000 for lost profit (row 1) 
change to folk (row 2) 
pay compensation of 2500 for lost profit + compensation of 5,000 for New Age’s soundproofing (row 3) 
pay compensation of 5000 for New Age’s soundproofing and change to folk (row 4) 
Clearly changing the name is the best (Row 2) Café is NOT liable to pay compensation CAFÉ PROFIT NEW AGE PROFIT TOTAL Neither firm mitigates 20,000 1,000 21,000 Café changes to Folk 14,000 + 6,000+ = 20,000 10,000 – 6,000+ = 4,000 24,000 New Age soundproofs 20,000 7500 – 5000 = 2500 22,500 Folk + Soundproof 14,000 10,000 – 5000 = 2500 19,000 New Age pays Café at least 6,000 (but not more than 7,500) to induce Café to change to Folk QUESTION 5 The Credit Insurance Company insurers credit risk for its clients. These policies compensate the policyholder if they lose money because of some credit event. It has two types of clients, high and low risks.  High risks have a 60% chance of a credit loss 
Low risks have a 30% chance of a credit loss Each policyholder starts with a wealth of 6 and will lose 5 if there is a credit loss. Each type, high and low, knows their own probability of loss. The insurance company does not know which is which. Assume there is only one of each type. Each policyholder has a utility function as follows U = (W)0.5 a. The insurer can offer a policy to any policyholder who wishes to buy. The policy will fully insure the loss of 5 at a premium of 2.5. How much expected profit will the insurer make if it offers this policy? b. The insurer can offer two policies to any policyholder who wishes to buy – policyholders can choose to buy either (or neither) policy. 
Policy A will fully insure the loss of 5 at a premium of 3.02. 
Policy B will pay only 1 million to the policyholder (so the remaining loss of 4 is borne by the policyholder), and the premium is 0.35 How much expected profit will the insurer make if it offers this menu of policies? ANSWER Part a. EU if no insurance HIGH 0.4(6)0.5 + 0.6(1)0.5 = 1.5798 LOW 0.7(6)0.5 + 0.3(1)0.5 = 2.0146 Max Premium High Risk will pay 1. 5798 = (6‐P)0.5 so P(max) = 3.5042 Max Premium Low Risk will pay 2.0146= (6‐P)0.5 so P(max) = `1.9412 Since the premium in part (a) is 2.5, then the high risk will buy and the low risk will not. Thus the insurer’s expected profit is PREM ‐ EL = 2.5 ‐ 0.6(5) = ‐ 0.5 (LOSS) Part b HIGH RISK EU Policy A : (6‐3.02)0.5 = 1.7263 Policy B : 0.4(6‐0.35)0.5 ‐ 0.6(6‐5+1‐0.35)0.5 = 1.7215 No Insurance: .4(6)0.5 + 0.6(1)0.5 = 1.5798 LOW RISK EU Policy A : (6‐3.02)0.5 = 1.7263 Policy B : 0.7(6‐0.35)0.5 ‐ 0.3(6‐5+1‐0.35)0.5 = 2.0492 No Insurance: 0.7(6)0.5 + 0.3(1)0.5 = 2.0146 So High buys policy A : expected profit = premium – EL = 3.02 – 0.6(5) = +0.02 So Low buys policy B : expected profit = premium – EL = 0.35 – 0.3(1) = +0.05 Total Profit is 0.05 + 0.02 = 0.07 (PROFIT) 6. The downstream part of McSinley consulting consists of a number of consultants who advise clients on business strategy. The demand for consults is P = 20,000 – 100Q where Q is the number of consults The downstream cost is TC = 10,000 + 4,000Q Now consultant are not mathematicians and each consult requires one unit of mathematical modeling. There is an upstream unit of McSinley that does this modeling and a price, P(M), is charged to the downstream per unit of math modeling. The total and marginal cost of a unit of math modeling is TC = 200Q2 and MC(M) = 400Q There is no outside market for math modeling, so the downstream can only buy from the upstream. And the upstream can sell only to the downstream. a. Suppose that the two units of McSinley behaved as separate profit maximizing firms. What would be the price of internal transactions for the purchase/sale of a unit of math modeling? How much profit does each unity make? How much profit is made in total for McSinley? b. You are asked to advice McSinley on how the transfer price should be set. What would you advise and how much profit will McSinley make if it follows your advice? ANSWER MR of downstream is 20,000 – 200Q Equate with MC 20,000 – 200Q = 4,000 + P(M) So the downstream demand is P(M) = 16,000 – 200Q We can now look at the upstream’s profit maximizing decision TR = P(M)Q = 16,000Q – 200Q2 And setting MR = MC 16,000 – 400Q = 400Q so Q= 20; P = 16,000 – 200(20) = 12,000 and PROFIT (M) = 20(12,000) – 200(20)2 = 160,000 Downstream will now maximize its profits given a modeling price of 12,000 20,000 – 200Q = 4,000 + P(M) 20,000 – 200Q = 4,000 + 12,000 So Q= 20; P = 20,000 – 100(20) = 18,000 and PROFIT = 20(18,000) – (10,000 + 4000(20)) – 20 (12,000) = 30,000 TOTAL PROFIT IS 160,000 + 30,000 = 190,000 PART B I would recommend transfer price = MC of upstream Thus when downstream maximizes profit it sets 20,000 – 200Q = 4,000 + 400Q So Q= 26.67; P = 20,000 – 100(26.67) = 17,333 and PROFIT = 26.67(17.333) – (10,000 + 4000(26.67)) – 200(26.67)2 = 203,333 TOTAL PROFIT HAS INCREASED FROM 190,000 to 203,333 QUESTION 7. Professor SHILTON is an English professor who teaches a course on Shakespeare and Milton. His unfortunate peculiarity is that he likes to show how clever he is by failing students in their Master Degree exam. He gets 
ONE UNIT OF PLEASURE FROM FAILING A STUDENT For her part, the student gets 
ONE UNIT OF PLEASURE FROM PASSING Thus the exam is a ZERO SUM GAME. It is not a popular course and there is only one student each year. Each year in the week before exams, the professor has to decide whether to write the exam on either Shakespeare or on Milton. Conversely, the student has to decide whether to study Shakespeare or Milton; she does not have time to study both. The following matrix shows the probabilities that the professor will be successful in failing the student (the first probability in the cell) student will be successful in passing the exam (the second probability in the cell) Professor sets exam on Shakespeare Professor sets exam on Milton Student studies Shakespeare 0.2 0.8 Student studies Milton 0.4 0.6 0.5 0.5 0.3 0.7 Work out the MIXED STRATEGY EQUILIBRIUM to this game. a. What probability will the professor choose to set the exam on Shakespeare b. What probability will the student choose to study Shakespeare? ***************************************************************************** ANSWER Professor must decide probability of setting exam on Shakespeare P S such that the student is indifferent between studying Shakespeare and studying Milton. Student’s expected payoff from studying Shakespeare = P S (0.8) + (1‐P S )(0.5) Student’s expected payoff from studying Milton = P S (0.6) + (1‐P S )(0.7) Equating these probabilities P S (0.8) + (1‐P S )(0.5) = P S (0.6) + (1‐P S )(0.7) So P S = 0.5 Student must decide probability of studying Shakespeare S S such that the professor is indifferent between setting the exam on Shakespeare or Milton. Professor’s expected payoff from setting Shakespeare = S S (0.2) + (1‐S S )(0.4) Professor’s expected payoff from setting Milton = S S (0.5) + (1‐S S )(0.3) Equating these probabilities S S (0.2) + (1‐S S )(0.4) = S S (0.5) + (1‐S S )(0.3) So S S = 0.25 BPUB 250
Spring 2009 Managerial Economics Final Exam
Instructions
1
2
3
4
5
6
7
Read over the whole exam, and do the easier questions first.
Show your work. No credit will be given if work is not shown.
No credit will be given for work shown on even-numbered pages. Use page
labeled as “overflow space” for work pertaining to that question only that
does not fit between questions.
Write legibly!
Initial the proctor’s class list when you hand in your exam.
Write your name and student I.D. number on every page of this exam in the
space provided.
Do not disassemble the exam.
Your Name:_____________________________________________________
Your
I.D. # _______________________________________________________________
Section number:_____________________________________________
Cohort name:_______________________________________________
Professor’s name:___________________________________________
DO NOT START UNTIL 6:00 PM.
Question 1 (30 points). Archer Daniels Midland (ADM) is the single domestic producer of Lysine, a livestock feedstuff ingredient. The demand for lysine in is given by ADM produces lysine using labor (L) and capital (K). Its production technology is given by and it pays market wages of and a rental of capital of . a) (10 points) What is ADM’s marginal cost of producing lysine? Derive the answer and show your work. b) (5 points) While there is one major foreign manufacturer of Lysine, ADM initially benefits from trade protection that grants it monopoly rights over the U.S. market. What price does ADM charge, how much lysine does it sell, and what are its profits? Assume that ADM incurs no fixed cost. (If you were not able to find ADM’s marginal cost in part a), you can assume that it is .) c) (15 points) ADM’s trade protection expires and the major foreign manufacturer enters the U.S. market. Its marginal costs (MC) and total costs (TC), respectively, are given by: The firms compete in Cournot fashion. How much does each firm produce and what is the resulting single market price? If ADM could hire a lobbying firm that guarantees the restoration of ADM’s monopoly rights, how much would ADM at most be willing to pay for the service? Question 2 (35 points). Kara’s utility function over wealth is U (W )  W and she has $10,000 saved up. Kara’s cousin, Andy, wants to buy a house on a contract that has to be executed today. Andy asks Kara to lend him the $10,000 for the home’s down payment and promises to compensate Kara by paying her $40,000 back in a week. Kara has known Andy for a long time, and estimates that there is a 40% chance that Andy will not pay anything back. On the upside, Kara reckons that there is a 60% chance that Andy will come up with the $40,000. The options for Kara are laid out in the following table: Lend Money Keep Money Andy repays (p = 0.6) $40,000 $10,000 Andy defaults (1‐p = 0.4) $0 $10,000 a) (5 points) What is the expected value of lending the $10,000 to Andy? b) (5 points) What is the expected utility of making the loan? c) (5 points) What is the certainty equivalent of the loan? d) (5 points) Should Kara lend the money? Explain why or why not. Lemon Investments, a financial services firm, offers “credit default swaps” (CDS) for these types of situations. Lemon’s CDS basically amounts to a type of insurance policy against default. If Andy doesn’t pay back, Lemon will reimburse the $10,000 to Kara and charge nothing. However, in exchange for that protection, Lemon will receive a share X of the net profits on the loan ($30,000) if Andy pays back. e) (15 points) What is the maximum percentage on the profits that Kara is willing to share in order to purchase the CDS from Lemon? Hint: Write down an expression for the new payoffs if Andy repays and if he doesn’t; then compare to next best alternative without the CDS Question 3 (35 points). The Administration is evaluating two proposals geared at curbing sulfur dioxide emissions. There are two groups of emitters, coal‐fired power plants (C) and oil-fired power plants (O), whose emissions will be governed by the proposal. Assume for simplicity that there is only one firm of each type. The marginal benefit to the coal‐fired plant of 1 unit of emissions, , (one ton of SO 2 emissions) is given by Similarly, Note that the MBE schedules are equivalent to marginal cost of abatement (MCA) schedules, where abatement is reduction in emissions, and the benefit of emissions is not having to pay for abatement. Each unit of emissions, while coming at no cost to the companies, imposes costs on society. Since coal and oil‐fired power plants typically do not operate in close proximity, the social cost of their SO 2 emissions does not compound. Instead, they generate separate costs on their local communities in the amount of: a) (10 points) In the absence of government intervention, how many units of emissions does each of the companies produce? What are the levels of emissions that would be socially optimal and how large is the deadweight loss from emissions generated by C and O each? Illustrate your answers in a graph that plots marginal benefits and costs on the y‐axis against emissions on the x‐axis. b) (8 points) Proposal 1 suggests imposing a simple emission standard of 20 units of emissions on each of the companies. Compute the total cost of this standard to each of the two companies. How large is the deadweight loss under this proposal? Show the regions corresponding to cost and deadweight loss in a new graph. c) (7 points) Proposal 2 is a cap‐and‐trade solution. It suggests granting each company 20 permits, each giving them the right to 1 unit of emissions, and allowing the companies to adjust their overall levels of emissions from the 20 units by trading permits amongst themselves. Propose a range in which the price of the first traded permit has to fall for both companies to be better off. d) (10 points) How many permits get ultimately traded and at what equilibrium price? Question 1 (ANSWERS).
(a) Find the cost-minimizing total cost:
min TC  r  K  w  L
subject to Q  K  L
At the optimal K and L levels, the MRTS = w/r:
0.5  K 0.5  L0.5 w

0.5  K 0.5  L0.5 r
K w
 
L r
w
K L
r
Substitute the expression back into the constraint:
Q
w
L
r
 L* 
 K* 
Q

w
r
r
Q
w
w
Q
r
Compute the associated total cost:
TC  r  w  Q  w  r  Q
r
w
 2 w r Q
Here, w=$10 and r=$22.5, so TC  30  Q and MC 
(b) ADM as a monopolist sets MR equal to MC:
dTC
 30 .
dQ
2
 Q  30  MC
100
 Q M  3,500
MR  100 
P M  $65
 M  (65  30)  3,500  $122,500
(c) Call ADM firm 1 and the foreign manufacturer firm 2. Demand in the market now
equals:
P  100 
Q1  Q2
100
ADM solves:
Q2 2Q1

 30  MC1
100 100
Q
Q
 70  2  1
100 50
MR1  100 
ADM's reaction function equals:
Q1  3,500 
Q2
2
The foreign manufacturer solves:
Q
2Q
Q
MR2  100  1  2  37  2  MC2
100 100
50
Q
Q
 63  1  2
100 25
Substitute in ADM's reaction function:
Q  Q

63   35  2   2
200  25

1 
 1
28   
  Q2
 25 200 
 Q2  800
ADM’s quantity choice:
800
Q1  3,500 
 3,100
2
2
Market price from the demand function:
1
P  100 
 (3,500  800)  $61
100
ADM’s new profit:
 D  (61  30)  3,100
 $96,100
ADM would at most be willing to pay $122.500 – $96,100 or $26,400 for its monopoly
rights.
3
Question 2 (ANSWERS)
a) EV= 0.6* 40,000 + 0.4 *0 = 24,000
b) EU (loan)  0.6  40,000  0.4  0  0.6  200  120
c) U (CE )  CE  120 , and so CE  1202  14, 400
d) Since CE>10,000 (the money she obtains for sure) she actually prefers to make the loan!
Another way to see this is to realize that U (10,000)  10,000  100  120
e) Note that the best alternative to Lemon’s CDS is lending the money without protection.
The relevant decisions are thus:
Lend Money (no insurance)
Lend Money and buy CDS
Andy repays (p = 0.6)
$40,000
$10,000+$30,000*(1-X)
Andy defaults (1-p = 0.4)
$0
$10,000
The expected utility of the CDS is:
EU (CDS )  0.6  40,000  30,000 X  0.4  10,000  0.6  40,000  30,000 X  40
The maximum X is such that the expected utilities of buying the CDS and going
uninsured are equal:
EU (CDS )  0.6  40,000  30,000 X  40  EU (loan)  120 , and therefore:
0.6  40,000  30,000 X  80 ;
0.36   40,000  30,000 X   80 2
40,000  30,000 X 
X
6, 400
 17,777.77
0.36
40,000  17,778
 0.74
30,000
Kara is willing to share up to 74% of net profits in order to eliminate the risk of default.
4
Question 3 (ANSWERS)
a) Graph of the Problem:

Since the companies do not incur private costs, they generate emissions at the point
where MBE=0.

The socially optimal level is at the point where MBE=MSC. Since C and O impose
damage on the residents of their local communities only, this entails:
5

The deadweight-loss to emissions os the excess social marginal cost not offset by a
marginal benefit from emissions (shaded in gray in the figure above)
b) The total cost of the standard equals the area underneath the MBE curve from E=20 to
Emax.
A standard of 20 coincides with the optimal emissions level for C. As a result, DWL C =0.
For O, the standard is too stringent, reducing E below its socially optimal level. Thus the
DWL is the MBE that exceeds the MSC for emissions from 20 to 40 (see graph below).
6
c) Note that
MBE C (E=20) = 5
MBE C (E=19) = 5.25
MBE O (E=20) = 14
MBE O (E=21) = 13.8
Thus if C were to sell its 20th permit to O, it would forego a marginal benefit of $5. O
would realize benefits of $13.8 from that permit. A price range that would make at least
one firm better off is a price between $5 and $13.8
Alternatively, we could compute the benefits foregone as the areas underneath the MBE
curves, between 19 and 20 for C and 20 and 21 for O (see graph below)
For C, the area is given by
MBE C (E=20 -> E=19) = 5+0.25/2 = 5 1/8
For O, the area is given by
MBE O (E=20 -> E=21) = 13.8+0.2/2 = 13.9
7
d) The firms trade number of permits, x, until
O thus buys all of the permits originally issued to C. The market clearing price is
=10
(A similar solution results from using the area-based approach from part c)
8
ANSWERS TO BPUB 250 FINAL 2010 Spring
Question 1 (Worth 15 Points Total): Jacob Sharkovsky is the founder of Sharkovsky,
Inc., a company specializing in the capture of live sharks. When attempting to capture a
shark, two outcomes are possible. One outcome is that Jacob manages to surprise the
shark and capture it alive. The other outcome is that a battle ensues, in which case Mr.
Sharkovsky has to kill the shark. These two outcomes occur with equal probabilities. The
market price of a live shark is $1,800, while the market price of a dead shark is $80.
Part a (Worth 3 Points): Suppose Jacob attempts to capture 70 sharks in a typical week.
What is his expected weekly revenue?
Jacob’s expected weekly revenue is:______________________
Part b (Worth 4 Points): Jacob’s son, Ahab, follows the family tradition and has
recently started capturing sharks himself. Unfortunately, Ahab, did not inherit his father’s
stealth capabilities, and finds it more difficult to catch the sharks by surprise.
When Ahab sails for a week of shark searching, two possible outcomes occur. With
probability of 0.7, he returns safely with 2 live sharks to be sold on the market. With
probability of 0.3, the sharks prove tricky and Ahab returns with 8 dead sharks, and also
incurs medical costs of $240.
Ahab’s utility (U) function is U = W0.5. His wealth (W) is equal to the revenue from
selling sharks, less any medical costs incurred.
What is Ahab’s expected utility?
Ahab’s expected utility is:___________________
Part c (Worth 4 Points): Jacob would like his son Ahab give up the shark pursuing, and
instead come work for him as an accountant. What is the minimal weekly wage that
must be offered to Ahab for Ahab to give up a week of shark capturing?
Ahab’s required minimal weekly wage is:_________________________
Part d (Worth 4 points). Jacob realizes he actually does not need an accountant and so
that option is off the table. So Ahab continues on with his shark catching career path.
Ahab now learns that he can obtain a shark radar called the “Shradar”. With this Shradar,
Ahab would be guaranteed to surprise the sharks (and hence capture them alive) all the
time and return safely, i.e., incur no medical expenses, with a weekly yield of 3 live
sharks. This state of the art radar can be rented for $2,900 per week. Should Ahab rent
the Shradar? (Convince us with numbers in the because section below).
Ahab (Should, Should Not) rent the Shradar
Circle One
because:_____________________________________________________
Question 2 (Worth 10 Points): Gino’s Steaks is a local restaurant that sells Philly
Cheese Steaks. Most customers purchase both a cheese steak and a soft drink; however,
Gino currently prices their cheese steaks separately and their drinks separately. They’ve
also estimated that different types of people have different reservation prices for cheese
steaks and soft drinks. They have also estimated that three different types of customers
exist and their estimates of each type’s reservation prices are shown in the table below.
Each person will demand no more than one cheese steak and one soft drink.
Demand Type
A
B
C
Cheese Steak
5
1.50
3
Soft Drink
1.40
2
1
It costs Gino’s a constant $1 to produce and serve a cheese steak and a constant $0.50 to
produce and serve a soft drink.
Given a choice of continuing their current price separately strategy or of switching
to a pure bundle (one cheese steak and one soft drink) strategy, what would be your
advice? For simplicity, assume there is one of each customer type.
I would advise them to:
(PRICE SEPARATELY, OFFER THE PURE BUNDLE)
CIRCLE ONE
What is Gino’s profit under your suggested pricing strategy?
Gino’s profit under your suggested pricing strategy = ______________________
Question 3 (Worth 25 Points Total): Two players, Letter and Number, play the
following game. Letter selects among strategies A, B, and C whereas Number selects
among strategies 1, 2, and 3:
A
B
C
1
0, 2
3, -1
2, 1
2
-1, 1
0, 5
1, 4
3
5, 3
2, 1
3, 6
Payoffs to Letter are written first in each cell, and payoffs to Number are written second
in each cell.
Part a (Worth 4 Points): Find any maximin equilibria.
The equilibria are: _________________________________________________
Part b (Worth 4 Points): Suppose that both players choose their strategies
simultaneously. Find any Nash equilibria
The Nash equilibria are: ______________________________________________
.
Part c (Worth 5 Points): Suppose Number moves first and Letter moves second. Find
any Nash equilibria.
The Nash equilibria are: ______________________________________________
Part d (Worth 5 Points): Suppose Letter moves first and Number moves second. Find
any Nash equilibria.
The Nash equilibria are: ______________________________________________
Part e (Worth 7 Points): Now suppose that players can costlessly write binding
contracts that commit them to playing some strategy, and that they can negotiate to make
payments to each other after the payoffs to the games have been realized. Both players
seek to maximize their net payoffs after compensation has taken place.
Part e1: If Number moves first and Letter moves second, what will be each
player’s net payoff, and what strategy will each player play? Explain.
Number’s strategy is: _________________________
Number’s net payoff is: ________________________
Letter’s strategy is: ___________________________
Letter’s net payoff is: _________________________
Part e2: If Letter moves first and Number moves second, what will be each
player’s net payoff, and what strategy will each player play? Explain.
Number’s strategy is: _________________________
Number’s net payoff is: ________________________
Letter’s strategy is: ___________________________
Letter’s net payoff is: _________________________
Question 4 (Worth 25 Points Total): Wal-Rus and Tar-Mart are the only two discount
department stores that serve the city of Cheltenham. Wal-Rus is the larger of the two
stores and it faces a demand of
QW = 28 – 2PW + PT
Tar-Mart’s demand is given by
QT = 14 – PT + PW
where Qi is the quantity demanded in store i when the unit price is Pi, where i = T, W
Assume that both stores incur no marginal cost, but have fixed costs (FC) of
FCW = $100 and FCT = $120
Part a (Worth 8 Points): If Wal-Rus and Tar-Mart simultaneously determine their
prices and units sold, what is Wal-Rus’ equilibrium price, quantity, and profit?
What is Tar-Mart’s equilibrium price, quantity, and profit?
Wal-Rus’
Price
= ______________
Quantity = ______________
Profit
= ______________
Tar-Mart’s Price
= ______________
Quantity = ______________
Profit
= ______________
Part b (Worth 9 Points): Wal-Rus and Tar-Mart merge. The merged company continues
to operate both of their Cheltenham stores, but coordinates the pricing decision for the
two stores. What optimal prices and quantities would the merged entity choose at
each store, and what would be its profit at each store?
Wal-Rus’
Price
= ______________
Quantity = ______________
Profit
= ______________
Tar-Mart’s Price
= ______________
Quantity = ______________
Profit
= ______________
Part c (Worth 8 Points): The merged company considers closing the smaller of its two
Cheltenham stores, the old Tar-Mart store, allowing it to save Tar-Mart’s fixed costs. The
company anticipates that this would increase demand at the old Wal-Rus store to
QW = 56 – 2PW
Assuming that Wal-Rus’ cost structure doesn’t change, what would be the store’s
new profit? Should the company proceed to close the Cheltenham Tar-Mart store
(convince us with numbers in the because section below)?
The store’s new profit = ____________________________________
The company (Should, Should Not) close the Cheltenham Tar-Mart store
Circle One
because:______________________________________________________
Question 5 (Worth 25 Points Total): “Ocean” is a seafood restaurant that offers its
customers a unique experience: dining while surrounded by aquariums containing live
sharks. Being a pioneer in this field “Ocean” currently enjoys monopoly power.
Part a (Worth 4 Points): “Ocean” faces a demand curve given by
P = 80 – (1/40)Q
where Q is the quantity of meals sold to customers when the price of an “Ocean” meal is
P. “Ocean’s” total cost (TC) function is given by
TC = 0.01Q2
How many meals should “Ocean” sell to maximize its profits? (A non-integer
number of meals is permitted as well as for price and profit numbers later in the
problem).
The number of meals “Ocean” should sell = __________________________
Part b (Worth 3 Points): What is the meal’s price, and “Ocean’s” profits?
The price of an “Ocean” meal should = ___________________________
Ocean’s profit maximizing profit = _______________________________
Part c (Worth 8 Points): Recent research of this evolving market has revealed that
demand has changed relative to the company’s original assessment and that customers
now belong in two main groups: “regular” customers and “students”. The demand of
“regular” customers is given by
PR = 85 – (1/20)QR
and the demand of “students” is given by
PS = 55 – (1/30)QS
where Qi is the number of meals demanded by demand type i when the price of a meal is
Pi, where i = R, S
Realizing this, “Ocean” now charges two prices: a “regular” price PR, and a “student”
price PS (with student ID). How many meals should “Ocean” sell to “regular”
customers, and how many meals should they sell to “students”? What would PR and
PS be?
The number of meals sold to “regular” customers should = _____________________
The price of a “regular” meal should = _____________________________
The number of meals sold to “student” customers should = _____________________
The price of a “student” meal should = _____________________________
Part d (Worth 10 Points): Inspired by “Ocean’s success, an entrepreneur opens a
competing restaurant—Shark in the Dark (SITD)---catering only to “students” (student
ID required). SITD’s meals are viewed by “students” to be perfect substitutes to
“Ocean’s”. SITD’s cost function is identical to “Ocean’s”, i.e., TC = 0.01Q2.
“Ocean” and SITD compete by setting quantities simultaneously. “Ocean” chooses QOR
(the quantity it sells to “regular” customers) and QOS (the quantity it sells to “students”).
SITD chooses QSS (the quantity it sells to “students”). The total quantity sold to
“students” is QS = QOS + QSS. What are the quantities of meals sold, i.e., QOR, QOS,
QSS, in the Nash equilibrium? IN THIS PART OF THE PROBLEM PLEASE
TAKE ALL CALCULATIONS OUT TO AT LEAST THE TEN THOUSANDTHS
DECIMAL PLACE (i.e., 4 to the right of the period).
The number of meals “Ocean” sells to “regular” customers = _________________
The number of meals “Ocean” sells to “student” customers = _________________
The number of meals SITD sells to “student” customers = _________________
Answer 1a: Every attempt to capture a shark results in either a live shark captured, which
sells for 1,800, or with a dead shark, that sells for 80. Since this happens with equal
probabilities, the expected revenue of every such attempt is
Expected Revenue = 0.5*1,800 + 0.5*80 = 900 + 40 = 940
Since Jacob repeats this 70 times, his expected revenue is 70*940 = 65,800
Answer 1b: With probability 0.7, Ahab sells 2 live sharks for revenue of 2*1,800 =
3,600. With probability 0.3, he sells 8 dead sharks for revenue of 8*80 = 640 but incurs
medical costs of 240 for a net return of 640 – 240 = 400. His expected utility is therefore
EU = 0.7*(3,600)0.5 + 0.3(400)0.5 = 0.7*60 + 0.3*(20) = 42 + 6 = 48
Answer 1c: We look for the certainty equivalent of a week of shark capturing. Ahab’s
expected utility from a week of shark capturing was found in 1b to be 48. The certain
accountant income W that would provide the same utility solves
W0.5 = 48  W = 482 = 2,304
So a weekly salary of $2,304 or above would be required to get Ahab to take the
accountant job.
Answer 1d: With the Shradar, Ahab’s weekly revenue would be 3*1,800 = 5,400.
Deducting the rental cost of the Shradar, his net weekly income is 5,400 – 2,900 =
$2,500. His utility would then be equal to
2,5000.5 = 50 > 48
Since his expected utility with the Shradar (50) is greater than without it (48), Ahab
should rent the Shradar.
Answer 2: If the cheese steaks and soft drinks are sold separately, the following results
PriceCheese Steak Cost Cheese Steak
5
1
1.5
1
3
1
PriceSoft Drink Cost Soft Drink
1.4
0.5
2
0.5
1
0.5
Profit/cheese steak # cheese steaks Profit
4
1
4 Best
0.5
3
1.5
2
2
4 Best
Profit/soft drink # soft drinks
0.9
2
1.5
1
0.5
3
Profit
1.8 Best
1.5
1.5
Thus, the best price separately strategy is to price cheese steaks at either 5 or 3 and price
soft drinks at 1.4. This yields profits of 4 + 1.8 = 5.8
If pure bundling is followed, the following profits result:
PriceBundle Cost Bundle Profit/bundle # bundles
6.4
1.5
4.9
1
3.5
1.5
2
3
4
1.5
2.5
2
Profit
4.9
6 Best
5
Thus, the best pure bundle strategy is to price the bundle at 3.5 which yields a profit of 6.
Since 6 > 5.8, pure bundling is the profit maximizing pricing strategy for Gino’s.
Answer 3a:
A
B
C
1
0, 2
3, -1
2, 1
2
-1, 1
0, 5
1, 4
3
5, 3
2, 1
3, 6
If Letter chooses A, then Min (0, -1, 5) = -1
If Letter chooses B, then Min (3, 0, 2) = 0
If Letter chooses C, then Min (2, 1, 3) = 1
Then if Letter Maxes the Mins, we have Max (-1, 0, 1) = 1 which means Letter does C
If Number chooses 1, then Min (2, -1, 1) = -1
If Number chooses 2, then Min (1, 5, 4) = 1
If Number chooses 3, then Min (3, 1, 6) = 1
Then if Number Maxes the Mins, we have Max (-1, 1, 1) = 1 which means number does
either 2 or 3.
So there are two maximin equilibria: C, 2 and C, 3
Answer 3b:
A
B
C
1
0, 2
3, -1
2, 1
2
-1, 1
0, 5
1, 4
3
5, 3
2, 1
3, 6
Using bold for Letter’s best strategy given a strategy by Number and underline for
Number’s best strategy given a strategy by Letter, we have the matrix above. The Nash
is the cell with both the bold and the underline. The Nash is where both players are
doing their best strategy, given what the other player is doing. So the Nash is A, 3
We could also solve this game by iterative dominance. Note the strategy 1 would never
be selected by Number (no underlines). So we can eliminate it.
2
3
A
-1, 1
5, 3
B
0, 5
2, 1
C
1, 4
3, 6
Now strategy B is never selected by Letter (no bolds). So we can eliminate it
So we left with
A
C
2
-1, 1
1, 4
3
5, 3
3, 6
We now see that Number has a dominate strategy to play 3, i.e., 3 > 1 and 6 > 4. Given
that Number plays 3, then Letter plays A, i.e. 5 > 3. The equilibrium is A, 3.
Answer 3c:
A
B
C
1
0, 2
3, -1
2, 1
2
-1, 1
0, 5
1, 4
3
5, 3
2, 1
3, 6
We know from above that Number will never choose strategy 1. If Number moves first
and picks strategy 2, Letter will respond and pick strategy C (since 1 > 0 > -1) and
Number will get 4. If Number moves first and picks strategy 3, Letter will respond and
pick strategy A (since 5 > 2 > 3) and Number will get 3. Since 4 > 3, Number will pick
strategy 2. Thus the equilibrium is C, 2
Answer 3d:
A
B
C
1
0, 2
3, -1
2, 1
2
-1, 1
0, 5
1, 4
3
5, 3
2, 1
3, 6
If Letter moves first and picks strategy A, Number will respond and pick strategy 3 (since
3 > 1 > 2) and Letter will get 5. If Letter moves first and picks strategy B, Number will
respond and pick strategy 2 (since 5 > 1 > -1) and Letter will get 0. If Letter moves first
and picks strategy C, Number will respond and pick strategy 3 (since 6 > 4 > 1) and
Letter will get 3. Since 5 > 3 > 1, Letter will pick strategy A. Thus the equilibrium is A,
3
Answer 3e1:
1
2
3
A
0, 2
-1, 1
5, 3
B
3, -1
0, 5
2, 1
C
2, 1
1, 4
3, 6
Since players can write contracts to compensate each other after receiving their payoffs,
the players seek to maximize the sum of their payoffs and then compensate each other to
make each other as well off as they would have been if they were not allowed to write
binding contracts to compensate each other.
If Number moves first and Letter moves second, we know from part c that the solution is
C,2 where Letter gets 1 and Number gets 4. Consider the alternative in which the two
players write a binding contract committing Letter to play strategy C and Number to play
strategy 3. The total payoff in that cell = 9 = 3 + 6 and exceeds the total payoff of any cell
of the other 8. Thus, the players will write a contract for Letter to play C and
Number to play 3 and then split the surplus by making payments to each other; any
outcome is sustainable as long as Letter gets net payoffs (after compensation) of at
least 1 and Number gets net payoffs (after compensation) of at least 4.
Answer 3e2: If Letter moves first and Number moves second, we know from part d that
the solution is A,3 where Letter gets 3 and Number gets 5. Consider the alternative in
which the two players write a binding contract committing Letter to play strategy C and
Number to play strategy 3. The total payoff in that cell = 9 = 3 + 6 and exceeds the total
payoff of any cell of the other 8. Thus, the players will write a contract for Letter to
play C and Number to play 3 and then split the surplus by making payments to each
other; any outcome is sustainable as long as Letter gets net payoffs (after
compensation) of at least 5 and Number gets net payoffs (after compensation) of at
least 3.
Answer 4a: If Wal-Rus and Tar-Mart choose prices simultaneously, they both maximize
profit.
Wal-Rus Profit = ΠW = TRW – FCW = PWQW – FCW = PW(28 – 2PW + PT ) – FCW
= 28PW – 2PW2 + PWPT – 100
Maximizing Wal-Rus’s profit with respect to PW gives
∂ΠW/∂PW = 28 – 4PW + PT = 0
Or
4PW = 28 + PT
Or
PW = 7 + 0.25PT
This is Wal-Rus’ reaction function.
Tar-Mart Profit = ΠT = TRT – FCT = PTQT – FCT = PT (14 – PT – PW) – FCT
= 14PT – PT2 – PWPT - 120
Maximizing Tar-Mart profit with respect to PT gives
∂ΠT/∂PT = 14 – 2PT + PW = 0
Or
2PT = 14 + PW
Or
PT = 7 + 0.5PW
This is Tar-Mart is reaction function.
Now substitute Wal-Rus’s reaction function into Tar-Mart’s reaction function and solve
for Tar-Mart’s price
Or
Or
PT = 7 + 0.5(7 + 0.25PT) = 7 + 3.5 + 0.125PT = 10.5 + 0.125PT
0.875PT = 10.5
PT = 12
Then substitute PT = 12 into Wal-Rus’ reaction function
PW = 7 + 0.25*12 = 7 + 3 = 10
Substituting PW = 10 and PT = 12 into Wal-Rus’ demand function gives
QW = 28 – 2*10 + 12 = 28 – 20 + 12 = 20
Wal-Rus’ total revenue is TRW = PWQW = 10*20 = 200
Wal-Rus’ profit is TRW – FCW = 200 – 100 = 100 = ΠW
Substituting PW = 10 and PT = 12 into Tar- Mart’s demand function gives
QT = 14 – 12 + 10 = 12
Tar-Mart’s total revenue is TRT = PTQT = 12*12 = 144
Tar-Mart’s profit is TRT – FCT = 144 – 120 = 24 = ΠT
Answer 4b: Wal-Rus and Tar-Mart now maximize profits jointly
Profit of merged firm = Π = TRW + TRT – FCW - FCT
= PW(28 – 2PW + PT ) – FCW + PT (14 – PT + PW) – FCT
= 28PW – 2PW2 + PWPT – 100 + 14PT – PT2 + PTPW – 120
= 28PW – 2PW2 + 2PWPT + 14PT – PT2 – 220
The first order conditions of maximizing profit with respect to PW and PT are
∂Π/∂PW = 28 – 4PW + 2PT = 0
∂Π/∂PT = 14 – 2PT + 2PW = 0
The first equation yields
Or
The second equation yields
Or
4PW = 28 + 2PT
PW = 7 + 0.5PT
2PT = 14 + 2PW
PT = 7 + PW
Substituting the PW equation into the PT equation gives
Or
Or
PT = 7 + (7 + 0.5PT) = 14 + 0.5PT
0.5PT = 14
PT = 28
Substituting PT = 28 into the equation for PW gives
PW = 7 + 0.5*28 = 7 + 14 = 21
Substituting PW = 21 and PT = 28 into Wal-Rus’ demand function gives
QW = 28 – 2*21 + 28 = 28 – 42 + 28 = 14
Wal-Rus’ total revenue is TRW = PWQW = 21*14 = 294
Wal-Rus’ profit is TRW – FCW = 294 – 100 = 194 = ΠW
Substituting PW = 21 and PT = 28 into Tar- Mart’s demand function gives
QT = 14 – 28 + 21 = 7
Tar-Mart’s total revenue is TRT = PTQT = 28*7 = 196
Tar-Mart’s profit is TRT – FCT = 196 – 120 = 76 = ΠT
The profits of both stores combined is 194 + 76 = 270
Answer 4c: Wal-Rus is now a monopolist. Their profit is
ΠW = PWQW – FCW = PW(56 – 2PW) – FCW = 56PW – 2PW2 – 100
To maximize profit with respect to PW set
or
or
dΠW/dPW = 56 – 4PW = 0
4PW = 56
PW = 14
Substituting PW = 14 into Wal-Rus’ demand function gives
QW = 56 – 2*14 = 56 – 28 = 28
Wal-Rus’ total revenue is TRW = PWQW = 14*28 = 392
Wal-Rus’ profit is TRW – FCW = 392 – 100 = 292 = ΠW
Since 292 > 270, the company is better off closing the Tar-Mart, saving on its fixed
cost, and diverting some of the store’s existing demand to the Wal-Rus store.
Answer 5a: Ocean’s total revenue is
TR = PQ = [80 – (1/40)Q]Q = 80Q – (1/40)Q2
Ocean’s marginal revenue is
MR = dTR/dQ = 80 – 0.05Q
Ocean’s total cost is TC = 0.01Q2
Ocean’s marginal cost is MC = dTC/dQ = 0.02Q
To maximize profit, Ocean will set MR = MC or
Or
Or
MR = 80 – 0.05Q = 0.02Q = MC
0.07Q = 80
Q = 1,142.857
Answer 5b: Substituting Q = 1,142.857 into the demand function gives
P = 80 – 0.025*1,142.857 = 80 – 28.571 = 51.429
Π = PQ – 0.01Q2 = (51.429)(1,142.857) – 0.01(1,142.857)2
= 58,775.51 – 0.01*1,306,122.45
= 58,775.51 – 13,061.22 = 45,714.29
Answer 5c: The total revenue from the regular customers is
TRR = PRQR = [85 – (1/20)QR]QR = 85QR – (1/20)QR2
Ocean’s marginal revenue from regular customers is
MRR = dTRR/dQR = 85 – 0.1QR
The total revenue from the student customers is
TRS = PSQS = [55 – (1/30)QS]QS = 55QS – (1/30)QS2
Ocean’s marginal revenue from student customers is
MRS = dTRS/dQS = 55 – 0.0667QS
To maximize profits Ocean must set MRR = MRS = MC
Let’s look at the left hand equality, i.e.,
Or
Or
MRR = 85 – 0.1QR = 55 – 0.0667QS = MRS
0.0667QS = -30 + 0.1QR
QS = -450 + 1.5QR
Ocean’s marginal cost was shown in 5a to be MC = 0.02Q. Since Q = QR + QS, then
Ocean’s MC is
MC = 0.02(QR + QS)
Substituting QS = -450 + 1.5QR in the MC equation gives
MC = 0.02(QR – 450 + 1.5QR) = 0.02(-450 + 2.5QR) = -9 + 0.05QR
To maximize profit, Ocean must set MRR = MC. Thus
Or
Or
MRR = 85 – 0.1QR = -9 + 0.05QR = MC
0.15QR = 94
QR = 626.667
Substituting QR = 626.667 into QS = -450 + 1.5QR gives
QS = -450 + 1.5*626.667 = -450 + 940 = 490
Substituting QR = 626.667 into the regular’s demand curve gives
PR = 85 – 0.05*626.667 = 85 – 31.333 = 53.667
Substituting QS = 490 into the student’s demand curve gives
PS = 55 – 0.0333*490 = 55 – 16.333 = 38.667
Answer 5d: Write the market demand for student meals as
PS = 55 – (1/30)QOS – (1/30)QSS
where QS = QOS + QSS
SITD’s total revenue will be
TRSITD = PSQSS = [55 – (1/30)QOS – (1/30)QSS]QSS
= 55QSS – (1/30)QSS2 – (1/30)QSSQOS
SITD’s marginal revenue will be
MRSITD = ∂TRSTID/∂QSS = 55 – 0.0667QSS – 0.0333QOS
SITD’s total cost is TCSITD = 0.01QSS2
SITD’s marginal cost is MCSITD = dTCSITD/dQSS = 0.02QSS
To maximize profit, SITD will set MRSITD = MCSITD or
MRSITD = 55 – 0.0667QSS – 0.0333QOS = 0.02QSS = MCSITD
Or
0.0867QSS = 55 – 0.0333QOS
Or
QSS = 634.615 – 0.385QOS
This is SITD’s reaction function.
Ocean’s profit function is
ΠO = TRR + TRS – TCO = PRQOR + PSQOS – TCO
= (85 – 0.05QOR)QOR + [55 – (1/30)QOS – (1/30)QSS]QOS – 0.01(QOR + QOS)2
= 85QOR – 0.05QOR2 + 55QOS – (1/30)QOS2 – (1/30)QOSQSS
– 0.01(QOR2 + 2QORQOS + QOS2)
= 85QOR – 0.05QOR2 + 55QOS – 0.0333QOS2 – 0.0333QOSQSS
– 0.01QOR2 – 0.02QORQOS – 0.01QOS2
= 85QOR – 0.06QOR2 + 55QOS – 0.0433QOS2 – 0.0333QOSQSS – 0.02QORQOS
Ocean will maximize its profits by choosing QOS and QOR such that
Or
Or
And
∂ΠO/∂QOR = 85 – 0.12QOR – 0.02QOS = 0
0.12QOR = 85 – 0.02QOS
QOR = 708.333 – 0.1667QOS
∂ΠO/∂QOS = 55 – 0.0867QOS – 0.0333QSS - 0.02QOR = 0
Substituting QSS = 634.615 – 0.385QOS and QOR = 708.333 – 0.1667QOS into the above
equation gives
Or
Or
55 – 0.08667QOS – 0.0333(634.615 – 0.385QOS) – 0.02(708.333 – 0.1667QOS)
= 55 – 0.08667QOS – 21.154 + 0.0128QOS - 14.167 + 0.00333QOS
= 19.679 – 0.0705QOS = 0
0.0705QOS = 19.679
QOS = 279.1
Substituting QOS = 279.1 into QSS = 634.615 – 0.385QOS gives
QSS = 634.615 – 0.385*279.1 = 634.615 – 107.343 = 527.273
Substituting QOS = 279.1 into QOR = 708.333 – 0.1667QOS gives
QOR = 708.333 – 0.1667*279.1 = 708.333 – 46.515 = 661.818
For Ocean to maximize profits, it must be the case that MROS = MROR
Substituting QOR = 661.818 into the regular’s marginal revenue function gives
MROR = 85 – 0.1*661.818 = 85 – 66.182 = 18.818
The marginal revenue to Ocean from selling to students is
MROS = 55 – 0.0667QOS – 0.0333QSS = 55 – 0.0667*279.1 – 0.0333*527.273
= 55 – 18.606 – 15.576 = 18.818
Finally, these MR must equal Ocean’s MC which is
MCO = 0.02(QOS + QOR) = 0.02(279.1 + 661.818) = 0.02*940.909 = 18.818
:
BPUB 250 Final Answers Spring 2011
Question 1 (Worth 24 points)
Ben owns a business selling baseball cards called Ben’s Baseball Cards (BBC). BBC’s
profits depend both on the state of the economy and how hard he works managing his
business. Ben does not mind working hard at all and is risk neutral, so his utility function
is given by UB(Π) = Π, where Π represents the profits of BBC. Profits and the
probabilities of each state of the economy are given in the table below.
Type of Economy
Bad (B)
Medium (M) Good (G)
Probability of Economy
30%
40%
30%
Low Effort of Manager
$4,000,000
$9,000,000
$16,000,000
High Effort of Manager
$9,000,000
$16,000,000
$25,000,000
Initially Ben runs the company himself.
Part (a) (Worth 10 points): What is Ben’s expected utility? Would Ben be willing to
purchase an insurance policy that pays him $5,000,000, in addition to his profits, if
there is a bad economy? If yes, calculate the maximum price he would pay for the
policy. If no, explain why in one sentence or prove mathematically. Show all
supporting work.
Ben’s expected utility is ____________________
Ben ( Would, Would Not) buy the insurance policy
CIRCLE ONE
If Ben would buy, the maximum price he would pay is $______________________
If Ben would not buy, the reason is ________________________________________
Part (b) (Worth 7 points): After a while, Ben gets tired of running BBC and decides to
hire someone to run BBC in his place. He is considering hiring Connor, but the problem
is that Ben knows that Connor does not like to work very hard. Ben knows that if he pays
Connor total compensation of w, Connor receives utility equal to UC(w) = w0.5 if Connor
exerts low effort and UC(w) = w0.5 – 75 if Connor exerts high effort. Ben cannot observe
Connor work, and Ben cannot tell how good the economy is going to be. All Ben
discovers are BBC’s profits (listed in the table above, which exclude any compensation to
Connor) at the end of the year.
Ben considers three compensation offers for Connor
(1) A fixed wage of $160,000 per year
(2) A wage of $90,000 per year plus a bonus of $160,000 if profits in the table above
are greater than or equal to $16,000,000
(3) A wage of $100,000 plus a bonus of 1% of the company’s profits in the table
above
Which compensation offer does Connor prefer? (Show the appropriate calculations)
Connor will prefer compensation offer [ (1), (2), (3) ]
CIRCLE ONE
because:_______________________________________________
Part (c) (Worth 7 points): Which compensation offer should Ben make to Connor?
(Show the appropriate calculations)
Ben should offer Connor compensation offer [ (1), (2), (3) ]
CIRCLE ONE
because:____________________________________________________
Question 1(a) Answer:
EULow = EΠLow = 0.3*4,000,000 + 0.4*9,000,000 + 0.3*16,000,000
= 1,200,000 + 3,600,000 + 4,800,000 = 9,600,000
EUHigh = EΠHigh = 0.3*9,000,000 + 0.4*16,000,000 + 0.3*25,000,000
= 2,700,000 + 6,400,000 + 7,500,000 = 16,600,000
Ben will work hard since 16,600,000 > 9,600,000
With the insurance payment (before the cost of the policy), Ben will now receive
9,000,000 if the economy is bad and he provides low effort and 14,000,000 if the
economy is bad and he provides high effort. With a policy of price P, Ben will net
9,000,000 – P with low effort and 14,000,000 – P with high effort.
Ben’s expected utility with low effort will be
EULow = EΠLow = 0.3*(4,000,000 + 5,000,000 – P) + 0.4*(9,000,000 – P)
+ 0.3*(16,000,000 – P)
= 0.3*(9,000,000 – P) + 0.4*(9,000,000 – P) + 0.3*(16,000,000 – P)
= 2,700,000 – 0.3P + 3,600,000 - 0.4P + 4,800,000 – 0.3P
= 11,100,000 – P
Ben’s expected utility with high effort will be
EULow = EΠLow = 0.3*(9,000,000 + 5,000,000 – P) + 0.4*(16,000,000 – P)
+ 0.3*(25,000,000 – P)
= 0.3*(14,000,000 – P) + 0.4*(16,000,000 –P) + 0.3*(25,000,000 – P)
= 4,200,000 – 0.3P + 6,400,000 – 0.4P + 7,500,000 – 0.3P
= 18,100,000 – P
Ben will work hard since 18,100,000 - P > 11,100,000 – P
Ben will be better off with the policy if 18,100,000 – P > 16,600,000
Or
1,500,000 > P
Thus Ben will be willing to pay up to 1,500,000 for the 5,000,000 policy
Question 1(b) Answer:
Under compensation offer 1, Connor’s expected utility if he exerts low effort is
EULow = 160,0000.5 = 400
Under compensation offer 1, Connor’s expected utility if he exerts high effort is
EULow = 160,0000.5 – 75 = 400 – 75 = 325
Since 400 > 325, Connor will exert low effort under compensation offer (1)
Under compensation offer 2, Connor’s expected utility if he exerts low effort is
EULow = 0.3*90,0000.5 + 0.4*90,0000.5 + 0.3(90,000 + 160,000)0.5
= 0.7*90,0000.5 + 0.3*250,0000.5
= 0.7*300 + 0.3*500 = 210 + 150 = 360
Under compensation offer 2, Connor’s expected utility if he exerts high effort is
EUHigh = 0.3*90,0000.5 + 0.4(90,000 + 160,000)0.5 + 0.3*(90,000 + 160,000)0.5 - 75
= 0.3*90,0000.5 + 0.7*250,0000.5 - 75
= 0.3*300 + 0.7*500 – 75 = 90 + 350 – 75 = 365
Since 365 > 360, Connor will exert high effort under compensation offer (2)
Under compensation offer 3, Connor’s expected utility if he exerts low effort is
EULow = 0.3*(100,000 + 0.01*4,000,000)0.5 + 0.4*(100,000 + 0.01*9,000,000)0.5
+ 0.3(100,000 + 0.01*16,000,000)0.5
= 0.3*(100,000 + 40,000)0.5 + 0.4*(100,000 + 90,000)0.5
+ 0.3*(100,000 + 160,000)0.5
= 0.3*(140,000)0.5 + 0.4*(190,000)0.5 + 0.3*(260,000)0.5
= 0.3*374.17 + 0.4*435.89 + 0.3*509.90
= 112.25 + 174.36 + 152.97 = 439.58
Under compensation offer 3, Connor’s expected utility if he exerts high effort is
EUHigh = 0.3*(100,000 + 0.01*9,000,000)0.5 + 0.4*(100,000 + 0.01*16,000,000)0.5
+ 0.3(100,000 + 0.01*25,000,000)0.5 - 75
= 0.3*(100,000 + 90,000)0.5 + 0.4*(100,000 + 160,000)0.5
+ 0.3*(100,000 + 250,000)0.5 - 75
= 0.3*(190,000)0.5 + 0.4*(260,000)0.5 + 0.3*(350,000)0.5 - 75
= 0.3*435.89 + 0.4*509.90 + 0.3*591.61 - 75
= 130.77 + 203.96 + 177.48 – 75 = 437.21
Since 439.58 > 437.21, Connor will exert low effort under compensation offer (3)
Connor will prefer compensation offer (3) because 439.58 > 400 > 365
Question 1(c) Answer: With compensation offer (1), Connor gives low effort, so Ben
makes expected profit of
0.3*4,000,000 + 0.4*9,000,000 + 0.3*16,000,000 – 160,000
= 1,200,000 + 3,600,000 + 4,800,000 – 160,000 = 9,440,000
With compensation offer (2), Connor gives high effort, so Ben makes expected profit of
0.3*9,000,000 + 0.4*16,000.000 + 0.3*25,000,000 – 90,000 – 0.7*160,000
= 2,700,000 + 6,400,000 + 7,500,000 – 90,000 – 112,000 = 16,398,000
With compensation offer (3), Connor gives low effort, so Ben makes expected profit of
0.99*(0.3*4,000,000 + 0.4*9,000,000 + 0.3*16,000,000) – 100,000
= 0.99(1,200,000 + 3,600,000 + 4,800,000) – 100,000
= 0.99(9,600,000) – 100,000
= 9,504,000 – 100,000 = 9,404,000
Ben prefers compensation offer (2) since 16,398,000 > 9,440,000 > 9,404,000
Question 2 (Worth 16 points)
Ed and Sarah each manage games on a boardwalk at the Jersey Shore. Their respective
profits depend on the price Ed charges for his game and the quality of prizes that Sarah
offers for winning her game. Ed can either charge a low (L), medium (M), or high (H)
price for playing his game, while Sarah can either offer cheap (1), average (2), or good
(3) quality prizes for playing her game. The payoffs to Ed and Sarah are shown in the
table below with Ed’s payoffs shown first in each matrix cell and Sarah’s payoffs shown
second.
Low (L)
Ed Medium (M)
High (H)
Sarah
Cheap (1) Average (2) Good (3)
( 4, 7)
(5, 8)
(2, 6)
( 1, 9)
(3, 9)
(4, 10)
(10, 8)
(2, 5)
(3, 15)
Initially, the game is played simultaneously
Part (a) (Worth 3 points): If both Ed and Sarah play maximin strategies, list all
maximin equilibria and show why they are maximin equilibria.
The maximin equilibria are:_________________________________________
because ___________________________________________________________
Part (b) (Worth 3 points): List all Nash equilibria and show why they are Nash
equilibria
The Nash equilibria are:_________________________________________
because ___________________________________________________________
Part (c) (Worth 3 points): Suppose the local government wants to discourage young
people from playing boardwalk games and so decides to pay Sarah $x if she chooses
Cheap (1) prizes but nothing if she chooses Average (2) or Good (3) quality prizes. Sarah
is aware of this payment when she makes her strategy choice. Find limits on the
possible values of x that make (high, cheap) a Nash equilibrium. Show all
supporting work
The limits on possible values of x to make (high, cheap) Nash are:____________
Part (d) (Worth 3 points): Now modify the original game (as in part (a)) so that Sarah
makes her choice first. What would each player’s action be and what would be the
outcome of the game? Show all supporting work.
Ed would charge ( Low, Medium, High )
CIRCLE ONE
because: _________________________________________________
Sarah would give ( Cheap, Average, Good ) quality prizes
CIRCLE ONE
because:____________________________________________________
Part (e) (Worth 4 points): Now modify the original game (as in part (a)) so that Ed
chooses the order of moves for the game (i.e., either Sarah goes first and then Ed OR Ed
goes first and then Sarah). Then the first player (as determined by Ed) moves followed by
the second. What order would Ed choose? What would each player’s action be and
what would be the outcome of the game? Show all supporting work.
Ed will choose ( Sarah goes first followed by Ed, Ed goes first followed by Sarah)
CIRCLE ONE
because ____________________________________________
As a result, Ed would charge ( Low, Medium, High )
CIRCLE ONE
because: _________________________________________________
and Sarah would give ( Cheap, Average, Good ) quality prizes
CIRCLE ONE
because:____________________________________________________
Question 2(a) Answer: Viewing the Table
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7)
(5,8)
(2,6)
Medium (M)
(1,9)
(3,9)
(4,10)
High (H)
(10,8)
(2,5)
(3,15)
Under Maximin, if Ed chooses low, he faces Min (4, 5, 2) = 2
if Ed chooses medium, he faces Min (1, 3, 4) = 1
if Ed chooses high, he faces Min (10, 2, 3) = 2
To maximize the minimums, Ed chooses Max (2, 1, 2) = 2
This entails Ed choosing either low or high
Under Maximin, if Sarah chooses cheap, she faces Min (7, 9, 8) = 7
if Sarah chooses average, she faces Min (8, 9, 5) = 5
if Sarah chooses good, she faces Min (6, 10, 15) = 6
To maximize the minimums, Sarah chooses Max (7, 5, 6) = 7
This entails Sarah choosing cheap
Thus there are two maximin solutions at (low, cheap) and (high, cheap)
Question 2(b) Answer: Viewing the Table
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7)
(5,8)
(2,6)
Medium (M)
(1,9)
(3,9)
(4,10)
High (H)
(10,8)
(2,5)
(3,15)
The bolds show Ed’s best payoffs given a prize quality of Sarah and the underlines show
Sarah’s best payoffs given a price of Ed in the above table
The Nashs occur in the cells with both a bold and an underline, i.e., (low, average)
and (medium, good)
Question 2(c) Answer: Viewing the Table from answer 2(b), in order for (high, cheap)
to be the Nash, Sarah must receive more than 8 if Ed prices low, more than 10 if Ed
prices medium, and more than 15 if Ed prices high.
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7)
(5,8)
(2,6)
Medium (M)
(1,9)
(3,9)
(4,10)
High (H)
(10,8)
(2,5)
(3,15)
Adding the x to Sarah’s cheap payoffs gives
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7+x)
(5,8)
(2,6)
Medium (M)
(1,9+x)
(3,9)
(4,10)
High (H)
(10,8+x)
(2,5)
(3,15)
Adding x > 7 (e.g., 7+) in the table gives
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7+7+)
(5,8)
(2,6)
Medium (M)
(1,9+7+)
(3,9)
(4,10)
High (H)
(10,8+7+)
(2,5)
(3,15)
or
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,14+)
(5,8)
(2,6)
Medium (M)
(1,16+)
(3,9)
(4,10)
High (H)
(10,15+)
(2,5)
(3,15)
With x > 7, (high, cheap) is the only cell with a bold and an underline and is the
Nash, i.e., (high, cheap)
Question 2(d) Answer: Viewing the Table
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7)
(5,8)
(2,6)
Medium (M)
(1,9)
(3,9)
(4,10)
High (H)
(10,8)
(2,5)
(3,15)
Sarah must reason backwards.
If she chooses cheap, Ed will price high (10 > 4 > 1) and Sarah will get 8.
If she chooses average, Ed will price low (5 > 3 > 2) and Sarah will get 8
If she chooses good, Ed will price medium (4 > 3 > 2) and Sarah will get 10
Sarah will choose good since 10 > 8. Ed will price medium and get 4
Question 2(e) Answer: From answer 2(d), we know that if Ed lets Sarah go first, that Ed
will get a payoff of 4.
Viewing the Table
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7)
(5,8)
(2,6)
Medium (M)
(1,9)
(3,9)
(4,10)
High (H)
(10,8)
(2,5)
(3,15)
Ed must reason backwards to figure out his payoff if he goes first.
If he chooses low, Sarah will choose average (8 > 7 > 6) and Ed will get 5.
If he chooses medium, Sarah will choose good (10 > 9) and Ed will get 4
If he chooses high, Sarah will choose good (15 > 8 > 5) and Ed will get 3
If Ed goes first, he will choose low since 5 > 4 > 3. Sarah will choose average and get 8
Since 5 > 4, Ed will choose that he goes first followed by Sarah. Ed will price low
and make 5 and Sarah will choose medium quality and make 8.
Question 3 (Worth 32 points): Bruce (B) and Patti (P) run competing cotton candy
stands. There is no difference in the quality of their products, but they do have different
total cost functions, which are given by
and
TC(qB) = 6qB + qB2
TC(qP) = 12qP
where qi is the number of units of cotton candy produced by producer i = B, P.
Bruce and Patti are the only two producers in their local area, and the demand for cotton
candy in the area is given by
Q(p) = 30 – 0.5p
where Q is the total units of cotton candy demanded when the price of a unit of cotton
candy is p. Non-integer number of units are permitted.
Part (a) (Worth 10 points): If Bruce and Patti determine quantities
SIMULTANEOUSLY, what quantities will they each choose? What will be the
market price? What profits will they each make? Show all supporting work.
Patti’s output will be _____________________units
Bruce’s output will be ______________________units
The market price will be $___________________
Patti’s profits will be $________________________
Bruce’s profits will be $________________________
Part (b) (Worth 7 points): If Bruce chooses his quantity FIRST and Patti follows,
what quantities will each choose? What will be the market price? What profits will
they each make? Show all supporting work.
Patti’s output will be _____________________units
Bruce’s output will be ______________________units
The market price will be $___________________
Patti’s profits will be $________________________
Bruce’s profits will be $________________________
Part (c) (Worth 10 points): Suppose that Patti and Bruce decide to get married, and
therefore intend to coordinate their production choices to maximize total profits for the
couple. Each will keep the profits earned from the quantity each produces at her/his own
factory. What quantity will each produce? What will be the market price? What
profits will they each make? Show all supporting work
Patti’s output will be _____________________units
Bruce’s output will be ______________________units
The market price will be $___________________
Patti’s profits will be $________________________
Bruce’s profits will be $________________________
Part (d) (Worth 5 points): Assume that the two of them only care about their own
profits and not about any other aspect of their marriage. Is marriage a Nash
equilibrium? If it is, explain your answer. If it is not, which partner would choose to
deviate? What would her and/or his profits be if she/he were the only one to deviate
from the collusive agreement in part (c)? (If both partners have incentive to deviate, it
is sufficient to show that one does). Show all supporting work.
Their marriage ( is, is not ) a Nash equilibrium
CIRCLE ONE
I answered “is” because _________________________________________
I answered “is not” and ( Patti, Bruce, Both ) will deviate from the agreement
CIRCLE ONE
and get profits of $______________________
Question 3(a) Answer: Rewrite the market demand as
Or
0.5p = 30 – Q
P = 60 – 2Q = 60 – 2qB – 2qP
Bruce’s total revenue is
TRB = pqB = (60 – 2qB – 2qP)qB = 60qB – 2qB2 – 2qBqP
Bruce’s marginal revenue is
MRB = ∂TRB/∂qB = 60 – 4qB – 2qP
Bruce’s marginal cost is
MCB = dTCB/dqB = 6 + 2qB
To maximize profit, Bruce will set MRB = MCB or
Or
Or
MRB = 60 – 4qB – 2qP = 6 + 2qB = MCB
6qB = 54 – 2qP
qB = 9 – (1/3)qP
This is Bruce’s reaction function
Patti’s total revenue is
TRP = pqP = (60 – 2qB – 2qP)qP = 60qP – 2qBqP– 2qP2
Patti’s marginal revenue is
MRP = ∂TRP/∂qP = 60 – 2qB – 4qP
Parri’s marginal cost is
MCP = dTCP/dqP = 12
To maximize profit, Patti will set MRP = MCP or
Or
Or
MRP = 60 – 2qB – 4qP = 12 = MCP
4qP = 48 – 2qP
qP = 12 – 0.5qB
This is Patti’s reaction function
Substituting qP = 12 – 0.5qB into Bruce’s reaction function gives
qB = 9 – (1/3)(12 - 0.5qB) = 9 – 4 + (1/6)qB = 5 + (1/6)qB
(5/6)qB = 5
qB = 6
or
or
Substituting qB = 6 into Patti’s reaction function gives
qP = 12 – 0.5*6 = 12 – 3 = 9
Q = qB + qP = 6 + 9 = 15. Substituting Q = 15 into the demand function gives
p = 60 – 2*15 = 60 – 30 = 30
Bruce’s total revenue = TRB = pqB = 30*6 = 180
Bruce’s total cost = TCB = 6*6 + 62 = 36 + 36 = 72
Bruce’s profit = ΠB = TRB – TCB = 180 – 72 = 108
Patti’s total revenue = TRP = pqP = 30*9 = 270
Patti’s total cost = TCP = 12*9 = 108
Patti’s profit = ΠP = TRP – TCP = 270 – 108 = 162
Question 3(b) Answer: Substitute Patti’s reaction function into the market demand
curve, i.e.,
P = 60 – 2qB – 2(12 – 0.5qB) = 60 – 2qB – 24 + qB = 36 – qB
This is the demand curve facing Bruce
Bruce’s total revenue is
TRB = pqB = (36 – qB)qB = 36qB – qB2
Bruce’s marginal revenue is
dTRB/dqB = MRB = 36 – 2qB
To maximize profit, Bruce will set MRB = MCB or
Or
Or
MRB = 36 – 2qB = 6 + 2qB = MCB
4qB = 30
qB = 7.5
Substituting qB = 7.5 into Patti’s reaction function gives
qP = 12 – 0.5*7.5 = 12 – 3.75 = 8.25
Q = qB + qP= 7.5 + 8.25 = 15.75
Substituting Q = 15.75 into the market demand function gives
p = 60 – 2*15.75 = 60 – 31.50 = 28.50
Bruce’s total revenue = TRB = pqB = 28.5*7.5 = 213.75
Bruce’s total cost = TCB = 6*7.5 + 7.52 = 45 + 56.25 = 101.25
Bruce’s profit = ΠB = TRB – TCB = 213.75 – 101.25 = 112.50
Patti’s total revenue = TRP = pqP = 28.5*8.25 = 235.125
Patti’s total cost = TCP = 12*8.25 = 99
Patti’s profit = ΠP = TRP – TCP = 235.125 – 99 = 136.125
Question 3(c) Answer The market demand curve is P = 60 – 2Q. So the total revenue of
the monopoly is
TR = pQ = (60 – 2Q)Q = 60Q – 2Q2
The marginal revenue of the monopolist is
MR = dTR/dQ = 60 – 4Q
View Bruce’s and Patti’s marginal costs in the figure below
$
MCB
12
MCP
6
3
Q
Equating the two marginal costs, i.e.,
MCB = 6 + 2qB = 12 = MCP
we see that the two MCs are equal when 2qB = 6 or qB = 3. For ouputs less than 3, Bruce
has the lower marginal cost and for outputs greater than 3, Patti has the lower marginal
cost. Thus Bruce should produce the first 3 units and Patti should produce the remainder
(if necessary). Thus, the monopoly’s marginal cost is
MC = 6 + 2Q for Q < 3 and MC = 12 for Q > 3
To maximize profit, the monopolist will set MR = MC. Let’s do this with MC = 12 first,
i.e.,
MR = 60 – 4Q = 12 = MC
Or
4Q = 48
Or
Q = 12
Since Q = 12 > 3, using MC = 12 was correct.
Let’s see what would happen if we used MC = 6 + 2Q, i.e.,
MR = 60 – 4Q = 6 + 2Q = MC
Or
6Q = 54
Or
Q=9
But MC = 6 + 2Q only holds if Q < 3. So this can’t be the correct marginal cost.
Bruce will produce qB = 3 and Patti will produce qP = 9 = Q – qB = 12 – 3
Substituting Q = 12 into the market demand curve gives
p = 60 – 2*12 = 60 – 24 = 36
Bruce’s total revenue = TRB = pqB = 36*3 = 108
Bruce’s total cost = TCB = 6*3 + 32 = 18 + 9 = 27
Bruce’s profit = ΠB = TRB – TCB = 108 – 27 = 81
Patti’s total revenue = TRP = pqP = 36*9 = 324
Patti’s total cost = TCP = 12*9 = 108
Patti’s profit = ΠP = TRP – TCP = 324 – 108 = 216
Question 3(d) Answer: Patti’s reaction function is qP = 12 – 0.5qB. Substituting qB = 3
into Patti’s reaction function gives
qP = 12 – 0.5*3 = 12 – 1.5 = 10.5
Q = qB + qP = 3 + 10.5 = 13.5
Substituting Q = 13.5 into the market demand curve gives
p = 60 – 2*13.5 = 60 – 27 = 33
Patti’s total revenue = TRP = pqP = 33*10.5 = 346.5
Patti’s total cost = TCP = 12*10.5 = 126
Patti’s profit = ΠP = TRP – TCP = 346.5 – 126 = 220.5
Since Patti made 216 under monopoly, she would wish to leave if Bruce stayed at output
3, i.e., 220.5 > 216. So Patti has an incentive to deviate.
Bruce’s reaction function is qB = 9 – (1/3)qP. Substituting qP = 9 into Bruce’s reaction
function gives
qB = 9 – (1/3)*9 = 9 – 3 = 6
Q = qB + qP = 6 + 9 = 15
Substituting Q = 15 into the market demand curve gives
p = 60 – 2*15 = 60 – 30 = 30
Bruce’s total revenue = TRB = pqB = 30*6 = 180
Bruce’s total cost = TCB = 6*6 + 62 = 36 + 36 = 72
Bruce’s profit = ΠB = TRB – TCB = 180 - 72 = 108
Since Bruce made 81 under monopoly, he would wish to leave if Patti stayed at output 9,
i.e., 108 > 81, so Bruce has an incentive to deviate.
So the marriage is not a Nash equilibrium
Question 4 (Worth 28 points): Cristiano has a monopoly in the production of
calculators. His marginal costs are given by
MC(q) = 8 + 2q
where q is the number of calculators produced by Cristiano. Cristiano has no fixed costs.
Market demand for calculators is given by
q(p) = 40 – 0.5p
where q is the quantity of calculators demanded when the price of a calculator is p.
Part (a) (Worth 5 points): What are the monopoly quantity, price, and Cristiano’s
profits? Show all supporting work
The monopoly quantity is _____________________units
The monopoly price is $_________________________
Cristiano’s monopoly profits are $__________________
Part (b) (Worth 5 points): What would the socially efficient quantity be? How much
larger is consumer surplus at the socially efficient quantity than at the single price
(part (a)) monopoly quantity? How much smaller would Cristiano’s profits be if he
produced the socially efficient quantity? Show all supporting work.
The socially efficient quantity is __________________units
The consumer surplus at the socially efficient quantity is $_________________
larger than the consumer surplus at the monopoly quantity
Cristiano’s profits at the socially efficient quantity are $_________________
smaller than his profits at the monopoly quantity
Part (c) (Worth 11 points): Now Cristiano notices that his student buyers have different
buying patterns than the rest of his customers (non-students), and he estimates that the
subset of his demand that represents students is
qS(p) = 16 – 0.25p
where qS is the quantity of calculators demanded by students when the price of a
calculator is p. Demand for the entire market is still as described above.
What price should Cristiano charge to each group, i.e., the students and the nonstudents? How do Cristiano’s profits and total quantity produced change compared
to the monopoly results in part (a)? Show all supporting work
The profit maximizing price for students is $_______________________
The profit maximizing price for non-students is $____________________
The difference in total quantity produced by Cristiano in this part of the problem
compared to the total quantity produced in part (a) is
__________________units
The profits made in this part of the problem are ( greater, less, the same)
CIRCLE ONE
by $______________________ than the profits made in part (a)
Part (d) (Worth 7 points): The city government estimates that each unit that Cristiano
sells causes damage to the environment. This damage increases linearly with Cristiano’s
total production; in particular, the sale of the qth unit causes damage equal to $5q.
What is the socially efficient quantity now? Suggest a government policy that
induces Cristiano to produce the socially efficient quantity. (Assume that Cristiano
can no longer identify students and that the government cannot tell Cristiano what
quantity to produce). Show all supporting work.
The socially efficient quantity is __________________units
I suggest that the government institute a policy of _______________________
Question 4(a) Answer: Rewrite the demand function as
Or
0.5p = 40 – q
p = 80 – 2q
The total revenue of the monopoly is
TR = pq = (80 – 2q)q = 80q – 2q2
The marginal revenue of the monopoly is
MR = dTR/dq = 80 – 4q
The firm will maximize profit where MR = MC or
Or
Or
MR = 80 – 4q = 8 + 2q = MC
6q = 72
q = 12
Substituting q = 12 into the demand curve gives
p = 80 – 2*12 = 80 – 24 = 56
If the firm’s marginal cost is MC = 8 + 2q, the firm’s variable cost must be VC = 8q + q2
such that
MC = dVC/dq = 8 + 2q
The VC is also the firm’s total cost since fixed costs are 0.
The firm’s total revenue = TR = 56*12 = 672
The firm’s total cost = 8*12 + 122 = 96 + 144 = 240
The firm’s profit is = Π = TR – TC = 672 – 240 = 432
Question 4(b) Answer: The socially efficient output is where p = MC or
or
or
p = 80 – 2q = 8 + 2q = MC
4q = 72
q = 18
Substituting q = 18 into the demand curve gives
p = 80 – 2*18 = 80 – 36 = 44
The firm’s total revenue = TR = 44*18 = 792
The firm’s total cost = 8*18 + 182 = 144 + 324 = 468
The firm’s profit is = Π = TR – TC = 792 – 468 = 324
Since the profits were 432 under monopoly and 324 at the socially optimal level, the
firm’s profits are 432 – 324 = 108 smaller.
Consumer surplus under monopoly is 0.5*(80 – 56)*12 = 0.5*24*12 = 144
The socially optimal consumer surplus is 0.5*(80 – 44)*18 = 0.5*36*18 = 324
Thus consumer surplus is 324 – 144 = 180 larger at the socially efficient level
Question 4(c) Answer: If student demanders are qS = 16 – 0.25p, then non-student
demanders (N) must be
qN = q – qS = 40 – 0.5p – (16 – 0.25p) = 24 – 0.25p
Rewrite the student demand as
Or
0.25p = 16 – qS
p = 64 – 4qS
Rewrite the non-student demand as
Or
0.25p = 24 – qN
p = 96 – 4qN
Thus our market demand of p = 80 – 2q describes the market when P < 64 or where q > 8
(note that when qN = 8, then p = 96 – 4*8 = 96 – 32 = 64. Since our solutions above of q
= 12 and q = 18 are greater than 8, the market demand curve was the correct demand
curve to use. If q is less than 8, then the market demand curve is just the non-student
demand curve, i.e., p = 96 – 4q.
The total revenue from the student market will be
TRS = pSqS = (64- 4qS)qS= 64qS – 4qS2
The marginal revenue in the student market is
MRS = dTRS/dqS = 64 – 8qS
The total revenue from the non-student market will be
TRN = pNqN = (96- 4qN)qN= 96qN – 4qN2
The marginal revenue in the non-student market is
MRN = dTRN/dqN = 96 – 8qN
To maximize profit, the firm will set MRN = MRS = MC. Let’s view MRN = MRS, i.e.,
MRN = 96 – 8qN = 64 – 8qS = MRS
Thus
Or
8qN = 32 + 8qS
qN = 4 + qS
The firm’s marginal cost is MC = 8 + 2q = 8 + 2qN + 2qS
Substitute qN = 4 + qS in the MC equation, i.e.,
MC = 8 + 2(4 + qS) + 2qS = 8 + 8 + 2qS + 2qS = 16 + 4qS
We know that MRS = MC for profit maximization, so
Or
Or
MRS = 64 – 8qS = 16 + 4qS = MC
12qS = 48
qS = 4
Substituting qS = 4 into qN = 8 + qS gives qN = 8 + 4 = 12. Note that q = qS + qN = 4 + 8
= 12, so that the quantity that the firm produces compared with the single price
monopoly doesn’t change.
Substituting qS = 4 into the student demand curve gives
pS = 64 – 4*4 = 64 – 16 = 48
The total revenue from the student market = TRS = pSqS = 48*4 = 192
Substituting qN = 8 into the non-student demand curve gives
pN = 96 – 4*8 = 96 – 32 = 64
The total revenue from the non-student market = TRN = pNqN = 64*8 = 512
The total revenue of the firm is TR = TRS + TRN = 192 + 512 = 704
The total cost is TC = 8q + q2 = 8*12 + 122 = 96 + 144 = 240
The firm’s profit is Π = TR – TC = 704 – 240 = 464.
Since profit under single price monopoly was 432 and under third degree price
discrimination is 464, profit has changed by 464 – 432 = 32
Question 4(d) Answer:.The socially efficient quantity is were MSB = MSC. The MSB is
just the demand curve of P = 80 – 2q. The MSC = MPC + externality cost, i.e.,
MSC = 8 + 2q + 5q = 8 + 7q
Thus
or
or
MSB = 80 – 2q = 8 + 7q = MSC
9q = 72
q=8
The firm will want to produce where its MR (80 – 4q) equals its MC (8 + 2q). To
internalize the externality, we’ll place a tax of T on each unit the firm produces. Thus the
firm’s MC becomes MC = 8 + 2q + T.
Now when the firm sets MR = MC, they set
Or
Or
MR = 80 – 4q = 8 + 2q + T = MC
6q = 72 – T
q = 12 – (1/6)T
We know that we want q = 8 to attain the socially efficient output. Setting q = 8 in the
equation above yields
Or
Or
8 = 12 – (1/6)T
(1/6)T = 4
T = 24
Substituting q = 8 into the demand curve gives P = 80 – 2*8 = 80 – 16 = 64. The only
buyers will be non-students.
The firm’s total revenue = TR = pqN = 64*8 = 512
The firm’s total private cost = TC = 8*8 + 82 = 64 + 64 = 128
The firm’s payment in taxes = TqN = 24*8 = 192
The firm’s profit = Π = TR – TC – TqN = 512 – 128 – 192 = 192
If the government imposed a standard of q = 8, the firm would price at 64 and make
profits of 384 and government tax revenue would be zero. Consumer surplus would be
the same under the tax or the standard (because the price to consumers under each would
be 64).
If the market was only the non-students, the firm would set MR = MC + T or
Or
MR = 96 – 8q = 8 + 2q + T
10q + T = 88
Since q must be 8, substituting q = 8 gives
Or
10*8 + T = 88
80 + T = 88
Or
T=8
Question 1 (Worth 24 points)
Ben owns a business selling baseball cards called Ben’s Baseball Cards (BBC). BBC’s
profits depend both on the state of the economy and how hard he works managing his
business. Ben does not mind working hard at all and is risk neutral, so his utility function
is given by UB(Π) = Π, where Π represents the profits of BBC. Profits and the
probabilities of each state of the economy are given in the table below.
Type of Economy
Bad (B)
Medium (M) Good (G)
Probability of Economy
30%
40%
30%
Low Effort of Manager
$4,000,000
$9,000,000
$16,000,000
High Effort of Manager
$9,000,000
$16,000,000
$25,000,000
Initially Ben runs the company himself.
Part (a) (Worth 10 points): What is Ben’s expected utility? Would Ben be willing to
purchase an insurance policy that pays him $5,000,000, in addition to his profits, if
there is a bad economy? If yes, calculate the maximum price he would pay for the
policy. If no, explain why in one sentence or prove mathematically. Show all
supporting work.
Ben’s expected utility is ____________________
Ben ( Would, Would Not) buy the insurance policy
CIRCLE ONE
If Ben would buy, the maximum price he would pay is $______________________
If Ben would not buy, the reason is ________________________________________
Part (b) (Worth 7 points): After a while, Ben gets tired of running BBC and decides to
hire someone to run BBC in his place. He is considering hiring Connor, but the problem
is that Ben knows that Connor does not like to work very hard. Ben knows that if he pays
Connor total compensation of w, Connor receives utility equal to UC(w) = w0.5 if Connor
exerts low effort and UC(w) = w0.5 – 75 if Connor exerts high effort. Ben cannot observe
Connor work, and Ben cannot tell how good the economy is going to be. All Ben
discovers are BBC’s profits (listed in the table above, which exclude any compensation to
Connor) at the end of the year.
Ben considers three compensation offers for Connor
(1) A fixed wage of $160,000 per year
(2) A wage of $90,000 per year plus a bonus of $160,000 if profits in the table above
are greater than or equal to $16,000,000
(3) A wage of $100,000 plus a bonus of 1% of the company’s profits in the table
above
Which compensation offer does Connor prefer? (Show the appropriate calculations)
Connor will prefer compensation offer [ (1), (2), (3) ]
CIRCLE ONE
because:_______________________________________________
Part (c) (Worth 7 points): Which compensation offer should Ben make to Connor?
(Show the appropriate calculations)
Ben should offer Connor compensation offer [ (1), (2), (3) ]
CIRCLE ONE
because:____________________________________________________
Question 1(a) Answer:
EULow = EΠLow = 0.3*4,000,000 + 0.4*9,000,000 + 0.3*16,000,000
= 1,200,000 + 3,600,000 + 4,800,000 = 9,600,000
EUHigh = EΠHigh = 0.3*9,000,000 + 0.4*16,000,000 + 0.3*25,000,000
= 2,700,000 + 6,400,000 + 7,500,000 = 16,600,000
Ben will work hard since 16,600,000 > 9,600,000
With the insurance payment (before the cost of the policy), Ben will now receive
9,000,000 if the economy is bad and he provides low effort and 14,000,000 if the
economy is bad and he provides high effort. With a policy of price P, Ben will net
9,000,000 – P with low effort and 14,000,000 – P with high effort.
Ben’s expected utility with low effort will be
EULow = EΠLow = 0.3*(4,000,000 + 5,000,000 – P) + 0.4*(9,000,000 – P)
+ 0.3*(16,000,000 – P)
= 0.3*(9,000,000 – P) + 0.4*(9,000,000 – P) + 0.3*(16,000,000 – P)
= 2,700,000 – 0.3P + 3,600,000 - 0.4P + 4,800,000 – 0.3P
= 11,100,000 – P
Ben’s expected utility with high effort will be
EULow = EΠLow = 0.3*(9,000,000 + 5,000,000 – P) + 0.4*(16,000,000 – P)
+ 0.3*(25,000,000 – P)
= 0.3*(14,000,000 – P) + 0.4*(16,000,000 –P) + 0.3*(25,000,000 – P)
= 4,200,000 – 0.3P + 6,400,000 – 0.4P + 7,500,000 – 0.3P
= 18,100,000 – P
Ben will work hard since 18,100,000 - P > 11,100,000 – P
Ben will be better off with the policy if 18,100,000 – P > 16,600,000
Or
1,500,000 > P
Thus Ben will be willing to pay up to 1,500,000 for the 5,000,000 policy
Question 1(b) Answer:
Under compensation offer 1, Connor’s expected utility if he exerts low effort is
EULow = 160,0000.5 = 400
Under compensation offer 1, Connor’s expected utility if he exerts high effort is
EULow = 160,0000.5 – 75 = 400 – 75 = 325
Since 400 > 325, Connor will exert low effort under compensation offer (1)
Under compensation offer 2, Connor’s expected utility if he exerts low effort is
EULow = 0.3*90,0000.5 + 0.4*90,0000.5 + 0.3(90,000 + 160,000)0.5
= 0.7*90,0000.5 + 0.3*250,0000.5
= 0.7*300 + 0.3*500 = 210 + 150 = 360
Under compensation offer 2, Connor’s expected utility if he exerts high effort is
EUHigh = 0.3*90,0000.5 + 0.4(90,000 + 160,000)0.5 + 0.3*(90,000 + 160,000)0.5 - 75
= 0.3*90,0000.5 + 0.7*250,0000.5 - 75
= 0.3*300 + 0.7*500 – 75 = 90 + 350 – 75 = 365
Since 365 > 360, Connor will exert high effort under compensation offer (2)
Under compensation offer 3, Connor’s expected utility if he exerts low effort is
EULow = 0.3*(100,000 + 0.01*4,000,000)0.5 + 0.4*(100,000 + 0.01*9,000,000)0.5
+ 0.3(100,000 + 0.01*16,000,000)0.5
= 0.3*(100,000 + 40,000)0.5 + 0.4*(100,000 + 90,000)0.5
+ 0.3*(100,000 + 160,000)0.5
= 0.3*(140,000)0.5 + 0.4*(190,000)0.5 + 0.3*(260,000)0.5
= 0.3*374.17 + 0.4*435.89 + 0.3*509.90
= 112.25 + 174.36 + 152.97 = 439.58
Under compensation offer 3, Connor’s expected utility if he exerts high effort is
EUHigh = 0.3*(100,000 + 0.01*9,000,000)0.5 + 0.4*(100,000 + 0.01*16,000,000)0.5
+ 0.3(100,000 + 0.01*25,000,000)0.5 - 75
= 0.3*(100,000 + 90,000)0.5 + 0.4*(100,000 + 160,000)0.5
+ 0.3*(100,000 + 250,000)0.5 - 75
= 0.3*(190,000)0.5 + 0.4*(260,000)0.5 + 0.3*(350,000)0.5 - 75
= 0.3*435.89 + 0.4*509.90 + 0.3*591.61 - 75
= 130.77 + 203.96 + 177.48 – 75 = 437.21
Since 439.58 > 437.21, Connor will exert low effort under compensation offer (3)
Connor will prefer compensation offer (3) because 439.58 > 400 > 365
Question 1(c) Answer: With compensation offer (1), Connor gives low effort, so Ben
makes expected profit of
0.3*4,000,000 + 0.4*9,000,000 + 0.3*16,000,000 – 160,000
= 1,200,000 + 3,600,000 + 4,800,000 – 160,000 = 9,440,000
With compensation offer (2), Connor gives high effort, so Ben makes expected profit of
0.3*9,000,000 + 0.4*16,000.000 + 0.3*25,000,000 – 90,000 – 0.7*160,000
= 2,700,000 + 6,400,000 + 7,500,000 – 90,000 – 112,000 = 16,398,000
With compensation offer (3), Connor gives low effort, so Ben makes expected profit of
0.99*(0.3*4,000,000 + 0.4*9,000,000 + 0.3*16,000,000) – 100,000
= 0.99(1,200,000 + 3,600,000 + 4,800,000) – 100,000
= 0.99(9,600,000) – 100,000
= 9,504,000 – 100,000 = 9,404,000
Ben prefers compensation offer (2) since 16,398,000 > 9,440,000 > 9,404,000
Question 2 (Worth 16 points)
Ed and Sarah each manage games on a boardwalk at the Jersey Shore. Their respective
profits depend on the price Ed charges for his game and the quality of prizes that Sarah
offers for winning her game. Ed can either charge a low (L), medium (M), or high (H)
price for playing his game, while Sarah can either offer cheap (1), average (2), or good
(3) quality prizes for playing her game. The payoffs to Ed and Sarah are shown in the
table below with Ed’s payoffs shown first in each matrix cell and Sarah’s payoffs shown
second.
Low (L)
Ed Medium (M)
High (H)
Sarah
Cheap (1) Average (2) Good (3)
( 4, 7)
(5, 8)
(2, 6)
( 1, 9)
(3, 9)
(4, 10)
(10, 8)
(2, 5)
(3, 15)
Initially, the game is played simultaneously
Part (a) (Worth 3 points): If both Ed and Sarah play maximin strategies, list all
maximin equilibria and show why they are maximin equilibria.
The maximin equilibria are:_________________________________________
because ___________________________________________________________
Part (b) (Worth 3 points): List all Nash equilibria and show why they are Nash
equilibria
The Nash equilibria are:_________________________________________
because ___________________________________________________________
Part (c) (Worth 3 points): Suppose the local government wants to discourage young
people from playing boardwalk games and so decides to pay Sarah $x if she chooses
Cheap (1) prizes but nothing if she chooses Average (2) or Good (3) quality prizes. Sarah
is aware of this payment when she makes her strategy choice. Find limits on the
possible values of x that make (high, cheap) a Nash equilibrium. Show all
supporting work
The limits on possible values of x to make (high, cheap) Nash are:____________
Part (d) (Worth 3 points): Now modify the original game (as in part (a)) so that Sarah
makes her choice first. What would each player’s action be and what would be the
outcome of the game? Show all supporting work.
Ed would charge ( Low, Medium, High )
CIRCLE ONE
because: _________________________________________________
Sarah would give ( Cheap, Average, Good ) quality prizes
CIRCLE ONE
because:____________________________________________________
Part (e) (Worth 4 points): Now modify the original game (as in part (a)) so that Ed
chooses the order of moves for the game (i.e., either Sarah goes first and then Ed OR Ed
goes first and then Sarah). Then the first player (as determined by Ed) moves followed by
the second. What order would Ed choose? What would each player’s action be and
what would be the outcome of the game? Show all supporting work.
Ed will choose ( Sarah goes first followed by Ed, Ed goes first followed by Sarah)
CIRCLE ONE
because ____________________________________________
As a result, Ed would charge ( Low, Medium, High )
CIRCLE ONE
because: _________________________________________________
and Sarah would give ( Cheap, Average, Good ) quality prizes
CIRCLE ONE
because:____________________________________________________
Question 2(a) Answer: Viewing the Table
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7)
(5,8)
(2,6)
Medium (M)
(1,9)
(3,9)
(4,10)
High (H)
(10,8)
(2,5)
(3,15)
Under Maximin, if Ed chooses low, he faces Min (4, 5, 2) = 2
if Ed chooses medium, he faces Min (1, 3, 4) = 1
if Ed chooses high, he faces Min (10, 2, 3) = 2
To maximize the minimums, Ed chooses Max (2, 1, 2) = 2
This entails Ed choosing either low or high
Under Maximin, if Sarah chooses cheap, she faces Min (7, 9, 8) = 7
if Sarah chooses average, she faces Min (8, 9, 5) = 5
if Sarah chooses good, she faces Min (6, 10, 15) = 6
To maximize the minimums, Sarah chooses Max (7, 5, 6) = 7
This entails Sarah choosing cheap
Thus there are two maximin solutions at (low, cheap) and (high, cheap)
Question 2(b) Answer: Viewing the Table
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7)
(5,8)
(2,6)
Medium (M)
(1,9)
(3,9)
(4,10)
High (H)
(10,8)
(2,5)
(3,15)
The bolds show Ed’s best payoffs given a prize quality of Sarah and the underlines show
Sarah’s best payoffs given a price of Ed in the above table
The Nashs occur in the cells with both a bold and an underline, i.e., (low, average)
and (medium, good)
Question 2(c) Answer: Viewing the Table from answer 2(b), in order for (high, cheap)
to be the Nash, Sarah must receive more than 8 if Ed prices low, more than 10 if Ed
prices medium, and more than 15 if Ed prices high.
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7)
(5,8)
(2,6)
Medium (M)
(1,9)
(3,9)
(4,10)
High (H)
(10,8)
(2,5)
(3,15)
Adding the x to Sarah’s cheap payoffs gives
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7+x)
(5,8)
(2,6)
Medium (M)
(1,9+x)
(3,9)
(4,10)
High (H)
(10,8+x)
(2,5)
(3,15)
Adding x > 7 (e.g., 7+) in the table gives
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7+7+)
(5,8)
(2,6)
Medium (M)
(1,9+7+)
(3,9)
(4,10)
High (H)
(10,8+7+)
(2,5)
(3,15)
or
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,14+)
(5,8)
(2,6)
Medium (M)
(1,16+)
(3,9)
(4,10)
High (H)
(10,15+)
(2,5)
(3,15)
With x > 7, (high, cheap) is the only cell with a bold and an underline and is the
Nash, i.e., (high, cheap)
Question 2(d) Answer: Viewing the Table
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7)
(5,8)
(2,6)
Medium (M)
(1,9)
(3,9)
(4,10)
High (H)
(10,8)
(2,5)
(3,15)
Sarah must reason backwards.
If she chooses cheap, Ed will price high (10 > 4 > 1) and Sarah will get 8.
If she chooses average, Ed will price low (5 > 3 > 2) and Sarah will get 8
If she chooses good, Ed will price medium (4 > 3 > 2) and Sarah will get 10
Sarah will choose good since 10 > 8. Ed will price medium and get 4
Question 2(e) Answer: From answer 2(d), we know that if Ed lets Sarah go first, that Ed
will get a payoff of 4.
Viewing the Table
Sarah
Cheap (1) Average (2) Good (3)
Ed
Low (L)
(4,7)
(5,8)
(2,6)
Medium (M)
(1,9)
(3,9)
(4,10)
High (H)
(10,8)
(2,5)
(3,15)
Ed must reason backwards to figure out his payoff if he goes first.
If he chooses low, Sarah will choose average (8 > 7 > 6) and Ed will get 5.
If he chooses medium, Sarah will choose good (10 > 9) and Ed will get 4
If he chooses high, Sarah will choose good (15 > 8 > 5) and Ed will get 3
If Ed goes first, he will choose low since 5 > 4 > 3. Sarah will choose average and get 8
Since 5 > 4, Ed will choose that he goes first followed by Sarah. Ed will price low
and make 5 and Sarah will choose medium quality and make 8.
Question 3 (Worth 32 points): Bruce (B) and Patti (P) run competing cotton candy
stands. There is no difference in the quality of their products, but they do have different
total cost functions, which are given by
and
TC(qB) = 6qB + qB2
TC(qP) = 12qP
where qi is the number of units of cotton candy produced by producer i = B, P.
Bruce and Patti are the only two producers in their local area, and the demand for cotton
candy in the area is given by
Q(p) = 30 – 0.5p
where Q is the total units of cotton candy demanded when the price of a unit of cotton
candy is p. Non-integer number of units are permitted.
Part (a) (Worth 10 points): If Bruce and Patti determine quantities
SIMULTANEOUSLY, what quantities will they each choose? What will be the
market price? What profits will they each make? Show all supporting work.
Patti’s output will be _____________________units
Bruce’s output will be ______________________units
The market price will be $___________________
Patti’s profits will be $________________________
Bruce’s profits will be $________________________
Part (b) (Worth 7 points): If Bruce chooses his quantity FIRST and Patti follows,
what quantities will each choose? What will be the market price? What profits will
they each make? Show all supporting work.
Patti’s output will be _____________________units
Bruce’s output will be ______________________units
The market price will be $___________________
Patti’s profits will be $________________________
Bruce’s profits will be $________________________
Part (c) (Worth 10 points): Suppose that Patti and Bruce decide to get married, and
therefore intend to coordinate their production choices to maximize total profits for the
couple. Each will keep the profits earned from the quantity each produces at her/his own
factory. What quantity will each produce? What will be the market price? What
profits will they each make? Show all supporting work
Patti’s output will be _____________________units
Bruce’s output will be ______________________units
The market price will be $___________________
Patti’s profits will be $________________________
Bruce’s profits will be $________________________
Part (d) (Worth 5 points): Assume that the two of them only care about their own
profits and not about any other aspect of their marriage. Is marriage a Nash
equilibrium? If it is, explain your answer. If it is not, which partner would choose to
deviate? What would her and/or his profits be if she/he were the only one to deviate
from the collusive agreement in part (c)? (If both partners have incentive to deviate, it
is sufficient to show that one does). Show all supporting work.
Their marriage ( is, is not ) a Nash equilibrium
CIRCLE ONE
I answered “is” because _________________________________________
I answered “is not” and ( Patti, Bruce, Both ) will deviate from the agreement
CIRCLE ONE
and get profits of $______________________
Question 3(a) Answer: Rewrite the market demand as
Or
0.5p = 30 – Q
P = 60 – 2Q = 60 – 2qB – 2qP
Bruce’s total revenue is
TRB = pqB = (60 – 2qB – 2qP)qB = 60qB – 2qB2 – 2qBqP
Bruce’s marginal revenue is
MRB = ∂TRB/∂qB = 60 – 4qB – 2qP
Bruce’s marginal cost is
MCB = dTCB/dqB = 6 + 2qB
To maximize profit, Bruce will set MRB = MCB or
Or
Or
MRB = 60 – 4qB – 2qP = 6 + 2qB = MCB
6qB = 54 – 2qP
qB = 9 – (1/3)qP
This is Bruce’s reaction function
Patti’s total revenue is
TRP = pqP = (60 – 2qB – 2qP)qP = 60qP – 2qBqP– 2qP2
Patti’s marginal revenue is
MRP = ∂TRP/∂qP = 60 – 2qB – 4qP
Parri’s marginal cost is
MCP = dTCP/dqP = 12
To maximize profit, Patti will set MRP = MCP or
Or
Or
MRP = 60 – 2qB – 4qP = 12 = MCP
4qP = 48 – 2qP
qP = 12 – 0.5qB
This is Patti’s reaction function
Substituting qP = 12 – 0.5qB into Bruce’s reaction function gives
qB = 9 – (1/3)(12 - 0.5qB) = 9 – 4 + (1/6)qB = 5 + (1/6)qB
(5/6)qB = 5
qB = 6
or
or
Substituting qB = 6 into Patti’s reaction function gives
qP = 12 – 0.5*6 = 12 – 3 = 9
Q = qB + qP = 6 + 9 = 15. Substituting Q = 15 into the demand function gives
p = 60 – 2*15 = 60 – 30 = 30
Bruce’s total revenue = TRB = pqB = 30*6 = 180
Bruce’s total cost = TCB = 6*6 + 62 = 36 + 36 = 72
Bruce’s profit = ΠB = TRB – TCB = 180 – 72 = 108
Patti’s total revenue = TRP = pqP = 30*9 = 270
Patti’s total cost = TCP = 12*9 = 108
Patti’s profit = ΠP = TRP – TCP = 270 – 108 = 162
Question 3(b) Answer: Substitute Patti’s reaction function into the market demand
curve, i.e.,
P = 60 – 2qB – 2(12 – 0.5qB) = 60 – 2qB – 24 + qB = 36 – qB
This is the demand curve facing Bruce
Bruce’s total revenue is
TRB = pqB = (36 – qB)qB = 36qB – qB2
Bruce’s marginal revenue is
dTRB/dqB = MRB = 36 – 2qB
To maximize profit, Bruce will set MRB = MCB or
Or
Or
MRB = 36 – 2qB = 6 + 2qB = MCB
4qB = 30
qB = 7.5
Substituting qB = 7.5 into Patti’s reaction function gives
qP = 12 – 0.5*7.5 = 12 – 3.75 = 8.25
Q = qB + qP= 7.5 + 8.25 = 15.75
Substituting Q = 15.75 into the market demand function gives
p = 60 – 2*15.75 = 60 – 31.50 = 28.50
Bruce’s total revenue = TRB = pqB = 28.5*7.5 = 213.75
Bruce’s total cost = TCB = 6*7.5 + 7.52 = 45 + 56.25 = 101.25
Bruce’s profit = ΠB = TRB – TCB = 213.75 – 101.25 = 112.50
Patti’s total revenue = TRP = pqP = 28.5*8.25 = 235.125
Patti’s total cost = TCP = 12*8.25 = 99
Patti’s profit = ΠP = TRP – TCP = 235.125 – 99 = 136.125
Question 3(c) Answer The market demand curve is P = 60 – 2Q. So the total revenue of
the monopoly is
TR = pQ = (60 – 2Q)Q = 60Q – 2Q2
The marginal revenue of the monopolist is
MR = dTR/dQ = 60 – 4Q
View Bruce’s and Patti’s marginal costs in the figure below
$
MCB
12
MCP
6
3
Q
Equating the two marginal costs, i.e.,
MCB = 6 + 2qB = 12 = MCP
we see that the two MCs are equal when 2qB = 6 or qB = 3. For ouputs less than 3, Bruce
has the lower marginal cost and for outputs greater than 3, Patti has the lower marginal
cost. Thus Bruce should produce the first 3 units and Patti should produce the remainder
(if necessary). Thus, the monopoly’s marginal cost is
MC = 6 + 2Q for Q < 3 and MC = 12 for Q > 3
To maximize profit, the monopolist will set MR = MC. Let’s do this with MC = 12 first,
i.e.,
MR = 60 – 4Q = 12 = MC
Or
4Q = 48
Or
Q = 12
Since Q = 12 > 3, using MC = 12 was correct.
Let’s see what would happen if we used MC = 6 + 2Q, i.e.,
MR = 60 – 4Q = 6 + 2Q = MC
Or
6Q = 54
Or
Q=9
But MC = 6 + 2Q only holds if Q < 3. So this can’t be the correct marginal cost.
Bruce will produce qB = 3 and Patti will produce qP = 9 = Q – qB = 12 – 3
Substituting Q = 12 into the market demand curve gives
p = 60 – 2*12 = 60 – 24 = 36
Bruce’s total revenue = TRB = pqB = 36*3 = 108
Bruce’s total cost = TCB = 6*3 + 32 = 18 + 9 = 27
Bruce’s profit = ΠB = TRB – TCB = 108 – 27 = 81
Patti’s total revenue = TRP = pqP = 36*9 = 324
Patti’s total cost = TCP = 12*9 = 108
Patti’s profit = ΠP = TRP – TCP = 324 – 108 = 216
Question 3(d) Answer: Patti’s reaction function is qP = 12 – 0.5qB. Substituting qB = 3
into Patti’s reaction function gives
qP = 12 – 0.5*3 = 12 – 1.5 = 10.5
Q = qB + qP = 3 + 10.5 = 13.5
Substituting Q = 13.5 into the market demand curve gives
p = 60 – 2*13.5 = 60 – 27 = 33
Patti’s total revenue = TRP = pqP = 33*10.5 = 346.5
Patti’s total cost = TCP = 12*10.5 = 126
Patti’s profit = ΠP = TRP – TCP = 346.5 – 126 = 220.5
Since Patti made 216 under monopoly, she would wish to leave if Bruce stayed at output
3, i.e., 220.5 > 216. So Patti has an incentive to deviate.
Bruce’s reaction function is qB = 9 – (1/3)qP. Substituting qP = 9 into Bruce’s reaction
function gives
qB = 9 – (1/3)*9 = 9 – 3 = 6
Q = qB + qP = 6 + 9 = 15
Substituting Q = 15 into the market demand curve gives
p = 60 – 2*15 = 60 – 30 = 30
Bruce’s total revenue = TRB = pqB = 30*6 = 180
Bruce’s total cost = TCB = 6*6 + 62 = 36 + 36 = 72
Bruce’s profit = ΠB = TRB – TCB = 180 - 72 = 108
Since Bruce made 81 under monopoly, he would wish to leave if Patti stayed at output 9,
i.e., 108 > 81, so Bruce has an incentive to deviate.
So the marriage is not a Nash equilibrium
Question 4 (Worth 28 points): Cristiano has a monopoly in the production of
calculators. His marginal costs are given by
MC(q) = 8 + 2q
where q is the number of calculators produced by Cristiano. Cristiano has no fixed costs.
Market demand for calculators is given by
q(p) = 40 – 0.5p
where q is the quantity of calculators demanded when the price of a calculator is p.
Part (a) (Worth 5 points): What are the monopoly quantity, price, and Cristiano’s
profits? Show all supporting work
The monopoly quantity is _____________________units
The monopoly price is $_________________________
Cristiano’s monopoly profits are $__________________
Part (b) (Worth 5 points): What would the socially efficient quantity be? How much
larger is consumer surplus at the socially efficient quantity than at the single price
(part (a)) monopoly quantity? How much smaller would Cristiano’s profits be if he
produced the socially efficient quantity? Show all supporting work.
The socially efficient quantity is __________________units
The consumer surplus at the socially efficient quantity is $_________________
larger than the consumer surplus at the monopoly quantity
Cristiano’s profits at the socially efficient quantity are $_________________
smaller than his profits at the monopoly quantity
Part (c) (Worth 11 points): Now Cristiano notices that his student buyers have different
buying patterns than the rest of his customers (non-students), and he estimates that the
subset of his demand that represents students is
qS(p) = 16 – 0.25p
where qS is the quantity of calculators demanded by students when the price of a
calculator is p. Demand for the entire market is still as described above.
What price should Cristiano charge to each group, i.e., the students and the nonstudents? How do Cristiano’s profits and total quantity produced change compared
to the monopoly results in part (a)? Show all supporting work
The profit maximizing price for students is $_______________________
The profit maximizing price for non-students is $____________________
The difference in total quantity produced by Cristiano in this part of the problem
compared to the total quantity produced in part (a) is
__________________units
The profits made in this part of the problem are ( greater, less, the same)
CIRCLE ONE
by $______________________ than the profits made in part (a)
Part (d) (Worth 7 points): The city government estimates that each unit that Cristiano
sells causes damage to the environment. This damage increases linearly with Cristiano’s
total production; in particular, the sale of the qth unit causes damage equal to $5q.
What is the socially efficient quantity now? Suggest a government policy that
induces Cristiano to produce the socially efficient quantity. (Assume that Cristiano
can no longer identify students and that the government cannot tell Cristiano what
quantity to produce). Show all supporting work.
The socially efficient quantity is __________________units
I suggest that the government institute a policy of _______________________
Question 4(a) Answer: Rewrite the demand function as
Or
0.5p = 40 – q
p = 80 – 2q
The total revenue of the monopoly is
TR = pq = (80 – 2q)q = 80q – 2q2
The marginal revenue of the monopoly is
MR = dTR/dq = 80 – 4q
The firm will maximize profit where MR = MC or
Or
Or
MR = 80 – 4q = 8 + 2q = MC
6q = 72
q = 12
Substituting q = 12 into the demand curve gives
p = 80 – 2*12 = 80 – 24 = 56
If the firm’s marginal cost is MC = 8 + 2q, the firm’s variable cost must be VC = 8q + q2
such that
MC = dVC/dq = 8 + 2q
The VC is also the firm’s total cost since fixed costs are 0.
The firm’s total revenue = TR = 56*12 = 672
The firm’s total cost = 8*12 + 122 = 96 + 144 = 240
The firm’s profit is = Π = TR – TC = 672 – 240 = 432
Question 4(b) Answer: The socially efficient output is where p = MC or
or
or
p = 80 – 2q = 8 + 2q = MC
4q = 72
q = 18
Substituting q = 18 into the demand curve gives
p = 80 – 2*18 = 80 – 36 = 44
The firm’s total revenue = TR = 44*18 = 792
The firm’s total cost = 8*18 + 182 = 144 + 324 = 468
The firm’s profit is = Π = TR – TC = 792 – 468 = 324
Since the profits were 432 under monopoly and 324 at the socially optimal level, the
firm’s profits are 432 – 324 = 108 smaller.
Consumer surplus under monopoly is 0.5*(80 – 56)*12 = 0.5*24*12 = 144
The socially optimal consumer surplus is 0.5*(80 – 44)*18 = 0.5*36*18 = 324
Thus consumer surplus is 324 – 144 = 180 larger at the socially efficient level
Question 4(c) Answer: If student demanders are qS = 16 – 0.25p, then non-student
demanders (N) must be
qN = q – qS = 40 – 0.5p – (16 – 0.25p) = 24 – 0.25p
Rewrite the student demand as
Or
0.25p = 16 – qS
p = 64 – 4qS
Rewrite the non-student demand as
Or
0.25p = 24 – qN
p = 96 – 4qN
Thus our market demand of p = 80 – 2q describes the market when P < 64 or where q > 8
(note that when qN = 8, then p = 96 – 4*8 = 96 – 32 = 64. Since our solutions above of q
= 12 and q = 18 are greater than 8, the market demand curve was the correct demand
curve to use. If q is less than 8, then the market demand curve is just the non-student
demand curve, i.e., p = 96 – 4q.
The total revenue from the student market will be
TRS = pSqS = (64- 4qS)qS= 64qS – 4qS2
The marginal revenue in the student market is
MRS = dTRS/dqS = 64 – 8qS
The total revenue from the non-student market will be
TRN = pNqN = (96- 4qN)qN= 96qN – 4qN2
The marginal revenue in the non-student market is
MRN = dTRN/dqN = 96 – 8qN
To maximize profit, the firm will set MRN = MRS = MC. Let’s view MRN = MRS, i.e.,
MRN = 96 – 8qN = 64 – 8qS = MRS
Thus
Or
8qN = 32 + 8qS
qN = 4 + qS
The firm’s marginal cost is MC = 8 + 2q = 8 + 2qN + 2qS
Substitute qN = 4 + qS in the MC equation, i.e.,
MC = 8 + 2(4 + qS) + 2qS = 8 + 8 + 2qS + 2qS = 16 + 4qS
We know that MRS = MC for profit maximization, so
Or
Or
MRS = 64 – 8qS = 16 + 4qS = MC
12qS = 48
qS = 4
Substituting qS = 4 into qN = 8 + qS gives qN = 8 + 4 = 12. Note that q = qS + qN = 4 + 8
= 12, so that the quantity that the firm produces compared with the single price
monopoly doesn’t change.
Substituting qS = 4 into the student demand curve gives
pS = 64 – 4*4 = 64 – 16 = 48
The total revenue from the student market = TRS = pSqS = 48*4 = 192
Substituting qN = 8 into the non-student demand curve gives
pN = 96 – 4*8 = 96 – 32 = 64
The total revenue from the non-student market = TRN = pNqN = 64*8 = 512
The total revenue of the firm is TR = TRS + TRN = 192 + 512 = 704
The total cost is TC = 8q + q2 = 8*12 + 122 = 96 + 144 = 240
The firm’s profit is Π = TR – TC = 704 – 240 = 464.
Since profit under single price monopoly was 432 and under third degree price
discrimination is 464, profit has changed by 464 – 432 = 32
Question 4(d) Answer:.The socially efficient quantity is were MSB = MSC. The MSB is
just the demand curve of P = 80 – 2q. The MSC = MPC + externality cost, i.e.,
MSC = 8 + 2q + 5q = 8 + 7q
Thus
or
or
MSB = 80 – 2q = 8 + 7q = MSC
9q = 72
q=8
The firm will want to produce where its MR (80 – 4q) equals its MC (8 + 2q). To
internalize the externality, we’ll place a tax of T on each unit the firm produces. Thus the
firm’s MC becomes MC = 8 + 2q + T.
Now when the firm sets MR = MC, they set
Or
Or
MR = 80 – 4q = 8 + 2q + T = MC
6q = 72 – T
q = 12 – (1/6)T
We know that we want q = 8 to attain the socially efficient output. Setting q = 8 in the
equation above yields
Or
Or
8 = 12 – (1/6)T
(1/6)T = 4
T = 24
Substituting q = 8 into the demand curve gives P = 80 – 2*8 = 80 – 16 = 64. The only
buyers will be non-students.
The firm’s total revenue = TR = pqN = 64*8 = 512
The firm’s total private cost = TC = 8*8 + 82 = 64 + 64 = 128
The firm’s payment in taxes = TqN = 24*8 = 192
The firm’s profit = Π = TR – TC – TqN = 512 – 128 – 192 = 192
If the government imposed a standard of q = 8, the firm would price at 64 and make
profits of 384 and government tax revenue would be zero. Consumer surplus would be
the same under the tax or the standard (because the price to consumers under each would
be 64).
If the market was only the non-students, the firm would set MR = MC + T or
Or
MR = 96 – 8q = 8 + 2q + T
10q + T = 88
Since q must be 8, substituting q = 8 gives
Or
10*8 + T = 88
80 + T = 88
Or
T=8
Spring 2012 BPUB 250 Final Exam
1. (30 minutes) Consider the market for steel. Among U.S. producers, the supply curve of steel is
(with quantities measured in tons):
Qs = 8P - 400 if P>=50 (and Qs = 0 if P<50).
The demand for steel in the U.S. market is Qd = 3200 – 4P if P<=800 (and Qd = 0 if P>800)
a. (4) Suppose initially there is no free trade. Solve for the equilibrium price and quantity in the
U.S. market.
At equilibrium, QS = 8P – 400 = 3200 – 4P = QD
12P = 3600
P* = 300, Q* = 2000
1
b. (4) Calculate both the consumer surplus and the producer surplus in this market.
CS = (800–300) × 2000 / 2 = 500,000
PS = (300–50) × 2000 / 2 = 250,000
c. (5) In reality, there is a world market for steel. Assume that U.S. consumers can purchase an
essentially unlimited quantity at a price of 240 per ton. Graph the total market supply curve in the
U.S. in this case and solve for the new equilibrium price and quantity.
With Free Trade, the supply curve will be
QS = 0 for P < 50
QS = 8P – 400 for 50 ≤ P ≤ 240
QS ≥ 1520 unlimited at P = 240
First assume that P* = 240, the equilibrium quantity will be Q* = 3200 – 4×240 = 2240 (this indeed lies
in the region Q ≥ 1520). So the new equilibrium price is P* = 240 and quantity is Q* = 2240
2
h the consum
mer surplus and the prooducer surpllus (among U
U.S. firms – you can
d. (5) Caalculate both
ignore fooreign firmss) in this maarket. How d
do these com
mpare with th
he “no tradee” values froom part
(b)? Com
mpare the reelative magn
nitudes of an
ny changes ((i.e. are the ggains to “win
nners” largeer than
losses too the “losers””).
CS = (8000–240) × 22240 / 2 = 6277,200
∆CS = 627200 – 5000000 = 127,2000
PS = (2440–50) × 15220 / 2 = 144,4400
∆PS = 1444400 – 2500000 = –105,6600
Consumeers are gaininng 127200 annd producerss are losing 105600. The ggain is greateer than the looss, so
overall thhe welfare inncreases by 127200 – 1055600 = 21,600 (yellow triaangle).
e. (6) Th
he governmeent imposes a quota of 400 tons on im
mported steeel to assist the U.S. steell industry.
Solve for the new eq
quilibrium p
price and qu
uantity.
With quoota, QS = (8P
P – 400) + 4000 = 3200 – 4P
4 = QD
12P = 32200
P* = 8000/3 ≈ 266.67,, Q* = 3200 – 4(800/3) = (9600 – 32000)/3 = 6400//3 ≈ 2133.33
US produucers producce 5200/3 ≈ 1733.33
1
and tthe governm
ment imports 4400 from the foreign prodducers.
3
s
from
m free-trade to the quotaa setting.
f. (6) Caalculate the cchange in prroducer and consumer surplus
(You can
n once again
n ignore anyy surplus to foreign
f
firm
ms). Does totaal surplus am
mong U.S. consumers
and U.S. producers increase or decrease?
CS = (8000–800/3) × ((6400/3) / 2 = 5120000/99 ≈ 568,888.889
∆CS = 568888.89 – 6627200 ≈ –588,311.11
So US coonsumers aree losing comppared to the ffree trade settting.
PSUS = (8800/3–50) × (5200/3) / 2 = 1690000/99 ≈ 187,777.778
∆PSUS = 187777.78 – 144400 = 443,377.78
So US prroducers are gaining com
mpared to the free trade seetting
DWL = ––58311.11 + 43377.78 = –14933.33
The totall surplus amoong U.S. connsumers and U.S.
U produceers decrease compared
c
to the free tradde setting.
4
5
minutes) Merrck is considering alternative pricingg strategies for its allerggy drug Singgulair.
2. (30 m
There arre two typess of consumeers of Singullair - those who
w take Sin
ngulair everyy day to man
nage their
symptom
ms (group E
E) and those who take Singulair onlyy when theirr symptoms fflare up (grooup F).
There arre 10 consum
mers in grou
up E and 30 consumers in group F aand the dem
mand for the number
of units of Singulairr for one perrson in each group is as follows:
QE = 12 – PE (for 0 < PE < 12)
QF = 8 – PF ((for 0 < PF < 8)
Assume that Merck
k has a constant marginaal cost of 2 oof producingg Singulair aand that therre are no
fixed cossts. Because of its patent protection
n for Singulaair, Merck iss a monopoliist for this product.
a. (5) Soolve for (in equation form
m) the mark
ket-level dem
mand for Sin
ngulair and graph
g
this (b
being sure
to label all interceptts, slopes, an
nd kink poin
nts).
QT = (122–P)×10 + (8–P)×30 = 3660–40P for 00<P<8
QT = (122–P)×10 = 1220 – 10P
for 88≤P<12
b. (5) Asssume now tthat Merck iis unable to distinguish b
between thee two types oof consumerss and thus
simply charges
c
one per-unit priice P. Solve for
f this proffit-maximizing price and
d for total prrofits.
Max πT = (9– QT/40)QT – 2QT
∂πT/∂QT = 9 – QT/20 – 2 = 0
QT = 1400, P = 5.5, πT = 140×(5.5–– 2) = 490
<Note>
If Merckk’s focuses onnly on the strrong type Gooup E,
QET = (12–P)×10 = 120 – 10PE → PE = 12 – QET/10
6
Max πE = (12– QET/10)× QET – 2QET
∂πE/∂QET = 12 – QET/5 – 2 = 0
QET = 50, PE = 7, πE = 50×(7– 2) = 250
So the overall profits are higher when selling Singulair to both groups.
c. (6) Suppose that Merck could distinguish between the two types of consumers and thus charge
each a separate per-unit price (PE and PF). Solve for these two profit-maximizing prices and
calculate Merck’s total profits. Remember that there are 10 consumers in group E and 30
consumers in group F.
Group E
QET = (12–P)×10 = 120 – 10PE → PE = 12 – QET/10
Max πE = (12– QET/10)× QET – 2QET
∂πE/∂QET = 12 – QET/5 – 2 = 0
QET = 50, PE = 7, πE = 50×(7– 2) = 250
Group F
QFT = (8–P)×30 = 240 – 30PF → PF = 8 – QFT/30
Max πF = (8– QFT/30)× QFT – 2QFT
∂πF/∂QFT = 8 – QET/15 – 2 = 0
QFT = 90, PF = 5, πF = 90×(5– 2) = 270
Therefore Merck’s total profits πT = πE + πF = 250 + 270 = 520
d. (4) Would the price in part b have been higher, lower, or the same if, instead of there being 10
people in group E and 30 in group F, there were 30 in group E and 10 in group F? Explain your
reasoning and no math is needed.
The price will be higher, because there are more people whose willingness to pay is higher. If Merck
could distinguish between the two types of consumers, the optimal price for Group E, PE = 7 is higher
than the optimal price for Group F, PF = 5. Merck cannot distinguish between the types in #2(b), so sets a
single price in between PE and PF. Now knowing that there are more people in Group E than in Group F,
Merck will set the single price to be closer to PE. (Perhaps to the extent that it might be more profitable
for Merck to just focus on Group E market and set P= PE )
7
e. (5) Merck is now considering a shift in pricing strategy – to a two-part tariff in which consumers
could pay an access fee F that would allow them to purchase Singulair at a per-unit price of P.
Assume for the moment that Merck has decided to set the per-unit price P at the marginal cost of 2
for all consumers. In this case, what is Merck’s profit-maximizing access fee and do both types of
consumers end up consuming Singulair? Remember that there are 10 consumers in group E and 30
consumers in group F.
If Merck only focuses on Group E, Access Fee = CS(Group E) = (12– 2)2 / 2 = 50. Since Merck only sells
to Group E, π = 50×10 = 500
If Merck sells to both Group E and F, Access Fee = CS(Group F) = (8– 2)2 / 2 = 18. Since Merck sells to
both groups, π = 18×40 = 720
Therefore Merck will set Access Fee = 18 and both types of consumers end up consuming Singulair.
f. (5) Now instead assume that Merck can freely choose both the access fee and the per-unit price.
Solve for Merck’s profit-maximizing two-part tariff (there are still 10 people in group E and 30 in
group F).
If Merck only focuses on Group E, the profits were π = 50×10 = 500.
If Merck sells to both Group E, F and chooses both the access fee and the per-unit price, P
Π = 40[(8– P)2 / 2] + (360 – 40P)×(P–2)
∂ Π /∂P = –40(8– P) + 360 – 40P – 40(P–2) = 0
P = 3, Π = 740
Therefore Merck will set unit-price = 3 and access fee = 12.5.
<Note>
In #2(e), we already know that serving both types yields more profits. If we have the flexibility in
choosing the access fee and the per-unit price to maximize the profit, Merck will definitely do better than
(or as well as) π = 720. Therefore in #2(f), we do not need to calculate the profits to compare the cases.
8
9
3. (30 minutes) Consider the market for desktop computer microprocessors, and assume that Intel
and AMD are the only two producers globally. They are attempting to penetrate the market of (a
fictitious country called) Narnia, which until now has been closed to international trade. The
government of Narnia is in the process of deciding whether to allow the firms to enter its market.
Consumers in Narnia consider the microprocessors made by Intel and AMD to be identical and
thus total output is QT = QI + QA. Assume that both of the firms have a constant marginal cost of
production – Intel’s is 400 per microprocessor and AMD’s is 480. Also assume that there are no
fixed costs. Total market demand in Narnia is Q = 1000 – 0.5P.
a. (7) Suppose that both companies are allowed to enter and that they choose their quantities
simultaneously. Solve for the Cournot-Nash equilibrium quantities and for the equilibrium price.
Intel takes
as given and solves,
400
max π
2000 2
2000 4
2
400 1600
4
(i)
AMD takes as given and solves,
480
max π
2000 2
2000 4
2
480 1520
4
2
2
0
0
(ii)
Combining (i) and (ii) together, we get
1600 760
280
Plugging this into (ii), we get
240
280 240 520
P 2000 2 520 960
4
b. (4) What is the maximum amount that each firm would be willing to pay to have access to this
market (assuming that their competitor will also have access)? Explain your reasoning.
The maximum amount that each firm would be willing to pay to have access to this market
(assuming that their competitor will also has access) is its profits under Cournot-Nash
equilibrium.
960 400
280 156800
960 480
240 115200
10
c. (6) The government is considering giving Intel the right to set its quantity first and then allowing
AMD to select its quantity. Solve for the quantity produced by each firm and the equilibrium price.
Since Intel moves first, we need to solve from AMD’s problem. We already solved AMD’s
.
problems in (a) and we know that AMD’s reaction function is
Now we solve Intel’s problem. Given AMD’s reaction function, Intel solves,
max π
2000
1240
2
2
400
400
1240
400
0
420
Plugging this into (ii), we get
. 170
420 170 590
P 2000 2 590 820
d. (6) How does consumer surplus compare in parts a and c? How does the maximum amount that
the government can raise from the companies in access fees compare in parts a and c? In both
cases, provide intuition for the source of any difference.
CS(PartA) =
2000
960
520
270400
CS(PartC) =
2000
820
590
348100
Consumer surplus is higher in PartC, because the total quantity supplied in the market increases (thus
market price decreases) when Intel chooses its quantity first. Intel has lower marginal costs, so when it
gains the right to set the quantity first, it has an incentive to produce much more.
From part (a) and (b) the maximum amount that the government can raise from the companies is π
156800
115200
272000
The maximum amount that the government can raise from part (c) is lower, because
π
420
820 400
176400
170
820 480
57800
176400 57800 234200
Because Intel produces much more, AMD now has to produce less, and earns much lower profits.
Attractiveness of having access to this market decreases for AMD. AMD’s willingness to pay for the
access is now much lower. (so much that its effect is even bigger than the increase in Intel’s willingness
to pay)
11
e. (7) In the U.S. market, consumers do not view Intel and AMD microprocessors as perfect
substitutes. Instead, the demand for each product depends on both prices as follows:
QI = 8000 – 4PI + PA
QA = 4000 – 2PA + PI
If Intel and AMD simultaneously set their prices, what will be the equilibrium price and quantity
for each firm? Marginal costs for both firms are the same as before (400 for Intel and 480 for
AMD) and there are still no fixed costs.
Intel takes as given and solves,
max π
8000 4
400
8000 8
1600 0
(i)
AMD takes as given and solves,
max π
4000 2
480
4000 4
960 0
(ii)
Combining (i) and (ii) together, we get
1200
1200
32
38400 4960
1398.71
Plugging this into (ii), we get
1589.68
8000
4
4000
2
12
13
Short answer questions (30 minutes)
4a. (10) Suppose that the market for tires is perfectly competitive and that Goodyear’s short-run
total cost function is TC = q3 – 6q2 + 36q + 72. What is the lowest price at which Goodyear would
choose to produce tires in the short run (i.e. choose q > 0 rather than q = 0)? Explain your
reasoning.
VC = q3 – 6q2 +36q
AVC = q2 – 6q + 36
∂AVC/∂q = 2q – 6 = 0
So AVC is the lowest when q=3 and the lowest price at which Goodyear would choose to produce tires
will be P=AVC(q=3) = 32 – 6×3 + 36 = 27. In the short-run, Goodyear produces a positive quantity as
long as the average variable cost is higher than the market price.
14
4b. (10) Suppose that Spencer has a utility function for wealth U(W) = log(W) and that he
maximizes his expected utility. He currently has wealth W = 100 dollars but faces a 20 percent
chance of losing 40 dollars and a 10 percent chance of losing 80 dollars (thus a 70 percent chance of
losing nothing). What is the maximum amount that Spencer would be willing to pay for a policy
that fully insured him (and thus paid 40 when he lost 40, 80 when he lost 80, and 0 when he lost 0)?
EU (No Insurance) = 0.2×log(60) + 0.1×log(20) + 0.7×log(100) = log(600.2 × 200.1 × 1000.7)
EU (Full Insurance) = log(100–P)
Spencer will buy the insurance as long as EU (Full Insurance) ≥ EU (No Insurance)
log(100–P) ≥ log(600.2 × 200.1 × 1000.7)
100–P ≥ 600.2 × 200.1 × 1000.7
P ≤ 100 – (600.2 × 200.1 × 1000.7) ≈ 23.134
<Note>
Both natural log and log10 are accepted.
15
Philadelphiaa is decidingg on the num
mber of firew
works to use in its 4th of July
J
4c. (10) The city of P
mal quantity.. In Philadellphia there aare three typ
pes of
parade aand hopes too pick the soocially optim
residentts (call them groups A, B,
B C) with a demand forr fireworks aas follows:
QA = 30 – P
QB = 20 – P
QC = 10 – P
There arre 10 consum
mers of typee A, 10 of typ
pe B, and 100 of type C (tthus 30 totall). The mark
ket for
firework
ks is perfectlly competitivve and the p
price per fireework is 2000. Solve for tthe social maarginal
benefit ffunction, graaph this (lab
bel interceptts, slopes, an
nd kink poin
nts), and solvve for the soocially
optimal number of fireworks.
f
SMB = ((30 – Q)×10 + (20 – Q)×110 + (10 – Q)×10 = 600 – 30Q
SMB = ((30 – Q)×10 + (20 – Q)×110 = 500 – 200Q
SMB = ((30 – Q)×10 = 300 – 10Q
Q
ffor 0≤Q≤10
f 10≤Q≤20
for
f 20≤Q≤30
for
From thee graph, we ccan see that S
SMC interseccts in the midddle range off SMB graph.
500 – 200Q = 200
Socially optimal num
mber of firew
works Q* = 155 ( this indeeed lies in the rregion 10≤Q
Q≤20 )
16
17
18
19
TOTAL POINTS
QUESTION 1: __________________
QUESTION 2: __________________
QUESTION 3: __________________
QUESTION 4: __________________
TOTAL:
__________________
20
2014 BEPP 250 Final Exam
120 points – 2 hours (There are 5 problems and ? pages.)
Full credit requires showing your work.
1. (20 points) Consider the two-player strategic form game presented below. Player 1
chooses between strategies A, B, and C, while player 2 chooses between strategies W,
X, Y, and Z. The first number in a cell is player 1’s payoff and the second number is
player 2’s payoff.
Player 2
Player 1
A
B
C
W
1,3
2,1
3,2
X
2,0
5,2
3,0
Y
5,1
2,0
3,2
Z
0,2
1,1
2,3
a. (6 points) Find the strategy pairs that are Nash equilibria. Make sure to
explain why they are Nash equilibria.
SOLUTION: There are two Nash equilibria: (B,X) and (C,Z).
(B,X) is a Nash equilibrium because, given player 2 uses X, B maximizes
player 1’s payoff: A yields a payoff of 2, B yields a payoff of 5, and C yields a
payoff of 3; and, given player 1 uses B, X maximizes player 2’s payoff: W
yields a payoff of 1, X yields a payoff of 2, Y yields a payoff of 0, and Z yields
a payoff of 1.
(C,Z) is a Nash equilibrium because, given player 2 uses Z, C maximizes
player 1’s payoff: A yields a payoff of 0, B yields a payoff of 1, and C yields a
payoff of 2; and, given player 1 uses C, Z maximizes player 2’s payoff: W
yields a payoff of 2, X yields a payoff of 0, Y yields a payoff of 2, and Z yields
a payoff of 3.
b.
(6 points) Find the strategies that survive the iterative elimination of strictly
dominated strategies.
SOLUTION: For player 1, no strategies are strictly dominated. For player 2,
Z strictly dominates Y and strategies W, X, and Z are not strictly dominated.
Having eliminated strategy Y for player 2, the reduced game is
Player 2
X
W
1
Z
Player 1
A
B
C
1,3
2,1
3,2
2,0
5,2
3,0
0,2
1,1
2,3
For player 1, strategies B and C strictly dominate A. No strategies are strictly
dominated for player 2. The reduced game is
Player 1
B
C
W
2,1
3,2
Player 2
X
5,2
3,0
Z
1,1
2,3
No strategies are strictly dominated. (While Z weakly dominates W, it does
not strictly dominate it.) Thus, the strategies that survive the iterative
elimination of strictly dominated strategies are B and C for player 1, and W, X,
and Z for player 2.
c.
(8 points) Now suppose players move sequentially. Player 1 chooses between
A, B, and C. Player 2 observes player 1’s choice and then chooses between W,
X, Y, and Z. Given a pair of choices, the payoffs are again as specified in the
payoff matrix. Find the subgame perfect Nash equilibria.
SOLUTION: Subgame perfect Nash equilibria can be derived through
backward induction. Consider player 2 for the three possible situations that he
can find himself: player 1 chose A, player 1 chose B, and player 1 chose C. If
player 1 chose A then player 2’s optimal action is W; if player 1 chose B then
player 2’s optimal action is X; and if player 1 chose C then player 2’s optimal
action is Z.
Given that strategy for player 2, player 1 knows that if she chooses A then
player 2 will choose W in which case player 1’s payoff is 1; if she chooses B
then player 2 will choose X in which case player 1’s payoff is 5; and if she
chooses C then player 2 will choose Z in which case player 1’s payoff is 2.
Hence, player 1 will optimally choose B. The unique SPE is: player 1’s
strategy is B, and player 2’s strategy is: if player 1 chose A then choose W, if
player 1 chose B then choose X, and if player 1 chose C then choose Z.
[Grading note: If a student states that the SPE is player 1 chooses B and player
2 chooses X (and if the analysis used in deriving it is correct) then they receive
7 out of 8 points.]
2
2. (36 points) Consider a monopolist that may face two consumers: A and B. Consumer
A has demand curve
𝐷! 𝑝 = 60 − 𝑝
which is shown in Figure 2a.
Consumer B has fixed demand of 20 units as long as price is no higher than 30, and
her demand is zero if price exceeds 30, as shown in Figure 2b.
The monopolist has constant marginal cost of 10.
3
a. (8 points) Suppose only consumer A exists so the monopolist only has
demand from consumer A. Find the price per unit that maximizes its profit.
SOLUTION: The monopolist’s profit function is
60 − 𝑝 (𝑝 − 10)
To maximize it, take the derivative with respect to price, set it equal to zero,
and solve for price:
60 − 2𝑝 + 10 = 0 → 𝑝!! = 35. Profit is
60 − 𝑝!! 𝑝!! − 10 = 60 − 35 35 − 10 = 625.
b. (8 points) Suppose only consumer A exists so the monopolist only has
demand from consumer A. Find the two-part tariff that maximizes its
profit. Recall that a two-part tariff is a fixed fee and a per unit price.
SOLUTION: With one consumer type, the optimal two-part tariff is a per unit
price equal to marginal cost and a fixed fee equal to consumer surplus. Thus,
the per unit price is 10 which generates consumer surplus of
60 − 10 60 − 10
!
!
= 1250.
Thus, the profit maximizing two-part tariff is a fixed fee of 1250 and a per
unit price of 10, which yields profit of 1250.
c.
(2 points) Suppose the monopolist has demand from both consumers A
and B. Draw a graph with the market demand curve.
SOLUTION:
4
d. (8 points) Suppose the monopolist has demand from both consumers A
and B and it offers the same per unit price to both consumers. Find the per
unit price that maximizes its profit
SOLUTION: First note that market demand is 0 if 𝑝 > 60, is 60 − 𝑝 if
30 < 𝑝 ≤ 60, and is 80 − 𝑝 if 𝑝 ≤ 30. Given that it prices above 30, in which
case demand is only from customer A, we know from part (a) that the optimal
price is 35 and profit is 625. If it prices at 30 then demand is 50 and profit is
80 − 30 30 − 10 = 1000.
Thus, the monopolist prefers to price at 30 (and sell to both consumers) than
to price at 35 (and sell only to consumer A). For prices less than 30, profit is
(80 − 𝑝)(𝑝 − 10)
If we take the derivative of it with respect to price, marginal profit equals
90 − 2𝑝 which is positive when 𝑝 < 30. Hence, profit is increasing for prices
lower than 30. Thus, a price of 30 yields higher profit than any price below 30.
In conclusion, the optimal price is 30 with profit of 1000.
e.
(10 points) Suppose the monopolist has demand from both consumers A and
B and it offers the same two-part tariff to both consumers. Find the two-part
tariff that maximizes its profit.
5
SOLUTION: Given a per unit price of 𝑝, consumer surplus for consumer A is
1
𝐶𝑆! 𝑝 = ( )(60 − 𝑝)!
2
when 𝑝 ≤ 60 (and is zero for higher prices), and consumer surplus for
consumer B is
𝐶𝑆! 𝑝 = 20(30 − 𝑝)
when 𝑝 ≤ 30 (and is zero for higher prices). Visually inspecting the demand
curves, it is clear that consumer surplus for consumer A is always strictly
greater than consumer surplus for consumer B when 𝑝 < 60 because
consumer A earns higher surplus on each unit and consumes more units. Thus,
given a per unit price 𝑝, if consumer B prefers to pay the fixed fee (which is
only true if 𝐶𝑆! 𝑝 is at least as great as the fixed fee) then consumer A will
also prefer to pay the fixed fee; that is, if
𝐶𝑆! 𝑝 ≥ 𝑓𝑖𝑥𝑒𝑑 𝑓𝑒𝑒
then
𝐶𝑆! 𝑝 ≥ 𝑓𝑖𝑥𝑒𝑑 𝑓𝑒𝑒
This means that the optimal two-part tariff is either the answer in part (b) (so
that only consumer A pays the fixed fee) or the fixed fee is set equal to the
consumer surplus for consumer B so that both consumers buy. We already
know the former yields profit of 1250. Let’s derive the optimal two-part tariff
when the fixed fee equals 𝐶𝑆! 𝑝 so that both consumers buy. In that case, the
monopolist’s profit is
2×20 30 − 𝑝 + 80 − 𝑝 𝑝 − 10
where the first term is the fixed fees collected from the two consumers and the
second term is the usage profit. Taking the derivative and setting it equal to
zero:
−40 + 80 − 2𝑝 + 10 = 0 → 𝑝∗ = 25
Profit is then
40 30 − 25 + 80 − 25 25 − 10 = 1025.
Given that this profit exceeds the profit of 1000 from selling only to consumer
A than the optimal two-part tariff is a fixed fee of 100 and a per unit price of
25.
6
3. (24 points) Consider a market with two firms – denoted firm 1 and firm 2 – offering
identical products. The inverse market demand curve is
𝑃 𝑄 = 180 − 2𝑄
where 𝑄 = 𝑞! + 𝑞! is total supply, 𝑞! is firm 1’s quantity, and 𝑞! is firm 2’s quantity.
Firm 1’s cost function is
𝐶! 𝑞! = 10𝑞!
and firm 2’s cost function is
𝐶! 𝑞! = 20𝑞! .
Firms are assumed to simultaneously choose quantities and each firm chooses its
quantity to maximize its profit. Price is set to clear the market (that is, price equates
demand with the total supply of the two firms).
a. (8 points) Find each firm’s best response function.
SOLUTION: Firm 1’s profit function is
180 − 2(𝑞! + 𝑞! ) − 10 𝑞!
Taking the derivative of it with respect to 𝑞! , setting it equal to zero, and
solving for 𝑞! gives us firm 1’s best response function:
170 − 4𝑞! − 2𝑞! = 0 → 𝑞! = 42.5 − 0.5𝑞!
Firm 2’s profit function is
180 − 2(𝑞! + 𝑞! ) − 20 𝑞!
Taking the derivative of it with respect to 𝑞! , setting it equal to zero, and
solving for 𝑞! gives us firm 2’s best response function:
160 − 2𝑞! − 4𝑞! = 0 → 𝑞! = 40 − 0.5𝑞! .
b. (8 points) Find the quantities and profits at a Nash equilibrium.
SOLUTION: We need to find a pair of quantities - 𝑞! and 𝑞! - such that each
firm is producing according to its best response function given the quantity of
the other firm:
𝑞! = 42.5 − 0.5𝑞!
and
7
𝑞! = 40 − 0.5𝑞! .
Plugging the second equation into the first gives
𝑞! = 42.5 − 0.5𝑞! = 42.5 − 0.5 40 − 0.5𝑞! → 𝑞! = 30
and then plugging this quantity into 𝑞! gives:
𝑞! = 40 − 0.5𝑞! = 40 − 0.5 30 = 25.
Thus, the Nash equilibrium has firm 1 producing 30 and firm 2 producing 25.
Firm 1’s profit is
180 − 2 30 + 25 − 10 30 = 1800
and firm 2’s profit is
180 − 2 30 + 25 − 20 25 = 1250.
c. (8 points) Now suppose firm 1 has a fixed cost of 2000 and firm 2 has a fixed
cost of 1000. This fixed cost can be avoided by producing zero. Find a Nash
equilibrium.
SOLUTION: Before netting out fixed costs, we derived in part (b) that firm
1’s variable profit is 1,800 and firm 2’s variable profit is 1,250 at the Nash
equilibrium. Given the fixed cost for firm 1 is 2,000 then its profit is -200 and
firm 2’s profit is 250 given its fixed cost is 1,000. It is not a Nash equilibrium,
however, for firm 1 to produce 30 and firm 2 to produce 25 because firm 1’s
profit is negative while it could earn 0 by producing 0.
Consider then firm 1 producing 0 and firm 2 producing the monopoly quantity
of 40. Firm 2’s quantity is optimal as long as its profit is non-negative which
is indeed the case as it profit is 3200 − 1000 = 2200. Given firm 2 produces
40, firm 1’s optimal quantity (given it produces a positive quantity) is
42.5 − 0.5 40 = 22.5
and its profit is
180 − 2 22.5 + 40 − 10 22.5 − 2000 = −987.5
Hence, firm 1 prefers to produce 0 to producing any positive quantity. It is
then a Nash equilibrium for firm 1 to produce 0 and firm 2 to produce 40.
4. (20 points) There are two firms, denoted A and B, selling identical goods and
competing in prices (that is, Bertrand competition). Each firm has constant marginal
8
cost. Firm A’s marginal cost is cA = 2 and firm B’s marginal cost is cB = 0. Assume
market demand for the good is
𝑄 = 10 − 𝑝
where 𝑄 is total market demand and 𝑝 is price. If firms set different prices then the
firm with the lower price has demand equal to market demand and the other firm has
zero demand. If firms set the same price then each receives half of market demand.
Firms are assumed to set prices simultaneously, and they can only choose price from
the following set: {0,1,2,3,4,5}.
a) (3 points) Suppose firms compete for only one period. Show that the following
strategy profile is a Nash Equilibrium: Firm A sets pA = 2 and firm B sets pB = 1.
SOLUTION: While not required, one can write the strategic form representation
of this one-shot game:
Firm B
Firm A
Prices
0
1
3
4
5
0
-10, 0
-20, 0
-20, 0
-20, 0
-20, 0
-20, 0
1
0 , 0
-4.5 , 4.5
-9 , 0
-9 , 0
-9 , 0
-9 , 0
2
0 , 0
0 , 9
0 , 8
0 , 0
0 , 0
0 , 0
3
0 , 0
0 , 9
0 , 16
3.5 , 10.5
7 , 0
7 , 0
4
0 , 0
0 , 9
0 , 16
0 , 21
6 , 12
12 , 0
5
0 , 0
0 , 9
0 , 16
0 , 21
0 , 24
7.5 , 12.5
2
There are two (pure-strategy) Nash Equilibria to this game: (2,1) and (3,2).
b) (4 points) Suppose firms compete for only one period. Are there other Nash
equilibria aside from pA = 2 and pB = 1? Explain your answer – a simple “Yes” or
“No” will not give you any credit.
SOLUTION: (3,2) is another Nash equilibrium.
9
c) (3 points) Suppose firms compete for only one period. Show that pA = 5 and pB =
5 is not a Nash equilibrium.
SOLUTION: Under the given strategy, firm A earns
(5 − 2) ∗
(10 − 5)
= 7.5
2
while firm B earns
5∗
(10 − 5)
= 12.5
2
Suppose firm A deviates by setting pA = 4 instead. This is a profitable deviation
since firm A gets
(4 − 2) ∗ (10 − 4) = 12 > 7.5 We can also prove that (5,5) is not a Nash Equilibrium by examining firm B.
Suppose firm B deviates by setting pB = 4 instead. This is a profitable deviation
since firm B gets
4 ∗ (10 − 4) = 24 > 12.5
Suppose now the firms play the Bertrand game for an infinite number of periods and
assume that both firms have a discount factor of 𝛿 where 0 < 𝛿 < 1. Consider the
following strategy:
Firm A:
In period 1:
Set pA = $5.
In period t, for t > 1: Set pA = $5 if pA and pB have always been $5 in the past.
Otherwise set pA = $2 forever.
Firm B:
In period 1:
Set pB = $5.
In period t, for t > 1: Set pB = $5 if pA and pB has always been $5 in the past. Otherwise
set pB = $1 forever.
10
d) (10 points) Find the smallest 𝛿 such that no firm has an incentive to deviate from
this strategy; that is, the strategy is a symmetric subgame perfect Nash
equilibrium.
SOLUTION: If both firms follow the suggested strategy, then they will price at
$5 forever. Per-period payoffs will then be 7.5 and 12.5 for firms A and B
respectively. Therefore
𝑃𝑟𝑜𝑓𝑖𝑡! =
!.!
!!!
and
!".!
𝑃𝑟𝑜𝑓𝑖𝑡! = !!!
Consider the history where every firm has set $5 in the past. The best one-stage
deviation for firm A is to set pA = 4 and get 12 in the current period. This triggers
the punishment hence profit from this deviation is just
!
𝑃𝑟𝑜𝑓𝑖𝑡! = 12 + 𝛿 !!! = 12
Thus for firm A, we need
7.5
> 12
(1 − 𝛿)
which implies 𝛿 > 0.375. Now for firm B, the best one-stage deviation is to set pB
= 4 and get 24 in the current period. Thus,
!
𝑃𝑟𝑜𝑓𝑖𝑡! = 24 + 𝛿 !!!
and so for firm B, we need
!".!
!!!
> 24 + 𝛿
!
!!!
which implies 𝛿 > 0.767. Since the punishment involves Nash equilibrium
strategies, any 𝛿 would suffice. Therefore the suggested solution is an SPNE as
long as 𝛿 > 0.767. [Grading note: Do not take any points off if a student fails to
note that the strategy prescribes an optimal action in the event of a punishment.]
5. (20 points) Two individuals, denoted A and B, are bidding for an item. A bidder can
either value the item $1 or $4. Bidders’ values are independently drawn with value $1
11
chosen with probability ¼ and value $4 with probability ¾. Each bidder knows her
own value but does not know the other bidder’s value. Regardless of the auction form
used, if both bidders submit the same bid then the auctioneer flips a coin to determine
who is the winning bidder so, in that situation, each bidder would have a ½ chance of
winning the auction.
a) (5 points) Suppose it is a second price sealed bid auction. Find the symmetric
Nash Equilibrium bidding strategy.
SOLUTION: Since this is a SPSB auction, symmetric Nash Equilibrium
consists of bidding one’s value for the item. Thus the symmetric Nash
equilibrium bidding strategy is to bid 4 when the value is 4 and bid 1 when the
value is 1.
b) (10 points) Suppose it is a first price sealed bid auction. Moreover, the two
bidders can only choose between the following two bidding strategies: (1)
shade one’s bid by 30% below one’s value (i.e. if v is a bidder’s value then bid
0.7v); and (2) shade one’s bid below one’s value by 50%. Show that shading
one’s bid below one’s value by 30% is a symmetric Nash equilibrium bidding
strategy.
SOLUTION: Suppose the rival bids by shading her bid 30% below her
valuation. That is, the rival bids $2.8 with probability ¾ and $0.7 with
probability ¼. Consider the optimal bidding strategy of the firm as a function
of valuation 𝑣. Expected payoff from bidding b is 𝑃𝑟(𝑤𝑖𝑛) ∗ (𝑣 − 𝑏). We
need to get the probability of winning, the surplus and expected payoff for
each possible 𝑏 and 𝑣.
Define 1{𝐴} = 1 if 𝐴 is true and 0 otherwise. The probability of winning the
auction is given by
¾ (1{𝑏 > 2.8} + ½ 1{𝑏 = 2.8}) + ¼ (1{𝑏 > 0.7} + ½ 1{𝑏 = 0.7}) If 𝑣 = 4:
12
Bid
Pr(win)
Surplus
Expected payoff
2.8
0.625
1.2
0.75
2
0.25
2
0.5
Bid
Pr(win)
Surplus
Expected payoff
0.7
0.125
0.3
0.0375
0.5
0
0.5
0
If 𝑣 = 1:
From the table we see that if 𝑣 = 4, it is optimal to bid 2.8, while if 𝑣 = 1,
it is optimal to bid 0.7. Therefore this is a symmetric Nash Equilibrium given
the restrictions on how firms can bid.
c) (5 points) Using your answers in (a) and (b), show that expected revenues of
the seller/auctioneer are equal under the two auction formats (when the final
number is rounded to the tenth decimal). (Recall that the probability of two
independent events - for example, the event that bidder A has value 4 and the
event that bidder B has value 1 - equals the product of the single event
probabilities – that is, the probability that bidder A has value 4 multiplied by
the probability that bidder B has value 1.)
SOLUTION: Under the SPSB auction, expected revenues are
Pr 𝑣𝐴 = 4 ∗ Pr 𝑣𝐵 = 4 ∗ 4 + Pr 𝑣𝐴 = 4 ∗ Pr 𝑣𝐵 = 1 ∗ 1
+ Pr 𝑣𝐴 = 1 ∗ Pr 𝑣𝐵 = 4 ∗ 1 + Pr 𝑣𝐴 = 1 ∗ Pr 𝑣𝐵 = 1 ∗ 1
=
9
3
3
1
36 7
∗ 4 + +
+
∗ 1 =
+
16
16 16 16
16 16
= 43/16 = 2.6875 = 2.7
Under the FPSB auction, expected revenues are
13
Pr 𝑣𝐴 = 4 ∗ Pr 𝑣𝐵 = 4 ∗ 2.8 + Pr 𝑣𝐴 = 4 ∗ Pr 𝑣𝐵 = 1 ∗ 2.8
+ 𝑃𝑟(𝑣𝐴 = 1) ∗ 𝑃𝑟(𝑣𝐵 = 4) ∗ (2.8) + 𝑃𝑟(𝑣𝐴 = 1) ∗ 𝑃𝑟(𝑣𝐵 = 1) ∗ (0.7) = 9
3
3
1
+
+
∗ 2.8 + ∗ 0.7 16 16 16
16
= 15/16 ∗ 2.8 + 0.7/16 = 2.66875 = 2.7
14
Name:___________________________
Penn ID #:___________________________
Fall 2013 BEPP 250 Final Exam
100 points --- 2 hours
4 questions, each question is worth 25 points, each question has 5 parts, each part is worth 5 points
Please carefully label all axes/intercepts in the graphs you plot!
1. (25 points)
Two firms A and B produce lumber. Demand for A’s lumber is = 100 − + ∗ , demand
for B’s lumber is = 100 − + ∗ . Firm A’s cost of producing lumber is ( ) = ∗
and firm B’s cost is ( ) = ∗ , where , are constants.
a. Interpret . For what values of (if any) are goods A and B complements? For what values of
(if any) are goods A and B substitutes? For what values of (if any) are goods A and B neither
complements nor substitutes?
ANSWER:
The demand parameter measures the substitutability between goods A and B. If > 0 the two
goods are substitutes, if < 0 they are complements, if = 0 they are neither substitutes nor
complements.
b. Suppose firms A and B compete with each other in a Bertrand price setting game. Write down
each firm’s objective function. Find each firm’s best response function. If firm B charges a higher
price, is firm A’s best response to charge a higher price, lower price or the same price? How does
this depend on ? What is the intuition?
ANSWER:
Π = ∗
− ∗
Π =( − )∗
Π = ( − ) ∗ (100 −
: = 1 ∗ (100 −
)
)+(
−
Similarly, firm B’s best response function is:
=
(100 − 2
=
+
∗
)+
∗
+
+
∗
∗
) ∗ (−1) = 0
=0
From firm A’s best response function, we see
∗
.
= . If the two goods are substitutes( > 0),
firm A should increase its price in response to an increase in price by B. If they are
complements( < 0) firm A should decrease its price in response to an increase in price by B. If
they are neither firm A should not change its price.
For the rest of the question let = and
=
= .
c. Find the Nash equilibrium prices. Find the corresponding quantities and profits. Plot the best
response functions with on the horizontal axis and on the vertical axis and label the Nash
Equilibrium.
ANSWER:
1
=
∗
=
=
=
∗
=
=
=
=
=
.
.
=
= 75.75 +
= 75.75 ∗ = 101 and = 101.
= 100 − + ∗
= 100 − 101 + 1 ∗ 101 = 100
= 100 − + ∗ = 100 − 101 + 1 ∗ 101 = 100
Π =
−
= 101 ∗ 100 − 1 ∗ 100 = 10000
Π =
−
= 101 ∗ 100 − 1 ∗ 100 = 10000
PB
300
Firm A
BR
200
Nash
Eqlm
100
Firm B
BR
50
100
150
200
PA
 100
d. Suppose each firm charges one dollar more than its equilibrium price in part c. What will happen
to each firm’s profits? Show mathematically and explain why this new set of prices is not sustainable
as an equilibrium.
ANSWER:
If =
= 101 + 1 = 102, then = 100 − +
= 100 = .
Π = 102 ∗ 100 − 1 ∗ 100 = 101 ∗ 100 = 10100
Π = 102 ∗ 100 − 1 ∗ 100 = 101 ∗ 100 = 10100
Under this new set of prices, both firms earn 100 dollars more than under the Nash equilibrium.
This set of prices cannot be sustained as a Nash equilibrium because if firm B sets = 102 then
firm A’s best response is =
=
=
= 101.5, which corresponds to
= 100 − +
= 100 − 101.5 + 102 = 100.5
Π = 101.5 ∗ 100.5 − 1 ∗ 100.5 = 100.5 = 10100.25
e. Suppose that firms A and B are still competing with each other in a price-setting game, but firm A
moves first and firm B observes and moves second. What are the Nash equilibrium prices in this
sequential move game? What are the corresponding quantities and profits?
Does a firm prefer to be a leader or follower?
ANSWER:
2
Name:___________________________
Penn ID #:___________________________
=
Firm A knows that B’s best response function is
Π = ∗
− ∗
Π = ( − 1) ∗ (100 − + )
Π = ( − 1) ∗ 100 − +
Π =(
FOC:
100 +
, hence its profit function is:
− 1) ∗ 100 +
= 100 +
−
+(
− 1) ∗ −
=0
+ =0
100 +
+ =
= 151
=
=
=
= 126
= 100 − 151 + 126 = 75
= 100 − 126 + 151 = 125
Π = 151 ∗ 75 − 1 ∗ 75 = 150 ∗ 75 = 11250
Π = 126 ∗ 125 − 1 ∗ 125 = 125 ∗ 125 = 15625
A firm prefers to be a follower in this sequential move price setting game.
3
2. (25 points) Tom has initial wealth
= 100. If there is a flood he will suffer losses = 20.
Tom’s utility from wealth is ( ) =
. Let the probability of a flood be =
.
a. What is Tom’s expected wealth? What is Tom’s expected utility of wealth given the risk of flood?
What is Tom’s certainty equivalent (i.e., the amount of wealth Tom would accept to avoid the flood
risk altogether)? Is Tom risk averse, risk neutral or risk loving? Explain.
ANSWER:
[ ] = (100 − 20) ∗
[ ( )] = (80) ∗
( ) = [ ( )]
+ 100 ∗
= 98
+ (100) ∗
≈ 9.89
(
) = 9.89
= (9.89) = 97.81
Tom is risk averse because his utility function is concave.
b. Suppose Tom can buy dollars of insurance for a premium of ∗ . If there is a flood, the
insurance company will pay Tom . Whether there is a flood or not, Tom must pay the premium.
Write down Tom’s expected utility if he purchases dollars of coverage at a premium of ∗ .
ANSWER:
[ ( )] =
∗ (80 −
∗ + ) +
∗ (100 −
∗ )
c. How much insurance will maximize Tom’s expected utility at a premium of ∗ ?
ANSWER:
[ ( )] =
∗ (80 −
∗ + ) +
∗ (100 −
FOC:
∗ (80 −
∗ + )
(80 −
∗ + )
∗ (1 − ) = 9 ∗ (100 −
∗ + )
∗
(80 −
= (100 −
80 + ∗ (1 − ) = (100 −
80 + ∗ (1 − ) = 100 ∗
∗ (1 − ) +
∗
=
(
)
∗ (1 − ) +
∗
∗ )
∗
∗ (100 −
∗ )
∗ )∗
−
= 100 ∗
∗ )
∗ ∗
− 80
∗
4
∗ )
∗( )
∗ (− ) = 0
Name:___________________________
Penn ID #:___________________________
d. How much insurance will Tom buy if insurance is actuarially fair = 0.1 (i.e. if the price per
dollar of insurance is equal to the probability of a loss)?
ANSWER:
= 20 Tom will fully insure.
e. What is the insurance company’s expected profit from selling dollars of insurance at a price of
per dollar of coverage? At what price will the insurance company make zero expected profit?
ANSWER:
[ Π] =
∗( ∗ − )+
If [Π] = 0, then =
∗( ∗ )=
∗ −
.
5
3. (25 points)
Firms provide power in units of Kilowatt Hours (KwH) to the Philadelphia region. All firms have
constant marginal cost of $2 per KwH. Fixed costs are zero. Consumers in the region have the
following demand schedule:
Q(P) = (20-P)/2
There is a constant marginal externality cost of pollution of $4 per KwH.
a. Suppose firms operate in a competitive market. What is the free market equilibrium price and
quantity? Plot this (Price on the vertical axis, quantity on the horizontal axis).
ANSWER:
=
= 2, (2) =
=
= 9.
P
20
15
10
5
0
0
2
4
6
8
10
Q
b. What is the socially optimal price and quantity (i.e., the price and quantity that account for the
externality of pollution)? Plot the socially optimal outcome in the graph in part a.
ANSWER:
=
+
= 2 + 4 = 6, (6) =
=
=7
P
20
15
10
5
0
0
2
4
6
8
10
6
Q
Name:___________________________
Penn ID #:___________________________
c. Suppose a monopolist (PECO) supplies all electricity to Philadelphia. What is the monopoly price
and quantity (ignoring the externality)? Add this to the graph in part a along with the marginal
revenue curve.
ANSWER:
Π = (20 − 2 ) ∗ − 2 ∗ = 20 − 2 − 2 = 18 − 2
= 18 − 4 = 0, = , = 20 − 2 ∗ = 11.
P
20
15
10
5
0
0
2
4
6
8
10
Q
d. Suppose for this part of the question that the monopoly takes the social cost into account in its
pricing and production decision. In that case, what would the monopoly price and quantity be? Plot
this outcome in a graph.
ANSWER:
Π = (20 − 2 ) ∗ − 6 ∗ = 20 − 2 − 6 = 14 − 2
= 14 − 4 = 0, = , = 20 − 2 ∗ = 13.
7
P
20
15
10
5
0
0
2
4
6
8
10
Q
e. What level of unit tax/subsidy on the monopolist would produce the socially optimal outcome?
(Hint: find the monopolist’s production as a function of the tax.) Is the optimal tax/subsidy
different from the social cost of production? Why?
Let the level of tax/subsidy be which will be negative for subsidy and positive for tax.
ANSWER:
Π = (20 − 2 ) ∗ − 2 ∗ −
= 18 − − 4 = 0, so =
= 20 − 2 − 2 ∗
and = 20 −
−
=
= (18 − ) − 2
We know from (b) that the socially optimal quantity is 7 and price is 6, so
The government should subsidize the monopolist $10 per KwH.
8
= 7 so = −10.
Name:___________________________
Penn ID #:___________________________
4. (25 points)
Two players, Letter and Number, play the following game. Letter selects among strategies A, B, and C,
whereas Number selects among strategies 1, 2, and 3:
1
2
3
A
(0,2)
(-1,1)
(5,3)
B
(3,-1)
(0,5)
(2,1)
C
(2,1)
(1,4)
(3,6)
Payoffs to Letter are written first in each payoff vector, and payoffs to Number are written second in each
payoff vector.
a. Suppose that both players choose their strategies simultaneously. Find all Nash equilibria.
ANSWER:
For Number, strategy 1 is dominated by strategy 3. Given that Number does not play strategy 1, then for
Letter, strategy B is dominated by strategy C. So we can effectively consider a game where Letter
chooses between A and C and Number chooses between 2 and 3. The Nash equilibrium outcome of this
game is where Letter plays A and Number plays 3, because then both players are doing their best strategy,
given what the other player is doing.
b. Suppose Number moves first and Letter moves second. Find any Nash equilibria.
ANSWER:
Strategy 1 is still dominated for Number, so Number does not play strategy 1. If Number plays strategy
2, Letter’s best response is to play C, and Number will get a payoff of 4. If Number plays strategy 3,
Letter’s best response is to play A, and Number will get a payoff of 3. So Number plays strategy 2 and
Letter plays C in the Nash equilibrium outcome.
c. Suppose Letter moves first and Number moves second. Find any Nash equilibria.
ANSWER:
If Letter plays A then Number’s best response is to play strategy 3, in which case Letter gets a payoff of
5. If Letter plays B then Number’s best response is to play strategy 2, in which case Letter gets a payoff
of 0. If Letter plays C then Number’s best response is to play strategy 3, in which case Letter gets a
payoff of 3. So the Nash equilibrium outcome is for Letter to play A and Number to play 3.
d. Now suppose that the game is going to be played 100 times, that both players know that it will be
played exactly 100 times, and that in each round of the game, Number moves first and Letter moves
second. Find any Nash equilibria.
ANSWER:
9
The fact that it is played 100 times does not change the Nash equilibrium in each round of the game
(despite the possibility of reciprocation). So the Nash equilibrium is for Number to play strategy 2 and
Letter to play strategy C in each round of the game.
e. Now suppose that the players can costlessly write binding contracts that commit them to playing some
strategy, and that they can negotiate to make payments to each other after the payoffs of the game have
been realized. Both players seek to maximize their net payoffs after compensation has taken place. If
Number moves first and Letter moves second, what will be each player’s net payoff, and what strategy
will each player play? If Letter moves first and Number moves second, what will be each player’s net
payoff, and what strategy will each player play? Explain.
ANSWER:
Since players can write contracts to compensate each other after receiving their payoffs, the players seek
to maximize the sum of their payoffs and then compensate each other to make each other as well off as
they would have been if they were not allowed to write binding contracts to compensate each other. If
Number moves first and Letter moves second, the Nash equilibrium in part c) above involves Number
playing 2 and Letter playing C, so that Number gets a payoff of 4 and Letter gets a payoff of 1. Consider
the alternative in which the two players write a binding contract committing Letter to play strategy C and
Number to play strategy 3. Then Number and Letter both do better than in the Nash equilibrium in part c),
and the sum of their payoffs is maximized at 9 (=6+3). Thus, players will write a contract for Letter to
play strategy C and Number to play strategy 3, and then split the surplus by making payments to each
other; any outcome is sustainable as long as Letter gets net payoffs (after compensation) of at least 1 and
Number gets net payoffs (after compensation) at least 4. If Letter moves first and Number moves second,
the Nash equilibrium in part d) above involves Letter getting a payoff of 5 and Number getting a payoff
of 3. Again, if the two players write a binding contract committing Letter to play strategy C and Number
to play strategy 3, then the sum of their payoffs is maximized at 9 (=6+3). This is only sustainable if
Number compensates Letter so that Letter receives at least 5 net of compensation. Any outcome is
sustainable as long as Number also receives at least 3 after compensating Letter. So the players will again
write a contract to commit Letter to play C and Number to play 3, and Number will compensate Letter in
such a way that Number receives at least 3 net of compensation, and Letter receives at least 5 net of
compensation.
10
2013 BEPP 250 Final Exam
120 points --- 2 hours
1. (30 points) Suppose that Marvel Comics is deciding whether to enter into the Chinese
market by printing its comics in Mandarin. However, at the same time Marvel’s rival, DC
Comics, is also deciding whether or not to enter this market. Suppose further that Marvel
and DC have identical cost functions given by
= + − where q is the number of comics printed per firm. Chinese demand for comics is given by
= − where P is the price of comics and Q is the sum of all comics printed (by both firms).
Marvel and DC are simultaneously deciding whether to enter the Chinese market. Assume
that if only one firm enters then it is a monopoly. But, if both firms enter then they each
understand their own market power (and, hence, play a Cournot game). Any firm that stays
out of the market gets zero profits (i.e. fixed costs are only incurred if the firm enters the
market).
a. If only Marvel enters, what profits will it earn? Similarly, if only DC enters what profits will it
earn? (5)
Since both firms are identical we can just solve for the profits and quantity of one firm and, by
symmetry, the profits and quantity of the other will be the same. We will solve for Marvel’s profits
and quantity.
Let ( , ) be Marvel’s profits when marvel produces and DC produces . We are
interested in solving for ( , 0). Doing so, however, first requires that we solve for Marvel’s
optimal quantity when DC produces zero. We can write Marvel’s profits as
)
( , 0) = ∗ ( ) − (22 + 80 − 5
)
= (140 − 20 ) ∗ ( ) − (22 + 80 − 5
To get the optimal quantity we take first order conditions
( , 0)
= 140 − 40 − 80 + 10 = 0 ⇒ = 2
plugging this back into the profit function yields
( , 0) = (140 − 20 ∗ 2) ∗ (2) − (22 + 80 ∗ 2 − 5 ∗ (2) ) = 38
DC’s profits and quantity when Marvel produces zero will be identical.
1
b. If both Marvel and DC enter, how much will each firm produce and what profits will each firm
earn? (5)
If both firms enter then they play a Cournot game. Again, since the firms are identical their profits
and quantities will also be the same. Solving for them requires that we know each firms best
response function.
Marvel’s profits can be expressed as
)
( , ) = ∗ ( ) − (22 + 80 − 5
)
= (140 − 20( + ) ∗ ( ) − (22 + 80 − 5
Taking the first order condition and solving for gives the best response function
( , )
60 − 20
= 140 − 40 − 20 − 80 + 10 = 0 ⇒ =
30
By symmetry, DC’s best response function is just
=
60 − 20
30
Setting the best response functions equal to each other gives the equilibrium quantities produced by
each firm
60 − 20
60
30 = 60 − 20 ∗ $
% ⇒ =
= 1.2 = 30
50
Firm profits are then just given by
= = '140 − 20 ∗ (1.2 + 1.2)( ∗ (1.2) − (22 + 80 ∗ (1.2) − 5 ∗ (1.2) ) = −0.4
c. Fill out the payoff matrix below for the entry game. Be sure to list the row player’s payoff first. (3)
Marvel Comics
Enter
Stay Out
Enter
-0.4 , -0.4
38 , 0
Stay Out
0 , 38
0,0
DC Comics
2
d. What is/are the Nash Equilibrium outcome(s) of this game? Explain your answer. (4)
There are two Nash equilibria (Enter, Stay Out) and (Stay Out, Enter) yielding payoffs (38 , 0) and
(0 , 38) respectively. At these equilibria, neither Marvel nor DC Comics has an incentive to deviate.
Now suppose Marvel chooses its quantity prior to DC doing so; that is, if both firms enter,
they play a Stackelberg game with Marvel as the leader. As before, if only one firm enters it
will be a monopoly and any firm that does not enter makes zero profits (i.e. fixed costs are
only incurred if the firm enters the market).
e. In this version of the game, if both Marvel and DC enter, how much will each firm produce and
what profits will each firm earn? [Hint: Pay attention to the economic feasibility of your answer, and
remember that Marvel wants to maximize its profits based on what it believes DC will do.] (6)
Since Marvel is first mover, it can take into account what DC’s best response will be when it chooses
its quantity in the first period. DC’s best response function is the same as before and so we can write
Marvels’ profit function as
)
( , ) = ∗ ( ) − (22 + 80 − 5
)
= (140 − 20( + ) ∗ ( ) − (22 + 80 − 5
60 − 20
)
= $140 − 20( +
% ∗ ( ) − (22 + 80 − 5
30
40
)
= $140 − 40 +
− 20 % ∗ ( ) − (22 + 80 − 5
3
Taking the first order condition yields Marvel’s optimal quantity choice
( , )
80
= 140 − 40 + − 40 − 80 + 10 = 0 ⇒ = 6
3
Plugging this quantity into DC’s best response function then gives
=
60 − 20 ∗ (6)
= −2
30
Remember, however, that firms cannot produce negative quantities (that is, there is a positivity
constraint). Looking at DC’s best response function
=
60 − 20
2
= 2 − 30
3
we can see that DC will produce zero comics if Marvel produces more than three comics. Thus
Marvel receives Stackelberg profits if it produces < 3 (i.e. profits where DC produces a positive
amount based on what Marvel produces) and monopoly profits (i.e. profits where DC produces
zero) if it produces ≥ 3.
3
As can be seen in the graph above, Marvel will choose to produce the smallest quantity that induces
DC to produce zero comics. That is, Marvel will choose to produce = 3 and DC will follow by
producing = 0.
Thus, total market quantity is + = 3 + 0 = 3 which yields individual firm profits of
= '140 − 20(3)( ∗ (3) − (22 + 80 ∗ (3) − 5(3) ) = 23
1
= '140 − 20(3)( ∗ (0) − $22 + 80 ∗ (0) − (0) % = −22
2
f. Fill out the payoff matrix below for this modified version of the entry game. (3)
Marvel Comics
Enter
Stay Out
Enter
-22 , 23
38 , 0
Stay Out
0 , 38
0,0
DC Comics
g. What is/are the Nash Equilibrium outcome(s) of this game? Explain your answer. (4)
There is now only one equilibrium which is for DC to Stay Out and Marvel to Enter yielding payoffs
of (0 , 38). At this equilibrium, neither Marvel nor DC Comics has an incentive to deviate.
4
2. (30 points) The typical policyholder of BankCo Insurance Group has a wealth of $500,000,
of which $400,000 represents the value of his or her house. There are two risk types in the
market for fire insurance: low-risk customers have a 25% chance that their house will be
destroyed by fire; high-risk customers have a 75% chance that their house will be destroyed
by fire. (Assume that there is one of each type.) This monopoly insurance company
considers offering two policies to potential policyholders in the market. Each group is
interested in buying at most one insurance policy.
Policy A: Offers partial coverage. It would pay only $40,000 to the policy holder in the event
that his or her house burns down. The state regulator requires that the premium for this
policy be set at $10,000.
Policy B: Offers full coverage. It would pay $400,000 to the policyholder if his or her house
burns down. The state regulator requires that the premium for this policy be set at $300,000.
Each policyholder maximizes expected utility given by the following utility function
,(-) = √where W is the wealth of the policyholder. The insurance company is risk neutral; thus, it is
interested only in maximizing expected profits.
a. Which insurance policy, if any would the low-risk customers buy? (6)
The expected utility of the low-risk policy holder if he/she does not insure equals
/012 = 0.253100,000 + 0.753500,000 ≈ 609.39
The expected utility of the low-risk policy holder if he/she chooses Policy A equals
/072 = 0.253100,000 + 40,000 − 10,000 + 0.753500,000 − 10,000 ≈ 615.14
The expected utility of the low-risk policy holder if he/she chooses Policy B equals
/082 = 0.253100,000 + 400,000 − 300,000 + 0.753500,000 − 300,000 ≈ 447.21
Therefore, the low-risk type would choose policy A since 615.14>609.39>447.21
b. Which insurance policy, if any would the high-risk customers buy? (6)
The expected utility of the high-risk policy holder if he/she does not insure equals
/019 = 0.753100,000 + 0.253500,000 ≈ 413.95
5
The expected utility of the high-risk policy holder if he/she chooses Policy A equals
/079 = 0.753100,000 + 40,000 − 10,000 + 0.253500,000 − 10,000 ≈ 445.42
The expected utility of the high-risk policy holder if he/she chooses Policy B equals
/089 = 0.753100,000 + 400,000 − 300,000 + 0.253500,000 − 300,000 ≈ 447.21
Therefore, the high-risk type would choose policy B since 447.21 > 445.42 > 413.95
c. What are the insurance company’s expected profits? (6)
If the low-risk type buys policy A and the high-risk type buys policy B then BankCo Insurance
would earn expected profits equal to
/ = 1(10,000) + 1(300,000) − :0.25(40,000) + 0.75(0); − :0.75(400,000) + 0.25(0);
= 10,000 + 300,000 − 10,000 − 300,000 = 0
Thus, the firm breaks even with the self-selecting menu of policies.
Now, due to a loophole in regulation, suppose that the insurer is free to charge any
premium it wishes for Policy B and, specifically, as a monopoly, would set the premium at
the maximum that the buying policy holder is willing to pay.
d. What is the maximum price (i.e. willingness to pay) that the insurer can charge for Policy B that
leaves the high type’s decision unchanged? (6)
The high-type buyer originally chose to purchase Policy B. We therefore want to find this type’s
reservation premium for policy B, which we find by equating the expected utility from policy B less
the reservation premium with the expected utility from this type’s next-best option, policy A:
/089 (=̅) = /079
0.753100,000 + 400,000 − =̅ + 0.253500,000 − =̅
= 0.753100,000 + 40,000 − 10,000 + 0.253500,000 − 10,000
0.753100,000 + 400,000 − =̅ + 0.253500,000 − =̅ ≈ 445.42
3500,000 − =̅ ≈ 445.42
500,000 − =̅ ≈ (445.42)
=̅ ≈ 301,604.28
Hence, the most that Capital Insurance could charge the high-risk types is slightly less than
$301,604.28.
(Note: Due to potential rounding error when making calculations, an answer of =̅ ≈ 301,601.02
will also be accepted.)
6
e. By how much would the firm’s expected profits increase? (6)
Now, we must check whether a premium of $301,604.28 for policy B changes the low-risk type’s
behavior. The expected wealth of the low-risk policyholder if he or she chooses policy B equals:
/082 = 0.253100,000 + 400,000 − 301,604.28 + 0.753500,000 − 301,604.28 ≈ 445.42
Thus, the low-risk type would still buy policy A since 615.14 > 609.39 > 445.42.
If the low-risk type buys policy A and the high-risk type buys policy B at their reservation premium,
then BankCo Insurance would earn expected profits equal to
/ = 1(10,000) + 1(301,604.28) − :0.25(40,000) + 0.75(0);
− :0.75(400,000) + 0.25(0); = 10,000 + 301,604.28 − 10,000 − 300,000
= 1,604.28 > 0
Hence, the firm earns $1,604.28 more in expected profits by charging the high-risk types their
reservation premium for policy B.
(Note: Due to potential rounding error when making calculations, an answer of /Π = $1,601.02
will also be accepted.)
7
3. (30 points) Suppose the U.S. market for cigarettes is characterized by the following market
demand and supply functions:
B = C
− D = −
+ E
a. Graph the market supply and market demand curves (make sure to label intercepts). Compute and
label the equilibrium price and quantity for this market. (4)
To find the equilibrium, just set market demand equal to market supply.
+ = +F → 475 − 50 = −45 + 30
80 = 520
∗ = 6.5, + ∗ = 150
The U.S. Environmental Protection Agency classifies secondhand smoke – a mixture of smoke from
the lighted end of cigarettes and smoke exhaled by smokers – as a “known human carcinogen.”
Exposure to secondhand smoke has been linked to cancer amongst non-smokers. The marginal
external cost of cigarette smoking is given by
HI =
You may assume that anyone who buys a cigarette then smokes the cigarette (i.e. consumers do not
just buy cigarettes and then keep them around unused).
b. Is this an example of a negative or positive externality? Is it a consumption or production externality?
Briefly explain your answer. (2)
This is an example of a negative (imposes a cost on others) consumption (because it’s imposed by
consumers of cigarettes) externality.
8
c. Write down the social marginal cost function (i.e. the relationship between quantity and social marginal
cost) and the social marginal benefit function (i.e. the relationship between quantity and social marginal
benefit). (4)
Because there is no production externality, the social marginal cost is simply equal to the private marginal
cost:
JKL = JL =
1
3
++
30
2
With a negative consumption externality, the social marginal benefit is equal to the private marginal
benefit minus the marginal external cost.
JKM = JM − J/L
1
19 1
− + − $ +%
=
50
2 50
19 1
=
− +
2 25
d. Graph the social marginal cost and social marginal benefit functions (make sure to label intercepts).
Compute and label the socially efficient quantity of cigarettes. (4)
(Note: It is not necessary to graph the private marginal benefit curve or label the curves - MSC, MPC,
MSB, MPB - to receive full credit).
To find the socially efficient quantity, just set the marginal social benefit equal to the marginal social cost:
JKL = JKM
1
3 19 1
++ =
− +
30
2
2 25
11
+ = 8 → + ∗ = 109.09
150
9
e. What is the deadweight loss from the presence of this externality? (5)
Deadweight loss is given by
1
NOP = (6.5 − 3.5)(150 − 109.09) = 61.36
2
f. What is the total damage (cost) from the externality at the competitive equilibrium? Is this more or less
than the total damage at the socially efficient quantity? (5)
The total damage from the externality at the competitive equilibrium is the area of the shaded region in
the figure below. It is equal to
QRSTU/VSWXYTUZS[LR\S
1
1
1
= (9.5 − 6.5)(300 − 0) − (9.5 − 6.5)(150 − 0) − (6.5 − 3.5)(300 − 150)
2
2
2
= 225
10
Another way to calculate this is by determining the area under the marginal external cost curve at the
competitive equilibrium quantity.
1
QRSTU/VSWXYTUZS[LR\S = (150)(3) = 225
2
This externality is negative so the total externality damage at the competitive equilibrium is more than the
total externality damage at the social optimum.
The government decides to charge cigarette buyers a per-unit tax for each cigarette they purchase.
g. What tax should they charge so that the competitive equilibrium quantity is the socially efficient
quantity? (6)
The government will charge a tax equal to the marginal external cost at the socially efficient quantity:
11
QTV = J/L(+ ∗ ) =
109.09
= 2.18
50
To see that this is true, we can check that the competitive equilibrium quantity when this tax is imposed is
the socially efficient quantity. At the competitive equilibrium:
+ = +F 475 − 50( + ]) = −45 + 30
475 − 50 − 50(2.18) = −45 + 30
80 = 410.91
= 5.14, + = 109.09
(Note: To receive full credit, it is sufficient to simply provide the calculation: QTV = J/L(+ ∗ ) = 2.18.)
12
4. (30 points) Greg, a risk-neutral concerned parent, is worried about his son Jacob who is having
trouble studying for his economics exam. Greg would like Jacob to be studious and decides to
incentivize Jacob by giving him a cash reward where the amount Jacob is paid depends on the
grade he receives.
Jacob can be either studious (exert costly high effort) or lazy (exert costless low effort) when
studying. Jacob is risk averse and only derives utility from money (i.e. he’s indifferent about which
grade he receives). His utility function is:
,(^) = _
√^ − ifJacobisstudious
ifJacobisstudious
ifJacobislazy
√^ifJacobislazy
where w is the cash he receives from his father, Greg.
Unfortunately, Greg is unable to observe what Jacob is doing in his own room or school (i.e. Greg
cannot tell if Jacob is being studious or lazy). Greg can only observe the grade Jacob receives when
he brings home his report card.
Jacob can either receive an A, B, or C letter grade for the exam. What grade Jacob receives is
partly a result of how much effort he puts in to studying and partly a result of luck. The table below
lists the probabilities of getting an A, B, or C letter grade both when Jacob is being studious and is
being lazy (both Greg and Jacob know these probabilities).
Studious
Lazy
A
0.4
0.2
B
0.4
0.5
C
0.2
0.3
Initially Greg considers a cash reward plan (call this “Original Plan”) where he pays Jacob $49 if
he receives an A, $16 if he receives a B, and $9 if he receives a C.
a. What is Greg’s expected payout to Jacob if Jacob is studious? If Jacob is lazy? (6)
If Jacob is studious, Greg’s expected payout to Jacob is given by
E:payout;F = 0.4(49) + 0.4(16) + 0.2(9) = $27.8
If Jacob is lazy, Greg’s expected payout to Jacob is given by
E:payout;2 = 0.2(49) + 0.5(16) + 0.3(9) = $20.5
b. Will Jacob choose to be studious or lazy if Greg promises to follow the Original Plan? Show your
work. (6)
To determine whether Jacob will choose to be studious or lazy, we must compare his expected utility
from being studious and from being lazy.
/0 F = 0.4(√49 − 1) + 0.4(√16 − 1) + 0.2(√9 − 1) = 4
13
/0 2 = 0.2√49 + 0.5√16 + 0.3√9 = 4.3
Since /0 2 > /0 F , Jacob will choose to be lazy under Plan A.
c. Given that Greg would like Jacob to be studious, does it ever make sense for him to pay Jacob a nonzero amount for receiving a C? For receiving a B? For receiving an A? Explain why or why not. (6)
It only ever makes sense to pay Jacob a non-zero amount for receiving an A. Greg should pay a zero
amount for receiving a B or a C.
Greg can only ensure that Jacob will be studious if his cash reward plan results in /0 F ≥ /0 2 . Note that
the probability of receiving a B or C is higher if Jacob is lazy so giving Jacob any cash if he receives a B
or C increases /0 2 more than it does /0 F and thus incentivizes him to be lazy. On the other hand, the
probability of receiving an A is higher if Jacob is studious so giving Jacob any cash if he receives an A
increases /0 F more than it does /0 2 and thus incentivizes him to be studious. Thus, Greg would be best
able to incentivize Jacob to be studious if he only pays him if he receives an A (and pays zero if Jacob
receives a B or C).
d. Given your answer to part c, what is the cheapest (lowest expected payout) cash reward plan (of the
form “pay Jacob $X if he receives an A, $Y if he receives a B, and $Z if he receives a C”) that Greg can
offer Jacob under which Jacob will choose to be studious. (6)
Given the answer to part c, Greg should choose a cash plan that pays some amount $X if Jacob receives
an A and pays $0 if Jacob receives a B or C. Then the cheapest such cash plan is the one pays the
minimum amount $X for receiving an A while still incentivizing Jacob to be studious (i.e. still resulting in
/0 F ≥ /0 2 ).
Thus to find the smallest amount of cash that Greg would have to pay Jacob for an A (while paying $0 for
a B or C), we must compare the expected utility Jacob receives from being studious and being lazy. Call
“X” the amount Greg pays Jacob for receiving an A. Jacob will choose to be studious so long as
/0 F ≥ /0 2
0.4'√p − 1( + 0.4'√0 − 1( + 0.2'√0 − 1( > 0.2'√p( + 0.5'√0( + 0.3'√0(
0.4√p − 1 ≥ 0.2√p
0.2√p ≥ 1
√p ≥ 5
p ≥ 25
Thus the cheapest cash reward plan under which Jacob will be studious is to pay Jacob $25 if he receives
an A, $0 if he receives a B, and $0 if he receives a C.
Greg’s expected payout is then
E:payout;F = 0.4(25) + 0.4(0) + 0.2(0) = $10.
14
Suppose that the probabilities of Jacob receiving an A, B, or C letter grade are as before except let
qr be the probability of Jacob receiving an A if he is studious and qs = . − qr be the probability
of Jacob receiving a B if he is studious.
A
=7
0.2
Studious
Lazy
B
0.8 − =7
0.5
C
0.2
0.3
e. Under the Original Plan, what is the minimum probability of receiving an A (=7 ) that Jacob must have
for him to have prefer to be studious. (6)
Jacob will be studious so long as
/0 F ≥ /0 2
=7 '√49 − 1( + (0.8 − =7 )'√16 − 1( + 0.2'√9 − 1( ≥ 0.2'√49( + 0.5'√16( + 0.3'√9(
6=7 + (0.8 − =7 )(3) + 0.2(2) ≥ 0.2(7) + 0.5(4) + 0.3(3)
6=7 + (0.8 − =7 )(3) ≥ 3.9
3=7 ≥ 1.5
=7 ≥ 0.5
Thus the probability of Jacob receiving an A must be at least 0.5 for him to have preferred being studious
under the Original Plan.
15