SOLUTION: Economics Demand Supply and Marginal Cost Questions

1. A publisher faces the following demand for the next novel of one of its popular authors:
Q
! = 400,000 − 2,000P
The author is paid $ 3 million to write the book and the marginal cost of publishing the book is a
constant at $25 per book.
!
FC
= $ 3,000, 000
!
MC
= $ 25
a) Derive the Marginal Revenue curve.
To derive the MR curve, we use the demand equation to derive the Revenue curve.
! = 400,000 − 2,000P
Q
2,000P
= 400,000 − Q
!
P
! =
400,000 − Q
2,000
! = 200 − 0.0005Q
P
! = PQ
R
R
! = (200 − 0.0005Q)Q
! = 200Q − 0.0005Q 2
R
!
MR
=
dR
dQ
!
MR
= 200(1) − 2(0.0005Q)
!
MR
= 200 − 0.001Q
b) Graph the marginal revenue, marginal cost, and demand curves. At what quantity do the
marginal revenue and marginal cost curves cross? What does this signify?
!
MR
= 200 − 0.001Q
!
MC
= 25
! = 200 − 0.0005Q
P
P
$200
MR
D
DWL
$25
MC
175,000 200,000
400,000
Q
In the graph, the red line represents the MR curve, the blue line presents the Demand curve, and
the green line represents the MC curve. The MR curve crosses the x-axis at Q = 400,000 while
the Demand curve crosses the x-axis at Q = 200,000. The triangle indicated by the DWL is equal
to the deadweight loss. The MR and MC curves cross at the point (175,000, $25) or when Q =
175,000 and P = $25. The point at which the MR and MC curves cross each other signify the
profit-maximizing point where the publisher should publish at the given quantity and price in
order to maximize their profit.
c) In your graph, shade the deadweight loss. Explain in words what this means.
P
$200
MR
D
MC
$25
175,000 200,000
400,000
Q
In the graph, the purple triangle represents the deadweight loss. The deadweight loss represents
the cost to society by market inefficiency. The deadweight loss signifies that the total surplus in
the economy is less than the total surplus if the market were to be competitive. This is due to the
fact that the monopolist produces less than the socially efficient level of output.
d) What quantity would a profit-maximizing publisher choose? What price would it charge?
Show how you derived these quantities.
The profit of the publisher will be maximized when MC = MR. Thus, we equate the two curves
and solve for Q to determine the quantity that would maximize the profit of the publisher.
!
MC
= MR
! = 200 − 0.001Q
25
!
0.001Q
= 200 − 25
Q
! =
175
0.001
Q
! = 175,000
From the calculation above, we see that profit is maximized when 175,000 books are produced.
Since the MC curve is constant at $25, the profit-maximizing price is equal to $25.
e) If the author were paid $2 million instead of $3 million to write the book, how would this
affect the publisher’s decision regarding the price to charge? Explain.
If the author were to be paid $2 million or any other amount other than the original price of $3
million, there would not affect the publisher’s decision regarding the price to charge. The amount
paid to the author is a fixed cost which will affect the total cost of producing books as well as the
profit, which would decrease; both the marginal revenue or marginal cost would not change. The
publisher would still be maximizing its profit by charging $25 per book regardless of the amount
they pay the author.
f) Suppose the publisher was not profit-maximizing but was concerned with maximizing
economic efficiency. What price would it charge for the book? How much profit would it make
at this price?
To maximize economic efficiency, the publisher should charge at a price equal to the marginal
cost, which is $25 per book. At this price, the publisher would earn negative profits which would
then be equal to the amount paid to the author, which is $3 million. To show this mathematically,
we derive the total cost curve and the profit curve.
TC
= (MC )Q + FC
!
!
TC
= 25Q + 3,000, 000
!
Prof
it, π = R − TC
! = 200Q − 0.0005Q 2 − (25Q + 3,000, 000)
π
! = − 0.0005Q 2 + 175Q − 3,000, 000
π
! (0) = − 0.005(0)2 + 175(0) − 3,000, 000
π
π
! (0) = − 3,000, 000
2. Based on market research, a monopolistic recording company obtains the following
information about the demand and production costs of its new CD.
!
Dem
a n d : P = 1000 − 2Q
!
Total
Re venu e : T R = 1000Q − 2Q 2
Margin
al Cost : MC = 100 +
!
1
Q
2
where Q indicates the number of copies sold and P is the price in cents.
a) Find the price and quantity that maximize the company’s profit.
To maximize the company’s profit, we equate the MR and MC curves and solve for Q. Then, we
substitute Q into either of the two curves to solve for P.
!
MR
=
dT R
dQ
!
MR
= 1000(1) − 2(2Q)
!
MR
= 1000 − 4Q
MC
= 100 +
!
1
Q
2
!
MR
= MC
1000
− 4Q = 100 +
!
1
Q
2
1
! Q + 4Q = 1000 − 100
2
9
! Q = 900
2
2
Q
! = 900
(9)
Q
! = 200
! = MR = 1000 − 4Q
P
P
! = 1000 − 4(200)
P
! = 1000 − 800
P
! = 200 ₵ = $ 2
Thus, the profit will be maximized when 200 copies of the CD are sold at a price equal to $2.
b) Find the price and quantity that would maximize social welfare.
To maximize the social welfare, we equate the demand curve and the MC curve and then solve
for Q. Using the Q, we then substitute it into either of the two curves to solve for P.
! = 1000 − 2Q
P
MC
= 100 +
!
1
Q
2
! = MC
P
1000
− 2Q = 100 +
!
1
Q
2
1
! Q + 2Q = 1000 − 100
2
5
! Q = 900
2
2
! = 900
Q
(5)
Q
! = 360
! = 1000 − 2(360)
P
! = 1000 − 720
P
P
! = 280₵ = $ 2.8
From the calculation above, we see that the social welfare is maximized when 360 units of the
CD are sold at the price of $2.8.
c) Calculate the deadweight loss from the monopoly.
To calculate the deadweight loss, we take the difference between the social welfare-maximizing
quantity and the profit-maximizing quantity.
Q
! social welfare − Qprofit = 360 − 200 = 160
Then, we take the difference between the demand when Q = 200 and the marginal cost when Q =
200.
! demand (200) − Qmarginal cost (200) = $6 − $ 2 = $ 4
Q
Finally, we calculate the deadweight loss by multiply the two quantities above and dividing the
product by two.
!
DW
L=
(160)($ 4)
= $ 320
2
Thus, the deadweight loss is equal to $320.
3. Bruce, Collen, and David are all getting together at Bruce’s house on Friday evening to play
their favorite game, Monopoly. They all love to eat sushi while they play. They all know from
previous experience that two orders of sushi are just the right amount to satisfy their hunger. If
they wind up with less than two orders, they all end up going hungry and don’t enjoy the
evening. More than two orders would be a waste, because they can’t manage to eat a third order
and the extra sushi just goes bad (although they receive no specific disutility from wasting food).
Their favorite restaurant, Chaya, packages its sushi in such large containers that each individual
person can feasibly purchase at most one order of sushi. Chaya offers takeout, but unfortunately
doesn’t deliver.
Suppose that each player enjoys $20 worth of utility from having enough sushi to eat on Friday
evening and 0 from not having enough to eat. The cost to each player of pick up an order of sushi
is $10. Unfortunately, the three have forgotten to communicate about who should be buying
sushi this Friday, and none of the players has a cell phone, so they must make independent
decisions of whether to buy (B) or not buy (NB) an order of sushi.
a) Write down this game in normal (table) form.
Options: B, NB
One buys
Two buy
All buy
(Bruce, Collen, David) (Bruce, Collen, David) (Bruce, Collen, David)
B, NB, NB
B, B, NB
No one buys
(Bruce, Collen,
David)
NB, B, NB
B, NB, B
NB, NB, B
NB, B, B
B, B, B
NB, NB, NB
b) Find all of the Nash equilibria in pure strategies.
From the table, we see that there are 8 pure strategies. We then transform the table such that the
number of orders is calculated with their corresponding total cost per strategy.
One buys
(Order, Cost)
Two buy
(Order, Cost)
1, $10
2, $20
1, $10
2, $20
1, $10
2, $20
All buy
(Order, Cost)
No one buys
(Order, Cost)
3, $30
0, $0
Then, we convert the table above into utility. When there is not equal to 2 orders, the utility is $0
while the utility is $20 when there are 2 orders.
One buys
(Utility)
Two buy
(Utility)
0
$20
0
$20
0
$20
All buy
(Utility)
No one buys
(Utility)
0
0
From the table above, we see that there are 3 pure strategies that will lead to Nash equilibria.
These are: (1) Bruce and Collen order, (2) Bruce and David order, and (3) Collen and David
order.
4. There are two distinct proposals, A and B, being debated in Washington. The Congress likes
proposal A, and the president likes proposal B. The proposals are not mutually exclusive; either
or both or neither may become law. Thus, there are four possible outcomes, and the rankings of
the two sides are as follows, where a larger number represents a more favored outcome.
Outcome
Congress President
A becomes law
4
1
B becomes law
1
4
Both A and B become law
3
3
Neither become law (status quo)
2
2
a) The moves in the game are as follows. First, the Congress decides whether to pass a bill and
whether it is to contain A or B or both. Then, the president decides whether to sign or veto the
bill Congress does not have enough votes to override a veto. Therefore, if the president vetoes
the bill, the status quo remains (i.e. neither A nor B become law). Draw the game tree, making
sure to label player’s nodes and branches and payoffs, and indicate the equilibrium derived using
backward induction.
President signs bill (5)
President vetoes bill (4)
Bill contains A
President signs bill (5)
Congress
Bill contains B
Bill contains A and B
President vetoes bill (4)
President signs bill (6)
President vetoes bill (4)
Bill contains neither A nor B
President signs bill (4)
From the game tree above, we see that the most favorable outcome
Presidentisvetoes
whenbill
the(4)
President signs
the bill that both contains A and B where the payoff is equal to 6.
b) Now suppose the rules of the game are changed in only one respect: the president is given the
extra power of a line-item veto. Thus, if the Congress passes a bill containing both A and B, the
president may choose not only to sign or veto the bill as a whole, but also to veto just one of the
two items. Show the new game tree and equilibrium.
President signs bill (5)
President vetoes bill (4)
Bill contains A
President signs bill (5)
Congress
Bill contains B
Bill contains A and B
President vetoes bill (4)
President signs bill (6)
President vetoes bill (4)
President signs A but vetoes B (5)
President signs B but vetoes A (5)
Bill contains neither A nor B
President signs bill (4)
President vetoes bill (4)
From the game tree above, we see that the most favorable outcome is still when the president
signs the bill containing A and B. Thus, equilibrium is maintained despite the new rule added.

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