MGMT 405 (Econ) Problem Set #6
Learning Team 9, Section B
Alanna McCarthy, Jing Gao, Dan Nagel, Andrew Na, Graham Bloom, Kevin Vaishnav
Problem Set 6 (100 points: 20 per problem)
Problem 1.
A. Find the market's equilibrium price and output.
Set demand and supply equations equal in terms of Q and solve.
200 - 0.2Q* = 100 + 0.3Q*
Q* = 200
P* = $160
B. Suppose the government imposes a tax of $20 per unit of output on all firms in the industry.
What effect does this have on the industry supply curve? Find the new competitive price and
output. What portion of the tax has been passed on to consumers via a higher price?
The industry supply curve will shift up / to the left to reflect the new tax; the new supply
curve is P = 100 + 20 + 0.3Qs = 120 + 0.3Qs. Set this new supply curve equal to the original
demand curve to calculate the new equilibrium.
200 - 0.2Q* = 120 + 0.3Q*
Change in price = $8
Q* = 160
P* = $168
portion passed on to consumers = 8/20 = 40%
C. Suppose a $20 per unit sales tax is imposed on consumers. What effect does this have on the
industry demand curve? Find the new competitive price and output. Compare this answer to your
findings in part 2.
The industry demand curve will shift down / to the left to reflect the new tax; the new
demand curve is P = 200 - 20 - 0.2Qd = 180 - 0.2Qd. Set this new demand curve equal to the
original supply curve to calculate the new equilibrium.
180 - 0.2Q* = 100 + 0.3Q*
But tax is $20, then Price is $168
The portion is 8/20 = 40%
0.5*Q* =80 Q*=160, P*=148,
D. The tax considered above is a specific (i.e. per unit) tax, can you find an ad-valorem tax that
would generate the same amount of revenues for the government? Using a diagram discuss
which one of these alternative tax schemes generates the highest dead-weight-loss.
An ad valorem or percentage tax will cause the supply curve to tilt or pivot to the left. This
is because the size of the tax increases with price. The diagram above shows the effect of a
specific tax, the diagram below shows the effect of an ad valorem tax
Price
Quantity
Tax rate
Option 1
168
160
11.9%
So, the Deadweight Loss is not changing whether the unit amount tax is imposed to Buyer
or Supplier, or unit rate tax(Ad-valorem tax) imposed to buyer.
(Figure Reference
http://www.tutor2u.net/economics/content/topics/marketsinaction/indirect_taxes.htm)
Scully Modification:
The demand curve is P=200-0.2Qd, Supply curve is P=100+0.3×Qs When there is ad valorem
tax, the new Supply curve’= (100+0.3Qs)(1+T),
So we need to get the new tax revenue equals the previous revenue which is (PbPseller)*Q=(168-148)*160=3200, So we need to get new (Pbuyer-Pseller)*Q=3200
Pbuyer=200-0.2Q,
Pseller=100+0.3Q,
so
[(200-0.2Q)-(100+0.3Q)]*Q=3200(1000.5Q)*Q=3200, 100Q-0.5Q^2-3200=0, Q^2-200Q+6400=0 Q1=160, Q2=40,
So accordingly, when Q1=160, Pbuyer=200-0.2*160-168, Pseller=100+0.3*160=148, so tax% =
(168-148)/148=13.51% which is the original unit sale tax rate.
When Q2=40, Pbuyer=200-0.2*40=192, Pseller=100+0.3Q=100+0.3*40=112,so tax% =(192112)112=71.42%
So the deadweight loss of tax per unit is (168-148)*(200-160)/2=400 and
The deadweight loss of ad-valorem tax is (192-112)*(200-40)/2=6400 and this is much
greater than the deadweight loss of tax per unit.
Please refer to the following two charts:
The deadweight loss of tax per unit:
The deadweight loss of ad-valorem tax:
Problem 2.
A. Calculate the firm's optimal price in the US. Show your work.
Pa = Price in America, Qa = Quantity in America
Optimal Price is where MR = MC.
Solve for P as a function of Q.
Qa = 2,000,000 - 20,000Pa
Pa = 100 - (Qa/20,000)
Insert Pa into the Total Revenue Equation to Solve for Total Revenue as function of Q.
TR = P x Q = TR = [100 - (Qa/20,000)] x Q = 100Q - [Qa^2/20,000]
Take the derivative of the TR equation to get Marginal Revenue
MR = 100 - (Qa/10,000)
Take the derivative of the TC equation to get the Marginal Cost
TCa = $40,000,000 + 5Qa , MC = 5
Optimal Price in America is where MC = MR
5 = 100 - (Qa/10,000) , Qa = 950,000
Insert Qa into the above equation for price: Pa = 100 - (Qa/20,000)
Pa = 100 - (950,000/20,000) , Pa = $52.5
B. What is the optimal price to charge in the Brazilian market? Show your work.
Pb = Price in Brazil
Qb = Quantity in Brazil
Optimal Price is where MR = MC.
Solve for P as a function of Q. : Qb = 200,000 - 3,000Pb, Pb = 66.67 - (Qb/3,000)
Insert Pb into the Total Revenue Equation to Solve for Total Revenue as function of Q.
TR = P x Q = TR = [66.67 - (Qb/3,000)] x Q = 66.67Q - [Qb^2/3,000]
Take the derivative of the TR equation to get Marginal Revenue
MR = 66.67 - (Qb/1,500)
Take the derivative of the TC equation to get the Marginal Cost
TCa = $1,000,000 + 5Qb, MC = 5
Optimal Price in Brazil is where MC = MR
5 = 66.67 - (Qb/1,500), Qb = 92,505
Insert Qb into the above equation for price: Pb = 66.67 - (Qb/3,000)
Pb = 66.67 - (92,505/3,000), Pb = $35.84
C. Explain why the problem of parallel imports, a form of arbitrage, may result from the
pricing structure you have calculated in the previous questions.
The price in Brazil is cheaper than in the US for the same product. This incentivizes
consumers to purchase the product in Brazil and sell in the US for a profit leveraging
arbitrage.
D. If it cost the company $1,500,000 annually to eliminate the problem of parallel imports,
should it do so? Explain!
Qa = 2,000,000 - 20,000Pa
+ Qb = 200,000 - 3,000Pa
Qu = (Qa + Qb) = 2,200,000 - 23,000Pu (Demand for United market Qu, Price : Pu)
MC = $5
Pu = (2,200,000-Qu)/23,000
R = (2,200,000Q - Q2) / 23,000
MR = (2,200,000 - 2Q) / 23,000 = 5
MR = MC
Qu = 1,402,500 should change to 1,042,500, Pu=50.326,
Profit of united market is $6,252,446, decreased by $724,637.68 if the parallel imports are
impossible. So, this company should not spend $1.5 million to eliminate the problem.
Problem 3.
A. Are there economies of scale for these coffee shops? Explain briefly.
To evaluate returns to scale, look at AC vs. Q
For SBUX: AC = TC/Q = 50k(1/Q) + 0.6
For CBTL: AC = 60k(1/Q) + 0.6
Since Q is in the denominator and AC goes down as Q goes up, we have increasing returns
to scale.
B. What is the optimal price for each of the two operators? Show your work.
Optimal price = price on demand curve where MR = MC
SBUX:
P = 2.273 - Q/220k
PQ = 2.273Q - Q^2/220k
MR = d(PQ)/dQ = 2.273 - Q/110k
MC = d(TC)/dQ = 0.60
Setting MR = MC, we get Q = 184,030 and plugging this into the demand curve, we get
P = $1.44 for SBUX
CBTL:
P = 2 - Q/200k
d(PQ)/dQ = 2 - Q/100k
MC = 0.60
Setting MR = MC, we get Q = 140,000 and plugging this into the demand curve, we get
P = $1.30 for CBTL
C. What is the maximum annual amount that the Airport Commission can get for this
franchise? Show your work.
The maximum annual amount the Airport Commission can extract is equal to Total Profit
= TR - TC = (P*Q) - TC
For SBUX:
= (1.44*184,030) - (50k + 0.6*184,030) = $104,585
For CBTL:
= (1.30*140,000) - (60k + 0.6*140k) = $38,000
Therefore, the maximum amount the Airport Commission can extract is $104,585
(assuming a blind bidding process)
Problem 4 – Lemon Cartel
A. Tabulate total, average and marginal costs for each firm for output levels between 1 and 5
cartons per month (i.e. for 1, 2, 3, 4, 5 cartons).
Output Level
Firm Cost Function
Total Cost
Average Cost
Marginal Cost
1
TC1 = 20 + 5Q^2
25.00
25.00
5.00
TC2 = 25 + 3Q^2
28.00
28.00
3.00
TC3 = 15 + 4Q^2
19.00
19.00
4.00
TC4 = 20 + 6Q^2
26.00
26.00
6.00
TC1 = 20 + 5Q^2
40.00
20.00
15.00
TC2 = 25 + 3Q^2
37.00
18.50
9.00
TC3 = 15 + 4Q^2
31.00
15.50
12.00
2
3
4
5
TC4 = 20 + 6Q^2
44.00
22.00
18.00
TC1 = 20 + 5Q^2
65.00
21.67
25.00
TC2 = 25 + 3Q^2
52.00
17.33
15.00
TC3 = 15 + 4Q^2
51.00
17.00
20.00
TC4 = 20 + 6Q^2
74.00
24.67
30.00
TC1 = 20 + 5Q^2
100.00
25.00
35.00
TC2 = 25 + 3Q^2
73.00
18.25
21.00
TC3 = 15 + 4Q^2
79.00
19.75
28.00
TC4 = 20 + 6Q^2
116.00
29.00
42.00
TC1 = 20 + 5Q^2
145.00
29.00
45.00
TC2 = 25 + 3Q^2
100.00
20.00
27.00
TC3 = 15 + 4Q^2
115.00
23.00
36.00
TC4 = 20 + 6Q^2
170.00
34.00
54.00
B. If the cartel decided to ship 10 cartons per month and set a price of $25 per carton, how
should output be allocated among the firms?
Orchard 1: 2 cartons
Orchard 2: 3 cartons
Orchard 3: 3 cartons
Orchard 4: 2 cartons
C. At this shipping level, which firm has the most incentive to cheat? Does any firm
not have an incentive to cheat?
Orchard 2 has the most incentive to cheat, while Orchards 3 and 4 have no incentive to
cheat by increasing their outputs as their marginal costs increase as output levels increase.
Problem 5 – Dayna’s Doorstops
A. What price should DD set to maximize profit? What output does the firm produce? How
much profit and consumer surplus does DD generate?
Q=10, P=35, Profit =200, Consumer Surplus 200, should change to 100
B. What would output be if DD acted like a perfect competitor and set MC = P? What profit
and consumer surplus would then be generated?
Q=15, P=25, Profit =125, Consumer Surplus 225
C. What is the deadweight loss from monopoly power in part A?
Dead weight loss=(35-15)*(15-10)/2=50
D. Suppose the government, concerned about the high price of doorstops, sets a maximum
price at $27. How does this affect price, quantity, consumer surplus and DD’s profits?
What is the resulting deadweight loss?
When there is price ceiling, we assume that the demand curve would not change because the
customer’s willing price has not change. However, the MR curve changes to a horizontal line of
MR=27 before the Output of 14(The intersection of MR and Demand). When output is greater
than 14, however, the MR curve is the MR=-55-4Q because no customers are willing to pay a
price greater than demand curve. When Output is greater than 14, DD will not produce as MR is
less than MC which leads to a loss.
Therefore, the price will be charged to 27 as this is the optimal point of profit maximizing and
Quantity is 14.
The Consumer surplus is (55-27)*14/2=196
The profit=TR-TC= P*Q-(100-5Q+Q^2)=27*14-(100-5*14+14^2)=378-226=152
Deadweight
Loss
2
E. Now suppose the government sets the maximum price at $23. How does this decision
affect price, quantity, consumer surplus, DD’s profit, and deadweight loss?
Q=14, P=23, Profit =96, Consumer Surplus 252, Deadweight Loss 2
When maximum price=23, same as D, the MR=23 because only at ceiling price, the profit is
maximized. The quantity is the intersection of MR and MC which is 14.
The Profit=TR-TC=23*14-(100-5*14+14^2)=96
The consumer surplus is pentagon area which is (55-27)*14/2+(27-23)*14=252
The dead weight loss is =(15-14)*(27-23)/2=2
F. Finally, consider a maximum price of $12. What will this do to quantity, consumer
surplus, profit and deadweight loss?
Q=8.5, P=12, Profit =-27.75, Consumer Surplus 293.25, Deadweight Loss 84.5
When the Price ceiling is 12, MR=12. When MR=MC=-5+2Q, the quantity is 12=-5+2Q, →
Q=8.5 and P=12
Profit=TR-TC=8.5*12-(100-5*8.5+8.5^2)= 102-129.75=- 27.75 because at this point, although
MR=MC and profit is maximized, MR is less than AC which is TC/Q=129.75/8.5=15.26, the
company would not operate here.
The consumer surplus is (55-38)*8.5/2+8.5*(38-12) =293.25
The dead weight loss is (38-12)*(15-8.5)/2 = 84.5