Download Answers to ECMC02 First Test, October 28, 2006

Answers to ECMC02 First Test, October 28, 2006
1. Demand is P = 50 - .015Q. Supply is P = 10 + .005Q. These intersect
where 50 - .015Q = 10 + .005Q, or 40 = .020Q, so Q* = 2000.
Substituting into the demand curve we have P* = 50 - .015(2000) =
$20. The consumer surplus is given by the area under the demand
curve but above the price line, which is (50 – 20) x 2000/2 = $30,000.
The correct answer is (I).
2. Producer surplus is given by the area below the price line and above
the MC or supply curve, which is (20 – 10) x 2000/2 = $10,000. The
correct answer is (D).
3. If the number of taxi rides per day is restricted by quota to 1000
rides, the going price will be P = 50 - .015(1000) = $35. The cost to
producers of providing this number of taxi rides is given by the
original supply curve, or P = 10 + .005(1000) = $15. The new producer
surplus is given by the area below the new price line and above the
supply curve, so it is [(35 – 15) x 1000] + (15 – 10) x 1000/2 = $22,500.
The correct answer is (H).
4. The quota restricts output to 1000 rides per day. The areas of
consumer surplus and producer surplus that originally existed beyond
1000 rides per day are lost to society because of the quota.
Therefore, the deadweight loss is (35 – 15) x (2000 – 1000)/2 =
$10,000. The correct answer is (D).
5. The new purchasing policy will expand producer surplus, but it will cost
a substantial amount of wasted government revenue. The net of these
two provides the measure of deadweight loss. First, we must
determine how much additional purchasing takes place. The question
says that price will be kept up at the “quota” level from Question 3,
which is $35. From the supply curve, we can see that this means that
$35 = 10 + .005Q, so that Q = 25/.005 = 5000 units. Private demand
for taxi rides at this price will be 1000 units, as we know from the
answers above. Therefore, the government will have to purchase
4000 taxi rides to keep price at this level. The additional producers
surplus will be 4000 x (35 – 20)/2 = $30,000. However, the wasted
cost of these taxi rides will be 4000 x $35 = $140,000. The net
amount of deadweight loss is therefore $110,000. The correct
answer is (U).
6. If demand is P = A – bQ, then MR = A – 2bQ. MC = dTC/dQ = c. The
monopoly firm will profit-maximize by setting A – 2bQ = c, so the
equilibrium quantity traded will be Q* = (A-c)/2b. The profitmaximizing price will be given by substituting this into the demand
curve, so P* = A – b((A-c)/2b) = A – A/2 + c/2 = A/2 + c/2 = (A + c)/2.
The correct answer is (F).
7. The amount of deadweight loss is given by comparing the monopoly
result with the perfectly competitive result. The perfectly
competitive result would be where P = MC or A – bQ = c, or Q = (A –
c)/b. The amount of lost production due to monopoly is therefore (A –
c)/b – (A – c)/2b. The amount of deadweight loss is [(A + c)/2 – c] x
[(A – c)/b – (A – c)/2b]/2 = [(A - c)/2] x [(A – c)/2b]/2 = (A – c)2/8b.
The correct answer is (C).
8. If b were twice as large, the denominator would be multiplied by 2.
This would decrease the deadweight loss by half (i.e., the more
inelastic the demand curve, the smaller the deadweight loss).
9. When MR = 0 with a linear demand curve, the demand curve is unit
elastic, or elasticity = -1. The correct answer is (O).
10. There are two distinct markets: Canada and China. In Canada, demand
is P = 1900 - .75Q. Therefore, MR = 1900 – 1.5Q. Since MC = 100, we
have 1900 – 1.5Q = 100 or Q* = 1200. Substituting into the demand
curve, we find P = 1900 - .75(1200) = $1000. The correct answer is
(W).
11. In China, the demand curve is P = 900 – .25Q. Therefore, MR = 900 –
.5Q. Since MC = 100, we have 900 – .5Q = 100, or Q* = 1600.
Substituting into the demand curve, we find P = 900 – 400 = $500.
The correct answer is (N).
12. If the widget producer is unable to discriminate between markets, he
must serve the entire market. We can find the joint market by
summing across the quantity dimension. We have: (Q1 = 2533.333 –
[4/3]P1) + (Q2 = 3600 – 4P2) = (Q = 6133.333 – [16/3]P). This can be
inverted to find P = 1150 - .1875Q. Therefore marginal revenue in the
joint market is MR = 1150 - .375Q. MC = 100, so profit maximization
is at 1150 - .375Q = 100 or Q* = 2800. Substituting into the joint
demand function, we find P = 1150 - .1875(2800) = $625. Profit
before markets are joined together (profit with price discrimination)
is [(1000 x 1200) + (500 x 1600)] – [500,000 + (100 x 2800)] =
$1,220,000. Profit after markets are joined together is [(625 x
2800) – (500,000 + (100 x 2800)] = $970,000. Profit falls by
$250,000 if the widget producer is unable to discriminate between
markets. The correct answer is (M).
13. In Canada, consumer surplus under price discrimination is [(1900 –
1000) x 1200]/2 = $540,000. Without price discrimination, the price
is $625. At this price, we can calculate using the Canadian demand
curve that 1700 units will be sold to Canadian consumers. The
consumer surplus will therefore be [(1900 – 625) x 1700]/2 =
$1,083,750. The rise in consumer surplus is $543,750. The correct
answer is (R).
14. In China, consumer surplus under price discrimination is [(900 – 500)
x 1600]/2 = $320,000. Without price discrimination, the price is
$625, and the quantity sold is 1100. The consumer surplus will,
therefore, be [(900 – 625) x 1100]/2 = $151,250. The fall in
consumer surplus by moving to the combined market is, therefore,
$168,750. The correct answer is (C).
15. Demand is P = 216 – 2Q, so MR = 216 – 4Q. MC = 12, so the profit
maximizing output is found where 216 – 4Q = 12, or where Q* = 51.
Substituting into the demand curve, we find that P = 216 – 2(51) =
$114. This is the joint monopoly result in this market. The correct
answer is (T).
16. With a Cournot duopoly, P = 216 – 2Q2 - 2Q1. For firm #1, MR = 216 –
2Q2 – 4Q1. Since MC = 12, we have 216 – 2Q2 – 4Q1 = 12 or 204 – 2Q2
= 4Q1. This gives us the reaction function of Firm #1, which is Q1 = 51
– .5Q2 . The reaction function of Firm #2 is similar: Q2 = 51 – .5Q1.
Substituting, we find that these two reaction functions can be
satisfied at only one point. Q1 = 51 – .5 (51 – .5Q1) or .75Q1 = 25.5. So
Q1* = 34 and Q2* = 34. Total output in the industry is 68, so P = 216 –
2(68) = $80. The correct answer is (P).
17. In the Stackelberg equilibrium, the first firm incorporates the
reactions of the second firm into its profit-maximizing decision and
firm #2’s output is no longer a constant from Firm #1’s point of view.
P = 216 – 2(51 – .5Q1) - 2Q1, which can be simplified to P = 114 - Q1 .
Therefore, MR1 = 114 - 2Q1 = 12 or Q1* = 51. From Firm #2’s reaction
function, we can calculate that Q2* = 25.5. Therefore, total output in
the Stackelberg industry is 76.5 and P = 216 – 2(76.5) = $63. The
correct answer is (M).
18. Under Bertrand assumptions, both firms will continue cutting price
until P = MC. The result would be a price of $12. The correct answer
is (B).
19. In a perfectly competitive equilibrium (or quasi-competitive
equilibrium as described in your text), P = $12 and Q = 102.
Compared to this, a Stackelberg equilibrium has deadweight loss of
(63 – 12) x (102 – 76.5)/2 = $650.25. Comparing the Cournot
equilibrium to the perfectly competitive equilibrium, we find a
deadweight loss of (80 – 12) x (102 – 68)/2 = $1156. The Cournot
equilibrium has a larger deadweight loss by $505.75. The correct
answer is (N).
20. The first firm still has a reaction function of Q1 = 51 – .5Q2 .
However, for firm 2, MR = 216 – 2Q1 – 4Q2. Since MC = 36, we have
216 – 2Q1 – 4Q2 = 36 or 180 – 2Q1 = 4Q2. This gives us the reaction
function of Firm #2, which is Q2 = 45 – .5Q1. These two are
consistent when Q1 = 51 – .5(45 – .5Q1 ). This happens when Q1 = 38.
Substituting this into Firm #2’s reaction function, we find that Q2 =
26. Total output is therefore 64 units. Substituting into the demand
curve, we have P = 216 – 2(64) = $88. The correct answer is (Q).
21. In the Stackelberg equilibrium, the first firm incorporates the
reactions of the second firm into its profit-maximizing decision and
firm #2’s output is no longer a constant from Firm #1’s point of view.
P = 216 – 2(45 – .5Q1) - 2Q1, which can be simplified to P = 126 - Q1 .
Therefore, MR1 = 126 - 2Q1 = 12 or Q1* = 57. From Firm #2’s reaction
function, we can calculate that Q2* = 16.5. Therefore, total output in
the Stackelberg industry is 73.5 and P = 216 – 2(73.5) = $69. The
correct answer is (N).
22. Lerner’s Index measures (P-MC)/P and is equal to the inverse of the
negative of elasticity of demand. If we calculate the equilibrium in
these three cases and the elasticity of demand at that equilibrium, it
is –2.333 for Firm #1, - 1.222 for Firm #2 and –1.4 for Firm #3.
Taking the inverse of elasticity, this will be largest for Firm #2,
which therefore has the greatest amount of market power. The
correct answer is (B).
23. The correct answer is (N).
24. The correct answer is (L).
25. With third-degree price discrimination, MR in the two market
segments will be equal. The formula for MR is MR = P(1 + [1/ED]). In
the first market MR is 20(1 + [1/-10]) = $18. Therefore, in the second
market segment, 18 = P(1 + [1/-4]) or P = (18 x 4)/3 = $24. The
correct answer is (F).
26. The total cost function faced by the dominant firm is TC0 = 0.125q02.
The total cost function of each of the ten price-taking (competitive
fringe) firms is TCi = 30qi + 1.25qi2. The demand function for the
market as a whole is P = 750 – .75QT where QT stands for the output
by all of the firms in the industry (both the dominant firm – price
leader – and the competitive fringe). The supply curve of the firms on
the competitive fringe can be calculated by summing the marginal cost
curves of these ten firms. We know that Q = 10qi, where Q is the
combined output of the competitive fringe, so that qi = 0.1Q. The
marginal cost of each of the firms on the competitive fringe is given
by MCi = 30 + 2.5qi, so the supply curve of each of these firms is given
by P = 30 + 2.5qi. Substituting in to find the combined supply curve
for the competitive fringe, we get P = 30 + 0.25Q.
To find the demand curve for the output of the dominant firm, we need
to subtract this from the demand curve for the total industry. A simple
way to find this demand curve for the dominant firm is to find the
intercept and then the slope. The intercept comes where the
competitive fringe’s output will completely exhaust all of the industry
demand, which comes where P = 750 – 0.75Q = 30 + 0.25Q, or where Q =
720, and P = 210. The slope will join this intercept and the point where
the dominant firm will have the entire industry demand curve to itself.
This latter point is where 30 = 750 – 0.75Q or at Q = 960. Therefore
the slope of this dominant firm demand curve is the rise over the run or
(210-30)/960 = .1875. Therefore the equation of the relevant part of
the dominant firm’s demand curve is P = 210 - .1875q0.
The dominant firm’s marginal revenue function will be MR = 210 - ..375q0.
Since MC0 = 0.25q0, we have 210 - ..375q0 = 0.25q0, or q0 = 336. The
profit maximizing price for the dominant firm will therefore be P = 210 .1875(336) = $147.
The supply function of the competitive fringe is P = 30 + 0.25Q. The
dominant firm has established the going price at $147. Therefore, the
quantity produced by the competitive fringe is given by 147 = 30 + 0.25Q,
or Q = 468. Since there are ten firms in the competitive fringe, each one
produces 46.8 units of output.
The price is $147 and the dominant firm produces 336 units of output.
The total cost function faced by the dominant firm is TC0 = 0.125q02.
The profit earned by the dominant firm is therefore (147 x 336) –
([0.125 x 336 x 336]) = $35,280. The profit earned by each of the
competitive fringe firms is 147 x 46.8 – [(30 x 46.8) + (1.25 x 46.8 x
46.8)] = $2737.80. This information allows us to fill in all of the cells in
the table below.
Dominant
Firm
Total of
competitive
fringe
Total of all
firms
Quantity
produced
Price
charged
Profit
Marginal
Revenue at
equilibrium
$35,280
Marginal
Cost of
production
at
equilibrium
$84
336
$147
468
$147
$27,378
$147
$147
804
N.A.
$62,658
N.A.
N.A.
$84
27(a). The demand curve for Type A consumers is P = 60 – .5Q. The
demand curve for Type B consumers is P = 80 - .5Q. The maximum
amount that Type A consumers will consume is 120 units; for Type B
consumers, this maximum is 160 units. If the monopolist was able to
separate these consumers and charge them a maximum price, it would
charge (60 x 120)/2 = $3600 to Type A consumers and (80 x 160)/2 =
$6400 to Type B consumers. Therefore the price and quantity for Type
A consumers would be 120 units at $3600 and for Type B consumers it
would be 160 units at $6400.
(b)
To encourage Type A consumers to just be willing to consume 120
units, the monopolist can charge them their maximum willingnessto-pay which is the entire area under the demand curve (60 x
120)/2 = $3600. However, Type B consumers could decide to
consume this “Type A” package once it is offered on the market.
If they did they would gain consumer surplus of [(80 x 160]/2 –
(3600) - (20 x 40)/2 = $6400 - $3600 - $400 = $2400 (this is
calculated by measuring the area under the entire Type B demand
curve up to 160 units of output and then subtracting areas A and C
from it. Because of this, any other package offered to Type B
consumers has to offer them at least this same amount of
consumer surplus, or else they will not select it. Therefore, if the
monopolist wishes to get Type B consumers to purchase 160 units
of output, the maximum it can charge is $4000 (i.e., the total
(c)
(d)
willingness to pay under the Type B demand curve, minus the
consumer surplus they need to be given). The two packages are,
therefore, 120 units at $3600 and 160 units at $4000.
The consumer surplus the Type B consumer would get, as calculated
above would be $2400.
By offering the Type A customers a somewhat lower quantity in
the Type A take-it-or-leave-it package, the monopolist will make
less profit off each Type A consumer. However, this will also mean
that each Type B consumer would get less consumer surplus by
choosing to consume the Type A package (it becomes less
attractive to Type B consumers). As a result, it is possible to
charge more to Type B consumers for the “Type B” package. What
is gained on Type B consumers will, up to a point, more than
compensate for the losses on Type A consumers.
Let’s try these different alternatives in order and calculate the results.
If the monopolist were to sell a quantity of 40 to Type A consumers,
Type A consumers would be willing to pay [(3600 – (40 x 80)/2] = $2000.
However, Type B consumers could decide to consume this “Type A”
package once it is offered on the market. If they did they would gain
consumer surplus of [(6400 – 2000 – (60 x 120)/2] = $800. Because of
this, any other package offered to Type B consumers has to offer them
at least this same amount of consumer surplus, or else they will not select
it. Therefore, if the monopolist wishes to get Type B consumers to
purchase 160 units of output, the maximum it can charge is 6400 – 800 =
$5600 (i.e., the total willingness to pay under the Type B demand curve,
minus the consumer surplus they need to be given). The monopolist would
make profit of: $2000 + $5600 = $7600.
If the monopolist were to sell a quantity of 60 to Type A consumers,
Type A consumers would be willing to pay [(3600 – (30 x 60)/2] = $2700.
However, Type B consumers could decide to consume this “Type A”
package once it is offered on the market. If they did they would gain
consumer surplus of [(6400 – 2700 – (50 x 100)/2] = $1200. Because of
this, any other package offered to Type B consumers has to offer them
at least this same amount of consumer surplus, or else they will not select
it. Therefore, if the monopolist wishes to get Type B consumers to
purchase 160 units of output, the maximum it can charge is 6400 – 1200 =
$5200 (i.e., the total willingness to pay under the Type B demand curve,
minus the consumer surplus they need to be given). The monopolist would
make profit of: $2700 + $5200 = $7900.
If the monopolist were to sell a quantity of 90 to Type A consumers,
Type A consumers would be willing to pay [(3600 – (15 x 30)/2] = $3375.
However, Type B consumers could decide to consume this “Type A”
package once it is offered on the market. If they did they would gain
consumer surplus of [(6400 – 3375 – (35 x 70)/2] = $1800. Because of
this, any other package offered to Type B consumers has to offer them
at least this same amount of consumer surplus, or else they will not select
it. Therefore, if the monopolist wishes to get Type B consumers to
purchase 160 units of output, the maximum it can charge is 6400 – 1800 =
$4600 (i.e., the total willingness to pay under the Type B demand curve,
minus the consumer surplus they need to be given). The monopolist would
make profit of: $3375 + $4600 = $7975.
As we saw earlier, the profit if the Type A package contains 120 units
would be $3600 + $4000 = $7600. Clearly, the best package to offer is
a Type A package of 90 units for $3375 and a Type B package of 160
units for $4600. The price charged to Type A consumers would be
$3375 for 90 books.
(e) As described above, the Type B package would contain 160 books for
$4600. The Type B consumer will get $1800 worth of consumer surplus.