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ISL233-MİKROİKTİSAT-UYGULAMA DERSİ-8
06.12.2011
44. The market for wheat consists of 500 identical firms, each with the total and marginal cost functions shown:
TC = 90,000 + 0.00001Q2
MC = 0.00002Q,
where Q is measured in bushels per year. The market demand curve for wheat is Q = 90,000,000 - 20,000,000P,
where Q is again measured in bushels and P is the price per bushel.
a.
b.
c.
Determine the short-run equilibrium price and quantity that would exist in the market.
Calculate the profit maximizing quantity for the individual firm. Calculate the firm's
short-run profit (loss) at that quantity.
Assume that the short-run profit or loss is representative of the current long-run prospects
in this market. You may further assume that there are no barriers to entry or exit in the
market. Describe the expected long-run response to the conditions described in part b.
(The TC function for the firm may be regarded as an economic cost function that captures
all implicit and explicit costs.)
Solution:
a.
Market supply is horizontal sum of individual firm supply (firms MC curve).
Firm's TC = 90,000 + 0.00001Q2
MC = 0.00002Q = P.
Solve for Q in terms of P to express as supply curve
P = 0.00002Q
Q = 50,000P
Market supply curve is horizontal sum of firm supply curve or N-times the firm supply curve
(N is the number of firms).
QS = 500(50,000)P
QS = 25,000,000P
equate QS and QD to determine price and quantity
25,000,000P = 90,000,000 - 20,000,000P
45,000,000P = 90,000,000
P = $2.00
Q = 25,000,000P
Q = 25,000,000(2)
Q = 50,000,000
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b.
To determine the firm's output, equate price and marginal cost - Firm's MC = 0.00002Q.
P = 2 = 0.00002Q
Q = 100,000
Firm's π = TR - TC
TR = 2.00(100,000)
TR = 200,000
TC = 90,000 + 0.00001Q 2
TC = 90,000 + 0.00001(100,000) 2
TC = 190,000
π = 200,000,000 - 190,000 = 10,000
c.
Firms are earning economic profit so we would expect entry to occur, causing the market
supply curve to shift rightward. As the market supply curve shifts rightward, price falls, which
in turn causes each firm to reduce its output. This will continue until we reach long-run
equilibrium at zero profit.
45. The marginal cost curves of six firms in an industry appear in the table below. If these firms behave
competitively, determine the market supply curve. Calculate the elasticity of market supply at $5.
Firm
Marginal cost
Firm #1
3q1 + 2
Firm #2
3q2 + 1.5
Firm #3
3q3 + 2.5
Firm #4
3q4 + 2
Firm #5
3q5 + 1.5
Firm #6
3q6 + 2.5
Solution: To determine each firm's individual supply, we need to solve for q when marginal cost is set
P − bi
equal to the market price. MC ( qi ) = 3qi + bi = P ⇒ qi =
. We can then add each firm's
3
individual supply together at each price to determine the market supply. This is done in the
following table:
Firm
Firm #1
Firm #2
Firm #3
Firm #4
Firm #5
Firm Supply
P−2
3
P − 1.5
3
P − 2.5
3
P−2
3
P − 1.5
3
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Firm #6
P − 2.5
3
Market
2P − 4
The market supply is the sum of all the firms’ quantity supplied at each price. As the table
Q = 2 P − 4.
indicates, the market supply is: S
At a price of $5, the quantity supplied is 6. So,
5 5
 ∆QS  P
ES = 
 Q = ( 2) 6 = 3 .
∆
P

 S
the point elasticity of supply at $5 is:
46. The long-run cost function for LeAnn's telecommunication firm is: C ( q ) = 0.03q 2 .
A local
telecommunication tax of $0.01 has been implemented for each unit LeAnn sells. This implies the marginal cost
function becomes: MC ( q, t ) = 0.06q + t. If LeAnn can sell all the units she produces at the market price of
$0.70, calculate LeAnn's optimal output before and after the tax. What effect did the tax have on LeAnn's output
level? How did LeAnn's profits change?
Solution: The profit maximizing output level is where the market price equals marginal cost (providing
the price exceeds the average variable cost). To determine the optimal output level, we need to
first equate marginal cost to the market price. That is,
2
MC ( q, 0 ) = 0.06q + ( 0 ) = P = 0.7 ⇔ q = 11 . The average variable cost at this output level
3
 2
 2
 2
is: AVC  11  = 0.03 11  = 0.35. Since P > AVC 11  , LeAnn will maximize profits at
 3
 3
 3
2
2
1
 2  
 2  
11 units. LeAnn's profits are: π = Pq − C ( q ) = 0.70  11  − 0.03  11   = 4 .
3
12
 3  
 3  
With
the
tax,
LeAnn's
optimal
output
level
requires:
MC ( q, 0.01) = 0.06q + ( 0.01) = P = 0.7 ⇔ q = 11.5. The average variable cost at this output
level is: AVC (11.5 ) = 0.03 (11.5 ) + .01 = 0.355. Since P > AVC (11.5 ) , LeAnn will maximize
profits
at
11.5
units.
{
LeAnn's
2
profit
with
the
tax
is:
}
π = Pq − C ( q ) = 0.70 (11.5 ) − 0.03 (11.5 ) + 0.01(11.5 ) = 3.9675. The tax reduces LeAnn's
output and profit.
47. The manufacturing of paper products causes damage to a local river when the manufacturing plant produces
more than 1,000 units in a period. To discourage the plant from producing more than 1,000 units, the local
community is considering placing a tax on the plant. The long-run cost curve for the paper producing firm is:
q2
C ( q, t ) =
+ tq, where q is the number of units of paper produced and t is the per unit tax on paper
1500
q
production. The relevant marginal cost curve is: MC ( q, t ) =
+ t. If the manufacturing plant can sell all of
750
its output for $2, what is the firm's optimal output if the tax is set at zero? What is the minimum tax rate
necessary to ensure that the firm produces no more than 1,000 units? How much are the firm's profits reduced
by the presence of a tax?
Solution: In the absence of a tax, we know the plant will maximize profits where marginal cost is equal
to the price (given average costs exceed the market price).
That is,
q
MC ( q, 0 ) =
+ ( 0 ) = 2 ⇒ q = 1,500. Thus, without a tax, we know the plant will produce
750
at a level that will cause damage to the river. The firms profits at this level are:
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 (1,500 )2

+ 0 (1,500 )  = 1,500.
 1,500

π = 2 (1,500 ) − 
To ensure that the plant doesn't go beyond
1,000 units of production, the community needs to make sure the firm's marginal cost is
equivalent to the market price at 1,000 units or less.
That is,
1000
1 2
MC (1000, t ) =
+ t = 2 ⇒ t = 2 − 1 = . A tax of 2 or greater will ensure the plant
3
750
3 3
will not produce beyond 1,000 units. If we set the tax rate at 2 , the firm's profits will be:
3
 (1, 000 ) 2 2

2
π = 2 (1, 000 ) − 
+ (1, 000 )  = 666 . Implementation of a tax equal to 2/3 will
3
3
 1,500

result in profits declining by 55.6%.
48. The elected officials in a west coast university town are concerned about the "exploitative" rents being
charged to college students. The town council is contemplating the imposition of a $350 per month rent ceiling
on apartments in the city. An economist at the university estimates the demand and supply curves as:
QD = 5600 - 8P
QS = 500 + 4P,
where P = monthly rent, and Q = number of apartments available for rent. For purposes of this analysis,
apartments can be treated as identical.
a.
b.
c.
Calculate the equilibrium price and quantity that would prevail without the price ceiling.
Calculate producer and consumer surplus at this equilibrium (sketch a diagram showing
both).
What quantity will eventually be available if the rent ceiling is imposed? Calculate any gains
or losses in consumer and/or producer surplus.
Does the proposed rent ceiling result in net welfare gains? Would you advise the town
council to implement the policy?
Solution:
a.
To calculate equilibrium set QD = QS and solve for P.
5600 - 8P = 500 + 4P
5100 = 12P
P = 425
Substitute P into QD to solve for Q
QD = 5600 - 8(425)
Q = 2200
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Q D = 5600 − 8P
P = 700 − 0.125Q D
Q S = 500 − 4P
P = 125 + 0.25Q
C.S. = area A
C.S. = 0.5(700 - 425) x 2200
C.S. = 302,500
P.S. = area B
P.S. = 0.5(425 - 125) x 2200
P.S. = 330,000
Sum of producer and consumer surplus is:
302,500 + 330,000 = 632,500
b.
Eventually the market will settle at the quantity supplied corresponding to $350 rent.
QS = 500 + 4(350)
QS = 1900
QD at P = 350
QD = 5600 - 8(350) = 2800
There will be a shortage of 900 apartments.
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Gain = Consumer surplus is area A
Area A = (425 - 350) x 1900 = 142,500
Area B = loss in consumer surplus
To find area B, first find consumer reservation price corresponding to an output of 1900.
P = 700 - 0.125(1900) = 462.50
Difference Q = 2200 - 1900 = 300
Area B = 0.5(462.50 - 425) x (2200 - 1900)
Area B = 5625
Loss in consumer surplus is 5625.
Area C is loss in producer surplus not offset by gain in consumer surplus.
Area C = 0.5(425 - 350) x (2200 - 1900)
Area C = 11,250
c.
Area A is a gain in consumer surplus, but it is offset by a loss in producer surplus. The net
changes are thus B (lost C.S.) and C (lost P.S.). The policy thus results in a deadweight loss.
The deadweight loss = lost C.S. + lost P.S. or 5625 + 11250 = 16,875.
Deadweight loss = 16,875
49. The demand and supply functions for pizza in the local market are: QD = 20, 000 − 833P and
QS = 5, 000 + 417 P. Calculate consumer and producer surplus in this market. If the minimum wage is increased
by $2 per hour, the new market supply curve becomes: QS′ = 4, 000 + 417 P. Calculate the loss in consumer and
producer surplus in the pizza market due to this change.
Solution:
First we must determine the market equilibrium quantity and price.
To do this, we set quantity demanded equal to quantity supplied and solve for
equilibrium price.
QD = 20, 000 − 833P = QS = 5, 000 + 417 P ⇒ P = 12.
At a price of $12, the quantity exchanged will be: 10,004.
The choke price (lowest price such that no units are transacted) is $24.
1
The consumer surplus is CS = ( 24 − 12 )10, 004 = 60, 024.
2
1
Producer surplus is PS = 12 ( 5,000 ) + (10, 004 − 5,000 )12 = 90, 024.
2
If the new minimum wage shifts market supply, the new equilibrium price is
QD = 20, 000 − 833P = QS′ = 4, 000 + 417 P ⇒ P = 12.80.
At a price of $12.80, the quantity exchanged will be: 9,337.6.
The choke price (lowest price such that no units are transacted) is $24.
1
The consumer surplus is CS ′ = ( 24 − 12.80 ) 9,337.6 = 52, 290.56.
2
1
Producer surplus is PS = 12.80 ( 4, 000 ) + ( 9,337.6 − 4, 000 )12.80 = 85,360.64.
2
The change in societal welfare in the pizza market due to the new minimum wage is:
∆W = ( CS ′ + PS ′ ) − ( CS + PS ) = 137, 651.2 − 150, 048 = −12,396.80.
The loss in welfare in the local pizza market is 12,396.80 or 8.3%.
SD_2011-2012/8