Download Tutorial 6 - Perfect Competition

Tutorial 6 - Perfect Competition
March 2014
Problem 1
In a small, but perfectly competitive market for pineapples, there are 8 identical
growers. Each grower has the following cost function:
C = 2 + 2q 2
where q is thousands of pounds of pineapples produced.
a) Write the firm’s supply curve, Si (qi )
A firm in a perfectly competitive market supplies a positive quantity when
the price is greater than the minimum average variable cost.
If the market price is greater than the minimum average variable cost, the
firm supplies the quantity that makes the marginal cost equal to the market
price, in order to maximize profits (equilibrium condition).
AV C =
2q 2
VC
=
= 2q
q
q
Thus, the minimum average variable cost is 0. For any positive price, the
firm is supplying a positive quantity.
To find the supply curve:
p = MC
MC =
dT C
= 4q
dq
p = 4q
Solving for q, we find the firm’s supply curve:
qiS =
1
p
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b) Write the equation for the market supply curve S(p).
To find the market supply curve, sum horizontally the individual firms’ supply curves.
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As firms are identical, we can multiply the individual firm’s supply curve by
the number of firms in the market.
1
QS = n × qiS = 8 × p = 2p
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c) Suppose the (inverse) market demand curve is
D1 : p(QD ) = 100 − 9.5QD
Solve for the equilibrium price and quantity.
Demand : p = 100 − 9.5QD
Supply : QS = 2p
In equilibrium, QS = QD :
p = 100 − 9.5 × 2p
p = 100 − 19p ⇒ 20p = 100
p=5
To find the quantity exchanged in this market in equilibrium, substitute the
equilibrium price in either the supply or the demand curve.
Q = 2 × 5 = 10
Each firm produces qi =
10
5
Q
=
=
n
8
4
d) In the long-run, will more growers enter the market or will existing growers
exit? Explain.
In the long run, firms can enter or exit the market, thus profits are zero. To
check whether this equilibrium is a long run one, we have to check profits.
Π = TR − TC
5
25
T R = p × qi = 5 × =
4
4
2
5
25
T C = 2 + 2qi2 = 2 + 2
=2+
4
8
Π=
25
25
9
−2−
= >0
4
8
8
Profits are positive, thus the equilibrium is not a long-run one. In the long
run more firms will enter the market.
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Alternative solution. Another strategy is to compare the price with the
average total cost of production.
AT C =
2
4
5
8 5
41
+ 2q = 2 × + 2 = + =
<p
q
5
4
5 2
10
The equilibrium price is higher than the average total cost. Thus profits are
positive.
e) Suppose instead the market demand curve is
D2 : p(QD ) = 60 − 9.5QD
Solve for the equilibrium price and quantity.
With the new demand curve:
p = 60 − 9.5 × 2p
p = 60 − 19p
⇒ 20p = 60
p=3
Q=2×3=6
q=
6
3
Q
= =
n
8
4
f) In the long run, will more growers enter the market or will existing growers
exit? Explain.
Π = TR − TC
3
9
TR = 3 × =
4
4
2
3
9
25
TC = 2 + 2
=2+ =
4
8
8
Π=
9 25
7
−
=− <0
4 16
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Profits are negative, so this is not a long-run equilibrium. Some firms will
exit the market.
g) What is the long-run equilibrium price at which there will be no entry or exit
in this market?
In the long run the equilibrium price will be equal to the minimum average
total cost (profits are zero).
As the marginal cost intersects the average total cost at its minimum, we
can find that minimum value of ATC setting MC = ATC.
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AT C =
2
+ 2q
q
M C = 4q
AT C = M C; ⇒
2q =
2
q
2
+ 2q = 4q
q
⇒q=1
Thus, the long-run equilibrium price is:
p = MC = 4 × 1 = 4
Problem 2
Suppose rice is produced under perfect competition. There are 1000 rice producers, and rice is homogeneous across all of them (think of 1000 as a really large
number of producers). The market demand for rice is given by:
1
QD = 20 − P
2
where P denotes the market price of rice, which all rice producers take as
given. All rice producers grow rice with the same technology, hence they all face
the same cost function.
T C(qi ) = F C + 1000qi2
Suppose each individual rice producer i (notice i = 1, 2, ..., 1000 is the index
of each producer), and F C denotes the fixed cost.
a) Find the equilibrium price and quantity in the rice market
Find the individual firm’s supply curve.
AV C =
1000qi2
= 1000qi
q
The minimum value is 0.
The firm produces a positive quantity for all possible prices.
To maximize profits,
p = MC
M C = 2000qi
p = 2000qi
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Individual firm’s supply curve:
qiS =
1
p
2000
To find the market supply curve, multiply this curve by the number of firms:
QS = n × qiS = 1000
1
1
p= p
2000
2
To find the market equilibrium, set QS = QD
1
1
p = 20 − P
2
2
p = 20
Q = 10
b) How much rice does each individual producer end up producing?
The quantity exchanged in the market is Q = 10
There are 1000 identical firms, thus the quantity supplied by the individual
firm is:
10
= 0.01
qi =
1000
c) Calculate the consumer surplus in the rice market
Figure 1: Problem 2, point c)
CS =
(40 − 20)10
= 100
2
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d) Suppose F C = 1. Is this market in a long-run equilibrium? Justify your
answer.
If F C = 1, total cost function is T C = 1 + 1000q 2
T R = 20 × 0.01 = 0.2
T C = 1 + 1000(0.01)2 = 1.1
Π = 0.2 − 1.1 = −0.9 < 0
Profits are negative, thus this market is NOT in a long-run equilibrium.
With negative profits, some firms will exit the market.
Problem 3
Consider a perfectly competitive market. In this market there are 10 firms,
characterized by the following cost function
T C(qi ) = 10qi
The production capacity of these firms is limited: the maximum amount produced
with the capacity installed is 10 units per firm.
The good can also be supplied by other 5 firms, less efficient, and characterized by the following cost function:
T C(qj ) = 15qi
and again these firms are able to produce 10 units each.
a) Suppose that the demand curve is:
QD = 180 − 12P
Find the equilibrium price and quantity.
M C1 = 10
The first group of firms has a marginal cost of 10 (constant!). Each firm can
offer 10 units at that price. Total supply at this price is thus 10 × 10 = 100.
The second group of firms is less efficient. Those firms are characterized by
a constant marginal cost equal to 15.
These firms can supply 10 units each and they are 5. Thus, total supply at
the price of 15 is 50. We have derived the market supply curve (see Figure
2).
To find the equilibrium draw the demand curve. The demand curve crosses
the supply curve at a price = 10.
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Figure 2: Problem 3
The equilibrium price is p = 10, the quantity demanded at this price is
QD = 180 − 12 × 10 = 60.
Only the firms of the first group (most efficient ones!) are serving the market.
The equilibrium price is equal to the marginal cost of these firms.
b) Suppose now that the demand curve becomes:
Q0D = 300 − 12P
Find the new equilibrium.
The new demand intersects the supply curve at p = 15.
At this price the quantity demanded is QD = 300 − 12 × 15 = 120.
The equilibrium price is 15. 100 units are supplied by the most efficient
firms, 20 units are supplied by the least efficient ones.
c) Compare the two situations in points a) and b). How is the price changed?
Why?
The price is set at the level of the marginal cost of the least productive firms.
Problem 4
a) Is the following statement true or false? Explain. In the short run, the supply
curve will be upward sloping in perfect competition
The statement is true. The supply curve is made up of the firms’ marginal
cost curves. In the short run marginal cost will be upward sloping because
of the principle of diminishing returns
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b) Is the following statement true or false? Explain. If firms in a perfectly
competitive industry are earning positive profits, the industry is not in long
run equilibrium
This statement is true. If firms in the industry are earning positive profits,
in the long run there will be entry. Entry will continue until the price has
fallen to the point that profits have been eliminated. If firms are earning
losses, there will be exit. Exit will continue until the price has risen to the
point where losses have been eliminated.
c) Assume the beer industry is perfectly competitive. What will happen to the
equilibrium price and quantity of beer and the firms’ profits in the short run
if the drinking age is lowered from 21 to 18? Explain your answer.
The market demand for beer increases. In the short run, equilibrium price
increase and quantity increases. The firm increases its output, profits become
positive (P > AC). The industry is not in a long run equilibrium because
firms are earning positive profits.
d) Is the following statement true or false? Explain. In the long run, the supply
curve will always be upward sloping
The statement is false. In long run, equilibrium price must be equal to the
minimum average cost because firms must earn zero profits. As a result the
long run supply curve, if the industry is characterized by constant costs, is
perfectly elastic at that price (horizontal).
e) Nate’s Car Wash is one firm in a perfectly competitive industry. What will
happen to price and output in the car wash industry if price of water increases
(assume the car wash industry has constant costs)?
The increase in the price of water shifts firms’ M C and AC curves up. This
causes the industry supply curve to shift inwards. Equilibrium price will
increase and output will decrease.
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