Homework #2. Perfect competition.
Setup
Market demand for __________________ is given by Q  P   108  P , and its cost function is given by
C  Q   4  3Q  0.25Q2
Must-know problems
1. Derive short run firm’s supply function
a) Find profit maximizing quantity as a function of market price
i) Write the short-run profit maximization rule
ii) Derive relation between market price and the profit maximizing quality
b) Find the shut down point
i) Write the shut down rule
ii) Find the shutdown price
c) Combine your answers to previous questions to determine firm’s short run supply function
d) Show the supply function on the graph below
i) Label the axis
ii) Label units
iii) Correctly label and identify (by the points on the axis) all relevant points
iv) Make sure the slopes are correct
1. Q
2. Find the short run market supply function for 5 identical firms with technologies described above
3. Find long run market supply function for 5 identical firms with technologies described above
a) Write the free entry condition for each firm (find price and quantity)
b) Write the long run market supply function for 5 firms
4. Find market equilibrium price and quantity
5. Given the market price, find each firm’s:
a) Output
b) Total cost
c) Profit
6. Given the market supply above is the market in a long run equilibrium? Explain.
If not find the number of firms n in long-run equilibrium
a) Write the long run equilibrium condition
b) Find out market quantity in the long run equilibrium
c) Using the quantity from the free entry condition find the number of firms in the long run
equilibrium (the correct answer is not necessarily an integer)
Additional problems
7. How would your answers change if there was another technology with cost function with the same
variable cost but twice higher fixed cost? Please, do not perform calculations, only provide qualitative
answers. For example: “the profit per firm will decrease”
8. Using the demand from the above setup, describe the equilibrium for a constant marginal cost
technology with a fixed cost, say C  Q   20  0.15Q . You may use the graph to illustrate your
answer
3. Q
9. How does the fixed cost affect the number of firms in the long run equilibrium?
10. Imagine a technology that has a constant marginal cost but only up to a certain quantity. If the firm
wants to increase the quantity beyond that point it will require additional investment. In other words:
2, Q  10
MC  Q   
, Q  10
How many firms would operate in the long run market equilibrium? What would be the price?
Quantity per firm?
Answers:
1. 𝑀𝐶 = 3 + .5𝑄 = 𝑃 → .5𝑄 = 𝑃 − 3 → 𝑄 = 2𝑃 − 6
a. 𝑄 = 2𝑃 − 6. We also need to check the shutdown condition. 𝐴𝑉𝐶 = 3 + .25𝑄
Shutdown occurs when 𝐴𝑉𝐶 = 𝑃 = 𝑀𝐶 → 3 + .5𝑄 = 3 + .25𝑄 → .25𝑄 = 0 → 𝑄 = 0 → 𝑃 = 3.
Therefore, the profit-maximizing quantity of Q is 2𝑃 − 6 when 𝑃 ≥ 3, and 𝑄 = 0 𝑤ℎ𝑒𝑛 𝑃 < 3.
i. P=MC.
ii. 𝑄 = 2𝑃 − 6 𝑜𝑟 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡𝑙𝑦 𝑃 = 3 + .5𝑄
b. Shutdown point occurs at Q=0 as shown in 1a.
i. Shutdown rule states that a firm will shut down if 𝑃 < 𝐴𝑉𝐶.
ii. Shutdown price is 3 as shown in 1a.
2𝑃 − 6 𝑤ℎ𝑒𝑛 𝑃 ≥ 3
c. As shown in 1 and 1a, the firm's short run supply function is 𝑄 = {
}.
0 𝑤ℎ𝑒𝑛 𝑃 < 3
10𝑃 − 30 𝑤ℎ𝑒𝑛 𝑃 ≥ 3
5(2𝑃 − 6) 𝑤ℎ𝑒𝑛 𝑃 ≥ 3
2. 𝑆5 = 5𝑆1 = {
}={
}.
0 𝑤ℎ𝑒𝑛 𝑃 < 3
5 ∗ 0 𝑤ℎ𝑒𝑛 𝑃 < 3
4
4
3. 𝐴𝐶 = + 3 + .25𝑄 = 𝑀𝐶 = 3 + .5𝑄 → − .25𝑄 = 0 → 4 − .25𝑄 2 = 0 → .25𝑄 2 = 4 → 𝑄 2 =
𝑄
𝑄
10𝑃 − 30 𝑤ℎ𝑒𝑛 𝑃 ≥ 5
}.
0 𝑤ℎ𝑒𝑛 𝑃 < 5
a. The free entry condition states that firms will enter when price is above minimum average cost
and will exit when price is below. This occurs at a price and quantity of 4 and 5, respectively, as
shown above.
10𝑃 − 30 𝑤ℎ𝑒𝑛 𝑃 ≥ 5
b. As shown above, 𝑆 = {
}.
0 𝑤ℎ𝑒𝑛 𝑃 < 5
4. 108 − 𝑃 = 10𝑃 − 30 → 11𝑃 = 138 → 𝑃∗ = 12.55 𝑎𝑛𝑑 𝑄 ∗ = 108 − 12.55 = 95.45
16 → 𝑄 = 4 → 𝑃 = 3 + .5(4) = 5. Therefore, 𝑆 = {
5. a. 𝑄𝑓 =
𝑄∗
5
=
95.45
5
= 19.09
b. 𝐶𝑓 = 4 + 3(19.09) + .25(19.092 ) = 152.38
c. 𝜋𝑓 = 𝑃 ∗ 𝑄 − 𝐶𝑓 = 12.55 ∗ 19.09 − 152.38 = 87.12
6. The market is not in the long run equilibrium because each firm is making positive profits. This will
induce other firms to enter the market. Those firms will continue to enter until the market price is
equal to the firms' average cost, which, as shown in 3a, happens when P=5 and Q=4. Therefore,
𝑆𝑀 = 108 − 5 = 103, where 𝑆𝑀 is the market supply. Since a total of 103 units are produced and
each firm is producing 4 individually, the number of firms in the market is
103
4
= 25.75. In other
words, you would expect 25 or 26 firms or, possibly, 25 firms and one small one in long run market
equilibrium.
a. The long run equilibrium condition is market price equals the minimum average cost of the
individual firm, or 𝑃 = 𝑀𝐶 = 𝐴𝐶𝑚𝑖𝑛 .
b. Market quantity, as shown in 6, is 103 units.
c. The number of firms, as shown in 6, is 25.75.
7. The profit per firm will obviously decrease because costs are higher. Since fixed costs are higher, this
will raise the market price which will cause quantity to drop. Also, the number of firms in the market
will decrease because output has been reduced. The output per firm will increase however, as
marginal cost doesn’t change, but average cost increases.
8. 𝑀𝐶 = .15 = 𝐴𝑉𝐶 𝑎𝑛𝑑 𝐴𝐶 =
20
𝑄
+ .15
𝑀𝐶 = 𝐴𝐶 → .15 =
20
𝑄
+ .15, which means that MC and AC never cross which means that AC has no
minimum. It just keeps decreasing. This means that it is never profitable to have more than one firm
in the market as this cost function will always exhibit economies of scale. Therefore, there will be
only one firm in the market which will produce the entire market output. Therefore, 𝑄 = 108 − 𝑃 =
108 − 𝑀𝐶 = 108 − .15 = 107.85 unless the firm acts as a monopoly, which would yield an entirely
different outcome.
9. When there are lower fixed costs, the average cost will shift down along the marginal cost curve
while the marginal cost curve will not change. This will lead to a lower market price in the long run,
which means that each individual firm will be producing less. Since price has fallen and demand has
not changed, it necessarily follows that market output has increased. Since total output is increasing
and individual firm output is decreasing, the number of firms in the market will increase.
10. Since marginal cost is constant and then suddenly becomes infinite at a fixed point, average cost will
slowly decrease until marginal cost becomes infinite, at which point average cost will become infinite
as well. (Adding the new, marginal, infinite cost to the old average cost yields an average cost of
infinite as well.) The long run price would be the minimum of average cost, which occurs right at the
𝐹
jump to infinite, which is at the firm’s production capacity. (In the example given, 𝑝 = 10 where F is
𝑄
fixed costs.) The number of firms in this market would be equal to 10 where Q is industry demand
since any individual firm can only produce a maximum of 10, and each firm would not produce less
than 10 because this would result in a higher average cost and less profits. My conclusions may not
be correct, however, because the long run takes account for time required to invest. This could
significantly change the marginal cost/supply function of each firm which would affect long-run,
industry equilibrium.