Perfect competition, game theory and monopoly

Solution Preview

See the attached file for complete solution. The text here may not print correctly. Thanks.

Q1:
Q=540-30P or P=18-Q/30
Initially let us assume that all 100 firms are identical, we will have Q=100q
P=18-10q/3
For perfect competition MR=P=18-10q/3
TVC=q^2
MC=2q
For profit maximization, equating MR=MC, we get
18-10q/3=2q
q=3.375 P=18-3.375*10/3=6.75
Now calculate the profit for individual firms
TC=20+q^2=20+3.375^2=31.39
TR=3.375*6.75=22.78
Profit = -8.61
Thus, each firm will incur a loss of -8.61
The first 50 firms (call Type 1), which have only $4 as sunk cost for their fixed cost, can decrease their losses from $8.61 to $4.00 by stopping their production. Thus, these firms will have two options either to produce and incur higher losses or not to produce.

For the remaining 50 (call Type 2), the entire $20 is sunk cost, so they will incur a loss of $20 if they stop production and hence it is better for them to keep producing.
If type 1 firms do not produce, there will be short supply in the market and the effective price for type 2 firms will earn be
P=18-5*3.375/3=12.375
TR=3.375*12.375=41.77
Profit = 10.38

Now, knowing fully well that it is not beneficial for the other 50 firms to be in the market and produce, the rest 50 firms may increase their quantity. If only 50 firms will produce, we will have
P=18-5q/3
For perfect competition MR=P=18-5q/3
TVC=q^2
MC=2q
For profit maximization, equating MR=MC, we get
18-5q/3=2q
q=4.91 P=18-4.91*5/3=9.82
Now calculate the profit for individual firms
TC=20+q^2=20+4.91^2=44.11
TR=4.91*9.82=48.21
Profit = 4.11
Thus, each firm producing the output will produce q=4.91 and sell at P=9.82 and will earn a profit of 4.11.

However, in a situation when type 1 produce 3.375 and type 2 produce 4.91, the ...