Question

Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by

C1=60Q1

and

C2=60Q2

where

Q1 is the output of Firm 1 and Q2 is the output of Firm 2. Price is determined by the following demand curve:

P=2700−Q where Q=Q1+Q2

Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium. (For all of the following, enter a numeric response rounded to two decimal places.)

When competing, each firm will produce ___ units of output.

In turn, each firm will earn profit of ___.

Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm's profit.

Each firm will produce ___ units of output.

In turn, each firm will earn profit of ___.

Suppose Firm 1 were the only firm in the industry. How would market output and Firm 1's profit differ from that found in above?

Firm 1 would produce ___ units of output.

Firm 1's would earn profit of ___.

Returning to the duopoly above, suppose Firm 1 abides by the agreement but Firm 2 cheats by increasing production. How many widgets will firm produce? What will be each firm's profits?

Firm 2 would cheat by producing ___ units of output.

As a consequence, Firm 1 would earn profit of ___.

Firm 2 would earn profit of ___.

Answer 1

C1 = 60 Q1

C2 = 60 Q2

P = 2700 - Q1 - Q2

TR1 = 2700Q1 - Q12 - Q1Q2

TR2 = 2700Q2 - Q22 - Q1Q2

Profit maximizing condition

Firm 1 : MR = MC

2700 - 2Q1 - Q2 = 60

or

2640 = 2Q1 + Q2.... (i)

Firm 2: MR = MC

2700 - 2Q2 - Q1 = 60

or

2640 = Q1 + 2Q2..... (ii)

solving (i) and (ii)

5280 = 2Q1 + 4Q2

2640 = Q1 + 2Q2

3Q2 = 2640

Q2 = 880

putting Q2 = 880 in (i)

2640 = 2Q1 + 880

2Q1 = 1760

Q1 = 880

Since outputs are same, Profit for Firm 1= Profit for firm 2: TR - C

Profit, = 2700Q1 - Q12 - Q1Q2 - 60 Q1

= 7,74,400 =

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Under cartel

P = 2700 - Q

TR = 2700 Q - Q2

TC = 60 Q

Profit maximizing condition

MR = MC

2700 - 2Q = 60

2Q = 2640

Q = 1320

each firm will produce half of the total output

Q1 = Q2 = 660

P = 2700 - 1320 = 1380

Profit for each firm will be

Profit, = 1380 (660) - 60 (660)

= 8,71,200

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When firm 1 is the only firm in the industry, i.e. monopoly

P = 2700 - Q

TR = 2700 Q - Q2

TC = 60 Q

Profit maximizing condition

MR = MC

2700 - 2Q = 60

2Q = 2640

Q = 1320

P = 2700 - 1320 = 1380

Profit = 1380*1320 - 60 * 1320

= 17,42,400 twice of what is produced under cartel.

***********

Firm 2 cheats

this implies that Q1 = 660

P = 2700 - (660 + Q2)

P = 2040 - Q2

Profit maximizing

TR2 = 2040 Q2 - Q22

MR = MC

2040 - 2Q2 = 60

1980 = 2Q2

Q2 = 990

P = 2700 - (990+660) = 1050

Profit, = 1050 * 660 - 60 * 660

= 990 * 660 = 6,53,400

Profit, = 1050 * 990 - 60 * 990

= 990 * 990 = 9,80,100

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