Firms A and B are duopolist producers of widgets. The cost function for producin

Question

Firms A and B are duopolist producers of widgets. The cost function for producing widgets is C(Q) = Q^2, with marginal cost MC = 2Q. The market demand function for widgets is Q^d = 40 - 0.5P, where Q measures thousands of widgets per year. Competition in the widget market is described by the Cournot model. What are the firms' equilibrium outputs? What is the resulting price? What do they each earn as profit? How does the price compare to marginal cost? How do the price and the two firms' joint profit compare to the monopoly price and profit?

Explanation / Answer

Qd = 40 - 0.5P

The inverse demand function is

P = 80 - 2Q

Here , Q = Q1 + Q2

P = 80 - 2(Q1 + Q2)

P = 80 - 2Q1 - 2Q2

Firm 1's reaction function:

TR1 P x Q1 = 80Q1 - 2Q12 - 2Q1Q2

MR1 = 80 - 4Q1 - 2Q2

MC = 2Q

For equilibrium MR1 = MC

80 - 4Q1 - 2Q2 = 2Q

80 - 4Q1 - 2Q2 = 2(Q1 + Q2)

80 - 4Q1 - 2Q2 = 2Q1 + 2Q2

Solving above equation for Q1:

Q1 = 13.33 - 0.67Q2 (firm 1's reaction function)

Similarly, firm 2's reaction function will be:

Q2 = 13.33 - 0.67Q1 (firm 2's reaction function)

Substituting the firm 2's reaction function in firm 1's reaction function, we get

Q1 = 13.33 - 0.67Q2 = 13.33 - 0.67(13.33 - 0.67Q1)

Q1 = 13.33 - 8.93 + 0.45Q1

So, 0.55Q1 = 4.4

Q1 = 4.4/0.55 = 8

Then, Q2 = 13.33 - 0.67Q1 = 13.33 - 0.67 x 8 = 7.97 ( it can be rounded to 8 units)

So, Q1 + Q2 = 8 + 8 = 16 units

Therefore, P = 80 - 2Q = 80 - 2 x 16 = $48

TC = Q2= 162 = $256

Profit of firm 1 = TR1 - TC

= P x Q1 - TC = 48 x 8 - 256 = $ 128

Profit of firm 2 = TR1 - TC

= P x Q2 - TC = 48 x 8 - 256 = $ 128

Total joint profit = $ 256

Here, price is greater than MC.

The monopoly Price & Qunatity is:

P = 80 - 2Q

TR = 80Q - 2Q2

MR = 80 - 4Q

MR = MC

80 - 4Q = 2Q

So, Q = 13.33

P = P = 80 - 2Q = P = 80 - 2 x 13.33 = $53.34

Profit = TR - TC

P x Q - TC

= 53.34 x 13.33 - 13.332

      = 711.02 - 177.68 = $ 533.34

Therefore, Monopoly profit is higher than Joint profit

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