Question

Demand in a market dominated by two firms (a Cournot duopoly) is determined according to: P = 300 – 4(Q₁ + Q₂), where P is the market price, Q₁ is the quantity demanded by Firm 1, and Q₂ is the quantity demanded by Firm 2. The marginal cost and average cost for each firm is constant; AC=MC = $68.

The cournot-duopoly equilibrium profit for each firm is _____.

Hint: Write your answer to two decimal places.

Answer 1

The equilibrium condition is (MR = MC).

Given the price function is as below:

P = 300 – 4(Q1 + Q2)

Now for the 1^st firm, TR1 = P × Q1 = 300Q1 – 4Q1^2 – 4Q1Q2

MR1 = Derivative of TR1 with respect to Q1

         = {300Q1^(1 – 1)} – {(4 × 2)Q1^(2 – 1)} – {4Q2Q1^(1 – 1)}

         = {300Q1^0} – {8Q1} – {4Q2Q1^0}

         = 300 – 8Q1 – 4Q2

Now as per the condition, MR1 = MC1

300 – 8Q1 – 4Q2 = 68

8Q1 + 4Q2 = 300 – 68

8Q1 + 4Q2 = 232 …………………….. (i)

Now for the 2^nd firm, TR2 = P × Q2 = 300Q2 – 4Q1Q2 – 4Q2^2

MR2 = Derivative of TR2 with respect to Q2

         = {300Q2^(1 – 1)} – {(4Q1Q2^(1 – 1)} – {(4 × 2)Q2^(2 – 1)}

         = {300Q2^0} – {4Q1} – {8Q2}

         = 300 – 4Q1 – 8Q2

Now as per the condition, MR1 = MC1

300 – 4Q1 – 8Q2 = 68

4Q1 + 8Q2 = 300 – 68

4Q1 + 8Q2 = 232 …………………….. (ii)

Now by solving (i) and (ii),

8Q1 + 4Q2 = 232 …………………….. (i)

4Q1 + 8Q2 = 232 …………………….. (ii) × 2

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8Q1 + 4Q2 = 232 …………………….. (i)

8Q1 + 16Q2 = 464 …………………… (ii)

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By subtracting

(8Q1 – 8Q1) + (4Q2 – 16Q2) = (232 – 464)

0 – 12Q2 = - 232

12Q2 = 232

Q2 = 232/12 = 19.33

Now by putting this value in (i),

8Q1 + 4Q2 = 232 …………………….. (i)

8Q1 + (4 × 19) = 232

8Q1 + 76 = 232

8Q1 = 232 – 76

Q1 = 156 / 8 = 19.33

Now Q1 and Q2 are to be placed in the price function to get the price,

P = 300 – 4(Q1 + Q2)

   = 300 – 4(19.33 + 19.33)

   = 300 – 4 × 38.66

   = 300 – 154.64

   = 145.36

Profit calculations:

Firm 1 profit = Total revenue – Total cost

                     = ($145.36 × 19.33) – ($68 × 19.33)

                     = 2,809.81 – 1,314.44

                     = $1,495.37

Firm 2 profit = Total revenue – Total cost

                     = ($145.36 × 19.33) – ($68 × 19.33)

                     = 2,809.81 – 1,314.44

                     = $1,495.37

Answer: profit of each firm is $1,495.37.