Two Firms Face A Demand Equation P = 200000 - 6q1 - Q2

Two Firms Face A Demand Equation Given By P200000 6q1q2 Where Q

Two firms face a demand equation given by: P=200,000 – 6(q1+q2) where q1 and q2 are the outputs of the two firms. The total cost equations for the two firms are given by: TC1=8000q1 and TC2=8000q2.

(A) If each of the firms sets its own output rate to maximize its profits, assuming that the other firm holds its rate of output constant, solve for the optimal output of each firm (q1 and q2), the optimal price (P*), and the profit of each firm.

Paper For Above instruction

In the competitive market scenario described, two firms are engaged in strategic decision-making regarding their output levels within a shared demand environment. The problem revolves around determining the Cournot equilibrium quantities, the resultant market price, and the individual profits accrued by each firm when both seek to maximize their profits while taking the other’s output as given. This analysis assumes that each firm acts rationally and aims to maximize profits independently, leading to the establishment of their best response functions and eventual equilibrium outputs.

Introduction

The Cournot model offers a framework to analyze oligopolistic markets where firms choose output quantities simultaneously, recognizing the strategic interdependence of their choices. Given the demand function P = 200,000 – 6(q1 + q2), total revenue for each firm depends on the market price and its output level, while their costs are linear with respect to output. The equilibrium analysis entails deriving the reaction functions for each firm by maximizing their respective profit functions and subsequently solving these equations simultaneously to find the equilibrium quantities, prices, and profits.

Deriving the Profit Functions

The profit function for Firm 1 (π1) is expressed as:

π1 = Revenue – Cost = q1 P – TC1 = q1 (200,000 – 6(q1 + q2)) – 8,000q1

Similarly, for Firm 2 (π2):

π2 = q2 * (200,000 – 6(q1 + q2)) – 8,000q2

Maximizing Each Firm’s Profit

To find the firm’s best response, we differentiate their profit function with respect to their own quantity and set the derivative equal to zero.

Firm 1’s best response

Differentiating π1 with respect to q1:

∂π1/∂q1 = 200,000 – 6(q1 + q2) – 6q1 – 8,000 = 0

Simplifying:

200,000 – 6q1 – 6q2 – 6q1 – 8,000 = 0

Combine like terms:

192,000 – 12q1 – 6q2 = 0

Rearranged as:

12q1 = 192,000 – 6q2

Therefore, the best response function for Firm 1 is:

q1* = (192,000 – 6q2)/12 = 16,000 – 0.5q2

Firm 2’s best response

Similarly, differentiating π2 with respect to q2:

∂π2/∂q2 = 200,000 – 6(q1 + q2) – 6q2 – 8,000 = 0

Simplify:

200,000 – 6q1 – 6q2 – 6q2 – 8,000 = 0

Combine like terms:

192,000 – 6q1 – 12q2 = 0

Rearranged as:

12q2 = 192,000 – 6q1

Thus, the best response function for Firm 2 is:

q2* = (192,000 – 6q1)/12 = 16,000 – 0.5q1

Solving the Simultaneous Equations for Equilibrium

To find the equilibrium quantities, substitute the best response functions into each other:

q1 = 16,000 – 0.5q2

q2 = 16,000 – 0.5q1

Replacing q2* in the first equation:

q1 = 16,000 – 0.5(16,000 – 0.5q1)

Expanding:

q1 = 16,000 – 8,000 + 0.25q1

Rearranging:

q1 – 0.25q1 = 8,000

0.75q1* = 8,000

q1* = 8,000 / 0.75 ≈ 10,666.67 units

Using q1 to find q2:

q2* = 16,000 – 0.5(10,666.67) ≈ 16,000 – 5,333.33 ≈ 10,666.67 units

Calculating the Market Price at Equilibrium

Plugging q1 and q2 into the demand function:

P* = 200,000 – 6(q1 + q2) = 200,000 – 6(10,666.67 + 10,666.67) = 200,000 – 6(21,333.33) = 200,000 – 128,000 ≈ 72,000

Profit Calculation for Each Firm

The profit for each firm at equilibrium is:

π1 = q1 (P – AC) = 10,666.67 (72,000 – 8,000) = 10,666.67 64,000 ≈ 682,666,880

π2 = same as π1 ≈ 682,666,880

Conclusion

The Cournot equilibrium yields each firm producing approximately 10,666.67 units, with the market price set at roughly $72,000. Both firms realize profits close to $683 million under these conditions. These results highlight strategic complementarities in the market and demonstrate the typical outcomes in oligopoly models involving quantity competition.

References

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