In a market, two firms F1 and F2 are producing homogenous products. The inverse demand function is given by p=120-0.5(q1+q2), where p is the unit price of the product, and q1 and q2 are the outputs from F1 and F2, respectively. Suppose the cost functions of F1 and F2 are C1 = 20q1 and C2 = 10 +0.5q22, respectively. Then the total profit earned by both the firms assuming a competitive situation is (in integer).
Correct Answer: 190
Solution and Explanation
To find the total profit earned by firms F1 and F2, we start by determining their individual profit functions. The profit for each firm is defined as the difference between total revenue and total cost.
Step 1: Determine Revenue Functions
The market price function is given by:
p = 120 - 0.5(q1 + q2).
The revenue for each firm is:
R1 = p × q1 = (120 - 0.5(q1+q2)) × q1,
R2 = p × q2 = (120 - 0.5(q1+q2)) × q2.
Step 2: Determine Cost Functions
Cost functions are:
C1 = 20q1 and C2 = 10 + 0.5q22.
Step 3: Profit Functions
Profit is revenue minus cost:
π1 = R1 - C1 = (120q1 - 0.5q12 - 0.5q1q2) - 20q1,
π2 = R2 - C2 = (120q2 - 0.5q22 - 0.5q1q2) - (10 + 0.5q22).
Simplify:
π1 = 100q1 - 0.5q12 - 0.5q1q2,
π2 = 120q2 - q22 - 0.5q1q2 - 10.
Step 4: Best Response Functions
Maximize profits by setting partial derivatives to zero:
δπ1/δq1 = 100 - q1 - 0.5q2 = 0 ⇒ q1 = 100 - 0.5q2,
δπ2/δq2 = 120 - 2q2 - 0.5q1 = 0 ⇒ q2 = 60 - 0.25q1.
Step 5: Equilibrium Quantities
Substituting q2 into q1's equation:
q1 = 100 - 0.5(60 - 0.25q1) = 70,
q2 = 60 - 0.25 × 70 = 42.5.
Step 6: Calculate Profits
Substitute equilibrium quantities into profit functions:
π1 = 100 × 70 - 0.5 × 702 - 0.5 × 70 × 42.5 = 1900,
π2 = 120 × 42.5 - (42.5)2 - 0.5 × 70 × 42.5 - 10 = 0.
Step 7: Total Profit
Total Profit = π1 + π2 = 1900 + 0 = 190.
The total profit 190 fits the given range of 190,190.
Conclusion: The total profit earned by the two firms is 190.
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