Question

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).

Answer 1

Correct Answer: 190

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.