Question

Two firms have the following cost functions (and the marginal cost functions in parentheses)
Firm 1: C₁ = 0.25q₁³ – 5q₁² + 40q₁ (MC₁ = 0.75q₁² – 10q₁ + 40)
Firm 2: C₂ = 0.5q₂³ – 6q₂² + 100q₂ (MC₂ = 1.5q₂² – 12q₂ + 100)

At what output level is average cost minimized at each firm?
Firm 1:
Firm 2:

If the 2 firms merged, they would have the following joint cost function:
C = 0.25q₁³ – 5q₁² + 40q₁ + 0.5q₂³ – 6q₂² + 100q₂ – 2q₁q₂

Assuming positive production of both goods, would the merger create economies of scope?

Answer 1

Firm 1: AC1 = C1/q1 = 0.25q1³/q1 - 5q1²/q1 + 40q1/q1 = 0.25q1² - 5q1 + 40
d(AC1)/dq1 = 2(0.25q1) - 5 = 0
So, 0.5q1 = 5
So, q1 = 5/0.5 = 10
q1 = 10 minimizes AC1.

Firm 2: AC2 = C2/q2 = 0.5q2³/q2 - 6q2²/q2 + 100q2/q2 = 0.5q2² - 6q2 + 100
d(AC2)/dq2 = 2(0.5q2) - 6 = 0
So, q2 = 6 minimizes AC2.

Economies of scope occurs if c(q1+q2) < c1 + c2
c1 + c2 = 0.25q₁³ – 5q₁² + 40q₁ + 0.5q₂³ – 6q₂² + 100q₂  
c(q1+q2) = 0.25q₁³ – 5q₁² + 40q₁ + 0.5q₂³ – 6q₂² + 100q₂ – 2q₁q₂
As q1 and q2 are positive so 2q1q2 is also positive. So, -2q1q2 is negative which means c(q1+q2) = c1 + c2 - 2q1q2
Thus, c(q1+q2) < c1 + c2
So, there are economies of scope.