What is the optimal input bundle lk of producing q 100

Question

1. Two firms currently produce the goods q₁ and q₂ separately. Their cost functions are C(q₁) = 25 + q₁, and C(q₂) = 45 + 2q₂. If the two firms merge, it is estimated that the merged firm can produce the two goods jointly with costs described by the function C(q₁, q₂) = 45 + 2q₁ + q₂. Are there scope economies in this case that would justify the merger?

2. A U.S. firm is considering moving its production to Mexico. Its estimated production function is q = L^0.5K^0.5. The U.S. factor prices are w_US = 10 and r_US = 10. The Mexican factor prices are w_MX = 5, r_MX = 10. Answer the following questions:

a. What is the optimal input bundle (L,K) of producing q = 100 units in U.S.?

b. What is the optimal input bundle (L,K) of producing q = 100 units in Mexico?

c. Will the U.S. firm achieve cost saving by moving its production to Mexico?