Question

You are the manager of DELL. Assume that DELL manufactures computers at two locations. The inverse demand function for DELL computers is P = 1000 – 4Q. The cost of producing computers at plant 1 is C1(Q1) = 10000 + 4Q_1^2 , and the cost at plant 2 is C2(Q2) = 8000 + 2Q_2^2. Determine the quantity and the price that maximize profits and the level of profits.

Answer 1

Profit = revenue – cost

= (1000 – 4Q1 – 4Q2)(Q1 + Q2) – 10000 – 4Q1^2 – 8000 – 2Q2^2

= 1000Q1 – 4Q1^2 – 4Q1Q2 + 1000Q2 – 4Q2^2 – 4Q1Q2 – 10000 – 4Q1^2 – 8000 – 2Q2^2

= 1000Q1 + 1000Q2 – 8Q1^2 – 6Q2^2 – 8Q1Q2

Profit maximizing quantities are found where marginal profits are zero

1000 – 16Q1 – 8Q2 = 0 and 1000 – 12Q2 – 8Q1 = 0

Q2 = 125 – 2Q1 and 1000 – 12*(125 – 2Q1) – 8Q1 = 0

1000 – 1500 + 24Q1 – 8Q1 = 0

Q1* = 31.25 and Q2* = 62.50

P* = 1000 – 4*(62.5 + 31.25) = $625

Profits = 1000*31.25 + 1000*62.50 – 8*(31.25^2) – 6*(62.5^2) – 8*31.25*62.50 = 46875