Question

Consider a monopoly facing inverse demand function ?(?) = 12 − ?, where ? = ?1 + ?2 denotes the monopolist’s production across two plants, 1 and 2. Assume that total cost in plant 1 is given by ??1 (?1 ) = (5 + 4?1)?1, while that of plant 2 is ??2 (?2 ) = [5 + (4 + ?)?2]?2, where parameter ? ≥ 0 represents plant 2’s inefficiency to plant 1. When ? = 0, the total (and marginal) cost of both plants coincide; but when ? > 0, plant 2 has a higher total and marginal cost than plant 1. a) Write down the monopolist’s joint profit maximization problem ? = ?1 + ?2. b) Find the optimal production in each plant. c) How does total optimal production change in the inefficiency of plant 2, ?? Answer this question by finding ?? ??. What is the optimal production in each plant if ? = 0?

Answer 1

p(Q) = 12 - Q

Q = q1 + q2

so p = 12 - q1 - q2

Now revenue of plant 1 = p*q1 = (12 - q1 - q2) * q1

Total Cost = (5 + 4?1) * ?1

So profit of plant 1 = ?1 = (12 - q1 - q2) * q1 - (5 + 4?1) * ?1

?1 = (7 - 5q1 - q2) * q1

Now revenue of plant 2 = p*q2 = (12 - q1 - q2) * q2

Total Cost = (5 + (4 + d)?2) * ?2

So profit of plant 2 = ?2 = (12 - q1 - q2) * q2 - (5 + (4 + d) ?2) * ?2

?2 = (7 - q1 - (5 + d) q2) * q2

So joint profit maximization equation

? = ?1 + ?2

or ? = (7 - 5q1 - q2) * q1 + (7 - q1 - (5 + d) q2) * q2

or ? = 7q1 - 5q1^2 - q1q2 + 7q2 - q1q2 - 5q2^2 - d*q2^2

or ? = 7q1 + 7q2 - 5q1^2 - 5q2^2 - 2q1q2 - d*q2 ^ 2

Marginal revenue of Plant 1 = 12 - 2q1 - q2

Marginal cost of Plant 1 = 5 + 8q1

At profit maximization, 12 - 2q1 - q2 = 5 + 8q1

or 10 q1 + q2 = 7.....i)

Marginal revenue of Plant 2 = 12 - q1 - 2q2

Marginal cost of Plant 1 = 5 + 2 * (4+d) * q2

At profit maximization, 12 - q1 - 2q2 = 5 + 2 * (4+d) * q2

or q1 + (10 + 2d) * q2 = 7.....ii)

From eqn 1 and ii

10 * (10 + 2d) q2 - q2 = 63

or 99q2 + 20d * q2 = 63

or q2 = 63 / (99 + 20d)

q1 = 7 - 63 * (10 + 2d) / (99 + 20d)

Total Optimal Production Q = q1 + q2 = 7 - 63 * (10 + 2d) / (99 + 20d) + 63 / (99 + 20d) =

Q = 7 - 63 * (9 + 2d) / (99 + 20d)

So  ?Q /  ?d = -1134 / (99 + 20d)^2

At d = 0,

optimal Production in each plant

q1 = 7 - 63 * (10 + 2d) / (99 + 20d) = 7 - 630 / 99 = 0.64

q2 = 63 / (99 + 20d) = 63/99 = 0.64

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