Suppose a monopolist has two manufacturing plants with total cost functions TC1(q1) = 3q1² + 10 and TC2(q2) = 6q2² + 20. If total market output is Q = q1 + q2, what is the plant-allocation rule for production allocation?

For a multi-plant monopolist, total cost for a given output is minimised when the marginal costs of the plants are equal: MC1 = MC2. If MC1 were higher than MC2, shifting one unit of production from plant 1 to plant 2 would reduce total cost. So there can be no gain from reallocation only when the two are equal. MC1 = d(TC1)/dq1 = 6q1 MC2 = d(TC2)/dq2 = 12q2 Setting MC1 = MC2: 6q1 = 12q2, so q1 = 2q2. Plant 1 has the flatter marginal cost curve, so it produces twice as much as plant 2. For example, if total output is 9, then q1 = 6 and q2 = 3, and MC1 = MC2 = 36. The fixed costs of 10 and 20 disappear on differentiation. They have no role in the allocation rule and matter only for the shutdown decision. Hence the correct option is (D).