Final answer:
To find the profit-maximizing level of output for the monopoly with two production plants, equate the marginal revenue and marginal cost equations and solve for the quantity for each plant.
Step-by-step explanation:
To find the profit-maximizing level of output for the monopoly, we need to equate the marginal revenue (MR) and marginal cost (MC). The marginal revenue can be found by taking the derivative of the demand equation concerning quantity (Q): MR = 500 - 20P. The marginal cost can be calculated by taking the derivative of the cost functions concerning quantity: MC = 0.1Q for C₁ and MC = 0.05Q for C₂.
To find the profit-maximizing level of output, we set MR = MC: 500 - 20P = 0.1Q₁ + 0.05Q₂. We also consider the constraint of total quantity: Q = Q₁ + Q₂ = 500 - 10P.
Solving these two equations simultaneously will give us the values of Q₁ and Q₂, which represent the profit-maximizing level of output for each production plant. The monopoly will then charge a price based on these quantities to maximize its profits.