Question

A monopolist sells in a market described by the inverse demand function ​p = 10 - 0.1Q ​ , where ​p is the price and ​Q ​ is the total quantity sold. The monopolist produce its output in two plants which have the cost functions ​C ​ 1 ​ = 0.25q ​ 1 ​ and ​C ​ 2 ​ = 0.5q ​ 2 ​ , where ​q ​ i ​ (i=1,2) is the output produced in plant i (of course, the total quantity produced is the same as the total quantity sold, ​Q = q ​ 1 ​ + q ​ 2 ​ ).

1) Write down the optimization problem for this question, as an unconstrained optimization problem with two variables, ​q ​ 1 ​ and ​q ​ 2 ​ .

2) Find the maximum profit obtained by the monopoly. Use the techniques you learned in this class.

Answer 1

Q = q1 + q2

Total Revenue Earned = P*Q = (10 - 0.1Q) * Q = 10Q - 0.1Q^2 = 10 * (q1 + q2) - 0.1 * ( q1 + q2)^2

Total Cost = C1 + C2 = 0.25q1 + 0.5q2

So total Profit = Total Revenue - Total Cost = 10 * (q1 + q2) - 0.1 * ( q1 + q2)^2 - (0.25q1 + 0.5q2)

So the optimization problem will be

Maximize π = 10 * (q1 + q2) - 0.1 * ( q1 + q2)^2 - (0.25q1 + 0.5q2)

MC1 = 0.25

MC2 = 0.5

So the monopolist will produce in Plant 1 only since there is no capacity constraint

Marginal Revenue = 10 - 0.2 * q1 (Since Q = q1 + q2 and marginal cost of plant 1 is lower. So production will be in Plant 1 only and hence q2 = 0)

So at profit maximization,

10 - 0.2q1 = 0.25

or 0.2q1 = 9.75

or q1 = 48.75

Maximum Profit = 10 * (q1 + q2) - 0.1 * ( q1 + q2)^2 - (0.25q1 + 0.5q2)

= 10 * 48.75 - 0.1*48.75^2 - 0.25*48.75 = 237.65

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