Question

If a monopolist produces its product in two factories, it must determine how to allocate production among those locations to minimize its total costs of production and thereby maximize its profits. It will do this by producing more at the factory with the lower marginal cost of production until the marginal cost of the last unit produced in each factory is equal: MC1 = MC2. If the marginal cost of producing at Factory 1 is MC1 = 2Q1, the marginal cost of producing at Factory 2 is MC2 = 4Q2, and the firm’s demand curve is p = 70 - Q, where Q = Q1 + Q2, how much will the monopolist produce in each of its two factories, and what price will it set for its product? How would your answer change if MC2 = 2Q2?

Perloff, Jeffrey M. Microeconomics, Global Edition, 8th Edition.

Answer 1

A).

Here the demand curve is given by, P=70-q, where “q=q1+q2”. So, the marginal benefit function is given by.

=> MB = 70 - 2*q = 70 - 2*q1 - 2*q2, => MB = 70 – 2*q1 – 2*q2. Now, the marginal cost functions are given by, “MC1=2*q1” and “MC2=4*q2”.

So, at the optimum “MR=MC1=MC2”.

=> MR=MC1, => 70-2*q1-2*q2 = 2*q1, => 4*q1 + 2*q2 = 70.

=> MR = MC2, => 70-2*q1-2*q2 = 4*q2, => 2*q1 + 6*q2 = 70.

So, by simultaneously solving the above two equation we have the solution, => “q1*=14” and “q2*=7”, => q=q1+q2=14+7=21, => P=70-q=70-21=49. So, the monopolist will produce “14 units” from “plant1” and “7 units” from “plan2”. So, the market clearing price is given by, “P=49”.

B),

Now, if “MC2=2*q2”, => MR = MC2, => 70-2*q1-2*q2 = 2*q2, => 2*q1 + 4*q2 = 70.

Similarly, MR=MC1, => 4*q1 + 2*q2 = 70.

So, by simultaneously solving the above two equation we have the solution, => “q1*=q2*=35/3=11.67”, => q=q1+q2=70/3 = 23.33, => P=70-23.33 = 70-23.33=46.67. So, the monopolist will produce “11.67 units” from both of the “plants”. So, the market clearing price is given by, “P=46.67”.