Question

 A firm operates two plants whose marginal cost schedules are 𝑀𝐶1 = 2 + 0.2𝑞1, 𝑀𝐶2 = 6 + 0.04𝑞2. It is a monopoly seller in a market where the demand schedule is 𝑝 = 66 − 0.1𝑞, where q is aggregate output and all costs and prices are measured in £. How much should the firm produce in each plant, and at what price should total output be sold, if it wishes to maximize profits

Answer 1

Give demand $P=66-0.1q$

Marginal revenue (MR) will have same intercept but twice the slope

$MR=66-0.2q$

MR=MC

Inverse of MC

$MC_1=2+0.2q_1\\MC-2=0.2q_1\\5MC_1-10=q_1$

$MC_2=6+0.04q_2\\MC_2-6=0.04q_2\\25MC-150=q_2$

Given $q=q_1+q_2$ by definition and $MC=MC_1=MC_2$ for profit maximization then by substituting the above inverse functions for q₁ and q₂ we get

$q=(5MC-0)+(25MC-150)\\q=30MC-160\\q+160=30MC\\ \frac{q+160}{30}=MC$

MC=MR

$\frac{q+160}{30}==66-0.2q\\q+160=1980-6q\\7q=1820\\q=260$

Substitute

$MC=\frac{q+160}{30}=\frac{260+160}{30}\\=\frac{420}{30}=14$

$MC_1=MC_2=MC=14\\q_1=5(14)-10=70-10=60\\q_2=25(14)-150=350-150=200$

$q_1+q_2=60+200=260=q\\p=66-0.1q=66-0.1(206)=66-26=40$

$q=260\\p=40$