Answer to Question #209388 in Microeconomics for kevin

Question #209388

Β A firm operates two plants whose marginal cost schedules are 𝑀𝐢1 = 2 + 0.2π‘ž1, 𝑀𝐢2 = 6 + 0.04π‘ž2. It is a monopoly seller in a market where the demand schedule is 𝑝 = 66 βˆ’ 0.1π‘ž, where q is aggregate output and all costs and prices are measured in Β£. How much should the firm produce in each plant, and at what price should total output be sold, if it wishes to maximize profits


1
Expert's answer
2021-06-22T15:48:37-0400

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"



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