Answer to Question #158066 in Microeconomics for S

Question #158066

(a) Determine by using calculus the best level of output of the firm by the marginal approach and

(b) find the total profit of the firm at this level of output.

Q3.

A perfectly competitive firm faces P = 4 and TC = Q3 - 7Q2 + 12Q + 5.


1
Expert's answer
2021-01-24T17:34:44-0500

(a) By the definition of the total revenue, we get:


TR=PQ=4Q.TR=PQ=4Q.

Let's find the marginal revenue:


MR=dTRdQ=ddQ(4Q)=$4.MR=\dfrac{dTR}{dQ}=\dfrac{d}{dQ}(4Q)=\$4.

Then, we can find the marginal cost:


MC=dTCdQ,MC=\dfrac{dTC}{dQ},MC=ddQ(Q37Q2+12Q+5)=3Q214Q+12.MC=\dfrac{d}{dQ}(Q^3-7Q^2+12Q+5)=3Q^2-14Q+12.

The best best level of output of the firm will be when MC=MR and the slope of the MC curve is positive:


3Q214Q+12=4,3Q^2-14Q+12=4,3Q214Q+8=0.3Q^2-14Q+8=0.

This quadratic equation has two roots Q1=23Q_1=\dfrac{2}{3} and Q2=4.Q_2=4.

Therefore, MC=MR at Q1=23Q_1=\dfrac{2}{3} and Q2=4.Q_2=4.

Let's find the slope of the MC curve:


dMCdQ=ddQ(3Q214Q+12)=6Q14.\dfrac{dMC}{dQ}=\dfrac{d}{dQ}(3Q^2-14Q+12)=6Q-14.

Let's substitute Q1Q_1 and Q2Q_2 into the equation of the slope of MC curve:


dMCdQ=62314=10,\dfrac{dMC}{dQ}=6\cdot\dfrac{2}{3}-14=-10,dMCdQ=6414=10.\dfrac{dMC}{dQ}=6\cdot4-14=10.

So, at Q2=4Q_2=4 the slope of the MC curve is positive, therefore, it is the best level of output of the firm and firm maximizes its totalprofits.

b) By the definition of the profit, we have:


Profit=TRTC,Profit=TR-TC,Profit=4Q(Q37Q2+12Q+5),Profit=4Q-(Q^3-7Q^2+12Q+5),Profit=Q3+7Q28Q5,Profit=-Q^3+7Q^2-8Q-5,Profit=(4)3+7(4)2845=$11.Profit=-(4)^3+7\cdot(4)^2-8\cdot4-5=\$11.

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