Answer to Question #158066 in Microeconomics for S

Question

(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.

Answer 1

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

$$TR=PQ=4Q.$$

Let's find the marginal revenue:

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

Then, we can find the marginal cost:

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

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

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

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

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

Let's find the slope of the MC curve:

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

Let's substitute $Q_1$ and $Q_2$ into the equation of the slope of MC curve:

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

So, at $Q_2=4$ 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:

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