Answer to Question #159695 in Microeconomics for Farah

Question

TR=150Q- 4Q2

TC= 40+10Q+3Q2

Qa) Calculate the profit maximizing level of output for a price making firm?

b)Calculate the amount of profit earned by the firm at profit maximizing level of output calculated in part 1.

c)Mathematically prove the slope of MR is twice the slope of AR.

Answer 1

(a) Let's find the marginal revenue:

$$MR=\dfrac{dTR}{dQ}=\dfrac{d}{dQ}(150Q-4Q^2)=150-8Q.$$

Then, we can find the marginal cost:

$$MC=\dfrac{dTC}{dQ},$$ $$MC=\dfrac{d}{dQ}(3Q^2+10Q+40)=6Q+10.$$

The profit maximizing level of output for a price making firm will be when MC=MR :

$$150-8Q=6Q+10,$$ $$14Q=140.$$ $$Q=10.$$

So, at level output $Q=10$ the price making firm maximizes its profit.

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

$$Profit=\pi=TR-TC,$$ $$\pi=150Q-4Q^2-(3Q^2+10Q+40),$$ $$\pi=-7Q^2+140Q-40,$$ $$\pi=-7\cdot(10)^2+140\cdot10-40=\$660.$$

c) The equation of the straight line can be writen as follows:

$$y=mx+c.$$

Then, the equation of the Average Revenue can be written as follows:

$$P=mQ+c,$$

here, $P$ is the price, $m$ is the slope of the curve, $Q$ is the quantity demanded and $c$ is the interception with $y$-axis.

By the definition of the Total Revenue, we get:

$$TR=PQ=(mQ+c)Q=mQ^2+cQ.$$

By the definition of the Marginal Revenue, we get:

$$MR=\dfrac{dTR}{dQ},$$ $$MR=\dfrac{d}{dQ}(mQ^2+cQ)=2mQ+c.$$

Let's compare the equations of the AR and MR curve. As we can see from the comparison, the slope of the MR curve is $2m$ and the slope of the AR curve is $m$. Therefore, the slope of MR is twice the slope of AR.