Question

Consider a firm facing a linear demand function and a constant marginal cost given below:

𝑝(𝑄)=𝑎−𝑏𝑄 and 𝑐(𝑄)=𝑐𝑄 where c is the constant marginal cost.

Find out the profit maximizing level output produced by the firm. Also, check the second order necessary condition for the same.

Answer 1

a) The profit maximizing level output produced by the firm occurs when MR=MC. Let's first find the total revenue:

$$TR=PQ=(a-bQ)Q=aQ-bQ^2.$$

Then, we can find the marginal revenue:

$$MR=\dfrac{dTR}{dQ}=a-2bQ.$$

By the definition of the marginal cost, we have:

$$MC=\dfrac{dTC}{dQ}=c.$$

Finally, we can find the profit maximizing level output produced by the firm:

$$a-2bQ=c,$$ $$Q=\dfrac{a-c}{2b}.$$

b) The second order condition can be witten as follows:

$$\dfrac{d^2TR}{dQ}-\dfrac{d^2TC}{dQ}<0,$$ $$\dfrac{d}{dQ}(a-2b)-\dfrac{d}{dQ}(c)<0,$$ $$-2b<0.$$

Therefore, as we can see, the second order condition is satisfied.