Question

A firm faces a demand curve given by the equation P = 80 – 2Q. Its marginal cost of production is $20 per unit.

a. Find the profit-maximizing price and quantity.

b. Suppose that the firm contemplates issuing a $10-off coupon. Assume that consumers who would purchase at a price $50 or more never redeem coupons. Consumers who do not purchase at $50 or more always redeem coupons. By how much would the firm’s profits change if it issues this kind of coupon?

Answer 1

A).

Consider the given problem here the demand curve is given by, “P=80-2Q”, => “MR=80-4Q”. Now, the “MC” is “$20 per unit”, => at the optimum “MR” must be same as “MC”.

=> MR = MC, => 80 - 4*Q = 20, => Q=60/4=15, => Q*=15. Now, at Q* the optimum P is given by, “P=80-2Q=80-2*15=50”, => P*=50. So, the optimum “P” and “Q” are given by, “$50” and “15 units” respectively.

B).

Now, if the firm contemplates issues “$10-off” coupons, => all the consumers those will collect the coupons will have to pay “$40”. Now, let’s assume that all the consumer who would purchase the good at “$50” will not collect the coupons. So, here the total output supplied by the producer is given by, “2Q=80-P=80-40=40”, => Q=20. Now, existing consumers are getting “15 units”, => new consumer will purchase “5 additional units”.

So, here the profit of the firm is given by.

=> A2 = 50*15 + 40*5 - 20*(15+5) = $550. Now, the initial profit is given by.

=> A1 = 50*15 – 20*15 = $450, => the change in profit is given by, “A2-A1=$100”. So, if the firm issues these kinds of coupons then the profit of the firm increases by “$100”.