Question

A new taproom is trying to figure out how many different microbrews to carry. They estimate every tap added will increase revenue, but with diminishing returns: each tap will increase profits by 20% less than the tap before. Revenue from the first tap is $10,000, adding the second is $10,000+$8000 =$18,000, the third tap is $18,000+$6,400=$24,400, and so on. If the fixed costs of the bar are $30,000 and each tap costs $400, how many taps should the bar install to maximize profit?

Answer 1

Let Price of tap = P and quantity of tap = Q

the total revenue decreases by 20%, and average fixed cost = $30000 and average variable cost = $400

Therefore, the demand and cost functions of the taproom can be given as below:

TC= 30000+400Q ..........(i)

and P = 10000+4/5Q ........(ii)

From (i), marginal cost(MC) = 400

and marginal revenue (MR) = d(PQ)/dQ= 10000+8/5Q

For profit maximization of a firm,

                                MR= MC

      or, 10000+8/5Q=400 or, Q = 6000, which is the total number of taps the bar should install to maximize the profits.