Suppose that you are a monopolist who produces gizmos, Z, with the total cost function C(Z) = F + 50Z, where F represents the firm's fixed cost. Your marginal cost is MC = 50. Suppose also that there is only one consumer in the market for gizmos, and she has the demand function P = 60 - Z.
a) If you use a constant per-unit price for gizmos, what price maximizes your profits? What is the smallest value of F such that you could earn positive profits at this price?
b) Suppose instead that you charge a per-unit price equal to marginal cost, that is, P = MC = 50. How many units would the customer purchase at this price? Illustrate your answer in a graph (featuring the individual demand curve and marginal cost).
c) Now consider charging the customer a "subscription fee" of S in addition to a usage fee. If you set the usage fee as in part (b), what is the largest fixed fee you could charge the consumer, while ensuring that she is willing to participate in this market?
d) For what values of F will you be able to earn positive profits if you follow the pricing strategy you outlined in part (c)? How does this relate to your answer in part (a)?
e) Suppose now that there are N consumers in the market for gizmos, each with the individual demand function P = 60 - Z. Expressing your answer in terms of N, how large can the fixed costs F be for you to still earn positive profits if you use the above nonlinear pricing strategy.