1. (a) Using calculus, derive the relationship between a monopolist's marginal revenue, the monopolists' price, and the price elasticity of demand. (b) Consider a monopolist who produces output at a constant marginal and average cost of $12. The price elasticity for the monopolist's product is 3. Use your answer to (a) to find the monopolist's profit maximizing price.
2. Bob's pharmaceutical company has developed a dew drug, Econosil, which helps students relax during economics exams.* A paten was obtained which entitles Bob to be the sold producer of Econosil. Bob's cost function for producing this new drug is given by C(Q) = 20Q, where Q is the number of capsules produced (in thousands). After extensive market research, Bob has determined the demand for his product is very high, and is given by the demand function P(Q) = 100 - Q. *Side effects may include nausea, hallucinations of enjoying economics, or premature death. (a) Find Bob's marginal revenue and marginal cost functions and plot them on the same graph, along with the demand function. (b) Assuming Bob is a profit-maximizing monopolist, how many capsules will Bob produce? What price will he charge? Indicate the profit-maximizing quantity (and price) on your graph. (c) Calculate Bob's profit as this level of output. Show your work. (d) If Econosil were instead a generic drug produced in perfectly competitive market, how many capsules would be produced, and what price would they be sold (assume the same cost and demand functions apply) (e) Finally, calculate the deadweight loss associated with monopoly production of Econosil and indicate this area on your graph. Hint: the area of a triangle is (1/2)base*height.
3. Use the demand and cost functions from Question 1 to answer the following questions, but here assume Bob practice first-degree price discrimination. (a) In this case, how many capsules would Bob produce? Show the profit on the diagram. Again, calculate the profit he would earn. (b) Of these two cases (standard monopoly pricing and fist degree price discrimination), which would result in the highest total surplus? Which would result in the highest consumer surplus? Explain.
4. Acme Widgets is a monopolist in the widget market. The widget market consists of 75 A's and 25 B's. The A's are willing to pay $90 for a high quality widget; the B's are willing to pay $200 for a high quality widget. Each consumer will either buy 1 high quality widget or no widgets. Acme can produce a high quality widget at zero cost. (a) Suppose initially Acme cannot engage in any form of price discrimination. Find Acme's profit maximizing price. How much profit will Acme earn? (b) Now suppose Acme can identify who is an A and who is a B and is able to engage in third degree price discrimination. How much profit will Acme earn?
5. College Park Airlines sells a single daily flight from College Park to Long Island to two types of passengers-business people, who have daily demand of QA = 260 - 0.4PA and University of Maryland students, who have daily demand of QB = 260 - 0.6 PB. The airline can easily distinguish students from business people, so they decide to charge them different prices (they also have to show a student ID to get the special rate-business passengers cannot pose as students). The cost of running each flight is $30,000 plus $100 per passenger. Answer the following questions, and show your work. (a) Find the marginal revenue function for each of College Park Airline's submarkets. (b) What price will the airline charge the students? What price will they charge business people? (c) How many of each type of customer will take the flight? (d) Under this pricing strategy, what will be College Park Airline's per-flight profit? (e) If third degree price discrimination is not allowed and only one price can be charged, what price will the airline charge? How many customers will take the flight altogether? What will be the College Park Airline's per-flight profit?