Help me with the following

3. Consider a monopolist with cost function c(y) = cy. with e > 0, facing demand function x(p) = ap-, where c> 0. a. Show that if & $ 1. then the monopolist's optimal price is not well defined. b. Assume that &> 1. Derive the monopolist's optimal price, quantity, and price-cost margin (p" -c)/p" . Calculate the resulting deadweight welfare loss. 4. Consider the jewelry market. The total demand by men is given by D_(p) =100-2p. and the total demand by women is given by D_(p)= 200-p. The cost of production is c per unit. a. Suppose the jewelry market is competitive. Find the equilibrium price and quantity sold. b. Suppose, instead, that the firm A is a monopolist of jewelry. If firm A is prohibited from discriminating (i.c. charging different prices to men and women), what is the profit-maximizing price and quantity. For what values of c will it choose to sell to both men and women? c. Suppose that firm A is allowed to discriminate. What prices does it charge? Does aggregate welfare rise or fall relative to the nondiscrimination regime? 5. One common way to price discriminate is to charge a lump sum fee to have the right to purchase a good, and then charge a per-unit cost for consumption of the good after that. The standard example is an amusement park where the firm charges an entry fee and a charge for the rides inside the park. Such a pricing policy is known as a two part tariff. Suppose that the cost of providing the service is c(x). If the monopolist sets a two part tariff, will it produce more or less than the efficient level of output?4. A country which does not tax cigarettes is considering the introduction of a $3 per pack tax. The estimated demand and supply lnctions for cigarettes are given as: QD = 160,000 10,000P Qs = - 20,000 + 5,000P, where Q = daily sales in packs of cigarettes, and P = price per pack. The country has hired you to provide the following information regarding the cigarette market and the proposed tax. a. What are the equilibrium values in the current environment with no tax? b. What price and quantity would prevail after the imposition of the tax? What portion of the tax would be borne by buyers and sellers respectively? c. Calculate the deadweight loss from the tax. Could the tax be justied despite the deadweight loss? What tax revenue will be generated? Intermediate Microeconomics School of Economics Huazhong University of Science and Technology Homework 3 [Total Points = 100 pts.] Notes: . Full credit will be given only when (1) your answers are correct and (2) you explain the steps or reasoning behind your answers. . Due date: Thursday, Dec 27th 2018, in class and in hard copy 1. Suppose a profit-maximizing monopolist has total cost and marginal cost as follow: TC = 0.103 + Q + 10 and MC = 0.2 Q + 1. It faces the demand curve Q =35 - 5P. (35 points) a) What are the price, output, and profit for this monopolist? b) Carefully draw the diagram that illustrates your answers. c) What are the equilibrium price, output, and total profit if this is a perfectly competitive market? d) Compare the results between monopoly and perfect competition. e) Calculate the consumer surplus, producer surplus, and deadweight loss if the firm acts as a monopolist. Illustrate your answer with a diagram. f) Calculate the consumer surplus, producer surplus, and deadweight loss if this is a perfectly competitive market. Illustrate your answer with a diagram. (You can use the same diagram above.) g) Does the firm perform economic efficient in each market? Explain. 2. Assume that a firm is uncertain about the future demand for its product and is considering four alternative plant sizes. The short-run average cost curves for the four plan sizes are given by SACi (small), SAC (medium), SAC3 (large), and SAC4 (extra-large). Illustrate your answer in a figure how can the firm choose the optimal plan size to build according to different output level. And then derive the long-run average cost curve of the firm. (20 points)PD [30 points -- Two-period Hotelling Model) You work as a reserves manager for 131ng Oil, Inc. You petroleum engineers and accountants have calculated your total reserves to be R = 169 million barrels. You have been asked to allocate that production over 2 years: R = 00 + (21 The demand function for crude is given by Q = 200 P Assume that marginal production costs are zero, the discount rate is 10%, and BigTex is a price taker in all markets in which it is involved. To solve problems like this, we assume that BigTex is a \"representative firm\" in the oil industry. [5) Set up and solve the two period maximization problem maximization that you will use to determine the optimal allocation of production between the two periods. Show the rst order conditions for output in the two periods and for the Lagrange multiplier, A. [5) Use the rst order conditions to derive the \"arbitrage condition" that describes the equilibrium allocation of production between the two periods. What happens if this condition is not met? [10) Solve for the equilibrium quanties and prices in both periods. [5) Draw a graph like those in chapter 12 of Dahl [or the versions that we covered in class) to show the optima] allocation of production across the two periods. What is the difference in the slopes of the inverse demand functions for crude oil in the two periods