26. Let the demand and cost functions of a multiplant monopolist be p = a - b(qı + q:), Ci = azq1 + Bigi, and C: = a292 + B2q3 where all parameters are positive. Assume that an autonomous increase of demand increases the value of a, leaving the other parameters unchanged. Show that output will increase in both plants with a greater increase for the plant in which marginal cost is increasing less fast.
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- . Suppose a monopolist producing Q units of output faces the demand curve P = 130 − 4Q. Its total cost when producing Q units of output is T C(Q) = F + 34Q+ 2Q2 , where F is a fixed cost. (a) For what values of F can a profit-maximizing firm charging a uniform price earn at least zero economic profit? (b) For what values of F can a profit-maximizing firm engaging in perfect first-degree price discrimination earn at least zero economic profit.5. A monopolist with cost function c(Q)=faces an inverse demand function given by P(Q)= √Q' (a) Find the elasticity of demand with respect to price. (b) Assuming that the monopolist uses MR = MC pricing rule, find his profit maximizing price, p", and output level, q". (c) Find the marginal cost at q" and calculate the Lerner index. (d) Does the monopolist's market power depend on his cost curve? In particu- lar, does it depend on a? Is your answer surprising?2. Suppose the home country open up to free trade and a foreign competitor enters the market. Assume thatthe foreign firm has the same cost structure as the home firm (the monopoly from the previous question). The demand for its product is given by the inverse demand function: P = 120 −QD. The company’s costs are: T C = 20Q+ 200 and MC = $20A) Derive the best response function for each firm (h-home and f-foreign)B) Find each firms’ output, the home market price, and each firms’ profit from the home market3. Now, suppose that in addition to the home country opening up to free trade, the foreign country has alsoopened up to free trade. As a result, both firms sell their product in both markets.A) Find each firms’ overall output, market price in each market, and each firms’ overall profitB) Explain what effect free trade has (relative to no trade) on the firms and consumers
- Suppose a monopoly market has a demand function in whichquantity demanded depends not only on market price (P) butalso on the amount of advertising the firm does (A, measuredin dollars). The specific form of this function isQ =(20 - P2) (1 + 0.1A - 0.01A2).The monopolistic firm’s cost function is given byC = 10Q + 15 + A.a. Suppose there is no advertising (A = 0). What outputwill the profit-maximizing firm choose? What market price will this yield? What will be the monopoly’sprofits?b. Now let the firm also choose its optimal level of advertising expenditure. In this situation, what output levelwill be chosen? What price will this yield? What will thelevel of advertising be? What are the firm’s profits in thiscase? Hint: This can be worked out most easily by assuming the monopoly chooses the profit-maximizing pricerather than quantity.1. A monopolist with cost function c(Q) = faces an inverse demand function given by P(Q) = (a) Find the elasticity of demand with respect to price. (b) Assuming that the monopolist uses MR = MC pricing rule, find his profit maximizing price, p", and output level, q™. (c) Find the marginal cost at q" and calculate the Lerner index. (d) Does the monopolist's market power depend on his cost curve? In particu- lar, does it depend on a? Is your answer surprising?3. Suppose the inverse demand function is linear: p(q) = 24-q The monopolist's cost %3D function is c(g)-0.5g*. Assume the monopolist must charge a uniform price. (a) Find the optimum monopoly price and quantity. Also calculate the deadweight loss. (b) Suppose the governmet can levy a lump-sum tax T (i.e., a fixed amount indepen- dent of produetion) and an excise tax t per unit of production on the monopolist. These taxes can be negative, in which case they are subsidies. The proceeds of these taxes can be transferred to consumers. The monopolist is always free to quit the market, in which case she does not have to pay any taxes. The government wants to maximize the ensumer welfare: Pind the optimum vaues of t and
- 2. Consider a pharmaceutical company considering research and development of a new drug. They estimate that demand for this new, innovative product (the only of its kind) is given by p = 200-q, and the firm knows that once the drug is developed it can produce as much as it would like at a constant marginal cost of $10 (this implies a cost function of c(q) = 10q). (a) What is the monopolist's profit-maximizing price and quantity produced, pm and qm if they develop the drug? Be sure to include a well-labeled diagram! (b) If the firm expects that the R&D will cost $10,000, should the firm pursue this product innovation? (c) What is the most the firm should be willing and able to spend on this innovation? (d) What is the socially optimal quantity of output, q*? (e) What is the value of the innovation to society if it was competitively/efficiently supplied? (f) Calculate the deadweight welfare loss we should expect if the firm produces the new drug as a monopoly.2. Suppose there is a monopolist sells good x with constant marginal cost c. Also, there are two consumers with money endowment m; (i=1,2) and utility functions: 'u + zx = 'n Uz = (3/2)x"² + m, m, = m, – px, a) Derive each individual's demand for good x b) Suppose now that the monopolist can only engage in indirect (2mnd degree) price discrimination and decides to offer take it or leave it contracts of the form (ri,X;), where r; is a lump-sum payment that entitles the consumer to purchase x; units of the good. i. Write down the 2 incentive compatibility constraints and the 2 participation constraints that the solution needs to satisfy Given the properties of the utility functions, which constraints are binding? iii. ii. Formulate the monopolist's profit maximizing problem given your answer to ii above and compute the optimal contracts that this monopolist should offer. iv. What are the consumption levels for both consumers? Which level of consumption is efficient and why? With your…Assume that two companies (A and B) are duopolists who produce identical products. Demand for the products is given by the following linear demand function:P = 200 - QA - QBwhere QA and QB are the quantities sold by the respective firms and P is the sellingprice. Total cost functions for the two companies areTCA = 1500 + 55QA + Q2ATCB = 1200 + 20QB + 2Q2BAssume that the firms act independently as in the Cournot model (i.e., each firmassumes that the other firm’s output will not change).a. Determine the long-run equilibrium output and selling price for each firm.b. Determine Firm A, Firm B, and total industry profits at the equilibrium solutionfound in Part (a).
- A monopolist's costs are given by: C(Q) = 4Q² + 10Q + 100 and it faces the demand function: P = 50 - Q What is the firm's profit-maximizing or loss-minimizing output level, and at what price will it sell this output? А. Would the firm be better off not producing anything, i.e., setting Q = 0, instead? Justify your answer carefully В.12) Suppose that a monopolist supplies a product in two distinct markets, LA and NY. The demand functions for the two markets are PLa = 65 – 3QLA and Pvy = 50 – 5QNY. The monopolist has a fixed cost of $10 and a constant marginal cost of $5 per unit. a. If segmenting is feasible, what are the profit-maximizing prices, quantities, and maximized profit? b. If segmenting is NOT feasible, what is the profit-maximizing price, quantity, and maximized profit? c. How much is the difference in total consumer surplus in the two cases? Which case makes consumers better off?4. Consider a monopolist with cost funetion c(q) = 20g, who is facing two con- sumers. The consumers' demand functions are given by 91 = 100 - p 92 = 300 – 3p. (a) Suppose the monopolist does not price discriminate. Find the monopolist's optimal pricing strategy, the resulting profit, and Lerner Index. (b) Suppose the monopolist can engage in third degree price discrimination. Find the monopolist's optimal pricing strategy and the resulting profit. Compare the profit with that found in part (a), and give an intuitive ex- planation. (c) Now, suppose the monopolist can produce the good at zero cost. That is clq) = 0. Find the monopolist's optimal second degree price discrimination strategy.