ECO101SolvedProblemsSinglePriceMonopolist

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD Solved Problems: Single-Price Monopolist Version Without Solutions 1. Ghirmay is a profit-maximizing monopolist constrained to charging the same price for each unit sold. He is also constrained to integer quantities. His fixed cost of production is $10 per period, and he faces a marginal cost of $2 for each unit he produces. Assume that all benefits accrue to the buyer and all costs are borne by Ghirmay. Quantity Price Total Producer Q P Effect Effect MR 1 Revenue MR 2 Surplus 1 $9.00 2 $8.00 3 $7.00 4 $6.00 5 $5.00 6 $4.00 7 $3.00 8 $2.00 9 $1.00 10 $0.00 Table 1: The demand schedule Ghirmay faces. You get to fill in the rest. (a) Table 1 gives the demand schedule facing Ghirmay’s firm. Complete the rest of the table. Calculate MR 1 (Marginal Revenue 1) by summing up the quantity and price effects. Calculate MR 2 (Marginal Revenue 2) by calculating the change in total revenue from selling one more unit. (b) What price does Ghirmay charge? (c) What are Ghirmay’s profits at the price that maximizes total surplus? Briefly explain. (d) You are in charge of regulating Ghirmay’s monopoly. Your goal is to get as much total surplus as possible under the constraint that Ghirmay’s profits are non-negative. Assuming prices must be in whole dollars, what is the maximal price you allow Ghirmay to charge? Briefly explain. 2. Demand characterized by P ( Q ) = MWTP ( Q ) = 200 - 2 Q , quantities need not be integers. For each of the following, calculate marginal revenue, and decompose marginal revenue into price and quantity effects. (a) Q = 25 for the single-price monopolist. (b) Q = 50 for the single-price monopolist. 20231101: Page 1 Single-Price Monopolist: Problems

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD (c) Q = 75 for the single-price monopolist. (d) Explain the relationship between quantity and size of the price effect for the single-price monopolist. (e) Assume total cost given by TC ( q ) = F +40 q . Calculate the following under the assump- tion of a monopolist constrained to charge the same price for each unit: • elasticity of demand at the profit-maximizing price; • the largest F where this monopoly is profitable; • consumer surplus; and • deadweight loss (assuming assumption BIG). (f) Assume instead that MC = 20 + Q 2 . Calculate the following under the assumption of a monopolist constrained to charge the same price for each unit: • consumer surplus; • producer surplus; and • deadweight loss (assuming assumption BIG). 3. Assume quantities must be integers. Demand schedule given by Figure 3, with marginal cost of production equal to $8 per unit. Q 1 2 3 4 5 6 7 8 9 10 11 P $32 $29 $27 $23 $21 $19 $17 $14 $11 $7 $3 Table 3: Demand schedule. (a) Assume a profit-maximizing monopolist constrained to charging the same price for each unit. For each unit from 1 to 11, calculate the quantity effect, the price effect and the marginal revenue. (b) Assume a profit-maximizing monopolist constrained to charging the same price for each unit. For each unit from 1 to 11, calculate the total revenue received from selling exactly that many units. Using this calculation, calculate marginal revenue again. (c) Assume a profit-maximizing monopolist constrained to charging the same price for each unit. Calculate PS, CS, TS and DWL. (Assume for any unit transacted, all societal benefits are captured by the consumer and all societal costs are borne by the producer.) 4. You are a monopolist facing a demand curve given by MWTP ( Q ) = 36 - 4 Q . Your cost function is C ( Q ) = F + 4 Q , where F is a per-period fixed cost. Assume that quantities need not be integers. (a) In dollars, what is your marginal cost? ( Hint: What are your total costs if you produce 0 units? What are your total costs if you produce 1 unit? The marginal cost is the difference between these two numbers.) (b) Assuming that you produce and are constrained to charging the same price to all cus- tomers, what quantity do you produce ( Q M ) and what price do you charge P M ? (c) Hopefully, you have found that price is above marginal cost for all Q M units. This is good. Hopefully it will be enough to cover those fixed costs! (Clearly, if fixed costs are really small, you will want to produce. Likewise, if fixed costs are super huge, you will 20231101: Page 2 Single-Price Monopolist: Problems

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD were stolen by passengers looking for cheap home decorating, let us assume that the quantity of advertisements the MTA 3 sold was less than the total space for advertising. True, False or Uncertain: The MTA could have increased profits by selling more advertisements. 8. At P = 10, the monopolist sells 100 units. At P = 9, the monopolist sells 115 units. TFU: If the profit-maximizing monopolist must choose either P = 10 or P = 9, she chooses P = 9. 9. At P = 10, the monopolist sells 100 units. At P = 11, the monopolist sells 95 units. TFU: If the profit-maximizing monopolist must choose either P = 10 or P = 11, she chooses P = 11. 10. You know two things: 1) you are a single-price monopolist; and 2) demand is characterized by P ( Q ) = 200 - Q 1000 . Your pricing manager suggests setting P = $75. TFU: You should fire your pricing manager. 11. A monopolist constrained to charging the same price for each unit has TC ( Q ) = 250 + 2 Q and faces demand MWTP ( Q ) = 12 - Q 8 . Assuming it is required to pay its fixed costs, what are its economic profits? Hint you will not get on a term test: Each time you increase quantity by one unit, but how much does total cost increase? This sounds conspicuously like marginal cost. 12. While constant MC (i.e., a natural monopoly) will usually lead to a monopoly, a monopolist can have increasing marginal cost. A monopolist constrained to charging the same price for each unit has MC ( Q ) = 40 + Q 4 and faces demand MWTP ( Q ) = 100 - Q 8 . (a) Find the profit-maximizing quantity and price. (Note that because marginal cost is not constant, we can not use our shortcuts to find either the monopolist’s price or quantity.) (b) At what per-period fixed cost does this monopolist earn zero economic profits. 13. Table 5 shows the supply and demand schedules in a particular labour market. Assume all benefits accrue to the buyer and all costs are borne by the supplier. There are two possible assumptions about the market. Assumption 1 Table 5 depicts the supply and demand schedules resulting from eight work- ers each willing to supply up to one unit of labour and eight firms each of which demands up to one unit of labour. Assumption 2 Table 5 depicts the supply and demand schedules resulting from eight work- ers each willing to supply up to one unit of labour and one firm demanding up to eight units of labour. The firm must pay each worker it hires the same wage. Q 1 2 3 4 5 6 7 8 MWTP $35.5 $30.5 $27.5 $25.5 $23.5 $21.5 $19.5 $17.5 MC $11 $12 $14 $16 $18 $20 $23 $26 Table 5: A labour market. 3 The Metropolitan Transportation Authority, the agency that runs the NYC subways. 20231101: Page 4 Single-Price Monopolist: Problems

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD (a) If Assumption 1 holds, what is the efficient quantity? (b) If Assumption 1 holds, what is an equilibrium wage? (c) If Assumption 1 holds, what is the largest equilibrium consumer surplus? (d) If Assumption 1 holds, what is the equilibrium deadweight loss? (e) If Assumption 2 holds, what is the efficient quantity? (f) If Assumption 2 holds, what is the equilibrium wage? (g) If Assumption 2 holds, what is the largest equilibrium consumer surplus? (h) If Assumption 2 holds, what is the equilibrium deadweight loss? (i) If under either Assumption 1 or 2 there is equilibrium deadweight loss, what price control maximizes total surplus? 14. You manage a software development firm that currently has 100 computer programmers, each of whom earns $100 per hour. A client offers you a project. Your only cost would be programmers. With 1000 hours of programmer time you will certainly complete it, and upon completion the client will pay you $125,000. Elvis say, “Even if you need to hire additional programmers, doing the project will be profitable.” Agree, Disagree or It Depends. 15. You manage a software development firm that develops software for financial services firms. Assume your only cost is wages for computer programmers. You currently hire 20 computer programmers, paying each $200,000 per year. If you start to develop software for healthcare firms, you would need to hire 10 programmers, paying each $225,000 per year. Healthcare software development guarantees you revenues of $2,500,000 per year. True, False or Un- certain: It is profitable to enter the market for healthcare software development. 16. Tigist works a total of 40 hours each each week. She can work as many hours as she likes as a programmer earning $40 per hour. She also does personal training, with demand for her services given by Q ( P ) = 60 - P 2 . For her personal training business, she rents a studio for $100 per week, but does not pay herself a wage. That is, each week she gets all of the personal training revenue left over after paying for the studio plus her earnings as a programmer. (a) Assume that Tigist incorrectly uses only explicit costs in calculating her optimal hourly rate for personal training. How much money does she make each week from all sources? (b) Assume that Tigist correctly calculates her optimal hourly rate for personal training. How much money does she make each week from all sources? (c) What is the most that she is willing to pay each week to rent a studio for her personal- training side hustle? 20231101: Page 5 Single-Price Monopolist: Problems