ECO101SolvedProblemsSinglePriceMonopolistSolutions

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University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD Solved Problems: Single-Price Monopolist Version With 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. Suggested Solution: See Figure 2. Note that Producer Surplus is just total revenue minus variable costs (i.e., the sum of variable costs). 20231101: Page 1 Single-Price Monopolist: Problems: Solutions

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD Quantity Price Total Producer Quantity Price Effect Effect MR 1 Revenue MR 2 Surplus 1 $9.00 $9.00 $0.00 $9.00 $9.00 $9.00 $7.00 2 $8.00 $8.00 -$1.00 $7.00 $16.00 $7.00 $12.00 3 $7.00 $7.00 -$2.00 $5.00 $21.00 $5.00 $15.00 4 $6.00 $6.00 -$3.00 $3.00 $24.00 $3.00 $16.00 5 $5.00 $5.00 -$4.00 $1.00 $25.00 $1.00 $15.00 6 $4.00 $4.00 -$5.00 -$1.00 $24.00 -$1.00 $12.00 7 $3.00 $3.00 -$6.00 -$3.00 $21.00 -$3.00 $7.00 8 $2.00 $2.00 -$7.00 -$5.00 $16.00 -$5.00 $0.00 9 $1.00 $1.00 -$8.00 -$7.00 $9.00 -$7.00 -$9.00 10 $0.00 $0.00 -$9.00 -$9.00 $0.00 -$9.00 -$20.00 Table 2: The completed table. (b) What price does Ghirmay charge? Suggested Solution: $6. From the table, we can see this in a couple of ways. First, recall that profits are the difference between producer surplus and fixed costs. Given that fixed costs are fixed, profit maximization requires choosing the price with the highest producer surplus: $6. Second, we could find the largest quantity where MR ≥ MC . This is Q = 4, where he charges P = $6. (c) What are Ghirmay’s profits at the price that maximizes total surplus? Briefly explain. Suggested Solution: To maximize total surplus, we want any and all units where marginal societal benefit MSB is at least as large as marginal societal cost MSC . Recall that the demand schedule gives marginal private benefit (i.e., MWTP ). Under the assumption that all benefits go to the buyer, we have MSB = MWTP , and under the assumption that all costs are borne by Ghirmay, we have MSC = MC . We thus want any and all units where MWTP ≥ MC = $2, meaning we want 8 units. Eight units are transacted when P = $2, which gives Ghirmay PS = $0. (That is, his variable costs are exactly equal to his revenues.) Given that he faces a fixed cost, his profits are π = PS - F = $ - $10 = - $10. (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. Suggested Solution: $4. Unfortunately, Ghirmay’s profits are negative when we maximize Total Surplus. Assum- ing we are below the efficient quantity, total Surplus is increasing as quantity increases, so as the regulator you want to find the largest quantity where Ghirmay’s profits are non-negative. In order for Ghirmay to have non-negative profits, his producer surplus must be at least as large as his fixed costs of $10. Therefore, you do not permit him to charge more than $4. He charges $4, and his producer surplus is $12 and his profits are $12-$10=$2. 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. 20231101: Page 2 Single-Price Monopolist: Problems: Solutions

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD (a) Q = 25 for the single-price monopolist. Suggested Solution: Via the “twice-as-steep” rule, we have marginal revenue MR ( Q ) = 200 - 4 Q , and thus MR (25) = 200 - 4 × 25 = 100. Quantity effect given by demand equation, and is thus equal to MWTP (25) = 200 - 2 × 25 = 150. Price effect is the difference between marginal revenue and quantity effect, and is thus -50. (b) Q = 50 for the single-price monopolist. Suggested Solution: Via the “twice-as-steep” rule, we have marginal revenue MR ( Q ) = 200 - 4 Q , and thus MR (50) = 200 - 4 × 50 = 0. Quantity effect given by demand equa- tion, and is thus equal to MWTP (50) = 200 - 2 × 50 = 100. Price effect is the difference between marginal revenue and quantity effect, and is thus -100. (c) Q = 75 for the single-price monopolist. Suggested Solution: Via the “twice-as-steep” rule, we have marginal revenue MR ( Q ) = 200 - 4 Q , and thus MR (75) = 200 - 4 × 75 = - 100. Quantity effect given by demand equation, and is thus equal to MWTP (75) = 200 - 2 × 75 = 50. Price effect is the difference between marginal revenue and quantity effect, and is thus -150. (d) Explain the relationship between quantity and size of the price effect for the single-price monopolist. Suggested Solution: This example shows that price effect increases in quantity. This makes sense: the more customers we are currently selling to, the more painful a price decrease in order to entice an additional customer. With P ( Q ) = MWTP ( Q ) = 200 - 2 Q , each 1 unit increase in quantity results in a $2 decrease in price. The price effect measures the effect of this decrease in price on marginal revenue. With Q = 25, we give this $2 discount to approximately 25 customers (price effect equal to $50), whereas with Q = 75 we must give this $2 discount to approximately 75 customers (price effect equal to 150). (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). Suggested Solution: As total costs increase by $40 for each one unit increase in quantity, MC = $40. The profit-maximizing quantity Q m equates marginal revenue (found using the “twice-as-steep rule”) and marginal cost, and the price P m results in a demand for exactly Q m units. 200 - 4 Q m = 40 160 = 4 Q m Q m = 40 P m = MWTP ( Q m ) = 200 - 2 × 40 = $120 Elasticity of demand at profit-maximizing price (using “calculus” method) is Δ Q Δ P P m Q m . As a one-unit change in quantity results in a two-unit change in price (i.e., Δ P Δ Q = 2), a one-unit change in price results in a one-half unit change in quantity (i.e., Δ Q Δ P = 1 2 ). 20231101: Page 3 Single-Price Monopolist: Problems: Solutions

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University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD Elasticity of demand is thus 1 2 × 120 40 = 3 2 . As advertised, the demand is elastic ( > 1) at the price maximizing monopoly profits. Figure 1: Monopolist facing constant marginal cost. Figure 1 depicts the monopoly. The monopolist’s profits will be non-negative when producer surplus is at least as large as fixed costs. Producer surplus is $80 × 40 = $3 , 200. As long as fixed costs are smaller than $3,200 per period, this is a profitable monopoly. Consumer surplus is 40 × $80 2 = $1 , 600. For deadweight loss, note that surplus is maxi- mized at Q effic = 80, meaning DWL is (80 - 40) × ($120 - $40) 2 = $1 , 600. (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). Suggested Solution: The profit-maximizing quantity Q m equates marginal revenue and cost, and the price P m results in a demand for exactly Q m units. 200 - 4 Q m = 20 + Q m 2 400 - 8 Q m = 40 + Q m 360 = 9 Q m Q m = 40 P m = MWTP ( Q m ) = 200 - 2 × 40 = 120 In order to calculate deadweight loss, we are going to need to calculate the efficient 20231101: Page 4 Single-Price Monopolist: Problems: Solutions

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD quantity. The efficient quantity Q effic equates marginal societal benefit and cost. 200 - 2 Q effic = 20 + Q effic 2 400 - 4 Q effic = 40 + Q effic 360 = 5 Q effic Q effic = 72 Figure 2: A monopoly with non-constant MC . Not drawn to scale. In Figure 2, consumer surplus is given by the area of triangle DEF, or 40 × 80 2 = 1600. Producer surplus is the area of the square CEFG (80 × 40 = 3200) plus the area of the triangle BCG ( 40 × 20 2 = 400) for a total of 3200+400=3600. Deadweight loss is the area of the triangle GFJ: 32 × 80 2 = 1280. Alternatively, we can take the difference between the maximal surplus available (the area of the triangle BEJ= 180 × 72 2 = 6480) and the surplus actually earned (1600+3600=5200), or 6480-5200=1280. 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. 20231101: Page 5 Single-Price Monopolist: Problems: Solutions

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD Suggested Solution: Let us look at Q = 6. For Q = 5, the profit maximizing monopolist would charge $21. To sell unit 6, she must drop the price to $19. The quantity effect is thus $19, the amount she receives for unit 6. For the price effect, we see that the monopolist would have sold 5 units at $21 had she not decided to sell unit 6. Thus for each of these 5 units, she loses $2 by selling unit 6, meaning the price effect is - $2 × 5 = $10. Marginal revenue is the summation of quantity and price effects, or $19 - $10 = $9. In Table 4, I perform this calculation for each 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 QE $32 $29 $27 $23 $21 $19 $17 $14 $11 $7 $3 PE $0 -$3 -$4 -$12 -$8 -$10 -$12 -$21 -$24 -$36 -$40 MR $32 $26 $23 $11 $13 $9 $5 -$7 -$13 -$29 -$37 TR $32 $58 $81 $92 $105 $114 $119 $112 $99 $70 $33 MR $26 $23 $11 $13 $9 $5 -$7 -$13 -$29 -$37 CS (P=19) $13 $10 $8 $4 $2 $0 $37 PS (P=19) $11 $11 $11 $11 $11 $11 $66 Table 4: Demand schedule, plus. (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. Suggested Solution: In Table 4, see the TR row and the second MR row. (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.) Suggested Solution: The profit maximizing firm sells all units whose marginal rev- enue is at least as large as marginal cost. In this case, we want all units whose marginal revenue is greater than or equal to $8, meaning we want to sell six units. If we wanted to make as much money as possible from selling exactly 6 units, we would want to charge $19. In Table 4, the row CS (P=19) calculates for each unit the consumer surplus when a the price is P = $19. The row PS (P=19) calculates for each unit the producer surplus for each item sold when P = $19 (i.e., for 6 units). Total surplus is thus $37+$66=$103. For DWL , we note that if we wanted to maximize total surplus, we would want to transact 9 units. The demand schedule gives MWTP , 1 and for each of the first 9 units societal benefits exceed societal costs and would result in positive surplus. Because we single-price monopolist restricts quantity in order to keep price high, we miss out on the surplus from units 7 ($17-$8=$9); 8 ($14-$8=$6) and 9 ($11-$8=$3). DWL is thus $9+$6+$3=$18. 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. 1 The fact that unit 9 is only purchased at a price of $11 or lower tells us that MWTP and thus MB must be $11. 20231101: Page 6 Single-Price Monopolist: Problems: Solutions

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University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD (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.) Suggested Solution: $4 (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 ? Suggested Solution: 36 - 8 Q | {z } MR ( Q ) = 4 |{z} MC ( Q ) Q M = 4 In order to sell 4 units, the profit-maximizing monopolist sets a price of P (4) = 36 - 4 * 4 = 20. (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 not want to enter this market even though you have a monopoly.) Assuming that you are constrained to charging the same price to all consumers, what is the maximum F at which your firm can make an economic profit? Suggested Solution: As the cost per item is 4, the monopolist makes 20 - 4 = 16 per unit, and makes 16 * 4 = 64 before fixed costs. Therefore, as long as fixed costs are less than 64, our monopolist is making an economic profit. (d) In a graph, depict: • the demand curve, • the marginal revenue curve, and • the marginal cost curve; and identify • the efficient quantity (labelled Q effic ), • the quantity selected by the monopolist (labelled Q M ), • the price charged by the monopolist (labelled P M ), and • the deadweight loss (labelled DWL). Suggested Solution: See figure 3. 20231101: Page 7 Single-Price Monopolist: Problems: Solutions

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD 2 4 6 8 4 8 12 16 20 24 28 32 36 Quantity 0 10 0 40 Demand MR MC Q M Price P M DWL Q e ff٠ c Figure 3: Your monopoly. (e) If you were setting a price to maximize total surplus as opposed to firm profits, how much more surplus could you create? Suggested Solution: We need to calculate the DWL: DWL = bh 2 = 4 * 16 2 = 32 . 5. Assume MWTP for the a Graphing Calculator iPhone app is MWTP ( Q ) = 10 - Q 10 , 000 . As- suming that Apple has already provided sufficient bandwidth for the App Store, the marginal cost of producing one more unit is $0.00. (a) What is the price that maximizes producer surplus? Suggested Solution: MB ( Q ) = MRQ = MC ( Q ) = 0 10 - Q 5 , 000 = 0 the twice-as-steep rule Q * M = 50 , 000 solving for PS maximizing quantity P * M = MWTP (50 , 000) = 10 - 50 , 000 10 , 000 = 5 solving for PS maximizing price (b) What is elasticity of demand at the price that maximizes monopolist profits? (Use the “calculus” method to calculate elasticity at a particular point.) Suggested Solution: 1. Even before we explicitly calculate it, note that we want the quantity where MR = MC = 0, which means we want to maximize total revenue. We know that total revenue is maximized when elasticity=1 . . . If you insist on calculating elasticity ( Δ Q Δ P P m Q m ), note that a one-unit change in quantity results in a 1/10,000 change in price Δ P Δ Q = 1 10 , 000 , meaning a 1 unit change in price results in a 10,000 unit change in quantity ( Δ Q Δ P 1 10 , 000 ). We thus have 10 , 000 × 5 50 , 000 = 1. Told you so. 20231101: Page 8 Single-Price Monopolist: Problems: Solutions

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD (c) Given that the Graphing Calculator app has already been written, what is the price that maximizes Total Surplus? Suggested Solution: P = 0 ensure that everyone who values it at marginal cost or higher purchase the app. (d) Instead of the price that maximizes producer surplus, the actual price is P = 1. What is the exact deadweight loss with a price of $1? Suggested Solution: At a price of $1, 90,000 consumers purchase, but 10,000 who value at least at marginal cost do not. These 10,000 would have received consumer surplus of 10 , 000 * 1 2 = 5 , 000. Thus the deadweight loss is 5,000. (e) Suppose Apple announced that starting today and continuing forever, the price of all iPhone apps will the the price you identified 5c, and this in no way affects marginal costs. Argue that this policy may not, in fact, maximize Total Surplus. (Hint: Think in the long run.) Suggested Solution: Given that an app has been produced, with marginal cost equal to $0.00, the price that delivers the optimal number of consumers is $0.00. Producer surplus is thus $0.00. Let me ask you, how much time are you going to devote to producing iPhone apps if you know that your revenue will be $0.00? While the surplus generated at P > 0 is less than maximal, it is a lot larger than if the app were never written. (f) Explain why with MC = 0, the single-price monopolist chooses the price where demand is unit elastic, whereas with MC > 0, the single-price monopolist chooses a price where demand is elastic. Suggested Solution: It might be helpful to look at a standard single-price monopolist graph (e.g., Figure 3). Let us start at the price/quantity where demand is unit elastic. At this point, marginal revenue will equal zero. 2 Let us now contemplate a slight reduction in quantity (i.e., a slight increase in price). Marginal revenue is positive for all units smaller than the quantity where marginal revenue equals zero. This means by not selling this unit, total revenue has decreased . So, was this quantity reduction a good thing? In the case where MC = 0 this quantity reduction was unprofitable, as you reduced revenues and did not reduce costs. We thus want to produce where MR = 0 (i.e., demand is unit elastic). In the case where MC > 0, this slight quantity reduction was profitable. Sure, my revenue decreased, but because I am producing less, my costs went down as well. At any point where my marginal revenue curve is beneath my marginal cost curve, quantity reductions will be profitable as the loss in revenue is smaller than the decrease in costs. I keep decreasing quantity until MR = MC . As I am now at a quantity where MR > 0, I am on the elastic portion of the demand curve. 6. Currently, patents in the U.S. and Canada last 20 years from the date on which the application for a patent is filed. The effective patent on a new prescription drug is about 8 years, as it takes about 12 years to test a new drug for safety and efficacy and get approval from the U.S. Food and Drug Administration or Health Canada. When the patent on a drug expires, any approved manufacturer can produce and sell the drug. (In general, when a drug comes off patent, there are many manufacturers who are approved to sell a generic version. You can assume a competitive market after the drug comes of patent.) True, False, or Uncertain: 2 This is a result you should know. 20231101: Page 9 Single-Price Monopolist: Problems: Solutions

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University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD A policy that increased the length of patents for new drugs from 20 to 32 years would result in a decrease in Total Surplus. Suggested Solution: Uncertain. There would definitely be static inefficiency. That is, for drugs which are already under development and would have its effective patent life increased from 8 to 20 years, this policy change would result in twelve more years of monopoly inef- ficiency. There may however be dynamic efficiencies. That is, there may be drugs that are not developed because the firm feels that 8 years of monopoly profits is insufficient to recover development costs, but might be profitable and therefore developed if the firm had 20 years of monopoly profits. 7. While riding the subway in NYC recently, I noticed that there were some advertisement locations where there no advertisements. While it might be the case that these advertisements 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. Suggested Solution: Well, if we assume that the MTA is profit maximizing, then the answer is of course false. Let’s see whether we can do better than this. First, let us assume that the market is perfectly competitive. 4 In this case, the MTA can sell as many advertisements as it wants at the market price, and thus the MTA is not maximizing profits. Second, let us assume that the MTA has market power. In particular, let us assume that it is a monopolist constrained to charging the same price for each unit with Q units of advertising space available. If it did not face space constraints, it would choose Q m the quantity where MR = MC . If its available advertising space is greater than Q m (i.e., Q > Q m ), it would only sell Q m units, meaning it would be profit maximizing to leave some spaces unsold (in order to keep the price high). If its available advertising space is less than Q m (i.e., Q < Q m ), it would sell Q at a price equal to MWTP ( Q ). In this case, there would be no unsold advertising locations. For those preferring numbers over variables, let us assume 100 advertising spaces per train. The MTA solves for the quantity maximizing profits ( Q m ) without considering that it only has room for 100 per train. If its profit maximizing quantity is less than 100 (e.g., 80), then it only sells 80 and leave 20 spaces blank. If its profit maximizing quantity is more than 100, in this case it sets the price so that exactly 100 units are sold. 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. Suggested Solution: Uncertain. While it is true that total revenues are greater under P = 9 (1035 versus 1000), it is only worthwhile to choose this lower price if the cost of producing these additional 15 units is less than 35. 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. Suggested Solution: True. Increasing price increases revenues (1045 versus 1000) and will not increase costs and quantity has decreased. 3 The Metropolitan Transportation Authority, the agency that runs the NYC subways. 4 For example, other forms of advertising are perfect substitutes. 20231101: Page 10 Single-Price Monopolist: Problems: Solutions

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD 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. Suggested Solution: True. As a single-price monopolist, you know that you price on the elastic portion of the demand curve. 5 With a linear demand curve, you know that demand is unit elastic at the midpoint of the demand curve: halfway between 0 and 200, or P = 100. For you linear demand curve, demand will be inelastic at prices less than the midpoint, such as P = $75. Profits must increase as you increase price from P = $75: total revenues increase as demand is inelastic and costs go down as quantity decreases. Your pricing manager is incompetent and should be fired. 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. Suggested Solution: The twice-as-steep rule gives us MR = 12 - Q 4 . We set MR ( Q M ) = MC ( Q M ), or 12 - Q M 4 = 2. Solving, we get Q M = 40. The price is thus MWTP ( Q M ) = MWTP (40) = 12 - 40 8 = $7. 6 We can calculate profits in two ways. First, total revenues are P M × Q M = $7 × 40 = $280. Total costs are TC ( Q M ) = TC (40) = 250+2 × 40 = $330. Profits are thus $280 - $330 = - $50. Alternatively, we can first calculate producer surplus. Because both price and marginal cost are constant, this will be a rectangle with height equal to P M - MC = $7 - $2 = $5 and width equal to Q M = 40 for a grand total of $200. This is the amount of money left over after paying variable costs. Unfortunately, fixed costs are $250, meaning a loss of - $50. 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.) Suggested Solution: The twice-as-steep rule gives us MR ( Q ) = 100 - Q 4 . Now, a 5 If marginal costs are zero, you then want to price at the unit elastic point. When marginal costs are zero, profit maximization implies revenue maximization. 6 Alternatively, because we have constant marginal cost, we could have just used the shortcut that tells us P M is halfway between the vertical-intercept of the demand curve (12) and the marginal cost (2). 20231101: Page 11 Single-Price Monopolist: Problems: Solutions

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD little math: MR ( Q M ) = MC ( Q M ) 100 - Q M 4 = 40 + Q M 4 400 - Q M = 160 + Q M 240 = 2 Q M Q M = 120 P M = MWTP ( Q M ) = 100 - 120 8 = $85 (b) At what per-period fixed cost does this monopolist earn zero economic profits. Suggested Solution: We calculate the amount of money the monopolist has left over after paying variable costs. That is, we calculate producer surplus: the area between the monopolist’s price and marginal cost. Figure 2 shows the case for the monopolist faced with increasing marginal cost. There will be a triangle with base equal to the difference between the marginal cost of the last unit ( MC (120) = $70) the vertical-axis intercept of the marginal-cost curve: 70-40=30. The height of the triangle is Q M = 120. The area is thus bh 2 = 120 × $30 2 = $1 , 800 . There is also a rectangle: base equal to Q M = 120, height equal to the difference between the monopolists price and the marginal cost of the last unit ($85 - $70 = $15), for another $1,800. Therefore, if per-period fixed costs are $3,600, economic profits are exactly equal to zero. (Likewise, if per-period fixed costs are less than $3,600, this monopolist earns positive 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. (a) If Assumption 1 holds, what is the efficient quantity? Suggested Solution: With MWTP giving MSB and MC giving MSC (as assumption BIG holds), MSB is at least as large as MSC for the first six units. 20231101: Page 12 Single-Price Monopolist: Problems: Solutions

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University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD (b) If Assumption 1 holds, what is an equilibrium wage? Suggested Solution: With a relatively large number of buyers and sellers, we are likely to get a competitive market. The competitive market will find a price equating quantity supplied and quantity demanded. MWTP is at least as large as MC for the first six units. Exactly six units are demanded if $19 . 5 ≤ P * ≤ $21 . 5. Exactly six units are supplied if $20 ≤ P * ≤ $23. Any price between $20 and $21.5 satisfies both of these conditions. (c) If Assumption 1 holds, what is the largest equilibrium consumer surplus? Suggested Solution: We get the largest equilibrium consumer surplus when we get the lowest equilibrium price, or P * = 20. Summing up consumer surplus for the each of the first six units give $15 . 5 + $10 . 5 + $7 . 5 + $5 . 5 + $3 . 5 + $1 . 5 = $44. (d) If Assumption 1 holds, what is the equilibrium deadweight loss? Suggested Solution: $0. The competitive market results in the efficient quantity, with the six units of labour provided by the workers with the lowest marginal societal benefit and consumed by the firms with the highest societal cost. (e) If Assumption 2 holds, what is the efficient quantity? Suggested Solution: With MWTP giving MSB and MC giving MSC (as assumption BIG holds), MSB is at least as large as MSC for the first six units. In other words, the efficient outcome is the same regardless of whether Assumption 1 or Assumption 2 holds. (f) If Assumption 2 holds, what is the equilibrium wage? Suggested Solution: The monopsonist is going to keep hiring as long as the marginal benefit is at least as large as the marginal (wage) cost. However, unlike a firm in a perfectly competitive labour market, it cannot hire as many workers as it wants at the current market wage. In order to hire an additional worker, it needs to offer a higher wage, and by assumption must also offer this higher wage to all workers who were willing to work at the previous 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 P ( Q s ) $11 $12 $14 $16 $18 $20 $23 $26 TWC $11 $24 $42 $64 $90 $120 $161 $208 MWC $11 $13 $18 $22 $26 $30 $41 $47 Table 6: The monopsonist. In Table 6, we develop the marginal (wage) cost. I have re-labelled the second row to P ( Q s ) to highlight that it is the labour-supply schedule. The third row is total wage cost (TWC). For example, the labour-supply schedule tells us that a wage of $18 is required in order to have five units of labour supplied, which means the total wage cost is 5 × $18 = $90 if five workers are hired. In the final row, we calculate marginal wage cost (MWC), the increase it total wage cost as we hire each additional worker. Keep hiring labour as long as the marginal benefit is at least as large as the cost, meaning the monopsonist hires four workers and offers the lowest wage at which exactly four workers accept, $ 16 . (g) If Assumption 2 holds, what is the largest equilibrium consumer surplus? 20231101: Page 13 Single-Price Monopolist: Problems: Solutions

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD Suggested Solution: Summing up consumer surplus when paying $16 for the each of the first six units gives ($35 . 5 - $16)+($30 . 5 - $16)+($27 . 5 - $16)+($25 . 5 - $16) = $55. This is greater than the consumer surplus under the assumption of a competitive labour market. (h) If Assumption 2 holds, what is the equilibrium deadweight loss? Suggested Solution: The monopsonist hires four workers while the efficient quantity is six workers. Hiring worker five would have created $23 . 5 - $18 = $5 . 5 in surplus. Hiring worker six would have created $21 . 5 - $20 = $1 . 5 in surplus. Deadweight loss is $5 . 5 + $1 . 5 = $7. (i) If under either Assumption 1 or 2 there is equilibrium deadweight loss, what price control maximizes total surplus? Suggested Solution: Under Assumption 2 (monopsony), there is deadweight loss without government intervention as the equilibrium quantity is less than the efficient quantity. The efficient quantity (six units) is both supplied and demanded for any wage between $21.5 and $20, meaning a minimum wage in this rate would maximize 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. Suggested Solution: It depends. If you hire programmers in a competitive market, it means that you can hire as many programmers as you want at $100 per hour. In this case, your costs will be $100,000, meaning you will profit. However, if you have monopsony power, you face an upward sloping supply curve for programmers. You will have to pay more than $100 per hour for these 1000 hours—and may have to increase the pay of the programmers you already use—which means your costs will be greater than $100,000. While the cost increse is greater than $100,000, it is uncertain whether it is more or less than $125,000. 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. Suggested Solution: (Likely) false. Just looking at healthcare, $2.25 million in costs, $2.5 million in revenues. It seems like profits of $250k. However, assuming there is just one market for computer programmers (as opposed to different markets for programmers who can develop for financial services and those who can develop for healthcare), your firm faces an upward-sloping labour supply curve. If you have to pay the same wage for every programmer you employ, you need to give your 20 current programmers a $25,000 per-year raise, which increases your costs by $500k. This turns your $250k gain into a $250k loss. However, this is assuming that you have to pay the same wage for every programmer you employ. This might not be the case if there are different markets for programmers who can develop for financial services and those who can develop for healthcare. Alternatively, even if there is one market for programmers and you have monopsony power, in theory you could pay 20231101: Page 14 Single-Price Monopolist: Problems: Solutions

University of Toronto Department of Economics ECO101: Principles of Microeconomics Robert Gazzale, PhD some programmers more than others. Good luck with paying your most recent hires more than your longtime employees! 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? Suggested Solution: Maximizing personal-training profits when she incorrectly uses $0 as her marginal cost. P ( Q ) = 120 - 2 Q MR ( Q ) = 120 - 4 Q 120 - 4 Q m | {z } MR = 0 |{z} MC Q m = 30 P m = P ( Q m ) = 120 - 2 × 30 P m = 60 Each week, her personal training puts 30 × $60 = $1800 into her bank account. Working as a personal trainer for 30 hours per week, she has 10 hours a week for programming. Programming puts 10 × $40 = $400. She is putting $2200 into her bank account each week, which means she is making $2100 each week after paying for her studio. (b) Assume that Tigist correctly calculates her optimal hourly rate for personal training. How much money does she make each week from all sources? Suggested Solution: Maximizing personal-training profits when she correct uses $40 as her marginal cost. 120 - 4 Q m | {z } MR = 40 |{z} MC Q m = 20 P m = P ( Q m ) = 120 - 2 × 20 P m = 80 Each week, her personal training puts 20 × $80 = $1600 into her bank account. Working as a personal trainer for 20 hours per week, she has 20 hours a week for programming. Programming puts 20 × $40 = $800. She is putting $2400 into her bank account each week, which means she is making $2300 each week after paying for her studio. (c) What is the most that she is willing to pay each week to rent a studio for her personal- training side hustle? Suggested Solution: If she just works as a computer programmer, she puts 40 × $40 = $1600 into her bank account each week. If she works the optimal number of hours as a personal trainer, she puts $2400 into her bank account each week. As long as her studio is $800 or less, the net amount of money from 20 hours a week at both programming and training is at least as large as 40 hours per week as a programmer. 20231101: Page 15 Single-Price Monopolist: Problems: Solutions

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