Managerial Economics 7th edition by William F. Samuelson, Stephen G. Marks solutions manual and test bank
CHAPTER TWO
OPTIMAL DECISIONS USING MARGINAL ANALYSIS
OBJECTIVES
1. To introduce the basic economic model of the firm
- The main focus is on determining the firm’s profit-maximizing level of output.
- The main assumption is that there is a single product (or multiple, independent products) with deterministic demand and cost.
2. To depict the behavior of price, revenue, cost, and profit as output varies.
3. To explain the notion of marginal profit (including its relationship to calculus) and show that maximum profit occurs at an output such that marginal profit equal zero
4. To reinterpret the optimality condition in terms of the basic components, marginal revenue and marginal cost.
5. To illustrate the uses of sensitivity analysis
TEACHING SUGGESTIONS
I. Introduction and Motivation
A. This is a “nuts and bolts” chapter. Because it appears up front in the text, it’s important to explain the motivation and assumptions. It is a good idea to remind students of the following points.
1) The model of the firm is deliberately simplified so that its logic is laid bare. Many additional complications will be supplied in later chapters. The key simplifications for now are:
• The model is of a generic firm. Although microchips are chosen to make the discussion concrete, there is no description of the kind of market or the nature of competition within it. The description and analysis of different market structures comes in Chapters 7 through 10.
• Profit is the sole goal of the firm; price and output are the sole decision variables.
• The description of demand and cost is as “bare bones” as it gets. The demand curve and cost function are taken as given. (How the firm might estimate these are studied in Chapters 4 through 6.)
B. In general, our policy is to use extended decision examples, different than the ones in the text, to illustrate the most important concepts. (Going over the same examples pushes the boredom envelope.) In the present chapter, we make an exception to this rule. It is important to make sure that students with different economic and quantitative backgrounds all get off roughly on the same foot. Reviewing a familiar example (microchips) makes this much easier.
II. Teaching the “Nuts and Bolts”
A. Graphic Overview. The text presents the revenue, cost, and profit functions in three equivalent forms: in tables, in graphs, and in equations. In our view, the best way to convey the logic of the relationships is via graphs. (The student who craves actual numbers can get plenty of them in the text tables.) Here is one strategy for teaching the nuts and bolts:
1. Using the microchip example, depict the demand curve, briefly note its properties and demand equation (in both forms).
2. Next focus on revenue, noting the tradeoff between price and quantity. Present and justify the revenue equation. Graph it and note its properties.
3. Repeat the same process with the cost function (reminding students about fixed versus variable cost). At this point, your blackboard graph should be a copy of Figure 2.8 (p. 46). Steps 1-3 should take no more than 20 minutes.
4. Since the gap between the revenue and cost curves measures profit, one could find the optimal output by carefully measuring the maximum gap (perhaps using calipers). Emphasize that marginal analysis provides a much easier and more insightful approach. Point out the economic meaning of marginal cost and marginal revenue. Note that they are the slopes of the respective curves.
5. Next argue (as on p. 47) that the profit gap increases (with additional output) when MR > MC but narrows when MR < MC. (On the graph, select quantities that are too great or too small to make the point.) Identify Q* where the tangent to the revenue curve is parallel to the slope of the cost function. In short, optimal output occurs where MR = MC.
B. Other Topics. The approach in part A provides a simple way of conveying the basic logic of marginal analysis using the components of MR and MC. Once this ground is covered, the instructor should emphasize other basic points:
1. The equivalence between MÏ€ = 0 and MR = MC.
2. Calculus derivations of MÏ€, MR, and MC.
3. The exact numerical solution for the microchip example.
4. The graphs of MR and MC and an exploration of comparative statics effects (shifts in the curves) and the effects on Q*.
C. Applications. Besides the applications in the text (pp. 45-49), the following problems are recommended: Problem 1 (a quick but important check), Problems 6, 7 and 9 (numerical applications), Problem 13 (the general solution), and Problem 14. (If the class has a good grasp of this last problem, nothing else will seem difficult.) The following question gets students thinking:
1. a. For five years, an oil drilling company has profitably operated in the state of Alaska (the only place it operates). Last year, the state legislature instituted a flat annual tax of $100,000 on any company extracting oil (or natural gas) in Alaska. How would this tax affect the amount of oil the company extracts? Explain.
b. Suppose instead that the state imposes a well-head tax, let’s say a tax of $10.00 on each barrel of oil extracted. Answer the questions of part a.
c. Finally, suppose that the state levies a proportional income tax (say 10% of net income). Answer the questions of part a. What would be the effect of a progressive tax?
d. Now suppose that the company has a limited number of drilling rigs extracting oil at Alaskan sites and at other sites in the United States. What would be the effect on the company’s oil output in Alaska if the state levied a proportional income tax as in part c?
Answer.
a. This tax acts as a fixed cost. As long as it remains profitable to produce in Alaska, the tax has no effect on the firm’s optimal output.
b. The well-head tax increases the marginal cost of extraction by $10.00 per barrel. The upward shift in MC means the new intersection of MR and MC occurs at a lower optimal level of output.
c. The income tax (either proportional or progressive) has no effect on the company’s optimal output. For instance, suppose that the company’s after-tax income is p = .9(R-C) under a 10% proportional tax. To maximize its after-tax income, the best the company can do is to continue to maximize its before-tax income. Another way of seeing this is to note that the tax causes a 10% downward shift in the firm’s MR and MC curves. With the matching shift, the new intersection of MR and MC is at the same optimal quantity as the old intersection.
d. When the firm operates in multiple states with limited drilling rigs, using a rig in Alaska means less oil is pumped (and lower profit is earned) somewhere else. There is an opportunity cost to Alaskan drilling. Thus, one can argue that before the tax, the company should have allocated rigs so as to equate marginal profits in the different states. With the tax, the marginal profit in Alaska is reduced, prompting the possible switch of rigs from Alaska to other (higher marginal profit) locations.
D. Mini-case: Apple Computer in the Mid-1990s
The mini-case reproduced on the next page provides a hands-on application of profit maximization and marginal analysis.
Answer
a. Clearly, the period 1994-1995 was marked by a significant adverse shift in demand against Apple due to major enhancements of competing computers: lower prices, better interfaces (Windows), sales to order (Dell), and more abundant software.
b. Setting MR = MC implies 4,500 - .3Q = 1,500, so Q* = 10,000 units and P = $3,000. Given 1994’s state of demand, Apple’s 1994 production strategy was indeed optimal.
c. In 1995, demand and MR have declined significantly. Now, setting MR = MC implies 3,900 - .3Q = 1,350, so Q* = 8,500 units and P = $2,625. Apple should cut its price and its planned output.
Apple Computer in the Mid 90s
Between 1991 and 1994, Apple Computer engaged in a holding action in the desktop market dominated by PCs using Intel chips and running Microsoft’s operating system.^{1}
In 1994, Apple’s flagship model, the Power Mac, sold roughly 10,000 units per month at an average price of $3,000 per unit. At the time, Apple claimed about a 9% market share of the desktop market (down from greater than 15% in the 1980s).
By the end of 1995, Apple had witnessed a dramatic shift in the competitive environment. In the preceding 18 months, Intel had cut the prices of its top-performing Pentium chip by some 40%. Consequently, Apple’s two largest competitors, Compaq and IBM, reduced average PC prices by 15%. Mail-order retailer Dell continued to gain market share via aggressive pricing. At the same time, Microsoft introduced Windows 95, finally offering the PC world the look and feel of the Mac interface. Many software developers began producing applications only for the Windows operating system or delaying development of Macintosh applications until months after Windows versions had been shipped. Overall, fewer users were switching from PCs to Macs.
Apple’s top managers grappled with the appropriate pricing response to these competitive events. Driven by the speedy new PowerPC chip, the Power Mac offered capabilities and a user-interface that compared favorably to those of PCs. Analysts expected that Apple could stay competitive by matching its rivals’ price cuts. However, John Sculley, Apple’s CEO, was adamant about retaining a 50% gross profit margin and maintaining premium prices. He was confident that Apple would remain strong in key market segments – the home PC market, the education market, and desktop publishing.
Questions.
1. What effect (if any) did the events of 1995 have on the demand curve for Power Macs?
Should Apple preserve its profit margins or instead cut prices?
2. a) In 1994, the marginal cost of producing the Power Mac was about $1,500 per unit, and a rough estimate of the monthly demand curve was: P = 4,500 - .15Q. At the time, what was Apple’s optimal output and pricing policy?
b) By the end of 1995, some analysts estimated that the Power Mac’s user value (relative to rival PCs) had fallen by as much as $600 per unit. What does this mean for Apple’s new demand curve at end-of-year 1995? How much would sales fall if Apple held to its 1994 price? Assuming a marginal cost reduction to $1,350 per unit, what output and price policy should Apple now adopt?
^{1} This account is based on J. Carlton, “Apple’s Choice: Preserve Profits or Cut Prices,” The Wall Street Journal, February 22, 1996, p. B1.
ADDITIONAL MATERIALS
I. Readings
R. Gibson, “Franchisee v. Franchiser,” The Wall Street Journal, February 14, 2011, p. R3.
R. Gibson, “Burger King Franchisees Can’t Have it Their Way,” The Wall Street Journal, January 21, 2010, p. B1.
M. Cieply, “For Movie Stars, the Big Money is now Deferred,” The New York Times, March 4, 2010, pp. A1, A3.
N. S. Riley, “Other People’s Money,” The Wall Street Journal, October 3, 2008, p. W11.
R. Chittum, “Price Points,” The Wall Street Journal, October 30, 2006, p. R7. (How providers of consumer services compare extra revenues and extra costs.)
T. H. Davenport, “Competing on Analytics,” Harvard Business Review, January 2006.
C. Oggier and E, Fragniere, and J. Stuby, “Nestle Improves its Financial Reporting with Management Science,” Interfaces, July-August, 2005, pp. 271-280.
J. Thomas, W. Reinartz, and V. Kumar, “Getting the Most out of your Customers,” Harvard Business Review, July-August, 2004, pp. 117-123.
A. M. Geoffrion and R. Krishnan, “Prospects for Operations Research in the E-Business Era,” Interfaces, March-April, 2001, pp. 6-36.
D. Ekwurzel and J. McMillan, “Economics Online,” Journal of Economic Literature, March 2001, pp. 7-10.
“Economics on the Net,” The Economist, March 13, 1999, p. 7.
W. Biddle, “Skeleton Alleged in the Stealth Bomber’s Closet,” Science, May 12, 1989.
II. Case
Colgate-Palmolive Co.: The Precision Toothbrush (9-593-064), Harvard Business School, 1993. Teaching Note (5-595-025). (Explores profit analyses of alternative launch strategies.)
III. Quips and Quotes
Small mistakes are the stepping stones to large failures.
There was an old saying about our small town. Our town’s population never changed. Every time a baby was born a man left town. (Does this say something about the balance of marginal changes at an optimum?)
The head of a small commuter plane service reported that as costs rose, the company’s breakeven point rose from 6 to 8 to 11 passengers. “I finally figured we were in trouble since our planes only have 9 seats.”
Quotes about economics in general:
If you laid all of the economists in the world end to end, they still wouldn’t reach a conclusion. (George Bernard Shaw)
An economist is a person who is very good with numbers but who lacks the personality to be an accountant.
The age of chivalry is gone; that of sophisters, economists, and calculators has succeeded. (Edmund Burke)
Please find me a one-armed economist so we will not always hear, “On the other hand . . .” (Herbert Hoover)
Answers to End-of-Chapter Problems
1. This statement confuses the use of average values and marginal values. The proper statement is that output should be expanded so long as marginal revenue exceeds marginal cost. Clearly, average revenue is not the same as marginal revenue, nor is average cost identical to marginal cost. Indeed, if management followed the average-revenue/average-cost rule, it would expand output to the point where AR = AC, in which case it is making zero profit per unit and, therefore, zero total profit!
2. The revenue function is R = 170Q - 20Q^{2}. Maximizing revenue means setting marginal revenue equal to zero. Marginal revenue is: MR = dR/dQ = 170 - 40Q. Setting 170 - 40Q = 0 implies Q = 4.25 lots. By contrast, profit is maximized by expanding output only to Q = 3.3 lots. Although the firm can increase its revenue by expanding output from 3.3 to 4.5 lots, it sacrifices profit by doing so (since the extra revenue gained falls short of the extra cost incurred.)
3. In planning for a smaller enrollment, the college would look to answer many of the following questions: How large is the expected decline in enrollment? (Can marketing measures be taken to counteract the drop?) How does this decline translate into lower tuition revenue (and perhaps lower alumni donations)? How should the university plan its downsizing? Via cuts in faculty and administration? Reduced spending on buildings, labs, and books? Less scholarship aid? How great would be the resulting cost savings? Can the university become smaller (as it must) without compromising academic excellence?
4. a. p = PQ – C = (120 - .5Q)Q - (420 + 60Q + Q2) = -420 + 60Q - 1.5Q2. Therefore, Mp = dp /dQ = 60 - 3Q = 0. Solving yields Q* = 20.
Alternatively, R = PQ = (120 - .5Q)Q = 120Q - .5Q2. Therefore, MR = 120 – Q. In turn, C = -420 + 60Q + Q2, implying: MC = 60 + 2Q. Equating marginal revenue and marginal cost yields: 120 - Q = 60 + 2Q, or Q* = 20.
b. Here, R = 120Q; it follows that MR = 120. Equating MR and MC yields: 120 = 60 + 2Q, or Q* = 30.
5. a. The firm exactly breaks even at the quantity Q such that p = 120Q - [420 + 60Q] = 0. Solving for Q, we find 60Q = 420 or Q = 7 units.
b. In the general case, we set: p = PQ - [F + cQ] = 0. Solving for Q, we have: (P - c)Q = F or Q = F/(P - c). This formula makes intuitive sense. The firm earns a margin (or contribution) of (P - c) on each unit sold. Dividing this margin into the fixed cost reveals the number of units needed to exactly cover the firm’s total fixed costs.
c. Here, MR = 120 and MC = dC/dQ = 60. Because MR and MC are both constant and distinct, it is impossible to equate them. The modified rule is to expand output as far as possible (up to capacity), because MR > MC.
6. a. If DVDs are given away (P = $0), demand is predicted to be: Q = 1600 - (200)(0) = 1,600 units. At this output, firm A’s cost is: 1,200 + (2)(1,600) =$4,400, and firm B’s cost is: (4)(1,600) = $6,400. Firm A is the cheaper option and should be chosen. (In fact, firm A is cheaper as long as Q > 600.)
b. To maximize profit, we simply set MR = MC for each supplier and compare the maximum profit attainable from each. We know that MR = 8 - Q/100 and the marginal costs are MCA = 2 and MCB = 4. Thus, for firm A, we find: 8 - QA/100 = 2, and so QA = 600 and PA = $5 (from the price equation). For firm B, we find QB = 400 and PB = $6. With Firm A, the station’s profit is: 3,000 - [1,200 + (2)(600)] = $600. With Firm B, its profit is 2,400 - 1,600 = $800. Thus, an order of 400 DVDs from firm B (priced at $6 each) is optimal.
7. a. The marginal cost per book is MC = 40 + 10 = $50. (The marketing costs are fixed, so the $10 figure mentioned is an average fixed cost per book.) Setting MR = MC, we find MR = 150 – 2Q = 50, implying Q* = 50 thousand books. In turn, P* = 150 –50 = $100 per book.
b. When the rival publisher raises its price dramatically, the firm’s demand curve shifts upward and to the right. The new intersection of MR and MC now occurs at a greater output. Thus, it is incorrect to try to maintain sales via a full $15 price hike. For instance, in the case of a parallel upward shift, P = 165 – Q. Setting MR = MC, we find: MR = 165 – 2Q = 50, implying Q* = 57.5 thousand books, and in turn, P* = 165 – 57.5 = $107.50 per book. Here, OS should increase its price by only $7.50 (not $15).
c. By using an outside printer, OS is saving on fixed costs but is incurring a higher marginal cost (i.e., printing cost) per book. With a higher marginal cost, the intersection of MR and MC occurs at a lower optimal quantity. OS should reduce its targeted sales quantity of the text and raise the price it charges per book. Presumably, the fixed cost savings outweighs the variable cost increase.
8. The fall in revenue from waiting each additional month is: MR = dR/dt = -8. The reduction in cost of a month’s delay is: MC = dC/dt = -20 + .5t. The optimal introduction date is found by equating MR and MC: -8 = -20 + .5t, which implies: .5t = 12 or t* = 24 months. The marketing manager’s 12-month target is too early. Delaying 12 more months sacrifices revenue but more than compensates in reduced costs.
9. a. The MC per passenger is $20. Setting MR = MC, we find 120 - .2Q = 20, so Q = 500 passengers (carried by 5 planes). The fare is $70 and the airline’s weekly profit is: $35,000 - 10,000 = $25,000.
b. If it carries the freight, the airline can fly only 4 passenger flights, or 400 passengers. At this lower volume of traffic, it can raise its ticket price to P = $80. Its total revenue is (80)(400) + 4,000 = $36,000. Since this is greater than its previous revenue ($35,000) and its costs are the same, the airline should sign the freight agreement.
10. The latter view is correct. The additional post-sale revenues increase MR, effectively shifting the MR curve up and to the right. The new intersection of MR and MC occurs at a higher output, which, in turn, implies a cut in price. (Of course, one must discount the additional profit from service and supplies to take into account the time value of money.)
11. p = -423 + 10.4P - .05P^{2} implies Mp = 10.4 - .1P. Setting Mp = 0, we obtain: 10.4 - .1P = 0, or P = $104 thousand. This is exactly the optimal price found earlier.
12. a. First note that if marginal cost and marginal benefit to consumers both increased by $25, the optimal output would not change since MR(Q*) = MC(Q*) implies that MR(Q*) + 25 = MC(Q*) + 25. The price would rise by $25 but, since marginal costs rise by $25, the firm’s total profits would remain the same. If marginal costs increased by more than $25, profits would fall. Thus the firm should not redesign when the increase in MC is $30.
b. If MC increases by $15 and MR increases by $25, the new intersection of the MR and MC occurs at a greater output. Output, price, and profit would all rise. Price, however, would rise by less than $25.
13. Setting MR = MC, one has: a – 2bQ = c, so that Q = (a - c)/2b. We substitute this expression into the price equation to obtain:
P = a - b[(a - c)/2b] = a - (a - c)/2 = a/2 + c/2 = (a + c)/2.
The firm’s optimal quantity increases after a favorable shift in demand - either an increase in the intercept (a) or a fall in the slope (b). But quantity decreases if it becomes more costly to produce extra units, that is., if the marginal cost (c) increases. Price is raised after a favorable demand shift (an increase in a) or after an increase in marginal cost (c). Note that only $.50 of each dollar of cost increase is passed on to the consumer in the form of a higher price.
*14. The Burger Queen (BQ) facts are P = 3 - Q/800 and MC = $.80.
a. Set MR = 0 to find BQ’s revenue-maximizing Q and P. Thus, we have 3 - Q/400 = 0, so Q = 1,200 and P = $1.50. Total revenue is $1,800 and BQ’s share is 20% or $360. The franchise owner’s revenue is $1,440, its costs are (.8)(1,200) = $960, so its profit is $480.
b. The franchise owner maximizes its profit by setting MR = MC. Note that the relevant MR is (.8)(3 - Q/400) = 2.4 - Q/500. After setting MR = .80, we find Q = 800. In turn, P = $2.00 and the parties’ total profit is (2.00 - .80)(800) = $960, which is considerably larger than $840, the total profit in part (a).
c. Regardless of the exact split, both parties have an interest in maximizing total profit, and this is done by setting (full) MR equal to MC. Thus, we have 3 - Q/400 = .80, so that Q = 880. In turn, P = $1.90, and total profit is: (1.90 - .80)(880) = $968.
d. The chief disadvantage of profit sharing is that it is difficult, time-consuming, and expensive for the parent company to monitor the reported profits of the numerous franchises. Revenue is relatively easy to check (from the cash register receipts) but costs are another matter. Individual franchisees have an incentive to exaggerate the costs they report in order to lower the measured profits from which the parent’s split is determined. The difficulty in monitoring cost and profit is the main strike against profit sharing.
15. a. The profit function is p = -10 - 48Q + 15Q^{2} - Q^{3}. At outputs of 0, 2, 8, and 14, the respective profits are -10, -54, 54, and -486.
b. Marginal profit is Mp = dp/dQ = -48 + 30Q - 3Q^{2} = -3(Q - 2)(Q - 8), after factoring. Thus, marginal profit is zero at Q = 2 and Q = 8. From part a, we see that profit achieves a local minimum at Q = 2 and a maximum at Q = 8.
Discussion Question
Suppose the firm considers expanding its direct sales force from 20 to, say 23 sales people. Clearly, the firm should be able to estimate the marginal cost of the typical additional sales person (wages plus fringe benefits plus support costs including company vehicle). The additional net profit generated by an additional sales person is a little more difficult to predict. An estimate might be based on the average profitability of its current sales force. A more detailed estimate might judge how many new client contacts a salesperson makes, historically what fraction of these contacts result in new business, what is the average profit of these new accounts, and so on. If the marginal profit of a sales person is estimated to be between $100,000 and $120,000 while the marginal cost is $85,000, then the firm has a clear-cut course of action, namely hire the additional 1, 2, or 3 employees.
Spreadsheet Problems
S1. a and b. Setting MR = MC implies: 800 – 4Q = 200 + Q. Therefore, Q* = 120 parts and P* = $560.
c. To confirm these values on a spreadsheet, we maximize cell F7 by changing cell B7. Maximum profit in cell F7 is $16,000.
S2. a. Given p = 20[A/(A+8)] –A, it follows that Mp = 20[8/(A+8)2] – 1. Setting Mp = 0 implies (A+8)2 = 160, or A* = $4.649 million.
b. Confirm this value on your spreadsheet by maximizing cell F7 by changing cell C7. Maximum profit in F7 is $2.702 million.
S3. a. To confirm these values on our spreadsheet, we maximize cell F7 by changing cell B7. The optimal sales volume is: Q* = 2.4 million units and the optimal price is P* = $210. Amazon’s margin on each reader is: 210 – 126 = $84, and its maximum profit (or, more precisely contribution) is $201.6 million.
b. We extend the spreadsheet by including contribution from sales of e-books ($100 per kindle sold) in cell G7 and add this to Kindle profit to compute total profit in cell H7. Maximizing total profit, we find the new optimal solution to be: Q* = 3.829 million units and P* = $160. (This price is close to current price levels for the Kindle.)
By lowering price, Amazon increases its Kindle sales. The increased profit from e-books more than makes up for reduced Kindle profit. Note that e-book profit is almost three times Kindle profit. Amazon’s total profit comes to some $513.0 million.
Appendix Problems
1. When tax rates become very high, individuals will make great efforts to shield their income from taxes. Furthermore, higher taxes will discourage the taxed activities altogether. (In the extreme case of a 100% tax, there is no point in undertaking income-generating activities.) Thus, a higher tax rate means a smaller tax base. Increasing the tax rate from zero, the revenue curve first increases, eventually peaks, and then falls to zero (at a 100% tax). Thus, the curve is shaped like an upside-down U.
2. a. B(t) = 80 - 100t. Therefore, R = 80t - 100t2. Setting MR = dR/dt = 0, we find: 80 - 200t = 0, or t = .4.
b. B(t) = 80 - 240t2. Therefore, R = 80t - 240t3. Setting MR = dR/dt = 0, we find: 80 - 720t2 = 0. Therefore, t2 = 1/9, or t = 1/3.
c. B(t) = 80 - 80t.5. Therefore, R = 80t - 80t1.5. Setting MR = dR/dt = 0, we find: 80 - 120t.5 = 0. Thus, t.5 = 2/3, or t = 4/9.
3. a. p = 20x - x2 + 16y - 2y2. Setting dp/dx = 0 and dp/dy = 0 implies x = 10 and y = 4.
b. The Lagrangian is L = p + z(8 - x - y). Therefore, the optimality conditions are: 20 - 2x - z = 0, 16 - 4y - z = 0, and x + y = 8. The solution is x = 6, y = 2, and z = 8.
c. The Lagrangian is L = p + z(7.5 - x - .5y). Therefore, the optimality conditions are: 20 - 2x - z = 0, 16 - 4y - .5z = 0, and x + .5y = 7.5. The solution is x = 6, y = 3, and z = 8.
File: Ch02; CHAPTER 2: Optimal Decisions Using Marginal Analysis
MULTIPLE CHOICE
1. According to the model of the firm, the management’s main goal is to:
a) increase revenue from sales.
b) maximize profit.
c) maximize its market share.
d) minimize its variable cost per unit.
e) maintain a steady and predictable growth in earnings.
ANSWER: b
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Easy
PAGE: 30
2. According to the law of demand, if a firm reduces the price of its good:
a) consumers in the market will demand more units of the good.
b) some consumers will exit the market.
c) consumers will demand fewer units than before the price cut.
d) the quantity of goods produced and sold by the firm will decline.
e) competing firms will increase prices.
ANSWER: a
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Easy
PAGE: 31
3. Which of the following is true of a firm that faces a downward sloping demand curve?
a) In order to sell more units, the firm needs to lower its price.
b) The total cost curve for the firm is also downward sloping.
c) The firm's total revenue and price are directly correlated.
d) The marginal revenue from each unit sold is constant.
e) The firm faces a constant marginal cost curve.
ANSWER: a
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Easy
PAGE: 31-32
4. The demand for a product is given by Q = 600 – 30P, where P = price and Q = quantity. At P = $15, the firm sells _____ units.
a) 100
b) 150
c) 300
d) 450
e) 600
ANSWER: b
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Medium
PAGE: 33
5. The demand for a product is given by P = 1,750 – 25Q, where P = price and Q = quantity. If the firm wishes to sell 50 units, each unit should be priced at _____.
a) $500
b) $400
c) $300
d) $200
e) $100
ANSWER: a
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Medium
PAGE: 33
6. A firm’s demand curve is given by Q = 800 – 2P, where P = price and Q = quantity. Therefore, its inverse demand equation is _____.
a) MR = 800 – 4P
b) P = 800 – 2Q
c) P = 400 – 0.5Q
d) P = 800 – 0.5Q
e) 800 = Q + 2P
ANSWER: c
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Medium
PAGE: 33-34
7. Suppose a firm's inverse demand function is P = 40 – 8Q. What is the firm's revenue function?
a) R = 40Q – 8Q^{2}
b) R = 40 – 16Q
c) R = –8Q
d) R = 40/Q – Q
e) R = 5 – Q
ANSWER: a
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Medium
PAGE: 35
The following table shows the total revenue (in dollars) and total cost (in dollars) from the production and sale of different units of a product.
Table 2-1
8. Refer to Table 2-1. What is the firm’s profit from the sale of the 3rd unit of the good?
a) $13
b) $11
c) $12
d) $39
e) $27
ANSWER: e
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Medium
PAGE: 37-38
9. Refer to Table 2-1. What is the marginal profit of the firm from the sale of the 2nd unit of the good?
a) $9
b) $3
c) $1
d) $5
e) $21
ANSWER: a
SECTION REFERENCE: Marginal Analysis
DIFFICULTY LEVEL: Medium
PAGE: 39
10. Suppose, at its current output level, a firm’s marginal profit is positive. Therefore, to maximize profit, it should:
a) decrease output until marginal profit is zero.
b) increase output because marginal revenue [MR] is less than marginal cost [MC].
c) increase both its output and its price.
d) increase output because MR is greater than MC.
e) increase output until it is producing at full capacity.
ANSWER: d
SECTION REFERENCE: Marginal Analysis
DIFFICULTY LEVEL: Medium
PAGE: 40
11. Suppose a firm’s profit is given by the equation p = –200 + 80Q – 0.2Q^{2}, where p = profit and Q = quantity. Which of the following is true?
a) The firm’s marginal profit [Mp] is given by the equation: Mp = 80 – 0.2Q.
b) The firm’s profit-maximizing output is Q = 400.
c) The firm’s profit-maximizing output is Q = 200.
d) The firm’s marginal profit [Mp] is given by the equation: Mp = 80 – 2Q.
e) The firm’s profit-maximizing output is Q = 800.
ANSWER: c
SECTION REFERENCE: Marginal Analysis
DIFFICULTY LEVEL: Medium
PAGE: 41-42
12. If a firm’s profit is given by p = 36Q^{2} – 360Q – 150, where p = profit and Q = quantity produced, then its optimal output is _____ units.
a) 12
b) 5
c) 2
d) 20
e) 36
ANSWER: b
SECTION REFERENCE: Marginal Analysis
DIFFICULTY LEVEL: Hard
PAGE: 41-42
13. What is the marginal revenue [MR] equation for a firm with the demand function P = a – bQ, where P = price and Q = quantity?
a) MR = b – Q
b) MR = a – 2bQ
c) MR = a + 2Q
d) MR = 2Q
e) MR = 2a + Q
ANSWER: b
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Medium
PAGE: 44
14. A firm’s total revenue function is given by R = 100 + 10Q + 2Q^{2}, where R = revenue and Q = quantity. Which of the following is true if Q = 10?
a) The firm’s total revenue is $400 and the marginal revenue is $10.
b) The firm’s marginal revenue is constant at $40.
c) The average revenue of the firm is $50.
d) The total revenue of the firm is $500.
e) The marginal revenue of the firm is $50.
ANSWER: e
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Medium
PAGE: 44
15. Which of the following correctly defines marginal revenue?
a) Marginal revenue is the price at which the firm sells the last unit of the good.
b) Marginal revenue is the change in revenue from a unit increase in the price of the good.
c) Marginal revenue is the additional revenue from a unit increase in output and sales.
d) Marginal revenue is the additional revenue earned from an increase in demand for the good.
e) Marginal revenue is the difference between price and marginal cost for the last unit sold.
ANSWER: c
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Easy
PAGE: 44
16. For a downward-sloping demand curve, the associated marginal revenue curve:
a) coincides with the demand curve.
b) lies below and is parallel to the demand curve.
c) has twice the slope as the demand curve.
d) is positive for all levels of sales.
e) is parallel to the quantity axis.
ANSWER: c
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Easy
PAGE: 44
The following table shows the total revenue (in dollars) and total cost (in dollars) from the production and sale of different units of a product.
Table 2-1
17. Refer to Table 2-1. What is the marginal revenue of the firm associated with the sale of the 5th unit of the good?
a) $55
b) $8
c) $7
d) $48
e) $4
ANSWER: c
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Medium
PAGE: 44
18. Refer to Table 2-1. What is the profit-maximizing level of output for the firm?
a) 3 units
b) 2 units
c) 1 unit
d) 5 units
e) 4 units
ANSWER: d
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Medium
PAGE: 45
19. Given that a firm's inverse demand function is P = 100 – 5Q and total cost is given by C = 550 + 10Q, what is the firm's profit-maximizing level of output?
a) 10 units
b) 15 units
c) 9 units
d) 8 units
e) 5 units
ANSWER: c
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Medium
PAGE: 45
20. Which of the following correctly defines marginal cost?
a) Marginal cost is the addition made to fixed cost when an extra unit is produced.
b) Marginal cost is the additional cost of producing an extra unit of output.
c) Marginal cost is the additional cost of increasing the scale of production in the long run.
d) Marginal cost is the difference between price and marginal revenue for the last unit sold.
e) Marginal cost is the same as the firm’s variable cost at all levels of output.
ANSWER: b
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Easy
PAGE: 45
21. Given the total cost equation for a firm, the marginal cost equation can be derived by:
a) dividing total cost by total output.
b) taking the first derivative of the cost function with respect to quantity.
c) dividing total variable cost by total output.
d) subtracting variable cost from the fixed cost at all levels of output.
e) multiplying the total cost equation by price.
ANSWER: b
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Easy
PAGE: 45
22. To maximize profit, the firm should set output at the level where:
a) the average cost per unit is minimized.
b) average revenue just equals average cost.
c) marginal cost equals zero.
d) marginal revenue is equal to marginal cost.
e) marginal revenue equals zero.
ANSWER: d
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Easy
PAGE: 45
23. Assume that a firm is producing at its profit-maximizing level of output. A decrease in the price of raw materials used in production is most likely to lead to:
a) an increase in quantity produced at an unchanged price.
b) a fall in the price of the good and an increase in the quantity produced.
c) a fall in both the price of the good and the quantity produced.
d) an increase in both the price of the good and the quantity produced.
e) a fall in the quantity produced of the good at an unchanged price.
ANSWER: b
SECTION REFERENCE: Sensitivity Analysis
DIFFICULTY LEVEL: Medium
PAGE: 50
24. A firm negotiates a new labor contract with a higher average hourly wage. What is the most likely effect of the higher wage on the firm's price and output?
a) Both price and output will not be affected.
b) Price will increase but output will not change.
c) Both price and output will increase.
d) Price will not change but output will decrease.
e) Price will increase but output will decrease.
ANSWER: e
SECTION REFERENCE: Sensitivity Analysis
DIFFICULTY LEVEL: Medium
PAGE: 50
25. Assume that a firm is producing at its profit-maximizing level of output. A decrease in fixed cost implies that:
a) marginal revenue will increase but marginal cost will decrease.
b) marginal revenue will not change but marginal cost will decrease.
c) neither average total cost nor marginal cost will change.
d) neither marginal revenue nor marginal cost will change.
e) both marginal revenue and marginal cost will decrease.
ANSWER: d
SECTION REFERENCE: Sensitivity Analysis
DIFFICULTY LEVEL: Medium
PAGE: 50
26. Due to an increase in the price of a competitor’s product, the demand for a firm’s product increases sharply. How is this most likely to affect the firm’s marginal revenue and marginal cost?
a) Marginal revenue will increase but marginal cost will decrease.
b) Both marginal revenue and marginal cost will not be affected.
c) Both marginal revenue and marginal cost will increase.
d) Marginal revenue will not change but marginal cost will increase.
e) Marginal revenue will increase but marginal cost will not change.
ANSWER: e
SECTION REFERENCE: Sensitivity Analysis
DIFFICULTY LEVEL: Medium
PAGE: 50-51
27. Assume that Burger King, a fast food chain, enters into a franchise agreement. The royalty paid to Burger King by the franchisee is calculated as a percentage of the franchisee’s revenue. Given that the franchisee faces a downward-sloping demand curve, which of the following is likely to be true?
a) The franchisee’s revenue-maximizing output will be greater than its profit-maximizing output.
b) To maximize revenue, Burger King will want the franchisee to produce at the level where total revenue is positive but falling.
c) The franchisee will produce at the level where the slope of the total revenue curve is zero in order to maximize profits.
d) The profit-maximizing level of output for the franchisee will be at the level where marginal revenue is lesser than marginal cost.
e) To maximize revenue, Burger King will want the franchisee to produce at the level where marginal revenue equals marginal cost.
ANSWER: a
SECTION REFERENCE: Sensitivity Analysis
DIFFICULTY LEVEL: Medium
PAGE: 53
SHORT ANSWERS
28. Are there any types of goods or situations where the law of demand does not hold? Explain.
ANSWER: The law of demand states that all other factors held constant, the higher the unit price of a good, the fewer the number of units demanded by consumers and, consequently, sold by firms. For certain goods, a high price is associated with a higher status or luxury, for example, a fancy wine or a designer bag. For such goods, a high price is seen as a sign of exclusivity, which means that the demand for these goods increases as price increases. These are called Veblen goods.
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Medium
PAGE: 31
29. What is the law of demand? How do managers use it in decision-making?
ANSWER: The law of demand states that all other factors held constant, the higher the unit price of a good, the fewer the number of units demanded by consumers and, consequently, sold by the firm. Managers use the demand curve as the basis for predicting the revenue consequences of alternative output and pricing policies.
SECTION REFERENCE: A Simple Model of the Firm
DIFFICULTY LEVEL: Easy
PAGE: 31-33
30. Carefully define marginal analysis, and explain how it is useful in managerial economics.
ANSWER: Marginal analysis is the process of considering small changes in a decision and determining whether such a change will improve the ultimate objective. The manager can follow a clear rule: Make a small move to a nearby alternative if and only if the move will improve one's objective. Keep moving until no further move will help.
SECTION REFERENCE: Marginal Analysis
DIFFICULTY LEVEL: Easy
PAGE: 38-40
31. Suppose that a firm operates in a competitive market where the commodity price is $15 per unit. The firm’s cost equation is C = 25 + 0.25Q^{2}, where C = total cost and Q = quantity.
(a) Find the profit-maximizing level of output for the firm. Determine its level of profit.
ANSWER: In a competitive market, revenue [R] = price × quantity = 15Q implying marginal revenue [MR] = ∂R/∂Q = $15. In turn, marginal cost [MC] = ∂C/∂Q = 0.5Q. Setting MR = MC, gives 15 = 0.5Q, or Q = 30 units. At Q = 30 units, R = $450, C =$250, and profit = $200.
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Medium
PAGE: 44-45
(b) Suppose that fixed costs increase to $75. Verify that this change in fixed costs does not affect the firm's optimal output.
ANSWER: The increase in fixed cost has no effect on MR or MC. MC = ∂C/∂Q = 0.5Q and MR = ∂R/∂Q = $15. Setting MR = MC, yields 15 = 0.5Q, or Q = 30 units. The firm's optimal level of output is unaffected. However, with the $50 rise in fixed cost, the firm's profit falls to $150.
SECTION REFERENCE: Sensitivity Analysis
DIFFICULTY LEVEL: Medium
PAGE: 50
32. The demand for a firm’s product is given by the equation: P = 36 – 0.2Q. The firm’s cost equation is given by C = 200 + 20Q.
(a) Determine the firm’s optimal quantity and price.
ANSWER: Marginal revenue [MR] = ∂R/∂Q = 36 – 0.4Q and marginal cost [MC] = ∂C/∂Q = $20. Setting MR = MC implies that the optimal output [Q*] = 40 units. From the price equation, it follows that the optimal price [P*] = 36 – (0.2)(40) = $28. Finally, profit is given by: p = $1,120 – 1,000 = $120.
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Medium
PAGE: 44-45
(b) Suppose that the firm’s costs change to C = 100 + 24Q. Determine the new optimal quantity and price. Explain why the results differ from the previous case.
ANSWER: With the new cost function, MC = $24. Setting MR = MC implies 36 – 0.4Q = 24, or Q* = 30 units. In turn, P* = 36 – (0.2)(30) = $30. Finally, profit is given by: p = $900 – $820 = $80. Here, the reduction in fixed cost has no impact on output, but the increase in marginal cost induces a smaller output quantity and a greater price.
SECTION REFERENCE: Sensitivity Analysis
DIFFICULTY LEVEL: Medium
PAGE: 50
33. A firm faces the demand curve, P = 80 – 3Q, and has the cost equation: C = 200 + 20Q, where P = price, C = total cost, and Q = quantity.
(a) Find the optimal quantity and price for the firm.
ANSWER: Profit is maximized by setting marginal revenue [MR] = marginal cost [MC]. From the price equation, MR = 80 – 6Q. Equating this with MC = $20 implies 80 – 6Q = 20, or the optimal output [Q*] = 10 units. In turn, the optimal price [P*] = 80 – (3)(10) = $50.
SECTION REFERENCE: Marginal Revenue and Marginal Cost
DIFFICULTY LEVEL: Medium
PAGE: 44-45
(b) Now suppose that the demand for the firm’s product changes to: P = 110 – 3Q. Find the new optimal quantity and price. Has there been an increase or a decrease in demand? Explain.
ANSWER: According to the new price equation, P = 110 – 3Q, MR = 110 – 6Q. Setting MR = MC implies 110 – 6Q = 20, or Q* = 15 units. In turn, P* = 110 – (3)(15) = $65. The increase in demand (in this case a parallel outward shift of the demand curve) has induced the firm to increase both its price and quantity.
SECTION REFERENCE: Sensitivity Analysis
DIFFICULTY LEVEL: Medium
PAGE: 50