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Unformatted text preview: University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 1 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements Note: Please don't memorize these solutions in the expectation that similar questions will appear on tests and exams. Instead, try to understand how to derive the answer as you'll be tested on techniques and applications, not on memorization. Moreover, tests and exams will cover topics and techniques that may not be in these practice problems. You are urged to go over all lectures, class notes and HWs thoroughly. As you may know from ECO 100, economics extensively uses marginal analysis (for example: "marginal revenue", "marginal cost" and "marginal utility"). This is because "thinking on the margin" is a powerful technique for solving problems, especially when there are no equations and graphs. Because we marginal analysis is used extensively in this course, you will practice "thinking on the margin" in the first three questions below. Question 1 In this question you will repeat the NPV example I gave in class. Ajax Private Equity (PAE) has unlimited funds to invest. Ajax's financial advisor, Madoff Advisors, has identified the following seven projects as investment opportunities: 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Project A B C D E F G Initial Investment $1m $0.4m $0.3m $0.1m $0.2m $0.2m $0.1m NPV $2m $1.4m $1.2m $0.6m $0.5m $0.3m $0.05m (a) Given that you have unlimited funds which project should you invest in? Answer: Since Ajax is lucky to have unlimited funds, not only can he afford the good things in life, he can and should invest in all seven projects since each of has a NPV > 0. (b) To his horror, Ajax discovers that he only has $1m. Unable to afford Madoff Advisors he instead engages an MBA from McGrill University in Montreal who tells Ajax that since he has $1m only he should invest in the feasible project (i.e. less than or equal to $1m outlay) with the highest NPV. Is he correct? Hint: look at the NPV per $ invested. Answer: No. As discussed in class and from ECO 100, when figuring where to allocate resources, you should think of the incremental gain not the total gain. To see this, note that if you invest in project A, with a $1m investment you will get a NPV of $2m. But you can do better than this by looking at the projects in terms of NPV per $ invested. 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Project A B C D E F G Initial Investment $1m $0.4m $0.3m $0.1m $0.2m $0.2m $0.1m Observe how project D has the highest NPV per dollar invested. Of the $1m, Ajax should allocate funds first to project D. Thereafter, with $0.9m left, he should invest in the project with the next highest NPV per dollar invested: project C. With investments in projects D and C he has $0.4m leftover. Using the logic of NPV per dollar Ajax should next invest in project B followed by project E. Together, projects D, C, B and E will require an investment of $1m and will yield a total NPV of $3.7m. Observe how thinking on the margin ("where should Ajax spend the next dollar?") is a superior analytical tool. Too bad they don't teach that at McGrill University. Question 2 (ECO 204 20082009 Test 4 question) In this question, you will again practice marginal analysis. AccountingMan (no relation to Eco man) owns an "accounting education in a hurry" business (recent customers include AIG, Bear Stearns, and Nortel). AccountingMan has a sales force of 16 people and is trying to decide how many salespeople to allocate to servicing existing large accounts and how many salespeople to allocate procuring new small accounts. AccountingMan contacts his cousin Economicsman who conducts brilliant market research. The results of the market research are reproduced in Table 1 below and shows the Marginal profits and total profits (in `000s of $) from the number of salespersons allocated to each type of account. (a) Fill in the total profits for the shaded cells in Table 1 below. NPV $2.0m $1.4m $1.2m $0.6m $0.5m $0.3m $0.05m NPV per $ Invested 2.0 3.5 4.0 6.0 2.5 1.5 0.5 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: For this question you have to remember what Marginal means. It literally means the additional profit from an extra salesperson. Given the Marginal 's I have shown the formula for in the shaded cells below. Hint: Don't forget the with 0 salespersons! I've given you the final answer you should get. Table 1 EconomicMan's Report to AccountingMan Large Accounts Small Accounts Number of Salespersons 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Marginal 80 80 80 80 80 70 70 70 40 40 40 40 40 40 40 40 0 0 + 5(80) = 400 400 + 3(70) = 610 930 Marginal 150 150 100 100 100 50 50 50 30 30 30 30 30 20 20 20 200 200 + 2(150) + 3(100) = 800 800 + 3(50) = 950 950 + 5(30) = 1,100 1,160 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (b) AccountingMan notes that with either 0 salespersons or 16 salespersons, the profits from large accounts are always greater than small accounts. He thinks all 16 salespersons should be allocated to large accounts. Is he right? Give a clear explanation. Answer: Note that: For the first 5 salespersons, the M for large accounts > M for small accounts. Hence, allocate these 5 salespersons to large accounts. Total salespersons used: 5 For the next 8 salespersons, the M for large accounts < M for small accounts. Hence, allocate these 8 salespersons to small accounts. Total salespersons used: 13 For the next 3 salespersons, the M for large accounts > M for small accounts. Hence, allocate these 3 salespersons to large accounts. Total salespersons used: 16 That is: 8 salespersons for large accounts and 8 salespersons to small accounts. I have explained the logic in the table below. 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Number of Salespersons 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Small Accounts Marginal 80 80 80 80 80 70 70 70 40 40 40 40 40 40 40 40 0 0 + 5(80) = 400 400 + 3(70) = 610 930 Marginal 150 150 100 100 100 50 50 50 30 30 30 30 30 20 20 20 Large Accounts 200 200 + 2(150) + 3(100) = 800 800 + 3(50) = 950 950 + 5(30) = 1,100 1,160 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Question 3 In this question you will practice marginal analysis for how to study for your exams. You have been procrastinating, watching TV, playing games and clubbing. One fine Monday, you wake up in the late afternoon of course and realize to your horror that you have an economics and an accounting test on Friday. Of course, you should spend every hour studying for these tests, but what is life without trips to Circa, Embassy, Wrong Bar and the Social? Given that you must club hop you calculate that you will have 5 hours to study for your tests. Your goal is to maximize the average grade across the two courses (which means you want to maximize the sum of the grades across the two courses). Your best guess for grades as a function of hours is given in Table 1 below: Study Hours 0 1 2 3 4 5 Economics Grade 70 78 83 88 90 92 Study Hours 0 1 2 3 4 5 Accounting Grade 75 81 85 87 89 90 How should you allocate 5 hours of studying between the two subjects? Answer: This is an allocation problem. Each of your five hours can be allocated to economics or accounting. You should allocate the hour to that subject in which the gain from studying is greater. Thus, you will calculate the marginal grade for each hour of studying economics and accounting: 7 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Study Hours 0 1 2 3 4 5 Economics Grade 70 78 83 88 90 92 Marginal Economics Grade Study Hours 0 Accounting Grade 75 81 85 87 89 90 Marginal Accounting Grade 8 5 5 2 2 1 2 3 4 5 6 4 2 2 1 Let's do the analysis hour by hour. 1st Hour: You can study economics and raise your total score by 8 while if that hour is spent studying accounting, the total score goes up by 6. Obviously you should study economics (shame on you for even thinking you should study accounting!). 2nd Hour: You can study economics and raise your total score by 5 but now if that hour is spent studying accounting (your first hour in fact studying accounting), the total score goes up by 6. Obviously you should study accounting. 3rd Hour: You can study economics and raise your total score by 5 while if that hour is spent studying accounting, the total score goes up by 4. Obviously you should study economics. 4th Hour: You can study economics and raise your total score by 5 while if that hour is spent studying accounting, the total score goes up by 4. Obviously you should study economics. 5th Hour: You can study economics and raise your total score by 2 while if that hour is spent studying accounting, the total score goes up by 4. Obviously you should study accounting. Here is the logic: 8 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Study Hours 0 1 2 3 4 5 Economics Grade 70 78 83 88 90 92 Marginal Economics Grade Study Hours 0 Accounting Grade 75 81 85 87 89 90 Marginal Accounting Grade 8 5 5 2 2 1 2 3 4 5 6 4 2 2 1 The breakdown of hours and increases in scores in each subject will be: Hour Study Economics Study Accounting 1 2 3 4 5 Yes (5) Yes (5) Yes (4) Yes (8) Yes (6) Your economics grade will be: 70 + 8 + 5 + 5 = 88 and your accounting score will be: 75 + 6 + 4 = 85. Note that if you hadn't studied you would've done better in accounting. But when you study you do better in economics. Now stop wasting time and do the next question! 9 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Question 4 (ECO 204 summer 2008 final exam question) In this question you will repeat and extend the elasticity example from lecture 1. Club Stereo currently charges a $10 cover charge (entrance fee). On average, customers have 1 drink (priced at $8/drink). The elasticity of demand with respect to cover charge price is 0.6. The club manager is debating whether to raise the cover charge from $10 to $11. (a) If the club seeks to maximize "cover" revenues, should it raise the cover charge from $10 to $11? If you say yes, indicate the percentage change in cover revenues. Answer: From ECO 100, you know that if E < 1, then raising price will raise revenues. Here, E = 0.6 so that raising the cover charge slightly see next question for why slightly matters will raise cover revenues. In fact, we can quantify the impact on cover revenues: Cover Revenues = Cover Charge*Club goers R = PQ From the Math Review, recall that: R = PQ log R = log (PQ) log R = log P + log Q Note: Now recall that if y = log x dy/dx = 1/x dy = dx/x Since y = log x we can express "dy" as "d log x" so that: d log x = dx/x Thus: d log R = d log P + d log Q 10 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain dR/R = dP/P + dQ/Q (dR/R)*100 = (dP/P)*100 + (dQ/Q)*100 % R = % P + % Q Now, cover charge has risen from $10 to $11 so that % P = 10%. What is % Q? Use the definition of Elasticity: E = % Q / % P = 0.6 % Q = 0.6 (% P) % Q = 0.6 (10%) % Q = 6% Thus: % R = % P + % Q becomes: % R = 10% 6% % R = 4% Thus, raising the cover charge will raise cover revenues by 4%. (b) If the club seeks to maximize total revenues, should it raise the cover charge from $10 to $11? If you say yes, indicate the percentage change in total revenues. Answer: From ECO 100, you know that if E < 1, then raising price will raise revenues. Here, E = 0.6 is the elasticity with respect to the cover charge not total charge; thus we cannot say right away whether cover charge should be raised from $10 to $11. What we can do is the following. Recognize: Total Revenues from cover charge and drinks = (Price of cover and drink)*(covers and drinks) R = PQ Now, because every clubber has one drink, the number of covers = number of drinks and so if 11 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain when the club raises the cover from $10 to $11, there will be 6% fewer cover and therefore 6% fewer drinks. Thus: % R = % P + % Q % R = % P 6% Now, cover charge has risen from $10 to $11 so that total price has risen from $10 + $8 = $18 to $11 + $8 = $19 or that % P = 5.6%. Thus % R = % P + % Q becomes: % R = 5.6% 6% % R = 0.4% Thus, raising the cover charge will lower total revenues by 0.4%. What's happening here is that raising the cover charge does lead to higher revenues at the door but the fewer clubbers results in fewer drinks sold, so that the lower revenues from drinks negates the higher revenues from the cover charge. Naturally, this leads to the next question. (c) What must the price of a drink be for the total revenues from cover charge and 1 drink to increase if the cover charge increases from $10 to $11? Show all steps and calculations. State any assumptions. Answer: The cover charge increased by {(11 10)/10}100 = 10%. From E = 0.6 this implies that there will be 0.6(10%) = 6% fewer club goers. The impact on total revenues is: R = PQ % R = %P + % Q % R = %P 6% For total revenues to increase (i.e. % R > 0) we require %P > 6%. Now, letting the price of the drink be $x: %P = 100{(11 + x) (10 + x)}/(10 + x) [I am using the "traditional" formula for percentage change; if you want, you can use the "arc" formula qualitatively, the answer is the same] 12 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain %P = 100/(10 + x) We want %P = 100/(10 + x) > 6 10 + x < 100/6 x < 100/6 10 x < $6.67 As long as drinks cost less than $6.67, raising the cover charge from $10 to $11 will raise total revenues. As a practice question, try repeating the question assuming clubbers consume 2 drinks on average. Question 5 (ECO 204 20072008 final exam question) In recent years, "vanity" license plates have become popular in many countries. Instead of pre assigned numbers, vanity plates have letterings chosen by drivers. For example, the plate could read "EcoBoy". Suppose Toronto City Hall experiments with vanity plates. It estimates demand for vanity plates to be given by the equation P = 100 10Q, where P is in dollars and Q is in `000s. The MC of manufacturing vanity plates is zero because as is the case in most countries vanity plates are made by prisoners. In the first year, vanity plates are priced at $60 each. In the second year, prices are lowered to $45. Toronto City Hall claims the lower prices was a resounding success as it led to higher revenues. In fact, City Hall managers propose lowering prices even further. Evaluate City Hall's pricing of vanity plates. In particular, would you advise Toronto to change their pricing strategy? Answer: If Toronto wants to maximize revenues from sales of vanity plates, it should simply set MR = 0: P = 100 10Q Use the short cut shown in the Math Econ Review that if the demand curve is linear, the MR curve has the same intercept and twice the slope: 13 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain MR = 100 20Q Setting MR = 0: 20Q = 100 Q = 100/20 = 5 P = 100 10(5) = $50 In fact, Toronto should NOT lower prices even further. City Hall managers are getting confused because the initial price of $60 is in the top half of the demand curve and the new price of $45 is in the lower half. The distance between $60 and the midpoint price is greater than the distance between the midpoint price and $45. Thus, when prices are lowered from $60, revenues rise as the price approaches the midpoint price and thereafter revenues fall as the price goes below the midpoint price. On the net, because $60 and $45 are not symmetric around the midpoint price, revenues increase fooling managers into thinking they should keep lowering prices. In fact, they should use MR = 0 and set the price at $50. This question was designed to test your understanding of elasticity. In ECO 100, you are taught that, for example, if E > 1, then lowering price leads to higher revenues. This statement is only true if the initial and final prices are both in the top half of the demand curve. If the prices are on different parts of the demand curve, then whether revenues rise or fall depends on what ratio of the price decline is in the top and bottom half of the demand curve. See figure below. 14 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain 15 ...
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