You will work on this in groups in your discussion section, then finish it on your own afterwards. In particular, you must write all of the solutions for your final version by your- self. Working together on the final write-up or copying another student's final answers is considered cheating. Similarly, copying an explanation from the textbook or the Internet is cheating. (If you do use an Internet resource to help you, say what resource you used.) You will upload a PDF with your answers to Canvas. You can write them neatly on paper and then scan them using Office Lens, Dropbox Scan, or another scanning app; write them neatly on a tablet (iPad, etc) if you have one; or type them in Word or similar. (Some of your answers may have pictures in them, so trying to do it all in Word may be tedious.) Because of Thanksgiving, this is the last written homework for the quarter. (In week 10, you'll have some time to discuss the project and/or review for the final in class, rather than written homework. Discussion section on Thanksgiving week is optional, and will be a combination of review and a chance to work together on homework or the project.) 1. CASE STUDY (The case study this week is long, so no extra practice questions this week. This case study was developed by Mike Price.) When a government agency taxes an industry, both the government and the industry have choices to make: for the government, how much to tax, and for the industry, how much of that tax to pass onto customers. Although it may seem optimal to increase the price of the product an amount equal to the tax, it may not actually maximize profit to do so, since it drives down consumer demand. The Tax Burden on Tobacco"l has relevant data summarized below regarding the his- torical pricing of cigarettes. Some selected years are given in the table below. Year Median? After-Tax Price Median? Tax ($ per package) ($ per package) 2004 3.74 1.17 2006 3.93 1.34 2008 4.35 1.59 2010 5.55 2.46 2012 5.65 2.50 2014 5.84 2.56 Data on sales of tobacco products from 2001 through 2014 is available on the US Federal Trade Commission's website, in Table 1A and 1B (beginning on page 10, the "units sold" column) of the report: https://bit.ly/350jZaJ (You can also find the report by Googling FTC cigarette report 2014). Things you will need: The table provided on this page of the project The data from Table 1A and/or 1B of the FTC link provided above That the pretax price is the median price minus the median tax per package That there are 20 cigarettes per package of cigarettes If your discussion section is with... Cruz Godar use data from 2012 and 2014 Elisha Hulbert use data from 2010 and 2012 Jacob Lebovic use data from 2008 and 2010 Bo Phillips use data from 2006 and 2008 Aaron Victorin-Vangerud use data from 2004 and 2006. (1) Say who your discussion leader is and, consequently, which years of data you are using. Cruz Godar (2) Assume that the demand is linear in terms of the pre-tax median price p of cigarettes. That is, the demand q in billions of packages) is given by q = mp + b for some constants m and b, where p is the pre-tax median price. Use the data from your two years (see the previous question) to find m and b. (Hint: the price of cigarettes each year is given the first page of the assignment. The number sold is given in the FTC report linked above. Together, those should give you two points on the line, which is enough to find an equation for the line.) (3) Write an equation for the revenue, R, obtained by cigarette companies as a function of p. (4) Estimates for the cost to produce cigarettes (including materials, manufacture, ad- vertising, etc) vary, so let's say that it costs $0.60 plus the last two digits of your student number (in cents) per package of cigarettes (e.g. if the last two numbers of your student number are 34 then the cost would be $0.60 + $0.34 = $0.94). Write a formula for the cost to produce q billion packages of cigarettes. (5) Recall that the demand q = f(p) is given by the linear function you found in (2). Use your cost and revenue formulas from the previous two questions to write a formula for the profit, P, as a function of price per package, p. (6) Now assume that in addition to production costs (which you included in the exercise above), the federal and state governments also tax the sale of cigarettes (which they do). If the net tax rate is t dollars per package, revise your equation for profit in the exercise above to include the company's requirement to pay these taxes. Your answer should contain both t and p. (Hint: if the previous cost per box was $0.94, for example, then the new cost per box with the additional tax of t dollars is 9.94+t dollars per box.) (7) What value of p maximizes the profit function P in the exercise above? Your answer will contain the constant t. (8) The government decides to be clever. Assuming that the cigarette companies will always price their product so that profit is maximized, the government can choose a value for t that also maximizes the total tax revenue gained from cigarette sales. (a) Write an equation for T, the total tax revenue gained from cigarette sales as a function of the tax rate, t. (Hint: write T in terms of t and p, then use the expression for p from (7), which maximizes the businesses' profit) (b) What tax rate t maximizes the total tax revenue T? (c) Consult the first page of the project and write the median taxes for the two years you studied. Is your answer to (8b) close to either of these values ? (d) Give two different reasons why the true combined state and federal taxes might be different from what you calculated. (9) The government has now determined how much it wants to tax the sale of cigarettes. What remains to be seen is how much the cigarette companies will increase the price of their product as a result. (a) If there were no tax, what would the cigarette company charge for a package of cigarettes in order to maximize its profit? (Hint: you found this formula in (7)) (b) Now assume that the tax rate t you found in (8b) is in effect. Again, use the result from (7) to find the price at which the cigarette company will obtain its maximum profit. (c) If the cigarette company passed on the entirety of the tax to the customers, then the answer to (9a) plus the tax rate t from (8b) would equal the answer in (9b). Does it? Again assuming the optimal tax rate from (8b) and that the company prices its cigarettes at the value in (9b), how much of the government's tax does the company actually pass on to the consumer? You will work on this in groups in your discussion section, then finish it on your own afterwards. In particular, you must write all of the solutions for your final version by your- self. Working together on the final write-up or copying another student's final answers is considered cheating. Similarly, copying an explanation from the textbook or the Internet is cheating. (If you do use an Internet resource to help you, say what resource you used.) You will upload a PDF with your answers to Canvas. You can write them neatly on paper and then scan them using Office Lens, Dropbox Scan, or another scanning app; write them neatly on a tablet (iPad, etc) if you have one; or type them in Word or similar. (Some of your answers may have pictures in them, so trying to do it all in Word may be tedious.) Because of Thanksgiving, this is the last written homework for the quarter. (In week 10, you'll have some time to discuss the project and/or review for the final in class, rather than written homework. Discussion section on Thanksgiving week is optional, and will be a combination of review and a chance to work together on homework or the project.) 1. CASE STUDY (The case study this week is long, so no extra practice questions this week. This case study was developed by Mike Price.) When a government agency taxes an industry, both the government and the industry have choices to make: for the government, how much to tax, and for the industry, how much of that tax to pass onto customers. Although it may seem optimal to increase the price of the product an amount equal to the tax, it may not actually maximize profit to do so, since it drives down consumer demand. The Tax Burden on Tobacco"l has relevant data summarized below regarding the his- torical pricing of cigarettes. Some selected years are given in the table below. Year Median? After-Tax Price Median? Tax ($ per package) ($ per package) 2004 3.74 1.17 2006 3.93 1.34 2008 4.35 1.59 2010 5.55 2.46 2012 5.65 2.50 2014 5.84 2.56 Data on sales of tobacco products from 2001 through 2014 is available on the US Federal Trade Commission's website, in Table 1A and 1B (beginning on page 10, the "units sold" column) of the report: https://bit.ly/350jZaJ (You can also find the report by Googling FTC cigarette report 2014). Things you will need: The table provided on this page of the project The data from Table 1A and/or 1B of the FTC link provided above That the pretax price is the median price minus the median tax per package That there are 20 cigarettes per package of cigarettes If your discussion section is with... Cruz Godar use data from 2012 and 2014 Elisha Hulbert use data from 2010 and 2012 Jacob Lebovic use data from 2008 and 2010 Bo Phillips use data from 2006 and 2008 Aaron Victorin-Vangerud use data from 2004 and 2006. (1) Say who your discussion leader is and, consequently, which years of data you are using. Cruz Godar (2) Assume that the demand is linear in terms of the pre-tax median price p of cigarettes. That is, the demand q in billions of packages) is given by q = mp + b for some constants m and b, where p is the pre-tax median price. Use the data from your two years (see the previous question) to find m and b. (Hint: the price of cigarettes each year is given the first page of the assignment. The number sold is given in the FTC report linked above. Together, those should give you two points on the line, which is enough to find an equation for the line.) (3) Write an equation for the revenue, R, obtained by cigarette companies as a function of p. (4) Estimates for the cost to produce cigarettes (including materials, manufacture, ad- vertising, etc) vary, so let's say that it costs $0.60 plus the last two digits of your student number (in cents) per package of cigarettes (e.g. if the last two numbers of your student number are 34 then the cost would be $0.60 + $0.34 = $0.94). Write a formula for the cost to produce q billion packages of cigarettes. (5) Recall that the demand q = f(p) is given by the linear function you found in (2). Use your cost and revenue formulas from the previous two questions to write a formula for the profit, P, as a function of price per package, p. (6) Now assume that in addition to production costs (which you included in the exercise above), the federal and state governments also tax the sale of cigarettes (which they do). If the net tax rate is t dollars per package, revise your equation for profit in the exercise above to include the company's requirement to pay these taxes. Your answer should contain both t and p. (Hint: if the previous cost per box was $0.94, for example, then the new cost per box with the additional tax of t dollars is 9.94+t dollars per box.) (7) What value of p maximizes the profit function P in the exercise above? Your answer will contain the constant t. (8) The government decides to be clever. Assuming that the cigarette companies will always price their product so that profit is maximized, the government can choose a value for t that also maximizes the total tax revenue gained from cigarette sales. (a) Write an equation for T, the total tax revenue gained from cigarette sales as a function of the tax rate, t. (Hint: write T in terms of t and p, then use the expression for p from (7), which maximizes the businesses' profit) (b) What tax rate t maximizes the total tax revenue T? (c) Consult the first page of the project and write the median taxes for the two years you studied. Is your answer to (8b) close to either of these values ? (d) Give two different reasons why the true combined state and federal taxes might be different from what you calculated. (9) The government has now determined how much it wants to tax the sale of cigarettes. What remains to be seen is how much the cigarette companies will increase the price of their product as a result. (a) If there were no tax, what would the cigarette company charge for a package of cigarettes in order to maximize its profit? (Hint: you found this formula in (7)) (b) Now assume that the tax rate t you found in (8b) is in effect. Again, use the result from (7) to find the price at which the cigarette company will obtain its maximum profit. (c) If the cigarette company passed on the entirety of the tax to the customers, then the answer to (9a) plus the tax rate t from (8b) would equal the answer in (9b). Does it? Again assuming the optimal tax rate from (8b) and that the company prices its cigarettes at the value in (9b), how much of the government's tax does the company actually pass on to the consumer