Department of Economics ECO 204 Microeconomic Theory for Commerce 2016-2017 (Ajaz) Test 1 Solutions YOU MAY USE A EITHER A PEN OR A PENCIL TO ANSWER QUESTIONS PLEASE ENTER THE FOLLOWING INFORMATION LAST Name (example: Trudeau) FIRST Name (example: Justin) STUDENT ID NUMBER (example: 20422015) PUT AN X BELOW THE SECTION WHICH YOU RE REGISTERED IN Tue 11 am 1 pm Tue 2 pm - 4 pm Wed 11 am 1 pm Wed 2 pm - 4 pm IMPORTANT NOTES Proceed with this exam only after getting the go-ahead from the Instructor or the proctor Do not leave during the first hour of the exam or the last 15 minutes of the exam You are not allowed to leave the exam room unattended. If you need to go to the washroom, please raise your hand and a proctor will accompany you to the washroom. You are allowed to go to the washroom ONLY. You are required to stop writing and turn your exam face down when asked to stop by the instructor or proctor at the end of the exam Please note that proctors will take down your name for academic offenses, which will be treated in accordance with the policies as published by the Faculty of Arts and Sciences. EXAM DETAILS Duration: 2 hours Total number of questions: 5 Total number of pages: XXX (including title page) Total number of points: 100 Please answer all questions. To earn credit you must show all calculations. This is a closed note and closed book exam. You may use a non-programmable calculator. Sharing is not allowed.. KEEP YOUR ANSWERS AS BRIEF AS POSSIBLE AND SHOW ALL NECESSARY CALCULATIONS GOOD LUCK! Page 1 of 14
QUESTION 1 [TOTAL 20 Points] (a) [10 Points] Give an example of a utility function where, due to a lower price of good 1, the total substitution effect of good 1 is always zero. Feel to illustrate your answer using a graph. Don t solve UMPs. Consider the utility function U = min(αq 1, βq 2 ). Here is the optimal choice before the price changes (when the budget set is (P 1,, Y)): A Now suppose the price of good 1 falls the optimal choice on the budget set (P 1,, Y) is: A C Page 2 of 14
Now imagine taking away income such that the original bundle A is once again affordable: A C The new optimal choice on budget line (P 1,, Y ) is bundle A! That is, the substitution effect is 0. B C Page 3 of 14
(b) [10 Points] Give an example of a utility function, where due to a lower price of good 1, the total income effect of good 1 is always zero. State all assumptions. Consider the utility function U = f(q 1 ) + q 2 where f(0) = 0, f (q 1 ) > 0, good 2 is the base good (i.e. = 1), and the consumer is in case D so that : MRS = P 1 f (q 1 ) = P 1 q 1 = f(p 1 ) The demand for good 1 does not depend on income. Hence, the income effect has to be zero. Page 4 of 14
QUESTION 2 [TOTAL 15 Points] According to a Globe and Mail article: At Netflix, The Picture is Darkening: Online Users can stream 16 two-hour movies for the same delivery cost to Netflix as a DVD Home delivery plan (2 DVDs a month) are priced at $8.99/month and has 82% margin Streaming-only plans would increase gross margins by five to eight percentage points above the roughly 82-per-cent margin on its $8.99-per-month by-mail plans. What is Netflix s cost of streaming a movie online and how much should it charge online customers to watch a movie online? Home delivery plans have an 82% margin which means: Assume TVC COGS Gross Margin Revenue Gross Margin Revenue = R COGS R R TVC R = 0.82 = 0.82 Assume constant returns to variable inputs (i.e. doubling variable inputs, doubles output): Thus: Gross Margin Revenue R TVC R = Pq AVCq Pq = P AVC P = $8.99 AVC $8.99 Average Home Delivery = 2 DVDs/month AVC = $1.6182 = $0.81 DVD 2 DVDs DVD AVC of 16 streams = AVC of 1 DVD Now, suppose Netflix can achieve 90% margins by streaming movies online: AVC = $0.81 = $0.05 Stream 16 Streams Stream = 0.82 AVC = $1.6182 Desired Gross Margin Best case scenario = 90% P stream 0.05 P stream = 0.90 P stream = $0.50/stream Page 5 of 14
QUESTION 3 [TOTAL 10 Points] Explain why the demands for goods 1 and 2 in a UMP are unaffected by uniform inflation. State all assumptions. Assume preferences never change. Suppose preferences stay constant. Then the set of affordable choices before inflation consists of all bundles on or below the budget line: q 2 Y A = (P 1,, Y) Y P 1 q 1 The intercepts of the budget line are Y P 1, Y while the slope of the budget line A is P 1. Now suppose there is x% uniform inflation. The new intercepts and slope will be: (1 + x)y (1 + x)y,, (1 + x)p 1 (1 + x)p 1 (1 + x) (1 + x) These are identical to the pre-inflation intercepts and slope. Uniform inflation does not impact the set of affordable choices so that the pre-inflation and post-inflation optimal choices are the same. Page 6 of 14
QUESTION 4 [TOTAL 10 Points] Between 1989 and 1994, Kodak s share of camera film fell from 76% to 70%. In 1994, Kodak s sales were growing at 3% a year while Fuji s, Kodak s closest rival, sales were growing at 15% a year. According to analysts: Fuji s gain.. attributed to keeping the line on price.. and.. Kodak tried to position [itself] as providing a superior film... The following table (from Consumer Report which opined that Customers tended to view film as a commodity ) shows quality ratings and price points of Kodak s and Fuji s flagship brands. What should Kodak do? State all salient assumptions and show all necessary calculations. Film Consumer Report Quality Rating (min = 0, = 100) Price ($ per roll) Fuji color Super G 94 $2.91 Kodak Gold Plus 93 $3.49 Kodak has lower quality than Fuji yet Kodak is has the higher price. Given that consumers view film as a commodity, Kodak should first attempt to change preferences via advertising. If this strategy fails, Kodak should charge a lower price than Fuji adjusting for perceptions. Let q 1 = Kodak and q 2 = Fuji and use the quality ratings for utility parameters: Currently, the budget line slope is: U = αq 1 + βq 2 = 93q 1 + 94q 2 P 1 = 3.49 2.91 = 1.2 To compete with Fuji, Kodak should price its film so that customers at the very least can buy either/or Fuji and Kodak (i.e. case D). For Case D: Kodak should charge a price of $2.85 or less. MRS = P 1 P 1 = 0.98 P 1 = 0.98 P 1 = 0.98($2.91) = $2.85 Page 7 of 14
QUESTION 5 [TOTAL 55 Points] Consider an economy in which all consumers allocate their income ($Y > 0) to consumption ($C) and savings ($S). A particular consumer has the following preferences: Here α, β > 0 and assume α + β = 1. U = C α S β (a) [5 Points] It s standard practice in micro to assume that consumers have rational preferences. Briefly explain rational preferences. To have rational preferences means that you have complete and transitive preferences. To have complete preferences means you must be able to rank any two bundles (say a, b) in the consumption set according: either a b, or b a, or a ~ b. To have transitive preferences means that your preferences are logical in the sense that any three bundles a, b, c in your consumption set, if you say a b and you say b c then you must say that a c. Page 8 of 14
(b) [20 Points] Solve the following UMP for C, S, marginal utility of income, marginal utility of the minimum expenditure on consumption (currently min consumption is zero), and the marginal utility of the minimum amount saved (currently min savings is zero): C,S Cα S β s. t. C + S = Y, C 0, S 0 State all salient assumptions, and show all necessary calculations. (You may, without proving them, use the rules for Cases B, C, D) The UMP is: C,S Cα S β s. t. C + S = Y, C 0, S 0 Notice that the minimum expenditure on consumption is currently zero) and the minimum amount saved is currently zero: C,S Cα S β s. t. C + S = Y, C C min = 0, S = S min 0 It s easier to work with the log linear Cobb-Douglas: C,S α ln C + β ln S s. t. C + S = Y, C C min = 0, S = S min 0 C,S α ln C + β ln S s. t. C + S Y = 0, C C min = 0, S S min = 0 C,S α ln C + β ln S s. t. C + S Y = 0, C + C min 0, S + S min 0 The Lagrange equation is: L = α ln C + β ln S λ 1 [C + S Y] λ 2 [ C + C min ] λ 3 [ S + S min ] C,S,λ 1,λ 2,λ 3 For the time being: C min = S min = 0 so that: L = α ln C + β ln S λ 1 [C + S Y] λ 2 [ C] λ 3 [ S] C,S,λ 1,λ 2,λ 3 L = α ln C + β ln S λ 1 [C + S Y] + λ 2 C + λ 3 S C,S,λ 1,λ 2,λ 3 The FOCs and KT conditions are: L C = α C λ 1 + λ 2 = 0 L S = β S λ 1 + λ 3 = 0 L λ 1 = [C + S Y] = 0 Page 9 of 14
Using the rules for cases B, C, D : λ 2 0, C 0, λ 2 C = 0 λ 3 0, S 0, λ 3 S = 0 Case B Case C Case D When MRS P 1 at q 1 = Y P 1, q 2 = 0 When MRS P 1 at q 1 = 0 q 2 = Y When neither cases B or C arise At solution: MRS = P 1 q 1 = Y P 1 q 1 = 0 q 2 = 0 q 2 = Y P 1 q 1 + q 2 = Y MU 1 P 1 = MU 2 λ 1 = MU 1 /P 1 λ 1 = MU 2 / λ 1 = MU 1 P 1 = MU 2 λ 2 = 0 λ 2 = MU 2 P 1 MU 1 0 λ 2 = 0 λ 3 = MU 1 P 1 MU 2 0 λ 3 = 0 λ 3 = 0 First, note that: MRS = MU 1 MU 2 = And in this question, the slope of the budget line is 1. α C β S = α β S C For Case B (C = Y, S = 0) to be the solution: MRS at Bundle B > Slope of the Budget line α β 0 Y > 1 0 > 1 Impossible so case B is not the solution Page 10 of 14
For Case C (C = 0, S = Y) to be the solution: MRS at Bundle C < Slope of the Budget line α β Y 0 > 1 < 1 Impossible so case C is not the solution The solution is Case D where: MRS at Bundle D = Slope of the Budget line α β S C = 1 We can solve for C, S by solving the following equations simultaneously: α β S C = 1 C + S = Y The solutions are: C = α Y = αy α + β S = Now, the Lagrange multipliers. By the envelope theorem: β α + β Y = βy L = α ln C + β ln S λ 1 [C + S Y] λ 2 [ C + C min ] λ 3 [ S + S min ] C,S,λ 1,λ 2,λ 3 Thus: dl dy = λ 1 dl dc min = λ 2 dl ds min = λ 3 dl * dy = λ 1 Page 11 of 14
By the FOCs and KT conditions: dl * dc min = λ 2 dl * ds min = λ 3 L * = α ln C + β ln S λ 1 [C + S Y] + λ 2 C U 0 0 + λ 3 S = U 0 Thus: du dy = λ 1 = Marginal Utility of Income du dc min = λ 2 = Marginal Utility of Mininum consumption du ds min = λ 3 = Marginal Utility of Mininum savings Since we re in case D, λ 2 = λ 3 = 0 so that the FOCs become: These imply: L C = α C λ 1 + 0 = 0 L S = β S λ 1 + 0 = 0 λ 1 = α C = α αy = 1 Y Thus: du dy = λ 1 = Marginal Utility of Income = 1 Y du dc min = λ 2 = Marginal Utility of Mininum consumption = 0 = 0 du ds min = λ 3 = Marginal Utility of Mininum savings = 0 = 0 Page 12 of 14
(c) [10 Points] Suppose the government requires each person to save at least S min = $5,000 a year. Consider the following UMP for a particular individual: C,S C0.8 S 0.2 s. t. C + S = $100,000, C 0, S S min What is the impact on her utility from an infinitesimal increase in S min? State all salient assumptions, and show all necessary calculations From above, we have: S = βy = β$100,000 = 0.2($100,000) = $20,000 Since $20,000 > $5,000 an infinitesimal increase in S min has no impact on her utility. Page 13 of 14
(d) [10 Points] [This question is independent of part (c)] Suppose the government requires each person to save at least S min a year. Consider the following UMP for a particular individual: C,S C0.8 S 0.2 s. t. C + S = $100,000, C 0, S S min What is the lowest value for S min such that the marginal utility from an infinitesimal increase in S min is negative? State all salient assumptions, and show all necessary calculations Anytime S min > $20,000 an increase in S min will reduce utility. Since $20,000 > $5,000 an infinitesimal increase in S min has no impact on her utility. Page 14 of 14