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ECO 3031 of 2Stony Brook UniversityFall 2016ASSIGNMENT 2Due: Thursday, September 29 in class.Instructions: Show all your work or points will be deducted.1. In this problem, we will explore an application of the consumer choice model to show how flexibleit is.Consider the following scenario. You have two midterm exams upcoming, and you have to decidehow to allocate your time. After eating, sleeping, exercising and maintaining some level of sociallife, you have 10 hours each day in which to study for your exams, i.e. 10 hours is your totalamount of study time. You have figured out that your GPA from your two courses, IntermediateMicroeconomic Theory and Biology takes the following formGPA Dp 4 p2 EC B ;7where E is the number of hours you devote to Economics and B are the hours you devote toBiology, which should be regarded as continuous variables. Suppose further that you only careabout your GPA.(a) Setup the constrained maximization problem and solve for the optimal allocation of studytime. You can solve this problem in two ways: You can either setup the Lagrangian1 or youcan use the fact that at an interior optimum, the price ratio should equal the Marginal Rateof Substitution. For the latter method, you need to think about what are the âpricesâ in thiscontext.(b) If you follow the optimal strategy you found in (a), what will be your GPA?(c) What is the value, measured in GPA units, of increasing your total amount of study time byone more hour? To work this out, you actually need to setup the Lagrangian and solve for at the optimum.(d) Now suppose that Intermediate Microeconomic Theory is a very hard class, so for every hourthat you study of it, you need to take a one hour break, i.e. if study for E hours Economics,the actual time that you need to spend is 2E . What is your optimal allocation of study timein this case? What is the value of increasing your total amount of study time?2. On a given evening, Mr. D.P.2 enjoys the consumption of cigars .c/ and brandy .b/ according tothe function.U.c; b/ D 20c c 2 C 18b 3b 2 :(a) How many cigars and glasses of brandy does he consume during an evening? (Cost is noobject to D.P.) Hint: This is an unconstrained maximization problem.1 Tobe covered in Thursdayâs Lecture.Pockets2 Deep1ECO 3032 of 2(b) Mr. D.P. recently had a medical check-up in which his doctor advised him that he shouldlimit the sum of glasses of brandy and cigars consumed to 5. How many glasses of brandyand cigars will he consume under these circumstances? Hint: Notice that D.P. can no longerconsume as many glasses of brandy and cigars as in (a).3. Mr. Odde Ball enjoys commodities x and y according to the utility functionqU.x; y/ D x 2 C y 2 :(a) Graph Mr. Ballâs indifference curve and its point of tangency with his budget constraint,given by px x C py y D I . What does the graph say about Mr. Ballâs behavior? Have youfound a true maximum?(b) Maximize Mr. Ballâs utility if px D $3, py D $4; and he has $50 to spend. Hint: It may beeasier here to maximize U 2 rather than U . Why will this not alter your results?4. Consider the following utility function (referred as a quasi-linear utility function as it is linear inthe second element):u.x; y/ D ln.x/ C y;with prices and income given by: px D 1; py 2 RC and I 2 RC .(a) For the special case of py D 2 and I D 1, show analytically and graphically that there is nopositive consumption choice .x ; y / 0 which satisfies the tangency condition:ux .x; y/pxD:uy .x; y/pyBriefly describe where the failure of the tangency condition stems from and what does itmean. Hint: Setup the tangency condition and solve it. What do you get? Is it affordable?(b) If we insist that .x ; y / 0; then describe, analytically and graphically, the optimal consumption choice for py D 2, I D 1. Hint: Think about corner solutions.(c) Now suppose that py 2 RC and I 2 RC are unspecified. Suppose that we donât insist onpositive values of either x or y . For which values of py and I do you get a nonsensical solutionin the sense that y < 0â¹5. Mr. A derives utility from martinis .m/ in proportion to the number he drinks:U.m/ D m:However, Mr. A is very particular about his martinis: He only enjoys them made in the exactproportion of two parts gin .g/ to one part vermouth .v/. Hence, we can rewrite Aâs utility functionasg U.m/ D U.g; v/ D min;v :2(a) Graph Mr. Aâs indifference curve in terms of g and v for various levels of utiliy. Show graphically that regardless of the prices of the two ingredients, Mr. A. will never alter the way hemixes martinis.(b) Calculate the demand functions for g and v . Notice that because the problem involves aperfect complements utility function, you cannot solve for utility-maximizing decisions byusing calculus.
