EC 03. & 03.3 Fall 04 Deniz Selman Bo¼gaziçi University Midterm Examination Solutions Monday 0 November 04. (5 pts) Defne is now in her senior year of high school and is preparing for her university entrance exam. Each week, she spends 40 hours at her high school, 6 hours at her dershane (cram school), 7 hours doing homework, 9 hours taking some form of transportation to and from her schools, and 6 hours preparing to leave for school. This means she has exactly 80 hours remaining in the entire week to do anything else. Her feasible options for things to do during these 80 hours are: sleeping (S) ; eating (E) and talking to her friends/family (F ). Let x (x S ; x E ; x F ) represent Defne s activity bundle during her 80 free hours, where one unit of an activity represents one hour spent doing it. Defne s weekly utility function from these activities is given by u (x) 4x S + 0x E x E + x F : (a) (5 pts) Write down Defne s consumer problem which she must solve in order to maximize her utility. (Hint: Defne has a time constraint rather than a budget constraint.) Soln: Defne s consumer problem is given by max x S ;x E ;x F 0 4x S + 0x E x E + x F s u b j. t o x S + x E + x F 80: (b) ( pts) Solve Defne s consumer problem which you wrote down in part (a). Denote the solution as x (x S ; x E ; x F ) : Soln: We can immediately see that sleeping always has a higher marginal utility (4) than that of spending time with friends/family () ; and the price of both activities is the same (since both prices are in terms of time and simply equal to one hour). Therefore Defne will not spend any time with her friends or family, and we can rewrite the problem as max x S ;x E 0 4x S + 0x E s u b j. t o x S + x E 80: x E Now, using the substitution method, we can substitute x S 80 choose x E in order to maximize the expression x E so that Defne simply needs to 4 (80 x E ) + 0x E x E 30 + 6x E x E: The FOC is 6 x E 0; and the necessary SOC holds (since the second derivative is negative: < 0), and so the FOC yields a maximum: x E 3: Substituting back into the time constraint yields x S 80 3 77; and so Defne s utility maximizing activity bundle is (x S; x E; x F ) (77; 3; 0) : (c) (8 pts) People sometimes say time is money. For this question, rede ne the concepts of normal and inferior by replacing income (in the standard de nition) with time. That is, a normal activity is one which Defne would spend more time doing if she had more time, and an inferior activity is one that she would spend less time doing if she had more time. For Defne, in her current situation as described in this problem, for each activity (sleeping (S) ; eating (E) and talking to her friends/family (F )), state and show whether the activity is normal, inferior, or time neutral.
Soln: If we replace the 80 in the consumer problem with m to represent time as income, the expression to maximize becomes 4 (m x E ) + 0x E x E 4m + 6x E x E; which still yields the same optimal amount of time spent eating (x E 3), whereas x S m 3 will increase as m increases. (x F 0 also still holds.) Therefore, we can conclude that eating and and talking to friends/family are both time neutral activities for Defne, whereas sleeping is a normal activity.. ( pts) Every evening at the dershane, the same meal (chicken) is served to the students for free (meaning the price of the chicken dinner is p C 0). Defne has the option to either eat the free dinner or buy a dürüm next door for some price p D > 0: (a) (4 pts) Suppose one day that Defne chooses to eat the chicken dinner. Can we tell by observing this behavior whether she on that day prefers the chicken dinner or dürüm more? Soln: No. It is possible that she simpy cannot a ord dürüm and is choosing the free chicken dinner for that reason. (b) (4 pts) Suppose one day that Defne chooses to buy a dürüm next door. Can we tell by observing this behavior whether she on that day prefers the chicken dinner or dürüm more? Soln: Yes. Defne could have chosen to eat the free chicken dinner but chose the dürüm. Therefore we know that she (at least weakly) prefers the dürüm to the chicken. (c) (3 pts) Let x (x C ; x D ) be the dinner bundle for Defne in one week, where x C is the number of free chicken dinners and x D is the number of dürüms she eats. Let u (x) be Defne s weekly utility function from these dinners. Suppose that Defne has a weekly budget of m > 0 for dinners and that she chooses to eat 4 free chicken dinners and purchase dürüm. That is, she chooses x (x C ; x D ) (4; ) in order to maximize her utility. iven this much information, for each of the following utility functions, say whether or not the function could possibly be Defne s utility function from dinners. Explain your answers brie y. i. u (x) x C + x D : Soln: When choosing (x C ; x D ) (4; ) ; Defne gets utility u (4; ) (4) + 9; while choosing the also feasible (5; 0) would yield u (5; 0) (5) + 0 0 > 9; and so this cannot be Defne s utility function. ii. u (x) x C + x D : Soln: Yes, this could be Defne s utility function. The marginal utility of dürüm is always higher than that of chicken, so with no budget constraint Defne would eat only dürüm, but it is possible that m p D so that she can only a ord one dürüm. In this case, with this utility function, she would in fact choose (x C ; x D ) (4; ) : iii. u (x) x C + x D: Soln: When choosing (x C ; x D ) (4; ) ; Defne gets utility u (4; ) 4 + 7; while choosing the also feasible (5; 0) would yield u (5; 0) 5 +0 5 > 7; and so this cannot be Defne s utility function. iv. u (x) ln (x C + ) + x D : Soln: Yes, this could be Defne s utility function. At Defne s choice of (x C ; x D ) (4; ) ; the @u @u marginal utility of dürüm, @x D ; is higher than that of chicken, @x C x C + 5 ; so with no budget constraint Defne could increase her utility by switching to less chicken and more dürüm, but it is possible that m p D so that she can only a ord one dürüm. In this case, with this utility function, she would in fact choose (x C ; x D ) (4; ) : 3. (40 pts) When Defne nally gets home and has some time to eat a snack, she goes shopping and chooses from three goods: z lay Soda (), green plums () and Perrier (P ). Her utility function from a goods bundle x (x ; x ; x P ) is given by u (x) 3 (x ) 3 (x ) 3 + x P : Prices are given by p (p ; p ; p P ) and Defne s income is given by m: To simplify notation, assume that p P : Throughout this entire question, assume that none of the solutions are corner solutions.
(a) (0 pts) Find Defne s ordinary demand functions x (p; m) (x (p; m) ; x (p; m) ; x P (p; m)) : Soln: Defne s consumer problem is given by The Lagrangian is The FOCs are max x ;x ;x P 0 3 (x ) 3 (x ) 3 + x P s u b j. t o p x + p x + x P m: L (p; m; x; ) 3 (x ) 3 (x ) 3 + x P + (m p x p x x P ) : @x (x ) 3 (x ) 3 p 0; (0.) @x (x ) 3 (x ) 3 p 0; (0.) @x P 0; (0.3) @ m p x p x x P 0: (0.4) Equation (0:3) yields ; which plugged into (0:) yields (x ) 3 (x ) 3 p 3 x p x x x p 3 x p 3 x ; (0.5) which then plugged into a similarly rearranged version of (0:) yields x p 3 x x which plugged back into (0:5) yields x p 3 p 3 x x p 3 p 6 x 4 p 3 p 6 x 3 x 3 p 3 p6 p p ; x p 3 x p 3 p p p 3 p p4 p p ; 3
which nally all plugged into (0:4) yields And so x P m p x p x m p p p m : p p p p p x (p; m) (x (p; m) ; x (p; m) ; x P (p; m)) p p ; p p ; m : p p (b) (0 pts) Find Defne s compensated Hicksian demand functions h (p; u ) (h (p; u ) ; h (p; u ) ; h P (p; u )) : (Continue to assume that none of the solutions are corner solutions.) Soln: Defne s expenditure minimization problem is given by The Lagrangian is The FOCs are min x ;x ;x P 0 p x + p x + x P s u b j. t o 3 (x ) 3 (x ) 3 + x P u : L (p; m; x; ) p x + p x + x P + u 3 (x ) 3 (x ) 3 x P : @x p (x ) 3 (x ) 3 0; (0.6) @x p (x ) 3 (x ) 3 0; (0.7) @x P 0; (0.8) @ u 3 (x ) 3 (x ) 3 x P 0: (0.9) From (0:8) we get and so (0:6) and (0:7) become identical to (0:) and (0:) from part (a), and so we again get x p and x p p ; which plugged into (0:9) yields p And so x P u 3 (x ) 3 (x ) 3 3 u 3 p p p p 3 u 3 p 3 p3 u 3 : p p 3 h (p; u ) (h (p; u ) ; h (p; u ) ; h P (p; u )) p p ; p p ; u 3 : p p 4
(c) (5 pts) Are z lay Soda and green plums net substitutes or net complements for Defne? Explain your answer. Soln: We have that @h (p; u ) @ @p @p p p < 0; and so z lay Soda and green plums are net complements. p p (d) (5 pts) What is the income e ect of a marginal increase in the price of z lay Soda on Defne s demand for Perrier? Soln: This income e ect is given by IE x (p; m) @x P (p; m) @m @ p p m @m p p p p () p p : (e) (5 pts) Are z lay Soda and Perrier net substitutes or net complements for Defne? Explain your answer. Soln: We have that @h P (p; u ) @p @ @p 3 p p > 0; and so z lay Soda and Perrier are net substitutes. u 3 p p (f) (5 pts) Is Perrier a i en good for Defne? Please answer either yes, no, or that we cannot tell from the information given, and explain your answer. Soln: No, it is not a i en good for Defne. Note that @x P (p; m) > 0; @m and so Perrier is a normal good for Defne, and we proved in lecture (using the Slutsky equation) that a normal good cannot be a i en good. 4. (4 pts) At the dershane, Defne takes a practice university entrance exam. Each question is multiple choice with ve choices. Each correct answer is worth point, each incorrect answer is worth 0:5 points, and each question left blank is worth zero points. There are 60 questions total on the exam. Defne s score x on the exam is therefore given by x c 0:5i; where c is the number of correct answers and i is the number of incorrect answers out of 60. 5
(a) (7 pts) If Defne were to answer every question by guessing completely randomly on each one, what would be the expected value of her exam score? Soln: The probability of answering a given question correctly is 5 ; and so the expected value of Defne s exam score would be E [x] 60 5 () + 4 5 ( 0:5) 0: (b) (7 pts) Whenever Defne has no idea how to answer a question (i.e. when she cannot eliminate any of the ve choices), she always chooses to leave that question blank. In terms of Defne s utility function for points on the exam, u (x) ; is Defne risk averse, risk loving, risk neutral, or are her risk preferences not revealed by this behavior? Explain your answer. Soln: Defne has revealed that her utility function u (x) satis es u (0) u 5 () + 4 5 ( 0:5) 5 u () + 4 5 u ( 0:5) ; and so she is risk averse. (Because it is possible that this inequality holds with equality, making Defne indi erent between guessing and not guessing, she could also be risk neutral. You will receive full credit if you explained that this could be the case.) 6