Boğaziçi University-Department of Economics

EC203 Microeconomics I-Fall 2016

Levent Yıldıran

PS2

1) (Chp 4) a) In a two-commodity competitive economy Ali has an income of 10TL and faces the prices px=py=1TL. His preferences are characterized by utility function U(x,y)= -x2-y2.

i) Draw this budget set.

ii) State whether the commodities are good or bad for him.

iii) Draw his indifference curves showing the direction in which they increase.

iv) Are his preferences convex or concave?

v) Mark his optimal bundle on the graph.

Repeat part a for

b) U(x,y)= y-x2 c) U(x,y)= xe-y d) U(x,y)= (x+y)2

e) U(x,y)= y-x f) U(x,y)= x2-y2 g) U(x,y)= y/x

2) (Chp 4) Ahmet likes to eat 2 slices of pizza with a glass of coke. He has 20L in his pocket and the prices of pizza and coke are 2L and 1L, respectively. His girl friend, Selin, is a naive person and what is good for Ahmet is good for her. That is why the closer she is to his favorite combination the happier she is. (She has enough money to reach his favorite combination).

a) Draw Ahmet's indifference curves, budget constraint and label the optimum point.

b) Draw Selin's indifference curve on which she consumes exactly 1 glass of coke and 4 slice of pizza.

3) (Chp 4) Use separate graphs to draw indifference curves for each of the following utility functions:

(a) U(x, y) = min{2x + y, 2y + x}

(b) U(x, y) =max{2x, y}

4) (Chp 4) A professor likes rules and symmetry, that’s why he has a strong preference for having exactly the same number of students in both sections of his course. He also likes highly populated classes. So, his utility function is U (x, y) = min(2x-y, 2y-x), where x is the number of students in the first section, and y is the number of students in the second section.

(a)Draw some of the indifference curves of this professor.

(b)Show the optimal combination of x and y, that makes him the happiest, if there are 150 students are taking his course.

5) (Chp 4) Devrim’s utility function is U(x, y) = xy. Yezdan’s utility function is U(x, y) = 1000xy. Hamit’s utility function is U(x, y) = −xy. İpek’s utility function is U(x, y) = −1/(xy + 1). Aydan’s utility function is U(x, y) = xy − 1000. Sinan’s utility function is U(x, y) = x/y. Gözde’s utility function is U(x, y) = x(y + 1). Which of these persons have the same preferences as Devrim?

6) (Chp 4) Asuman consumes goods x and y. Her indifference curves are described by the formula y = k/(x + 3). Higher values of k correspond to better indifference curves. Which of the below is true? a. Angela prefers bundle (8, 9) to bundle (9, 8). b. Angela likes good y and hates good x. c. Angela prefers bundle (11, 9) to bundle (9, 11). d. Angela likes good x and hates good y. e. More than one of the above statements are true.

7) (Chp 4) Nick's indifference curves are circles, all of which are centered at (12, 12). Of any two indifference circles, he would rather be on the inner one than the outer one. Which of the below is true? a. Nick’s preferences are not complete. b. Nick prefers (16, 17) to (10, 10). c. Nick prefers (10, 17) to (10, 10). d. Nick prefers (8, 8) to (17, 21). e. More than one of the above statements are true.

8) (Chp 4) A consumer has preferences represented by the utility function U(x1,x2 )=10(x12+2x1x2 + x2)−50. Explain why for this consumer goods 1 and 2 are perfect substitutes.

9) (Chp 4) If we graph Mahinur's indifference curves with avocados on the horizontal axis and grapefruits on the vertical axis, then whenever she has more grapefruits than avocados, the slope of her indifference curve is -2. Whenever she has more avocados than grapefruits, the slope is -1/2. Mahinur would be indifferent between a bundle with 22 avocados and 37 grapefruits and another bundle that has 37 avocados and

a. 27 grapefruits.

b. 32 grapefruits.

c. 17 grapefruits.

d. 22 grapefruits.

e. 24.5 grapefruits.

10) (Chp 4) Scholastica is taking a class from Professor Chaos. Professor Chaos gives two tests in this course and determines a student’s grade as follows. He determines the smaller of the following two numbers: half of the score on the first test (which is a relatively easy test) and the total score on the second test. He gives each student a numerical score equal to the smaller number and then ranks the students. Scholastica would like to be ranked as high as possible in Professor Chaos’s rankings. If we represent her score on the first exam on the horizontal axis and her score on the second exam on the vertical axis, then her indifference curves

a. are L-shaped with kinks where the two exam scores are equal.

b. have sections with a slope -2 and sections with a slope 1/2.

c. are positively sloped.

d. are L-shaped with kinks where the exam 1 score is twice the exam 2 score.

e. are straight lines with a slope of -1/2.

11) (Chp 5) Mehmet consumes only apples and Eti Form. His utulity function is U(x,y) = x4y5 , where x is the number of apples consumed and y is the number of Eti Form consumed. Mehmet’s income is 72, Px=2 and Py=3. Find his optimum consumption bundle (x,y)?

12) (Chp 5) Melek consumes wine and cheese. Wine is sold in an unusual way. There is only one supplier, and the more wine you buy from her, the higher price you have to pay per bottle. In fact y bottle of wine will cost Melek y2 dollars. Cheese is sold in the usual way at a price of 2 dollars per unit. Melek’s income is 20 dollars and her utility function is U(x, y) = x + 2y where x is her consumption of cheese and y is her consumption of wine.

(a) Sketch Melek’s budget set.

(b) Sketch some of her indifference curves and label the point that she chooses.

13) (Chp 5) Çağatay’s utility function is U(x, y) = x+2y, where x is consumption of good X and y is consumption of good Y. His income is 2. The price of good Y is 2. The cost per unit of good X depends on how many units he buys. The total cost of x units of good X is x1/2. Find the bundle (x, y) that maximizes his utility both graphically and by lagrangian method?

14) (Chp 5) Ayse received a tape recorder as a birthday gift and is not able to return it. Her utility function is U(x, y, z) = x + f(y)z1/2 where z is the number of tapes she has, y is the number of tape recorders she buys and x is the amount of money she has left to spend. f(y) = 0 if y < 1 and f(y) = 12 if y ≥1. The price of tape is 3 and she can easily afford to buy dozens of tapes. How many tapes will she buy?

15) (Chp 5) Emrah has strictly convex preferences for cherries and grapes. His utility function over cherries, C, and grapes, G, is defined by the function U(C,G)=C1/4 +G1/4.Let PC and PG be the prices of cherries and grapes respectively. W denotes Emrah’s income that is available for him to spend on the two goods.

(a)Write out Emrah’s utility maximization problem. (b)Set up the Lagrangian and solve for the first order condition. (c) Solve for Emrah’s demand functions for cherries and grapes. (d)Find Emrah’s utility function in terms of PC and PG and W.

16) (Chp 5) There are two agents 1 and 2 with utility functions U1(x1,y1)= x12+2x1y1 and U2(x2,y2)= x2+y22. Both agents have incomes of 180. Px=5 and Py=6. But, Py increases to 10 for any additional units of y in excess of 5 units.

(a) Draw the budget constraint (which is the same for both agents).

(b) What are the optimum consumption bundles for these two agents?

agent 2 will choose to spend all her money on x?

17) (Chp 5) Zeynep has utility U(x, y) = min{4x, 2y}. Write down her demand function for x as a function of the variables w, px and py; where w is her income, px is the price of x and py is the price of y.

18) (Chp 5) Laura’s preferences between golf and tennis are represented by U(g,t) = gt where g is the number of rounds of golf and t is the number of the tennis matches she plays per week. She has $24 per week to spend on these sports. A round of golf and tennis match each cost $4. She used to maximize her utility subject to this budget. She decided to limit the time she spends on these sports to 16 hours a week. A round of golf takes 4 hours. A tennis match takes 2 hours. As a result of this additional constraint, how are her choices changed?

19) (Chp 5) Harmon’s utility function is U(x1,x2)=x1x2. His income is $100, the price of good 2 is p2=4. Good 1 is priced as follows. The first 15 units cost $4 per unit and any additional units cost $2 per unit. What consumption bundle does Harmon choose?

20) (Chp 5) Kamil has 42 TL, which he decides to spend on commodities X and Y. Commodity X costs 13 TL per unit and commodity Y costs 12 TL per unit. He has the utility function U(x, y) = 3x2 + 2y2, where x is the quantity of commodity X he consumes and y is the quantity of commodity Y he consumes. How much of commodity X and how much of commodity Y will he choose to consume?

21) (Chp 5) Rollo would love to have a Mercedes. His preferences for consumption in the next year are represented by a utility function U(x, y), where x=0 if he has no Mercedes and x=1 if he has a Mercedes for the year and where y is the amount of income he has left to spend on other things. If U(0,y)=y1/2 and U(1,y)=(10/9)y1/2 and if Şahap’s income is $50,000 a year, how much would he be willing to pay per year to have a Mercedes?

22) Gorkem has the utility function U(x,y)=x-(1/y). Price of x is $4 and the price of y is $1. If his income is $30, how much of the goods will he consume?

23) Rengin has the utility function U(x,y) = (x+1)(y+4). Does she have convex preferences? The price of y is 1. Rengin spends all of her income to buy 6 units of y and no x. From these facts we can tell that the price of x must be at least how much?

24) Albin’s utility function is U(x,y) = min{x+2y,3x+y} where x is butter and y is milk. The price of butter is 4 and the price of milk is 5. What utility will Albin have if he was given to consume 4 units of butter and 3 units of milk? What would it cost to Albin to buy the cheapest bundle to reach the same utility level?