a. Show the budget set containing all of the commodity bundles that the following individuals can afford.

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1 Chapter. To buy a commodity one has to pay with money and a certain amount of ration cards. Suppose that we have two commodities A and B. The price on each commodity is krona, but in order to buy A you also need two ration cards and to buy B you need 4 ration cards. a. Show the budget set containing all of the commodity bundles that the following individuals can afford. Income Ration cards Individual 9 kr 4 Individual 6 kr 4 b. Show that individual will not spend his entire income. c. Another person also has an endowment of 4 ration cards, but he is allowed a discount on commodity A. The price of A is 0.5 kr. He spends all his income and uses all his ration cards. What is his largest possible income? Chapter 3 3. Draw some indifference curves for Jens, who loves money but hates work. Assume strictly convex preferences. (Horizontal axis: work and vertical axis: income) 3. Johanna likes pancakes with jam. After eight pancakes, she gets tired of pancakes and eating more pancakes makes her less happy. However, she always prefers more jam to less. Show Johanna s preferences (horizontal axis: jam and vertical axis: pancakes) if: a. Johanna dislikes her parents complaining about her not eating everything put on her plate. b. She does not care about her parents complaints. 3.3 Mary Granola consumes two goods, grapefruits and avocados. If she has more grapefruits than avocados, her marginal rate of substitution is ( grapefruits for avocado), otherwise it is /. Draw two indifference curves for Mary. (Horizontal axis: avocado, vertical axis: grapefruit) Does Mary have convex preferences? Strictly convex preferences? 3.4 Ronny Rigid likes to eat lunch at noon, but he is prepared to eat earlier or later if he is sufficiently compensated for it. Draw a few of Ronny's indifference curves for money spend on "all other goods" and dining time.

2 3.5 Paul is currently consuming 0 cheeseburgers and 0 Cokes a week. A typical indifference curve for Paul is depicted below. Coke Cheeseburgers a. Would Paul accept to trade one cheeseburger for one extra Coke? b. Would he accept to trade one Coke to get an extra cheeseburger? c. What is the maximum number of cheeseburgers he would give up at an exchange rate of cheeseburgers for Coke? d. At what rate of exchange would Paul be willing to stay put at his current consumption level? 3.6 Markus is happiest when he has 8 cookies and 4 glasses of milk per day. When he has more of either food, he gets sick. When he has less than his favourite combination, giving him more makes him better off. His mother makes him drink 6 glasses of milk and only allows him 4 cookies per day. One day, Markus's sadistic sister made him eat 3 cookies and only gave him glass of milk. Although Markus complained later to his mother, he had to admit that he liked the diet that his sister forced on him better than what his mother demanded. Draw some indifference curves for Markus that are consistent with this story.

3 Chapter 4 4. Tomas utility function is U(x, y) = xy a. Suppose that Tomas originally consumed 4 units of x and units of y. Draw an indifference curve through this point. b. Show that if he is indifferent between the bundles (x0, y0) and (x, y) then if you doubled the amount of each good in each bundle, the new bundles will also be regarded as indifferent. 4. Martin has the utility function U = x + y a. What is the name for this kind of preferences? b. Draw an indifference curve passing through the point x = 9, y = 0 and an indifference curve passing through the point x = 6, y = 0. c. Does the same hold as for Tomas in 4..b? 4.3 Martha Modest has preferences represented by the utility function U = xy 00 and Bertha Brassy has preferences represented by U = 000x y. Draw two indifference curves for each one and compare their preferences. The first curve must pass through the point x = 8 and y =. The second curve must pass through the point x = 6 and y = Bo has preferences represented by the utility function U(x, y) = x + y a. Draw a few of his indifference curves. b. Does Bo have convex preferences? 4.5 Sture has the utility function U(x, y) = min(x + y, y + x). Draw the indifference curve that passes through the point (6,3).

4 Chapter 5 5. Fredrik's utility function is U = x(y+). Derive the demand for y as a function of income and prices. 5. Now return to Ronny Rigid in chapter 3. His favourite diner has adopted the following policy: if you show up for lunch t hours before or after noon, you get to deduct t euro from your bill. If you come early or late enough, lunch is free. Assume that Ronny has 5 euro and that lunch at noon costs 5 euro. Illustrate Ronny's budget set in a graph, where the horizontal axis measures the time of the day he eats lunch, and the vertical axis measures the amount of money that he will have to spend on things other than lunch. Draw some of Ronny s indifference curves that show that his optimal lunch time is pm. 5.3 The market price for cheese is 50 kr/kg, and the market price for marmalade is 40 kr/kg. Leif consumes only marmalade at these prices. Suppose that Leif has an income of 00 kr. Draw a picture (horizontal axis: cheese, vertical axis: marmalade) that illustrates Leif's budget set and indifference curve consistent with the above description. What can we say about his marginal rate of substitution? Is it greater/less than? Is it necessary for Leif's indifference curve to have the same slope as his budget line? 5.5 Find the demand functions for Bill: U(x,x ) = x x 3 Buster: U(x, y)=x /5 y 3/5 Ben: U(x, y) = (x+) (y+) 3 Barbara: U(x,x ) =3x +x Beth: U(x, y)=min{x, y}

5 Solutions Chapter. PA =, ration cards PB =, 4 ration cards. Total income ind = 9 kr, Rationing cards=4 a. We can solve this by seeing that we have two restrictions, one for income so that A+B=9 income budget line A=9-B and one for ration cards A+4B=4 A=-B Both restrictions must apply, if we draw these then; A Rationing card constraint 9 Budget constraint 6 9 Because both restrictions must apply the budget set is bounded by the inner (darkened) line. b. Individual will not spend his entire income since the budget set is completely bounded by the ration card restriction. 6 A Rationing card constraint Budget constraint 6 6 c. PA = 0,5. We know that he spends all of his income and uses all his ration cards. This means that the choice must lie on the ration card restriction. According to this restriction he can (at the extreme), consume either 6 units of B or units of A. Thus his income must be: PA * A = 0,5* = 6 or PB * B = *6 = 6. Thus his income must be equal to 6. (Any point, not only extremes on the ration card restriction line may be taken)

6 Chapter 3 3. Work is a bad. This means that if we give Jens more work, we have to have to give him some extra income to compensate him. Income Work 3. a. If we give her more than 8 pancakes, the pancakes are bads for Johanna. This means that we have to give her some extra jam to compensate her (jam being always a good ). Pancake 8 More jam compensate Jam b. In this case more than 8 pancakes doesn't change Johannas' utility. (she just leaves them on her plate) Pancake 8 Jam

7 3.3 Draw a 45 line that reflects the situation where the consumption of avocado is equal to the consumption of dg grapefruit. To the left of this line the slope of the indifference curve is - and to the right of this line the da slope is -. Grapefruit 45 Avocado She has convex preferences, but not strictly convex preferences. Since she is indifferent to some of the weighted averages (when taken on only one side of the 45 o line). Convex preferences: Averages are preferred to extremes. If (x, x) (y, y) ==> (tx + [-t]y, tx + [-t]y) (x, x) where 0 < t <. Strict convexity means that the weighted average of the two indifferent bundles is strictly preferred to the two extreme bundles.

8 3.4 If we change the lunch time from noon, we have to compensate him with other goods. Other goods (income) Lunch time 3.5 dcoke The slope ( ) is currently 0,5 ( burgers for Coke). dburgers a. Yes, since he is willing to pay with burgers for Coke. b. No, he would require at least cheeseburgers to give up one coke. c. 0-0 = 0 burgers. (to the level when he has 0 burgers and 5 cokes) d. burgers for Coke. 3.6 a. We have three different points: (cookies, milk) (8,4) (3,) (4,6) We have a case of satiation: there is some overall best bundle, at (8,4). Milk Mother Sister 6 4 x Cookies

9 Chapter 4 4. U(x,y) = xy a. U = 4* = 48 ; which implies that the indifference curve must move through this point and all others where U(x,y)=xy=48; so that the equation for the indifference curve in the x/y plane will be ==> y = x 48, ex: U(,48)=U(,4)=U(8,6)=(48,)=48 etc. graphically the indifference curve for U=48 is: 60 y( x ) x b. If we have that (x0,y0) (x,y), then we also must have that x0 y0 = x y (all according to the utility function). Now we double the amount of each good: U(xo,y0) = xo y0 = 4 x0 y0 U(x,y) = x y = 4 x y ==> 4 x0 y0 = 4 x y (the utility function is homogeneous) 4. U = x + y a. Quasilinear preferences. The indifference curves are vertical transfers of one another. b. The general equation for the indifference curves is: y=u- x; When x=9, y=0, then U(9,0)=3 and the indifference curve for U=3 is y=3- x, while for U(6,0)=4 the indifference curve is y=4- x x y U c. Martin is indifferent between (9,0) and (4,). Now we double the amount of each good: U(8,0) = ,4 and U(8,) 4,83 ==> The same does not hold as for Thomas. Observe that the utility functions are not homogeneous. 4.3 UM = xy 00 UB = 000x y 00U ==> y = M ==> y = x U B 000x U x y U x y 0,6 8 56' ,6 0,6

10 0, ' ,6 ie: for positive values of U as is in this case, the utility functions are monotonic transformations and have the same indifference curves even though the numbers of the curves are different. U B = U M * U = x + y ==> y = U - x 7 Set U to arbitrary values, graphing the indifference curves we get for example: y x b. No, concave preferences. 4.5 For Sture the utility is given by the combination (x + y or y + x) that gives the lowest value. x + y y + x U U(6,3) 6 + *3 = 3 + *6 = 5 (the first) U(4,4) 4 + *4 = 4 + *4 = (both) U(8,) 8 + * = + *8 = 8 (the first) U(,8) + *8 = * = (the second) An easy way to solve this is to sketch the curves for U= separately, ie y=6-0,5x and y=-x. The outer bound must be the relevant indifference curve (because more is better than less, then increasing either x or y from the lower curve will lead to a utility >) y x

11 Chapter 5 5. The problem can be written as Max U = x(y+) s.t m = Py y + P x x Can be solved using ex: Lagrange or the necessary condition that MU x MUy = P x, for the latter: Py First we derive the marginal utilities: du = MU x = y + dx du = MU y = x dy Set the marginal rate of substitution equal to the slope of the budget line: dy MUx MRS = = dx MUy = P x Py y+ x = P x Py x = P y(y+) Px (solve for x) (insert this into the budget restriction) m = Py y + Px P y(y+) Px (Px is cancelled out) m = Py y + py y + Py = Py y + Py (solve for y) y = m - P y = Py m - (insert this into the expression for x) P y x = P y ( m P y P ) X = m Px + P y Px Thus demand for y is: y = m - P y The demand for x is: x = m Px + P y Px

12 5. Income = 5 Price at = 5 Discount: t euro per hour. (here t=) kr Cheese: 50 kr/kg Marmalade: 40 kr/kg Lunch time a. The slope of the budget line dm dc P c 50 ==> PM 40 = -,5 If Leif wishes to buy kg cheese, he must abstain from,5 kg marmalade. But as he is not willing to buy any cheese, his marginal rate of substitution must be is less than,5 (indifference curves are flatter than the budget line). In this case we have a boundary (corner) solution, and it need not be so that the slope of the budget line is equal to the slope of the indifference curve as is generally the case with an interior solution. marmelade,5 Max indifference curve obtainable Budget line cheese

13 5.5 Bill: u(x,y) = x y 3 Use for example Lagrange. Max x y 3 s.t P x x+p y y=m L= x y 3 +( P x x+p y y-m) Set dl dl dl 0 and solve dx dy d From the first two we get y = 3 P x x Py (substitute this into the budget restriction) m = Py 3 P x x Py + P x x = 3P x x + Px x = 5 P x x (solve for x) x = m 5Px (substitute this into the expression for y) y = 3 P x m Py 5Px = 6m 0Py = 3m 5Py Buster: u(x,y) = x /5 y 3/5 Same demand functions as for Bill. The utility function is just a monotonic transformation of Bill s. Check for yourself! What you put into the budget constraint is the same as above. Ben: u(x,y) = (x+) (y+) 3 This may be solved using Lagrange or by using the necessary condition. du = (x+) (y+) 3 du = (x+) 3(y+) dx dy (x+) (y+) 3 (x+) 3(y+) = P x (simplify) Py (y+) 3(x+) = P x Py (solve for y) y = P x Py 3(x+) - 4 y = P x Py 3(x+) - 4 (substitute this into the budget restriction) m = Px x + Py ( P x Py 3(x+) - 4 ) (simplify) m = Px x + Px 3(x+) - Py m = Px x + 3 P x x + 3 P x - Py (solve for x)

14 Px x + 3 P x x = m - 3 P x + Py 5 P x x = m - 3 P x + Py (divide both sides by 5 P x) x = m 5 Px - 3 Px 5 Px + Py 5 Px x = m 5 Px Py 5 Px (substitute this into the expression for y) y = P 3( m x 5 Px Py Py Px + ) - y = P x 3 m Py 5 Px - P x 3 3 Py 5 + P x 3 4 Py Py 5 Px + P x 3 Py - (simplify) y = m 6 Py 0 - P x 9 Py P x 3 Py - y = m 3 Py y = m 3 Py Px Py Px Py Barbara perfect substitutes Two cases: i) If the budget line is flatter than the indifference curve she consumes only x (when p /p <3/ or p <3p ). ii) If the budget line is steeper than the indifference curve she consumes only x x x m p any 0 m p any 0 number between 0 and number between 0 and m/ p m/ p when p when p when p when p when p when p 3 p.5 p.5 p p p p or p.5 p

15 Beth perfect complements always consumes x y m p x p y

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