A. a Typical Person S Indifference Curve Between Food and Garbage. Put Food on the Y-Axis

Economics 331 Fall 2007Homework #3: Key1. For each of the following cases, draw a set of indifference curves. Indicate in which direction utility increases. Provide a brief explanation with your diagrams. a. A typical person’s indifference curve between food and garbage. Put food on the y-axis.FoodGarbageUtility increase to the north-west since you get more food and less garbage. The indifference curves slope up since you need to be compensated for taking more garbage with more food. The slope of the indifference curve increases: for each additional food you acquire, you would be willing to take a smaller and smaller amount of garbage to remain indifferent. b. Oscar the Grouch’s indifference curve for food and garbage. Oscar loves garbage but has no use for food. Put food on the y-axis.FoodGarbage Oscar’s utility increases as you move east. For a given amount of food, Oscar gets more utility as he gets more garbage. The indifference curves are vertical lines since food is a neutral good; holding the amount of garbage fixed, more or less food neither increases/decreases utility. c. Eve likes apples but has no use for pears. Put pears on the y-axis.Same logic as part (b). d. Adam eats beef ribs and pork ribs, but likes beef ribs twice as much as pork ribs. Put pork on the y-axis.PorkBeefUtility increases in the northeast direction since more of both beef and pork increase utility. The goods are perfect substitutes with a MRS of 2: Adam is willing to give up 2 pork ribs for 1 beef rib. e. Holly likes to give large parties, and has a strong preference for having the same number of men and women at the party. Draw two sets of indifference curves under the following conditions: i. extra men or women are useless but they don’t detract from the party.This is just an example of goods that are perfect complements: the ratio of M to W is 1:1. ii. Holly’s utility function is given by: U(W,M) = min {2W-M, 2M-W}MenM=WWomenUtility increases in the northeast direction. The “legs” of the indifference curves both pivot in to the M=W line. Note that one leg has a slope of 2 and the other a slope of 1/2 . This reflects the MRS when there are more men than women at the party and vice versa. 2. You have $200, which you spend on tea and coffee. The price of coffee is $10/lb and the price of tea is $4/lb. Consider the following two government programs: a. On a large diagram depict the following. Make sure you draw your diagram to scale or you may have problems.- in black: draw the original budget constraint and shade in the feasible set (the set of points that are affordable). - in red: show what happens to the feasible set under the simple rationing program. - in blue: show what happens to the feasible set under the ration coupon program.For the simple ration program, the feasible set is the area on an underneath the area bounded by the red line. For the ration coupon proram, the feasible set is the area on an underneath the line ABC. Note that although the ration coupon set is larger, a portion of the set for the simple constraint does lie above the ration coupon set. b. Assuming that your preferences are typical, which rationing program would you prefer? Explain and depict your answer on the diagram.It depends. If you preferences are such that you have a relatively strong preference for coffee over tea or tea over coffee, you would prefer the ration coupon program: see U1 and U2. Otherwise, your utility curve will look like U3 and you will consume at the corner of the simple constraint budget set. c. Assume that your preferences can be described by the utility function U(T,C) = T3/4C1/4 . Which program would you prefer and how much of each good would you purchase?Consider the ration coupon program first. You know that you have to satisfy both the money constraint and the coupon constraint. Which one is relevant for this problem depends on the preferences of tea over coffee. Given the utility function, there is a stronger preference for tea, so it makes more sense to find your utility maximization point by using the coupon constraint. max U(T,C) = T3/4C1/4 subject to 3C + 6T = 1201. Find the MRS:-1/4 ¼ 3/4 -3/4 MRS = MUT/MUC = ¾ T C / ¼ T C = 3C/T2. Set the MRS = PT/PC3C/T = 6/3 → 3C = 2T → T = 1.5C3. Plug into the budget constraint:6(1.5C) + 3C = 120 → 12C = 120 → C* = 10 so T* = 15This is point A on the 2nd diagram with utility U1.(Note that if you used the money constraint first you would have found T*=37.5 and C*=5. This point is not feasible since it lies outside the feasible set).Although this is an optimal point for the coupon program, this person can do better under the simple constraint program. They could consume 15 tea and 14 coffee (U2), which is clearly preferable to 15T and 10C. 42.54037.5 Ration coupon constraint3532.53027.525 EE 22.5 U1 F FO 20 C A 17.515 B 12.510 U3 7.552.5 U2 Regular constraint C 0 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 42.5 45 47.5 50 52.5 TEA42.54037.53532.53027.525 EE 22.5 F FO 20 C17.51512.5107.55 U2 2.5 U10 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 42.5 45 47.5 50 52.5 TEA 3. Peter, Paul and Mary go to the grocery store where they can buy steak (S) and beer (B). The price of steak is $5 and the price of beer is $1. They each have $50 to spend. a. Peter’s preferences can be described by the utility function U(S,B) = S1/2B1/2. How many steaks and beers will he buy? What is his total utility? max U(S,B) = S1/2B1/2 s.t 5S + 1B = 50-1/2 ½ 1/2 -1/2 MRS = MUs/MUB = ½ S B / ½ S B = B/SMRS = PS/PB → B/S = 5/1 → B = 5S5S + 1(5S) = 50 → 10S = 50 → S* = 5 → B* = 25Utility = 101/2 501/2 = 11.18Note at this point you spend half of your income on each good, which is consistent with the utility function. b. Paul’s preferences can be described by the utility function U(S,B) = S3/4B1/4. How many steaks and beers will he buy? What is his total utility? max U(S,B) = S3/4B1/4 s.t 5S + 1B = 50-1/4 1/4 3/4 -3/4 MRS = MUs/MUB = ¾ S B / 1/4 S B = 3B/SMRS = PS/PB → 3B/S = 5/1 → B = (5/3)S5S + 1(5/3 S) = 50 → (20/3)S = 50 → S* = 7.5 → B* = 12.5Utility = 7.53/412.51/4 = 8.52Note at this point you spend $37.50 ($12.50) on steak (beer), which is 75% (25%) of your total income. c. Mary’s preferences can be described by the utility function U(S,B) = S1/5B4/5. How many steaks and beers will she buy? What is her total utility? max U(S,B) = S1/5B4/5 s.t 5S + 1B = 50-4/5 4/5 1/5 --1/5 MRS = MUs/MUB = 1/5 S B / 4/5 S B = B/4SMRS = PS/PB → B/4S = 5/1 → B = 20S5S + 1(20S) = 50 → S* = 2 → B* = 40Utility = 21.97 d. For each person, what is the marginal rate of substitution when the number of steaks is equal to the number of beers. What does this tell you about each person’s relative preferences for steak and beer? Is this consistent with your answers from (a-c)?PERSON MRS IN GENERAL MRS WHEN S=B PETER B/S 1 PAUL 3B/S 3 MARY B/4S ¼The figure below shows each person’s indifference curve going through 10S and 10B (at this point each person has a utility of 10). The slope of a line tangent to each indifference curve at this point is the MRS. Peter for example, is willing to give up 1 B to get 1 more S. Paul on the other hand, is willing to give up 3 B to get 1 more steak; Mary ¼ B. Therefore, Paul has the strongest preference for steak since he is willing to give up the most beer for one more steak. Mary has the weakest preference for steak since she is willing to give up the least beer.Note that this is consistent with each person’s utility function. The weight placed on steak in Paul’s utility function, 3/4 , is the most of all three people; Mary’s is the least.You should also be able to see that the steeper the utility function, the stronger the preference for the good on the x-axis.40353025201510 MARYPETER 5PAUL0 0 5 10 15 20 25 30 35 40Note that you didn’t have to find the MRS for each person where S=B: that was just for convenience. You could have found each person’s MRS where S=2B. Then Peter, Paul and Mary’s MRS’ would be ½ , 3/2, and 1/8 respectively. Again, Peter has the strongest preference for steak; Mary the weakest. 4. Little Miss Muffet sat on her tuffet eating her curds and whey. She will only eat her curds and whey if there is one unit of curd for two units of whey. Muffet has $20 to spend. a. Suppose that curds cost $1 per unit and whey cost $0.75 per unit. Draw a diagram that depicts her budget constraint and indifference curve.See below. b. How many units of each good will she consume to maximize her utility?Muffet’s problem is to max U(C,W) = min {2C,W} s.t. 1c + .75W = 20We know that she consumes C and W a 1:2 ratio or 2C = 1W.We also know that 1C + 0.75W = 20.Therefore, 1C + 0.75(2C) = 20 → 2.5C = 20 → C* = 8 → W* = 16 c. How many units of each good will she consume to maximize her utility if the price of curds falls to $0.50 and the price of whey rises to $1.50?Now her budget constraint is .5C + 1.5W = 20, but her consumption at the optimum still satisfies 2C = 1W.Therefore, .5C + 1.5(2C) = 20 → 3.5C = 20 → C* = 5.714 → W* = 20 d. Write out an equation for Muffet’s demand for whey as a function of her income I, the price of curds, Pc, and the price of whey, Pw. Explain/show how you got it.In general the budget constraint is: PWW+ PCC = I Given the consumption ratio: W = 2C → C = ½ W PWW+ PC( ½ W) = I→ W* = I / [PW + ½ PC]You can check this to make sure it works. In part (b), Pw = 0.75 and PC = 1.00.Therefore, W* = 20 / [0.75 + ½ x 1.00] = 16. 5. Jack and Diane both spend $24 per week on videos and movies. When the price of videos and movies are both $4, they each rented 3 videos and bought 3 movie tickets. Suppose the price of videos fell to $2 and the price of a movie ticket rose to $6. Now, Diane rents 6 videos and buys 2 movie tickets. Jack, however, buys 1 movie ticket and 9 videos. a. Is Diane better or worse off after the price change? Support your answer. b. Is Jack better or worse off after the price change? Support your answer.You know that their new budget constraint contains the same combination of videos and movies that they bought at last week (3 of each). Since we assume tastes don’t change, this tells use that neither of them can be worse this period than last period. Otherwise they would have stuck with the same combination. If we assume that preferences are typical in the sense that the indifference curves are convex, we know that there is an indifference curve tangent to the initial consumption point, but curves underneath the new consumption point (the figure drawn applies to Diane). Since the new consumption point lies on a higher indifference curve, they both must be better off.7654 S E I V OM INITIAL FOR JACK AND 3 DIANE2 NEW DIANE1 NEW JACK0 INITIAL CONSTRAINT NEW CONSTRAINT 0 3 6 9 12 15 VIDEOSNote that it is possible to draw a curve that is tangent to the original (3,3) point and then cross to the left of Diane’s new point. The problem with this is that when you draw Diane’s new indifference curve it would intersect the old one, which is not possible.

A. a Typical Person S Indifference Curve Between Food and Garbage. Put Food on the Y-Axis

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