How do I do number 3a?
Homework # 3 1. Each day Paul, who is in third grade, eats lunch at school. He only likes Twinkies (T) and Orange Slice (O), and these provide him a utility of U T , O TO . a) If Twinkies cost $0.10 each and Orange Slice costs $0.25 per cup, how should Paul spend the $1 his father gives him in order to maximize his utility? b) The school decides to discourage Twinkie consumption and therefore raises the price of Twinkies from $.10 to $0.50. How many Twinkies and cups of Orange Slice will Paul buy after the price change? c) Paul's father does not like to see his son suffer, so he decides to increase Paul's lunch budget so that he can be as happy after the price change as he was before. How much additional money would his father have to give him for lunch every day to achieve this goal? 2. A young connoisseur has $300 to spend to build a small wine cellar. She enjoys two vintages in particular: an expensive 2013 French Bordeaux WF and a less expensive 2016 California wine WC . Her preferences over the two wines can be characterized by the utility function U WF , WC WF2 / 3WC1/ 3 . a) Suppose the French and California wines are priced at $20 and $4 per bottle, respectively. How much of each wine should our young connoisseur purchase? b) When she arrives at the wine store, our young wine connoisseur discovers that due to a decline in the value of the Euro, the price of the French wine has fallen to $10 per bottle. If the price of the California wine remains stable at $4, how much should our friend purchase to maximize utility in the face of these new conditions? Is she better, worse, or equally well off after the price change? 3. Suppose an individual has perfect-complements preferences that can be represented by the utility function U x , y min 3x ,2 y . Furthermore, suppose that she faces a standard linear b g budget constraint, with income denoted by m and prices denoted by px and p y , respectively. a) Derive the demand functions for x and y. b) How does demand for the two goods depend on the prices, px and p y ? Explain. 4. Kristin's has $20 to spend on two soft-drinks, Poke (P) and Cepsi (C), that she enjoys. The two drinks are very similar, so Kristin's preferences for them can be represented by the following utility function: U P, C 4 P 3C . a) If the price of Poke and Cepsi is $1 and $2 per can, respectively, how much of each will Kristin choose to buy? b) Suppose the price of Cepsi falls from $2 to $1.50. What will happen to Kristin's optimal consumption of Poke and Cepsi? Explain your answer carefully. c) Describe what will happen to Kristin's optimal consumption if the price of Cepsi continues to fall. 5. A consumer faces a standard linear budget constraint and has preferences that can be represented by the following utility function: U x , y x 2 ln y . a) Suppose that we have an interior solution. Derive the demand functions for x and y. Denote the price of x by px , the price of y by p y , and income by m. b) Is it necessarily the case that the optimal consumption is interior the way we assumed in part a)? If so, why is that? If not, derive the non-interior solution. b g 6. Arthur consumes hot dogs (H) and buns (B). His utility function looks as follows: U H , B min 13H ,13B . Arthur has $60 to spend. A hot dog costs $2 and a bun costs $1. a) How much of the two goods will Arthur consume? b) Suppose the price of a hot dog increases to $3. How many of each good will Arthur buy now? c) Comment on how the price change affects Arthur's mix of hot dogs and buns. 7. There are two things that Adana likes: chips (C) and guacamole (G). The prices of the two goods are $5 for a bag of chips and $20 for a jar of guacamole. Adana has $400 to spend and her preferences are captured by the utility function U C , G C 4G12 . a) Derive Adana's optimal consumption bundle of chips and guacamole. b) Adana gets a pay increase, so now she can afford to spend $600 on the two goods. At the same time, the price of chips increases to $15, while that of guacamole falls to $15. How much of the two goods will Adana consume after these changes? 8. Carina uses paper to write. There are two kinds of paper. One kind, denoted by F, has pretty flowers at the top. The other, denoted by S, has a picture of Spiderman. Carina's utility function for the two types of paper looks as follows: U F , S 2 F 5S . A package of flowered paper costs $5, while one with Spiderman costs $10. Carina has $120 to spend on paper. a) Does Carina like one type of paper more than the other? Explain. b) What quantities of the two types of paper would Carina choose to buy? c) Suppose the price of flowered paper falls to $3. How much of the two types of paper would Carina buy now? 9. Jane likes bread (B) and cheese (C). But she is very particular about her eating. She must have exactly two slices of cheese on every piece of bread. She has $12 to spend and a piece of bread and a slice of cheese each costs $1. a) Write down a utility function that represents how Jane feels about bread and cheese. Explain why this utility function is an accurate model of her preferences. b) Derive Jane's optimal consumption bundle of bread and cheese. c) Suppose the price of cheese falls to 50 cents. Derive Jane's new consumption bundle. 10. Luca eats red (R) and green (G) peppers. A utility function representing his preferences over the two goods is U R, G 32 R 16G . In Palermo where Luca lives, the prices of red and green peppers are 3 and 1, respectively. He has 30 in his pocket. a) Derive Luca's optimal consumption bundle of red and green peppers. b) The price of green peppers increases to 3. Derive Luca's new consumption bundle
