Suppose the consumer's utility u: X → R is strictly conver i.e. such that any x,y EX with r y and any 0 < X < 1 have Au(x) + (1 − λ)u(y) > u(λx + (1 − X)y). (a) Show that the consumer will do one of the following: Buy nothing. . Spend all their money on good 1. . Spend all their money on good 2. [Hint: The previous question could be helpful.] (b) Suppose that, in addition to benig strictly convex, u is increasing. Explain why the consumer will spend all their money on one good.

Suppose the consumer's utility u: X → R is strictly conver i.e. such that any x,y EX with r y and any 0 < X < 1 have Au(x) + (1 − λ)u(y) > u(λx + (1 − X)y). (a) Show that the consumer will do one of the following: Buy nothing. . Spend all their money on good 1. . Spend all their money on good 2. [Hint: The previous question could be helpful.] (b) Suppose that, in addition to benig strictly convex, u is increasing. Explain why the consumer will spend all their money on one good.

Exploring Economics

8th Edition

ISBN:9781544336329

Author:Robert L. Sexton

Publisher:Robert L. Sexton

Chapter10: Consumer Choice Theory

Section: Chapter Questions

Problem 9P

See similar textbooks

Similar questions

Would you expect total utility to rise or fall with additional consumption of a good? Why?
3. Tony is throwing a party at his Fraternity and is trying to choose what booze to buy. A bottle of vodka has three times the alcohol as a six-pack of beer. Assume that Tony only cares about the total amount of alcohol in his basket. (use vodka on the X-axis and beer measured in six-pack on the Y-axis) a) Devise a utility function to represent these preferences. b) Suppose a bottle of vodka costs $40, a six-pack of beer costs $10, and the budget is $200. Write the budget constraint. c) Solve Tony's utility maximization problem and find the optimal combination. d) Suppose that a bottle of vodka cost has increased to $50. What will be his new optimal combination?
1- If Isabelle's marginal utility for pizza is 10 and her marginal utility for salads is 2, then Isabelle will choose to eat five times more (a) salads than pizzas . (b) Isabelle will be ready to exchange five salads for a pizza. (c) Both of the above answers are correct. (d) None of the answers above is not correct. 2- Consider a consumer who consumes two goods, good 1 and good2. We denote by p1 the price of the good1 and by p2 the price of the good2. We denote by Um1 the marginal utility of good 1 for the consumer and by Um2 its marginal utility of good 2. The quantity of good 1 that the consumer must give up in order to be able to consume an additional unit of good 2 is equal to: (a) -p1 / p2 (b) -p2 / p1 (c) -um1/ um2 -Um2 / Um1. 3- Consider a consumer who consumes two goods, good 1 and good 2. We denote by q1 his consumption of good 1 and by q2 his consumption of good 2. Graphically, we measure the quantity of good 1 on the abscissa and the quantity of good 2 on the order. Suppose…
4. Liam initially has a weekly allowance of €24 to spend on eggs (at €0.40 per egg) and beans (at €0.80 per tin). Liam will only consume 4 eggs with each tin of beans and never changes this proportion. (a) What is the utility maximising amount of eggs and beans that Liam buys each week? (b) Suppose the price of eggs rise to €0.60 per egg but the price of beans falls to €0.60 per tin. At the same time, Liam's weekly allowance rises to €30. How much of each good will Liam now buy? Compare your answer to that in (a) and explain the difference, if there is one. Use a diagram to complement your answer.
The table below represents how Marco feels about chocolate candy bars. a. Fill in the missing values for total utility and marginal utility. Instructions: Enter your answers as a whole number. Chocolate Candy Bars and Marco's Utility Chocolate Candy Total Utility Bars (utils) 0 0 1 27 2 3 4 5 6 (Click to select) 60 ♦ 76 Marginal Utility (utils) 27 19 Suppose Marco currently has two candy bars. You tell Marco you will give him either a soda, which gives him 28 utils of happiness, or two additional candy bars. b. Which is he likely to prefer? 6 1
2. Preferences:a. Suppose Wendy consumes 10 sandwiches and 10 cups of coffee over the course of a week.Her utility at the end of the week is equal to10. Now suppose Amanda consumes the same bundle, but her utility at theend of the week is 20. Can we say that Amanda is twice as happy as Wendy?Why or why not. b. Alex prefers 3 slices of cake to 4 slices,4 slices of cake to 5 slices, and 3 slices of cake to 5 slices. What is so strange about Alex? Are Alex's preferences transitive? Is Alex an irrational consumer? Explain your answers briefly.
. Answer all parts (a) (c) of this question. (a) Consider an agent whose preferences over any couple (x1, x2), where 2₁ ER+ and x2 € R+, e.g., apples and oranges, is such that she prefers the bundle that is closest to having the same number of apples and oranges. Write a utility function u: R² → R+ which represent these preferences. A politician remarks "Our recent increases in the wage rates of teachers has been a total suc- cess! The shortage of teachers has been reduced drastically. Another, similar wage increase should eliminate this shortage entirely" (b) Explain and illustrate in a diagram what is meant by "income effects" and "substitution effects" of a wage rate change. (c) Explain and illustrate how you would model the labour supply decision of a potential teacher. Do you agree that the wage increase will increase the labour supply in this case? Carefully outline the assumptions underlying your argument.
3. Susan has the following utility function U(x,y) = xy, where x represents food and y represents other goods. Susan has £20 (budget m) to spend on these two goods. Suppose that the initial price of x (Px) is £1 per unit, and that of y (Py) is £1 per unit. Note that MUx=y and MUy=x. (i) How many units of x and y will Susan purchase? Assuming that Px varies and m and Py stay constant at the values £20 and £2, respectively, what is the individual demand function for x? (ii) What are the characteristics of the individual demand function for x? What is its aggregate counterpart, and what is its economic importance?
8. Show that if strict preferences > over three outcomes x, y and z, do not satisfy transitivity, so that x > y > z > x, then these preferences cannot be represented by a utility function.
6. My utility over goods 1 and 2 is given by: u(x1,x2)=min{x1/a,x2/b} Where x1is the quantity of good 1 I consume and x2 is the quantity of good 2 I consume. Assuming I purchase only goods 1 and 2... If a = 2, b = 5, the price of good 1 is $2/unit, the price of good 2 if $7/unit and my income is $5, how much good 2 will I consume to maximize my utility? (Note: The answer may not be a whole number, so round your answer to the nearest hundredth) (Note: The numbers may change between questions, so read carefully)
4.. If the preferences are concave, will the consumer ever consume both of the goods together?
True or false with reasoning: 1) _______When we claim that utility can be ordinally measured, we assume that the consumer is able to measure the total and marginal utility received when one extra unit of a commodity is consumed. 2)_______If MRS between two goods is constant, then having more of one good without having more of the other does not increase utility. 3)_______Marginal Utility increases until total utility is at a maximum and then marginal utility decreases.