With utility function u(x1, x2) = √x1 + x2, has 100 dollars to spend on goods 1 and 2.(a) The prices are p1 = 1 and p2 = 4. Find the optimal bundle x∗1 , x∗2. B)Keep the price of good 1 as p1 = 1 and let p2 be a parameter, p2 = p > 0. Find the optimal bundle as a function of p, x1(p), x2(p) and compute the derivatives x1′ (p), x2′ (p).

With utility function u(x1, x2) = √x1 + x2, has 100 dollars to spend on goods 1 and 2.(a) The prices are p1 = 1 and p2 = 4. Find the optimal bundle x∗1 , x∗2. B)Keep the price of good 1 as p1 = 1 and let p2 be a parameter, p2 = p > 0. Find the optimal bundle as a function of p, x1(p), x2(p) and compute the derivatives x1′ (p), x2′ (p).

Microeconomics A Contemporary Intro

10th Edition

ISBN:9781285635101

Author:MCEACHERN

Publisher:MCEACHERN

Chapter6: Consumer Choice And Demand

Section: Chapter Questions

Problem 6QFR

See similar textbooks

Similar questions

A consumer has preferences over two goods, denominated by x and y, given by the utility function U(x,y) = min{αx,y} with α > 0. The prices of the good are px = 2 and py = 5. The consumer has an income of I > 0. (a) Provide an intuition for this utility function. Specifically, are these goods substitutes or complements? If x is bicycle tires and y is bicycle frames, what is the value for α? (b) For what values of α will the consumer demand (i.e., Walrasian demand) more x than y. (c) For what values of α will the consumer spend more on x than on y (given her Walrasian demands).
Consider the utility function U(x,y) = xy, which describes the enjoyment Emi gets from consuming tacos (x) and sandwiches (y) over a period of 1 week. a) Does Emi like both tacos and sandwiches? Does she like variety? b) Let Emi have budget I=$24, and let prices be Px=$2, Py=$4. Find Emi's optimal basket of goods x and y. Is this an interior or a corner solution? c) What will happen if tacos (good x) go on sale for $1? Find the new optimal bundle.
You consume music (M) and concert tickets (C). Your utility function is U(M, C) = M1/4C3/4. The marginal utility for concert tickets, MUC is MUC =3/4C-1/2M1/4 and the marginal utility for music, MUM is MUm = 1/4C3/4M-3/4 (a) Calculate MRSMC using only the given marginal utilities. (b) Solve for the utility of bundle A where M = 16 and C = 16. Solve for your utility at bundle B where M = 128 and C = 8. Are the utilities the same? (c) Calculate MRSMC at bundle A and at bundle B. Are they the same? (d) Are your indifference curves convex? Draw the ICs. Make sure to label the quantities of the consumption bundles, the axis, and the MRS at those bundles.
A consumer’s utility only depends on the consumption of goods A and B according to the following Cobb-Douglass utility function: U(A, B) = A1/4 B 3/4. The price of goods A and B are $20 and $40, respectively. The consumer has a budget of $1200 that he can use to consume the two goods. a. Write down the budget constraint and plot it. b. Calculate the optimal bundle and maximized utility for the consumer. c. A new tax of $10 is imposed on the price of good B. Compute the new optimal bundle of good A and B for the same consumer. What is the utility loss due to the tax? d. Show that the consumer would prefer a lump sum income tax that raises the same revenue as the tax on good B. Note:- Do not provide handwritten solution. Maintain accuracy and quality in your answer. Take care of plagiarism. Answer completely. You will get up vote for sure.
Peter's preferences over two goods, x and y, are represented by the utility function u(x, y) = y + 2x. a) Peter is currently consuming bundle A = (2,4) with 2 units of good x and 4 units of good y. Calculate his current level of utility from consuming this bundle. b) Write the expression the indifference curve representing Peter's current level of utility (i.e., the one you found in part (a). Next draw this indifference curve. c) By looking at the indifference curve you drew in part (b), answer the following questions: Does Peter like good x? Good y? Explain. What can you say about the marginal rate of substitution of good x for y, MRSxy? Is it positive? Negative? Constant? Increasing? Decreasing? Interpret/explain your answer in terms of the tradeoffs Peter is willing to make between goods to keep the same utility level. d) On the same graph you drew in part (b), draw the indifference curve for a utility level of 10. Plot and label in the graph bundles B = (1,2), C = (1,6), and D =…
Suppose a consumer has a utility function given by U(X,Y) = MIN(X, 2Y). The consumer has $90 to spend (M = $90). The Price of good Y is Py = $1. a) Draw the consumer's Price Consumption Curve for the following prices for good X: Px = $1, Px = $2, Px = $2.5. For each set of prices, make sure to accurately draw each budget constraint, and also identify the consumer's optimal bundle on each budget constraint. Make sure your graph is neat, accurate, scaled properly, and fully labeled to receive full credit.
A consumer with I dollars budget has the utility u(x,y) = x(y+1) over amounts of cake (x) and ice cream (y) she consumes. The prices are px , py respectively. (a) Derive her demand for cake (x), as a function of prices px , py and her budget I. (b) Looking at the demand function in (a), is cake a normal good or an inferior good? Are cake and ice cream complements or substitutes? (c) Calculate the (i) (own) price, (ii) income, (iii) cross- price elasticity of demand for cake at the point where I = 80, px =10, py = 20.
Suppose the utility function of U(x1, x2) = x11/2x21/2 and the budget constraint of p1x1+p2x2=m. Let’s assume that p1=$1.5, p2=$2, and m=$60. Find the optimal bundle. Also, specify the optimal bundle.
I. You only consume two goods and your preferences are represented by utility function U(x₁, x₂) = (x0.5 + x2.5)² for x₁ > 2 and x₂ > 0. 1. Find your marginal utility for each good. Simplify your answers. Based on your answers, are your preferences strictly monotonic? Why or why not? 2. Do your preferences obey the law of diminishing marginal utility for each good? Why or why not? If not, what is the behavior of your marginal utility for each good? Use calculus and words to answer this question.
Consider a consumer who has utility function U(x,y)= x + 2y, where x is the amount of good 1, and y is the amount of good 2, and income of 90. Suppose the price of good 1 is 5 dollars per unit, and the price of good 2 is 15 dollars per unit. What is the consumer's quantity demanded of each good? I.e. which consumption bundle (x,y) that the consumer can afford maximizes their utility? O(4.5,3) O (20,6) (18,0) O (0,6)
Biwei’s utility function U=4XY, where X is consumption of beer and Y is consumption of pizza. For this utility function, the marginal utility of X is MUx = 4Y; the marginal utility of Y is MUY = 4X. 1) Suppose Y = 3. Calculate Biwei’s utility for X = 2, 3, 10, and 11. For a given level of Y, does good X display diminishing marginal utility? 2) Suppose X = 3. Calculate Biwei’s utility for Y = 2, 3, 10, and 11. For a given level of X, does good Y display diminishing marginal utility?
Ricky has utility function u=x'y. This implies that MUx=2xy. MUy=x². His income is 100. The price of y is 10. (a) Find his demand for x at price 20. (b) Find his demand for x at price 30. (c) Write down his demand function for x: that is, write down his demand for x as a function of the price of x.