1-aAnswer following questions by using the Cobb-Douglas utility function u=xx and assuming that the price of good 1 is pi and good 2 is p2. Assume 0<a<1, pi>0, p2>0, and income I>0. Suppose that we consider a tax on housing. We can analyze the impact of tax byanalyzing the consumption of square feet of housing (x1) and the dollars of"other goods consumption" (x2). Suppose that t dollar of tax is imposed persquare feet of housing. Show graphically and explain in words, how much thetotal tax revenue (T), the consumer would be willing to pay to remove such a tax(L), and the dead weight loss would be by drawing the indifference curve of theconsumer.

To analyze the impact of a tax on housing using the Cobb-Douglas utility function, follow these…

Answered: 1-a Answer following questions by using…

Microeconomics

13th Edition

ISBN:9781337617406

Author:Roger A. Arnold

Publisher:Roger A. Arnold

Chapter7: Consumer Choice: Maximizing Utility And Behavioral Economics

Section: Chapter Questions

Problem 5QP

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1 Consider a consumer who consumes only two goods, 21 and z2. His utility function is u (21, 22) = In z1 + 2. where In represents the natural logarithm. 1.a Derive the consumer's MRS of good 1 for good 2 using calculus to calculate his marginal utility from z and his marginal utility from z2. 1.b If the price of z1 is $2 per unit and the price of z2 is $4 per unit, and the consumer's income is $100, What is the equation of this consumer's budget line? 1.c What are the optimal consumption choices, z and z5, for this consumer? Show your work.
In a simple but delicious world, Joey eats only sandwiches, s, and jam, j. He has a Cobb-Douglas utility function ?(?,?) = ??!"#?#, where 0 < ? < 1 and ? > 0. The price of jam is ?$, the price of sandwiches is ?%, and Joey has a monthly budget ? to spend on lunch. a. Explain why you can safely use a simpler Cobb-Douglas utility function, ?(?,?), to represent Joey’s preferences, which is the same as ?(?,?) except for replacing N with 1. b. Transform ?(?,?) by taking natural logs and bringing down exponents. Explain why it is useful to do this for a Cobb-Douglas utility function, but not for a quasi-linear utility function. c. Use ln(?(?,?)) and the substitution method to derive the formulas for Joey’s optimal amount of jam, j*, and sandwiches, s*, to buy and consume per month. d. What fraction of his income does Joey spend on jam, and what fraction on sandwiches?
2. Suppose John's utility function is UX,Y) = 4/XY , where X is consumption of beer and Y is consumption of pizza. For this utility function, ( Calculus Kahuna say: MUx = 2Y/X and MUy = 2\X/Y ) a) Suppose Y=3. Calculate John's utility for X=4,9,16, and 25. For a given level of Y, does good X display diminishing marginal utility?
4. If we have the utility function, U(x)=(x−7)^2, and we know that there is an indifference set around x = 4 and an upper contour set of x = 6. How to graphically represent this function? I'm confused about how to draw the upper contour set on the graph? I know the concept that upper contour is the area over the convex line, but don't know how to put it on the graph. Could you draw a simple graph to help me identify the upper contour?
Suppose Carlos has to choose between purchasing jewelry and water. Which of the following is the utility-maximizing rule that Carlos should follow while choosing the optimal quantities of these two goods? (Note: In the answer options that follow, MU stands for "marginal utility.") O (MU of Water) x (Price of Water) = (MU of Jewelry) x (Price of Jewelry) O MU of Water = MU of Jewelry MU of Jewelry %3D Price of Water MU of Water Price of Jewelry MU of Jewelry Price of Jewelry MU of Water %3D Price of Water Since water costs little and jewelry is expensive, it must follow that when people choose their optimal quantities of water and jewelry to purchase, the marginal utility they receive from the last piece of jewelry they buy is than the marginal utility they receive from the last gallon of water they buy.
4 Emily goes camping on the weekend and eats marshmallows (M) and crackers (C) there. She wants to match 1 marshmallow with 2 crackers, but does not eat in different proportion (i.e., perfect com- plement). The price of a marshmallow is 10 cents and the price of a cracker is 5 cents, and she spend up to $3 in total. a) Write down a utility function which represents Emily's preferences over marshmallows and crackers. b) How many marshmallows and crackers is she going to consume? (Hint: the first-order conditions will not help; draw the budget constraint and the indifference curves and look at the highest one that intersect the budget constraint). c) What is the exact value of Emily's indirect utility?
Mariah consumes music downloads (M) and concert tickets (C). Her utility function is giveh by U = 0.5M? + 2C². a. Using calculus, what is the marginal utility of music downloads? b. Using calculus, what is the marginal utility of concert tickets? Write the equation for the MRSMC (i.e., the MRS of C for M; M is increasing, C is С. decreasing). d. Would the bundles of (M = 4 and C = 1) and (M = 2 and C = 2) be on the same %3D %3D %3D %3D indifference curve? How would you know? e. Calculate the MRSMC when M = 4 and C =1 and when M = 2 and C = 2. Explain why the %3D %3D values of MRSMC differ across these two bundles. f. Based on your answers above, are Mariah's indifference curves convex? Explain using the values calculated. er this
1. Is it impossible for two indifference curves to intersect one another? why? 2. What does it mean that preferences are complete? Now would the real-life implica- tions of this assumption look like?
Claire has the following utility function: ??(??, ??) = ??1/2??1/2 and faces the budget constraint: ?? = ?????? + ??????. Suppose ?? = 120, ???? = 1 and ???? = 4. Find the optimal ??and ??
2. Eren's two main hobbies are taking vacations overseas (V) and eating expensive meals (M). His utility function is given as: U(V,M) = V²M Last year, the average price of taking a vacation overseas was US$200 and the average price of an expensive meal is $50. However, due to supply problems in Onions, the average price of an expensive meal rose to $75. The average price of a vacation did not change. His income, which is $1500, did not change. a. b. C. Calculate the change in consumer surplus from consuming the expensive meals considering the price change (Hint: you need to compare his optimal consumption bundle before and after the price change to get the change in CS). Suppose that the Department of Welfare wants to know how much should be given to Eren to offset his change un utility due to the price increase of an expensive meal. Calculate the compensative variation (CV). Calculate for the equivalent variation (EV) for the price change.
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.
A consumer with income I=120 facing prices pX = 4 and pY = 8 for two goods X and Y (for each good she prefers more to less) chooses optimally to consume 12 units of X. If the prices change and now pX = 6 and pY = 4, what is the possible range for her optimal X consumption? (like, x >/ 7 or 10 >/ x >/7…etc. Use indifference curve analysis on a graph to reason about the possible locations of the new optimal bundle.)

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