2. Consider an individual who not only consumes goods x and y, but also needs a mini-mum amount of x = xo to survive. Her (Stone-Geary) utility function is given byU(x, y) = (x − xo)¹/2y¹/2Let prices be pa, Py and income is I > Prxo. (b) How do the ratios p/I and pyy/I (i.c., budget shares) change as income increases

The individuals maximise their utility at point where the slope of the indifference curve ( Marginal…

Answered: (b) How do the ratios prx/I and pyy/I…

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(b) How do the ratios prx/I and pyy/I (i.c., budget shares) change

Exploring Economics

8th Edition

ISBN:9781544336329

Author:Robert L. Sexton

Publisher:Robert L. Sexton

Chapter10: Consumer Choice Theory

Section: Chapter Questions

Problem 3P

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2. Consider an individual who not only consumes goods x and y, but also needs a mini- mum amount of x= xo to survive. Her (Stone-Geary) utility function is given by U(x, y) = (x − x)¹/2y¹/2 Let prices be Pa, Py and income is I > Prxo.
2. Consider an individual who not only consumes goods x and y, but also needs a mini- mum amount of x = xo to survive. Her (Stone-Geary) utility function is given by U(x, y) = (x — xo)¹/²y¹/² Let prices be pa, Py and income is I > Pro. (a) Show that this individual maximizes utility by spending (I-Pro) + Pro on good rand(I - Pro) on good y. Provide an economic interpretation for this result. [Hint: It might help to define z = x - xo and I = I - Pro. Then find the optimal z (and therefore x = z + xo) and y]
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.
Adrienne consumes three goods, x, y, and z. Her consumption preference is given by the utility function: u(x, y, z) = x*y} z#, (Pz-Py, Pz) where x, y, and z are the quantities she consumes (respectively). Let p = be the prices of the goods. Answer the following questions, and justify your answers carefully and mathematically. (a) Last year p = sumption bundle last year. (1, 10, 10) and Adrienne's income was 35. Find her optimal con- (b) This year, the prices of x and z have both risen 3.5% while the price of y has risen 7%, and Adrienne got an income raise of 3%. Find the (approximate) percentage changes of her consumption in the three goods due to these changes in prices and income. Does her raise adequately compensate her for the changed prices? What additional percentage change of her income would be needed if we were to restore Adrienne to last year's exact utility level?
Question 2 David spends his budget on chocolate and chip. His utility function is given by ?(?1,?2)= 2?1?2, where ?1 is the number of chocolates he consumers per week, and ?2 is the number of chips he buys per week. A chocolate costs 10 SEK, and a chip costs 20 SEK. David’s weekly budge for consuming on these two goods is 120 SEK. (1) What is David’s budge line? Draw the budget line on a graph with chocolate amounts on the horizontal axis and chip amounts on the vertical axis. Write explicitly at which points budget line crosses the axis. (2) What is David’s marginal utilities for the two goods, respectively? What is his marginal rate of substitution between the two goods? (3) What is David’s optimal choice? Calculate the numerical answer for the optimal bundle. Also draw an indifference curve for David on the same graph as question(1) and show the optimal bundle.
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.)
QUESTION 1 For the utility function U = Qx0.50Qy(1-0.50) and the budget 122 = 8Qx + 14Qy find the CHANGE in optimal consumption of Y if the price of Xincreases by a factor of 1.7. Please enter your response as a positive number with 1 decimal and 5/4 rounding (e.g. 1.15 1.2, 1.14 = 1.1).
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VI. Utility Maximization 3. Consider an individual with the following utility function, U(x, x2) = ax5x95. The current price %3D of x, is 2 and the current price of x, is 4. 1. What is this consumer's budget constraint? 2. Calculate the optimal consumption of x, and x, if the consumer has income m.
Suppose a consumer has utility function U(X, Y) = X2/3Y1/3. Suppose the consumer has M =$90 to spend and the prices of goods X and Y are PX = $15 and PY = $6 . a. Carefully express the consumer’s choice problem, using the given information (this is whereyou write out the max operator, the choice variables, the objective function, and the budget constraint). b. Compute the absolute value of the consumer’s marginal rate of substitution, and inspect it todetermine the shape of the consumer’s indifference curves: C-shaped, linear, )-shaped, or some other shape.To show your work, neatly use the arrow argument, increasing X (↑) and decreasing Y (↓) to see whether|MRS(X, Y)| is diminishing along an indifference curve. c. If the indifference curves are C-shaped write out the budget line and the equal slopes conditionthat characterize an interior solution to the consumer’s choice problem. Use the particulars for the givenconsumer. Solve these conditions to find the interior solution. On the…
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7. Suppose U = x₁x2 and the budget constraint is given as x₁ + 1 = B, solve for x and 1-α 8. Suppose U = xx and our budget is given as p₁x₁+P2x2 = m, where m is our income. Solve for solve for x and x₂. 9. Suppose now we want to minimize our expenditure while achieving a certain amount of utility. So say our budget is given as P₁x₁ + P2x2, which we want to minimize, while our utility x and x = u, where u is the level of utility we would like to achieve. Solve for x and x₂.