Could you solve last 3 questions please? (a)_Suppose we have preferences U(X, Y) = X² Y².. What is the utility at the bundle X = 5 and Y = 20?Create a table then graph/sketch the indifference curve through (5, 20). Interpret the MRS(with MRS = Y/X) at this bundle. <(b) _Let prices be Px = $5, Py = $5 and income M = $1000. Draw/sketch the budget constraint.Explain what the slope of the budget line means in economic terms.<(c) State the consumer's utility maximization problem and express this in words. What are the twoconditions (equations) that identify the optimum? Sketch this in a figure and explain.<(d) Use the information in (a) and (c) to show that demands are: X = ½ M/PX and Y = ½ M/PY.<(e) For the income and prices in (b), what is optimal X, Y, and utility? Show your work.(f) Suppose Px falls to $4 and Py rises to $7 but income stays at $1000. Does consumer utility riseor fall? Show and explain. <(g) Calculate the Compensating Variation that ensures the consumer is no worse off nor better offwith these price changes. Show and explain your work.

Compensating variation: When the price of a good changes and hence the consumption bundle, that new…

(e) For the income and prices in (b), what is optimal X, Y, and utility? Show your work.< (f) Suppose Px falls to $4 and Py rises to $7 but income stays at $1000. Does consumer utility rise or fall? Show and explain. (g) Calculate the Compensating Variation that ensures the consumer is no worse off nor better off with these price changes. Show and explain your work.<

(e) For the income and prices in (b), what is optimal X, Y, and utility? Show your work.< (f) Suppose Px falls to $4 and Py rises to $7 but income stays at $1000. Does consumer utility rise or fall? Show and explain. (g) Calculate the Compensating Variation that ensures the consumer is no worse off nor better off with these price changes. Show and explain your work.<

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.7P

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(a) Suppose we have preferences U(X, Y) = min [X, 3Y]. Graph/sketch the indifference curve through the bundle (X = 30, Y = 30). What is the utility of (30, 30) and explain why the indifference curves look the way they do. (b) What does the Marginal Rate of Substitution tell us about preferences? < (c) Why is the Marginal Rate of Substitution not applicable in this example? < (d) What do we mean by a composite good? What does this composite good look like with these preferences? Show and explain.< (e) State the consumer's maximization problem and express this in words.< (f) Now let Px = 10, Px= 20 and income M = 2000. Find optimal X*, Y*, and the resulting Utility (U*). Show your work. < (g) Now let Py = 15. How does optimal consumption (X*.Y*) and utility (U*) change relative to (e)? Explain in simple terms and show in a diagram.<
(a) Suppose we have preferences U(X, Y) = 10X²/³ Y¹/3, Create a table and graph/sketch the indifference curve through the bundle X = 30 and Y = 30.< (b) The Marginal Rate of Substitution is MRSxy=-2Y/X. For the bundle (X= 30, Y = 30), calculate and then interpret what the value of the MRS means.< (c) Cobb-Douglas preferences are strictly convex. What does this imply about the MRS as we move along the indifference curve? Explain/discuss (you may want to draw a picture). < (d) What are the two conditions (equations) that identify the optimum given these preferences and the consumer's budget constraint? Sketch this in a figure and explain.< (e)_From (d) we can show that optimal demands are: X=½ M/PX and Y = ½ M/Px. (you do not have to derive these, just use the equations I have given you.) Calculate optimal demands (X*, Y*) and utility if Px = 10, Px= 5 and income M = 1200. < (f)_Suppose Px falls to Px = 8 but Py and M are unchanged (Px = 5 and M = 1200). Calculate the new optimal demands…
(a)_Suppose we have preferences U(X, Y) = X² Y².. What is the utility at the bundle X = 5 and Y = 20? Create a table then graph/sketch the indifference curve through (5, 20). Interpret the MRS (with MRS = Y/X) at this bundle. < (b) _Let prices be Px = $5, Py = $5 and income M = $1000. Draw/sketch the budget constraint. Explain what the slope of the budget line means in economic terms.< (c) State the consumer's utility maximization problem and express this in words. What are the two conditions (equations) that identify the optimum? Sketch this in a figure and explain.< (d) Use the information in (a) and (c) to show that demands are: X = ½ M/PX and Y = ½ M/PY.< (e) For the income and prices in (b), what is optimal X, Y, and utility? Show your work. (f) Suppose Px falls to $4 and Py rises to $7 but income stays at $1000. Does consumer utility rise or fall? Show and explain. < (g) Calculate the Compensating Variation that ensures the consumer is no worse off nor better off with these price…
2. Sally consumes two goods, X and Y. Her utility function is given by the expression U = 3XY². The current market price for X is £10, while the market price for Y is £5. Sally's current income is £500. (a) Sketch a set of two indifference curves for Sally in her consumption of X and Y. (b) Write the expression for Sally's budget constraint. Graph the budget constraint and determine its slope. (c) Determine the X, Y combination which maximises Sally's utility, given her budget constraint. (d) Calculate the impact on Sally's optimum market basket of an increase in the price of X to $15. What will happen to her utility as a result of the price increase?
5. Consider a consumer whose utility function isu(x,y) = sqrt(xy) (MRS(x,y)=y/x)a. Assume the consumer has income $120 and initially faces the prices px = $1 and py = $1. Howmuch x and y would they buy? Draw the budget constraint and the demands. b. Next, suppose the price of x were to increase to $2. How much would they buy now? Draw thisin the same figure.c. Decompose the total effect of the price change on demand for x into the substitution effect and theincome effect. That is, determine precisely how much of the change is due to each of thecomponent effects. (Hint: See the lecture notes for the two properties that determine the locationof “z”, the reference point for distinguishing the income and substitution effects.)
2. Sally consumes two goods, X and Y. Her utility function is given by the expression U = 3XY². The current market price for X is £10, while the market price for Y is £5. Sally's current in come is £500. (a) Sketch a set of two indifference curves for Sally in her consumption of X and Y. (b) Write the expression for Sally's budget constraint. Graph the budget constraint and determine its slope. (c) Determine the X,Y combination which maximises Sally's utility, given her budget constraint. (d) Calculate the impact on Sally's optimum market basket of an increase in the price of X to $15. What will happen to her utility as a result of the price increase?
4. Assume the prices of r and y are 1 and income is 10. For the following two utility functions, find the quantities of r and y consumed at the con- sumer's optimum. (Here, it may be helpful to solve by drawing out some indifference curves.) (a) U(r, y) = Min(x, 2y) (b) U(x, y) = 2x + y (c) Suppose the price of a rises to 3. What are the new optima in each case. What are the substitution effects?
1. A consumer has a utility function defined over two goods X and Y. Let the quantity of Good X be x ≥ 0 and the quantity of Good Y be y ≥ 0. The utility function is given below: u(x, y) = xy + 2y. Assume that the consumer has income m and that prices are px and py. (a) Explain whether the preferences underlying this utility function satisfy completeness and transitivity. = (b) Explain whether the preferences underlying this utility function satisfy monotonicity and convexity (c) Find the consumer's Marshallian demands for Good X and Good Y at prices px > 0 and py > 0. (d) Show that goods X and Y are normal goods and explain whether either good is a substitute for the other. (e) Assume that px = 10, Py = 5 and m = 100. Suppose that px increases to p how much of the change in demand for Good X is via the substitution effect and how much is via the income effect? Note: You may assume an interior solution (i.e. x > 0 and y > 0). = 15,
1. A consumer has an income of $3024 to spend each day. The only two goods the consumer is interested in purchasing are goods A and B. The marginal-utility schedules for these two goods are shown in the table below. The price of B does not change and is $378. The marginal utility per dollar from B is also shown in the table. But the price of A varies as shown in the table. The marginal utility per dollar from A when the price of A is $1512 and $756 is shown in the following table. Good A Good B Quantity MU MU/$1512 MU/$756 MU MU/$378 1 24 2 15 3 12 4 8 5 6 6 4 48 32 24 16 8 4 ||||| ||||| |||
Mylie's total utility from singing the same song over and over is 70 utils after one repetition, 110 utils after two repetitions, 140 utils after three repetitions, 160 utils after four repetitions, 120 utils after five repetitions, and 60 utils after six repetitions, as shown in the table below. a. Write down her marginal utility for each repetition. Instructions: Enter your answers as a whole number. If you are entering any negative numbers be sure to include a negative sign (-) in front of those numbers. Repetitions 0 1 2 3 4 5 6 Total Utility Marginal Utility 0 70 110 140 160 120 60 b. Once Mylie's total utility begins to decrease, does each additional singing of the song hurt more than the previous song or less than the previous song? O More than the previous song O Less than the previous song
Give examples by drawing a graph of each of the following. Please draw your graphs neatly using a ruler to draw the axes and label your graphs completely. In each case draw ar least three indiffer- ence curves and indicate clearly the direction of increasing prefer- ence. (i) the preferences being convex but not strictly convex. (ii) the preferences being strictly increasing but not convex.
2. Utility is given by U(x1; x2) = x1x2. Marginal utilities are MU1 = x2, and MU2 = x1. The price of x1 is $1, theprice of x2 is $2, and income is $40. The price of x2 falls to $1. (c) What is the substitution effect of the price change? Illustrate on the graph. (d) What is the actual optimal consumption choice after the price change? Illustrate on the graph andlabel that choice C. (e) What is the income effect of the price change? Illustrate on the graph.