Question 1 For the following utility functions (3 pts each for a, b, and c): •...
Question 1
For the following utility functions (3 pts each for a, b, and c):
• Find the marginal utility of each good at the point (5, 5) and at the point (5, 15)
• Determine whether the marginal utility decreases as consumption of each good increases (i.e., does the utility function exhibit diminishing marginal utility in each good?)
• Find the marginal rate of substitution at the point (5, 5) and at the point (5, 15)
• Discuss how MRSXY changes as the consumer substitutes X for Y along an indifference curve
• Derive the equation for the indifference curve where utility is equal to a value of 50
• Graph the indifference curve where utility is equal to a value of 50, labelling at least three points on the curve
a. U(X,Y) = 5X + 6Y
b. U(X,Y) = (X1/4)*(Y3/4)
c. U(X,Y) = X*Y1/2
Homework Answers
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Question 1
For the following utility functions (3 pts each for a, b, and
c):
•...
For each of these utility functions, b. Compute the MRS. c. Do these tastes have...
For each of these utility functions,
b. Compute the MRS.
c. Do these tastes have diminishing marginal rates of
substitution? Are they convex?
d. Construct an indifference curve for each of these functions
for utility numbers U1 = 10 , U2 = 100 , U3 = 200 .
e. Do these utility functions represent different preference
orderings?
1. Consider the following utility functions: (i) U(x,y)- 6xy, (ii) U(x,y)=(1/5)xy, MU,--y and MU,--x ii) U(x,y)-(2xy)M 8xy2 and MUy -8x2y MU,-6y and...
2. Show that each of the following utility functions has a diminishing MRS. Do they exhibit...
2. Show that each of the following utility functions has a diminishing MRS. Do they exhibit constant, increasing, or decreasing marginal utility? Is the shape of the marginal utility function an indicator of the convexity of indifference curve? a. (2) U(X,Y) = XY b. (2) U(X,Y) = x2y2 c. (2) U(X,Y) = In X + In Y
Assume that an individual’s preferences is represented by the following utility function: ?(?, ?) = (?^1/3)*(y^2/3)...
Assume that an individual’s preferences is represented by the following utility function: ?(?, ?) = (?^1/3)*(y^2/3) a. What could you tell about the type of x and y? (“good” , “bad” or a “neuter”) b. Derive the equation for his/her indifference curve for utility level of 100? c. Derive marginal utility of x and marginal utility of y as a function of x,y. d. Does goods x and y exhibit diminishing marginal utility, constant marginal utility, or increasing marginal utility?...
For each of the following utility functions (a-d), answer the following questions (i-iv) i) Calculate MUz...
For each of the following utility functions (a-d), answer the following questions (i-iv) i) Calculate MUz and MUy ii) Do the consumer 's preferences exhibit a diminishing marginal utility for each good? iii) Calculate MRSa,y iv) Do the consumer's preferences exhibit a diminishing marginal rate of substitution of x? (al U (x,y) = 3x2/5y3/5 (b) (x,y)= 2x2/3+y (c) (x,y)= 12x3/2y1/2 3
2. For each of the three utility functions below, answer these questions: Does the marginal utility...
2. For each of the three utility functions below, answer these questions: Does the marginal utility of good x diminish, remain constant, or increase as the consumer buys more x, holding good y constant? Justify. Does the MRS of x for y diminish, remain constant, or increase as the consumer substitutes good y for more of good x to the right along an indifference curve? Justify. . . a. U(x, y)xyb, where 0 < a < 1 and b> 0...
2. For each of the three utility functions below, answer these questions: Does the marginal utility...
2. For each of the three utility functions below, answer these questions: Does the marginal utility of good x diminish, remain constant, or increase as the consumer buys more x, holding good y constant? Justify. Does the MRS of x for y diminish, remain constant, or increase as the consumer substitutes good y for more of good x to the right along an indifference curve? Justify. . . a. U(x, y)xyb, where 0 < a < 1 and b> 0...
For each of the following utility functions: Calculate MUx, MUy, MRSx,y; determine whether or not the...
For each of the following utility functions: Calculate MUx, MUy, MRSx,y; determine whether or not the property of “more is better” is satisfied for both goods; determine whether or not the marginal utility of x diminishes, remains constant, or increases as the consumer buys more x; determine whether or not the marginal rate of substitution diminishes, remains constant, or increases as the consumer substitutes x for y along an indifference curve; and sketch the graph of a typical indifference curve....
5. Consider the utility function U(x, y) = 2/x + y. 1) Is the assumption that...
5. Consider the utility function U(x, y) = 2/x + y. 1) Is the assumption that "more is better” satisfied for both goods? 2) What is MRS, for this utility function? 3) Is the marginal rate of substitution diminishing, constant, or increasing in x as the consumer substitutes x for y along an indifference curve? 4) Will the indifference curve corresponding to this utility function be convex to the origin, concave to the origin, or straight lines? Explain.
U (x, y) = 4x11/6 y1/6 . a)Calculate MUx and MUy. b) Do the consumer’s preferences...
U (x, y) = 4x11/6 y1/6 . a)Calculate MUx and MUy. b) Do the consumer’s preferences exhibit a diminishing marginal utility for each good? c) Calculate MRSx,y. d) Do the consumer’s preferences exhibit a diminishing marginal rate of substitution of x?
For the following utility functions, a. Find the marginal rate of substitution. b. Derive the equation...
For the following utility functions, a. Find the marginal rate of substitution. b. Derive the equation for the indifference curve where utility is equal to a value of 100. c. Plot the indifference curve where utility is equal to a value of 100. (1) u(x1, x2) = x1x2; (2) u(x1, x2) = x1x2 + 10x2; (3) u(x1, x2) = x12 + x2
For each of these utility functions,
b. Compute the MRS.
c. Do these tastes have diminishing marginal rates of
substitution? Are they convex?
d. Construct an indifference curve for each of these functions
for utility numbers U1 = 10 , U2 = 100 , U3 = 200 .
e. Do these utility functions represent different preference
orderings?
1. Consider the following utility functions: (i) U(x,y)- 6xy, (ii) U(x,y)=(1/5)xy, MU,--y and MU,--x ii) U(x,y)-(2xy)M 8xy2 and MUy -8x2y MU,-6y and...
2. Show that each of the following utility functions has a diminishing MRS. Do they exhibit constant, increasing, or decreasing marginal utility? Is the shape of the marginal utility function an indicator of the convexity of indifference curve? a. (2) U(X,Y) = XY b. (2) U(X,Y) = x2y2 c. (2) U(X,Y) = In X + In Y
For each of the following utility functions (a-d), answer the following questions (i-iv) i) Calculate MUz and MUy ii) Do the consumer 's preferences exhibit a diminishing marginal utility for each good? iii) Calculate MRSa,y iv) Do the consumer's preferences exhibit a diminishing marginal rate of substitution of x? (al U (x,y) = 3x2/5y3/5 (b) (x,y)= 2x2/3+y (c) (x,y)= 12x3/2y1/2 3
2. For each of the three utility functions below, answer these questions: Does the marginal utility of good x diminish, remain constant, or increase as the consumer buys more x, holding good y constant? Justify. Does the MRS of x for y diminish, remain constant, or increase as the consumer substitutes good y for more of good x to the right along an indifference curve? Justify. . . a. U(x, y)xyb, where 0 < a < 1 and b> 0...
2. For each of the three utility functions below, answer these questions: Does the marginal utility of good x diminish, remain constant, or increase as the consumer buys more x, holding good y constant? Justify. Does the MRS of x for y diminish, remain constant, or increase as the consumer substitutes good y for more of good x to the right along an indifference curve? Justify. . . a. U(x, y)xyb, where 0 < a < 1 and b> 0...
5. Consider the utility function U(x, y) = 2/x + y. 1) Is the assumption that "more is better” satisfied for both goods? 2) What is MRS, for this utility function? 3) Is the marginal rate of substitution diminishing, constant, or increasing in x as the consumer substitutes x for y along an indifference curve? 4) Will the indifference curve corresponding to this utility function be convex to the origin, concave to the origin, or straight lines? Explain.
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