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For the two good utility function: U = x^2y^2

a) Is the assumption that more is better satisfied with both goods?

b) Does this consumer's preferences exhibit diminishing marginal utility of X?

c) Does this consumer's preferences exhibit diminishing marginal utility of Y?

d) What is the MRSx,y?

e) Is the MRSx,y diminishing with respect to x? Use math to verify your answer.

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Darryn D'Souza
Darryn D'SouzaLv10
28 Sep 2019

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Related questions

Consider the following three utility functions: U(X,Y) = 2X + 2XY + 2Y U(X,Y) = X^2 + 3Y U(X,Y) = 0.5ln(X) + 0.5ln(Y) a. Is the assumption that more is better satisfied for both goods? Justify. (6 points) b. Does the marginal utility of good X diminish, remain constant, or increase as the consumer buys more X, holding good Y constant? Justify. (6 points) c. Is the MRS of X for Y diminishing, constant, or increasing as the consumer substitutes good Y for more of good X along an indifference curve? Justify. (6 points)

For the following utility function,


* Find the marginal utility of each good.


* 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..


* 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 100.


* Graph the Indifference curve where Utility is equal to a value of 100.


a. U(X,Y) = 5X + 2Y


b. U(X,Y) = X^0.33 Y^0.67


c. U{X,Y) = 10X^0.5 + 5Y

PROBLEM 1 (This one is designed to give you lots of math practice!)

Consider the following individual utility functions:

Adam Smith: U(x,y) = xy

Jeremy Bentham: U(x,y) = xayb (Cobb-Douglas utility function)

Alfred Marshall: U(x,y) = ln x + ln y

John M. Keynes: U(x,y) = x + yb (Quasi-linear utility function)

Joan Robinson: U(x,y) = aX + bY (linear; perfect substitutes)

a. Solve for the MRS(XY) for each of the gentlemen (i.e., all but Joan Robinson). Show that Mr. Smith, Mr. Bentham, and Mr. Marshall all have diminishing MRS(XY), but that if a=b=2, their preferences exhibit constant, increasing, and decreasing marginal utility respectively. (Fortunately, diminishing MRS is the condition that guarantees convex indifference curves, so we can solve for their optimal consumption!)

 

b. What is the general formula for Mrs. Robinsons (constant) MRS?

c. Find demand functions for x and y for Smith, Bentham, Marshall, and Keynes. Assume that each man has income I and that the prices of x and y are px and py respectively.

d. Now assume that a=1/2, b = 1/4, I = 600, px = 10 and py = 20. Use these values to find the utility-maximizing quantities of x and y consumed by each man.

e. Find an expression for the share of income Mr. Bentham and Mr. Marshall each devote to the consumption of good x, where this share, sx, is defined as

(px x)/I.

f. Find expressions for the income elasticity of demand for good x for Mr. Bentham and Mr. Marshall. Find expressions for the (own) price elasticity of demand for good y for these two men.

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