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29 Nov 2019

Consider the following utility functions:
U(x, y) = xy
U(x,y) = 3x + 6y
U(x,y) = 20x1/2+ 2y
a. Fixing y at 4, what is the marginal utility of X when x is 10, 20, and 30 for each utility function?

b. Do these utility functions exhibit diminishing marginal utility of X? Explain.

c. For each utility function, calculate the marginal rate of substitution of good X for good Y (i.e. -
marginal value of X, measured in units of good Y) associated with the bundle (x, y) = (10, 4).

d. Do these utility functions exhibit diminishing marginal rates of substitution? Explain.

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Related questions

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!)

Ā 

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.

5. For each of the following utility functions, draw the three indierence curves that correspond to the three stated utility levels, labeling each curve with the corresponding utility

level. Label any intercepts or \kinks" in the line to fully identify the position of the curve.

(a) U(X; Y ) = minf3X; Y g; utility levels = 3; 6; 9

(b) U(X; Y ) = minf3X; 2Y g; utility levels = 6; 12; 18.

(c) U(X; Y ) = 5X + 2Y ; utility levels = 10; 15; 20.




6. For each of the following utility functions, derive MUX; MUY , and the MRS. Show your

steps.

(a) U(X; Y ) = X2Y

(b) U(X; Y ) = X(1/3) Y(2/3)

(c) U(X; Y ) = 4rootXY

(d) U(X; Y ) = XY + 3X + 2Y

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