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Economics
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a) U = xy

b) U = (xy)^1/3

c) U = min(x,y/2)

d) U = 2x + 3y

e) U = x^2 y^2 + xy

All homogeneous utility functions are homothetic. Are any of the above functions homothetic but not homogeneous? Show your work.

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Joshua Stredder
Joshua StredderLv10
28 Sep 2019
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Related questions

Jill buys two goods: X and Y and has the following utility function: U(X,Y) = 2X0.25Y0.75

Please show all work

a. Find the equation for her generalized demand curve for X*(Px,Py,I)

b. Find the equation for her generalized demand curve for Y*(Py,Px,I)

c. Are the above generalized demand functions homogeneous of degree 0? (Show your work)

d. Is X a normal good?

e. Is X a gross substitute or complement to Y?

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.

A cost of living index is introduced in the previous exercise. Suppose the consumer’s direct utility function is u(x1, x2) = √x1 + x2.

(a) Let base prices be p0 = (1, 2), base income be y0 = 10, and suppose p1 = (2, 1). Compute the index I.

(b) Let base and final period prices be as in part (a), but now let base utility be u0. Show that the value of the index I will vary with the base utility.

(c) It can be shown that when consumer preferences are homothetic, I will be independent of the base utility for any prices p0 and p1. Can you show it?

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