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Consider 2 consumers, each with endowment (100, 220).

Biff has utility function u(x, y)= xy
Baff has utility function U(x, y)= ln x +4 ln y

Compare their choices and utility when facing interest rate 10% per period versus facing interest rate 25% per period.

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Tod Thiel
Tod ThielLv2
9 Aug 2018

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Related questions

Please explain the steps to get to the answers. I already have just the answers.

Adam and Eve are consumers in a 2-consumer, 2-good exchange economy. The price of X is Px, and the price of Y is $1. Each one has an initial endowment of (X=6, Y=6); therefore, each one has a "budget line" of X*Px+ Y*Py = 6*Px + 6. Adam's utility function is U = XY. The expression for Adam's marginal rate of substitution (i.e.   ) is Y/X, which at his initial endowment of (6, 6) has the value of 1. Adam's demand for X as a function of the price of X is X = 3 + 3/PX.

If Eve's utility function is,  

find: (i) the expression for Eve's marginal rate of substitution.

(ii) the value of Eve's marginal rate of substitution at her initial endowment.

(iii) Eve's demand for X as a function of the price of X.

(iv) What is the price of X in equilibrium?

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.

Consider an individual making choices over two goods, x and y with prices px = 2 and py = 6, and who has income I = 50.

(a) If the individual's preferences can be represented by the utility function u(x; y) = 2x+5y how much of each good does the individual buy in order to maximize his/her utility and what is his/her utility level? If py decreases to $4 (all other things staying the same), what are the individual'­s new utility-maximizing choices, and what is his/her new utility level? Explain.

(b) If the individual's preferences can be represented by the utility function u(x; y) = min( x2 ; y), how much of each good does the individual buy in order to maximize his/her utility, and what is his/her utility level? If py decreases to $4 (all other things staying the same), what are the individual'­s new utility-maximizing choices, and what is his/her new utility level? Explain.

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