2. For each of the following, find the optimal consumption bun dle (a) U = xy , P x = 2 ,P y = 5 , I = 40 (b) U = x 1 2 y 1 2 , P x = 8 , P y = 4 , I = 60 (c) U = 3 x + y 2 3 , P x = 6 , P y = 8 , I = 30

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Jamar FerryLv2

2 Jul 2019

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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) = x^ay^b (Cobb-Douglas utility function)

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

John M. Keynes: U(x,y) = x + y^b (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 p_x and p_y respectively.

d. Now assume that a=1/2, b = 1/4, I = 600, p_x = 10 and p_y = 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, s_x, is defined as

(p_x 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.

aquamarinescorpion997

With the following condition: UxxUx^2-2UxyYxUy+uyyUy^2<0, check the convexity of the indifference curves for each of the following functions and describe the precise relationship between diminishing marginal utility and quasi-concavity for each case.

A. U(x,y)=3x+y

B. U(x,y)=(x.y)^1/2

C. U(x,y)=(x)^1/2+y

D. U(x,y)=(x^2-y^2)^1/2

E. U(x,y)=(xy)/(x+y)

(^1/2 used inplace for square root)