Exercise 3.
Robinson is the only person in an island, and has an endowment of80 units of X and 140 units of Y. His utility function is:

U = ln(X) + ln(Y)

a) Find the marginal utility of X and Y
b) Find the marginal rate of substitution when Robinson consumesall his endowment
c) Suppose now that Robinson finds that he is not alone on theIsland, so now he can trade
goods with other people. Find the relative prices PX/PY for whichRobinson decides to
sell X and to buy Y.

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Prachi DabasLv10

28 Sep 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

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.