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How to tell if preferences are Cobb-Douglas... How can you look at a function (i.e. U(X, Y) = ln X + 2ln Y) and tell it's C.D.? Explain in simple terms.

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Related questions

If preferences over goods x and y can be represented by a Cobb-Douglas utility function, the income consumption curve is necessarily upward sloping.

True or False?

Suppose that a consumer's preferences over two goods, x and y, are represented by a Cobb-Douglas utility function:

U(x,y) = x1-iĀ² yiĀ²

a. Find the expenditure function of the consumer, and the compensated demand functions.

b. Suppose that iĀ² = 1/4 and that initially a consumer is in equilibrium with px = 1,py = 1, I = 100. What are the demands for xand y.

c. What is the minimum increase in income necessary for the consumer to be as well offunder price px = 1, py = 2, as she/he was at prices px = 1, py = 1?

Explain why the percentage increase is smaller than the increase inthe CPI for the consumer that you derived in part c.

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.

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