Consider the following individual utility functions:
Adam Smith: U(x,y) = xyJeremy 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. Robinson’s (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 pyrespectively.
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.