3.10

a. Show that the CES function X

s Y s

O T

+

'T is homothetic. How does the MRS depend on the ratio F/X?

b. Show that your results from part

(a) agree with Example 3.3 for the case 6 = 1 (perfect substitutes) and 5 = 0 (Cobb-Douglas).

c. Show that the MRS is strictly diminishing for all values of 8 < 1.

d. Show that if X= Y, the MRS for this function depends only on the relative sizes of a and j8.

e. Calculate the MRS for this function when Y/X= .9 and Y/X= 1.1 for the two cases 8 = .5 and 8 = — 1. What do you conclude about the extent to which the MRS changes in the vicinity of X = F? How would you interpret this geometrically?