3.12 CES utility a. Show that the CES function a xd d b yd d is homothetic. How does the MRS depend on the

3.12 CES utility

a. Show that the CES function a xd d þ b yd d

is homothetic. How does the MRS depend on the ratio y/x?

b. Show that your results from part

(a) agree with our discussion of the cases d ¼ 1 (perfect substitutes) and d ¼ 0 (Cobb–

Douglas).

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

d. Show that if x ¼ y, the MRS for this function depends only on the relative sizes of a and b.

e. Calculate the MRS for this function when y/x ¼ 0.9 and y/x ¼ 1.1 for the two cases d ¼ 0.5 and d ¼ %1. What do you conclude about the extent to which the MRS changes in the vicinity of x ¼ y? How would you interpret this geometrically?