3.12 Ces utility a. Show that the CES function x + y is homothetic. How does the MRS depend on

3.12 Ces utility a.

Show that the CES function

α

x

δ

δ +β

y

δ

δ

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

b.

c.

d.

e.

Show that your results from part

(a) agree with our discussion of the cases δ = 1 (perfect substitutes) and δ = 0 (Cobb–Douglas).

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

Show that if x = y, the MRS for this function depends only on the relative sizes of α and β.

Calculate the MRS for this function when y/x =

0.9 and y/x = 1.1 for the two cases δ = 0.5 and

δ = −1. What do you conclude about the extent to which the MRS changes in the vicinity of x =

y? How would you interpret this geometrically?