CES utility with weights a. Show that the CES function x 1 y

CES utility with weights

a. Show that the CES function

α

x δ

δ 1 β

y δ

δ

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 δ 5 1 (perfect substitutes) and

δ 5 0 (Cobb–Douglas).

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

δ , 1.

d. Show that if x 5 y, the MRS for this function depends only on the relative sizes of α and β.

e. Calculate the MRS for this function when y/x 5 0.9

and y/x 5 1.1 for the two cases δ 5 0.5 and δ 5 21.

What do you conclude about the extent to which the MRS changes in the vicinity of x 5 y? How would you interpret this geometrically?