6.10 Separable utility A utility function is called separable if it can be written as Ux, y U1x U2 y, where Ui 0

6.10 Separable utility A utility function is called separable if it can be written as Uðx, yÞ ¼ U1ðxÞ þ U2ð yÞ, where Ui 0 > 0, Ui 00 < 0, and U1, U2 need not be the same function.

a. What does separability assume about the cross-partial derivative Uxy? Give an intuitive discussion of what word this condition means and in what situations it might be plausible.

b. Show that if utility is separable then neither good can be inferior.

c. Does the assumption of separability allow you to conclude definitively whether x and y are gross substitutes or gross complements? Explain.

d. Use the Cobb–Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.