Properties of the Indirect Utility Function: Note: The indirect utility function

Question

Properties of the Indirect Utility Function:

Note: The indirect utility function v(p,y) = u(x(p,y))

Suppose that we did not require u(x) to be strictly increasing. (For example, suppose the utility function had a bliss point, or that the bundles were not goods but ”bads”.) How, if at all, must the following properties be amended?

i.) Continuity

ii.) Homogeneous of degree zero in (p,y)

iii.) Strictly increasing in y

iv.) Decreasing in p

v.) Quasiconvex in (p,y)

vi.) Satisfies Roy's Identity.

Support your argument and illustrate any claims with a two-good case.

Explanation / Answer

i) Continuity will still be required because a utility function is not continuous then we will not be able to find the optimal value. So, we need continuity even if u() is not strictly increasing.

ii) It will always be homogeneous of degree 0 in p and y because homogeneous of degree 0 says that by changing prices and wealth by the same amount your optimal bundle will remain same and this is indeed true.

iii) Now, this property needs a little amendment because if the good is bad then my optimal bundle will not be strictly increasing in y. Now we should replace this with non-increasing in y.

iv) Again, if the good that we are considering is bad then may be increasing in p.

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