Definition
For a given UMP, the utility value at the optimum is the indirect utility function v(p, w). It is the highest utility we can have.
v(p, w) = u(x(p,w))
p: price vector
w: wealth
Proposition
Suppose u is continuous and locally non-satiated, the indirect utility function v(p, w) is:
(1) Homogeneous of degree 0
(2) Strictly increasing in w, an non-increasing in 
(3) Quasi-convex: 
is convex (set) for all 
(4) Continuous in p and w.
Proposition
Suppose preference are strictly convex and the indirectly utility function is differentiable. Then by Roy's identity:
