Definition
Hicksian demand function, or compensated demand function: the optimal commodity level that minimizes the cost given the lower bound of utility u. It is denoted as h(p, u). It is the EMP equivalent of the UMP's Marshallian demand x(p, w)
Note: we cannot observe hicksian demand because we cannot observe u.
Properties
- h(p, u) is homogeneous of degree 0 in p:
for all α > 0. - h(p, u) has no excess utility: For all
, u(x) = u. - If
is convex, h(p, u) is a convex set; if is strictly convex, h(p, u) is a strictly convex set
Proposition
Suppose h(p, u) is a singleton, then for all p' and p'', it is the case 
Proposition
(Shepard's lemma)
Suppose preference are strictly convex, then 
, or:
Proposition
Suppose preference are strictly convex and h(p, u) is continuously differentiable, then:
is a negative semi-definite matrix- is symmetric
Note: is the derivative of Hicksian demand h(p, u), with respect to p (price vector).
Proposition
Suppose preference are strictly convex. Then for all (p, w), and u = v(p, w), we have:
Or:
Implication: the matrix of Hicksian price derivatives is equal to the Slutsky matrix
From rational preference, we know this matrix must be negative semi-definite, symmetric and satisfy S(p, w) p = 0