Definition
Supply correspondence (optimal production correspondence) at p:
Interpretation
y(p) is the production vector that produces the maximum profit (profit function), given p. Or y(p) is the solution to profit maximization problem.
y(p) lies on the iso-profit line (a line along which all points generate equal profits)
y(p) can be a vector, a set, or not exist (when 
)
Properties of y(p)
(1) y(p) is homogeneous of degree 0
(2) If Y is convex, then for all p, y(p) is convex.
(3) If Y is strictly convex, 
, then y(p) is a singleton
(4) Law of supply: for any 
, it is the case that:
Proposition
If y(p) is single valued and continuously differentiable then 
is symmetric and positive semi-definite, with 