I am maximizing a function $f(x,z)$ on $x$ ($z$ is treated a parameter in the maximization). The function $f$ is strictly concave on both variables.


I know how to use the envelope theorem for the first derivative. But I am interested in knowing if the function $f$ evaluated at the maximum, i.e., $f(x^\*(z),z)$, where $x^\*(z)$ is the solution of the maximization, is also a concave function of $z$.


Is there any theorem for this?



Envelope theorem for second derivative
Yes, the resulting function is concave in $z$.
This operation is sometimes called *partial maximization* and is known to preserve concavity (strict concavity is not required).


More precisely, if $f(x,z)$ is a concave function, then the partially maximized function $g(z)$


$g(z) = \sup\_{x} f(x,z)$ 


is also concave.


This can be found for example in the *Convex Optimization* [book](https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf) of Boyd and Vandenberghe (see Chapter 3.2.5, written for the slightly different but equivalent operation of partial minimization of a convex function).


