I am interested in the concavity in $p$ of the indirect utility function $$V(p,W)=max_{x,y,z} pf_1(x,y)+(1-p)f_2(x,z)$$ under the constraint $$x+py+(1-p)z=W$$
where $0<p<1$ and where $f_1,f_2$ are strictly increasing and concave. For example, this could be a problem of state-contingent resources allocation (insurance, etc...).
I would like to know whether it is true that $V$ is convex in $p$, or what conditions are required for this to be the case.
My first guess is that both the objective and the constraint are linear in $p$. Therefore, for a fixed $W$, the function $p\mapsto V(p,W)$ is the pointwise maximum of a collection of linear functions, thus it should be convex.
However, I am not sure I can apply that reasoning when the optimization problem is constrained, as I have here. For example, the same reasoning might lead us to conclude that $V$ is convex in $W$. But when $f_1$ and $f_2$ are concave, we can show that $V$ is concave in $W$... so the pointwise maximum argument seems not to work here...
Any pointers appreciated.
