# Convexity of indirect utility in probabilities

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 $$0 and where $$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.

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