How can I show convexity of this value function?

Question

I have set up an optmization problem as follows:

$$V(A)=\max_{l, C} \quad u(C,l)$$

Where the only constraint is as follows:

$$C=f(l,A)$$

Here $u$ is the utility function which captures social welfare. $f$ is the production function that produces consumption $C$.

$l$ represents labour and so the utility function $u$ is decreasing in the $l$ input, while the production function $f$ is increasing in $l$. Clearly, $u$ is increasing in terms of consumption $C$. Finally, $A$ is a state variable, which is a negative input to the production function, i.e. more $A$ equals less production.

Here I assume the utility function captures the usual concavity/convexity properties, capturing decreasing marginal benefit of good things and increasing marginal cost of bad things. I also assume $f$ is a continuous function.

Now my question is: can we conclude that this value function is convex in terms of $A$?

My approach to this has been to use the following theorem:

Suppose $V$ is the upper envelope of convex functions i.e. $V (a) = \max_{b} v(a, b)$ where $v(·, b)$ is a convex function for each $b$, then $V$ is convex.

However, I am not entirely sure how to apply it here, any ideas?

Answer 1

Suppose that $u(C,l)=\sqrt{C}-l^2$ and $f(l,A)=\big(l+g(A)\big)^2$, where $g$ is any function of $A$ that is not convex.

Then $$u\big(f(l,A),l\big)=l+g(A)-l^2.$$ The optimal labor supply is given by $1/2$. So the value function is given by

$$V(a)=1/4+g(A).$$ Since $g$ is not convex, the value function is not convex either. Clearly, you need more assumptions on $f$.