# What assumptions can be made to ensure convexity in this optimization problem?

This question is a continuation of the question I asked at:

How can I show convexity of this value function?

Where I came to the conclusion that more assumptions are required to show that the following welfare value function is convex in terms of $$A$$:

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

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 follow up question is what kind of assumptions should be made on $$f$$ in order to ensure the value function is convex in terms of $$A$$?

Assume that $$f$$ is convex in $$A$$ and that $$u$$ is convex and increasing in $$c$$. We also assume that the optimization problem has a solution.
Consider $$A_1$$ and $$A_2$$ and let $$\alpha \in [0,1]$$. Let: $$(c_\alpha, \ell_\alpha) \in \arg\max_{c, \ell} u(c,\ell) \text{ s.t. } c = f(\ell, \alpha A_1 + (1-\alpha) A_2).$$
Note that $$V(\alpha A_1 + (1-\alpha)A_2) = u(c_\alpha, \ell_\alpha)$$.
By convexity of $$f$$ in $$A$$: \begin{align*} c_\alpha &= f(\ell_\alpha, \alpha A_1 + (1-\alpha) A_2),\\ &\le \alpha \underbrace{f(\ell_\alpha, A_1)}_{=c_1} + (1-\alpha) \underbrace{f(\ell_\alpha, A_2)}_{=c_2}. \end{align*}
By definition $$c_1 = f(\ell_\alpha, A_1)$$ and $$c_2 = f(\ell_\alpha, A_2)$$ satisfy the constraints for $$A_1$$ and $$A_2$$, so: $$V(A_1) \ge u(c_1, \ell_\alpha) \text{ and } V(A_2) \ge u(c_2, \ell_\alpha).$$
Then using montonicity of $$u$$ in $$c$$ (and $$c_\alpha \le \alpha c_1 + (1-\alpha) c_2)$$ together with convexity of $$u$$ in $$c$$ gives: \begin{align*} V(\alpha A_1 + (1-\alpha) A_2) &= u(c_\alpha, \ell_\alpha),\\ &\le u(\alpha c_1 + (1-\alpha) c_2, \ell_\alpha),\\ &\le \alpha u(c_1,\ell_\alpha) + (1-\alpha) u(c_2, \ell_\alpha),\\ &\le \alpha V(A_1) + (1-\alpha) V(A_2). \end{align*}