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:


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 $A$?


1 Answer 1


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: $$ \begin{align*} V(\alpha A_1 + (1-\alpha) A_2) &= u(c_\alpha, \ell_\alpha),\\ &\le u(\alpha c_1 + (1-\alpha) c_2, \ell_\alpha),\\ &\le \alpha u(c_1,\ell_\alpha) + (1-\alpha) u(c_2, \ell_\alpha),\\ &\le \alpha V(A_1) + (1-\alpha) V(A_2). \end{align*} $$


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