4
$\begingroup$

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 $A$?

$\endgroup$

1 Answer 1

1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.