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

CC BY-SA 4.0
2024-02-24T10:15:29.337Z
2024-02-24T10:15:29.337Z
tdm
<p>This question is a continuation of the question I asked at:</p>
<p><a href="https://economics.stackexchange.com/q/51150/40943">How can I show convexity of this value function?</a></p>
<p>Where I came to the conclusion that more assumptions are required to show that the following welfare value function is convex in terms of <span class="math-container">$A$</span>:</p>
<p><span class="math-container">$$V(A)=\max_{l, C} \quad u(C,l)$$</span></p>
<p>Where the only constraint is as follows:</p>
<p><span class="math-container">$$C=f(l,A)$$</span></p>
<p>Here <span class="math-container">$u$</span> is the utility function which captures social welfare. <span class="math-container">$f$</span> is the production function that produces consumption <span class="math-container">$C$</span>.</p>
<p><span class="math-container">$l$</span> represents labour and so the utility function <span class="math-container">$u$</span> is decreasing in the <span class="math-container">$l$</span> input, while the production function <span class="math-container">$f$</span> is increasing in <span class="math-container">$l$</span>. Clearly, <span class="math-container">$u$</span> is increasing in terms of consumption <span class="math-container">$C$</span>. Finally, <span class="math-container">$A$</span> is a state variable, which is a negative input to the production function, i.e. more <span class="math-container">$A$</span> equals less production.</p>
<p>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 <span class="math-container">$f$</span> is a continuous function.</p>
<p>Now my follow up question is what kind of assumptions should be made on <span class="math-container">$f$</span> in order to ensure the value function is convex in terms of <span class="math-container">$A$</span>?</p>

<macroeconomics><utility><optimization><dynamic-programming><dynamic-optimization>
What assumptions can be made to ensure convexity in this optimization problem?
L. Johnson
# What assumptions can be made to ensure convexity in this optimization problem?

(Asked by: username_0 on 2022-04-19)

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

[How can I show convexity of this value function?](https://economics.stackexchange.com/q/51150/40943)

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 $A$?

**Tags:** macroeconomics, utility, optimization, dynamic-programming, dynamic-optimization


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## 1 answers

### Answer by username_1 on 2024-02-24. Score: 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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