<p>Suppose that <span class="math-container">$u(C,l)=\sqrt{C}-l^2$</span> and <span class="math-container">$f(l,A)=\big(l+g(A)\big)^2$</span>, where <span class="math-container">$g$</span> is any function of <span class="math-container">$A$</span> that is not convex.</p>
<p>Then <span class="math-container">$$u\big(f(l,A),l\big)=l+g(A)-l^2.$$</span>
The optimal labor supply is given by <span class="math-container">$1/2$</span>. So the value function is given by</p>
<p><span class="math-container">$$V(a)=1/4+g(A).$$</span> Since <span class="math-container">$g$</span> is not convex, the value function is not convex either. Clearly, you need more assumptions on <span class="math-container">$f$</span>.</p>

CC BY-SA 4.0
2022-04-19T14:04:16.960Z
Thank you for you response. I understand what you are saying. So in order to have V(A) be convex in terms of A would the production function f need to be convex in terms of A? Or are there some other assumptions that tend to be made if one wishes to ensure this property?
L. Johnson
CC BY-SA 4.0
2022-04-19T15:24:49.493Z
@L.Johnson I think something like that, but I'm not sure what exactly is needed. The point of the example is kinda that without any restriction on $f$, the assumption of decreasing marginal benefit in $C$ is meaningless.
Michael Greinecker
CC BY-SA 4.0
2022-04-20T19:34:36.850Z
I see what you are saying. Would it make a difference if I assumed that the production function f was concave in the A input? Or would I then still have to make assumptions about u?
L. Johnson
CC BY-SA 4.0
2022-04-20T20:41:39.513Z
@L.Johnson Concavity might be enough; I'm not sure about that.
Michael Greinecker
CC BY-SA 4.0
2022-04-19T13:17:11.157Z
2022-04-19T13:17:11.157Z
Michael Greinecker
<p>I have set up an optmization problem as follows:</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 question is: can we conclude that this value function is convex in terms of <span class="math-container">$A$</span>?</p>
<p>My approach to this has been to use the following theorem:</p>
<p>Suppose <span class="math-container">$V$</span> is the upper envelope of convex functions i.e. <span class="math-container">$V (a) = \max_{b} v(a, b)$</span>
where <span class="math-container">$v(·, b)$</span> is a convex function for each <span class="math-container">$b$</span>, then <span class="math-container">$V$</span> is convex.</p>
<p>However, I am not entirely sure how to apply it here, any ideas?</p>

<utility><optimization><public-economics><dynamic-programming><dynamic-optimization>
How can I show convexity of this value function?
L. Johnson
# How can I show convexity of this value function?

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

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?

**Tags:** utility, optimization, public-economics, dynamic-programming, dynamic-optimization


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## 1 answers

### Answer by username_1 on 2022-04-19. Score: 5 | Accepted answer

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 $f$.

> **Comments:**
> **username_0**: Thank you for you response. I understand what you are saying. So in order to have V(A) be convex in terms of A would the production function f need to be convex in terms of A? Or are there some other assumptions that tend to be made if one wishes to ensure this property?  
> **username_1**: @L.Johnson I think something like that, but I'm not sure what exactly is needed. The point of the example is kinda that without any restriction on $f$, the assumption of decreasing marginal benefit in $C$ is meaningless.  
> **username_0**: I see what you are saying. Would it make a difference if I assumed that the production function f was concave in the A input? Or would I then still have to make assumptions about u?  
> **username_1**: @L.Johnson Concavity might be enough; I'm not sure about that.  

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