<p>Consider the optimization problem
<span class="math-container">$$
V(\gamma) \ \equiv \ \max_x \ f(x,\gamma) \quad \text{s.t.} \quad g(x,\gamma) \leq 0
$$</span></p>
<p>Roughly, the Benveniste-Scheinkman theorem implies that if <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are smooth enough and if <span class="math-container">$f$</span> is concave and <span class="math-container">$g$</span> is convex, then <span class="math-container">$V$</span> is differentiable.</p>
<p>Are there assumptions on <span class="math-container">$f$</span> and <span class="math-container">$g$</span> that ensure <span class="math-container">$V$</span> is twice differentiable?</p>

CC BY-SA 4.0
2024-03-04T21:05:57.107Z
The title is about the policy function, but the question in the body is about the value function. Could you clarify?
Michael Greinecker
CC BY-SA 4.0
2024-03-05T02:03:10.657Z
Sorry for the typo and thank you for clarifying. I updated the title. I am interested in the value function.
John Sturm
CC BY-SA 4.0
2024-03-05T08:53:13.247Z
This might be relevant: [Araujo(1991) The once but not twice differentiability of the Policy function](https://doi.org/10.2307/2938372) or this one [Santos (1991): Smoothness of the Policy function in discrete time models](https://doi.org/10.2307/2938371).
tdm
<optimization><real-analysis>
When is a value function twice differentiable?
John Sturm
# When is a value function twice differentiable?

(Asked by: username_0 on 2024-03-04)

Consider the optimization problem
$$
V(\gamma) \ \equiv \ \max\_x \ f(x,\gamma) \quad \text{s.t.} \quad g(x,\gamma) \leq 0
$$

Roughly, the Benveniste-Scheinkman theorem implies that if $f$ and $g$ are smooth enough and if $f$ is concave and $g$ is convex, then $V$ is differentiable.

Are there assumptions on $f$ and $g$ that ensure $V$ is twice differentiable?

**Tags:** optimization, real-analysis

> **Comments:**
> **username_1**: The title is about the policy function, but the question in the body is about the value function. Could you clarify?  
> **username_0**: Sorry for the typo and thank you for clarifying. I updated the title. I am interested in the value function.  
> **username_2**: This might be relevant: [Araujo(1991) The once but not twice differentiability of the Policy function](https://doi.org/10.2307/2938372) or this one [Santos (1991): Smoothness of the Policy function in discrete time models](https://doi.org/10.2307/2938371).  

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## 0 answers

