<p>They are related and usually fall into the same discussion, but as @Alecos mentions in the comments, the two theorems show different things.</p>

<p>I suppose the connection that you're after is the fact that if
the derivative
$$
\left .\frac{\partial f(x, a)}{\partial a} \right |_{x=x(a)} 
$$
exists, then because differentiability implies continuity, you might be able
to get part of theorem of the maximum out of it. However, to compare and contrast two theorems you mustn't just look at the results. You need to look at the assumptions as well. For example, the theorem of the maximum doesn't assume any sort of differentiability. The envelope theorem does (at least some forms of it). In any case, the assumptions that go into each are different (some stronger, some weaker).</p>

<p>Also, there is this. The envelope theorem
doesn't tell you that anything about the control function. Therefore, you definitely wont be able to get the result that $C^*$ is upper hemicontinuous. </p>

CC BY-SA 3.0
2015-02-20T12:07:04.130Z
2015-02-20T12:07:04.130Z
jmbejara
<p>Quoting the OP from a comment</p>

<blockquote>
  <p><em>What sort of conditions on utility function and constraint enables us
  to apply envelope theorem only after we established the continuity of
  value function by Berge's theorem?
  <a href="http://people.hss.caltech.edu/~pbs/expfinance/Readings/Lucas1978.pdf" rel="nofollow noreferrer">people.hss.caltech.edu/~pbs/expfinance/Readings/Lucas1978.pdf</a></em></p>
</blockquote>

<p>In the referenced Lucas (1978) paper, Proposition 1 establishes that</p>

<p><img src="https://i.sstatic.net/NHtC3.jpg" alt="enter image description here"></p>

<p>where $v(z,y;p)$ is the value function, and $(i)$ is its definition. So it appears that it is the continuity of the Price function that is singled out as a condition here, but earlier in the paper Lucas defines the Utility function as a non-negative function which is</p>

<blockquote>
  <p><em>continuously differentiable, bounded, increasing and strictly concave</em></p>
</blockquote>

<p>Proposition 2 of the paper establishes the differentiability of the value function, without requiring further assumptions.</p>

CC BY-SA 3.0
2015-02-20T15:14:50.617Z
2015-02-20T15:14:50.617Z
Alecos Papadopoulos
<p>Berge's theorem states</p>

<blockquote>
  <p>Let $X \in \mathbb R^m, \Theta \in \mathbb R^n $, $f : X \times \Theta \to \mathbb R$ be a jointly continuous function, $C : \Theta  \rightrightarrows X$ be a continuous(both upper and lower hemicontinuous) compact-valued correspondence.The maximized value function and maximizer are
  $$V(\theta) := \max_{x \in X}f(x, \theta)$$
  $$C^\ast(\theta):=\{x \in C(\theta) \mid f(x, \theta )=V(\theta)\}$$
  Then $V : \Theta \to \mathbb R$ is continuous and $C^\ast :\Theta \rightrightarrows  X$ is upper hemicontinuous.</p>
</blockquote>

<p>According to Varian's Microeconomic Analysis(1992), page 490, Envelope theorem is simply:</p>

<blockquote>
  <p>$$\frac{dM(a)}{da}=\frac{\partial f(x,a)}{\partial a}\mid_{x=x(a)}$$</p>
</blockquote>

<p>$x(a)$ is the maximizer of $f(\cdot, a)$.</p>

<p>It seems to me Envelope theorem entails Berge's theorem, but the derivation looks way simpler. Is there a relationship between the two?</p>

CC BY-SA 3.0
2015-02-19T13:45:15.517Z
It does not look like the two are preoccupied with the same target. Berge's establishes properties of the value function and of the set of maximizers. Envelope is concerned with showing what the effect of varying a parameter is... perhaps you could elaborate on the kind of connection between the two that intrigues you.
Alecos Papadopoulos
CC BY-SA 3.0
2015-02-20T14:15:07.947Z
@AlecosPapadopoulos Appologize for the vagueness of my question. Now I found out this queston stemmed from my vague memory of proposition 2 in Lucas(1978). Now I can formulate it more precisely. What sort of conditions on utility function and constraint enables us to apply envelope theorem only after we established the continuity of value function by Berge's theorem?  http://people.hss.caltech.edu/~pbs/expfinance/Readings/Lucas1978.pdf
Epicurus
CC BY-SA 3.0
2015-02-20T14:24:05.277Z
I don't think you necessarily need to "establish continuity of the value function" in order to use the envelope theorem. It think the key part is the point about the control $C^*$. See Theorem 2 on the Wikipedia page. There, continuity of V is a result. In any case, the Wikipedia page states the theorems in full. It'll tell you what you need to assume to use the theorem. http://en.wikipedia.org/wiki/Envelope_theorem
jmbejara
<mathematical-economics><academic-graduate><optimization>
Is there a way to link Berge's theorem of maximum to Envelope theorem?
Epicurus
# Is there a way to link Berge's theorem of maximum to Envelope theorem?

(Asked by: username_0 on 2015-02-15)

Berge's theorem states

> Let $X \in \mathbb R^m, \Theta \in \mathbb R^n $, $f : X \times \Theta \to \mathbb R$ be a jointly continuous function, $C : \Theta \rightrightarrows X$ be a continuous(both upper and lower hemicontinuous) compact-valued correspondence.The maximized value function and maximizer are
> $$V(\theta) := \max\_{x \in X}f(x, \theta)$$
> $$C^\ast(\theta):=\{x \in C(\theta) \mid f(x, \theta )=V(\theta)\}$$
> Then $V : \Theta \to \mathbb R$ is continuous and $C^\ast :\Theta \rightrightarrows X$ is upper hemicontinuous.

According to Varian's Microeconomic Analysis(1992), page 490, Envelope theorem is simply:

> $$\frac{dM(a)}{da}=\frac{\partial f(x,a)}{\partial a}\mid\_{x=x(a)}$$

$x(a)$ is the maximizer of $f(\cdot, a)$.

It seems to me Envelope theorem entails Berge's theorem, but the derivation looks way simpler. Is there a relationship between the two?

**Tags:** mathematical-economics, academic-graduate, optimization

> **Comments:**
> **username_1**: It does not look like the two are preoccupied with the same target. Berge's establishes properties of the value function and of the set of maximizers. Envelope is concerned with showing what the effect of varying a parameter is... perhaps you could elaborate on the kind of connection between the two that intrigues you.  
> **username_0**: @username_1 Appologize for the vagueness of my question. Now I found out this queston stemmed from my vague memory of proposition 2 in Lucas(1978). Now I can formulate it more precisely. What sort of conditions on utility function and constraint enables us to apply envelope theorem only after we established the continuity of value function by Berge's theorem?  http://people.hss.caltech.edu/~pbs/expfinance/Readings/Lucas1978.pdf  
> **username_2**: I don't think you necessarily need to "establish continuity of the value function" in order to use the envelope theorem. It think the key part is the point about the control $C^*$. See Theorem 2 on the Wikipedia page. There, continuity of V is a result. In any case, the Wikipedia page states the theorems in full. It'll tell you what you need to assume to use the theorem. http://en.wikipedia.org/wiki/Envelope_theorem  

---

## 2 answers

### Answer by username_2 on 2015-02-20. Score: 6

They are related and usually fall into the same discussion, but as @username_1 mentions in the comments, the two theorems show different things.

I suppose the connection that you're after is the fact that if
the derivative
$$
\left .\frac{\partial f(x, a)}{\partial a} \right |\_{x=x(a)}
$$
exists, then because differentiability implies continuity, you might be able
to get part of theorem of the maximum out of it. However, to compare and contrast two theorems you mustn't just look at the results. You need to look at the assumptions as well. For example, the theorem of the maximum doesn't assume any sort of differentiability. The envelope theorem does (at least some forms of it). In any case, the assumptions that go into each are different (some stronger, some weaker).

Also, there is this. The envelope theorem
doesn't tell you that anything about the control function. Therefore, you definitely wont be able to get the result that $C^\*$ is upper hemicontinuous.


---

### Answer by username_1 on 2015-02-20. Score: 4

Quoting the OP from a comment

> *What sort of conditions on utility function and constraint enables us
> to apply envelope theorem only after we established the continuity of
> value function by Berge's theorem?
> [people.hss.caltech.edu/~pbs/expfinance/Readings/Lucas1978.pdf](http://people.hss.caltech.edu/~pbs/expfinance/Readings/Lucas1978.pdf)*

In the referenced Lucas (1978) paper, Proposition 1 establishes that

![enter image description here](https://i.sstatic.net/NHtC3.jpg)

where $v(z,y;p)$ is the value function, and $(i)$ is its definition. So it appears that it is the continuity of the Price function that is singled out as a condition here, but earlier in the paper Lucas defines the Utility function as a non-negative function which is

> *continuously differentiable, bounded, increasing and strictly concave*

Proposition 2 of the paper establishes the differentiability of the value function, without requiring further assumptions.


---

