# Is there a way to link Berge's theorem of maximum to Envelope theorem?

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?

• 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 Feb 19 '15 at 13:45
• @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? people.hss.caltech.edu/~pbs/expfinance/Readings/Lucas1978.pdf – Epicurus Feb 20 '15 at 14:15
• 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. en.wikipedia.org/wiki/Envelope_theorem – jmbejara Feb 20 '15 at 14:24

## 2 Answers

They are related and usually fall into the same discussion, but as @Alecos 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.

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

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

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.