<p>First, let's consider a standard &quot;economic interpretation&quot; of the <strong>envelope theorem for unconstrained maximization</strong>:</p>
<p>if <span class="math-container">$x^*(b)$</span> solves <span class="math-container">$\max_x f(b,x)$</span>, then <span class="math-container">$\frac{d f(b,x^*(b))}{d b_k} = \frac{\partial f(b,x^*(b))}{\partial b_k},$</span> i.e., the total derivative of <span class="math-container">$f$</span> w.r.t. <span class="math-container">$b_k$</span> equals the partial derivative of <span class="math-container">$f$</span> w.r.t. <span class="math-container">$b_k$</span>.</p>
<p>The <em>economic interpretation (in the given problem)</em>: Let <span class="math-container">$\pi^*(p,w) = p f(x^∗) − w·x^*$</span> be the maximized value of profits given output price <span class="math-container">$p$</span> and input price vector <span class="math-container">$w$</span>. Then the <span class="math-container">$i$</span>'th input demand function is <span class="math-container">$x^*_i(·) = −\frac{\partial \pi^*(·,·)}{\partial w_i}$</span>, known as <strong>Hotelling's Lemma</strong>, after Harold Hotelling, (1895–1973). This derivative is negative: if the price of the input increases, then the firm's maximal profit decreases.</p>
<hr />
<p>Second, let's consider a standard &quot;economic interpretation&quot; of the <strong>envelope theorem for constrained maximization</strong>:</p>
<p>if <span class="math-container">$x^*(b), \lambda^*$</span> solves <span class="math-container">$\max_x f(b,x)$</span> s.t. <span class="math-container">$h^j(b,x) = 0$</span>, <span class="math-container">$j= 1, ...m,$</span> then</p>
<p><span class="math-container">$$\frac{d f(b,x^*(b))}{d b_k} = \frac{\partial f(b,x^*(b))}{\partial b_k} + \sum_{j=1}^m \lambda_j^*(b)\frac{\partial h^j(b,x^*(b))}{\partial b_k}, $$</span></p>
<p>i.e., the total derivative of <span class="math-container">$f$</span> w.r.t. <span class="math-container">$b_k$</span> equals the partial derivative of <span class="math-container">$f$</span> w.r.t. <span class="math-container">$b_k$</span>, plus the <span class="math-container">$\lambda^*$</span>-weighted sum of the partial derivatives of the <span class="math-container">$h^j$</span>'s w.r.t. <span class="math-container">$b_k$</span>.</p>
<p>The <em>economic interpretation (in the given problem)</em>: Let <span class="math-container">$\hat c(\bar q, p, w) = w·\hat x$</span> be the minimized level of costs given prices <span class="math-container">$(p,w)$</span> and output level <span class="math-container">$\bar q$</span>. Then the <span class="math-container">$i$</span>'th conditional input demand function is <span class="math-container">$\hat x_i(·) = −\frac{\partial \hat c(·,·,.)}{\partial w_i}$</span>,  known as <strong>Shepard's Lemma</strong>, after Ronald Shephard  (1912-1982). The partial derivatives of the expenditure function with respect to the prices of goods equal the Hicksian demand functions for the relevant goods.</p>

CC BY-SA 4.0
2020-08-28T00:03:13.643Z
The "unconstrained" example $\max \pi $ is not exactly unconstrained. There is a constrain $x\geq 0$. The discussion (and the Envelope Theorem) only applies to interior solutions.
Michael
CC BY-SA 4.0
2020-08-28T05:29:24.110Z
Shouldn't these be "the economic interpretation (in the given problem)" rather than "**the** economic interpretation"? Though these are the most famous ones, there are other constrained and unconstrained optimization problems in economics.
Giskard
CC BY-SA 4.0
2020-08-28T07:37:52.210Z
Agree with @Giskard. A general economic interpretation is not tied to a particular example/application. The unconstrained theorem says, economically, that since the indirect objective function is the objective function at a maximum, the endogenous variables are solutions to an optimization problem, therefore they have zero marginal effect on the objective function by FOC. For the constrained version, the partial derivatives w.r.t. the endogenous variables at binding constrains also enter (multiplied by the marginal value/Lagrange multiplier of those constraints).
Michael
CC BY-SA 4.0
2020-08-28T08:16:00.340Z
@Michael "*the endogenous variables [...] have zero marginal effect*" Not sure if I would call this statement economic :)
Giskard
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2020-08-28T08:19:13.603Z
@Giskard "...at a maximum...the endogenous variables...have zero marginal effect...by FOC".
Michael
CC BY-SA 4.0
2020-08-28T08:20:55.487Z
@Michael I do not question the statement's veracity, but it seems to be math, not economics. Which is fine, I was just perplexed by your insertion of the word "*economically*".
Giskard
CC BY-SA 4.0
2020-08-28T08:24:44.510Z
@Giskard What would you suggest...?
Michael
CC BY-SA 4.0
2020-08-28T08:27:27.110Z
@Michael Dropping that single word? It is not important though, and you cannot edit old comments anyway.
Giskard
CC BY-SA 4.0
2020-08-28T08:28:55.860Z
@Giskard Eh ok...if one must keep that word, what would you suggest?
Michael
CC BY-SA 4.0
2020-08-28T08:31:28.657Z
@Michael I am not sure what you mean. My whole point is that I would not keep the word. But as I wrote before, it is fine, no edits are necessary.
Giskard
CC BY-SA 4.0
2020-08-28T08:34:18.140Z
@Giskard I mean how would you answer the question "...economic interpretation of the envelope theorem?"
Michael
CC BY-SA 4.0
2020-08-28T08:36:11.093Z
@Michael The question is poorly phrased. As [emeryville pointed out](https://economics.stackexchange.com/questions/39436/is-there-an-economic-interpretation-of-the-envelope-theorem/39438?noredirect=1#comment60013_39436) the theorem has economic applications, not interpretations.
Giskard
CC BY-SA 4.0
2020-08-27T18:57:20.467Z
2021-12-04T07:15:59.303Z
2021-12-04T07:15:59.303Z
emeryville
<p>I wonder if envelope theorem has also some hidden economic interpretation. For example, Lagrangian multiplier in economics can have interpretation of 'shadow price' which is useful economic concept.</p>
<p>The envelope theorem seems like something that should have some deeper economic interpretation as it is about examining how function adjusts when parameters change given that we are at some optimum. The envelope theorem says that only the direct effects of a change in an exogenous variable need be considered, even though the exogenous variable may enter the maximum value function indirectly as part of the solution to the endogenous choice variables. But I am not sure if this makes sense. Is there any economic concept attached to the envelope theorem?</p>

CC BY-SA 4.0
2020-08-27T17:33:52.103Z
could you edit your post with a description of what the envelope theorem is so that we can recall similar concepts from economics
develarist
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2020-08-27T17:38:56.957Z
is this okay or do you want me to copy equations from textbook?
WilliamT
CC BY-SA 4.0
2020-08-27T18:12:14.777Z
Can we consider Hotelling's and Shephard's lemmas as economic interpretations of the unconstrained envelope theorem and the constrained envelope theorem, respectively, rather than simple applications?
emeryville
<mathematical-economics>
Is there an economic interpretation of the envelope theorem?
WilliamT
# Is there an economic interpretation of the envelope theorem?

(Asked by: username_0 on 2020-08-27)

I wonder if envelope theorem has also some hidden economic interpretation. For example, Lagrangian multiplier in economics can have interpretation of 'shadow price' which is useful economic concept.

The envelope theorem seems like something that should have some deeper economic interpretation as it is about examining how function adjusts when parameters change given that we are at some optimum. The envelope theorem says that only the direct effects of a change in an exogenous variable need be considered, even though the exogenous variable may enter the maximum value function indirectly as part of the solution to the endogenous choice variables. But I am not sure if this makes sense. Is there any economic concept attached to the envelope theorem?

**Tags:** mathematical-economics

> **Comments:**
> **username_1**: could you edit your post with a description of what the envelope theorem is so that we can recall similar concepts from economics  
> **username_0**: is this okay or do you want me to copy equations from textbook?  
> **username_2**: Can we consider Hotelling's and Shephard's lemmas as economic interpretations of the unconstrained envelope theorem and the constrained envelope theorem, respectively, rather than simple applications?  

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## 1 answers

### Answer by username_2 on 2020-08-27. Score: 9 | Accepted answer

First, let's consider a standard "economic interpretation" of the **envelope theorem for unconstrained maximization**:

if $x^\*(b)$ solves $\max\_x f(b,x)$, then $\frac{d f(b,x^\*(b))}{d b\_k} = \frac{\partial f(b,x^\*(b))}{\partial b\_k},$ i.e., the total derivative of $f$ w.r.t. $b\_k$ equals the partial derivative of $f$ w.r.t. $b\_k$.

The *economic interpretation (in the given problem)*: Let $\pi^\*(p,w) = p f(x^∗) − w·x^\*$ be the maximized value of profits given output price $p$ and input price vector $w$. Then the $i$'th input demand function is $x^\*\_i(·) = −\frac{\partial \pi^\*(·,·)}{\partial w\_i}$, known as **Hotelling's Lemma**, after Harold Hotelling, (1895–1973). This derivative is negative: if the price of the input increases, then the firm's maximal profit decreases.

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Second, let's consider a standard "economic interpretation" of the **envelope theorem for constrained maximization**:

if $x^\*(b), \lambda^\*$ solves $\max\_x f(b,x)$ s.t. $h^j(b,x) = 0$, $j= 1, ...m,$ then

$$\frac{d f(b,x^\*(b))}{d b\_k} = \frac{\partial f(b,x^\*(b))}{\partial b\_k} + \sum\_{j=1}^m \lambda\_j^\*(b)\frac{\partial h^j(b,x^\*(b))}{\partial b\_k}, $$

i.e., the total derivative of $f$ w.r.t. $b\_k$ equals the partial derivative of $f$ w.r.t. $b\_k$, plus the $\lambda^\*$-weighted sum of the partial derivatives of the $h^j$'s w.r.t. $b\_k$.

The *economic interpretation (in the given problem)*: Let $\hat c(\bar q, p, w) = w·\hat x$ be the minimized level of costs given prices $(p,w)$ and output level $\bar q$. Then the $i$'th conditional input demand function is $\hat x\_i(·) = −\frac{\partial \hat c(·,·,.)}{\partial w\_i}$, known as **Shepard's Lemma**, after Ronald Shephard (1912-1982). The partial derivatives of the expenditure function with respect to the prices of goods equal the Hicksian demand functions for the relevant goods.

> **Comments:**
> **username_3**: The "unconstrained" example $\max \pi $ is not exactly unconstrained. There is a constrain $x\geq 0$. The discussion (and the Envelope Theorem) only applies to interior solutions.  
> **username_4**: Shouldn't these be "the economic interpretation (in the given problem)" rather than "**the** economic interpretation"? Though these are the most famous ones, there are other constrained and unconstrained optimization problems in economics.  
> **username_3**: Agree with @username_4. A general economic interpretation is not tied to a particular example/application. The unconstrained theorem says, economically, that since the indirect objective function is the objective function at a maximum, the endogenous variables are solutions to an optimization problem, therefore they have zero marginal effect on the objective function by FOC. For the constrained version, the partial derivatives w.r.t. the endogenous variables at binding constrains also enter (multiplied by the marginal value/Lagrange multiplier of those constraints).  
> **username_4**: @username_3 "*the endogenous variables [...] have zero marginal effect*" Not sure if I would call this statement economic :)  
> **username_3**: @username_4 "...at a maximum...the endogenous variables...have zero marginal effect...by FOC".  
> **username_4**: @username_3 I do not question the statement's veracity, but it seems to be math, not economics. Which is fine, I was just perplexed by your insertion of the word "*economically*".  
> **username_3**: @username_4 What would you suggest...?  
> **username_4**: @username_3 Dropping that single word? It is not important though, and you cannot edit old comments anyway.  
> **username_3**: @username_4 Eh ok...if one must keep that word, what would you suggest?  
> **username_4**: @username_3 I am not sure what you mean. My whole point is that I would not keep the word. But as I wrote before, it is fine, no edits are necessary.  
> **username_3**: @username_4 I mean how would you answer the question "...economic interpretation of the envelope theorem?"  
> **username_4**: @username_3 The question is poorly phrased. As [username_2 pointed out](https://economics.stackexchange.com/questions/39436/is-there-an-economic-interpretation-of-the-envelope-theorem/39438?noredirect=1#comment60013_39436) the theorem has economic applications, not interpretations.  

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