<p>With subjects like Hotelling's lemma precise notation is crucial.</p>

<p>The function $\Pi(a,b,c,d)$ is defined as
$$
\Pi(a,b,c,d) \triangleq a \cdot c - b \cdot d.
$$
Now if we want to make this the profit function
$$
\Pi (p,w,y(p,w),x(p,w)) = p \cdot y(p,w) - w \cdot x(p,w),
$$
we should use the input where $y(p,w)$ and $x(p,w)$ are solutions of the constrained optimization problem
$$
\begin{align*}
\max_{y,x} \ &amp; p \cdot y - w \cdot x \\
s.t. \ &amp; f(x) = y 
\end{align*}
$$</p>

<p>There is difference between 
$$
\frac{\text{d}\Pi (p,w,y(p,w),x(p,w))}{\text{d}p}
$$
and
$$
\frac{\partial\Pi (p,w,y(p,w),x(p,w))}{\partial p}.
$$</p>

<p>You seem to be doing
$$
\frac{\text{d}\Pi (p,w,y(p,w),x(p,w))}{\text{d}p}
=
y(p,w) + p\frac{\partial y(p,w)}{\partial p} -w \frac{\partial x(p,w)}{\partial p}
$$
whih is not trivially the same as
$$
\frac{\partial \Pi (p,w,y(p,w),x(p,w))}{\partial p}
=
y(p,w).
$$</p>

<hr>

<p>A trivial example to illustrate the difference:</p>

<p>Let the function $A(x,y) \triangleq x \cdot y$ measure the area of a rectangle with sides the length of $x$ and $y$. Then the partial derivate w.r.t. $x$
$$
\frac{\partial A(x,y)}{\partial x} = y
$$
measures how much the size of the rectangle would increase if I were to increase size x.
Now imagine that I am interested in squares specifically, so I look at rectangles where if one size has length $x$, the length $y$ of the other size is $y(x) = x$. Then the function becomes
$$
A(x,y(x)) = x \cdot y(x) = x^2.
$$
If I now consider how much the size of the square would increase if I were to increase only one side, the answer is still
$$
\frac{\partial A(x,y(x))}{\partial x} = y(x) = x.
$$
(This is the definition of partial derivatives.)
But if I want to consider how much the area of the square would change if I were to increase the parameter determining the length of both sides I should take the full derivative
$$
\frac{\text{d} A(x,y(x))}{\text{d} x} = \frac{\text{d} x^2}{\text{d} x} = 2x.
$$
Notice that in this case
$$
\frac{\text{d} A(x,y(x))}{\text{d} x} \neq \frac{\partial A(x,y(x))}{\partial x}.
$$
<a href="https://en.wikipedia.org/wiki/Envelope_theorem#Applications_to_producer_theory" rel="nofollow noreferrer">Hotelling's lemma</a> tells you that some types of constrained optimization problems are magic because there this equation holds.</p>

CC BY-SA 3.0
2018-04-27T20:05:01.630Z
2018-04-28T12:06:44.493Z
2018-04-28T12:06:44.493Z
Giskard
<p>You are basically on the right track but might have been carried away by not writing down function arguments. The profit function $\pi$ has arguments $p$ and $w$; don't drop them.</p>

<p>In Hotelling's lemma, $y$ is the output. On the other hand, you write $y$ for the production function. Let's make things clear by writing neutrally $f$ for the production function. The profit-maximizing input choice is $x(p,w)$, so the quantity produced is $f\big(x(p,w)\big)$, the correct meaning of $y$ in Hotelling's lemma. </p>

<p>Hotelling's lemma then becomes
$$\frac{\partial \pi(p,w)}{\partial p}=f\big(x(p,w)\big).$$
The profit function is given by 
$$\pi(p,w)=pf\big(x(p,w)\big)-wx(p,w).$$ </p>

<p>Taking the derivative with respect to $p$ gives us
$$\frac{\partial \pi(p,w)}{\partial p}=f\big(x(p,w)\big)+pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial p}-w\frac{\partial x(p,w)}{\partial p}.$$
This is pretty much what you got there above. </p>

<p>Now $x(p,w)$ is by definition a maximizer of $pf(x)-wx$. We have the first order condition 
$pf'(x^*)=w$. Therefore, 
$$pf'\big(x(p,w)\big)-w=0,$$
which implies $$pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial p}-w\frac{\partial x(p,w)}{\partial p}=0$$
This reduces our expression for ${\partial \pi(p,w)}/{\partial p}$ to Hotelling's lemma. </p>

<p>This tell's us also what the important message of Hotelling's lemma is: Indirect effects don't matter. A marginal change in prices has a direct effect on profits since the output gets multiplied by a different number. There is also an indirect effect that comes from firms adjusting how much to produce. Hotelling's lemma just tells us that the latter effect is zero for a profit-maximizing firm, which need not do much adjusting. At a profit maximum, the marginal effect of adjusting production on profits is zero.</p>

CC BY-SA 3.0
2018-04-29T02:31:56.240Z
"$pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial p}-w\frac{\partial x(p,w)}{\partial p}=0$
This reduces our expression for ${\partial \pi(p,w)}/{\partial p}$ to Hotelling's lemma." This is astounding. I'm simulating this result and am amazed with it.
EconJohn
CC BY-SA 3.0
2018-04-27T21:57:06.903Z
2018-04-27T21:57:06.903Z
Michael Greinecker
<p>Hotelling's lemma is stated as:</p>

<p>$$\frac{\partial \pi}{\partial p}=y$$</p>

<p>knowing however that on the more basic level, output $y$ is determined by the input(s) $x(p,w)$,let the profit function be defined as:</p>

<p>$$\pi=py(x(p,w))-wx(p,w)$$</p>

<p>taking the derivative with respect to $p$</p>

<p>$$\frac{\partial\pi}{\partial p}=y(x(p,w))+p\frac{\partial y(x(p,w))}{\partial x(p,w)}\frac{\partial x(p,w)}{\partial p}-w \frac{\partial x(p,w)}{\partial p}$$</p>

<p>Wouldn't this be a more accurate definition of hotelling's lemma?</p>

<p>I'm speculating considering some of the critsisim I read on hotelling's lemma in applied work. Namely: <a href="https://core.ac.uk/download/pdf/7043885.pdf" rel="nofollow noreferrer">Duality, Optimization, and Microeconomic Theory: Pitfalls for the Applied Researcher</a> </p>

CC BY-SA 3.0
2018-04-27T19:52:03.503Z
The wikipedia page for Hotelling's Lemma is abysmal, but the one for the [Envelope theorem](https://en.wikipedia.org/wiki/Envelope_theorem) is good, you should read that one.
Giskard
<microeconomics><mathematical-economics>
Modifying Hotelling's lemma: Is this valid?
EconJohn
# Modifying Hotelling's lemma: Is this valid?

(Asked by: username_0 on 2018-04-27)

Hotelling's lemma is stated as:

$$\frac{\partial \pi}{\partial p}=y$$

knowing however that on the more basic level, output $y$ is determined by the input(s) $x(p,w)$,let the profit function be defined as:

$$\pi=py(x(p,w))-wx(p,w)$$

taking the derivative with respect to $p$

$$\frac{\partial\pi}{\partial p}=y(x(p,w))+p\frac{\partial y(x(p,w))}{\partial x(p,w)}\frac{\partial x(p,w)}{\partial p}-w \frac{\partial x(p,w)}{\partial p}$$

Wouldn't this be a more accurate definition of hotelling's lemma?

I'm speculating considering some of the critsisim I read on hotelling's lemma in applied work. Namely: [Duality, Optimization, and Microeconomic Theory: Pitfalls for the Applied Researcher](https://core.ac.uk/download/pdf/7043885.pdf)

**Tags:** microeconomics, mathematical-economics
> **Comments:**
> **username_1**: The wikipedia page for Hotelling's Lemma is abysmal, but the one for the [Envelope theorem](https://en.wikipedia.org/wiki/Envelope_theorem) is good, you should read that one.  

---

## 2 answers

### Answer by username_2 on 2018-04-27. Score: 4 | Accepted answer

You are basically on the right track but might have been carried away by not writing down function arguments. The profit function $\pi$ has arguments $p$ and $w$; don't drop them.

In Hotelling's lemma, $y$ is the output. On the other hand, you write $y$ for the production function. Let's make things clear by writing neutrally $f$ for the production function. The profit-maximizing input choice is $x(p,w)$, so the quantity produced is $f\big(x(p,w)\big)$, the correct meaning of $y$ in Hotelling's lemma.

Hotelling's lemma then becomes
$$\frac{\partial \pi(p,w)}{\partial p}=f\big(x(p,w)\big).$$
The profit function is given by
$$\pi(p,w)=pf\big(x(p,w)\big)-wx(p,w).$$

Taking the derivative with respect to $p$ gives us
$$\frac{\partial \pi(p,w)}{\partial p}=f\big(x(p,w)\big)+pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial p}-w\frac{\partial x(p,w)}{\partial p}.$$
This is pretty much what you got there above.

Now $x(p,w)$ is by definition a maximizer of $pf(x)-wx$. We have the first order condition
$pf'(x^\*)=w$. Therefore,
$$pf'\big(x(p,w)\big)-w=0,$$
which implies $$pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial p}-w\frac{\partial x(p,w)}{\partial p}=0$$
This reduces our expression for ${\partial \pi(p,w)}/{\partial p}$ to Hotelling's lemma.

This tell's us also what the important message of Hotelling's lemma is: Indirect effects don't matter. A marginal change in prices has a direct effect on profits since the output gets multiplied by a different number. There is also an indirect effect that comes from firms adjusting how much to produce. Hotelling's lemma just tells us that the latter effect is zero for a profit-maximizing firm, which need not do much adjusting. At a profit maximum, the marginal effect of adjusting production on profits is zero.

> **Comments:**
> **username_0**: "$pf'\big(x(p,w)\big)\frac{\partial x(p,w)}{\partial p}-w\frac{\partial x(p,w)}{\partial p}=0$ This reduces our expression for ${\partial \pi(p,w)}/{\partial p}$ to Hotelling's lemma." This is astounding. I'm simulating this result and am amazed with it.  

---

### Answer by username_1 on 2018-04-27. Score: 1

With subjects like Hotelling's lemma precise notation is crucial.

The function $\Pi(a,b,c,d)$ is defined as
$$
\Pi(a,b,c,d) \triangleq a \cdot c - b \cdot d.
$$
Now if we want to make this the profit function
$$
\Pi (p,w,y(p,w),x(p,w)) = p \cdot y(p,w) - w \cdot x(p,w),
$$
we should use the input where $y(p,w)$ and $x(p,w)$ are solutions of the constrained optimization problem
$$
\begin{align\*}
\max\_{y,x} \ & p \cdot y - w \cdot x \\
s.t. \ & f(x) = y
\end{align\*}
$$

There is difference between
$$
\frac{\text{d}\Pi (p,w,y(p,w),x(p,w))}{\text{d}p}
$$
and
$$
\frac{\partial\Pi (p,w,y(p,w),x(p,w))}{\partial p}.
$$

You seem to be doing
$$
\frac{\text{d}\Pi (p,w,y(p,w),x(p,w))}{\text{d}p}
=
y(p,w) + p\frac{\partial y(p,w)}{\partial p} -w \frac{\partial x(p,w)}{\partial p}
$$
whih is not trivially the same as
$$
\frac{\partial \Pi (p,w,y(p,w),x(p,w))}{\partial p}
=
y(p,w).
$$

---

A trivial example to illustrate the difference:

Let the function $A(x,y) \triangleq x \cdot y$ measure the area of a rectangle with sides the length of $x$ and $y$. Then the partial derivate w.r.t. $x$
$$
\frac{\partial A(x,y)}{\partial x} = y
$$
measures how much the size of the rectangle would increase if I were to increase size x.
Now imagine that I am interested in squares specifically, so I look at rectangles where if one size has length $x$, the length $y$ of the other size is $y(x) = x$. Then the function becomes
$$
A(x,y(x)) = x \cdot y(x) = x^2.
$$
If I now consider how much the size of the square would increase if I were to increase only one side, the answer is still
$$
\frac{\partial A(x,y(x))}{\partial x} = y(x) = x.
$$
(This is the definition of partial derivatives.)
But if I want to consider how much the area of the square would change if I were to increase the parameter determining the length of both sides I should take the full derivative
$$
\frac{\text{d} A(x,y(x))}{\text{d} x} = \frac{\text{d} x^2}{\text{d} x} = 2x.
$$
Notice that in this case
$$
\frac{\text{d} A(x,y(x))}{\text{d} x} \neq \frac{\partial A(x,y(x))}{\partial x}.
$$
[Hotelling's lemma](https://en.wikipedia.org/wiki/Envelope_theorem#Applications_to_producer_theory) tells you that some types of constrained optimization problems are magic because there this equation holds.


---

