<blockquote>
<p>What is the economic interpretation of the identity (apart from giving
the law supply for free)?</p>
</blockquote>
<p>A more basic question is <strong>why does one get the expected sign for the law of supply for free?</strong> In other words, why should <span class="math-container">$\frac{\partial}{\partial p}q(p, w)$</span> be positive?</p>
<p>For the law of demand, the sign that one would like is <strong>not</strong> for free. One has to introduce convexity into the problem, i.e. preference needs to be quasi-concave. Then, by Envelope Theorem, one has
<span class="math-container">$$
x^l(p, u) = D_p e(p,u) \Rightarrow D_p x^l(p, u) = D^2_p e(p,u),
$$</span>
where <span class="math-container">$x^l$</span> is the Hicksian demand and <span class="math-container">$e(p,u)$</span> the minimum expenditure function.
By quasi-concavity of preference, <span class="math-container">$e(p,u)$</span> is concave. This give the negative-semidefiniteness of the substitution matrix <span class="math-container">$D_p x^l(p, u)$</span>.</p>
<p>Here we get <span class="math-container">$\frac{\partial}{\partial p}q(p, w) \geq 0$</span> <em>for free</em>, because we get convexity for free for the firm's problem, unlike for the consumer's problem. <span class="math-container">$\pi$</span> is by definition a maximum of affine functions, therefore convex.
Symmetry is a consequence of convexity, which can be related to transitivity of preferences in the case of consumer's problem. For the firm's problem, I don't know an economic interpretation.</p>
<p>The derivation can be much shorter. Similar to the demand case,
<span class="math-container">$$
x(p,w) = - D_w \pi(p,w) \Rightarrow D_w x(p, w) = - D^2_w \pi(p,w),
$$</span></p>
<p>For an interpretation, the identity can be re-written as (say <span class="math-container">$n = 1$</span>)
<span class="math-container">$$
pq \left( \frac{p}{q} \frac{dq}{dp} \right) + wx \left( \frac{w}{x} \frac{dx}{dw}  \right) = 0,
$$</span>
i.e revenue times output price elasticity and cost times input price elasticity must offset each other.</p>

CC BY-SA 4.0
2021-10-31T12:02:51.223Z
There might be a typo: the profit function $\pi$ is convex, not concave.
tdm
CC BY-SA 4.0
2021-11-04T08:05:27.233Z
@tdm Right, convex not concave.
Michael
CC BY-SA 4.0
2021-10-31T10:53:12.393Z
2021-11-04T08:06:35.297Z
2021-11-04T08:06:35.297Z
Michael
<p>Notice that you don't need this entire proof to show that <span class="math-container">$\dfrac{\partial q(p,w)}{\partial p} \ge 0$</span>.</p>
<p>As <span class="math-container">$\pi(p,w)$</span> is convex, then we immediately have by the envelope theorem that:
<span class="math-container">$$
0 \le \frac{\partial^2 \pi(p,w)}{\partial p^2} = \frac{\partial q(p,w)}{\partial p}.
$$</span></p>
<p>One, maybe more math intuition for the proof is that it readily follows from the fact that the optimal supply function <span class="math-container">$q(p,w)$</span> and factor demand functions <span class="math-container">$x_i(p,w)$</span> are homogeneous of degree zero in output and input prices:
<span class="math-container">$$
\begin{align*}
&amp;q(tp,tw) = q(p,w),\\
&amp;x_i(tp,tw) = x_i(p,w).
\end{align*}
$$</span>
Euler's theorem gives:
<span class="math-container">$$
\begin{align*}
&amp;p\frac{\partial q(p,w)}{\partial p} + \sum_j w_j \frac{\partial q(p,w)}{\partial w_j} = 0 \tag{1}\\
&amp;p \frac{\partial x_i(p,w)}{\partial p} + \sum_j w_j \frac{\partial w_i(p,w)}{\partial w_j} = 0 \tag{2}
\end{align*}
$$</span>
From <span class="math-container">$(1$</span>) and using symmetry of price effects, we have:
<span class="math-container">$$
p \frac{\partial q(p,w)}{\partial p} - \sum_j w_j \frac{\partial x_j(p,w)}{\partial p} = 0
$$</span>
Then if we use <span class="math-container">$(2)$</span> to substitute out the terms <span class="math-container">$\dfrac{\partial x_j(p,w)}{\partial p}$</span> we get the final result:
<span class="math-container">$$
p \frac{\partial q(p,w)}{\partial p} + \sum_{j} \sum_i \frac{w_j w_i}{p} \frac{\partial x_i(p,w)}{\partial w_j} = 0
$$</span></p>
<blockquote>
<p>What is the economic interpretation of the identity (apart from giving the law supply for free)? It seems to relate the self-price elasticity of output with the cross-price elasticities of inputs?</p>
</blockquote>
<p>I don't know if the identity has a clear economic intuition. Equivalently, one could  ask for the economic intuition behind the fact that the cross price elasiticities of Hicksian demands are symmetric? We know it readily follows from Young's theorem and the envelope theorem, but in general I don't think there is a clear economic intuition for this.</p>
<p>For the current derivation we the envelope theorem and Young's theorem (to get symmetry) and homgeneity of degree zero of the supply and factor demands. But I don't think there is a clear economic intution.</p>
<p>Expressing the identity in elasticity form, we have:
<span class="math-container">$$
r \varepsilon^q_p + \sum_i c_i \sum_j \varepsilon^{x_i}_{w_j} = 0,
$$</span>
where <span class="math-container">$r$</span> is the revenue (<span class="math-container">$r = p q(p,w)$</span>) and <span class="math-container">$c_i$</span> is the cost for factor <span class="math-container">$i$</span>: (<span class="math-container">$c_i = w_i x_i(p,w)$</span>). Next, <span class="math-container">$\varepsilon^q_p$</span> is the output price elasiticity of supply and <span class="math-container">$\varepsilon^{x_i}_{w_j}$</span> is the factor <span class="math-container">$j$</span> input price elasticity for factor <span class="math-container">$i$</span> demand.</p>

CC BY-SA 4.0
2021-10-31T08:30:20.590Z
2021-10-31T08:30:20.590Z
tdm
<p>Consider a profit maximizing firm that is perfectly competitive in both input and output markets. It takes <span class="math-container">$n$</span> inputs <span class="math-container">$ x \in \mathbb{R}^n_+ $</span> and produces <span class="math-container">$f(x)$</span> units of output, where <span class="math-container">$ f: \mathbb{R}^n_+ \to \mathbb{R}_+ $</span> is the production function. From now on, assume <span class="math-container">$f$</span> has all the necessary regularity conditions whenever the proof requires.</p>
<p><strong>Theorem</strong></p>
<p>Suppose at each input price <span class="math-container">$w \in \mathbb{R}^n_{++}$</span> and output price <span class="math-container">$p &gt; 0$</span>, the profit maximization problem</p>
<p><span class="math-container">$$ \pi(p, w) = \max\{ p f(x) - w \cdot x : x \in \mathbb{R}^n_+ \} $$</span></p>
<p>has a unique solution <span class="math-container">$x(p, w)$</span>. Let <span class="math-container">$ q(p, w) = f(x(p, w)) $</span> be the corresponding output supply function, then</p>
<p><span class="math-container">$$ \frac{\partial q}{\partial p}(p, w) = -\frac{1}{p^2} w^T D_w x(p, w) w \geq 0 $$</span></p>
<p><strong>Proof:</strong></p>
<p>The first-order condition for profit maximization is:</p>
<p><span class="math-container">$$ p \frac{\partial f}{\partial x_i}(x(p, w)) = w_i \quad \forall i $$</span></p>
<p>Differentiate <span class="math-container">$ q(p, w) = f(x(p, w)) $</span> with respect to <span class="math-container">$p$</span>:</p>
<p><span class="math-container">$$ \frac{\partial q}{\partial p}(p, w) = \sum_{i=1}^n \frac{\partial f}{\partial x_i}(x(p, w)) \frac{\partial x_i}{\partial p}(p, w) = \sum_{i=1}^n \frac{w_i}{p} \frac{\partial x_i}{\partial p}(p, w) $$</span></p>
<p>To evaluate <span class="math-container">$ \partial x_i / \partial p $</span>, apply the envelop theorem to the profit function <span class="math-container">$\pi$</span> twice:</p>
<p><span class="math-container">$$ \frac{\partial x_i}{\partial p} = \frac{\partial}{\partial p} \left( -\frac{\partial \pi}{\partial w_i} \right) = -\frac{\partial}{\partial w_i} \frac{\partial \pi}{\partial p} = -\frac{\partial q}{\partial w_i} $$</span></p>
<p>Differentiate <span class="math-container">$ q(p, w) = f(x(p, w)) $</span> with respect to <span class="math-container">$w_i$</span> and use the first-order condition again:</p>
<p><span class="math-container">$$
\begin{align*}
\frac{\partial x_i}{\partial p} = -\frac{\partial q}{\partial w_i} &amp;= -\sum_{j=1}^n \frac{\partial f}{\partial x_j}(x(p, w)) \frac{\partial x_j}{\partial w_i}(p, w) \\
&amp;= -\sum_{j=1}^n \frac{w_j}{p} \frac{\partial x_j}{\partial w_i}(p, w)
\end{align*}
$$</span></p>
<p>Substitute the above back to <span class="math-container">$ \partial q / \partial p $</span>:</p>
<p><span class="math-container">$$
\begin{align*}
\frac{\partial q}{\partial p}(p, w) &amp;= \sum_{i=1}^n \frac{w_i}{p} \left( -\sum_{j=1}^n \frac{w_j}{p} \frac{\partial x_j}{\partial w_i}(p, w) \right) \\
&amp;= -\frac{1}{p^2} \sum_{i,j=1}^n w_i \frac{\partial x_j}{\partial w_i}(p, w) w_j
\end{align*}
$$</span></p>
<p>We are almost done! To get the required matrix form, we need <span class="math-container">$ D_w x = ( \partial x_j / \partial w_i )_{ij} $</span> to be a symmetric matrix. But this follows from the envelop theorem again:</p>
<p><span class="math-container">$$ D_w x = \left( \frac{\partial x_j}{\partial w_i} \right)_{ij} = \left( -\frac{\partial^2 \pi}{\partial w_i \partial w_j} \right)_{ij} = -D^2_w \pi $$</span></p>
<p>where <span class="math-container">$ D^2_w \pi $</span> is the Hessian matrix of <span class="math-container">$ \pi $</span> with respect to <span class="math-container">$w$</span>. Thus, we have the required identity:</p>
<p><span class="math-container">$$ \frac{\partial q}{\partial p}(p, w) = -\frac{1}{p^2} w^T D_w x(p, w) w $$</span></p>
<p>Finally, <span class="math-container">$\pi$</span> is a convex function, so <span class="math-container">$ D^2 \pi = - D_w x $</span> is positive semi-definite. But this means <span class="math-container">$ - w^T D_w x(p, w) w $</span> is a positive semi-definite quadratic form, so <span class="math-container">$ \partial q / \partial p \geq 0 $</span>. This is just the <strong>law of supply</strong>.</p>
<p><strong>Question:</strong></p>
<p>What is the economic interpretation of the identity (apart from giving the law supply for free)? It seems to relate the self-price elasticity of output with the cross-price elasticities of inputs?</p>

<microeconomics><mathematical-economics><supply-and-demand><elasticity>
What is the economic interpretation of $ \partial q / \partial p = - (1/p^2) w^T (D_w x) w $ in profit maximization?
user141240
# What is the economic interpretation of $ \partial q / \partial p = - (1/p^2) w^T (D_w x) w $ in profit maximization?

(Asked by: username_0 on 2021-10-30)

Consider a profit maximizing firm that is perfectly competitive in both input and output markets. It takes $n$ inputs $ x \in \mathbb{R}^n\_+ $ and produces $f(x)$ units of output, where $ f: \mathbb{R}^n\_+ \to \mathbb{R}\_+ $ is the production function. From now on, assume $f$ has all the necessary regularity conditions whenever the proof requires.

**Theorem**

Suppose at each input price $w \in \mathbb{R}^n\_{++}$ and output price $p > 0$, the profit maximization problem

$$ \pi(p, w) = \max\{ p f(x) - w \cdot x : x \in \mathbb{R}^n\_+ \} $$

has a unique solution $x(p, w)$. Let $ q(p, w) = f(x(p, w)) $ be the corresponding output supply function, then

$$ \frac{\partial q}{\partial p}(p, w) = -\frac{1}{p^2} w^T D\_w x(p, w) w \geq 0 $$

**Proof:**

The first-order condition for profit maximization is:

$$ p \frac{\partial f}{\partial x\_i}(x(p, w)) = w\_i \quad \forall i $$

Differentiate $ q(p, w) = f(x(p, w)) $ with respect to $p$:

$$ \frac{\partial q}{\partial p}(p, w) = \sum\_{i=1}^n \frac{\partial f}{\partial x\_i}(x(p, w)) \frac{\partial x\_i}{\partial p}(p, w) = \sum\_{i=1}^n \frac{w\_i}{p} \frac{\partial x\_i}{\partial p}(p, w) $$

To evaluate $ \partial x\_i / \partial p $, apply the envelop theorem to the profit function $\pi$ twice:

$$ \frac{\partial x\_i}{\partial p} = \frac{\partial}{\partial p} \left( -\frac{\partial \pi}{\partial w\_i} \right) = -\frac{\partial}{\partial w\_i} \frac{\partial \pi}{\partial p} = -\frac{\partial q}{\partial w\_i} $$

Differentiate $ q(p, w) = f(x(p, w)) $ with respect to $w\_i$ and use the first-order condition again:

$$
\begin{align\*}
\frac{\partial x\_i}{\partial p} = -\frac{\partial q}{\partial w\_i} &= -\sum\_{j=1}^n \frac{\partial f}{\partial x\_j}(x(p, w)) \frac{\partial x\_j}{\partial w\_i}(p, w) \\
&= -\sum\_{j=1}^n \frac{w\_j}{p} \frac{\partial x\_j}{\partial w\_i}(p, w)
\end{align\*}
$$

Substitute the above back to $ \partial q / \partial p $:

$$
\begin{align\*}
\frac{\partial q}{\partial p}(p, w) &= \sum\_{i=1}^n \frac{w\_i}{p} \left( -\sum\_{j=1}^n \frac{w\_j}{p} \frac{\partial x\_j}{\partial w\_i}(p, w) \right) \\
&= -\frac{1}{p^2} \sum\_{i,j=1}^n w\_i \frac{\partial x\_j}{\partial w\_i}(p, w) w\_j
\end{align\*}
$$

We are almost done! To get the required matrix form, we need $ D\_w x = ( \partial x\_j / \partial w\_i )\_{ij} $ to be a symmetric matrix. But this follows from the envelop theorem again:

$$ D\_w x = \left( \frac{\partial x\_j}{\partial w\_i} \right)\_{ij} = \left( -\frac{\partial^2 \pi}{\partial w\_i \partial w\_j} \right)\_{ij} = -D^2\_w \pi $$

where $ D^2\_w \pi $ is the Hessian matrix of $ \pi $ with respect to $w$. Thus, we have the required identity:

$$ \frac{\partial q}{\partial p}(p, w) = -\frac{1}{p^2} w^T D\_w x(p, w) w $$

Finally, $\pi$ is a convex function, so $ D^2 \pi = - D\_w x $ is positive semi-definite. But this means $ - w^T D\_w x(p, w) w $ is a positive semi-definite quadratic form, so $ \partial q / \partial p \geq 0 $. This is just the **law of supply**.

**Question:**

What is the economic interpretation of the identity (apart from giving the law supply for free)? It seems to relate the self-price elasticity of output with the cross-price elasticities of inputs?

**Tags:** microeconomics, mathematical-economics, supply-and-demand, elasticity


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## 2 answers

### Answer by username_2 on 2021-10-31. Score: 4 | Accepted answer

Notice that you don't need this entire proof to show that $\dfrac{\partial q(p,w)}{\partial p} \ge 0$.

As $\pi(p,w)$ is convex, then we immediately have by the envelope theorem that:
$$
0 \le \frac{\partial^2 \pi(p,w)}{\partial p^2} = \frac{\partial q(p,w)}{\partial p}.
$$

One, maybe more math intuition for the proof is that it readily follows from the fact that the optimal supply function $q(p,w)$ and factor demand functions $x\_i(p,w)$ are homogeneous of degree zero in output and input prices:
$$
\begin{align\*}
&q(tp,tw) = q(p,w),\\
&x\_i(tp,tw) = x\_i(p,w).
\end{align\*}
$$
Euler's theorem gives:
$$
\begin{align\*}
&p\frac{\partial q(p,w)}{\partial p} + \sum\_j w\_j \frac{\partial q(p,w)}{\partial w\_j} = 0 \tag{1}\\
&p \frac{\partial x\_i(p,w)}{\partial p} + \sum\_j w\_j \frac{\partial w\_i(p,w)}{\partial w\_j} = 0 \tag{2}
\end{align\*}
$$
From $(1$) and using symmetry of price effects, we have:
$$
p \frac{\partial q(p,w)}{\partial p} - \sum\_j w\_j \frac{\partial x\_j(p,w)}{\partial p} = 0
$$
Then if we use $(2)$ to substitute out the terms $\dfrac{\partial x\_j(p,w)}{\partial p}$ we get the final result:
$$
p \frac{\partial q(p,w)}{\partial p} + \sum\_{j} \sum\_i \frac{w\_j w\_i}{p} \frac{\partial x\_i(p,w)}{\partial w\_j} = 0
$$

> What is the economic interpretation of the identity (apart from giving the law supply for free)? It seems to relate the self-price elasticity of output with the cross-price elasticities of inputs?

I don't know if the identity has a clear economic intuition. Equivalently, one could ask for the economic intuition behind the fact that the cross price elasiticities of Hicksian demands are symmetric? We know it readily follows from Young's theorem and the envelope theorem, but in general I don't think there is a clear economic intuition for this.

For the current derivation we the envelope theorem and Young's theorem (to get symmetry) and homgeneity of degree zero of the supply and factor demands. But I don't think there is a clear economic intution.

Expressing the identity in elasticity form, we have:
$$
r \varepsilon^q\_p + \sum\_i c\_i \sum\_j \varepsilon^{x\_i}\_{w\_j} = 0,
$$
where $r$ is the revenue ($r = p q(p,w)$) and $c\_i$ is the cost for factor $i$: ($c\_i = w\_i x\_i(p,w)$). Next, $\varepsilon^q\_p$ is the output price elasiticity of supply and $\varepsilon^{x\_i}\_{w\_j}$ is the factor $j$ input price elasticity for factor $i$ demand.


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### Answer by username_1 on 2021-10-31. Score: 2

> What is the economic interpretation of the identity (apart from giving
> the law supply for free)?

A more basic question is **why does one get the expected sign for the law of supply for free?** In other words, why should $\frac{\partial}{\partial p}q(p, w)$ be positive?

For the law of demand, the sign that one would like is **not** for free. One has to introduce convexity into the problem, i.e. preference needs to be quasi-concave. Then, by Envelope Theorem, one has
$$
x^l(p, u) = D\_p e(p,u) \Rightarrow D\_p x^l(p, u) = D^2\_p e(p,u),
$$
where $x^l$ is the Hicksian demand and $e(p,u)$ the minimum expenditure function.
By quasi-concavity of preference, $e(p,u)$ is concave. This give the negative-semidefiniteness of the substitution matrix $D\_p x^l(p, u)$.

Here we get $\frac{\partial}{\partial p}q(p, w) \geq 0$ *for free*, because we get convexity for free for the firm's problem, unlike for the consumer's problem. $\pi$ is by definition a maximum of affine functions, therefore convex.
Symmetry is a consequence of convexity, which can be related to transitivity of preferences in the case of consumer's problem. For the firm's problem, I don't know an economic interpretation.

The derivation can be much shorter. Similar to the demand case,
$$
x(p,w) = - D\_w \pi(p,w) \Rightarrow D\_w x(p, w) = - D^2\_w \pi(p,w),
$$

For an interpretation, the identity can be re-written as (say $n = 1$)
$$
pq \left( \frac{p}{q} \frac{dq}{dp} \right) + wx \left( \frac{w}{x} \frac{dx}{dw} \right) = 0,
$$
i.e revenue times output price elasticity and cost times input price elasticity must offset each other.

> **Comments:**
> **username_2**: There might be a typo: the profit function $\pi$ is convex, not concave.  
> **username_1**: @username_2 Right, convex not concave.  

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