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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?

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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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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.

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    $\begingroup$ There might be a typo: the profit function $\pi$ is convex, not concave. $\endgroup$
    – tdm
    Oct 31 '21 at 12:02
  • $\begingroup$ @tdm Right, convex not concave. $\endgroup$
    – Michael
    Nov 4 '21 at 8:05

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