<ol>
<li><p>In my classes, they’re called Marshallian or uncompensated demands, as in consumer theory.</p>
</li>
<li><p>The partial derivative on the right hand side means to directly differentiate the demand for <span class="math-container">$z$</span> (the optimal <span class="math-container">$z$</span> level) with respect <span class="math-container">$w_x$</span> (the price of the other factor, <span class="math-container">$x$</span>).</p>
</li>
</ol>
<p>In consumer theory for goods <span class="math-container">$x,y$</span> with prices <span class="math-container">$p_x,p_y$</span>, complement and substitute goods are defined as:</p>
<ul>
<li>Complements</li>
</ul>
<p><span class="math-container">$\frac{\partial x}{\partial p_y} &lt; 0$</span></p>
<p>Here an increase in the price of good <span class="math-container">$y$</span> makes a lower amount of <span class="math-container">$y$</span> to be demanded as it is more expensive, and hence a lower amount of <span class="math-container">$x$</span> is demanded as <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are usually consumed together.</p>
<ul>
<li>Substitutes</li>
</ul>
<p><span class="math-container">$\frac{\partial x}{\partial p_y} &gt; 0$</span></p>
<p>Here an increase in the price of good <span class="math-container">$y$</span> makes a lower amount of <span class="math-container">$y$</span> to be demanded as it is more expensive, and a higher amount of <span class="math-container">$x$</span> is demanded as the consumer substitutes good <span class="math-container">$y$</span> for the relatively cheaper alternative good <span class="math-container">$x$</span>.</p>
<ul>
<li>Unrelated goods</li>
</ul>
<p><span class="math-container">$\frac{\partial x}{\partial p_y} = 0$</span></p>
<p>A change in price in good <span class="math-container">$y$</span>, and hence its quantity demanded, has no effect over the demand for good <span class="math-container">$x$</span>.</p>
<p>Here you’re working with the analogous notion of complements/substitutes/unrelated goods (here production factors) in firm theory.</p>

CC BY-SA 4.0
2023-05-06T20:05:55.177Z
2023-05-06T20:05:55.177Z
Nicolas Torres
<p><span class="math-container">$\pi(x,z) = p(a\ln(x) + b\ln(z)) - w_xx - w_zz$</span></p>
<p><strong>Question 1:</strong></p>
<p>Using the first order conditions, we get:</p>
<p><span class="math-container">$x = \frac{pa}{w_x}$</span>
<span class="math-container">$z = \frac{pb}{w_z}$</span></p>
<p>What do we call these Input demand functions as a function of price i.e. are they just un-compensated demand functions as in consumer theory? . I.e. When it's in terms of output, we call them <strong>contingent</strong></p>
<p><strong>Question 2</strong></p>
<p>I'm asked <em>&quot;What is the relationship between <span class="math-container">$x$</span> and <span class="math-container">$z$</span>, are they complements or substitutes&quot;</em>. The answer is</p>
<blockquote>
<p>These inputs <span class="math-container">$x$</span> and <span class="math-container">$z$</span> are not related because : <span class="math-container">$\pi_{w_z,w_x} = - \frac{\partial z}{\partial w_x} = 0$</span></p>
</blockquote>
<p>Could someone kindly explain what is going on here?
I believe the <span class="math-container">$\pi_{w_z,w_x}$</span> tells us how the marginal profit from a change in the input cost of <span class="math-container">$z$</span> changes when the cost of input <span class="math-container">$x$</span> changes. But I'm not sure i understand the partial derivative on the right hand side. It looks a bit like the chain rule for implicit differentiation.</p>

<mathematical-economics><production-function><optimization><profit-maximization>
Mixed Partial Derivatives in Profit Function
CormJack
# Mixed Partial Derivatives in Profit Function

(Asked by: username_0 on 2023-05-06)

$\pi(x,z) = p(a\ln(x) + b\ln(z)) - w\_xx - w\_zz$

**Question 1:**

Using the first order conditions, we get:

$x = \frac{pa}{w\_x}$
$z = \frac{pb}{w\_z}$

What do we call these Input demand functions as a function of price i.e. are they just un-compensated demand functions as in consumer theory? . I.e. When it's in terms of output, we call them **contingent**

**Question 2**

I'm asked *"What is the relationship between $x$ and $z$, are they complements or substitutes"*. The answer is

> These inputs $x$ and $z$ are not related because : $\pi\_{w\_z,w\_x} = - \frac{\partial z}{\partial w\_x} = 0$

Could someone kindly explain what is going on here?
I believe the $\pi\_{w\_z,w\_x}$ tells us how the marginal profit from a change in the input cost of $z$ changes when the cost of input $x$ changes. But I'm not sure i understand the partial derivative on the right hand side. It looks a bit like the chain rule for implicit differentiation.

**Tags:** mathematical-economics, production-function, optimization, profit-maximization

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## 1 answers

### Answer by username_1 on 2023-05-06. Score: 2 | Accepted answer

1. In my classes, they’re called Marshallian or uncompensated demands, as in consumer theory.
2. The partial derivative on the right hand side means to directly differentiate the demand for $z$ (the optimal $z$ level) with respect $w\_x$ (the price of the other factor, $x$).

In consumer theory for goods $x,y$ with prices $p\_x,p\_y$, complement and substitute goods are defined as:

* Complements

$\frac{\partial x}{\partial p\_y} < 0$

Here an increase in the price of good $y$ makes a lower amount of $y$ to be demanded as it is more expensive, and hence a lower amount of $x$ is demanded as $x$ and $y$ are usually consumed together.

* Substitutes

$\frac{\partial x}{\partial p\_y} > 0$

Here an increase in the price of good $y$ makes a lower amount of $y$ to be demanded as it is more expensive, and a higher amount of $x$ is demanded as the consumer substitutes good $y$ for the relatively cheaper alternative good $x$.

* Unrelated goods

$\frac{\partial x}{\partial p\_y} = 0$

A change in price in good $y$, and hence its quantity demanded, has no effect over the demand for good $x$.

Here you’re working with the analogous notion of complements/substitutes/unrelated goods (here production factors) in firm theory.


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