<blockquote>
<p>If the cost function is linear, and it is simply equal to (,,)=⋆+⋆, where ⋆ and ⋆ are the conditional demand functions (depending on  as well), why the MC is not linear?</p>
</blockquote>
<p>The cost function is linear in <span class="math-container">$l$</span> and in <span class="math-container">$k$</span>, but it need not be linear in <span class="math-container">$y$</span>, that is,  <span class="math-container">$l(y, *)$</span> and <span class="math-container">$k(y, *)$</span> are not, in general, linear functions of <span class="math-container">$y$</span>.
The marginal cost is the derivative of <span class="math-container">$C(W, R, y )$</span> with respect to <span class="math-container">$y$</span>, not with respect to <span class="math-container">$l$</span> or <span class="math-container">$k$</span>.</p>
<p>Remember that when calculating the marginal cost you use the chain rule, which involves the derivatives of <span class="math-container">$l$</span> and <span class="math-container">$k$</span> with respect to <span class="math-container">$y$</span>.</p>
<p>Formally:</p>
<p><span class="math-container">$MC= \frac{\partial (,,) }{\partial y}= W \frac{\partial  l(y, *)}{\partial y}+R \frac{\partial k(y,*)}{\partial y}$</span></p>
<p>As for the equation in red, you can proceed as follows:</p>
<p><a href="https://i.sstatic.net/QQf0H.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QQf0H.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-09-05T13:20:55.100Z
2022-09-05T21:23:59.767Z
2022-09-05T21:23:59.767Z
BakerStreet
<p>I'm studying basic micro, and I did not get how such a result is possible.</p>
<p>According to what I studied, the marginal cost is simply the partial derivative of the cost function with respect to the output <span class="math-container">$y$</span>. If the cost function is linear, and it is simply equal to <span class="math-container">$ C(W,R,y) = W l^\star +R k^\star$</span>, where <span class="math-container">$l^\star$</span> and <span class="math-container">$k^\star$</span> are the conditional demand functions (depending on <span class="math-container">$y$</span> as well), why the MC is not linear?
What I have done is to solve the cost minimization problem and to take the ratio of the optimal solutions, and then solving for one of the two input factors. After that, I substituted into the production function and solving for the other input factor, I find the conditional demand function of one input and then I find the conditional demand function of the other one. In the end I simply subsituted into the cost function and derived with respect to <span class="math-container">$y$</span>.  However this does not give me the result below in red.</p>
<p><a href="https://i.sstatic.net/H4Hva.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/H4Hva.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/oSOtV.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/oSOtV.png" alt="enter image description here" /></a></p>
<p>Can you show me how to get the result in red?</p>

<microeconomics><mathematical-economics>
Marginal Cost and lagrange multiplier
Molo
# Marginal Cost and lagrange multiplier

(Asked by: username_0 on 2022-09-03)

I'm studying basic micro, and I did not get how such a result is possible.

According to what I studied, the marginal cost is simply the partial derivative of the cost function with respect to the output $y$. If the cost function is linear, and it is simply equal to $ C(W,R,y) = W l^\star +R k^\star$, where $l^\star$ and $k^\star$ are the conditional demand functions (depending on $y$ as well), why the MC is not linear?
What I have done is to solve the cost minimization problem and to take the ratio of the optimal solutions, and then solving for one of the two input factors. After that, I substituted into the production function and solving for the other input factor, I find the conditional demand function of one input and then I find the conditional demand function of the other one. In the end I simply subsituted into the cost function and derived with respect to $y$. However this does not give me the result below in red.

[![enter image description here](https://i.sstatic.net/H4Hva.png)](https://i.sstatic.net/H4Hva.png)

[![enter image description here](https://i.sstatic.net/oSOtV.png)](https://i.sstatic.net/oSOtV.png)

Can you show me how to get the result in red?

**Tags:** microeconomics, mathematical-economics


---

## 1 answers

### Answer by username_1 on 2022-09-05. Score: 2 | Accepted answer

> If the cost function is linear, and it is simply equal to (,,)=⋆+⋆, where ⋆ and ⋆ are the conditional demand functions (depending on as well), why the MC is not linear?

The cost function is linear in $l$ and in $k$, but it need not be linear in $y$, that is, $l(y, \*)$ and $k(y, \*)$ are not, in general, linear functions of $y$.
The marginal cost is the derivative of $C(W, R, y )$ with respect to $y$, not with respect to $l$ or $k$.

Remember that when calculating the marginal cost you use the chain rule, which involves the derivatives of $l$ and $k$ with respect to $y$.

Formally:

$MC= \frac{\partial (,,) }{\partial y}= W \frac{\partial l(y, \*)}{\partial y}+R \frac{\partial k(y,\*)}{\partial y}$

As for the equation in red, you can proceed as follows:

[![enter image description here](https://i.sstatic.net/QQf0H.png)](https://i.sstatic.net/QQf0H.png)


---

