<p>Assume a labor-intensive production function:</p>

<p><span class="math-container">$Q(L)=L^\beta$</span></p>

<p>Find demand of labor that maximizes profits (unconditional demand <span class="math-container">$L(p,w,r)$</span>), the demand of labor that minimizes costs (unconditional demand <span class="math-container">$L(q,w,r))$</span>, the supply function <span class="math-container">$(Q(L(p,w,r))$</span>), and the benefits <span class="math-container">$(\pi(L(p,w,r))$</span> if:</p>

<p><strong>a)</strong> <span class="math-container">$\beta&gt;1$</span></p>

<p><strong>b)</strong> <span class="math-container">$\beta=1$</span></p>

<p>For the first case (increasing returns to scale) I am able to Maximize profits by getting the First Order Conditions, etc., and finding <span class="math-container">$L(p,w,r))$</span>:</p>

<p><span class="math-container">$L(p,w,r))=(w/p\beta)^{1/\beta-1}$</span></p>

<p>Thus, by replacing <span class="math-container">$L(p,w,r))$</span> into the benefits:</p>

<p><span class="math-container">$\pi(L(p,w,r)=p*(w/p\beta)^{\beta/\beta-1}-w*(w/p\beta)^{1/\beta-1}$</span> (It can be simplified by doing more algebra).</p>

<p>By replacing <span class="math-container">$L(p,w,r))$</span> into the production function:</p>

<p><span class="math-container">$Q(L(p,w,r)=(w/p\beta)^{\beta/\beta-1}$</span></p>

<p>Finally, the conditioned (on a production of "q") labor demand is given by:</p>

<p><span class="math-container">$L(q,w,r)=q^{1/\beta}$</span></p>

<p>However, for the second case (constant returns to scale) I am not able to get the First Order Conditions and so on for obtaining the benefits, the supply function, etc. Thus, I would like to know how to address this problem.</p>

<p>I would appreciate any help,</p>

<p>Thanks in advance</p>

CC BY-SA 4.0
2019-02-13T17:19:51.687Z
You'll probably have to consider corner solutions for the case where $\beta=1$. Also, why is labor demand a function of $r$ (presumably rent) when capital is not required in production?
Herr K.
CC BY-SA 4.0
2019-02-13T19:58:39.237Z
You're right! My bad, I just wrote the standard form but sure, it only depends on $w$ and $p$, in the case of non-conditioned labor demand. For a single-input production function, corner solutions just refer to the case when $L(p,w)=0$?
F. Sánchez
CC BY-SA 4.0
2019-02-13T20:14:08.697Z
Depending on the ratio of $p/w$, corner solution could be $0$ or the upper bound on labor supply.
Herr K.
CC BY-SA 4.0
2024-01-19T09:43:07.393Z
So, your profit function is $(p-w)L$: As long as $p>w$, $L = +\infty$. As long as $p < w$, $L = 0$. If $p = w$, $L = x \in \mathcal{R^+}$, where x is any real and positive number.
Matteo Bulgarelli
<microeconomics><firm>
Profit Maximization and Returns to Scale
F. Sánchez
# Profit Maximization and Returns to Scale

(Asked by: username_0 on 2019-02-13)

Assume a labor-intensive production function:

$Q(L)=L^\beta$

Find demand of labor that maximizes profits (unconditional demand $L(p,w,r)$), the demand of labor that minimizes costs (unconditional demand $L(q,w,r))$, the supply function $(Q(L(p,w,r))$), and the benefits $(\pi(L(p,w,r))$ if:

**a)** $\beta>1$

**b)** $\beta=1$

For the first case (increasing returns to scale) I am able to Maximize profits by getting the First Order Conditions, etc., and finding $L(p,w,r))$:

$L(p,w,r))=(w/p\beta)^{1/\beta-1}$

Thus, by replacing $L(p,w,r))$ into the benefits:

$\pi(L(p,w,r)=p\*(w/p\beta)^{\beta/\beta-1}-w\*(w/p\beta)^{1/\beta-1}$ (It can be simplified by doing more algebra).

By replacing $L(p,w,r))$ into the production function:

$Q(L(p,w,r)=(w/p\beta)^{\beta/\beta-1}$

Finally, the conditioned (on a production of "q") labor demand is given by:

$L(q,w,r)=q^{1/\beta}$

However, for the second case (constant returns to scale) I am not able to get the First Order Conditions and so on for obtaining the benefits, the supply function, etc. Thus, I would like to know how to address this problem.

I would appreciate any help,

Thanks in advance

**Tags:** microeconomics, firm
> **Comments:**
> **username_1**: You'll probably have to consider corner solutions for the case where $\beta=1$. Also, why is labor demand a function of $r$ (presumably rent) when capital is not required in production?  
> **username_0**: You're right! My bad, I just wrote the standard form but sure, it only depends on $w$ and $p$, in the case of non-conditioned labor demand. For a single-input production function, corner solutions just refer to the case when $L(p,w)=0$?  
> **username_1**: Depending on the ratio of $p/w$, corner solution could be $0$ or the upper bound on labor supply.  
> **username_2**: So, your profit function is $(p-w)L$: As long as $p>w$, $L = +\infty$. As long as $p < w$, $L = 0$. If $p = w$, $L = x \in \mathcal{R^+}$, where x is any real and positive number.  

---

## 0 answers

