<p>I'm not sure but it seems to me that the logistic function <span class="math-container">$\frac{e^{x}}{1+e^{x}}$</span> could serve your purpose. </p>

<p>You may need to scale it as its output falls between 0 and 1, but it does have an analytical derivative that you can then use to solve for the labour demand function.</p>

CC BY-SA 4.0
2020-06-15T17:37:12.737Z
You would have to shift it to the right (and down) to make $f(0)=0$ in the convex part. As soon as you do this, the FOC is no longer solvable analytically.
VARulle
CC BY-SA 4.0
2020-06-15T20:15:43.030Z
Is it? When replacing x with x-4 in both numerator and denominator and subtracting a small constant I can still make $f(0)=0$, take the FOC w.r.t. x and solve analytically for x as a function of w and p. Solution looks far from nice of course, but does exist as such.
Maarten Punt
CC BY-SA 4.0
2020-06-16T10:37:04.367Z
Upon trying once more to solve it I succeeded (and found my mistake in the previous try), so yes, you are right and I herewith revoke my earlier comment!
VARulle
CC BY-SA 4.0
2020-06-14T09:18:10.757Z
2020-06-14T09:18:10.757Z
Maarten Punt
<p>One of the simplest specification I can think of (and for which the first order condition can be solved analytically in <span class="math-container">$L$</span>) is:</p>

<p><span class="math-container">$$
y=\left\{ 
\begin{array}{ccc}
L^{\alpha} &amp;  &amp; L\leq L_{e} \\ 
L_{e}^{\alpha}+g\left( L-L_{e}\right)  &amp;  &amp; L&gt;L_{e}%
\end{array}%
\right. 
$$</span>
with <span class="math-container">$g\left( L-L_{e}\right) =(L-L_{e})^\beta$</span> and <span class="math-container">$\alpha\geq1$</span> and <span class="math-container">$0&lt;\beta&lt;1,$</span> or with <span class="math-container">$g\left( L-L_{e}\right)=\beta\ln \left( 1+L-L_{e}\right)$</span>.</p>

CC BY-SA 4.0
2020-06-15T20:18:22.247Z
+1 so much more elegant than mine. How does one deal with the derivative at $L=L_e$ though? Not sure whether it matters...
Maarten Punt
CC BY-SA 4.0
2020-06-15T21:45:55.830Z
Yes, you are right, the production function is continuous but not differentiable at $L_e$. This is not so harmful, because the input demand function is not continuous at $L_e$ either. Most tricks allowing to smooth the production function, unfortunately make the first order condition difficult to solve analytically.
Bertrand
CC BY-SA 4.0
2020-06-14T13:45:07.073Z
2020-06-15T05:32:37.673Z
2020-06-15T05:32:37.673Z
Bertrand
<p>Let say that a firm produces a commodity using only one input (i.e. Labor if we suppose to be in the very short run). Then we have a general production function of the following form <span class="math-container">$y=f(L)$</span>, for <span class="math-container">$L≥0$</span>, be the output obtained when <span class="math-container">$L$</span> units of Labor are employed. </p>

<p>We further assume that the first derivative (i.e. firm’s marginal product) is always <span class="math-container">$&gt;0$</span>, but ( assuming that f is twice differentiable), <span class="math-container">$f''(L)≥0$</span> in <span class="math-container">$[0,c]$</span>, and <span class="math-container">$f''(L)≤0$</span> in <span class="math-container">$[c,∞)$</span>. Hence, <span class="math-container">$f(L)$</span> is first convex and then concave, with <span class="math-container">$c$</span> as an inflection point.</p>

<p>When this is the case, contrary to the standard case of a concave production technology (with <span class="math-container">$f''(L)$</span> always <span class="math-container">$&lt;0$</span>), firms do not always make profits if they follow the standard rule for profit maximization (i.e. <span class="math-container">$\frac{dF(L)}{dL}=w$</span>). Indeed, with IR, there are real wage levels that equal marginal returns for which labor costs exceed revenues. The value of output is less than labor costs for all employment levels lower than a critical employment level (let say <span class="math-container">$Lc$</span>), that is the level where average productivity maximizes, or, where marginal productivity equals average productivity. For this reason the curve for demand of labor is first decreasing and then at a certain point falls to zero.</p>

<p><a href="https://i.sstatic.net/sidx0.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/sidx0.jpg" alt="enter image description here"></a></p>

<p>What can be a specific functional form suitable to represent this situation and that allows me to derive a curve of demand for work by firms with these characteristics?</p>

CC BY-SA 4.0
2020-06-14T02:30:24.230Z
Do you need a functional form which allows you to derive the labor demand curve analytically, or would it be enough to be able to compute it numerically?
VARulle
<microeconomics><mathematical-economics><labor-economics><production-function>
Convenient S-shaped production function (i.e. with IRS and DRS) to derive a discontinuous demand for labor
Alessandro
# Convenient S-shaped production function (i.e. with IRS and DRS) to derive a discontinuous demand for labor

(Asked by: username_0 on 2020-06-13)

Let say that a firm produces a commodity using only one input (i.e. Labor if we suppose to be in the very short run). Then we have a general production function of the following form $y=f(L)$, for $L≥0$, be the output obtained when $L$ units of Labor are employed.

We further assume that the first derivative (i.e. firm’s marginal product) is always $>0$, but ( assuming that f is twice differentiable), $f''(L)≥0$ in $[0,c]$, and $f''(L)≤0$ in $[c,∞)$. Hence, $f(L)$ is first convex and then concave, with $c$ as an inflection point.

When this is the case, contrary to the standard case of a concave production technology (with $f''(L)$ always $<0$), firms do not always make profits if they follow the standard rule for profit maximization (i.e. $\frac{dF(L)}{dL}=w$). Indeed, with IR, there are real wage levels that equal marginal returns for which labor costs exceed revenues. The value of output is less than labor costs for all employment levels lower than a critical employment level (let say $Lc$), that is the level where average productivity maximizes, or, where marginal productivity equals average productivity. For this reason the curve for demand of labor is first decreasing and then at a certain point falls to zero.

[![enter image description here](https://i.sstatic.net/sidx0.jpg)](https://i.sstatic.net/sidx0.jpg)

What can be a specific functional form suitable to represent this situation and that allows me to derive a curve of demand for work by firms with these characteristics?

**Tags:** microeconomics, mathematical-economics, labor-economics, production-function
> **Comments:**
> **username_1**: Do you need a functional form which allows you to derive the labor demand curve analytically, or would it be enough to be able to compute it numerically?  

---

## 2 answers

### Answer by username_2 on 2020-06-14. Score: 2

I'm not sure but it seems to me that the logistic function $\frac{e^{x}}{1+e^{x}}$ could serve your purpose.

You may need to scale it as its output falls between 0 and 1, but it does have an analytical derivative that you can then use to solve for the labour demand function.

> **Comments:**
> **username_1**: You would have to shift it to the right (and down) to make $f(0)=0$ in the convex part. As soon as you do this, the FOC is no longer solvable analytically.  
> **username_2**: Is it? When replacing x with x-4 in both numerator and denominator and subtracting a small constant I can still make $f(0)=0$, take the FOC w.r.t. x and solve analytically for x as a function of w and p. Solution looks far from nice of course, but does exist as such.  
> **username_1**: Upon trying once more to solve it I succeeded (and found my mistake in the previous try), so yes, you are right and I herewith revoke my earlier comment!  

---

### Answer by username_3 on 2020-06-14. Score: 1

One of the simplest specification I can think of (and for which the first order condition can be solved analytically in $L$) is:

$$
y=\left\{
\begin{array}{ccc}
L^{\alpha} & & L\leq L\_{e} \\
L\_{e}^{\alpha}+g\left( L-L\_{e}\right) & & L>L\_{e}%
\end{array}%
\right.
$$
with $g\left( L-L\_{e}\right) =(L-L\_{e})^\beta$ and $\alpha\geq1$ and $0<\beta<1,$ or with $g\left( L-L\_{e}\right)=\beta\ln \left( 1+L-L\_{e}\right)$.

> **Comments:**
> **username_2**: +1 so much more elegant than mine. How does one deal with the derivative at $L=L_e$ though? Not sure whether it matters...  
> **username_3**: Yes, you are right, the production function is continuous but not differentiable at $L_e$. This is not so harmful, because the input demand function is not continuous at $L_e$ either. Most tricks allowing to smooth the production function, unfortunately make the first order condition difficult to solve analytically.  

---

