Convenient S-shaped production function (i.e. with IRS and DRS) to derive a discontinuous demand for labor

Question

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.

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?

Answer 1

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.

Answer 2

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 $g\left( L-L_{e}\right)=\beta\ln \left( 1+L-L_{e}\right)$.