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

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?

• 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? Jun 14 '20 at 2:30

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.

• 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. Jun 15 '20 at 17:37
• 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. Jun 15 '20 at 20:15
• 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! Jun 16 '20 at 10:37

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

• +1 so much more elegant than mine. How does one deal with the derivative at $L=L_e$ though? Not sure whether it matters... Jun 15 '20 at 20:18
• 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. Jun 15 '20 at 21:45