Is optimal labour zero when (i) capital fixed and (ii) elasticity of substitution less than 1?

Question

I am getting some "weird results". I find that in a CES, with short term fixed capital and elasticity of substitution smaller than one, it is optimal for firms to hire zero labour, which seems at odd with the idea that the marginal product of labour is always positive (except in the extreme case of Leontief, where $\rho = -\infty$).

Assume a simple CES production function:

$$Y = A\left( \alpha L^{\rho} + (1-\alpha)K^{\rho} \right)^{\frac{1}{\rho}}$$

Assume that capital is fixed in the short term, so the firm only optimises over $L$. Importantly, the firm must pay the cost of capital anyway. Therefore, as long as $p>0$, the firm is always better off by producing something. Thus, the key question is how much labour to hire.

Under competitive markets, optimal labour comes from:

\begin{equation} \frac{\partial Y}{\partial L} \equiv \alpha A^{\rho} \left(\frac{Y^*}{L^*}\right)^{1-\rho} = \left(\frac{w}{p}\right) \end{equation}

Importantly, notice that MPL is (i) always positive, (ii) for $\rho<1$ (including negative values) and $L^*=0$, it is infinity!. In other words, whenever there is imperfect substitution of factors, and the wage rate is non-infinity, you want to produce with some labour.

Replacing output into the above yields a single equation with only one unknown ($L_i^*$), which solution is:

$$ L^* = K \Omega^{\frac{1}{\rho}} $$

where

$$ \Omega = \left(\dfrac{(1-\alpha)}{\left(\frac{w}{A p \alpha}\right)^{\frac{\rho}{1-\rho}}-\alpha}\right) $$

Now, discard the parameterisations where $\Omega<0$. These are corner solutions where $L^*=0$.

So far so good. Except for some strange result when $\rho<0$. To see this, let us replace optimal labour into the production function. After some rearranging, you get:

$$ Y^* = AK\left(\alpha\Omega + (1-\alpha)\right)^{\frac{1}{\rho}} $$

Now, compare this with the case of zero labour. Let us call this $Y_0$. Here, output is:

$$ Y_0 = AK(1-\alpha)^{\frac{1}{\rho}} $$

Now, we want to know under which parameterisation the firm is better off producing with zero labour. Profits with optimal labour and with zero labour are respectively:

$$ \pi = pY^* - rK - wL^* $$

$$ \pi_0 = pY_0 - rK $$

Thus, the firm will produce with no labour iff $\pi_0> \pi$, which is:

$$ wL^* > p(Y-Y_0) $$

So, if the labour costs do not compensate for extra profits, the firm will choose $L^*=0$. Here is where the problem arises: for $\rho<0$, $Y-Y_0$ is always negative! Which means firm will produce with zero labour.

To see this, let us find the conditions under which $\Delta Y=Y-Y_0<0$. This is,

$$ AK\left[\left( \alpha \Omega + (1-\alpha)\right)^{\frac{1}{\rho}} - (1-\alpha)^{\frac{1}{\rho}} \right] <0 $$

$$ \left( \alpha \Omega + (1-\alpha)\right)^{\frac{1}{\rho}} < (1-\alpha)^{\frac{1}{\rho}} $$

Noting that $\rho<0$ will invert the components, you get:

$$ \frac{1}{\left( \alpha \Omega + (1-\alpha)\right)} < \frac{1}{1-\alpha} $$

Which is clearly true for:

$$ 0< \alpha\Omega $$

So, for "normal" parameterisations where $\Omega>0$, we can see that $Y-Y_0<0$ is always true.

I have confirmed this in R. The following code produces delta in output = -15:

rm(list=ls())

rho       <- -0.5
alpha     <- .6
A <- 1
p <- 1
w <- 0.7

K <- 2.756729
omega = (1-alpha)/((w/(alpha*A*p))^(rho/(1-rho))-alpha)
L = K*omega^(1/rho)

ll = alpha*L^rho
kk = (1-alpha)*K^rho
tr = (1/rho)
Y = A*(ll+kk)^tr
Y0 = A*(kk)^tr

delta_output = Y-Y0

delta_profits = p*(Y-Y0)-w*L

if I change $\rho$ to 0.5, you get the desired result (that using labour does increases output).

You can also see the plot of delta output here. It is clear that the delta output is always negative, and asymptotically zero for $x=\infty$, which is when $L^*=0$.

Answer 1

Ok, rather embarrassing, but the problem was in the formula for $Y_0$. In effect, for $\rho>0$, it is true that

$$ Y_{_{L=0}} = AK(1-\alpha)^{\frac{1}{\rho}} $$

However, for the case of $\rho<0$, the limit of $Y$ is zero. The proof is trivial. Rewrite the CES as:

$$ Y = AL\left(\alpha + (1-\alpha)\left(\frac{L}{K}\right)^{-\rho} \right)^{\frac{1}{\rho}} $$

Then, it is trivial to see that:

$$ \lim_{L\rightarrow 0} Y = A0 \: \alpha^{\frac{1}{\rho}} = 0 $$

With this, we can see that $\Delta Y = Y > 0$. This solves the inconsistency.

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PS: as it turns out, the marginal product of labour at $L=0$ is not infinity. Apparently, the Inada conditions do not hold for the CES with $\rho<0. Actually, the marginal product of labour seem to be zero! (check slide 15 here). This is a major revelation to me.

Answer 2

Starting from

$$Y = A\left( \alpha L^{\rho} + (1-\alpha)K^{\rho} \right)^{\frac{1}{\rho}}$$

$$...\implies MP_L = \alpha A\cdot\left( \alpha + (1-\alpha)\left(\frac {K}{L}\right)^{\rho} \right)^{\frac{1}{\rho}-1}$$

assume that $\rho <0$ (which gives an elasticity of substitution smaller than unity). Then we can write

$$MP_L = \frac A {\left( \alpha + (1-\alpha)\left(\frac {L}{K}\right)^{|\rho|} \right)^{\frac{1}{|\rho|}+1}}$$

So

$$L\to 0 \implies MP_L \to \frac A { \alpha ^{\frac{1}{|\rho|}+1}} >0 $$

although not infinite.