Is optimal labour zero when (i) capital fixed and (ii) elasticity of substitution less than 1?
Ok, rather embarrassing, but the problem was in the formula for $Y_0$. In effect, for $\rho&gt;0$, it is true that

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

However, for the case of $\rho&lt;0$, 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 $\Delta Y = Y &gt; 0$. This solves the inconsistency.



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&lt;0. Actually, the marginal product of labour seem to be zero! (check slide 15 here). This is a major revelation to me.