  Microwave kinetic inductance detectors (MKIDs) are thin film, cryogenic,
superconducting resonators. Incident Cooper pair-breaking radiation increases
their kinetic inductance, thereby measurably lowering their resonant frequency.
For a given resonant frequency, the highest MKID responsivity is obtained by
maximizing the kinetic inductance fraction $\alpha$. However, in circuits with
$\alpha$ close to unity, the low supercurrent density reduces the maximum
number of readout photons before bifurcation due to self-Kerr non-linearity,
therefore setting a bound for the maximum $\alpha$ before the noise equivalent
power (NEP) starts to increase. By fabricating granular aluminum MKIDs with
different resistivities, we effectively sweep their kinetic inductance from
tens to several hundreds of pH per square. We find a NEP minimum in the range
of $25\; \text{aW}/\sqrt{\text{Hz}}$ at $\alpha \approx 0.9$, which results
from a tradeoff between the onset of non-linearity and a non-monotonic
dependence of the noise spectral density vs. resistivity.
