We use new mid-infrared (mid-IR) photometry from the Spitzer Space Telescope to study the relations between low-frequency radio luminosity density , mid-IR (12 μm rest frame) luminosity and optical emission-line ([O ii]) luminosity L]O ii], for a complete sample of z∼ 1 radio galaxies from the 3CRR, 6CE, 6C*, 7CRS and TOOT00 surveys. The narrow redshift span of our sample (0.9 < z < 1.1) means that it is unbiased to evolutionary effects. We find evidence that these three quantities are positively correlated. The scaling between and L[O ii] is similar to that seen in other active galactic nuclei samples, consistent with both and L[O ii] tracing accretion rate. We show that the positive correlation between and implies that there is a genuine lack of objects with low values of at high values of . Given that traces accretion rate, while traces jet power, this can be understood in terms of a minimum accretion rate being necessary to produce a given jet power. This implies that there is a maximum efficiency with which accreted energy can be chanelled into jet power and this efficiency is of the order of unity.