Tradeoff between low-noise and low-power design

Can anyone help me to understand why this sentence below (which I had find it in a paper) is correct?

Designers must address the tradeoff between low-noise and low-power designs of the amplifier.For an ideal thermal-noise-limited amplifier with a constant bandwidth and supply voltage, the power of the amplifier scales as $1/(vn^2)$ where $vn$ is the input-referred noise of the amplifier.

But I had read in a lot of books that refer that the thermal noise given by this expression $P = K \cdot T \cdot \Delta f= \dfrac{vn^2}{4 \cdot R}$ so power and $vn$ are proportional. So which statement is correct?

• It doesn't make a lot of sense to me without more context. I mean, one can always add gain stages (after a low noise first stage) if you need more power. – George Herold Jun 30 '17 at 18:42