Strange question. Let's try a thought experiment. Let's say you have a resistor(R) from input to output, and your output is infinite impedance. Infinite bandwidth, and current consumption is zero. Now let's say you put a cap across the load (you now have a low pass filter). At very high frequencies your current is Vin/R; at very low frequencies current =zero (again). So your current is a function of your INPUT frequency, not your bandwidth. I think you need to analyze your particular circuit. And what do you mean by 'empirical relationship'? By definition, empirical means by determined solely by experimentation or observation, NOT theory.