Does the counterweight in a space elevator need to be accelerated?

Suppose you have a space elevator and you use it to lift weight to orbit. Obviously this is pulling down on the counterweight. So doesn't this mean that you need to apply appropriate acceleration to the counterweight so it does not keep moving inward? If this were the case the energy still needs to be applied somewhere. Or does the rotation of the Earth provide that energy (Earth's rotation is decreased slightly)?

When you attach an object to the tether and then ascend the tether the tension in the tether above the object remains as it would without the object, and the tension of tether below the object is reduced by $F=m\,(a+g)$ Where a is usually small and g is a function of height. The energy to advance the object upward with this force must be supplied by the object pulling itself along tether. The force of the object pulling itself along the tether means that the tension in the tenther while there's no object ascending must be greater than this reduction (assuming a flexible tether than cannot sustain compression forces)