Based on personal experience, I agree with Zach's conclusion that the dedicated water boiling kettle is more effcient than the stove.

My 1500W electric kettle boils a liter of water about 2-1/2 times faster than a covered pan on my stove's 2100W element (4 minutes versus 10 minutes)... and more than 3 times faster than the stove's smaller 1600W element).

My pan just covers the entire large heating element if I place it carefully, and has considerable overlap over the smaller element.

I believe the heating element in the kettle has much more efficient heat transfer to the water - while I can't see the element itself,I can see a raised bump at the bottom of the kettle where the heating element is, and I assume that it's mechanically bonded to the metal bottom of the kettle allowing for very efficient heat transfer. Further, the plastic bottom of the kettle doesn't get extremely hot, so the bottom of the element must be insulated to keep most of the heat against the water. And since the bottom of the kettle is not vented, there's not much opportunity for convection to pull heat away from the heating element.

The stove, on the other hand, relies on the weight of the pan to make contact with the element, and the spiral element may not be completely level, so some parts of the element are in better contact than others. Since the bottom of the element is exposed, it is free to radiate some of the heat down into the stove drip pan. Also, there seem to be some convection air currents carrying heat away, since if I hold my hand a few inches over the pan while heating water, it becomes uncomfortably warm in a short period of time.

While I agree that there is some heat lost in heating up both the kettle and the the pan, the specific heat of water is about 8 times greater than stainless steel, so the energy wasted in heating up the 24 ounce kettle is around the same energy needed to heat up 3 ounces of water.

Based on personal experience, I agree with Zach's conclusion that the dedicated water boiling kettle is more effcient than the stove.

My 1500W electric kettle boils a liter of water about 2-1/2 times faster than a covered pan on my stove's 2100W element (4 minutes versus 10 minutes)... and more than 3 times faster than the stove's smaller 1600W element).

My pan just covers the entire large heating element if I place it carefully, and has considerable overlap over the smaller element.

I believe the heating element in the kettle has much more efficient heat transfer to the water - while I can't see the element itself,I can see a raised bump at the bottom of the kettle where the heating element is, and I assume that it's mechanically bonded to the metal bottom of the kettle allowing for very efficient heat transfer. Further, the plastic bottom of the kettle doesn't get extremely hot, so the bottom of the element must be insulated to keep most of the heat against the water. And since the bottom of the kettle is not vented, there's not much opportunity for convection to pull heat away from the heating element.

The stove, on the other hand, relies on the weight of the pan to make contact with the element, and the spiral element may not be completely level, so some parts of the element are in better contact than others. Since the bottom of the element is exposed, it is free to radiate some of the heat down into the stove drip pan. Also, there seem to be some convection air currents carrying heat away, since if I hold my hand a few inches over the pan while heating water, it becomes uncomfortably warm in a short period of time.

While I agree that there is some heat lost in heating up both the kettle and the the pan, the specific heat of water is about 8 times greater than stainless steel, so the energy wasted in heating up the 24 ounce kettle is around the same energy needed to heat up 3 ounces of water.

Of course, it all depends on what type of appliances you're using. I have a stove with traditional spiral elements, if I had an inductive cooktop, it's likely that it would be more efficient than the electric kettle since more of the stove's energy would be going into the cooking pan. A glass cooktop with heating elements or infrared elements might also change the equation. As would a gas stove.