G SXTY's got it. It's not that the engine is more efficient at altitude, but that the engine is best matched to the aircraft at altitude. The machine is bloody overpowered at takeoff, is working hardest at top of climb, and happiest in cruise. :ok:

Then consider an underpowered aircraft - say a twin with one engine out or a trijet with one or two engines out, or a quadjet with two engines out. Assuming that the inoperative engines are properly shut down, they burn no fuel and no fuel is leaking.

After the plane has descended to its ceiling and continues on a cruise... the remaining engines are working harder than in normal cruise, therefore burning more fuel each, but the inoperative engines burn none at all.

Does the aircraft then lose efficiency and therefore range? Does a plane cover longer distance with all engines operative cruise or some engines out cruise, given the same fuel load?

The operating engines may actually be at a more optimum operating point (better SFC), although that's a big MAYBE.

But with less total thrust available, a lower cruise altitude is forced on the machine, which ruins any advantage. The higher air density means more thrust available, but more fuel burned too.

Which is probably because the jet engines are less efficient at lower TAS and therefore generally less efficient at lower altitude. If a plane descends below the optimum cruise altitude for a jet then I suspect that whatever the plane does - running all engines at less than optimum thrust, or shutting some down and running the others at the optimum thrust - the plane would be less efficient in burning more fuel per distance, and therefore lose range for a given fuel reserve.

Bear in mind too that at lower altitude with 1 or more shut down, the fuel burn PER HOUR may be lower, which is advantageous when you're killing time in a loiter. Patrol aircraft often use this to extend their loiter duration.

However, this means relatively less distance traveled, which is bad news if your job is to get from point A to point B in good time.

Indeed. Back at the efficiency. The total time that a plane stays airborne with a given amount of fuel may indeed be longer if the plane flies low and slow. But the total distance covered is probably less.

Which is not just a matter of "good time". It also is bad news if the job is to get to point B or another runway at all, any time, rather than ditch or crash off-airport after fuel exhaustion short of diversion.

How much is the range of a typical airliner diminished if it has to stay below 3000 metres or so because it cannot be pressurized?