Well you might not be reading in the right direction.
Many factors affect the power requirements of a jet engine with regards to efficiency.
The power required from a jet engine is proportional to the acceleration required, drag developed, and the designed cruising speeds.
Before going into great details, one should consider a basic law.
1)Force is equal to mass x acceleration.
(Mass being a quantity that is constant)
If you apply this to a stationary aircraft at high power settings, W the weight of air will be accelerated from rest to V2Ft/Sec.
The thrust developed will be WV2Lbs./G. Because the aircraft is not moving, the "propulsive efficiency" will be zero.
If you allow the aircraft to move, it will attain a certain velocity, call it V1Ft/sec. the thrust formula changes to W(V2-V1)Lbs./G
The air that now passes into the engine has a velocity. (Actually, it is the forward movement of the aircraft that give the static airmass a velocity)Assuming the resultant velocity of this air relative to the aircraft is still V2Ft/sec. the acceleration of the airmass will be
V2-V1Ft./sec. If the "mass" flow remains the same, the thrust will be W(V2-V1)Lb./G
Developed Thrust will decrease as altitude increases, but efficiency increases as altitude increases, Why?
At sea level, the power available from one pound of air is increased if the temperature is lowered. The cubic volume of measured air will have more weight due to the increased density.
2) As stated by Mr. lovell, Ram effect maybe the most important single effect concerning the efficiency of a Jet engine.
Pressure increases significantly above 250 kts. This increases the "Mass" flow through the engine per unit of fuel.
Again, Mass X acceleration = Thrust.
As altitude is increased, Pressure, Density, and temperature decrease. At a constant RPM (Or N1) the pressure ratio of the compressor and the temperature rise within the engine remains constant irrespective of altitude. Due to the reduced density, fuel will have to be reduced because high EGT's are possible due to the fuel/air mixture. As the OAT decreases with altitude, more fuel can be burned without exceeding the max EGT's. Sound confused? The increase in temperature results from an increase in change of momentum of the gases passing through the engine. One pound of air now produces more thrust at altitude because there is an improved expansion ratio across the turbine which increases the efficiency of the engine at altitude!!! Don't forget, one pound of air at altitude will have a much larger volume.
(The FCU again detects this and reduces the FF accordingly, increasing efficiency)
As the aircraft gains altitude,the compressor loads will decrease due to the loss of density and the engine could overspeed if f. f. is not adjusted.
(This is part of the FCU function again)
An increase of temperature at altitude would would cause a loss of power due to the reduced air density (Less weight per volume)and more fuel would have to be added to develope the same power.
At altitude, the over all thrust is less mainly due to the reduced air density. Since drag is relative to density, the TOTAL drag will be also reduced and less power will be required for the same TAS.
If thrust decreases with an increase in altitude at the same rate as the decrease in total drag, then the TAS will remain constant at all altitudes. But as altitude is gained, the drag curve reduces and less power is required for the same TAS.
All this increases the compressor efficiency and it therefore absorbs less "work" (defined as heat) from the turbines!
I'm probably not making sense, but it works!
3) More power to maintain the same IAS.
You will definately require an increase of power to maintain the same IAS at altitude. But remember that TAS and not IAS is the important factor for fuel efficiency. Remember the formula,
SAR=TAS/FF= NM per Lb. Fuel.
Most large transport aircraft could not attain the higher IAS at altitude due to the limiting Vmo and Mmo.