Should I use low or high temperature to use a stove energy efficiently? If I use low temperatures a lot of energy will be wasted with time, but using high temperature is not energy efficient as well. So what's more economic?
This answer won't cover all cases, but when you are cooking with water it starts boiling at roughly 100 degrees and won't get any hotter (unless you boil it dry). Unless you're steaming, cooking a reduction or need the vigorous motion of water boiling hard, you may as well turn down the element at that point to save energy.
Often foods don't cook until quite a high temperature is reached, so if you only slowly bring the food up to the required temperature then more energy will be lost. Once the temperature is adequate though, any excess heat is essentially wasted (and may burn the food or make it less healthy too). It's clearly best to aim to bring the temperature up quickly, but not overheat.
There are other ways to reduce the energy used; for example using a minimal quantity of water to boil or steam requires less energy to bring up to temperature than using an excessive amount.
Probably not answering the question, but I cannot resist.
It is increadable that you understand that you cook by temperature rather than by applying the energy. Maintaining the temperature, no matter how high, ideally, does not consume any energy whatsoever (likewise the body, once accelerated, maintains its speed itself. Only heat up is energy consuming. Maintaining does not consume anything. You can boil for free!). That is a law of heat from Galileo. You need an accelerator just to counteract the resistance (aka energy leakage/loss). The faster you move, the more you leak per second. This is another law of heating/cooling from Newton
It says that the higher is temperature, the faster you loose the energy, exponentially faster. You see the contradiction between the laws because your pan is not ideal. Galileo was first to notice that if you boil with ideal lid, your pan will keep its temperature forever, for free. You can boil for free! Wait, I repeat myself.
The bottom line is that you should always use a lid and probably wrap your pan into towels. The latter can be risky. I recommend therefore reaching the water boiling temperature quickly first, turn off the heater and cover everything with with towels. This will insulate your cooking-optimal temperature pan and you won't waste any temperature/energy at all and it is safe. It is also optimal ib terms of the fact that, as the dish cools down, it is prepared and once preparation completes, your dish has optimal temperature for consumption (in contrast to those slaves of evil who are convinced that beauty and good taste = energy waste, enjoy their fresh dish scalding). The real beauty is energy saving, when things go smoothly themselves.
Such stewing is a best way to boil pasta, eggs and various crops. The latter should be soaked in advance. Soaking = boiling at room temperature. It is free (costs no money and does not increase world entropy whatsoever). Yes, you should boil the crops and beans at room temperature first. Boil them normally, as described above, after soaking. It will be a huge saver. I think that thinly filmed potatoes will go either.
Finally, boil with as few water as possible and never dispose stem-hot water into the drain. Boil something else or use it to clean fat or recuperate. Washing steam-hot water down the drain is a (entropic) crime against the God. I normally boil the crops such that after soaking, there is left some water in the pan and it is partially evaporated and further soaked into beans during stewing. So, nothing is wasted.
But your question is interesting. It re-raises my old question, why do kitchen bosses do not produce the thermically insulated pans to enable zero-energy cooking?