Since oil is found in limited amounts it makes sense to think eventually we will hit a peak in oil production. Evidently to produce x quantity of oil at the surface there must be x multiplied by a functio of y volume of oil left underground. We must describe x as a function of y which likely depends on the actual environment in which the oil is found. Since there are many different environments this would be sigma x = sigma of many different functions of many different y's. sigma y would be the amount left in the ground at any given time.
The above seems to me to be rock solid math. Therefore we can conclude the decay function is likely exponential but may turn hyperbolic over time. This implies we will never run out of oil but reaching peak oil is an absolute certainty.
Since the y volumes of oil are found in different environments and the oils are of different qualities then we can also pose a law which states some kinds of environments and some kinds of oil will reach peak production before others do. This seems banal but it seems to point that the most desirable oils coming from the environments which are easier to exploit will peak first.
Therefore we can also conclude the oil price influences the peaks for these different oils. And this implies with a near absolute certainty that oil prices will rise as the peak is approached.
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