I started from the somewhat iconoclastic hypothesis that the different primary energy sources are commodities competing fora market, like different brands of soap or different processes to make steel, so that the rules of the game may after all be the same. These rules are best described by Fischer and Pry, and can be [summarised] in saying that the fractional rate at which a new commodity penetrates a marketis proportional to the fraction of the market not yet covered.
F'/F = a(1-F)
He then shows some examples from actual data, and says
All of them refer to competition between two products. In the case of energy we have three or four energy sources competing most of the time and it is mathematically impossible that [the sum of all the F's equal 1], so I had to extend the treatment slightly with the extra stipulation that one of the fractions is defined as the difference to one of the sum of the others. This fraction follows approximately an equation of [the above type] most of the time, but not always. It finally shows saturation and change in coefficients. The fraction dealt with in this way corresponds to the oldest of the growing ones. The rule can be expressed in the form: First In, First Out.
With two products, with market shares F and G adding up to 1, one can write
F'/F = a G
which implies
F' = a F G
and, given that F' + G' = 0,
G'/G = -a F
So, in the case of two commodities the Fischer-Pry model is compatible with F + G = 1. Now, suppose we have F1, F2, F3, ... Then, Marchetti is right that
F1'/F1 = a1 (1 - F1) F1'/F2 = a2 (1 - F2) ... Fn'/Fn = an (1 - Fn)
is incompatible with F1 + ... + Fn = 1. However, realising that 1 - F1 = F2 + F3 + F4 + ... + Fn, one can see that the "proper" generalisation of the Fischer-Pry model to n > 2 commodities is
F1'/F1 = a12 F2 + a13 F3 + ... + a1n Fn F2'/F2 = a21 F1 + a23 F3 + ... + a2n Fn ... Fn'/Fn = an1 F1 + an2 F2 + ... + an{n-1} F{n-1}
which is compatible with F1 + ... + Fn = 1 as long as a{ij} = - a{ji}.
The coefficients a{ij} are a measure of the rate at which the ith commodity takes market share away from the jth one (or loses market share if the coefficient is negative).
So, question: do we have the necessary data to fit this extended model and compare it with Marchetti's simpler one? Can the last politician to go out the revolving door please turn the lights off?
It would be interesting to use the data prior to, say 1975, to see the result. I don't have data prior to 1965, but for Oil, Coal and Natural Gas they likely exist.
Do you want to dig deeper into this? We could try to find that data. Vencit omnia veritas.
dX1 = a X2 dt dX2 = -a X1 dt
describes a circle of constant radius, the stochastic (Ito) differential equation
implies
d(X1^2 + X2^2) = 2X1 dX1 + 2 X2 dX2 + dX1 dX1 + dX2 dX2 = a^2 (X1^2 + X2^2) dt
or
d(R^2) = a^2 R^2 dt
that is, R^2 grows exponentially with time in the stochastic case, even though it is constant in the ordinary case.
So, the secular 2% increase in total energy use might be "explained" by interpreting the extended Fisher - Pry - Marchetti model as a stochastic (Ito) differential equation. Can the last politician to go out the revolving door please turn the lights off?
the stochastic (Ito) differential equation dX1 = a X2 dt dX2 = -a X1 dt implies d(X1^2 + X2^2) = a^2 (X1^2 + X2^2) dt
d(X1^2 + X2^2) = a^2 (X1^2 + X2^2) dt
dX1 = X2 (a dt + s dz1) dX2 = X1(- a dt + s dz2) implies d(X1^2 + X2^2) = a^2 s^2 (X1^2 + X2^2) dt + X1 X2(s1 dz1 + s2 dz2)
d(X1^2 + X2^2) = a^2 s^2 (X1^2 + X2^2) dt + X1 X2(s1 dz1 + s2 dz2)
a{ij} = [(dFi/Fi) Fj - Fi(dFj/Fj)] / [sum{k} Fk^2]
I will apply it to the BP+FAO data to see what the coefficients look like over the past 40 years. Can the last politician to go out the revolving door please turn the lights off?
