Here is a simplified description of the situation which is to be modelled:
- $C_p$ units of coal is needed to produce $1$ unit of electricity
- $P_c$ units of electricity is needed to mine $1$ unit of coal
- $C_p \lt 1$, $P_c \lt 1$, i.e. both coal and power can be produced infinitely
The problem:
Given $C_0$ units of coal initially to bootstrap the production.
How much electricity is to be produced in order to have $C$ units of coal ($C > C_0$)?
How many units of coal will be burned while this production (or how many units of coal totally will be extracted)?
Question:
My guts feeling tells me it can be modelled with a system of linear equations, but I seem to be stuck, I only end up with infinite series.
Huge thanks in advance for any help.
Here's how you can model this:
Denote electricity at time $t$ by $e_t$ and coal at time $t$ by $c_t$. You obtain the following system of linear equations: $$\begin{pmatrix}c_{t+1}\\e_{t+1}\end{pmatrix} = \begin{pmatrix}0 & 1/P_c\\1/C_p & 0\end{pmatrix}\begin{pmatrix}c_{t}\\e_{t}\end{pmatrix},$$ or $$\begin{pmatrix}c_{t}\\e_{t}\end{pmatrix} = \begin{pmatrix}0 & 1/P_c\\1/C_p & 0\end{pmatrix}^t\begin{pmatrix}c_{0}\\e_{0}\end{pmatrix},$$
(This assumes that at each point in time all available coal will be transformed into electricity and all available electricity will be transformed into coal)