Lots about decoupling recently and I just wanted to muse through an simple example using an input-output approach. Let be a vector of emissions per dollar of a sector's output, A a matrix of technical coefficients, L the Leontief matrix, and f a vector of final demands.

Now, let us state some useful things. The product ELF gives emissions, and intensity can be obtained from that. Therefore we have the following where I(t) is the emissions intensity and EM(t) the emissions.

Let us assume that sector one is less emissions intensive than sector two (e.g. e1 < e2). From the emissions equation the first obvious way to reduce emissions is to reduce final demands in the more emissions intense sector and shift that to the less emissions intense sector.

The second way is more complex. We might imagine that the toy economy is reorganized so that the technical coefficients themselves change so that there is less production dependency on the more emissions intense sector (e.g. changing the a coefficients in the A matrix)

Finally, we might imagine that the emissions intensity coefficients e1 and e2 might be decreased by better technology. Given a level of total final demand, we have these ways to reduce emissions. Assuming an emissions budget B the game becomes the following differential equation.

The above equation states that the budget B is reduced per unit time according to the magnitude of emissions. For some total final we can slow the reduction by shifting demand to the less intensive sector, changing the technical coefficients, and finding ways to reduce e1 and e2.

Now, obviously today we are used to growth in GDP percentage wise be year so lets make our final demands more interesting. We assume some initial period final demands grow exponential given as:

We can now restate the emissions budget problem as shown in the image. One particularly simple solution to the below problem of staying in the budget is to reduce the emissions coefficients. In this simple toy model lets just imagine an energy transition reduces these.

We might imagine some similar exponential decay time paths for the magnitudes of these intensity parameters that reduce the emissions per time period sufficiently rapidly to keep the budget from being exhausted. Making some assumptions these could be found with some algebra.

Huzzah. But now the question. We have a plausible mechanism for reducing emissions but what happens if we started this scenario over but with coefficients denoting energy per dollar of output. In this case we are a bit stuck.

Basic physical intuition demands that such coefficients cannot be made arbitrarily small or else we are doing things in an economy expending vanishingly small quantities of energy per dollar output. So if we assume some minimum possible values for e1 and e2 what happens?

Even if we move final demands entirely over to the less energy intensive sector, and inexplicably managed to make all the technical coefficients zero (leading to a Leontief matrix = identity matrix) this won't be enough.

At some point, maybe rather far down the road, the inexorable logic of the exponential growth of final demands induces (even at minimum energy coefficients) a growing energy usage that eventually gets vey large.

While its certainly not a stretch to imagine we can decouple emissions from growth, it is rather more dubious that we can from energy use. Does this matter? Perhaps, the point though is that decoupling for an exponentially growing economy leads to weird things.

For intensities that are implausible to make arbitrarily small it seems difficult to imagine what the decoupling story means.

Note well: obviously there is no energy budget analogue of the emissions budget, that was unclear on my part.

Also, this thread on something work adjacent is a clear indication of just how dull even my procrastination techniques have become. Ill be insufferable as a person by the end of this latest Ontario lockdown.