This paper really annoyed me last night.
The assumptions made don't even render a world that looks remotely similar to what we have, and that abstracting away of messy complexity actually makes the central characteristics of the problem harder to wrap your head around, even if it does make it easier to reframe into a paper that'd look good in an academic journal.
Case in point:
You don't need higher math to understand that human beings are fundamentally lossy in nature. Perfect replication of the information required to sustain the current cumulative knowledge level doesn't happen because there is no guarantee that any one person that knows a thing can successfully communicate it to someone else in a way that can be understood, and that in the process of doing so, that that information isn't mutated in such a manner that something important isn't lost. Children amuse themselves with this phenomena all the time. It's called a game of "Telephone".
Given an acknowledgement of this, the tendency for population decline to result in problems for a world that has been architected around the guarantee of population and economic growth is immediately intuitive.
As death removes older generations, the knowledge they had and failed to clearly replicate dies with them. Each new member of the population birthed in is a new crank on the slot machine of information propagation. Their life circumstances might allow them to understand that bit of arcane knowledge that no one else was able to do anything with because it didn't make sense to them because the constellation of shared lived experience between communicator and communicated necessary for comprehension wasn't there. New idea formulation, or old idea reformulation and propagation becomes reliable and less likely to snowball out of control to irrevocable loss as long as you keep throwing new blood at it.
If there is no new blood though, the numbers quickly begin to favor the life experiences required to underpin all the minutiae to sustain the corpus of knowledge as is not reoccurring in total across the entire breadth of the population, and you start to find your smaller remaining population further and further isolated from the practical experiences that made the linguistic representations of knowledge written down understandable. The knowledge then becomes useless, a dependency in the infrastructure that enables efficient physical completion of labor is lost, thus your population's output of tasks done per unit time drops, thus more work must be done just to keep things running, and so on and so forth until basic principles are again reformulated, and efficiencies from a chance rediscovery of the necessary experience underpinning a lynchpin technology can be realized again.
Just as information transfer is lossy, so is physical labor. If you have 1 million tasks that must be done in a day to maintain a stable level of output, and you decrease the number of agents available to do the tasks, the math simply works out that more work has to be done per agent to maintain output in a steady state, and we all know that the work output of a human being isn't trivially scaled up without ballooning the prerequisite knowledge required to do the task (I.e. skills at using tools, skills to teach the skills to use the tools, the tools to make the tools, the skills to operate the tools to make the tools, etc...) all of which themselves become tasks added on to the overall workload of your dwindling population who are working harder and harder to maintain the highest level of output possible.
The other issue not even touched on is the desirability of continued high growth regimes in the presence of a near tipping-point constraint violation to which a remediary measure has not been formulated. I.e. Excess carbon footprint. If you stay in your high growth regime in a wager to get that innovation faster from a new member, you're still templating your current footprint of resource use, and accelerating reaching your tipping point, with no guarantee you'll find it in time. Decline is a natural outcome in an organism facing this type of environmental dynamic constraint to buy time for other (in this paper, unrepresented, but which one can represent as decreased demand to output more of the destabilizing factor by your population, or natural processes operating to free up buffer space by things like carbon recapture and sequestration) processes time to shift the dynamics back in favor of supporting a new high growth regime.
I have the utmost respect for those that can trivially converse in the higher maths; I'll admit I had to crack open some books to understand what the heck they were getting at and whether it even made sense. I draw the line though where doing your analysis as they did takes you right out of reality into the atrophied realm of post-facto narrativization.
Those equations are meaningless to most readers, whereas putting things in terms of tasks to be completed, and characterizing overall knowledge level management as what it is, a bunch of overhead tasks draws the same Senate picture that it took them mucking around in control theory to even tangentially articulate. It started a conversation yes, but by constraining it to the realm of finance and economics, and leaving out the rest of the world you run afoul of that great aphorism that "problems cannot be solved with the same level of thinking that created them in the first place", which is exactly what this paper appears to be an example of doing.
Gah!