📈 增长模型中的误区：文章指出，许多增长模型倾向于使用指数曲线，但这种模型往往忽略了增长的内在局限性。作者认为，指数增长是一种误导性的表述，因为它假设增长可以无限持续，而现实中所有增长都受到资源限制。

📉 S曲线才是增长的本质：作者解释说，真正的增长过程更符合S曲线的形态。S曲线描述了从早期缓慢增长到快速增长，再到增长放缓直至饱和的完整生命周期。这种形态反映了增长过程中资源获取、利用和耗尽的动态变化。

🌐 宏观视角下的增长：尽管微观层面的增长（如企业、科技）有明显的周期性，但从宏观历史来看，人口、能源消耗、GDP等似乎一直在指数级增长。然而，作者认为这只是长期内多个S曲线叠加的结果，最终也会达到极限。

⏳ 增长的终结是必然的：文章的核心论点是，所有增长终将结束。无论是在有限的空间内还是在更广阔的宇宙尺度上，增长都依赖于可用的资源。当关键资源（如能源、材料、效率）耗尽时，增长的积极反馈循环就会转变为消极循环，导致增长停止。

💡 建模的严谨性：作者呼吁在进行增长建模时，必须认识到增长的有限性。模型应包含对增长相关输入因素及其消耗率的考量。如果模型假设增长可以无限持续，那么现实世界最终会证明其错误。

Published on September 3, 2025 5:20 PM GMT

A couple times a week I read something about growth, usually about AI, that includes a chart that looks like this:

And I get immediately annoyed, because this is a zoomed in view of a chart that should look like this:

Mathematically there's of course nothing wrong with exponential curves and functions, and they are obviously different from the S-curves and sigmoid functions that produce them. But when used to model growth, exponentials are always the wrong model, because no growth lasts forever.

The limitations of growth are a simple consequence of the limited resources available within the Hubble volume—or a more confined space, like a planet or country or city—where the growth is happening. Positive feedback loops eventually become negative when the key resources fueling the growth run out, be that exhausting the supply of energy, materials, efficiency, or other underutilized resources from which there are gains to be made.

We know growth is limited on a small scale. Every tech boom runs its course. Every improvement in market efficiency exhausts opportunities for arbitrage. Every meme saturates culture to the limit of the number of minds that will adopt it. But what about on the big scale?

If we look back at history, it feels exponential. Everything goes up and to the right when we zoom out: population, energy extraction, GDP, everything! Sure seems like the growth can't end.

But we know in theory that growth must always end. Long-run growth only looks exponential because we’re in the middle of a long process of stacking S-curves on top of each other. But eventually we will run out of S-curves to stack, and growth will end.

Now, I'm not saying we're near the limits of our growth yet. In fact, compared to the astronomically large amount of growth that is possible, our growth to date looks like a flat line. But, all growth does end, and if we are to accurately model the world, we must account for this fact.

So, the next time you try to model growth, remember that it will end. Your model is incomplete so long as it fails to include a measure of the causally relevant inputs to growth and their rate of exhaustion. If you posit that growth will continue indefinitely, the world will eventually prove you wrong.