本文深入探讨了杰文斯悖论，并从边际主义经济学的角度进行了分析。文章指出，效率的提升并非成本曲线的移动，而是生产函数的改变。当生产函数f(x)从x变为2x时，资源x的价值增加，导致其消费量增加。除非效用函数U(y)在达到一定程度后迅速降低，否则杰文斯悖论普遍成立。文章通过数学推导，得出杰文斯悖论成立的条件是需求的价格弹性小于-1。这为理解资源效率提升带来的实际影响提供了新的视角。

📈 效率提升并非成本曲线移动，而是生产函数f(x)的改变，例如从f(x)=x变为f(x)=2x。

💡 杰文斯悖论的核心在于，当f'(x)增加时，资源x的价值U(f(x))随之增加，导致对x的消耗也增加，因为其边际收益曲线向右移动。

📉 只有当效用函数U(y)在达到一定程度后迅速降低，即“need not greed™”时，杰文斯悖论才可能不成立。例如，当世界对AI的需求有限时，提高效率反而可能减少计算资源的使用。

🧮 通过数学推导，杰文斯悖论成立的条件可归结为：需求的价格弹性（PED）小于-1，即MB(y)/(yMB′(y)) < -1。

Published on January 27, 2025 11:04 PM GMT

Recent discourse surrounding Jevon’s paradox provides an opportunity to flesh out the precise intuitions behind marginalist economics and how concepts like “elasticity” arise naturally from agent/graph-based first principles.

Say you have resource $x$ (e.g. compute) with cost $C\left(x\right)$ that produces good y (e.g. AI), with utility function $U\left(x\right)$ via production function $y=f\left(x\right)$.

The market maximizes $U\left(f\right(x\left)\right)-C\left(x\right)$ — note that the derivatives $U^{\prime }\left(f\right(x\left)\right)f^{\prime }\left(x\right)$ and $C^{\prime }\left(x\right)$ are exactly the MB and MC curve for $x$, so this is just intersecting demand and supply for $x$.

Here is the explanation on Wikipedia and other places … this is not actually wrong (as we’ll see), but if you think about it, it really looks wrong!

Contrary to what the above figure appears to show, an efficiency improvement is not a shift of $C\left(x\right)$ at all! It is a shift of $f\left(x\right)$, e.g. from the function $f\left(x\right)=x$ to $f\left(x\right)=2x$.

Jevon's paradox is then quite intuitive — you're increasing how valuable $x$ is ($U\left(f\right(x\left)\right)$ increases) when $f^{\prime }\left(x\right)$, so of course the consumption of $x$ will increase. Its MB curve is literally shifting to the right!

Unless if y is something you only have "need not greed™" for, i.e. if $U\left(y\right)$ diminishes really quickly after a point. If the world only needs 200kg of AI, and has no use for anymore, then you can now make that with much less compute and you’re done.

So e.g. with $U\left(y\right)=\log (1+y)$, $C\left(x\right)=x^2$ you see Jevon’s paradox. Pictured below: $U\left(ax\right)-C\left(x\right)$ as $a$ increases — you can see that the argmax increases.

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But with this $U\left(y\right)$:

We have the following behaviour:

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We can generalize this: in general, Jevon’s paradox holds—

$⟺$ $\arg maxU\left(ax\right)-C\left(x\right)$ increases as $a$ increases

$⟺$ $U\left(ax\right)-C\left(x\right)$ rises faster with $a$ for larger $x$ than for smaller $x$

$⟺$ $\frac{\partial }{\partial a}\left[U\right(ax)-C(x\left)\right]$ increases with $x$

$⟺$ $x{U}^{\prime }\left(ax\right)$ increases with $x$.

$⟺$ $U^{\prime }\left(ax\right)+ax{U}^{\prime \prime }\left(ax\right)>0$

$⟺$ $MB\left(y\right)+yM{B}^{\prime }\left(y\right)>0$

$⟺$ $MB\left(y\right)/\left(yM{B}^{\prime }\right(y\left)\right)<-1$

$⟺$ $PED\left(y\right)<-1$

Discuss