**10 November 2022**

*C. Hinsley*

Economists often make the assumption that the efficiency of resource consumption for a given market will drive less overall resource consumption. However, the opposite is often true: cheaper goods may elicit increased demand, outweighing the gain in efficiency. This effect is called the *Jevons paradox,* seen in coal consumption at the beginning of the Industrial Revolution as well as in traffic flows — adding new alternative roadways has occasionally been observed to increase congestion.

Below, I give an example of a dynamical system that is capable of exhibiting the Jevons paradox.

$r$: Resource consumption rate

$s$: Current supply of goods

$d$: Demand (or consumption of goods) rate

$\alpha$: Manufacturing efficiency

$$ \dot{s} = \alpha r - d $$

$$ \dot{d} = u(s, d) $$

We refer to the nonlinear function $u$ as the *adoption rate of goods*. The Jevons paradox can be precisely defined in this system: The paradox is observed when $\alpha$ increases, but so too does $r$ due to a dependence of $d$ on $s$. That is, for some $\alpha, \Delta\alpha > 0$ and initial conditions $s(0), \dot{s}(0), d(0)$, the averaged long-run variable

$$ \langle r \rangle := \lim_{t\to\infty} \frac1t\int_0^t r(\tau)\ d\tau $$

for the system with manufacturing efficiency $\alpha$ (denoted by $\langle r \rangle_\alpha$) is lesser than that for the system with manufacturing efficiency $\alpha+\Delta\alpha$ (which we denote by $\langle r \rangle_{\alpha+\Delta\alpha}$). Do note that the presence of a rebound (or rebound effect, which we have prior referred to as Jevons paradox) in a system depends solely on the function $u(s)$. A natural task, then, is to determine for what functions $u$ a rebound may occur:

$$ \langle r \rangle_{\alpha+\Delta\alpha} > \langle r \rangle_\alpha $$

$$ \lim_{t\to\infty}\frac1t\int_0^t (r_{\alpha+\Delta\alpha}(\tau) - r_\alpha(\tau))\ d\tau > 0 $$

$$
\lim_{t\to\infty} \frac1t\int_0^t \left(\frac{\dot{s}*{\alpha+\Delta\alpha}(\tau) + d*{\alpha+\Delta\alpha}(\tau)}{\alpha+\Delta\alpha} - \frac{\dot{s}*\alpha(\tau) + d*\alpha(\tau)}{\alpha}\right) \text{d}\tau > 0
$$

$$
\lim_{t\to\infty} \frac1t \int_0^t \left(\alpha\dot{s}*{\alpha+\Delta\alpha}(\tau) + \alpha d*{\alpha+\Delta\alpha}(\tau) - (\alpha+\Delta\alpha)\dot{s}*\alpha(\tau) - (\alpha+\Delta\alpha)d*\alpha(\tau)\right) \text{d}\tau > 0
$$

Using the fact that $s_{\alpha+\Delta\alpha}(0) = s_\alpha(0)$, this gives the following: