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: