**Definition A2.**

$$\text{DR} = 1 + \eta\_{\ell}^{E} \tag{A3}$$

*where DR represents direct rebound and* η*<sup>E</sup>* <sup>ε</sup> *represents the elasticity of energy demand with respect to e*ffi*ciency. Definitions A1 and A2 are equivalent, and we explain in detail below.*

Equation (A1) can be rewritten as:

$$S = \varepsilon E \tag{A4}$$

Total differentiation of Equation (A4) after applying natural logarithms is:

$$\frac{\text{d}S}{\text{S}} = \frac{\text{d}E}{E} + \frac{\text{d}\varepsilon}{\varepsilon} \tag{A5}$$

If energy efficiency improvement does not result in an increase in energy services, then d*S* = 0. Equation (A5) is simplified to:

$$\frac{\mathbf{d}E}{E} = -\frac{\mathbf{d}\varepsilon}{\varepsilon} \tag{A6}$$

then:

$$
\eta\_{\varepsilon}^{E} = \frac{\mathbf{d}E}{\mathbf{d}\varepsilon} \frac{\varepsilon}{E} = -1 \tag{A7}
$$


Energy efficiency at the household level is often unobservable, so Definition A2 is also rarely used for empirical research.
