**Definition A3.**

$$\text{DR} = -\eta\_{P\_E}^E \tag{A8}$$

where η*<sup>E</sup> PE* represents the elasticity of energy demand with respect to energy price.

Equation (A8) is equivalent to Equation (A3), and the derivation process of Equation (A8) will be described in detail below.

*Energies* **2019**, *12*, 2069

According to Equation (A1), the relationship between energy service price (or energy service cost) and energy price (or energy cost) is:

$$P\_{\mathbb{S}} = P\_{\mathbb{E}} / \mathfrak{s} \tag{A9}$$

where *PS* represents energy service price and *PE* represents energy price.

Combined with Equations (A1), and (A9), Equation (A3) can be rewritten as:

$$\begin{array}{l} \text{DR} \quad = 1 + \eta\_{\varepsilon}^{\text{E}} = 1 + \frac{\partial \ln \text{E}}{\partial \ln \varepsilon} = 1 + \frac{\partial \ln \langle \mathcal{S}/\varepsilon \rangle}{\partial \ln \varepsilon} = 1 + \left(\frac{\partial \ln \mathcal{S}}{\partial \ln \varepsilon} - 1\right) \\\ = \frac{\partial \ln \mathcal{S}}{\partial \ln P\_{\mathcal{S}}} \frac{\partial \ln P\_{\mathcal{S}}}{\partial \ln \varepsilon} = \frac{\partial \ln \mathcal{S}}{\partial \ln P\_{\mathcal{S}}} \frac{\partial \ln \langle P\_{\mathcal{E}}/\varepsilon \rangle}{\partial \ln \varepsilon} \\\ = \frac{\partial \ln \mathcal{S}}{\partial \ln P\_{\mathcal{S}}} \left(\frac{\partial \ln P\_{\mathcal{E}}}{\partial \ln \varepsilon} - 1\right) \end{array} \tag{A10}$$

Because nominal energy prices are not affected by energy efficiency, therefore ∂ ln *PE*/∂ ln ε = 0. Equation (A10) is simplified to:

$$\text{DR} = -\frac{\partial \ln S}{\partial \ln P\_S} = -\frac{\partial S}{\partial P\_S} \frac{P\_S}{S} = -\eta\_{P\_S}^S \tag{A11}$$

where η*<sup>S</sup> PS* represents the elasticity of energy service demand with respect to energy service price.

Combined with Equations (A1), and (A9), η*<sup>S</sup> PS* can be rewritten as:

$$\begin{array}{lcl} \eta\_{P\_S}^S &= \frac{\partial \ln S}{\partial \ln P\_S} = \frac{\partial \ln S}{\partial \ln P\_E} \frac{\partial \ln P\_E}{\partial \ln P\_S} = \frac{\partial \ln(\epsilon E)}{\partial \ln P\_E} \frac{\partial \ln(\epsilon P\_S)}{\partial \ln P\_S} \\\ &= \left(\frac{\partial \ln \epsilon}{\partial \ln P\_E} + \frac{\partial \ln E}{\partial \ln P\_E}\right) \left(\frac{\partial \ln \epsilon}{\partial \ln P\_S} + \frac{\partial \ln P\_S}{\partial \ln P\_S}\right) \end{array} \tag{A12}$$

Assuming that energy efficiency is exogenous, then <sup>∂</sup> ln <sup>ε</sup> <sup>∂</sup> ln *PE* <sup>=</sup> 0 and <sup>∂</sup> ln <sup>ε</sup> <sup>∂</sup> ln *PS* <sup>=</sup> 0. Equation (A12) is simplified to:

$$
\eta\_{P\_S}^S = \left(0 + \frac{\partial \ln E}{\partial \ln P\_E}\right) (0 + 1) \ = \frac{\partial \ln E}{\partial \ln P\_E} = \eta\_{P\_E}^E \tag{A13}
$$

The electricity efficiency here mainly refers to energy efficiency ratio of household appliances, which is usually determined by the technical level of the manufacturer. Consumers can only improve the utilization efficiency of household appliances. In fact, some researches point out that higher efficiency may only be achieved by purchasing more expensive new equipment in China, so the electricity efficiency is exogenous.

Combined with Equations (A10), (A11) and (A13), Equation (A3) can be rewritten as:

$$\text{DR} = 1 + \eta\_{\varepsilon}^{E} = -\eta\_{P\_{E}}^{E} \tag{A14}$$

Equation (A14) indicates that the price elasticity could be an ideal proxy indicator of direct rebound effect with other variables controlled.

Definition A3 allows identification of the rebound effect. Due to the ease of data acquisition, most empirical studies have adopted Definition A3.

The above analysis shows that the direct rebound effect is mainly related to the falling of real energy price induced by improvement in energy efficiency. However, the power price has both rising and falling periods in the real economy, and the impact of rising and falling price on the electricity demand is not completely reversible. Generally, price elasticity during price rise period is greater than during the price decline period. Direct use of Definition A3 will overestimate the direct rebound effect, and it is necessary to decompose the price, so the price decomposition method is not used to identify the rebound effect. However, it is introduced to improve the accuracy of measurement results.
