Energy Efficiency and Rebound Effects: An Econometric Analysis of Energy Demand in the Commercial Building Sector

Abstract

It is widely recognized that the adoption of energy saving innovations can induce an increase in the usage of the corresponding technologies and thus can possibly increase energy consumption. Among other concerns is that uncertainties regarding the magnitude of this “rebound effect” can deter policy makers from promoting energy efficiency. This paper analyzes the rebound effects of the adoption of energy efficient technologies in commercial buildings. Based upon a structural model of technology adoption and subsequent energy demand at the building level, the empirical results are that energy efficiency can reduce electricity use by about 35 % and natural gas consumption by about 50 %.

Appendix: Derivation of the Modified Heckman Selection Model

The following is the modified Heckman selection model. The energy consumption of a building we can observe is a combination of two components:

$$\begin{aligned} Q_i =Q_i ^{1}T_i +Q_i ^{0}(1-T_i ), \end{aligned}$$

where $Q_i ^{1}$ is the energy consumption of a building if it adopts an energy efficient technology; $Q_i ^{0}$ is the consumption if a building does not adopt the technology. $T_{i}=1$ means a building adopts the technology.

The expected value of the energy consumption is

$$\begin{aligned} E(Q_i )=E(Q_i \left| {T_i =1} \right. )\Pr (T_i =1)+E(Q_i \left| {T_i =0} \right. )\Pr (T_i =0) \end{aligned}$$

From equation $Q_i =\mu +\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} +\theta _i T_i +\varepsilon _i$, we have

$$\begin{aligned} Q_i ^{1}&= \mu +\theta +\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} +\varepsilon _i +\nu _i\\&= \mu ^{1}+\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} +\varepsilon ^{1}_i\\ Q_i ^{0}&= \mu +\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} +\varepsilon _i =\mu ^{0}+\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} +\varepsilon ^{0}_i \end{aligned}$$

where $\mu ^{1}=\mu +\theta ,\,\varepsilon ^{1}_i =\varepsilon _i +\nu _i ,\,\mu ^{0}=\mu ,\,\varepsilon ^{0}_i =\varepsilon _i$

Thus

$$\begin{aligned} Q_i&= \mu ^{0}+\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} +(\mu ^{1}-\mu ^{0})T_i +\varepsilon ^{0}_i +(\varepsilon ^{1}_i -\varepsilon ^{0}_i )T_i\\ E(Q_i \left| {T_i =1} \right. )&= \mu ^{1}+\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} +E(\varepsilon ^{1}_i \left| {T_i =1} \right. ) \\ E(Q_i \left| {T_i =0} \right. )&= \mu ^{0}+\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} +E(\varepsilon ^{0}_i \left| {T_i =0} \right. ) \end{aligned}$$

Assume that $\varepsilon ^{1}_i ,\,\varepsilon ^{0}_i$ and $\omega _i$ are all normally distributed random variables with mean zero and covariance matrix as follows:

$$\begin{aligned} \left[ {{\begin{array}{l@{\quad }l@{\quad }l} {\sigma _{\varepsilon ^{1}}^2 }&{} &{} \\ {\sigma _{\varepsilon ^{1}\varepsilon ^{0}} }&{} {\sigma _{\varepsilon ^{0}}^2 }&{} \\ {\sigma _{\varepsilon ^{1}\omega } }&{} {\sigma _{\varepsilon ^{0}\omega } }&{} {\sigma _\omega ^2 } \\ \end{array} }} \right] \end{aligned}$$

Then

$$\begin{aligned} E(\varepsilon _i^1 \left| {T_i =1} \right. )&= E\left( \varepsilon _i^1 \left| {\omega _i \ge -(\Psi _0 +\Psi _1 C_{j(i)} +\Psi _2 Z_{j(i)} +\Psi _3 X_i +\Psi _4 Y_{j(i)} +\Psi _5 U_i )} \right. \right) \\&= \sigma _{\varepsilon ^{1}\omega } \frac{\phi (\Psi _0 +\Psi _1 C_{j(i)} +\Psi _2 Z_{j(i)} +\Psi _3 X_i +\Psi _4 Y_{j(i)} +\Psi _5 U_i )}{\Phi (\Psi _0 +\Psi _1 C_{j(i)} +\Psi _2 Z_{j(i)} +\Psi _3 X_i +\Psi _4 Y_{j(i)} +\Psi _5 U_i )} \\ E(\varepsilon _i^o \left| {T_i =0} \right. )&= E\left( \varepsilon _i^0 \left| {\omega _i \le -(\Psi _0 +\Psi _1 C_{j(i)} +\Psi _2 Z_{j(i)} +\Psi _3 X_i +\Psi _4 Y_{j(i)} +\Psi _5 U_i )} \right. \right) \\&= -\sigma _{\varepsilon ^{0}\omega } \frac{\phi (\Psi _0 +\Psi _1 C_{j(i)} +\Psi _2 Z_{j(i)} +\Psi _3 X_i +\Psi _4 Y_{j(i)} +\Psi _5 U_i )}{1-\Phi (\Psi _0 +\Psi _1 C_{j(i)} +\Psi _2 Z_{j(i)} +\Psi _3 X_i +\Psi _4 Y_{j(i)} +\Psi _5 U_i )} \end{aligned}$$

Define

$$\begin{aligned} a_i&= 1-\Phi (\psi _0 +\psi _1 C_{j(i)} +\psi _2 Z_{j(i)} +\psi _3 X_i +\psi _4 Y_{j(i)} +\psi _5 U_i ) \\ b_i&= \Phi (\psi _0 +\psi _1 C_{j(i)} +\psi _2 Z_{j(i)} +\psi _3 X_i +\psi _4 Y_{j(i)} +\psi _5 U_i )=1-a_i \\ c_i&= \varphi (\psi _0 +\psi _1 C_{j(i)} +\psi _2 Z_{j(i)} +\psi _3 X_i +\psi _4 Y_{j(i)} +\psi _5 U_i ) \end{aligned}$$

where $a_{i}$ is the probability that building $i$ does not adopt the energy efficient technology; $b_{i}$ is the probability that building $i$ adopts the technology; $c_{i}$ can be interpreted as the probability that the latent variable $S_{i}$ takes on a certain value. These three variables are estimated from the probit model.

The expectation of the energy demand we observe is

$$\begin{aligned} E(Q_i )&= E(Q_i \left| {T_i =1} \right. )\Pr (T_i =1)+E(Q_i \left| {T_i =0} \right. )\Pr (T_i =0) \\&= \mu ^{1}b_i +(\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} )b_i +\sigma _{\varepsilon ^{1}\omega } c_i +\mu ^{0}a_i \\&\quad +\,(\beta \overline{p_i }+\,\gamma ngp_i +\delta X_i +\eta Y_{j(i)} )a_i -\sigma _{\varepsilon ^{0}\omega } c_i \end{aligned}$$

Because $a_i +b_i =1$

$$\begin{aligned} E(Q_i )=\mu ^{0}+(\mu ^{1}-\mu ^{0})b_i +(\sigma _{\varepsilon ^{1}\omega } -\sigma _{\varepsilon ^{0}\omega } )c_i +\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} \end{aligned}$$

Thus the final form of the modified Heckman selection model that estimates the average of the price elasticties across all buildings is

$$\begin{aligned} E(Q_i )=\mu ^{0}+\theta b_i +(\sigma _{\varepsilon ^{1}\omega } -\sigma _{\varepsilon ^{0}\omega } )c_i +\beta \overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} \end{aligned}$$

Now let us look at the model that can allow the estimation of two price elasticities: one average elasticity for buildings that adopt energy efficient technologies and one for buildings that do not.

Assume that $\beta$ in Eqs. (9) and (10) are different:

$$\begin{aligned} Q_i ^{1}&= \mu ^{1}+\beta ^{1}\overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} +\varepsilon ^{1}_i\\ Q_i ^{0}&= \mu ^{0}+\beta ^{0}\overline{p_i } +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} +\varepsilon ^{0}_i \end{aligned}$$

Then

$$\begin{aligned} E(Q_i )&= E(Q_i \left| {T_i =1} \right. )\Pr (T_i =1)+E(Q_i \left| {T_i =0} \right. )\Pr (T_i =0) \\ \!&= \!\left[ \mu ^{1}\!+\!\beta ^{1}\overline{p_i } \!+\!\gamma ngp_i \!+\!\delta X_i \!+\!\eta Y_{j(i)} \!+\!\sigma _{\varepsilon ^{1}\omega } \frac{c_i }{b_i }\right] b_i \!+\!\left[ \mu ^{0}\!+\!\beta ^{0}\overline{p_i } \!+\!\gamma ngp_i \!+\!\delta X_i \right. \\&\quad \left. +\,\eta Y_{j(i)} -\sigma _{\varepsilon ^{0}\omega } \frac{c_i }{a_i }\right] a_i \end{aligned}$$

Thus the final form of the modified Heckman selection model which allows estimating two price elasticities is

$$\begin{aligned} E(Q_i )=\mu ^{0}+\theta b_i +(\sigma _{\varepsilon ^{1}\omega } -\sigma _{\varepsilon ^{0}\omega } )c_i +\beta ^{1}\overline{p_i } b_i +\beta ^{0}\overline{p_i } a_i +\gamma ngp_i +\delta X_i +\eta Y_{j(i)} \end{aligned}$$