Economic analysis of the energy-efficient household appliances and the rebound effect

Abstract

Many strategies, such as improving energy efficiency, were identified as solutions to reduce energy consumption and carbon emissions. Nonetheless, the presence of a rebound effect could lead to a decrease in potential energy savings and carbon reductions resulting from technological advances in energy consumption. This study focuses on direct and indirect rebound effects on households’ behavior. We examine the situation where consumers demand two types of energy services and explore how their choices are affected by changes in the efficiency of providing these services—and, importantly, the consequent implications for energy use. We employ a (narrowly construed) general equilibrium methodology in an attempt to provide a complete picture of the interactions in play in a theoretically confined setting. We limit the general equilibrium problem to two categories of energy appliances but include consideration of the production side of the equation and consequent budget implications, thus “closing” the system in a general equilibrium sense. We find that rebound magnitudes (both indirect and direct) are large.

Appendices

Appendix A

Recall the Eqs. 11 and 27

$$E_{g}=\left( \frac{R}{P_{E}}\right)\left( \frac{\alpha\beta A}{\alpha+\beta A}\right) $$

$$ A=\left[\frac{\alpha\theta}{\beta(1-\theta)} \right]^{\frac{1}{1+\rho}} \left[\frac{e_{q}}{e_{g}} \right]^{\frac{\rho}{1+\rho}} $$

Then,

$$\begin{array}{@{}rcl@{}} \upsilon^{e_{g}}_{E_{g}}&=& \frac{\partial E_{g}}{\partial e_{g}} \frac{e_{g}}{E_{g}} \end{array} $$

(119)

$$\begin{array}{@{}rcl@{}} &=&\left( \frac{\alpha \beta R}{P_{E}}\right)\left( \frac{\displaystyle\frac{\partial A}{\partial e_{g}}(\alpha+\beta A)-A \beta \displaystyle\frac{\partial A}{\partial e_{g}}}{(\alpha+\beta A)^{2}} \right)\left( \frac{e_{g}}{E_{g}}\right)\\ \end{array} $$

(120)

Or

$$\begin{array}{@{}rcl@{}} \frac{\partial A}{\partial e_{g}}&=& \left( \frac{\theta}{1-\theta} \right)^{\frac{1}{1+\rho}}e_{q}^{\frac{\rho}{1+\rho}} \left( \frac{\rho}{1+\rho} \right)\left( -\frac{1}{{e_{g}^{2}}}\right) \left( \frac{1}{e_{g}}\right)^{\frac{\rho}{1+\rho}-1} \end{array} $$

(121)

$$\begin{array}{@{}rcl@{}} &=&-\left( \frac{\rho}{1+\rho} \right) \frac{A}{e_{g}} \end{array} $$

(122)

Next, insert the Eq. 122 into (120)

$$\begin{array}{@{}rcl@{}} \upsilon^{e_{g}}_{E_{g}}&=& \left( \frac{\alpha \beta R}{P_{E}}\right)\left( \frac{-\displaystyle\frac{\rho}{1+\rho}\; \displaystyle\frac{A}{e_{g}}\;(\alpha+\beta A)+A \beta \displaystyle\frac{\rho}{1+\rho} \displaystyle\;\frac{A}{e_{g}}}{(\alpha+\beta A)^{2}} \right)\\ &&\times\left( \frac{e_{g}}{E_{g}}\right) \end{array} $$

(123)

$$\begin{array}{@{}rcl@{}} &=&\left( \frac{R}{P_{E}}\right) \left( \frac{\alpha \beta A}{\alpha+\beta A} \right)\left( \displaystyle \frac{-\displaystyle \frac{\rho}{1+\rho}\displaystyle\frac{\alpha}{e_{g}}}{\alpha+\beta A} \right)\left( \frac{e_{g}}{E_{g}}\right) \end{array} $$

(124)

$$\begin{array}{@{}rcl@{}} &=& \left( \frac{E_{g}}{e_{g}}\right) \left( -\frac{\rho}{1+\rho} \right)\left( \frac{\alpha}{\alpha+\beta A} \right) \left( \frac{e_{g}}{E_{g}}\right) \end{array} $$

(125)

$$\begin{array}{@{}rcl@{}} &=&\left( -\frac{\rho}{1+\rho} \right)\left( \frac{\alpha}{\alpha+\beta A} \right) \end{array} $$

(126)

Appendix B

Recall the Eq. 12

$$ E_{q}=\left( \frac{R}{P_{E}}\right)\left( \frac{\alpha\beta}{\alpha+\beta A}\right) $$

Then,

$$\begin{array}{@{}rcl@{}} \upsilon^{e_{g}}_{E_{q}}&=& \frac{\partial E_{q}}{\partial e_{g}} \frac{e_{g}}{E_{g}} \end{array} $$

(127)

$$\begin{array}{@{}rcl@{}} &=& \left( \frac{\alpha \beta R}{P_{E}}\right)\left( \frac{-\beta \displaystyle\frac{\partial A}{e_{g}}}{(\alpha+\beta A)^{2}} \right)\left( \frac{e_{g}}{E_{q}}\right) \end{array} $$

(128)

$$\begin{array}{@{}rcl@{}} &=& \left( \frac{R}{P_{E}}\right) \left( \frac{\alpha \beta} {\alpha+\beta A} \right) \left( \frac{\frac{\rho}{1+\rho} \frac{\beta A}{e_{g}}}{\alpha+\beta A} \right) \left( \frac{e_{g}}{E_{q}}\right) \end{array} $$

(129)

$$\begin{array}{@{}rcl@{}} &=& \left( \frac{E_{q}}{e_{g}}\right) \left( \frac{\rho}{1+\rho} \right)\left( \frac{\beta A}{\alpha+\beta A} \right) \left( \frac{e_{g}}{E_{q}}\right) \end{array} $$

(130)

$$\begin{array}{@{}rcl@{}} &=& \left( \frac{\rho}{1+\rho} \right)\left( \frac{\beta A}{\alpha+\beta A} \right) \end{array} $$

(131)

Appendix C

Recall the Eq. 28

$$G=\left( \frac{R}{P_{E}}\;\frac{\alpha\beta A}{\alpha+\beta A}\right) e_{g}$$

Then,

$$\begin{array}{@{}rcl@{}} \upsilon^{e_{g}}_{G}&=&\frac{\partial G}{\partial e_{g}} \frac{e_{g}}{G}=\left( \frac{R \alpha \beta}{P_{E}}\right) \end{array} $$

(132)

$$\begin{array}{@{}rcl@{}} &&\times\left( \frac{\left( -\displaystyle\frac{\rho}{1+\rho} \displaystyle\frac{A}{e_{g}} e_{g}+A\right)(\alpha+\beta A )-Ae_{g}\beta\left( -\displaystyle\frac{\rho}{1+\rho}\displaystyle\frac{A}{e_{g}} \right)}{(\alpha+\beta A)^{2}} \right)\\ &&\times\left( \frac{e_{g}}{G}\right)= \left( \frac{R \alpha \beta}{P_{E}}\right)\left( \frac{\displaystyle\frac{A}{1+\rho} (\alpha+\beta A )+A^{2}\beta \displaystyle\frac{\rho}{1+\rho}}{(\alpha+\beta A)^{2}} \right) \left( \frac{e_{g}}{G}\right)\\&=&\left( \frac{R \alpha \beta}{P_{E}}\right) \left( \displaystyle\frac{A}{(1+\rho)(\alpha-\beta A)} +\frac{A^{2}\beta \rho}{(1+\rho)(\alpha-\beta A)^{2}} \right)\\ &&\times\left( \frac{R\alpha \beta}{P_{E}} \frac{A e_{g}}{\alpha+\beta A} \right) e_{g}\\ &=&\displaystyle\frac{1}{1+\rho}+\displaystyle\frac{A \beta\rho}{(1+\rho)(\alpha+\beta\rho)} \\&=&\left( \displaystyle\frac{1}{1+\rho} \right)\left( 1+ \displaystyle\frac{A \beta \rho}{\alpha+\beta A}\right) \end{array} $$

(133)

$$\begin{array}{@{}rcl@{}} \upsilon^{e_{g}}_{G}-1&=&\displaystyle\frac{1}{1+\rho} +\left( \frac{\rho}{1+\rho}\right) \left( \displaystyle\frac{A \beta \rho}{\alpha+\beta A}\right) -1 \end{array} $$

(134)

$$\begin{array}{@{}rcl@{}} &=& \frac{\alpha+\beta A+\rho \beta A-(1+\rho)(\alpha+\beta A)}{(1+\rho)(\alpha+\beta A)} \end{array} $$

(135)

$$\begin{array}{@{}rcl@{}} &=&-\left( \displaystyle\frac{\rho}{1+\rho}\right)\left( \displaystyle\frac{\alpha}{\alpha+\beta A}\right) \end{array} $$

(136)

$$\begin{array}{@{}rcl@{}} &=& \upsilon^{e_{g}}_{E_{g}} \end{array} $$

(137)

Appendix D

Recall the Eq. 30

$$P_{G}=\frac{P_{E}}{\alpha e_{g}} $$

Then,

$$ {\nu}^{e_{g}}_{P_{G}}=\frac{\partial P_{G}}{\partial e_{g}}\frac{e_{g}}{P_{G}} $$

(138)

$$ = -\frac{P_{E}}{\alpha {e_{g}^{2}}}\frac{e_{g}}{P_{G}} $$

(139)

$$ = -\frac{P_{E}}{\alpha {e_{g}^{2}}} \; \frac{e_{g}}{\frac{P_{E}}{\alpha e_{g}}}=-1 $$

(140)

Appendix E

Recall the Eq. 19

$$D_{G}=\frac{R\beta A}{\alpha+\beta A} $$

Then,

$$\begin{array}{@{}rcl@{}} \nu^{e_{g}}_{D_{G}}&=& \frac{\partial D_{G}}{\partial e_{g}}\frac{e_{g}}{D_{G}} \end{array} $$

(141)

$$\begin{array}{@{}rcl@{}} &=& R\beta \left[\displaystyle\frac{\displaystyle\frac{-\rho}{1+\rho}\;\displaystyle\frac{A}{e_{g}}(\alpha+\beta A) +A\beta\displaystyle\frac{\rho}{1+\rho}\;\displaystyle\frac{A}{e_{g}}}{(\alpha+\beta A)^{2}}\right]\left( \frac{e_{g}}{D_{G}}\right) \end{array} $$

(142)

$$\begin{array}{@{}rcl@{}} &=&\left[ R\beta\frac{\rho}{1+\rho}\displaystyle\frac{A}{e_{g}}\left( \frac{\beta A-\alpha-\beta A} {(\alpha+\beta A)^{2}} \right)\right]\left( \frac{e_{g}}{D_{G}}\right) \end{array} $$

(143)

$$\begin{array}{@{}rcl@{}} &=& \left( \frac{R\beta A}{\alpha+\beta A}\right) \left( \frac{1}{e_{g}}\right) \left( \frac{\rho}{1+\rho}\right)\left( \frac{- \alpha}{\alpha+\beta A}\right)\left( \frac{e_{g}}{D_{G}}\right) \end{array} $$

(144)

$$\begin{array}{@{}rcl@{}} &=&-\left( \displaystyle\frac{\rho}{1+\rho}\right)\left( \displaystyle\frac{\alpha}{\alpha+\beta A}\right) \end{array} $$

(145)

Appendix F

Recall the Eq. 29

$$Q= \left( \frac{R}{P_{E}}\;\frac{\alpha\beta}{\alpha+\beta A}\right)e_{q} $$

Then,

$$\begin{array}{@{}rcl@{}} \nu^{e_{g}}_{Q}&=&\frac{\partial Q}{\partial e_{g}}\frac{e_{g}}{Q} \end{array} $$

(146)

$$\begin{array}{@{}rcl@{}} &=& \left( \frac{R\alpha\beta e_{q}}{P_{E}} \right) \left( \frac{\displaystyle\frac{\rho}{1+\rho}\beta \displaystyle\frac{A}{e_{g}}}{(\alpha+\beta A )^{2}}\right) \left( \frac{e_{g}}{Q} \right) \end{array} $$

(147)

$$\begin{array}{@{}rcl@{}} &=& \left( \frac{R \alpha\beta}{\alpha+\beta A}\;\frac{e_{q}}{P_{E}}\right) \left( \frac{\rho}{1+\rho} \right) \left( \frac{\beta}{\alpha+\beta A}\right) \left( \frac{A}{e_{g}}\right) \left( \frac{e_{g}}{Q} \right) \end{array} $$

(148)

$$\begin{array}{@{}rcl@{}} &=& \left( \frac{\rho}{1+\rho}\right) \left( \frac{\beta A}{\alpha+\beta A}\right) \end{array} $$

(149)

Appendix G

Recall the Eq. 21

$$D_{Q}=\frac{R \alpha}{\alpha+\beta A} $$

Then,

$$\begin{array}{@{}rcl@{}} \nu^{e_{g}}_{D_{Q}}&=&\frac{\partial D_{Q}}{\partial e_{g}} \frac{e_{g}}{D_{Q}} \end{array} $$

(150)

$$\begin{array}{@{}rcl@{}} &=& R\alpha \left( \frac{\beta \displaystyle \frac{\rho}{1+\rho} \displaystyle \frac{A}{e_{g}}} {(\alpha+\beta A)^{2}}\right)\left( \frac{e_{g}}{D_{Q}} \right) \end{array} $$

(151)

$$\begin{array}{@{}rcl@{}} &=& \left( \frac{R \alpha} {\alpha+\beta A}\right) \left( \frac{\rho}{1+\rho}\right)\left( \frac{\beta A}{\alpha+\beta A} \right)\left( \frac{1}{e_{g}}\right)\left( \frac{e_{g}}{D_{Q}} \right) \end{array} $$

(152)

$$\begin{array}{@{}rcl@{}} &=&\left( \frac{\rho}{1+\rho}\right)\left( \frac{\beta A}{\alpha+\beta A}\right) \end{array} $$

(153)