The Effect of Rebate and Loan Incentives on Residential Heat Pump Adoption: Evidence from North Carolina

Abstract

Electrification can promote deep decarbonization to tackle climate change with a cleaner power grid. Electric heat pumps provide a feasible and energy-efficient way to replace fossil-fuel furnaces for space heating. Rebate and loan programs are the two most widely used incentives for residential heat pump installations in the U.S. This study compares the impacts of rebate and loan incentives on residential air-source heat pump adoption in North Carolina. First, our results show that the rebate program ($300–$450 per system) increases the adoption density by 13% in a year. Second, we find that the rebate program is more effective in promoting heat pump adoption for average consumers than two loan programs (annual loan interest rate: 9%, 3.9%). Third, we find the rebate program is less effective for low-income households than high-income households. Lastly, we compare the rebate with the loan programs in terms of cost-effectiveness and we find the rebate program is more cost-effective under certain circumstances.

Appendices

Appendix A: The Density of Air-Source Heat Pumps by the State in the U.S. in 2020

State	HP density (number/10 K persons)	State	HP density (number/10 K persons)
North Carolina	903.25	South Dakota	22.23
Virginia	865.64	Iowa	21.41
Maryland	659.31	Massachusetts	15.81
South Carolina	572.16	Wyoming	15.28
Kentucky	332.33	DC	13.25
Washington	310.77	Arkansas	13.13
Oregon	244.66	Maine	11.41
Delaware	200.33	Minnesota	11.26
Tennessee	168.09	Connecticut	11.16
Nebraska	137.33	Mississippi	10.53
Kansas	127.99	Alabama	5.72
Georgia	108.06	New Hampshire	5.52
Pennsylvania	100.43	Rhode Island	4.02
Idaho	98.43	Michigan	3.63
Indiana	90.60	West Virginia	2.28
Oklahoma	81.68	Texas	2.00
Ohio	67.60	Colorado	1.95
Florida	60.05	North Dakota	1.24
Arizona	54.53	California	1.13
Montana	44.50	Alaska	0.98
Utah	42.02	New Mexico	0.93
Nevada	39.47	Wisconsin	0.24
Missouri	28.55

Data source: ZTRAX database.

Appendix B: The Correlation Between Heat Pump Adoption and Other Factors Using the National County-Level Data

See Table

Table 6 The correlation between heat pump adoption rate and other factors using national county-level data and an OLS model

6.

Appendix C: The Distribution of Electric Utilities in North Carolina

See Figs

Fig. 7

The distribution of electric utilities in North Carolina. Figure source: Carolina country, https://www.carolinacountry.com/media/zoo/images/serviceareas_df2b99d1ada9ee3658733481c4b1f227.jpg

.7 and

Fig. 8

The distribution of three borderlines (Duke Energy—Rutherford; Duke Energy—Union Power; Duke Energy—Haywood) in the three studies. Figure source: Carolina country, https://www.carolinacountry.com/media/zoo/images/serviceareas_df2b99d1ada9ee3658733481c4b1f227.jpg

8.

Appendix D: The Heat Pump Adoption Density Before and After the Rebate in Sample 1

Figure 

Fig. 9

The evolution of heat pump adoption density in the Duke Energy utility area (treatment group) and in the Rutherford utility area (control group) in sample 1. The adoption density is computed within a 500 m × 500 m’ grid. The green bars show the ten percentile and the ninety percentile of the adoption densities

9 shows the evolution of heat pump adoption density (within 500 m × 500 m) across three years in the treatment and control groups in sample 1, respectively. For each year, we use all the observations within the year to calculate the average, the ten percentile, and the ninety percentile, which are plotted in the figure. Table

Table 7 The descriptive statistics of heat pump adoption density across years in sample 1

7 presents the detailed numbers of these descriptive statistics across these three years. In the treatment group and in 2016 before the introduction of the rebate policy, the average adoption density is 0.19. After the rebate policy, the average adoption density in the treatment group increased to 0.35. Also, we can find that the adoption density of heat pumps increased in both treatment and control groups. Thus, we utilized the DID method to compare the changes of adoption density before and after the rebate policy in the Duke Energy utility area (treatment group) with those in the Rutherford utility area (control group).

Appendix E: Estimates of the Rebate Effect Using a Simple 2-Periods DID Model

We conduct a simple 2-periods DID model to estimate the rebate program effect using observations on 07-31-2017 and 12-30-2018, which are the last period before the treatment and the last period available, respectively. Results are shown in Table

Table 8 The effect of rebate program on heat pump adoption using DID with two periods (07–31-2017 and 12-30-2018)

8, which apply the same specifications as in Tables 3 and 4 in our manuscript. All the coefficients are positive and statistically significant, which are consistent with our baseline estimations.

Appendix F: Estimates of the Rebate Effect Using Cox Proportional Hazards Model

In our setting, the installation of a heat pump for an individual household can be regarded as a hazardous event. We focus on a period from 2017-11-02 to 2018-12-30 right after the introduction of Duke Energy’s rebate program. We drop the houses that have installed the heat pump on 2017-11-02 from our sample and those remaining houses did not install the heat pump at the beginning of our study period. Since we can only observe house features from 4 periods (including 2017-11-02, 2018-01-07, 2018-08-05, 2018-12-30), we are not able to know the exact adoption dates of heat pumps. Thus, we suppose that if a house did not have a heat pump in one period and it had a heat pump in the following period, the house installed the heat pump in this following period. The survival time in the Cox model is supposed to be from 2017-11-02 to the date of heat pump adoption. In addition, we consider a censored proportional hazards model. Those houses that had not installed the heat pump at the end of our study period are regarded as “alive” or “censored” on 12-30-2018. Then, we fit the following model:

$${\text{h}}\left( {\text{t}} \right)_{ic} = h_{0} \left( t \right)_{ic} \times {\text{exp}}\left( {\alpha D_{i} + {{\varphi }}_{c} } \right)$$

where ${\text{t}}$ represents the survival time; ${\text{h}}\left( {\text{t}} \right)_{ic}$ is the hazard function for household ${\text{i}}$ in census block group ${\text{c}}$, meaning the hazardous risk at time ${\text{t}}$, or the probability of a hazardous event at time ${\text{t}}$; $h_{0} \left( t \right)_{ic}$ is the baseline hazard function when all the covariates equal to zero; $D_{ic}$ is the treatment group dummy, which takes value one for households in the Duke Energy utility area and takes value zero otherwise; ${{\varphi }}_{c}$ is the census block group fixed effects, which can control for the time-fixed census demographic features at the block group level.

Estimation results are presented in Table

Table 9 Estimates of the rebate effect using Cox proportional hazards model

9. We apply two different specifications by not adding or adding the census block group fixed effects. In both columns, the coefficients of the treatment dummy are positive significantly, suggesting that the rebate program is positively associated with the event probability (the probability of adopting a heat pump). In other words, the rebate program can significantly increase the adoption of heat pumps, which is consistent with our baseline estimations.

Appendix G: Tobit Estimates of the Rebate Effect

Using our first sample across two periods (2016-3-22 and 2018-08-05), we apply a Tobit model to estimate the effect of the rebate program. In the Tobit model, we regress the first differences of heat pump adoption density on a dummy variable indicating the Duke Energy utility area. Estimation results are shown in Table

Table 10 Tobit estimates of the rebate effect

10. In column (2), we added the ZIP code fixed effects while we do not include them in column (1). The coefficients are significant and positive in both specifications. Based on column (2), results show the rebate program can increase the heat pump adoption density by 0.027 significantly in a one-year period, which is consistent with our baseline estimates.

Appendix H: Estimates of Rebate Effects Using DID with Spatial Regression Discontinuity

We further combine the DID approach with the spatial Regression Discontinuity (RD) method to estimate the local effect of the rebate on heat-pump adoption at the borderline based on the first sample. RD has been widely used by empirical studies to evaluate policy impacts by comparing the differences in the outcome around the cutoff. Subjects receive the treatment on one side of the cutoff while receiving no treatment on the other side of the cutoff. Since subjects at the cutoff share the same probability of receiving the treatment, RD approximates a natural experiment at the cutoff region. In our study, we follow two steps to estimate the rebate effect using the DID-RD approach and only use the variables at the individual household level. First, we compute the first difference of the dependent variable, namely the dummy variable indicating heat pump adoption status for each household. Second, using the computed first differences as the outcome, a local linear regression discontinuity model is applied to estimate the local average treatment effect (LATE) of the rebate on heat-pump adoption. Intuitively, we compare the changes of heat-pump adoption statuses of households who are infinitely close to one side of the border with those of households who are infinitely close to the other side of the border. Although the RD-DID estimator is more local, the estimator is less vulnerable to unobserved confounders due to the spatial proximity of the observations. Treatment and control groups at the cutoff should be indifferent except for the treatment status, so there should be no differential trends of heat pump adoption at the cutoff between the treatment and control groups if the treatment had not been in place. The parallel trend assumption of the DID is met at the cutoff. The time window in the RD-DID analysis is from 03/22/2016 to 08/05/2018. A local linear regression with a triangular kernel function, which has been a standard choice for RD estimations (Imbens and Lemieux 2008; Lee and Lemieux 2010), is applied as follows:

$$\Delta Y_{{i, t_{1} ,t_{2} }} = \alpha + \beta D_{i} + \gamma \left( {X_{i} - c} \right) + \delta D_{i} \left( {X_{i} - c} \right) + \varepsilon_{i}$$

(13)

where $\Delta Y_{{i, t_{1} ,t_{2} }}$ is the first difference of a binary variable indicating whether household i installs a heat pump in time t. $D_{i}$ is a binary treatment group variable, which equals to one if household i is located in the Duke Energy utility and if $X_{i} \ge c$. $X_{i}$ is the running variable, measuring the distance to the borderline of household i. When household i is in the Duke Energy utility (treatment group), $X_{i} > 0$; when household i is in the Rutherford utility (control group), $X_{i} < 0$. $c$ is the treatment cut off, which equals zero. $\varepsilon_{i}$ is the error term. A triangular kernel function on the distance of each household to the borderline is applied to compute the weights when approximating the regression functions below and above the cutoff (Calonico et al. 2014). Since the performance of point estimators and confidence intervals in RD is sensitive to the specific bandwidth selected, we use the mean squared error (MSE)-optimal approach (Imbens and Kalyanaraman 2012) to compute a data-driven bandwidth for the RD point estimator. However, the MSE-optimal bandwidth selector usually leads the confidence interval inferences biased (Calonico et al. 2014). To ensure robust confidence intervals, we use the robust bias-corrected (RBC) methods (Calonico et al. 2014, 2020) to compute the bandwidth for confidence interval inference. The $\beta$ is the coefficient of our interest, which measures the local average treatment effect of the rebate program on heat pump adoption.

First, we present graphical evidence for the rebate’s effect. Figure 

Fig. 10

Regression discontinuity plots (The distance to the borderline is above 0 at the side of Duke Energy utility and is less than 0 at the side of Rutherford utility)

10 shows the regression discontinuity plots with a polynomial regression function of order 3 and a local linear regression function, respectively. The polynomial order of 3 is chosen by the Bayesian information criterion (BIC). We regress the outcome variable (the first differences in heat pump adoption status) on the running variable using global polynomial specification allowing up to order 6. We compare the BIC of each model, and the model of polynomial order 3 is the best-fit model with the lowest BIC. We use the integrated mean square error (IMSE)-optimal evenly spaced method (Calonico et al. 2015) to compute the number of bins (31 bins in plot a and 15 bins in plot b), which are assigned to calculate the sample averages within bins in both plots. The bandwidth is 5 miles in plot a and is 1 mile in plot b. Both plots demonstrate that the outcome variable (first differences in heat pump adoption status at the household level) jumps at the cutoff (borderline) and is higher at the side of the Duke Energy utility which implemented the rebate program, which provides suggestive evidence that the rebate program can increase the probability of installing a heat pump.

The bandwidths in the above RD plots are artificially set by researchers, which provides informal graphical evidence of the treatment effect. We then provide formal statistical evidence using a data-driven RD-DID approach. We utilize a nonparametric local polynomial estimator, which has been a standard choice for RD estimations (Imbens and Lemieux 2008; Lee and Lemieux 2010). A local linear regression with a triangular kernel function is applied. We use the MSE-optimal approach (Imbens and Kalyanaraman 2012) to compute the bandwidth employed for the RD point estimator, and a 0.867-miles bandwidth is chosen, leading to an effective sample size of 9,411 households. We use the robust bias-corrected (RBC) methods (Calonico et al. 2014, 2020) to compute the bandwidth for confidence interval interference and a 1.802-miles bandwidth is chosen. Results show that the estimated coefficient of β is 0.051, with a standard error of 0.025 and a p-value of 0.046. We reject the null hypothesis that there are no differences in heat pump adoption changes between the two utilities at the cutoff at a 95% confidence level. The rebate program can significantly increase the probability of installing a heat pump by 5.1% for households at the borderline. We conduct a McCrary (2008) manipulation test using local polynomial density estimation following Cattaneo et al. (2018)’s approach. The final test T statistic is 0.507, with a p-value of 0.61. There is no statistical evidence of systematic manipulation of the running variable (distance to the borderline) in our RD design.

In addition, for the RD-DID analysis in sample 1, a potential robustness check could be to consider just adoption outcomes in period 2 (08-05-2018) and then control for household characteristics in period 2. This would allow us to avoid any housing characteristic changes over time that may affect the result in the RD-DID. In this robustness check, we utilize a nonparametric local polynomial estimator, and a local linear regression with a triangular kernel function is applied. Since some variables of building characteristics have too many missing values (See the Table

Table 11 Descriptive statistics of building characteristics in sample 1

11), we only controlled two building characteristics (building year built and the number of stories). We use the MSE-optimal approach (Imbens and Kalyanaraman 2012) to compute the bandwidth employed for the RD point estimator, and a 0.788-miles bandwidth is chosen, leading to an effective sample size of 5,989 households. We use the robust bias-corrected (RBC) methods (Calonico et al. 2014, 2020) to compute the bandwidth for confidence interval interference and a 1.589-miles bandwidth is chosen. The result shows that the estimated coefficient of β is 0.00476, with a standard error of 0.03134 and a p-value of 0.879. The estimated effect is close to zero and insignificant, which could be due to two factors. First, we are not able to control for a sufficient number of variables of building characteristics in the RD model since there are so many missing observations of building characteristics. Second, the RD estimator in period 2 could be biased from the true rebate effect, since there might be other factors jumping at the cutoff along with the treatment (the rebate incentive). For instance, the two utilities may have different energy efficiency incentives. Our baseline RD-DID approach has the advantage of ruling out all the time-invariant confounding factors (including the time-invariant energy efficiency incentives). As for the time-variant confounders, although the building characteristics may change over time, the time trends of changing building features should be consistent between the treated and control houses at the cutoff borderline, since the probability of being treated at the cutoff is the same for each house at the cutoff. Thus, we think the RD-DID approach better reflects the treatment effect.

Appendix I: Ruling Out the Potential Influences of Unbalanced Covariates in Sample 3

Although the third sample is overall balanced, there are two imbalanced observable covariates, namely population median age and building’s year remodeled. To fully rule out the potential influences of the two imbalanced covariates, we conduct a robustness test by directly controlling the covariates in our differential-trends model. We use the first differences of the heat pump adoption statuses at the individual household level as the outcome variable. A regression model is applied as follows:

$$\Delta Y_{{i, t_{1} ,t_{2} }} = \beta D_{i} + X_{i} + \varepsilon_{i}$$

where $\Delta Y_{{i, t_{1} ,t_{2} }}$ is the first difference of a binary variable indicating whether household i installs a heat pump in time t. $D_{i}$ is a binary treatment group variable, which equals to one if household i is located in the Duke Energy utility. $X_{i}$ is the time-invariant observable unbalanced covariant. $\varepsilon_{it}$ is the error term. Estimated results are shown in Table

Table 12 Robustness tests by adding unbalanced covariates

12. The estimations are insensitive to adding the control variables (population median age and building’s year remodeled).

Appendix J: The Impact of Electricity Price on Heat Pump Adoption Rate in North Carolina

We estimate the impact of electricity prices on heat pump adoption rate based on about 1 million residential buildings across 26 Electric Membership Cooperatives of North Carolina. Figure 

Fig. 11

The distribution of Electric Membership Cooperatives in North Carolina

11 plots the distribution of these EMCs. We apply the following two-way fixed effect model using yearly panel data from 2016 to 2019:

$$Y_{iut} = \beta P_{ut} + \varphi_{i} + \vartheta_{t} + \varepsilon_{iut}$$

where $Y_{it}$ is the heat pump adoption rate within a 1 km × 1 km grid i and on year t. We calculate the yearly heat pump adoption rate based on four assessments of ZTRAX database which were conducted on 02/03/2017, 01/07/2018, 12/30/2018, 01/02/2020, respectively. $P_{ut}$ is the average electricity residential price (cents) of utility u on year t. Different utilities have different electricity prices. We calculate the yearly average residential electricity price by dividing cooperative’s residential total revenue by residential sales based on EIA 861 forms from 2016 to 2019. The EIA 861 forms were obtained from the website of Annual Electric Power Industry Report, U.S. Energy Information Administration. $\varphi_{i}$ is the individual grid fixed effects. The heterogeneous incentives for heat pumps provided by different cooperatives can be controlled by the individual grid fixed effects. $\vartheta_{t}$ is the year fixed effects controlling for common time trend of heat pump adoption rate. We cluster the standard error at the individual grid level.

Table

Table 13 The impact of electricity price on heat pump adoption rate in North Carolina

13 presents the estimation result. The coefficient of the electricity price variable is negative significant. One cent of decrease in residential electricity price can lead to the heat pump adoption rate increased by 0.0016 in North Carolina.

Appendix K: The Amounts and Percentage of Personal Loan and Credit Card Debt in the U.S

See Table

Table 14 The amounts and percentage of personal loan and credit card debt in the U.S

14.

Appendix L: The Heterogeneous Effects of Rebate on Heat Pump Adoption by Income Quartiles

See Fig. 

Fig. 12

The heterogeneous effects of the rebate program on heat pump adoption rate (share of households with heat pumps within a 500 m × 500 m grid) by income quartiles using DID approach (in the figure, circles are point estimates, and error bars are 95% confidence intervals)

12.

Appendix M: Cost-Effectiveness Analysis

In this section, we compare the rebate program with loan programs in terms of cost-effectiveness. We use the amount of dollars spent per heat pump adopted caused by the incentive to measure the cost-effectiveness. The cost-effectiveness ($\sigma$) of the incentive program is computed by the following equation:

$$\sigma = \frac{{k \cdot \left( {n + \Delta } \right) \cdot c}}{\Delta } = c + \frac{kn}{\Delta }c$$

where $\Delta$ is the heat pump adoption growth induced by the policy; $n$ is the heat pump natural adoption growth without the incentives (if the incentives had not been in place); $k$ is the proportion of residents who apply for the incentive in residents with new heat pump installations; $c$ is the program cost paid for each application (including the natural adoption of the heat pump). For the rebate program, ${c}_{R}$ equals the rebate amount, which is $300 to $450. For the loan program, $c$ equals the difference in the loan principal and the present value of repayments, and is computed according to the following equation:

$$c_{L} = P - \mathop \sum \limits_{i = 1}^{5} \frac{{P \cdot \left( {r_{APR} /12} \right)}}{{1 - \left( {1 + r_{APR} /12} \right)^{ - n} }} \cdot \frac{12}{{\left( {1 + r_{S\& P} } \right)^{i} }}$$

where ${c}_{L}$ is the loan program cost for each application; $P$ is the amount of loan principal; $r_{APR}$ is the annual interest rate of the loan; $r_{s\& p}$ is the S&P 500 (a stock market index) annualized return rate from 01.01.2015 to 01.01.2020 (a five-year period), which is 10.08%; $n$ is the number of monthly repayments. Here, we consider a common amortization loan program, which spreads out a loan into a series of monthly fixed payments. The two loan programs in our study are both five-year loan programs. We assume every loan applicant applies for an $8,000 five-year loan program for the heat pump, and then calculate the sum of the present value of each year’s repayment using the S&P 500 annualized return rate as the discount rate. The discount rate should reflect the return of alternative investments for loan lenders. Using the above equation, $c_{L}$ equals $461 for the loan program with 9% APR, and $c_{L}$ equals $1,328 for the loan program with a 3.9% APR. Thus, we find that $c_{R} < c_{L}$. Based on our previous estimations, the rebate’s effect is larger than the other two loan programs, so $\Delta_{R} > \Delta_{L}$. If we assume that the application proportions of the rebate program and the loan program are the same ($k_{R} = k_{L}$), we have:

$$c_{R} + \frac{n}{{\Delta_{R} }} \cdot c_{R} < {c_{L} + \frac{n}{{\Delta_{L} }} \cdot c_{L} \quad {\varvec{s}}.{\varvec{t}}.\quad c_{R} < c_{L} ,\Delta_{R} } > \Delta_{L}$$

$$\sigma_{R} \left\langle {\sigma_{L} \quad {\varvec{s}}.{\varvec{t}}.\quad c_{R} < c_{L} ,\Delta_{R} } \right\rangle \Delta_{L}$$

As a result, the rebate program is more cost-effective than the loan program if the same proportions of residents applied for the rebate and loan programs. In other words, the amount of dollars spent per heat pump adopted induced by the rebate program is less than that by the loan programs. Note that we only consider monetary loss and do not take into the costs of program administration for both utilities and consumers. The administration costs of the loan program could be much higher than the rebate program because loans require more administration work such as credit history checks, application paperwork, and a series of repayments. Also, the funding-raising costs of the loan program are higher than the rebate program. For example, utilities need to give $8,000 to each applicant in a loan program, while they only need to give $450 to each applicant in a rebate program. If we consider these extra costs, the loan program should be much less cost-effective compared to the rebate program.

Policymakers should take several important factors into account when investigating the cost-effectiveness of incentives: the natural adoption growth without the intervention of incentives, the percentage of consumers who will apply for the incentives, and the causal effect of the incentives on the technology adoption. If the natural adoption growth without the intervention of incentives is high and those residents also apply for the rebate and loan programs, it would lead to the free-riding problem. With a high amount of freeriding, policymakers and utilities are better off not having incentives since this will lead to a high cost of the rebate and loan programs. Incentives (rebate and loan) targeted for specific marginal consumers whose willingness to pay is at the bottom edge of upfront costs could be applied to improve the cost-effectiveness of the incentives.

The rebound effects after installing heat pumps (Winther and Wilhite 2015) may reduce the expected effect of incentives for heat pumps on decarbonization. Households can get energy bill savings from installing the energy efficiency technology and use these savings to increase energy consumption in other areas or within the same area for increased comfort. A study finds a rebound effect of 20% for consumers after replacing direct electric heating with air source heat pumps in Denmark (Gram-Hanssen et al. 2012). In the US, the rebound effect after heat pump adoption may also exist and reduce the expected effects of incentives, since increased electricity consumption produces air and carbon pollutions. However, the rebound effect will not diminish the effect of heat pumps on decarbonization with a carbon-free power grid in the future.

Since we only estimate the rebate’s effect based on sample 1 and we do not know the absolute value of the rebate and loans’ effects in samples 2 and 3, we can only compute the absolute value of cost-effectiveness for the rebate program in sample 1. The computed cost-effectiveness of the rebate program in sample 1 based on a one-year time window is $2,914–$3,921 per heat pump if we assume every resident who installed the heat pump applied for the rebate. The cost-effectiveness is much larger than the rebate amount ($300–$450) because the natural adoption growth in that region is quite high.

It is also important to discuss the relationship between the cost of instruments (or the cost-effectiveness) and the cost of carbon. We find two challenges to conducting this analysis. First, it is difficult to calculate the absolute value of the cost-effectiveness of the rebate and loan programs. In reality, we are not able to observe the $k$, and the $n$ varies a lot across different sites or samples. Thus, it is difficult for us to compute the absolute value of cost-effectiveness. Also, results will depend on site selections because of the large variations of the cost-effectiveness across different sites.

Second, the avoided costs of carbon associated with the switches from typical natural gas furnaces to air source heat pumps can be time-variant since it depends on the carbon emissions from electricity generation. Although currently coal and natural gas power plants are still in service, more and more clean energy sources (e.g., wind and solar power) will replace the traditional fossil fuel sources in the near future, which generate zero carbon emissions. The avoided costs of carbon will be much larger in the future with more clean energy sources in the power grid. These clean energy sources are expanding very quickly in these years. As a result, it is hard for us to predict the future carbon emissions of electricity generation in this paper and the avoided costs of carbon associated with switching to heat pumps in the near future.

To summarize, there are two major challenges to comparing the costs of policies with the avoided costs of carbon. To address these challenges, we make the following assumptions: (1) every resident who installed the heat pump applied for the rebate; (2) under the sustainable development scenario, the electricity generation produces zero carbon emissions. Based on the first assumption, we estimated the cost-effectiveness of the rebate program, which is $2,914–$3,921 per heat pump (see Sect. 8). Based on the second assumption, the avoided costs of carbon are the social costs of carbon emissions from a typical natural gas furnace. According to Vaishnav and Fatimah (2020)’s study, the average annual CO₂ emission from one typical natural gas furnace in North Carolina (NC) is 4,266 lbs. We then use the following equation to calculate the present value of the lifetime social carbon costs of the natural gas furnace (Ω):

$$\Omega = \mathop \sum \limits_{t = 2020}^{t + n} e \cdot P_{t}$$

where $e$ is the annual CO₂ emissions from a typical natural gas furnace in NC. $\mathrm{P}$ is the social cost or carbon price of CO₂ in year ${\text{t}}$, and we apply the values of social cost of CO₂ by years (from 2021 to 2025) provided by the US Environmental Protection Agency (2016). $n$ is the number of years that the natural gas furnace can operate for and we suppose the natural gas furnace is installed in 2020 and it will be used for 25 years. After calculation, Ω equals 3400 (in 2021 dollars). So, we can find that the Ω is within the range of the cost-effectiveness of the rebate program in sample 1. In other words, the costs of the rebate program in sample 1 can be compensated by the avoided carbon costs associated with the switches from natural gas furnaces to heat pumps. The cost-effectiveness of rebate in sample 1 is much larger than the rebate amount ($300–$450) because the natural adoption growth in that region is quite high. In regions with lower natural adoption growth of heat pumps (or, less “free riders”), the costs of the policy can be much lower. The similar calculation for the loan program is not possible since we cannot estimate the absolute value of the loans’ effects in samples 2 and 3, due to data limitation.

Appendix N: Contemporaneous Energy Efficiency Incentives Programs

We check the incentives for energy efficiency products in our sample areas from the Database of State Incentives for Renewables and Efficiency (DSIRE). Based on the ZIP code, we can identify all the energy efficiency and renewables incentives in our sample areas. In our samples, we find 48 incentives in total since 2000. Among these 48 incentives, only one incentive program (in addition to the heat pump rebate and loan programs) changed over time within the period of our study (2016-03-22 to 2020-01-02), which is Duke Energy Solar Rebate Program (initiated from 09/24/2018). The effect of the solar rebate program could be one potential confounding factor in our study. To check the potential correlation between solar PV adoption and heat pump adoption, we utilized the 2015 Residential Energy Consumption Survey (RECS) database (EIA 2020, https://www.eia.gov/consumption/residential/data/2015/), which is a U.S. nationwide representative sample of households. Using the RECS data, we conducted a Probit model by regressing the dummy of solar PV adoption on the dummy of heat pump adoption. The result shows that the marginally increased probability of installing a heat pump for a household with solar PV is only 1.3% (p-value < 0.01), which indicates that the effect of solar PV adoption on heat pump adoption can be very small, and the effect of the solar rebate program on heat pump adoption will be much smaller or even negligible.