Optimal dynamic electricity consumption function estimation: an institutional experimental evidence from Guangzhou, China (1949-2016)

Abstract This research demonstrates from a dynamic optimal perspective that electricity consumption for a metropolitan area is a function of economic output, electricity consumption habits, and electricity demand management reform. The empirical results include: (1) an unidirectional Granger causality exists linking economic output to electricity consumption; (2) given electricity consumption habits under the context of the electricity demand management reform, an economic output increase of 1% results in the increase of electricity consumption by 0.22%, and (3), after demand management has been implemented, economic output continues to increase electricity consumption, but at a lower rate than prior to reform. These empirical results imply that the ‘conservation hypothesis’ is upheld over the long-run at the regional level in Guangzhou from 1949 to 2016.


Introduction
One of the surprising discoveries in electricity economics over the past twenty years has been the relationship between electricity consumption and economic output. The concept of a consumption function dates back to the origin of Keynesian macroeconomics where The General Theory of Employment, Interest, and Money emphasized the central importance of consumption. A consumption function reflects the relationship between consumption and economic output . As summarized in section 2, there is an extensive literature which has estimated relationships between electricity consumption and national income (as measured by gross domestic production (GDP)). This literature, however, show contradictory evidence across different regions and countries around the world. One reason why there are different empirical results on this relationship is that an electricity consumption function has not yet been developed in the literature. Thus, this study derives a theoretical relationship between electricity consumption and economic output by solving an optimal intertemporal income problem.
In addition to a limited exploration of theory concerning electricity consumption function, electricity demand management reform is another seldom researched aspect of electricity markets. In terms of the metropolitan areas in China, they face electricity power shortages on a constant basis (Pollitt et al., 2017). One response to these shortages has been to implement reform of electricity markets. Since electricity consumption changes with implementation of demand management, this type of institutional reform matters in the electricity consumption function. Electricity demand management reform could be a key driver of affordable and efficient electricity consumption through expanding economic growth.
The example of interest in this research is Guangzhou, where a prolonged process of electricity market reform has occurred since 1985. Up until 1984, consumption of electricity was measured on a community basis (not individual household), so that household payments for electricity reflected average usage among all households in the community. Starting in 1985, however, individual household metering of electricity consumption began so that payments reflect household level consumption. This demand management reform transformed electricity usage from a community (public) to an individual (private) good for residential customers (households), thereby producing an incentive for households to save electricity. This change required that electric grids be adjusted to the "ammeter sole use system" to help alleviate shortages of electricity. In addition, a schedule of peak rates for commercial and industrial customers was designed to encourage reductions in electricity consumption and alleviate electricity shortages. Specifically, during peak demand periods, a charge from 1.3 to 1.5 times the basic electricity rate is levied.
Based upon these considerations, the three objectives of this study are to: (1) establish a theoretical basis for an electricity consumption function using optimal control theory, (2) empirically examine the relationship between electricity consumption and economic output in Guangzhou, China, and (3) introduce electricity demand management reform into both the theoretical and empirical models. This research is based upon the perspective that with rising income, consumers are more likely to afford electronic appliances, such as televisions, refrigerators, washing machines, computers, and air conditioners, thus increasing the demand for electricity (Huang et al., 2018). This perspective leads to three questions: 1. What is a theoretically appropriate relationship between electricity consumption and economic output? 2. As economic output increases, how much more electricity consumption will occur? 3. How does electricity demand management impact electricity consumption?
The contributions of this paper include: (1) development of an inter-temporal optimization model that connects electricity consumption to economic output based upon electricity consumption habits, which has not been discussed in the literature previously, and (2) design of a natural experiment using kink regression discontinuity approach to investigate the effect of the 1985 electricity demand management reform on electricity consumption.
This section provides a brief introduction of the background and motivation for this research. In section 2, literatures are reviewed on a general consumption function and the nexus between electricity consumption and economic output. In section 3, a theoretical framework is presented that applies optimal control theory to derive a metropolitan electricity consumption function. Section 4 introduces time series econometric methods to test the unit root and cointegration for the data utilized in the analysis. Empirical results will be addressed in section 5. Finally, section 6 presents conclusions and further discussion.

Literature on general consumption function
The nature of the relationship between electricity consumption and economic output in economic theory can be expressed as an electricity consumption function where electricity consumed is a function of income or wealth. In the economic literature since Friedman (1957), many economists have conducted theoretic and empirical research on consumption functions (Gorman, 1964). Spiro (1962) finds that if income is to remain permanently constant, the desired stock of wealth will ultimately be accumulated and therefore consumption would equal net income. The Zellner consumption function (Zellner, 1957) fits well but gives rather low estimate of the long run marginal propensity to consume and a rather high and hard to interpret coefficient for the liquid assets variable (Griliches et al., 1962).
In terms of consumption function theory development (Zellner & Geisel, 1970), Thompson (1967) asserts and demonstrates an equivalence that exists between the utility function and standard aggregate consumption function. Baxter and Moosa (1996) propose a split consumption expenditures on non-durable items between 'basic needs' and other expenditure. Recently, economists have gradually transitioned into empirical research on consumption function from theoretical modeling. Next, we discuss empirical literature on electricity consumption functions. Table 1 provides a summary of recent literature on the nexus between electricity consumption and economic output for different regions and countries around the world. There are four main hypotheses that relate electricity consumption to economic output. The first is the conservation hypothesis which implies a unidirectional Granger causality running from economic output to electricity consumption. In contrast, the growth hypothesis postulates a unidirectional Granger causality running from electricity consumption to economic output. The third hypothesis (called feedback) is contemplates a bidirectional Granger causality such that electricity consumption and economic output mutually influence each other. Finally, the fourth hypothesis is one of neutrality with no direct Granger causal links between electricity consumption and economic output .

Literature on the relationship between electricity consumption and GDP
Based upon Table 1 review of the literature, the nexus between electricity consumption and economic output has been extensively studied, but the evidence so far is contradictory and inconclusive (Stern et al., 2018). Most of the scholars utilize national level data without consideration of any institutional factors (like demand management reform). In addition, there has been limited attention in the economic literature devoted to investigating the effect of total income combined with electricity demand management reform on electricity consumption in metropolitan area. Thus, our approach includes a theoretical model to investigate the relationship between electricity consumption and gross product (total income) that empirically examines this relationship using data from the metropolitan level (Guangzhou, China) while incorporating electricity demand management reform into the model.

Theoretic model
Our first objective is to derive a metropolitan electricity consumption function that rests upon a theoretical basis of optimally allocating government expenditures for electricity infrastructure in order to maximize a metropolitan's inter-temporal total income (Y). This objective is based on an assumption that competition between regions motivates metropolitan officials to maximize a metropolitan's inter-temporal total income. Chinese metropolitans compete against each other for performance rankings and metropolitan officials' careers are linked to their performance with the most popular performance indicator being GDP (Xu, 2011). In addition, according to Wagner's Law, an increase in total income in the society has a positive effect on government spending (K onya & Abdullaev, 2018). Hence, in order to increase their share of the economy, officials consider maximizing total income in their metropolitan as an incentive .
Based upon this assumption, the inter-temporal metropolitan total income function (M) can be expressed as below: where h represents a social discount rate and T represents metropolitan government's planning period. With constrained optimization, the first constraint is based on an income accounting identity: where consumption is C t , I t denotes all investment (private and public) that is outside the electricity generation industry, and G t denotes annual government investment on electricity infrastructure, all in year t. Investment represents the change in the economy's stock of capital: where d denotes depreciation rate of capital. Hence, plug (3) into (2), we obtain: which transforms to: The production function for Y t is assumed to be represented in Cobb-Douglas (C-D) form: where A t denotes technology level, K t is capital, E t is electricity utilized in production, and L t is labor. The parameters of Equation (6) (a, b, c, d) are each restricted to between zero and one. An aggregate C-D production function is assumed here to help ensure well behaved solutions. Production represents the total supply of metropolitan goods and services. A change of capital stock can be derived from (5) and (6) as: The second constraint comes from the capacity of electricity production due to available infrastructure. To express this constraint, we use F t to represent value of electricity infrastructure at year t and G t to represent annual investment on electricity infrastructure. So, the value of electricity infrastructure at year t þ 1 (F tþ1 ) is composed of G t and the value of remaining electricity infrastructure. A convenient way of modeling the latter is to assume that the value of remaining electricity infrastructure at the end of year t is (F t À pF t Þ, where the electricity infrastructure depreciation rate is p: Therefore, we define changes in the value of electricity infrastructure as Finally, the amount of electricity consumption, E t , is treated as the functional of the capacity of electricity infrastructure and other factors (e t ¼ e t ðD t ÞÞ involving electricity demand management reform ðD t Þ, so that E t ðF t , e t D t ð ÞÞ ¼ fF t e t D t ð Þ, where f is the transfer coefficient representing what percentage of stock of electricity generated from electricity infrastructure can be effectively used, and e t will be transferred as the form of the error term in the econometric model. Therefore, we have the following set-up for an optimal control problem: and C t are exogenous variables, G t is the control variable, K t and F t are state variables. Finally, the electricity consumption function is derived from the optimal control solution to this problem and shown in Equation (10). Details about this derivation provided in the Appendix.

Econometric methods and data
To estimate an electricity consumption function, the first step is to conduct the unit root tests without and with break data (Figure 1). If the variables are stationary at level, we can run the OLS regression directly, since there is no spurious issue in that case. However, if the variables are found to be non-stationary at level, the process is to continue to conduct the unit root tests for all variables at first difference. When variables are stationary at first difference, we can further conduct cointegration tests by Johansen and ARDL approaches. If there is cointegration relationship among variables, the spurious problem will be solved and then the Granger Causality tests and Kink Discontinuity Regression method can be conducted. Based on Chow and Niu (2015), we do not involve any time dummy variables, interaction terms containing time dummy variable, or lagged variables in the unit-root and cointegration tests.

Unit root and cointegration tests
Because standard Granger causality tests should be conducted on stationary time series or cointegration with unit root process, we first test the unit roots of all variables to confirm the stationary properties of each variable. This is achieved by using the Augmented Dickey-Fuller (ADF) test (Dickey and Fuller 1979;Mackinnon, 1996). Expressing the time series variables of lnE t and lnY t as X t , the Augmented Dickey-Fuller (ADF) relationship is: where D is the difference operator, l is the auto-regressive lag length that must be large enough to eliminate possible serial correlation in b i , a 0 is a constant, a 1 is the coefficient of interest, a 2 is the coefficient on a time trend, and u t is the error term. In addition, Phillips and Perron (1988) propose an alternative (nonparametric) method of controlling for serial correlation when testing for a unit root called the PP test: However, when there are any structural breaks in the data, the ADF test is biased towards a spurious acceptance of non-stationarity because of misspecification bias and size distortions. A Perron test allows for a one-time change in structure occurring at time T B (1< T B <T, T is the number of observations). The model that is considered in this test is one which allows for an exogenous change in the level of the series： where DT t Ã ¼ t -T B if t > T B and 0 otherwise. The null hypothesis implies that the data are non-stationary. In this test, the alternative is taken as trend-stationary with a terminal at time T. The breakpoint choice is correlated with the data utilized and cannot be considered as independent of the data. Zivot and Andrews test (Zivot & Andrews, 1992) addresses this issue by estimating the structural break data endogenously instead of considering an exogenous break date. We estimate the following equations for the Zivot and Andrews test with the endogenous location of the breakpoint k¼ T B / T: The Johansen multivariate cointegration test (Johansen, 1995) takes the following form as below: Another way to verify the cointegration relationship is to apply an ARDL model (Pesaran et al., 2001), if none of the series are I(2). The ARDL(p, q) model used in this study is expressed as:

Granger causality test and kink discontinuity regression
Although the Johansen cointegration test and the ARDL approach to cointegration explore whether the time-series data are cointegrated, they do not reveal the causality directions between ln E t and ln Y t . For this purpose, we use the Granger causality (Granger, 1969) as shown in Equations (18) and (19): In order to design an experiment to investigate the causal effect of electricity demand management reform program on electricity consumption, we use the KRD (Kink Regression Discontinuity) approach for robustness analysis (Card et al., 2015). The idea of regression discontinuity design is that there is a continuous variable lnY t (assignment variable) which determines the treatment variable D t by a cutoff. The random distribution of samples in a small neighborhood d À l, d þ l ½ of lnE t is regarded as "quasi experiment". By estimating LATE (Local Average Treatment Effect), it is possible to identify whether the dependent variable (lnE t ) has a cutoff at A generalization of electricity consumption function based on Equation (10) allows different trend function for E lnE 0t lnY t ½ and E lnE 1t lnY t ½ : Modeling both of these variables as conditional expectation functions (CEFs) results in Equations (20 and (21): To derive a regression model that can be used to estimate the causal effect of interest, we use the fact that D t is a deterministic function of lnY t to write Substituting regression for conditional expectations, then we have The electricity consumption function in regression discontinuity reduced form is expressed in Equation (24): where a 0 ¼ b 0 À b 01 lnY 1985 , a 1 ¼ b 01 , a 2 ¼ b 2 , a 3 ¼ Àðb 11 Àb 01 ÞlnY 1985 þ b 3 , and a 4 ¼ b 11 À b 01 :

Background
With over 2,100 years of history, Guangzhou is a major commercial center in south China . Guangzhou is the third largest metropolitan area in China, after Beijing and Shanghai, and the largest city in south central China (Yang et al., 2013). As the capital of Canton Province, it is located within 120 km of both Hong Kong and Macau. Because Guangzhou is adjacent to Hong Kong, which was a colony of the Britain from 1842 to 1997 and is a typical metropolitan market economy, the Chinese central government allowed Guangzhou be an experimental metropolitan in terms of institutional reforms. Thus, Guangzhou has become the commercial and free trade center of south China (Bercht, 2013).

Data
The Guangzhou Statistical Division provides the most complete and longest duration time series dataset (from 1949 to 2016) among all the metropolitan areas in China.
We utilize this time series data to estimate a metropolitan electricity consumption function. Table 2 lists the variables, their definitions and summary statistics for all variables included in the sample. According to the form of electricity consumption function in Equation (10), the dependent variable is the natural logarithmic of metropolitan electricity consumption (ln E). The main independent variable is the natural logarithmic of metropolitan economic output (ln Y).

ADF test and PP test
ADF test and PP test are applied to detect the possible presence of unit roots in ln Y t and ln E t . The null hypothesis of unit root can be rejected in favor of the alternative hypothesis of no unit root when the p-value is small . Table 3 indicates that no variable is stationary in their levels since the p-values for each variable are greater than 10%. On the other hand, ln Y t and ln E t are stationary process in   Table 4 and Table 5, respectively. They show that non-stationary processes are found in all series at level but variables are found to be stationary at first difference. This confirms that ln Y t and ln E t are integrated at I(1). Note: 1) ÃÃÃ p < 0.01, ÃÃ p < 0.05, Ã p < 0.1; 2) C, L, and T represent the constant, and lag length, time trend, respectively. Source: Stata and R software.    Table 6 shows the lag order in Johansen test is one. Based upon unit root test results, integration of variables is of the same order so that we continue to test whether these variables are cointegrated over the sample period . The Johansen cointegration test in Table 7 shows the trace statistic for non-cointegrating equations (29.0565) is greater than the 5% critical value (20.2618), but not for the at most one cointegrating equation (p-value 0.2781 is greater than 10%). This test rejects the hypothesis of none cointegration and indicates that there is at least one cointegrating equation at the 5% significance level, demonstrating there is a long-run relationship between ln Y t and ln E t in Guangzhou. The results of the bound test show there exists a long run relationship exists between ln Y t and ln E t , because their F-statistic (10.1886) are higher than the upperbound critical value (5.5800) at the 1% level (Table 8). This implies that the null hypothesis of no cointegration between ln Y t and ln E t is rejected, when ln E t is dependent variable. Table 9 reports the Granger causality analysis between lnY t and lnE t based on Vector Error Correction Model. Only in the long run, there is a unidirectional Granger causality from lnY t to lnE t since the related p-value of ECT t-1 (0.0048) is less than a 1% level. Moreover, the coefficient of ECT t-1 is negative and significant. Furthermore, this Granger Causality demonstrates that the evidence from Guangzhou supports the conservation hypothesis. This result is inconsistent with the finding that confirms the Granger Causality running from electricity consumption per capita to economic output per capita in the short run for Guangzhou . However, the latter neglects the further discussion on the Granger Causality test for long-run.  Moreover, the empirical results in Table 8 just reflect Granger Causality between electricity consumption and economic output, which means a variable ln Y t is useful in forecasting another variable ln E t (past values of ln Y t should contain information that helps predict ln E t above and beyond the information contained in past values of ln E t alone) but this does not imply that ln Y t actually causes ln E t , in terms of causal inference. To investigate the actual causal effect between electricity consumption and economic output, we need to continue to conduct the causal inference by using KRD approach.

Regression results incorporating demand management reform using the KRD approach
Because electricity demand management reform in Guangzhou started in 1985, the natural log of real GDP in 1985GDP in (lnY 1985 serves as a cutoff to compare electricity consumption prior to and after this date. Therefore, the ln Y t prior to 1985 are not exposed to reform while ln Y t in 1985 and all years thereafter are exposed to reform. Table 10 demonstrates that the local Wald estimator with one bandwidth during the period 1949 to 2016 is significantly positive, which confirms that the natural log of real GDP on 1985 is the cutoff, statistically. Table 11 illustrates that the total marginal effects of ln Y t are significantly positive both prior to 1985 (0.3774) and after 1985 (0.2207 ¼ 0.3774 À 0.1567). This result means that while real GDP increases electricity consumption throughout the entire time-period in Guangzhou, after demand management reform is implemented, its impact is lessened. Since the total marginal effect of ln Y t represents the income elasticity of electricity demand, we find that electricity consumption under the context of electricity demand management reform increases by 0.2207%, for an 1% change in economic output. Therefore, these results also confirm that there is true causality relationship running  from economic output to electricity consumption, which is consistent with the empirical results from Granger Causality Test in Table 9. Moreover, since the coefficient of D t Ã ln Y t is negative and statistically significant, after the electricity demand management reform, we conclude that the income elasticity of electricity consumption is lower after reform. we attribute this result to the electricity demand management reform where electricity use has transitioned from a community (public) good to individual (private) good for the residential customers (households). With this reform, it is presumed that the consumers prefer to purchase more energy-efficient appliances (such as compact fluorescent lamp or light emitting diode lamp) as their income increases after reform.
This reform created incentives for households to save electricity ). In addition, the regime of peak period rate also constrains commercial and industrial customers' demand for electricity. Finally, the coefficient of ln E t-1 is positive and statistically significant (Table 11). This result means that previous electricity consumption habits have a "path dependence" effect on current electricity consumption, because the consumer has formed the habits of consuming electricity.

Conclusions
To analyze the nexus between electricity consumption and metropolitan economic output in Guangzhou City, China, we develop a theoretical framework utilizing an inter-temporal, constrained optimization model of societal income with government investment in electricity infrastructure. This model also featured aspects of electricity consumption habits by consumers and electricity demand management reform. A natural experiment design with a kink regression discontinuity method is utilized to evaluate the electricity consumption function after reform. Therefore, a metropolitan electricity consumption function is derived and estimated including GDP, electricity consumption habits, and electricity demand management reform in this study.
Previous studies have explored the nexus between electricity consumption and GDP by only examining empirical relationships without developing an underlying theoretical basis for this relationship (Ghosh, 2002;Ikegami & Wang, 2016;. Although some empirical researchers have examined Granger causality between electricity consumption and GDP, they do not provide an underlying theoretical explanation for the logic of such a linkage between economic output and electricity consumption. Based on the theoretical hypotheses developed from our model, the empirical results demonstrate three findings: (1) unidirectional Granger causality running from economic output to electricity consumption, (2) given electricity consumption habits under the context of the electricity demand management reform, an economic output increase of 1% results in the increase of electricity consumption by 0.22% (the income elasticity demand of electricity), and (3), after demand management has been implemented, economic output continues to increase electricity consumption, but at a lower rate than prior to reform. These empirical results imply that the 'conservation hypothesis' is upheld over the long-run at the regional level in Guangzhou. In addition, it is instructive that, electricity consumption is the consequence of income growth.
This study is also helpful in balancing the relationship between electricity use and economic reform. Especially, the experience of electricity demand management reform in Guangzhou provides evidence that the "ammeter sole use system" improves the electricity use efficiency (units of electricity use per unit of GDP), because individual households pay for electricity that they actually use.
Different from the conventional research on economic impact of energy use (Collins et al., 2012), the literature on the electricity-growth nexus is dominated by empirical research (Payne, 2010). However, these are variability of causality results, particularly across sample periods, sample sizes, and model specification (Smyth, 2013). Further research in these areas may shed light on regional variations in the functional form of electricity consumption.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work is financially supported by National Ten Thousand Talent Program [W02070352] and Chinese National Funding of Social Sciences [19FJYB050].
From (A2), I obtain: From (A4), I obtain: According to the condition that each profit maximizing firm should hire any input up to the point at which the input's marginal contribution to production is equal to the marginal cost of hiring any input, we assume that there are n units of homogeneous firms in the metropolitan, so Y t ¼ ny t and K t ¼ nk t , where y t and k t denote each firm's output and capital input at year t, respectively. Hence, each firm's marginal productivity of capital should be identical to the interest rate that is supposed to be social discount rate which reduces to: According to the general solution of the first order difference equation, we get where W is an initial value in the solution of the first order difference equation of u t , so it is a constant. Based on (A6), we get Plug (A16) into (A13): So, we obtain: Let (A17)-b Ã (A18): Hence, from (A19), the reduced form of metropolitan electricity consumption is where b 0 ¼ ðB 0 þ B 1 Þð1 À bÞ, b 1 ¼ B 2 ð1 À bÞ, b 2 ¼ b and e t ¼ e t -be tÀ1 ¼ hðe t D t ð ÞÞ: Equation (A20) means that the current electricity consumption is determined by the previous electricity consumption and the current real GDP, and other factors in error term. Here, previous electricity consumption can be considered as the electricity consumption habits.
Finally, if we assume e t ¼ hðe t D t ð ÞÞ ¼ b 3 D t þ v t , where v t denotes the unobservable factors excluding electricity demand management reform, then the electricity consumption function is as below: