### The definitions of relative energy prices and energy consumption

The essence of relative energy prices is the relationship among prices. Taking the definition of relative energy prices at home and abroad (Wei and Lin 2007; Yang 2009)^{Footnote 4} as a reference, we consider the relative energy price as a comparison between energy prices and the general price level from the perspective of inflation costs. Thus, we do not consider relative prices based on other factor inputs such as capital and labor or through price comparisons among different types of energy (Wing 2008). Moreover, regarding energy consumption, there are also two levels of analysis: total energy consumption and energy consumption intensity.

Based on the above analysis, we first affirm the linkage between energy prices and the general price level. According to Lin and Wang (2009), the range of price variations can be expressed as

$$\Delta P_{j} = \sum\limits_{i = 1}^{n} {m_{ij} \Delta P_{i} } ,\quad {\text{j}} = 1,2, \ldots ,{\text{n}},$$

(1)

where ∆*P*
_{
i
} and ∆*P*
_{
j
} are the price variations of commodity *i* and *j*, respectively; the price variation of commodity *j* is influenced by the price variations of all the commodities related to *j*; m_{
ij
} is the direct consumption coefficient of commodity *I*; and *j* is an input–output table. This equation can be rewritten as

$$\Delta P_{n - 1,1} = \sum\limits_{n = 1}^{N} {m_{n,n - 1} \Delta P_{n,1} }$$

(2)

where ∆*P*
_{n} is the range of variation in energy prices. As for the original order n in the direct consumption coefficient matrix M in the input–output table, transposing the order n − 1 in the direct consumption coefficient matrix after removing row n and column n, we have

$$\Delta P_{n - 1,1} = \sum\limits_{n = 1}^{N} {\mathop {\left[ {(E - M_{n - 1} )^{ - 1} } \right]^{T} m}\nolimits_{n,n - 1} } \Delta P_{n,1}$$

(3)

where *E* is an n × n identity matrix.

To obtain the general price level, a price index \(\overline{\Delta P}\) is generally constructed as a weighted average of the prices of related commodities. Thus, the impact of energy prices on the general price level can be expressed as

$$\overline{\Delta P} = \sum\limits_{n = 1}^{N} {\mathop \theta \nolimits_{n - 1,1} } \left( {\sum\limits_{n = 1}^{N} {\mathop {\left[ {(E - M_{n - 1} )^{ - 1} } \right]^{T} m}\nolimits_{n,n - 1} } \Delta P_{n,1} } \right)$$

(4)

where *θ*
_{
n−1} is the proportion of commodity *n* − 1 in the price index compilation.

This analysis shows that rising energy prices inevitably lead to increases in other related product prices and finally increase the possibility of inflation. Inflation can be regarded as a type of policy cost because it deviates from the target of macro-regulation and will increase the costs of the central bank’s reaction to inflation (Fiore et al. 2006). In addition, both unexpected as well as expected inflation generates costs for society (Han 2004), including menu costs, shoe-leather costs and welfare costs, among others (Chiu and Molico 2010; Lee 2013; Nakata 2014). In fact, almost any cost is the definite result of a constantly increasing general price level that may lead to low efficiency in the economy in general (Heer and Sussmuth 2007; Bick 2010; Schneider 2014). Therefore, in this paper, we do not focus on the inflation cost itself but consider inflation (a constant increase of the general price level) as a comprehensive policy cost, which means that achieving energy savings by increasing energy prices is bound to create an inflation cost in practice.

Based on the foregoing, we define \(relative\;energy\;prices = \frac{energy\;prices}{general\;price\;level}\); here both the relative energy prices and energy prices are domestic indices. Thus, we can use research on the effects of monetary policy as references. Generally speaking, the basic coefficient is BCE = GDP growth indexation/CPI, where GDP growth indexation is profit and CPI is the cost of monetary policy. Using this equation as a reference, we consider that energy prices can impact both energy consumption and the general price level. The former represents profit, while the latter represents the cost of the pricing policy. Then, the ratio of the two components is the effect of the energy price lever, which includes consideration of energy conservation. In addition, Zhang et al. (2014a, b) show that once inflation is considered, a central bank’s monetary policy that aims to control inflation may also influence energy prices, which means that there is a transmission chain through “energy prices–price level–energy prices” in practice. Moreover, excluding the general price level in examining energy prices not only may reflect the real variation in energy prices but also may consider the cost of price lever regulation and even the endurance of the economy’s price system, which is more instructive in practice. It is through this mechanism that Amano (1990) indicates that rising real energy prices may result in energy savings. In general, the inflation cost in this paper fundamentally includes two aspects: first, according to the analysis described above, inflation itself is a cost of pricing policy; second, inflation may result in many costs, such as shoe-leather costs, menu costs and tax distortions. In this paper, we do not focus on the specific costs in detail. We only assume that all costs are reflected by the general price level. Then, the question becomes, how can the general price level be excluded in examining energy prices? Using the relationship between nominal variables and real variables (such as the nominal and real interest rate, the nominal and real product price index, and nominal and real energy prices) as a reference (Yamada 2002; Shen and Wang 2000; Yang 2009),^{Footnote 5} we first consider the difference between energy prices and the general price level. However, the difference between the two indices may be negative, which prevents the use of the natural logarithm and influences the models in empirical research, and the absolute value of the difference does not reflect rising and falling price variations. Therefore, the ratio between the two indices can reasonably be adopted. Based on the foregoing, we utilize relative prices to reflect the real energy prices such that the inflation cost of energy price variations is excluded, and the effects of relative energy prices on energy consumption are therefore analyzed.

### Theoretical relationship between energy price and energy consumption

According to general commodities theory, the relationship between energy prices and energy consumption essentially belongs to the “price-demand” research framework. This relationship can be interpreted based on two components, the factors involved and market equilibrium. As for the factors involved, energy prices are important factors in production, and an increasing energy price can thus result in increasing costs for related products, whereas a high energy consumption industry may accelerate its industrial transition and ultimately lead to decreasing energy consumption. Furthermore, to operate continuously and maintain a profit margin and a sustainable competitive advantage, enterprises may increase their technology input and actively search for alternative energy sources. As for market equilibrium, energy demand in the industrialization process is rigid, and energy supply and demand is imbalanced. Thus, rising energy prices can stimulate energy enterprises to expand the scale of their production and sales, which is bad for energy conservation.

In the real economy, energy is both a factor input and also is general merchandise. Energy prices can regulate energy consumption by influencing the economy in the aggregate, the economic structure, and energy efficiency. A correlation between energy prices and the economy in the aggregate has been confirmed by many scholars, although whether it is positive or negative is inconsistent and depends on different economic conditions (Jin et al. 2009; Berk and Yetkiner 2014; Bretschger 2015). As for the path of the economic structure, Hu et al. (2008) indicate that energy prices not only can lead to changes in the economic structure but also can influence the internal industrial structure. This influence can be confirmed through the theory of sector transfer. As for energy prices and energy efficiency, rising energy prices generally stimulate technological innovation and decrease energy consumption, which was summarized in the introduction of this paper. Finally, it must be stressed that because energy intensity is the ratio of total energy consumption to GDP—and because energy prices can impact both of these components (at least theoretically)—energy prices may certainly influence energy intensity.

### The basic model

#### The direct effects of relative energy prices on energy consumption

Many factors affect energy consumption. Existing studies mainly conclude that factors influencing energy consumption include both productive factors, such as output, the industrial structure, and technological efficiency, and consumptive factors, such as population, and per capita wealth. Based on the Kaya equation and LMDI decomposition (Ang and Liu 2007; Ma and Stern 2008; Ang 2015), we have:^{Footnote 6}

$$CO_{2} \,emissions = C_{e} \times E_{i} \times Y_{p} \times P_{o}$$

(5)

where *C*
_{
e
}, *E*
_{
i
}, *Y*
_{
p
} and *P*
_{
o
} are the carbon content of the energy, energy intensity, GDP per capita and population, respectively. This paper mainly focuses on the productive factors, and then we consider the carbon content of energy, energy intensity and GDP. ^{Footnote 7} In addition, studies of factors affecting the carbon content of energy, energy intensity and GDP (we cited the related studies in the background section) mainly focus on technology innovation and the industrial and energy structure; meanwhile, the energy price is taken as a driving factor of technology innovation and the industrial and energy structure. Based on the above analysis, we finally have:

$$Energy\,{\kern 1pt} consumption = Y \times E \times S \times T \times P$$

(6)

where *Y*, *E*, *S*, *T* and *P* are the aggregate output, the energy structure, the industrial structure, the technological level and energy prices (relative energy prices in this paper). These parameters have been observed to directly affect energy consumption. The direct effect model is shown in Eq. (7). To eliminate heteroskedasticity and directly obtain elasticities, natural logarithms of the variables are used in the model. *C*
_{1} and *C*
_{2} represent total energy consumption and energy intensity, respectively.

$$lnC_{i} = \alpha + \beta_{1} lnY + \beta_{2} lnS + \beta_{3} lnE + \beta_{4} lnT + \beta_{5} lnP$$

(7)

From a practical perspective, the most significant flaw of Model (7) is that price is treated as the central mechanism of resource allocation. This model aims to measure the direct influence of the related variables, especially relative energy prices, on energy consumption. Moreover, prices could also affect energy consumption through their influence on variables such as the aggregate output, the industrial structure, and the energy structure. Thus, the key is to highlight the regulatory effects of relative energy prices on energy consumption.

#### The regulatory effects of relative energy prices on energy consumption

Based on the analysis in the section of “Theoretical relationship between energy price and energy consumption” and fully considering the regulation of relative energy prices on energy consumption, the model might be rewritten as (8):

$$\begin{aligned} lnCi &= \alpha + \beta 1ln(P \times Y) + \beta 2ln(P \times S) + \beta 3ln(P \times E) \\ &\quad + \beta 4lnT + \beta 5lnP + \beta 6lnY + \beta 7lnS + \beta 8lnE \end{aligned}$$

(8)

In Eq. (8), the relative price of energy might influence energy consumption by regulating related variables. The cross-product items represent the regulatory process. They aim to measure the indirect effect of relative energy prices on energy consumption through the influences of intermediary variables such as aggregate output, the industrial structure, and the energy structure. Obviously, there is significant multicollinearity in this model under the general regression method. However, in view of their economic significance, the above variables must be included in the model. Thus, the ridge regression method is utilized to analyze the actual operations.

#### The time-varying effects of relative energy prices on energy consumption

Models (7) and (8) capture the effects of energy prices on average energy consumption. However, the influence of energy prices on energy consumption exhibits a remarkable time-varying characteristic with variations in the macroeconomic environment and policy conditions. Thus, a state-space model (SSM) may better reflect the dynamic nature of these relationships. It should be noted, however, that SSMs have difficulties with multicollinearity. Therefore, the regulating variables are not included in Model (9). Among the variables included in the empirical study, some must be removed based on causality tests. The model aims to capture the dynamic effect (time-varying coefficients) of relative energy prices on energy consumption.

$$\begin{aligned} lnC_{it} & = C + \beta_{1,t} lnY_{t} + \beta_{2,t} lnS_{t} + \beta_{3,t} lnE_{t} + \beta_{4,t} lnT_{t} + \beta_{5,t} lnP_{t} \mu_{t} \\ \beta_{1,t} & = C(1) \times \beta_{1,t - 1} + \varepsilon_{1,t} ,\beta_{2,t} = C\left( 2 \right) \times \beta_{2,t - 1} + \varepsilon_{2,t} \\ \beta_{3,t} & = C(3) \times \beta_{3,t - 1} + \varepsilon_{3t} ,\beta_{4,t} = C\left( 4 \right) \times \beta_{4,t - 1} + \varepsilon_{4t} \\ \beta_{5,t} & = C(3) \times \beta_{5,t - 1} + \varepsilon_{5t} \\ \end{aligned}$$

(9)