The DSGE model includes six economy sectors: the household sector, the intermediate goods sector, the final goods sector, the fossil fuel technology sector, the renewable energy technology sector, and the government sector. Figure 1 illustrates the basic structure of the DSGE model, where the dashed arrows indicate flows of materials while the solid arrows denote flows of funds. The household sector is the owner of the factors of production, and it supplies labor, capital, and energy to the various production sectors and receives income. Energy technologies produced by the fossil fuel technology sector and the renewable energy technology sector also enter the production process of the intermediate goods sector. The intermediate goods sector produces intermediate goods for supply to the final goods sector. The final goods sector produces final goods, which are supplied to the household sector for consumption, and pays profits to the household sector. The government sector collects revenue through taxes, partly to purchase final products, and partly to transfer payments to the household sector. We also incorporate pollutant emission, energy productivity, and carbon reduction policy components into the DSGE model framework. In this section, we will discuss this framework in detail.
The framework of the DSGE model.
The most significant difference between this study and previous studies is the introduction of the energy technology sector in the DSGE framework, which is subdivided into the fossil energy technology sector and the renewable energy technology sector. This not only allows us to explore the heterogeneous effects of different energy technologies on economic development and carbon emission reduction, but also to simulate the impacts of carbon emission reduction policies on different energy technologies.
Households
The household sector is postulated to be homogeneous, with ownership of capital (\(K_t\)), labor (\(L_t\)), fossil fuel (\(M_t^FF\)), and renewable energy (\(M_t^RE\)). According to Xiao et al. (2018), the representative agent derives positive utility from consumption but negative utility from both labor and energy usage. The goal of the representative household is to maximize its lifetime utility as Eq. (1).
$$\max \ell =\mathop\sum \limits_t=0^\infty \beta ^t\left\\mathrmln\,C_t-\fracL_t^1+\gamma 1+\gamma -\frac\chi (M_t^RE)^1+\upsilon 1+\upsilon -\frac[(1-er_t)\mu M_t^FF]^1+\nu1+\nu\right\$$
(1)
where \(er_t\) is the proportion of emission reductions, \(0\,< \,\beta \,< \,1\) is the discount factor, \(\gamma \ge 0\), \(\upsilon \ge 0\) and \(\nu \ge 0\) are the inverse of the elasticity of labor supply, renewable energy supply, and fossil fuel supply, \(\chi \,> \,0\) is the coefficient of the disutility of the renewable energy, and μ is the emission coefficient. Consequently, the intertemporal budget constraint is expressed in terms of goods as Eq. (2).
$$\beginarraycP_tC_t+P_tI_t+P_tB_t\le (1-\tau _t^K)R_tK_t-1+(1-\tau _t^L)W_tL_t+(1-\tau _t^FF)P_t^FFM_t^FF\\ +\,P_t^REM_t^RE+(1+R_t^B)B_t-1+Tr_t\endarray$$
(2)
The representative household owns the firms, and payments for capital (\(R_t\)), labor (\(W_t\)), fossil fuel (\(P_t^FF\)), and renewable energy (\(P_t^RE\)) are received by the representative household from the intermediate goods sector, fossil fuel technology sector, and renewable energy technology sector. And households need to pay a percentage of taxes to the government when it receives payment from capital (\(\tau _t^K\)), labor (\(\tau _t^L\)), and fossil fuel (\(\tau _t^FF\)). In addition, the government pays a lump-sum transfer (\(Tr_t\)) to the representative household and levies taxes at varying rates on factor incomes. After receiving income and transfer payments, the household allocates these resources towards consumption (\(C_t\)), investment (\(I_t\)) and financial activities such as purchasing government bonds (\(B_t\)), and the household receives interest at bond interest rate (\(R_t^B\)).
According to Dixit and Pindyck (1994), we incorporate Generalized Quadratic (GQ) investment adjustment costs into our model, which are a crucial component of contemporary DSGE models (Smets and Wouters, 2007). At the period t, the representative household holds capital and makes investments. The investment expression is as Eq. (3).
$$K_t+1=(1-\delta _K)K_t+\left[1-\frac\vartheta 2\left(\fracI_tI_t-1-1\right)^2\right]I_t$$
(3)
where \(\delta _K\) donates the capital depreciation rate, \([1-\tfrac\vartheta 2(\tfracI_tI_t-1-1)^2]I_t\) refers to the capital adjustment cost, and \(\vartheta\) represents the corresponding capital adjustment cost coefficient.
Enterprises
Energy technology sector
In the DSGE framework, our model separates the production sector into the technology production sector and the good production sector, and further divides energy technology module into fossil fuel technology producers and renewable energy technology producers. Fossil fuel technology innovation will improve the efficiency of fossil fuel burning, thereby reducing carbon emissions, and improving production efficiency. Meanwhile, the innovation of renewable energy technology can reduce its costs, enhance its comparative advantage to a certain extent, and improve the usage of renewable energy by complementary effect to achieve the purpose of optimizing the energy consumption structure. Energy technology is the output of the two energy technology sectors and the production factors of the intermediate goods sector. In our research, we concentrate on one representative enterprise j.
We draw inspiration from Rivera-Batiz and Romer (1991), and establish the production function for the energy technology sector as a Cobb-Douglas production function, encompassing both capital and labor inputs. The representative fossil fuel technology producer donates the capital \(K_t^FF\) and the labor \(L_t^FF\) to produce fossil fuel technology, the production function of the representative fossil fuel technology producer is as Eq. (4). The objective of the representative fossil fuel technology producer is to maximize its profit as Eq. (5).
$$TE_t^FF(j)=A_t^T[K_t^FF(j)]^\alpha _FF[\eta _t^LL_t^FF(j)]^1-\alpha _FF$$
(4)
$$\max \varPi _t^FF(j)=P_t^TEFF(j)TE_t^FF(j)-W_tL_t^FF(j)-R_tK_t^FF(j)$$
(5)
where \(P_t^TEFF\) is the price of the fossil fuel technology, \(A_t^T\) is the energy technology research and development efficiency, and it follows an AR(1) process as Eq. (6).
$$\beginarraycc\mathrmln\,A_t^T-\,\mathrmln\,A^T=\rho _AT\,\mathrmln\,A_t-1^T-\rho _AT\,\mathrmln\,A^T+\varepsilon _t,A^T & \varepsilon _t,A^T\mathop \sim \limits^i.i.d.N(0,\sigma _AT^2)\endarray$$
(6)
Similarly, the representative renewable energy technology producer donates the capital \(K_t^RE\) and the labor \(L_t^RE\) to produce renewable energy technology. The production function of the representative fossil fuel technology producer is as Eq. (7). The objective of the representative renewable energy technology producer is to maximize its profit as Eq. (8).
$$TE_t^RE(j)=A_t^T[K_t^RE(j)]^\alpha _RE[\eta _t^LL_t^RE(j)]^1-\alpha _RE$$
(7)
$$\max \varPi _t^RE(j)=P_t^TERE(j)TE_t^RE(j)-W_tL_t^RE(j)-R_tK_t^RE(j)$$
(8)
where \(P_t^TERE\) represents the price of the renewable energy technology. The energy technologies produced by two energy technology sectors continue to participate in the production chain of intermediate goods as factors of production.
Intermediate goods sector
According to Dixit and Stiglitz (1977), there exists a multitude of final goods producers who operate within a perfectly competitive market and rely on intermediate goods to craft their final goods. These final goods producers employ a production function that consistently exhibits constant returns to scale. In contrast, the intermediate goods producers engage in competition under monopolistic conditions within their respective product markets, with no control over factor prices. This agent employs labor \(L_t^Y(j)\), use capital \(K_t^Y(j)\), and purchases fossil fuel \(M_t^FF(j)\) and renewable energy \(M_t^RE(j)\) to manufacture intermediate goods with the Cobb-Douglas technology. The production function of the representative intermediate goods producer is as Eq. (9).
$$Y_t(j)=A_t^Y[K_t^Y(j)]^\alpha _Y[\eta _t^LL_t^Y(j)]^\Delta _Y[\eta _t^FFM_t^FF(j)]^\sigma _Y[\eta _t^REM_t^RE(j)]^1-\alpha _Y-\Delta _Y-\sigma _Y$$
(9)
where \(A_t^Y\) means the total factor productivity (TFP), which represents the level of technology in intermediate goods production. And it also follows the AR(1) process as Eq. (10).
$$\beginarraycc\mathrmln\,A_t^Y-\,\mathrmln\,A^Y=\rho _A^Y\,\mathrmln\,A_t-1^Y-\rho _A^Y\,\mathrmln\,A^Y+\varepsilon _t,A^Y & \varepsilon _t,A^Y\mathop \sim \limits^i.i.d.N(0,\sigma _A^Y^2)\endarray$$
(10)
According to Jorgenson (1984), energy efficiency is closely linked to the amount of energy technology input. Therefore, we introduce fossil fuel efficiency \(\eta _t^FF\) and renewable energy efficiency \(\eta _t^RE\) into the production function. Following “learning by doing” (LBD) proposed by Arrow (1962), we assume that the efficiency of energy inputs is closely related to the energy technology used in production.
$$\eta _t^FF=\lambda _t(TE_t^FF)^\omega _FF-1$$
(11)
$$\eta _t^RE=\lambda _t(TE_t^RE)^\omega _RE-1$$
(12)
where \(\lambda _t\) represents the variable denoting the enhancement of energy technology efficiency throughout the LBD approach. And it adheres the AR(1) process as Eq. (13).
$$\beginarraycc\mathrmln\,\lambda _t-\,\mathrmln\,\lambda =\rho _\lambda \,\mathrmln\,\lambda _t-1-\rho _\lambda \,\mathrmln\,\lambda +\varepsilon _t,\lambda & \varepsilon _t,\lambda \mathop \sim \limits^i.i.d.N(0,\sigma _\lambda ^2)\endarray$$
(13)
Consequently, the production function of the representative intermediate goods producer can be expressed as Eq. (14).
$$Y_t(j)=A_t^Y[K_t^Y(j)]^\alpha _Y[\eta _t^LL_t^Y(j)]^\Delta _Y[\lambda _t(TE_t^FF)^\omega _FF-1M_t^FF(j)]^\sigma _Y[\lambda _t(TE_t^RE)^\omega _RE-1M_t^RE(j)]^1-\alpha _Y-\Delta _Y-\sigma _Y$$
(14)
The representative intermediate goods enterprise emits pollutants, and μ is the emission coefficient. Under the pressure of environmental regulations and pollutant emission costs, the enterprise makes efforts to reduce emissions, where the emission reduction ratio is denoted as \(er_t(j)\), whose size depends on the input of fossil fuel technology \(TE_t^FF\). Therefore, the emission reduction ratio \(er_t(j)\), pollutant emissions \(Z_t(j)\) and emission reductions \(RE_t(j)\) can be expressed as follows:
$$er_t(j)=\phi \cdot TE_t^FF(j)$$
(15)
$$Z_t(j)=\mu (1-er_t(j))M_t^FF(j)$$
(16)
$$RE_t(j)=\mu \cdot er_t(j)M_t^FF(j)$$
(17)
According to the research of Nguyen (2023), the connection between emission reductions and the cost associated with those reductions can be approximately expressed as a quadratic function, which is illustrated as Eq. (18).
$$CE_t(j)=\phi _0+\phi _1RE_t(j)+\phi _2(RE_t(j))^2=\phi _0+\phi _1\mu \cdot er_t(j)M_t^FF(j)+\phi _2\mu ^2\cdot [er_t(j)]^2(M_t^FF(j))^2$$
(18)
The labor efficiency coefficient (\(\eta _t^L\)) is closely related to the pollutant stock (\(ST_t\)), and according to Xiao et al. (2018) and Heutel (2012), we set the labor efficiency shown as Eq. (19), where \(\eta _0\), \(\eta _1\) and \(\eta _2\) are the damage function parameters. Due to the accumulation process of pollutant stock over time, it is assumed that the pollutant stock of any two periods follows the following relationship shown in Eq. (20), where \(\delta _Z\) represents the depreciation rate of pollutant stock.
$$\eta _t^L=1-(\eta _0+\eta _1ST_t+\eta _2ST_t^2)$$
(19)
$$ST_t=(1-\delta _Z)ST_t-1+Z_t$$
(20)
The total production cost faced by enterprises, in addition to the cost of capital, labor, energy, and energy technology, also includes the cost of reducing carbon emissions. Therefore, the Lagrange function for maximizing the profits faced by enterprises in the production process can be determined as Eq. (21).
$$\beginarrayl\beginarrayl\max \mathop\prod \limits_t^Y(j)=\fracP_t(j)P_tY_t(j)-\fracW_tP_tL_t^Y(j)-\fracR_tP_tK_t^Y(j)-\fracP_t^FFP_tM_t^FF(j)-\fracP_t^REP_tM_t^RE(j)\\ \qquad\qquad\quad-\fracP_t^TEFFP_tTE_t^FF-\fracP_t^TEREP_tTE_t^RE-\fracP_t^ZP_t[\mu (1-er_t(j))M_t^FF(j)]-CE_t(j)\endarray\\ \beginarraylls.t. \,Y_t(j)=A_t^Y[K_t^Y(j)]^\alpha _Y[\eta _t^LL_t^Y(j)]^\Delta _Y[\eta _t^MM_t^FF(j)]^\sigma _Y[\eta _t^MM_t^RE(j)_t]^1-\alpha _Y-\Delta _Y-\sigma _Y\endarray\endarray$$
(21)
To determine price adjustments, we adopt the method of Calvo (1983), positing that intermediate goods firms can only modify their nominal prices in reaction to a stochastic signal. The possibility of such price adjustments occurring in any given period is determined by the parameter \(1-\omega\), where \(\omega\) signifies the level of price rigidity prevalent in the economy.
$$\mathop\max \limits_P_t(j)\varPhi =E_t\mathop\sum \limits_i=0^\infty (\beta \omega )^i\fracU^\prime (C_t+i)U^\prime (C_t)Y_t+i\left[\fracP_t(j)P_t+i\left(\fracP_t+iP_t(j)\right)^\phi -MC_t+i\left(\fracP_t+iP_t(j)\right)^\phi \right]$$
(22)
Final goods sector
The representative final goods producer uses \(Y_t(j)\) units of each intermediate good \(j\in [0,1]\) to manufacture the final good \(Y_t\), following Dixit and Stiglitz (1977) assumption of constant returns to scale, competitive firms produce \(Y_t\) using a Constant Elasticity of Substitution (CES) technology, where \(\varphi\, >\, 1\) denotes the elasticity of substitution between intermediate goods.
$$Y_t=\left[\int _0^1Y_t(j)^\frac\varphi -1\varphi dj\right]^\frac\varphi \varphi -1$$
(23)
The representative final goods producer seeks to maximize profitability by determining \(Y_t(j)\) and \(Y_t\), as expressed by Eq. (24).
$$\mathop\max \limits_Y_t(j)P_t\left[\int _0^1Y_t(j)^\frac\varphi -1\varphi dj\right]^\frac\varphi \varphi -1-\int _0^1Y_t(j)P_t(j)dj$$
(24)
where \(P_t\) represents the price of the final good, and \(P_t(j)\) represents the price of the intermediate good \(j\). The first-order condition generates the demand function for the intermediate goods are shown in Eq. (25) and Eq. (26), and we can clearly see that the price of final goods \(P_t\) is also a reflection of the price level.
$$Y_t(j)=\left(\fracP_t(j)P_t\right)^-\varphi Y_t$$
(25)
$$P_t=\left[\int _0^1P_t(j)^1-\varphi dj\right]^\frac11-\varphi $$
(26)
Government
The financing of public consumption \(G_t\) comes from the taxes of labor, capital and fossil fuel, as well as the fees charged for pollutant emission permits. And the government adjusts lump-sum transfers \(Tr_t\) in a passive manner to ensure budget equilibrium in each period. So, the government budget constraint can be represented as Eq. (27).
$$P_tG_t+(1+R_t-1^B)B_t-1+Tr_t\le \tau _t^LW_tL_t+\tau _t^KR_tK_t+\tau _t^MP_t^FFM_t^FF+P_t^ZZ_t+B_t$$
(27)
Aggregation and market clearing
Following Calvo (1983), we define the price dispersion as given in Eq. (28), and the production function can be written as Eq. (29).
$$V_t=(1-\psi )\left(\fracP_t^\ast P_t\right)^-\varphi +\psi \left(\fracP_tP_t-1\right)^\varphi V_t-1$$
(28)
$$Y_t=A_t^Y(K_t^Y)^\alpha _Y(\eta _t^LL_t^Y)^\Delta _Y(\eta _t^MM_t^FF)^\sigma _Y(\eta _t^MM_t^RE)^1-\alpha _Y-\Delta _Y-\sigma _Y(V_t)^-1$$
(29)
To characterize the long-term equilibrium of the model, this study assumes that the commodity market in the model system is in long-term equilibrium, the market-clearing condition is defined as Eq. (30).
$$Y_t=C_t+I_t+G_t+CE_t$$
(30)
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