The DSGE mannequin consists of six economic system sectors: the family sector, the intermediate items sector, the ultimate items sector, the fossil gas know-how sector, the renewable power know-how sector, and the federal government sector. Determine 1 illustrates the fundamental construction of the DSGE mannequin, the place the dashed arrows point out flows of supplies whereas the strong arrows denote flows of funds. The family sector is the proprietor of the elements of manufacturing, and it provides labor, capital, and power to the varied manufacturing sectors and receives revenue. Power applied sciences produced by the fossil gas know-how sector and the renewable power know-how sector additionally enter the manufacturing strategy of the intermediate items sector. The intermediate items sector produces intermediate items for provide to the ultimate items sector. The ultimate items sector produces ultimate items, that are equipped to the family sector for consumption, and pays income to the family sector. The federal government sector collects income via taxes, partly to buy ultimate merchandise, and partly to switch funds to the family sector. We additionally incorporate pollutant emission, power productiveness, and carbon discount coverage elements into the DSGE mannequin framework. On this part, we are going to focus on this framework intimately.

Probably the most important distinction between this examine and former research is the introduction of the power know-how sector within the DSGE framework, which is subdivided into the fossil power know-how sector and the renewable power know-how sector. This not solely permits us to discover the heterogeneous results of various power applied sciences on financial growth and carbon emission discount, but in addition to simulate the impacts of carbon emission discount insurance policies on totally different power applied sciences.

### Households

The family sector is postulated to be homogeneous, with possession of capital (({Okay}_{t})), labor (({L}_{t})), fossil gas (({M}_{t}^{FF})), and renewable power (({M}_{t}^{RE})). In response to Xiao et al. (2018), the consultant agent derives constructive utility from consumption however adverse utility from each labor and power utilization. The aim of the consultant family is to maximise its lifetime utility as Eq. (1).

$$max ell =mathop{sum }limits_{t=0}^{infty }{beta }^{t}left{{mathrm{ln}},{C}_{t}-frac{{L}_{t}^{1+gamma }}{1+gamma }-frac{chi {({M}_{t}^{RE})}^{1+upsilon }}{1+upsilon }-frac{{[(1-e{r}_{t})mu {M}_{t}^{FF}]}^{1+nu}}{1+nu}proper}$$

(1)

the place (e{r}_{t}) is the proportion of emission reductions, (0,< ,beta ,< ,1) is the low cost issue, (gamma ge 0), (upsilon ge 0) and (nu ge 0) are the inverse of the elasticity of labor provide, renewable power provide, and fossil gas provide, (chi ,> ,0) is the coefficient of the disutility of the renewable power, and *μ* is the emission coefficient. Consequently, the intertemporal finances constraint is expressed when it comes to items as Eq. (2).

$$start{array}{c}{P}_{t}{C}_{t}+{P}_{t}{I}_{t}+{P}_{t}{B}_{t}le (1-{tau }_{t}^{Okay}){R}_{t}{Okay}_{t-1}+(1-{tau }_{t}^{L}){W}_{t}{L}_{t}+(1-{tau }_{t}^{FF}){P}_{t}^{FF}{M}_{t}^{FF} +,{P}_{t}^{RE}{M}_{t}^{RE}+(1+{R}_{t}^{B}){B}_{t-1}+T{r}_{t}finish{array}$$

(2)

The consultant family owns the companies, and funds for capital (({R}_{t})), labor (({W_{t}})), fossil gas (({P}_{t}^{FF})), and renewable power (({P}_{t}^{RE})) are acquired by the consultant family from the intermediate items sector, fossil gas know-how sector, and renewable power know-how sector. And households must pay a proportion of taxes to the federal government when it receives cost from capital (({tau }_{t}^{Okay})), labor (({tau }_{t}^{L})), and fossil gas (({tau }_{t}^{FF})). As well as, the federal government pays a lump-sum switch ((T{r}_{t})) to the consultant family and levies taxes at various charges on issue incomes. After receiving revenue and switch funds, the family allocates these assets in direction of consumption (({C}_{t})), funding (({I}_{t})) and monetary actions comparable to buying authorities bonds (({B}_{t})), and the family receives curiosity at bond rate of interest (({R}_{t}^{B})).

In response to Dixit and Pindyck (1994), we incorporate Generalized Quadratic (GQ) funding adjustment prices into our mannequin, that are a vital part of latest DSGE fashions (Smets and Wouters, 2007). On the interval *t*, the consultant family holds capital and makes investments. The funding expression is as Eq. (3).

$${Okay}_{t+1}=(1-{delta }_{Okay}){Okay}_{t}+left[1-frac{vartheta }{2}{left(frac{{I}_{t}}{{I}_{t-1}}-1right)}^{2}right]{I}_{t}$$

(3)

the place ({delta }_{Okay}) donates the capital depreciation price, ([1-tfrac{vartheta }{2}{(tfrac{{I}_{t}}{{I}_{t-1}}-1)}^{2}]{I}_{t}) refers back to the capital adjustment value, and (vartheta) represents the corresponding capital adjustment value coefficient.

### Enterprises

#### Power know-how sector

Within the DSGE framework, our mannequin separates the manufacturing sector into the know-how manufacturing sector and the great manufacturing sector, and additional divides power know-how module into fossil gas know-how producers and renewable power know-how producers. Fossil gas know-how innovation will enhance the effectivity of fossil gas burning, thereby decreasing carbon emissions, and bettering manufacturing effectivity. In the meantime, the innovation of renewable power know-how can cut back its prices, improve its comparative benefit to a sure extent, and enhance the utilization of renewable power by complementary impact to realize the aim of optimizing the power consumption construction. Power know-how is the output of the 2 power know-how sectors and the manufacturing elements of the intermediate items sector. In our analysis, we focus on one consultant enterprise *j*.

We draw inspiration from Rivera-Batiz and Romer (1991), and set up the manufacturing operate for the power know-how sector as a Cobb-Douglas manufacturing operate, encompassing each capital and labor inputs. The consultant fossil gas know-how producer donates the capital ({Okay}_{t}^{FF}) and the labor ({L}_{t}^{FF}) to supply fossil gas know-how, the manufacturing operate of the consultant fossil gas know-how producer is as Eq. (4). The target of the consultant fossil gas know-how producer is to maximise its revenue as Eq. (5).

$$T{E}_{t}^{FF}(j)={A}_{t}^{T}{[{K}_{t}^{FF}(j)]}^{{alpha }_{FF}}{[{eta }_{t}^{L}{L}_{t}^{FF}(j)]}^{1-{alpha }_{FF}}$$

(4)

$$max {varPi }_{t}^{FF}(j)={P}_{t}^{TEFF}(j)T{E}_{t}^{FF}(j)-{W}_{t}{L}_{t}^{FF}(j)-{R}_{t}{Okay}_{t}^{FF}(j)$$

(5)

the place ({P}_{t}^{TEFF}) is the worth of the fossil gas know-how, ({A}_{t}^{T}) is the power know-how analysis and growth effectivity, and it follows an AR(1) course of as Eq. (6).

$$start{array}{cc}{mathrm{ln}},{A}_{t}^{T}-,{mathrm{ln}},{A}^{T}={rho }_{AT},{mathrm{ln}},{A}_{t-1}^{T}-{rho }_{AT},{mathrm{ln}},{A}^{T}+{varepsilon }_{t,{A}^{T}} & {varepsilon }_{t,{A}^{T}}mathop{ sim }limits^{i.i.d.}N(0,{sigma }_{AT}^{2})finish{array}$$

(6)

Equally, the consultant renewable power know-how producer donates the capital ({Okay}_{t}^{RE}) and the labor ({L}_{t}^{RE}) to supply renewable power know-how. The manufacturing operate of the consultant fossil gas know-how producer is as Eq. (7). The target of the consultant renewable power know-how producer is to maximise its revenue as Eq. (8).

$$T{E}_{t}^{RE}(j)={A}_{t}^{T}{[{K}_{t}^{RE}(j)]}^{{alpha }_{RE}}{[{eta }_{t}^{L}{L}_{t}^{RE}(j)]}^{1-{alpha }_{RE}}$$

(7)

$$max {varPi }_{t}^{RE}(j)={P}_{t}^{TERE}(j)T{E}_{t}^{RE}(j)-{W}_{t}{L}_{t}^{RE}(j)-{R}_{t}{Okay}_{t}^{RE}(j)$$

(8)

the place ({P}_{t}^{TERE}) represents the worth of the renewable power know-how. The power applied sciences produced by two power know-how sectors proceed to take part within the manufacturing chain of intermediate items as elements of manufacturing.

#### Intermediate items sector

In response to Dixit and Stiglitz (1977), there exists a mess of ultimate items producers who function inside a superbly aggressive market and depend on intermediate items to craft their ultimate items. These ultimate items producers make use of a manufacturing operate that constantly displays fixed returns to scale. In distinction, the intermediate items producers interact in competitors beneath monopolistic circumstances inside their respective product markets, with no management over issue costs. This agent employs labor ({L}_{t}^{Y}(j)), use capital ({Okay}_{t}^{Y}(j)), and purchases fossil gas ({M}_{t}^{FF}(j)) and renewable power ({M}_{t}^{RE}(j)) to fabricate intermediate items with the Cobb-Douglas know-how. The manufacturing operate of the consultant intermediate items producer is as Eq. (9).

$${Y}_{t}(j)={A}_{t}^{Y}{[{K}_{t}^{Y}(j)]}^{{alpha }_{Y}}{[{eta }_{t}^{L}{L}_{t}^{Y}(j)]}^{{Delta }_{Y}}{[{eta }_{t}^{FF}{M}_{t}^{FF}(j)]}^{{sigma }_{Y}}{[{eta }_{t}^{RE}{M}_{t}^{RE}(j)]}^{1-{alpha }_{Y}-{Delta }_{Y}-{sigma }_{Y}}$$

(9)

the place ({A}_{t}^{Y}) means the overall issue productiveness (TFP), which represents the extent of know-how in intermediate items manufacturing. And it additionally follows the AR(1) course of as Eq. (10).

$$start{array}{cc}{mathrm{ln}},{A}_{t}^{Y}-,{mathrm{ln}},{A}^{Y}={rho }_{{A}^{Y}},{mathrm{ln}},{A}_{t-1}^{Y}-{rho }_{{A}^{Y}},{mathrm{ln}},{A}^{Y}+{varepsilon }_{t,{A}^{Y}} & {varepsilon }_{t,{A}^{Y}}mathop{ sim }limits^{i.i.d.}N(0,{sigma }_{{A}^{Y}}^{2})finish{array}$$

(10)

In response to Jorgenson (1984), power effectivity is intently linked to the quantity of power know-how enter. Subsequently, we introduce fossil gas effectivity ({eta }_{t}^{FF}) and renewable power effectivity ({eta }_{t}^{RE}) into the manufacturing operate. Following “studying by doing” (LBD) proposed by Arrow (1962), we assume that the effectivity of power inputs is intently associated to the power know-how utilized in manufacturing.

$${eta }_{t}^{FF}={lambda }_{t}{(T{E}_{t}^{FF})}^{{omega }_{FF}-1}$$

(11)

$${eta }_{t}^{RE}={lambda }_{t}{(T{E}_{t}^{RE})}^{{omega }_{RE}-1}$$

(12)

the place ({lambda }_{t}) represents the variable denoting the enhancement of power know-how effectivity all through the LBD method. And it adheres the AR(1) course of as Eq. (13).

$$start{array}{cc}{mathrm{ln}},{lambda }_{t}-,{mathrm{ln}},lambda ={rho }_{lambda },{mathrm{ln}},{lambda }_{t-1}-{rho }_{lambda },{mathrm{ln}},lambda +{varepsilon }_{t,lambda } & {varepsilon }_{t,lambda }mathop{ sim }limits^{i.i.d.}N(0,{sigma }_{lambda }^{2})finish{array}$$

(13)

Consequently, the manufacturing operate of the consultant intermediate items producer could be expressed as Eq. (14).

$${Y}_{t}(j)={A}_{t}^{Y}{[{K}_{t}^{Y}(j)]}^{{alpha }_{Y}}{[{eta }_{t}^{L}{L}_{t}^{Y}(j)]}^{{Delta }_{Y}}{[{lambda }_{t}{(T{E}_{t}^{FF})}^{{omega }_{FF}-1}{M}_{t}^{FF}(j)]}^{{sigma }_{Y}}{[{lambda }_{t}{(T{E}_{t}^{RE})}^{{omega }_{RE}-1}{M}_{t}^{RE}(j)]}^{1-{alpha }_{Y}-{Delta }_{Y}-{sigma }_{Y}}$$

(14)

The consultant intermediate items enterprise emits pollution, and *μ* is the emission coefficient. Beneath the stress of environmental rules and pollutant emission prices, the enterprise makes efforts to cut back emissions, the place the emission discount ratio is denoted as (e{r}_{t}(j)), whose measurement is determined by the enter of fossil gas know-how (T{E}_{t}^{FF}). Subsequently, the emission discount ratio (e{r}_{t}(j)), pollutant emissions ({Z}_{t}(j)) and emission reductions (R{E}_{t}(j)) could be expressed as follows:

$$e{r}_{t}(j)=phi cdot T{E}_{t}^{FF}(j)$$

(15)

$${Z}_{t}(j)=mu (1-e{r}_{t}(j)){M}_{t}^{FF}(j)$$

(16)

$$R{E}_{t}(j)=mu cdot e{r}_{t}(j){M}_{t}^{FF}(j)$$

(17)

In response to the analysis of Nguyen (2023), the connection between emission reductions and the fee related to these reductions could be roughly expressed as a quadratic operate, which is illustrated as Eq. (18).

$$C{E}_{t}(j)={phi }_{0}+{phi }_{1}R{E}_{t}(j)+{phi }_{2}{(R{E}_{t}(j))}^{2}={phi }_{0}+{phi }_{1}mu cdot e{r}_{t}(j){M}_{t}^{FF}(j)+{phi }_{2}{mu }^{2}cdot {[e{r}_{t}(j)]}^{2}{({M}_{t}^{FF}(j))}^{2}$$

(18)

The labor effectivity coefficient (({eta }_{t}^{L})) is intently associated to the pollutant inventory ((S{T}_{t})), and in accordance with Xiao et al. (2018) and Heutel (2012), we set the labor effectivity proven as Eq. (19), the place ({eta }_{0}), ({eta }_{1}) and ({eta }_{2}) are the harm operate parameters. As a result of accumulation strategy of pollutant inventory over time, it’s assumed that the pollutant inventory of any two intervals follows the next relationship proven in Eq. (20), the place ({delta }_{Z}) represents the depreciation price of pollutant inventory.

$${eta }_{t}^{L}=1-({eta }_{0}+{eta }_{1}S{T}_{t}+{eta }_{2}S{T}_{t}^{2})$$

(19)

$$S{T}_{t}=(1-{delta }_{Z})S{T}_{t-1}+{Z}_{t}$$

(20)

The full manufacturing value confronted by enterprises, along with the price of capital, labor, power, and power know-how, additionally consists of the price of decreasing carbon emissions. Subsequently, the Lagrange operate for maximizing the income confronted by enterprises within the manufacturing course of could be decided as Eq. (21).

$$start{array}{l}start{array}{l}max mathop{prod }limits_{t}^{Y}(j)=frac{{P}_{t}(j)}{{P}_{t}}{Y}_{t}(j)-frac{{W}_{t}}{{P}_{t}}{L}_{t}^{Y}(j)-frac{{R}_{t}}{{P}_{t}}{Okay}_{t}^{Y}(j)-frac{{P}_{t}^{FF}}{{P}_{t}}{M}_{t}^{FF}(j)-frac{{P}_{t}^{RE}}{{P}_{t}}{M}_{t}^{RE}(j) qquadqquadquad-frac{{P}_{t}^{TEFF}}{{P}_{t}}T{E}_{t}^{FF}-frac{{P}_{t}^{TERE}}{{P}_{t}}T{E}_{t}^{RE}-frac{{P}_{t}^{Z}}{{P}_{t}}[mu (1-e{r}_{t}(j)){M}_{t}^{FF}(j)]-C{E}_{t}(j)finish{array} start{array}{ll}s.t. ,{Y}_{t}(j)={A}_{t}^{Y}{[{K}_{t}^{Y}(j)]}^{{alpha }_{Y}}{[{eta }_{t}^{L}{L}_{t}^{Y}(j)]}^{{Delta }_{Y}}{[{eta }_{t}^{M}{M}_{t}^{FF}(j)]}^{{sigma }_{Y}}{[{eta }_{t}^{M}{M}_{t}^{RE}{(j)}_{t}]}^{1-{alpha }_{Y}-{Delta }_{Y}-{sigma }_{Y}}finish{array}finish{array}$$

(21)

To find out value changes, we undertake the tactic of Calvo (1983), positing that intermediate items companies can solely modify their nominal costs in response to a stochastic sign. The opportunity of such value changes occurring in any given interval is set by the parameter (1-omega), the place (omega) signifies the extent of value rigidity prevalent within the economic system.

$$mathop{max }limits_{{P}_{t}(j)}varPhi ={E}_{t}mathop{sum }limits_{i=0}^{infty }{(beta omega )}^{i}frac{U^{prime} ({C}_{t+i})}{U^{prime} ({C}_{t})}{Y}_{t+i}left[frac{{P}_{t}(j)}{{P}_{t+i}}{left(frac{{P}_{t+i}}{{P}_{t}(j)}right)}^{phi }-M{C}_{t+i}{left(frac{{P}_{t+i}}{{P}_{t}(j)}right)}^{phi }right]$$

(22)

#### Ultimate items sector

The consultant ultimate items producer makes use of ({Y}_{t}(j)) items of every intermediate good (jin [0,1]) to fabricate the ultimate good ({Y}_{t}), following Dixit and Stiglitz (1977) assumption of fixed returns to scale, aggressive companies produce ({Y}_{t}) utilizing a Fixed Elasticity of Substitution (CES) know-how, the place (varphi, >, 1) denotes the elasticity of substitution between intermediate items.

$${Y}_{t}={left[{int }_{0}^{1}{Y}_{t}{(j)}^{frac{varphi -1}{varphi }}djright]}^{frac{varphi }{varphi -1}}$$

(23)

The consultant ultimate items producer seeks to maximise profitability by figuring out ({Y}_{t}(j)) and ({Y}_{t}), as expressed by Eq. (24).

$$mathop{max }limits_{{Y}_{t}(j)}{P}_{t}{left[{int }_{0}^{1}{Y}_{t}{(j)}^{frac{varphi -1}{varphi }}djright]}^{frac{varphi }{varphi -1}}-{int }_{0}^{1}{Y}_{t}(j){P}_{t}(j)dj$$

(24)

the place ({P}_{t}) represents the worth of the ultimate good, and ({P}_{t}(j)) represents the worth of the intermediate good (j). The primary-order situation generates the demand operate for the intermediate items are proven in Eq. (25) and Eq. (26), and we will clearly see that the worth of ultimate items ({P}_{t}) can be a mirrored image of the worth stage.

$${Y}_{t}(j)={left(frac{{P}_{t}(j)}{{P}_{t}}proper)}^{-varphi }{Y}_{t}$$

(25)

$${P}_{t}={left[{int }_{0}^{1}{P}_{t}{(j)}^{1-varphi }djright]}^{frac{1}{1-varphi }}$$

(26)

### Authorities

The financing of public consumption ({G}_{t}) comes from the taxes of labor, capital and fossil gas, in addition to the charges charged for pollutant emission permits. And the federal government adjusts lump-sum transfers (T{r}_{t}) in a passive method to make sure finances equilibrium in every interval. So, the federal government finances constraint could be represented as Eq. (27).

$${P}_{t}{G}_{t}+(1+{R}_{t-1}^{B}){B}_{t-1}+T{r}_{t}le {tau }_{t}^{L}{W}_{t}{L}_{t}+{tau }_{t}^{Okay}{R}_{t}{Okay}_{t}+{tau }_{t}^{M}{P}_{t}^{FF}{M}_{t}^{FF}+{P}_{t}^{Z}{Z}_{t}+{B}_{t}$$

(27)

### Aggregation and market clearing

Following Calvo (1983), we outline the worth dispersion as given in Eq. (28), and the manufacturing operate could be written as Eq. (29).

$${V}_{t}=(1-psi ){left(frac{{P}_{t}^{ast }}{{P}_{t}}proper)}^{-varphi }+psi {left(frac{{P}_{t}}{{P}_{t-1}}proper)}^{varphi }{V}_{t-1}$$

(28)

$${Y}_{t}={A}_{t}^{Y}{({Okay}_{t}^{Y})}^{{alpha }_{Y}}{({eta }_{t}^{L}{L}_{t}^{Y})}^{{Delta }_{Y}}{({eta }_{t}^{M}{M}_{t}^{FF})}^{{sigma }_{Y}}{({eta }_{t}^{M}{M}_{t}^{RE})}^{1-{alpha }_{Y}-{Delta }_{Y}-{sigma }_{Y}}{({V}_{t})}^{-1}$$

(29)

To characterize the long-term equilibrium of the mannequin, this examine assumes that the commodity market within the mannequin system is in long-term equilibrium, the market-clearing situation is outlined as Eq. (30).

$${Y}_{t}={C}_{t}+{I}_{t}+{G}_{t}+C{E}_{t}$$

(30)