2.1 Final good sector

At each date $t$ , the economy produces a flow $Y(t)$ of final good using a “renewable-based” input $Y_{r}(t)$ and a ”fossil-based” input $Y_{f}(t)$ , according to the CES aggregate production function

(1) \begin{equation} Y(t)=\left [ Y_{r}(t)^{(\varepsilon -1)/\varepsilon }+Y_{f}(t)^{(\varepsilon -1)/\varepsilon }\right] ^{\varepsilon /(\varepsilon -1)}, \end{equation}

where $\varepsilon \in (0,+\infty)$ is the elasticity of substitution between the renewable and fossil-based inputs.

The renewable and fossil-based subsectors respectively produce $Y_{r}(t)$ and $Y_{f}(t)$ according to the production functions

(2) \begin{align} Y_{r}(t) & =R^{1-\alpha }\int _{0}^{1}A_{ri}(t)^{1-\alpha }x_{ri}(t)^{\alpha }\,di, \end{align}

(3) \begin{align} Y_{f}(t) & =F(t)^{1-\alpha }\int _{0}^{1}A_{fi}(t)^{1-\alpha }x_{fi}(t)^{\alpha }\,di, \end{align}

with $0\lt \alpha \lt 1$ .Footnote ⁷ Note that, as in the standard DTC literature (following, e.g., Acemoglu et al. Reference Acemoglu, Aghion, Bursztyn and Hemous2012), we consider here Cobb-Douglas technologies; Lemoine (Reference Lemoine2024) generalizes this formulation by considering CES functions. $A_{r}$ is the stock of renewable-related knowledge and $A_{f}$ is the stock of fossil-related knowledge.

For each sector $j=r,f$ , there is a continuum of sector-specific intermediate goods indexed by $i\in \left [ 0,1\right]$ : $x_{ji}(t)$ denotes the quantity of intermediate good $i$ used in sector $j$ . For example, an intermediate good in the fossil-based sector $j=f$ may be a type of conventional gas engine; an intermediate good in the renewable-based sector $j=r$ may be a type of solar panel.

To produce the output $Y_{r}(t)$ , the renewable-based sector also uses a flow $R$ of renewable energy supposed to be non-polluting (e.g., biofuels, solar, and wind energies). For simplicity, we assume that this flow is constant over time as if it was produced from a constant flow of renewable labor energy. The fossil-based sector uses a flow of fossil resource, $F(t)$ , that we describe in more detail below (Subsection 2.3).

2.2 R&D and intermediate goods

2.2.1 Law of motion of knowledge

We show here that the law of motion of knowledge usually used in growth theory can be deduced from two assumptions: first, the probability of an innovation is a strictly concave function of the labor devoted to research; second, once an innovation occurs, the jump in quality of any intermediate good depends on the labor used in research and on the initial level of quality which prevails at the time of the innovation.

As previously mentioned, technical change is directed in the sense of Acemoglu (Reference Acemoglu2002): there are two R&D sectors, one “renewable-oriented” and one “fossil-oriented,” respectively improving the renewable and the fossil-based production sectors. These two R&D sectors are Schumpeterian as in Aghion and Howitt (Reference Aghion and Howitt1992): they improve the quality level $A_{ji}(t)$ of intermediate goods used in each sector. In each R&D sector $j=r,f$ , a number $L_{ji} (t)$ of atomistic scientists are dedicated to improving the quality level $A_{ji}(t)$ of intermediate good $i\in \lbrack 0,1]$ .

First, we assume that the instantaneous probability that these scientists produce an innovation at date $t$ is

(4) \begin{equation} \eta _{j}L_{ji}(t)^{\beta },\text{ with }0\lt \beta \lt 1. \end{equation}

In this probability, $\eta _{j}$ is a time-invariant and sector-specific parameter. As we will see below, due to the assumption of strict concavity in these stepping-on-toes effects, the research sectors are simultaneously active both at the interior steady state and outside of it (see Greaker et al. Reference Greaker, Heggedal and Rosendahl2018 for a similar assumption).

Second, in case such a success occurs in sector $j$ , the quality level $A_{ji}(t)$ rises by

(5) \begin{equation} \gamma A_{ji}(t)L_{ji}(t)^{\sigma },\text{ with }\gamma \gt 0\text{ and } \sigma \gt 0,\text{ } \end{equation}

so that the new version of the associated intermediate good is more productive; otherwise, that is absent any such success, $A_{ji}(t)$ remains unchanged. Thus, at any date $t\geq 0$ , given the contemporaneous quality level $A_{ji}(t)$ and the number of scientists $L_{ji}(t)$ , it can be established that the expected instantaneous rise in $A_{ji}(t)$ is given by the law of motion (see Appendix A for detailed computation) $\dot {A}_{ji}(t)=\gamma \eta _{j}L_{ji}(t)^{\beta +\sigma }A_{ji}(t),\ \forall j=r,f,\ \forall i\in \lbrack 0,1],$ with $A_{ji}(0)=A_{jk}(0)$ for all $j=r,f$ and any $i,k\in \lbrack 0,1]$ . In what follows, we will focus on the particular case where $\sigma =1-\beta$ , which yields the law of motion of knowledge that is standard in the fully endogenous growth literature:

(6) \begin{equation} \dot {A}_{ji}(t)=\gamma \eta _{j}L_{ji}(t)A_{ji}(t). \end{equation}

For reasons of simplicity, we have chosen to study a fully endogenous growth model. This assumption is undoubtedly debatable. In Appendix A, we explain that the methodology presented above can be generalized. In particular, we show that it is possible to introduce fishing-out effects in research, which leads to a semi-endogenous growth model in which Equation (6) becomes $\dot {A}_{ji}(t)=\gamma \eta _{j}L_{ji} (t)^{\beta }A_{ji}(t)^{\varphi },$ $0\lt \beta \lt 1$ , $\varphi \leq 1$ (see Kruse-Andersen, Reference Kruse-Andersen2023 for an analysis in this context). The complete analysis of such a specification of our model, particularly the study of non-steady state dynamics, is beyond the scope of the present paper.

2.2.3 Intermediate goods

At each date $t\geq 0$ , the intermediate good $x_{ji}(t)$ of quality level $A_{ji}(t)$ is produced according to the linear production function

(7) \begin{equation} x_{ji}(t)=\frac {1}{\psi }y_{ji}(t),\ j=r,f,\ i\in \left [ 0,1\right], \end{equation}

with $\psi \gt 0$ and where $y_{ji}(t)$ is an amount of final good.

2.3 The fossil resource

The production of the fossil-based sector’s output $Y_{f}(t)$ requires a flow $F(t)$ of fossil (polluting and non-renewable) resource. At each date $t$ , any flow of fossil resource use $F(t)$ yields a corresponding flow of carbon emissions. As we assume away abatement activities such as carbon sequestration, the use of fossil resource $F$ is proportional to carbon emissions. We will thus use the same variable, $F,$ for resource use and emissions in the remainder of the paper.

2.3.1 Interpretation of the Hotellian fossil resource

At each date, $F(t)$ is costlessly extracted from a limited stock $S(t)$ . We have $S(t)=S(0)-\int _{0}^{t}F(x)dx$ . By time-differentiating this equation, one obtains the law of motion of this stock of resource:

(8) \begin{equation} \dot {S}(t)=-F(t). \end{equation}

Here, we make the Hotellian assumption following which, at each date $t$ , the remaining cumulated resource extraction has an upper bound: $\int _{t}^{\infty }F(x)dx\leq S(t)$ at each date $t$ . This assumption of a physical limit to total fossil extraction is common in the growth and climate literature (including recent contributions, see, e.g., Hassler et al. Reference Hassler, Krusell and Olovsson2022 or Sriket and Suen, Reference Sriket and Suen2022) and it allows us to relate our results to common findings of the classical theoretical models. However, this assumption is criticized. Indeed, it is now commonly admitted that the whole stock of available fossil resource will not be exploited. Fully exhausting the existing reserves would yield an overall stock of carbon emissions that would create a climate disaster (IPCC, 2022). A significant share of the physically available stock of resource is thus likely to remain locked up in situ.Footnote ⁸

The Hotellian assumption can nevertheless be justified as follows. As avoiding a climate disaster implies to maintain the accumulated stock of carbon emissions below a certain level—commonly referred to as the carbon budget, this generally imposes a maximal stock of cumulated fossil resource extraction (which could be implemented with a cap-and-trade system allowing intertemporal tradeFootnote ⁹ ). Indeed, the rise in global temperature is largely determined by cumulative carbon emissions (see e.g., Dietz and Venmans, Reference Dietz and Venmans2019, who show that increases in temperature are a linear function of the accumulated stock of emissions). If we denote the accumulated stock of carbon by $Z(t)$ and the overall carbon ceiling above which a climate disaster occurs by $\bar {Z}$ , the carbon budget is given by $B(t)=\bar {Z}-Z(t)$ . The total quantity of resource that can be extracted after date $t$ , $\int _{t}^{\infty }F(x)dx$ , also represents the total carbon emissions over the period. If we assume that the carbon stock decay is nil, we have $Z(t)=Z(0)+\int _{0}^{t}F(x)dx$ for all $t$ , where $Z(0)$ is the initial stock of carbon—which one can see as its pre-industrial level. Thus, $Z(\infty)=Z(0)+\int _{0}^{t}F(x)dx+\int _{t}^{\infty }F(x)dx=Z(t)+\int _{t}^{\infty }F(x)dx\leq \bar {Z}$ implies $\int _{t}^{\infty }F(x)dx\leq B(t)$ . Hence, we can interpret $S(t)$ as the carbon budget of the economy: avoiding a climate disaster implies an upper bound to the cumulated extraction of the fossil resource.Footnote ¹⁰

Examining the intertemporal profile of carbon emissions is crucial, even in a framework where the carbon budget is asymptotically consumed but never exceeded. Indeed, in this case as well, the timing of emissions is determined by economic trade-offs. First, the standard trade-off between present and future consumption, and a second intertemporal trade-off focused on the timing of resource use and its climatic consequences. Consider the case in which cumulative emissions from time 0 to infinity equal the carbon budget. A time profile where all resource extraction and associated emissions are postponed to a distant future results in minimal climate impact but a significant decrease in consumption for the first generations; conversely, a time profile in which emissions at time 0 consume the entire budget, with no subsequent emissions, favors short-term consumption yet has a strong environmental impact on future generations. Indeed, climate science literature demonstrates that within a non-exceedance carbon budget scenario, higher short-term emissions—despite reducing future emissions—impose greater environmental costs,Footnote ¹¹ notably because of longer exposition to extreme climatic conditions. While catastrophic climate thresholds are avoided by not exceeding the carbon budget, elevated short-term emissions still exacerbate environmental degradation, including biodiversity loss, sea-level rise, ice melt, and forest dieback, with impacts that intensify when emissions are concentrated in the near term.

2.4 Representative household and basic constraints

The instantaneous utility function of the representative infinitely-lived household depends on both consumption, $C(t)$ , and the stock of carbon, $Z(t)$ .Footnote ¹² The intertemporal utility function is given by:

(9) \begin{equation} U=\int _{0}^{+\infty }\left [ \ln \big (C(t)\big)-\omega \left [ Z(t)\right] \right] e^{-\rho t}\,dt,\text{ }\rho \gt 0\text{ and }\omega \geq 0\text{.} \end{equation}

$\omega \left [ Z(t)\right]$ is the instantaneous disutility of the stock of carbon at date $t$ , with $d\omega \left [ Z(t)\right] /dZ(t)\gt 0$ . $\rho$ is the rate of time preferences.Footnote ¹³

At each date, a constant flow $L\gt 0$ of skilled labor is used by the two competing R&D sectors. Normalizing $L$ to unity, we have

(10) \begin{equation} \int _{0}^{1}L_{ri}(t)\,di+\int _{0}^{1}L_{fi}(t)\,di=1,\ \forall t\geq 0. \end{equation}

In this model, there is no explicit inclusion of unskilled labor in the production of final goods (including resource production). We make this modeling choice to keep the model tractable, allowing us to obtain closed-form solutions, and analyze the transition outside the steady state. Consequently, our study does not feature a research-production trade-off, a recurring characteristic in models with DTC models with climate considerations (see, for example, Casey, Reference Casey2024 or Hassler et al. Reference Hassler, Krusell and Olovsson2022). Nevertheless, we maintain an analytical framework that enables us to focus on our main objective: studying the impact of technological shocks on the direction of technical progress as well as on the carbon emissions path.

Last, the final good produced at each date $t\geq 0$ is either used for consumption or the production of renewable and fossil-related intermediate goods:

(11) \begin{equation} Y(t)=C(t)+\int _{0}^{1}y_{ri}(t)di+\int _{0}^{1}y_{fi}(t)di,\ \forall t\geq 0\text{.} \end{equation}

3.1 Final good sector

The final sector maximizes the following profit function:

(12) \begin{align} \pi _{Y}(t) & =\left [ Y_{r}(t)^{(\varepsilon -1)/\varepsilon }+Y_{f} (t)^{(\varepsilon -1)/\varepsilon }\right] ^{\varepsilon /(\varepsilon -1)}\nonumber \\ &\quad -\sum \limits _{j=r,f}\int _{0}^{1}p_{ji}(t)x_{ji}(t)\,di-p_{R}(t)R-p_{F} (t)F(t). \end{align}

The first-order conditions for the choice of input $F(t)$ and of the quantities $x_{ri}(t)$ and $x_{fi}(t)$ of renewable and fossil-related intermediate goods are:

(13) \begin{align} p_{F}(t) & =(1-\alpha)Y(t)^{1/\varepsilon }Y_{f}(t)^{(\varepsilon -1)/\varepsilon }/F(t)\equiv MP_{F}(t). \end{align}

(14) \begin{align} p_{ri}(t) & =\alpha Y(t)^{1/\varepsilon }Y_{r}(t)^{-1/\varepsilon }\left [ \frac {RA_{ri}(t)}{x_{ri}(t)}\right] ^{1-\alpha }\equiv MP_{x_{ri}} (t)\text{,} \end{align}

(15) \begin{align} p_{fi}(t) & =\alpha Y(t)^{1/\varepsilon }Y_{f}(t)^{-1/\varepsilon }\left [ \frac {F(t)A_{fi}(t)}{x_{fi}(t)}\right] ^{1-\alpha }\equiv MP_{x_{fi} }(t)\text{. } \end{align}

Each condition equates the input price to its marginal productivity and gives the production sector’s inverse demand for this input.

3.2 Intermediate goods sectors

We assume that, at each date $t\geq 0$ , all intermediate goods $x_{ji}(t)$ , $j=r,f$ , $i\in \lbrack 0,1]$ , are monopolistically supplied. By Equation (7), producing a quantity $x_{ji}(t)$ of intermediate good requires an amount $\psi x_{ji}(t)$ of final good. The profit derived from this activity thus writes

(16) \begin{equation} \pi _{ji}(t)=x_{ji}(t)\left [ p_{ji}(t)-\psi \right], \end{equation}

where $p_{ji}(t)$ is given by (14) and (15). In this context, monopoly prices $p_{ji}(t)$ exhibit a mark-up above the marginal cost $\psi$ :

(17) \begin{equation} p_{ji}(t)=\frac {\psi }{\alpha },\ \forall j=r,f,\ \forall i\in \lbrack 0,1]. \end{equation}

They are time-invariant, as well as independent of the sector $j=r,f$ to which they are dedicated and of the type of intermediate good $i\in \lbrack 0,1]$ , as is standard.

3.3 R&D sectors

In Acemoglu et al. (Reference Acemoglu, Aghion, Bursztyn and Hemous2012), where time is modeled as discrete, the innovator achieves a monopoly position that lasts for the shortest time interval in their analytical framework: one period. We adapt this assumption by considering that the monopoly position endures for a duration $dt$ . Specifically, when an innovation—generating a new, more productive type of intermediate good $x_{ji}(t)$ , with $j=r,f$ , and $i\in (0,1)-$ occurs at time $t\geq 0$ , the innovator benefits exclusively from the profit $\pi _{ji}(t)dt$ obtained from the sale of the new intermediate good over the interval $(t,t+dt)$ . This assumption of an infinitesimal patent length is a theoretical simplification that allows us to derive tractable results in both steady-state and transitional dynamics.Footnote ¹⁴ In the real world, it approximates industries characterized by rapid technological obsolescence or high competition, where firms can expect only brief periods of market exclusivity before competitors enter or new technologies emerge—such as technology or software, where the lifecycle of innovations is short. Thus, the value of an innovation at date $t$ —and therefore the value of the associated patent—is equal to the flows of profits earned over the period $(t,t+dt)$ . We have:

(18) \begin{equation} V_{ji}(t)=\pi _{ji}(t)dt, \end{equation}

where it follows from the analysis of the previous section (Equation (16) with equilibrium price (17)) that $\pi _{ji}(t)$ can be expressed as a linear function of $x_{ji}(t)$ only:

(19) \begin{equation} \pi _{ji}(t)=\frac {(1-\alpha)\psi }{\alpha }x_{ji}(t). \end{equation}

We know that at each date $t\geq 0$ , $L_{ji}(t)$ scientists are dedicated to improving the quality level of intermediate good $i\in \lbrack 0,1]$ used in sector $j=r,f$ , each having the instantaneous probability $\eta _{j} L_{ji}(t)^{\beta }$ of being the successful innovator. For any $j=r,f$ , and any $i\in \lbrack 0,1]$ , the total profit from R&D activities thus writes

(20) \begin{equation} \pi _{R\&Dji}(t)=\eta _{j}L_{ji}(t)^{\beta }V_{ji}(t)-w(t)L_{ji} (t), \end{equation}

where $V_{ji}(t)$ is given by Equation (18).

Maximizing this profit leads to the first-order condition $\beta \eta _{j} L_{ji}(t)^{\beta -1}V_{ji}(t)=w(t),$ which implies

(21) \begin{equation} \eta _{r}L_{ri}(t)^{\beta -1}V_{ri}(t)=\eta _{f}L_{fi}(t)^{\beta -1} V_{fi}(t)\text{.} \end{equation}

This equation simply means that the equilibrium marginal productivity of scientists is equalized everywhere they are allocated. $V_{ji}(t)$ is obtained from (18) and (19):

(22) \begin{equation} V_{ji}(t)=\frac {\psi (1-\alpha)}{\alpha }x_{ji}(t)dt. \end{equation}

As in the literature, the value of an innovation depends on two effects (see Acemoglu, Reference Acemoglu2002). First, the price effect, represented by the term $\frac {\psi (1-\alpha)}{\alpha }$ , which depends on both the production cost of the intermediate good (see Equation (7)) and parameter $\alpha$ , which characterizes the inverse demand function of this good (see Equations (14) and (15)). Second, the market size effect, that is, the quantity of the intermediate good $x_{ji}(t)$ sold.

From Equations (14), (15) and (17), we have $x_{ri}(t)=x_{r}(t)$ and $x_{fi} (t)=x_{f}(t),\forall i\in \lbrack 0,1].$ Then, from Equations (19) and (22), we also have $\pi _{ji}(t)=\pi _{j}(t)=\frac {(1-\alpha)\psi }{\alpha }x_{j}(t)$ and $V_{ji} (t)=V_{j}(t)=\pi _{j}(t)dt,\forall$ $j=r,f$ $,$ and $\forall$ $i\in \lbrack 0,1].$ Consequently, the first-order condition $\beta \eta _{j} L_{ji}(t)^{\beta -1}V_{j}(t)=w(t)$ implies $L_{ji}(t)=$ $L_{j}(t),\forall j=r,f,$ and $\forall i\in \lbrack 0,1].$

Since the ratio of intermediate-good uses, i.e., the relative market size, $x_{r}(t)/x_{f}(t)$ plays a central role in the following analysis, we will henceforth use the compact notation

\begin{equation*} \frac {x_{r}(t)}{x_{f}(t)}\equiv X(t). \end{equation*}

We observe here that the relative value of innovations is equal to the relative market size: $V_{r}(t)/V_{f}(t)=\pi _{r}(t)/\pi _{f} (t)=X(t)$ (see Equation (22)). The relative price effect does not appear in this equation because we have assumed that the production cost of intermediate goods is identical in both sectors and that the demand functions for these two types of goods from the final sector have the same price elasticity.

Furthermore, we have:

(23) \begin{equation} \left [ \frac {L_{r}(t)}{L_{f}(t)}\right] ^{1-\beta }=\frac {\eta _{r}x_{r} (t)}{\eta _{f}x_{f}(t)}\equiv \frac {\eta _{r}}{\eta _{f}} X(t). \end{equation}

This equation highlights that the direction of technical change, i.e., the relative effort in R&D, is determined by the relative market size, $X(t)$ . The interpretation is the following. An increase in the ratio of intermediate goods’ uses, $x_{r}(t)/x_{f}(t)\equiv X(t)$ , raises the relative profitability of the production of intermediate goods in the renewable-based sector (see Equation (19)). Research in the renewable-oriented sector thus becomes more profitable (see Equation (18)). As a result, renewable-oriented research drains relatively more labor: $L_{r}(t)/L_{f}(t)$ increases. This means, for example, that an increase in the use of wind turbines (for instance, due to a subsidy), while the use of coal-fired power plants remains fixed or decreases, reallocates the overall research effort in favor of renewable-oriented innovation.

Using Equations (10) and (23), we have:

(24) \begin{equation} L_{r}\left [ X(t)\right] =\frac {1}{\left [ \frac {\eta _{r}}{\eta _{f} }X(t)\right] ^{\frac {1}{\beta -1}}+1}. \end{equation}

Observe that $d\left [ L_{r}(X(t))\right] /d\left [ X(t)\right] \gt 0$ : $L_{r}(t)$ is an increasing function of $X(t)$ with an upper bound equal to $1$ . Equation (23) showed that the relative effort in renewable-oriented R&D, $L_{r}(t)/L_{f}(t)$ , is an increasing function of the intermediate goods ratio $X(t)$ . Since the total amount of labor in R&D in the economy is fixed, the total effort devoted to renewable-oriented R&D is also increasing with $X(t)$ .

With the earlier-made assumption that quality levels $A_{ji}(0)$ are initially equal across intermediates $i\in \lbrack 0,1]$ for each sector $j=r,f$ , the law of motion (6) implies that the quality levels $A_{ji}(t)$ in the symmetric equilibrium obey the same average dynamics within the renewable and fossil-oriented sectors. We have $A_{ji}(t)=A_{j} (t)$ , $j=r,f$ and $i\in \lbrack 0,1]$ . Thus Equation (6) writes:

(25) \begin{equation} \dot {A}_{j}(t)=\gamma \eta _{j}L_{j}(t)A_{j}(t),\ \forall j=r,f. \end{equation}

Finally, Equations (6) and (10) yield:

(26) \begin{equation} g_{A_{r}/A_{f}}\left [ X(t)\right] =\gamma \eta _{r}\left [ L_{r}\left [ X(t)\right] \right] -\gamma \eta _{f}\left [ 1-L_{r}\left [ X(t)\right] \right] \text{,} \end{equation}

where $L_{r}\left [ X(t)\right]$ is given by Equation (24).

Observe that the growth rate of the knowledge ratio $g_{A_{r}/A_{f}}\left [ X(t)\right]$ , that is, the direction of technical change (see, e.g., Acemoglu, Reference Acemoglu2002), is an increasing function of the effort in renewable-oriented R&D. From Equation (24), we can deduce that the direction of technical change is also an increasing function of the ratio of intermediate goods $X(t)$ . In other words, by making renewable-oriented R&D relatively more profitable, an increase in $X(t)$ draws labor from fossil-oriented research and reallocates it to renewable-oriented R&D; as a consequence, fossil-related knowledge $A_{f}(t)$ grows less fast and renewable-related knowledge $A_{r}(t)$ grows faster.

3.4 Resource sector

On the competitive natural resource market, we assume extraction costs to be nil. The maximization of the profit function

$\int _{t}^{+\infty }p_{F}(s)F(s)e^{-\int _{t}^{s}\bar {r}(u)du}ds$ , subject to $\dot {S}(s)=-F(s)$ , $S(s)\geq 0$ , $F(s)\geq 0,$ $s\geq t$ , yields the standard decentralized equilibrium “Hotelling rule” (Hotelling, Reference Hotelling1931):

(27) \begin{equation} g_{p_{_{F}}}(t)=\bar {r}(t)\text{ for all }t\text{.} \end{equation}

As usual, the transversality condition is $\lim _{t\rightarrow +\infty }S(t)=0$ .

3.5 Representative household

The household’s consumption-saving arbitrage determines the growth rate of consumption $C(t)$ . The maximization of the intertemporal utility function (9), subject to any intertemporal budget constraint arising under a perfect financial market, yields the standard Ramsey-Keynes condition: $g_{C}(t)=\bar {r}(t)-\rho.$ In Appendix C, we show that consumption is a linear function of output, from which we deduce:

(28) \begin{equation} g_{C}(t)=g_{Y}(t)=\bar {r}(t)-\rho. \end{equation}

4.1 Dynamics of carbon emissions and the intermediate-good ratio

Lemma 1 presents two key results. First, it expresses the dynamics of carbon emissions $F(t)$ by relating their growth rate to the growth rate of the intermediate-good ratio, $X(t)$ . Second, it provides the differential equation that governs the dynamics of $X(t)$ .

Lemma 1. The growth rate of carbon emissions $g_{F}(t)$ is a function of the level of the intermediate-good ratio $X(t)$ and of its growth rate $gX(t)$ :

(29) \begin{equation} g_{F}(t)=-\rho -\frac {X(t)}{1+X(t)}g_{(Y_{r}/Y_{f})^{\frac {\varepsilon -1}{\varepsilon }}}(t)=-\rho -\frac {X(t)}{1+X(t)}g_{X}(t)\text{.} \end{equation}

The differential equation that allows to study the dynamics of the intermediate-good ratio $X(t)$ is the following:

(30) \begin{align} g_{X}\left [ X(t)\right] & =\frac {1-\alpha }{\frac {\varepsilon } {\varepsilon -1}-\frac {\alpha +X(t)}{1+X(t)}}\left [ \rho +g_{A_{r}/A_{f}}\left [ X(t)\right] \right], \end{align}

with $g_{A_{r}/A_{f}}\left [ X(t)\right]$ given by Equation ( 26 ).

Let us briefly comment on the dynamics of $X(t)$ , which are characterized by Equation (30) and graphically illustrated in the phase diagram in Figure 1. The properties of this equation depend on the substitutability of the two sectors, that is, the value of parameter $\varepsilon$ .Footnote ¹⁵

Figure 1.

Phase diagram in the cases $\varepsilon \gt 1$ and $\varepsilon \lt 1$ .

If $\varepsilon \gt 1$ , we show in Appendix D.3 (Case A) that function $g_{X}(.)$ is an increasing function of $X(t)$ with a negative lower bound and a positive upper bound (see Figure 1). This implies that there exists an interior steady-state level $X_{SS}\gt 0$ such that $g_{X}(X_{SS})=0$ . Since $g_{X}\left [ X(t)\right] \gt 0$ for all $X(t)\gt X_{SS}$ and $g_{X}\left [ X(t)\right] \lt 0$ for all $X(t)\lt X_{SS}$ , this steady state is unstable. If $X(t)\lt X_{SS}$ , $X(t)$ decreases down to $0.$ If $X(t)\gt X_{SS}$ , $X(t)$ increases indefinitely.

If $\varepsilon \lt 1$ , we show in Appendix D.3 (Case B) that $g_{X}(.)$ is a decreasing function of $X(t)$ with a negative lower bound and a positive upper bound (see Figure 1). Thus, there exists $X_{SS}$ such that $g_{X}(X_{SS})=0$ . Here, since $g_{X}\left [ X(t)\right] \lt 0$ for all $X(t)\gt X_{SS}$ and $g_{X}\left [ X(t)\right] \gt 0$ for all $X(t)\lt X_{SS}$ , this steady state is stable.

In the two cases ( $\varepsilon \gt 1$ and $\varepsilon \lt 1$ ), the common interior steady state $X_{SS}\gt 0$ is thus such that $g_{A_{r}/A_{f}}(X_{SS})=-\rho$ (see Equation (30)). Then, $X_{SS}$ is an implicit solution of $g_{A_{r}/A_{f}}(X_{SS})=-\rho$ and Equation (24) $.$ Since $g_{X}(X_{SS})=0$ , from Equation (29), we also have $g_{F}(t)=-\rho$ in the steady state.

In Appendix D.3, we provide further comments on the dynamics of $X(t)$ . We analyze the overall (steady-state and non-steady-state) dynamics of the economy below in Subsection 4.2.

4.2 General equilibrium: definition and characterization

Before analyzing the impact of technology shocks, we first provide a definition and a characterization of the general equilibrium of the decentralized economy. In particular, we show that the entire dynamics of the economy are essentially determined by the dynamics of $X$ .

We now sum up the characterization of the general equilibrium in the following lemma.

Lemma 2. The equilibrium of the decentralized economy is characterized as follows.

1. The dynamics of variables $X(t)$ , $F(t)$ , $L_{r}(t)$ , $A_{r}(t)$ and $A_{f}(t)$ are given by Equations ( 30 ), ( 29 ), ( 24 ) and ( 25 ) respectively.

2. In the interior steady state, equilibrium quantities, growth rates, and prices are given by:

a) quantities and growth rates

\begin{align*} L_{r} & =\frac {\gamma \eta _{f}-\rho }{\gamma (\eta _{r}+\eta _{f})},\text{ } L_{f}=\frac {\gamma \eta _{r}+\rho }{\gamma (\eta _{r}+\eta _{f})},\text{ }F(t)=F(0)e^{-\rho t},\text{ }C(t)=(1-\alpha {{}^2})Y(t),\\ x_{r}(t) & =\frac {\alpha {{}^2} Y(t)}{\psi (1+H)},\text{ }x_{f}(t)=\frac {\alpha {{}^2} Y(t)}{\psi (1+1/H)},\text{ }X_{SS}\equiv \frac {x_{r}(t)}{x_{f}(t)}=\frac {1} {H},\text{ where }H=\frac {\eta _{r}}{\eta _{f}}\left ( \frac {L_{f}}{L_{r} }\right) ^{1-\beta },\\ \frac {A_{r}(t)}{A_{f}(t)} & =\frac {F(t)}{R}X_{SS}^{\frac {\varepsilon (1-\alpha)+\alpha }{(1-\alpha)(\varepsilon -1)}},\text{ }g_{A_{r}/A_{f} }(t)=g_{F}(t)=-\rho \text{,}\\ g_{Y} & =g_{C}=g_{Y_{r}}=g_{Y_{f}}=g_{x_{r}}=g_{x_{f}}=g_{A_{r}}=\gamma \eta _{r}L_{r},\text{ and }g_{A_{f}}=\gamma \eta _{f}L_{f}=\gamma \eta _{r} L_{r}+\rho. \end{align*}

b) prices

\begin{align*} & p_{j}=\frac {\psi }{\alpha },\text{ for }j=r,f,\text{ }p_{F}(t)=p_{F} (0)e^{\bar {r}t},\text{ }\bar {r}=\gamma \eta _{r}L_{r}+\rho,\\ & \text{ and }w(t)=\frac {\beta \psi (1-\alpha)\eta _{j}}{\alpha }L_{j}^{\beta -1}x_{j}(t),\text{ for }j=r,f. \end{align*}

If $\varepsilon \gt 1$ , this steady state is unstable. If $\varepsilon \lt 1$ , it is stable.

We now provide a more detailed analysis of the dynamics of the economy, as presented in Lemma2.

5.1 The renewable and fossil-based production sectors are gross complements: $\varepsilon \lt 1$

In the case where the renewable and fossil-based production sectors are gross complements ( $\varepsilon \lt 1$ ), a technology shock at date $t$ has an impact on short-term carbon emissions that is presented in Proposition 1 below.

All formal elements regarding this case are presented in Appendix F. Here, we identify the transmission channel between the technological shock and its climate impact and we provide the main intuitions that are required to interpret the results. For simplicity, we focus on the case of a green technology shock.

As $\varepsilon \lt 1$ , the steady state $X_{SS}$ is stable (see Lemma2). We thus assume that the initial state of the economy is $X_{1}(t^{\ast })=X_{SS}$ . First, recall that before the technology shock, the ratio of intermediate-good marginal productivities, $MP_{x_{r}}/MP_{x_{f}}$ , is equal to one. At the date $t^{\ast }$ of the shock (where $A_{r}/A_{f}$ increases), $MP_{x_{r}}/MP_{x_{f}}$ gets lower since $\varepsilon \lt 1$ (see Equation (D.4)). Then, firms reduce $X$ to restore the equality of this ratio to one; we have $X_{2}(t^{\ast })\lt X_{1}(t^{\ast })$ . We know that the two R&D sectors are financed by profits made on the sales of intermediate goods. Therefore, the change in $X$ has a direct effect on this financing: it causes a reallocation of labor between the two sectors of research. That is the basis of the dynamics impacts of the green technology shock that we now examine. Formally, the decrease in $X$ induces a reduction of $L_{r}/L_{f}$ (see Equation (23)), that is, labor in renewable-oriented R&D is reduced, which promotes fossil-oriented R&D. This result is analogous to the reduction in energy efficiency R&D observed in Casey (Reference Casey2024) in his study of the rebound effect.

The last steps concern the impact of these instantaneous modifications on the dynamics of variables $X$ , $MP_{F}$ , and $F$ . For $t\gt t^{\ast }$ , the reduction of $L_{r}/L_{f}$ leads to a decrease in the growth rate of the knowledge stock ratio, $A_{r}/A_{f}$ : formally, we have $g_{A_{r}/A_{f}}(t)\lt -\rho$ —see Equation (26). Hence, this modifies the dynamics of $X$ (see Equation (30)): $g_{X_{2}}(t)$ is positive (the economy is on the left side of the phase diagram). This also changes the dynamics of $MP_{F}$ (see Equation (31)). In trajectory 1 (i.e., in steady state), we have $g_{MP_{F1}}(t)=\bar {r} (t)-\rho -g_{F1}(t)$ (since $g_{X1}(t)=0$ ). In trajectory 2, we have $g_{MP_{F2}}(t)=\bar {r}(t)-\rho -g_{F2}(t)-\frac {X_{2}(t)}{1+X_{2}(t)} g_{X2}(t)$ . Assume that, as a first step, firms in the final sector do not modify the dynamics of the resource (i.e., $g_{F2}(t)=g_{F1}(t)$ for any $t\gt t^{\ast }$ ). Since $g_{X2}(t)$ is positive, we have $g_{MP_{F2} }(t)\lt g_{MP_{F1}}(t)$ , for any $t\gt t^{\ast }$ . Thus, after the shock, firms anticipate that the resource becomes less profitable tomorrow (i.e., in the long term) than today (i.e., in the short term); they thus decide to use more resource today and less tomorrow. In other words, the growth rate of resource use (and hence the growth rate of carbon emissions), $g_{F}(t)$ , gets lower at any date $t$ (see Equation (29)). This is why short-term carbon emissions increase.

5.2 The renewable and fossil-based production sectors are gross substitutes: $\varepsilon \gt 1$

In the case where the renewable and fossil-based production sectors are gross substitutes ( $\varepsilon \gt 1$ ), a technology shock at date $t^{\ast }$ has an impact on short-term carbon emissions that is presented in Proposition 2 below and summarized in Table 1.

Table 1.

Impact of a green technology shock on short-term carbon emissions when $\varepsilon \gt 1$

Here also, we focus on the case of a green technology shock.

We know that, at the date $t^{\ast }$ of the shock (where $A_{r}/A_{f}$ increases), the ratio $MP_{x_{r}}/MP_{x_{f}}$ gets higher (the reasoning presented in the previous subsection goes in opposite ways since $\varepsilon \gt 1$ ). In this case, $X$ jumps up: $X_{2}(t^{\ast })\gt X_{1}(t^{\ast })$ .

The interior steady state being unstable since $\varepsilon \gt 1$ , two initial situations are possible, as shown in Subsection 4.2: either $X_{SS} \lt X_{1}(t^{\ast })$ (pre-shock R&D is renewable-oriented) or $X_{1}(t^{\ast })\lt X_{SS}$ (pre-shock R&D is fossil-oriented). In the first case ( $X_{SS}\lt X_{1}(t^{\ast })$ ), we necessarily have $X_{ss}\lt X_{1}(t^{\ast })\lt X_{2}(t^{\ast })$ after the shock. In the second case ( $X_{1}(t^{\ast })\lt X_{SS}$ ), we can have either $X_{1}(t^{\ast })\lt X_{ss}\lt X_{2}(t^{\ast })$ , that is, the shock makes R&D renewable-oriented, or $X_{1}(t^{\ast })\lt X_{2} (t^{\ast })\lt X_{ss}$ , that is, the shock is not sufficient and leaves R&D fossil-oriented. We now analyze the dynamics of the economy by distinguishing between the two cases that can occur after the shock.

5.3 The production technology of output is Cobb-Douglas: $\varepsilon =1$

The result (short-term emissions are not modified) is demonstrated in Appendix F.

The essential point is that at the date $t^{\ast }$ of the shock, the marginal productivities of the renewable and fossil-related intermediate goods are affected similarly. That implies that the ratio of intermediate-good marginal productivities, $MP_{xr}/MP_{xf}$ , and therefore the variable $X$ , are unchanged; formally, we have $MP_{xr}(t)/MP_{xf}(t)=X(t)=1$ , and thus $g_{X}(t)=0$ , for all $t\gt t^{\ast }$ (see Equations (D.1) and (D.2)).

For $t\gt t^{\ast }$ , since $X$ is unchanged, there is no reallocation of labor in R&D: formally, $L_{r}/L_{f}$ is not modified. Therefore, the knowledge-stock ratio $A_{r}/A_{f}$ is at a higher level, but its dynamics, characterized by the rate of growth $g_{A_{r}/A_{f}}$ , are unchanged.

Finally, since the dynamics of $X$ are not affected, the dynamics of the marginal productivity of the resource, $g_{MP_{F}}$ , are also unchanged. That is why there is no change in the rate of resource extraction over time and, therefore, no change in short-term emissions. Formally, in Equation (31), we have $g_{X}(t)=0$ , $g_{F}(t)=-\rho$ , and thus $g_{MP_{F}}(t)=\bar {r}(t)$ .

In summary, the green technological shock (i.e., the increase in $A_{r}/A_{f}$ ) leads to an increase in the levels of $Y$ (see Equation (1)), $C$ , $x_{r}$ and $x_{f}$ (see Appendix C), but it does not affect the dynamics of these variables. It also does not affect the dynamic use of fossil resource, implying that short-term carbon emissions are unchanged.