Time Scales of the Low-Carbon Transition: A Data-Driven Dynamic Multi-Sector Growth Model

2.1 Constant-Technology Cross-Dual Dynamics

The pioneering work of Morishima (1981) and Goodwin (1983) contributed an early classification of microeconomic adjustment mechanisms (Table 1) and a thorough study of their dynamics, using Lyapunov functions and gradient processes, of particular importance as it highlights the complex dynamical relation between micro- and macroeconomic adjustment processes (Flaschel et al. 1997). As Flaschel notes (1997), the first two adjustment processes are a description of a proportional autocontrol mechanism through cross-effects between prices and quantities, which Morishima identified as cross-dual dynamics (Morishima 1981).

The Flaschel–Semmler cross-dual model of multi-sector growth under linear production (without fixed capital investment)^[1] and constant technology describes complex price p _i and quantity x _i oscillations over time around the equilibrium values of N prices p* (a row vector) and N quantities x (a column vector) of the Sraffa–von Neumann system (Table 2) (Sraffa 1960; Von Neumann 1945). Production is linear, with matrix A as inputs and matrix B as outputs, with positive values.^[2]

By the Perron–Frobenius theorem, unique positive equilibrium values p* and x* solve for the positive gross rate of return or expansion rate R > 1. The expansion rate R is the inverse of the unique largest positive real eigenvalue of the matrix of input–output coefficients A/B, equilibrium prices p* are its associated positive row eigenvector, and equilibrium output x* is the associated positive column eigenvector: p*B is the equilibrium unit revenue and Rp*A is the equilibrium unit cost p*A in relation to the expansion rate R (that is, costs including normal profits):^[3]

As in the Goodwin model of capital accumulation (1982), the coupled price and quantity oscillations of cross-dual dynamics are of the Lotka–Volterra predator–prey form, but in this, case profits (prices) play the role of the prey and profit-seeking investment (quantities) plays the role of the predator (Gandolfo 1971, ch. III.3). In the baseline model of cross-dual adjustment, industry market prices, relative quantities, and profitability gravitate around their equilibrium values, ultimately determined by the technological structure in line with the von Neumann–Sraffa input–output model (Sraffa 1960; Von Neumann 1945). Normal growth, around which industry growth gravitates, is zero, focusing on out-of-equilibrium dynamics of relative prices and quantities.

Following the law of excess demand, market prices p will decline (rise) if supply Bx is greater (smaller) than demand:

Following the law of excess profitability, quantity x _i will rise (decline) if unit revenues pB are greater (smaller) than unit costs times R, RpA, since capital will flow out of the sectors with below-normal profitability into the sectors with above-normal profitability:

where x ̇ x is the column vector of the growth rates in relative quantities, p ̇ p is the row vector of the growth rates in relative prices, and δ _p and δ _x are diagonal matrices with N positive adjustment coefficients (so they can also be understood as vectors).

Cross-duality in its simplest form may give rise to dynamic stability, yet not necessarily the conventional “asymptotic” convergence to competitive economic equilibrium to the single-point rest state with uniform profitability across sectors of production (Flaschel and Semmler 1987). In contrast, cross-duality generally reflects the classical theme of dynamic gravitation as envisioned by the great classical economists (Adam Smith, David Ricardo, Karl Marx, and John Stuart Mill), that is, of ceaseless over- and undershooting of sectoral prices, quantities, and profitability around their long-run equilibrium “natural values” as centers of gravity determined by Leontief linear technology (Duménil and Lévy 1987; Goodwin 1983; Sraffa 1960).^[4] A stylized description of the classical industry cycle is as follows: if an industry shows above (below) profitability, profit-seeking firms flow in (out), raising (lowering) supply with respect to demand until the latter is above (below) the former, triggering an increase (decrease) in prices through the Walrasian mechanism to adjust the supply–demand imbalance, but also lowering (raising) industry profitability, closing the cycle, and starting anew.

In contrast, the Keynesian tradition focuses instead on short-run dual dynamics (see Table 1), characterized by imperfect competition, oligopolistic markup price-setting, and barriers to capital entry, while investment operates within a longer time scale (Morishima 1981; Semmler 1984). In the short run, oligopolistic firms set prices as markups over costs, while varying supply through inventories in response to demand conditions. Classical cross-dual dynamics can thus be considered unsatisfactory in the short run, suggesting dynamic adjustments without cross-effects between prices and quantities where price and quantity movements may even be independent of each other for a while, what Morishima termed “dual dynamics” (Flaschel et al. 1997; Morishima 1981).

As an illustrative example of classical gravitation, constant-technology dynamics are simulated for the 2003 US direct requirements matrix with seven sectors (Figure 1).^[5] The nature of the linear-response model can be appreciated in Figure 2, which shows the same stylized dynamics as a scatterplot in a two-dimensional plane. In this case, the x axis refers to the imbalance in quantities and prices (as excess unit profits), and the y axis corresponds to the change in prices and quantities, so that all observations for a specific industry will be located on a line with the slope of its corresponding adjustment coefficient.

Figure 1:

Constant-technology dynamics for the 2003 US direct requirements matrix, seven sectors. Dashed horizontal lines indicate equilibrium values for profitability r*, prices p*, quantities x*, and aggregate growth g* = 0 (as the model deals with relative prices and quantities). Each colored line refers to the time evolution of market prices, quantities, profitability, and growth for each of the seven sectors. Oscillations vary in amplitude and frequency depending on the adjustment parameters chosen.

Figure 2:

Relative price and quantity changes with respect to excess demand and excess unit profit. The linear slopes correspond to the manually selected adjustment parameters δ _p and δ _x . The law of excess profitability shows a slight departure from strict linearity due to the stability adjustment γ. Dashed color lines indicate the linear regressions for each sector.

Such general systems of micro-economic adjustment processes can be re-purposed as highly stylized dynamic macroeconomic models of multi-sector growth that can be empirically calibrated without much problems using real training data, as each law of motion translates to a single reaction coefficient for each particular industry (Vallès Codina 2023). The econometric estimation of the cross-dual adjustment parameters using real training data is described in Appendix B. Once the linear adjustment coefficients are estimated for both the law of excess demand and the law of excess profitability, simulations of the dynamical process of technical substitution of specific industries can be implemented using the cross-dual model of multi-sector growth with process innovation and extinction. Keynesian dual short-run dynamics, which can further stabilize classical oscillations, can also be explored in a more general setup (Vallès Codina 2023).

2.2 Process Innovation and Extinction

In a subsequent contribution (Flaschel and Semmler 1992), Flaschel and Semmler propose an extension based on their classical competitive process of dynamical adjustment of the model of technical change presented in Silverberg (1984).^[6] The Goodwin model assumes neutral, exponential, disembodied technical progress, under fixed coefficients production, with fluctuating unemployment regulating changes in the level of real wages. Instead of disembodied technical progress, the contribution by Silverberg presents an economy with a fixed production process and then proceeds by examining the stability of the resulting equilibrium state when a second production process embodied in a new capital good with different technical coefficients is introduced. For a detailed description of the dynamic process of technical substitution, we refer to the original contribution by Flaschel and Semmler (1992).

Our framework refers to a different tradition of the study of technical change based on set theory and duality theory (for an example see Morishima (1981) or Mittnik et al. (2014)). In contrast, the conventional theory on directed technical change models it as continuous increases in the capital-to-labor ratio (i.e., capital deepening) or in total factor productivity in the aggregate production function (Acemoglu et al. 2012; Golosov et al. 2014). Instead, we consider a one-off process of technical substitution in an input–output context where innovation is the result of a more efficient process taking over a less efficient one due to its cost advantage. In particular, the focus is on modeling technical change as a dynamic process of within-sector technical substitution in the form of the innovation of more cost-efficient, less carbon-intensive processes of production that explicitly takes over and extinguishes older processes, following the work of Mittnik et al. (2014).

3.1 Carbon Phase-Out/Green Phase-In without Policy and with Carbon Pricing

In the absence of fiscal policy, technological differentials in unit costs drive profitability and growth of green innovation and carbon extinction. Unit costs are:

with corresponding market prices,

profitability,

and profitability deviations from equilibrium profitability r*:

where R is the maximum rate of expansion (equation (1)). For modeling convenience, carbon and green processes j = c, g feature input–output proportional coefficients and thus constant proportional unit costs, which can be summarized by the single technical parameter θ:

In such situation, since green and carbon processes produce the same output and thus share a common price p t c = p t g in the model, cost advantage in production θ, the ratio in unit capital costs between the green and carbon processes, is the only parameter regulating their profitability differentials,

In the standard theory of capital deepening and technical change, capital deepening is separated from technical change, where the latter is measured by increasing total factor productivity. We rather follow Kaldor and Mirrlees (1962), who suggested that it is hard to distinguish between those two drivers of productivity, since capital deepening mostly brings in new technology using more productive inputs. We thus measure endogenous technical change as decreasing costs of production by the explicit introduction of newer processes of production, as indicated by the parameters κ and θ. In this model, profitability differentials, regulated by cost differences summarized by θ, drive the inflow and outflow of profit-seeking investments and corresponding variations in quantities: investments will flow into and expand green output faster only if it bears a cost advantage over the carbon process.^[9]

In the context of carbon pricing, cost advantage θ is most relevant, as it is the parameter that internalizes the carbon price in the production costs of each process and will increase the profitability differential of the green process over more carbon-intensive ones. In order to see this, emission intensities of output ϵ ^j can be defined in terms of CO₂ tons emitted divided by quantity produced x ^j for j = c, g. By definition, ϵ ^c ≫ ϵ ^g . Unit costs with carbon pricing are now

where π is the carbon price and ϵ ^j π are the carbon costs faced by processes j = c, g where ϵ ^c π ≫ ϵ ^g π ∼ 0. The new cost advantage including carbon pricing will then be

for ϵ ^g ∼ 0. Internalized carbon costs ϵ ^c π must be of the order of unit production costs κ t c to have a substantial impact on the profitability differential.

In order to keep track of the dynamics of the process of carbon phase-out and green phase-in, it is convenient to define the output ratio between the green and carbon processes:

The growth rate differential between carbon and green processes is thus dependent on their technical parameter θ, output investment ratio σ _t , and the profitability differential ratio. Using equations (3), (7), and (12),

as well as the λ ratio of the quantity adjustment coefficients for the carbon and green sectors, defined as

Using equation (13), output ratio σ _t has evolution rule (without time subscripts for simplicity):

Lower-cost green technology (θ < 1) will thus ensure higher profitability r t g > r t c and growth rate g t g > g t c ; therefore, lower costs in production alone will eventually induce the phase-out of the carbon process. Although at the current moment green sources are increasingly outcompeting carbon ones with a cost advantage θ between 0.7 and 1.1 (IRENA 2020), there is no guarantee that profitability differentials only driven by technological differences in costs will drive the speed of decarbonization fast enough to fall within UN IPCC time targets, even if carbon pricing is considered. The point at which the low-carbon transition can be considered “successful” is the time when the green process overtakes the carbon process so that their output ratio σ _t reaches and surpasses a critical value σ ^ above 1. Thus, the success of the low-carbon transition not only depends on the growth rate differentials between the carbon and green processes but also their initial output ratio σ ₀ that corresponds to the initial investment in green technology.

3.2 Carbon Phase-Out/Green Phase-In with Tax–Subsidy Policy

The target-consistent path approach for carbon pricing first imposes time targets for the duration of the low-carbon transition as given, established by policy-makers following the advice of the climate science. The welfare problem of the policy maker thus consists of finding the required policy set (τ, ρ, σ ₀) given carbon phase-out time target t*, that is, a percentage carbon tax rate τ on real carbon output x ^c , raising a tax revenue of τx ^{c [10]} is that is added to green investment x ^g at subsidy rate ρ, starting from initial investment ratio σ ₀) to phase out carbon, σ t → σ ^ at target value σ ^ within duration time t → t*, with associated carbon emissions E ^ :

Taking advantage of the cross-dual adjustment dynamics, the policy maker employs the tax–subsidy mix to expand the profitability and growth differentials between the green and carbon processes. Instead of taxing carbon content as in carbon pricing, the tax levied is on the volume produced by the carbon-intensive process, as in Mittnik and Semmler (2022). Green and carbon outputs using the tax–subsidy mix (where the hat notation distinguishes the variable with and without policy) thus becomes:

Output proportion with tax–subsidy policy τ, ρ becomes:

Profitability for the carbon and green sectors internalizes the tax–subsidy policy:

While the negative contribution of tax τ is linear on the carbon sector, its positive effects on green profitability depend on the fraction τ ρ σ t , i.e., they are the largest when carbon output is much larger than green output (σ _t ∼ 0), that is, at the beginning of the introduction of the policy, and they are multiplied by cost advantage θ.

The profitability differential once the policy is introduced can be then computed in terms of cost advantage θ, output proportion σ _t , and tax rate τ:

which shows how a tax–subsidy policy can reinforce cost-induced profitability and growth differentials (when θ < 1) or even offset them (when θ > 1). Growth rate differential with policy τ, ρ with and without policy can be compared:

Once again, the additive presence of the ratio τ ρ σ t shows that the policy to direct technical change toward decarbonization is the most effective at the earliest stages of the phase-in (i.e., σ _t ∼ 0) when the subsidy rate is nonzero. This result, confirmed by the simulations, shows the relevance of green subsidies (ρ > 0) in kickstarting and mobilizing private funds for decarbonizing the economy, in line with recent studies (Deleidi, Mazzucato, and Semieniuk 2020; Heine et al. 2019; Semmler et al. 2021). However, a tax rate τ on real output alone can already accelerate substantially the phase-out of the carbon sector without any green subsidies ρ = 0, even if green cost advantage is lower (θ > 1), although there is no guarantee decarbonization may occur in time to achieve the UN IPCC targets.

4.1 Specific Policy Scenarios of the Low-Carbon Transition

The simulations can be conceived as a computational, stylized “thought experiment” to identify what are the policy dimensions most relevant in the domain of the low-carbon transition, that is, whether carbon prices or green investment. Simulations study the phase-out of the carbon process and phase-in of the green process within particular carbon-intensive industries by time t*, of relative capital intensities θ, under a carbon tax rate τ, a green subsidy rate ρ, and initial investment ratio σ ₀. The specific carbon-intensive industries targeted for substitution are “Manufacture of coke and refined petroleum products” (C19), “Electricity, gas, steam, and air conditioning supply” (D), and “Mining and Quarrying” (B), which feature positive adjustment coefficients for seven sectors in five of the selected countries: Germany, France, Italy, Japan, and the Netherlands. Apart from this one-off process of substitution, no other forms of technological change occur as the growth model operates on constant technology.^[11] At the current moment, green energy is increasingly outcompeting carbon energy: the cost of green energy can be considered to be between 0.7 and 1.1 times the cost of carbon energy – this is the range of values chosen for θ, which can also be thought to internalize a carbon price (IRENA 2020). For OECD economies, the current share of final energy consumption in renewable sources over carbon sources is around 5 % (Upadhyaya 2010), which is the benchmark value that is taken for the initial output ratio σ ₀ in the simulations, which can also be increased by the issuance of green bonds. For simplicity, target carbon–green output ratio is σ ^ = 1 , which implies that the low-carbon transition is successfully achieved when green output is equal to carbon-intensive output. Each timestep can be considered as one year, given the time dimension included in the adjustment coefficients, which were computed from yearly data.

Specific simulations of the low-carbon transition for particular policy scenarios under different chosen parameters in order to obtain a first qualitative, stylized comparison of their impact on carbon phase-out and green phase-in. For each of the selected policy scenarios and target industries, Figure 3 shows the time of carbon phase-out and green phase-in, i.e., the duration of the low-carbon transition when target output ratio is 1, highlighting that, although a percentage tax on carbon-intensive volumes substantially accelerates decarbonization, it should be accompanied by green investment in order to effectively meet the 2018 UN IPCC targets, as carbon taxes alone may be too high to be politically feasible. While the flagship UN Emissions Gap Report of 2021 recently urged the pledging nations to halve carbon emissions in the next 8 years, the simulation results are compared with slightly less drastic duration targets for decarbonization of the 2018 UN IPCC report of 2018 (Masson-Delmotte et al. 2018):

• 16 years for a 66 % chance of avoiding a temperature increase of 1.5 °C,

• 23 years for a 50 % chance of avoiding a temperature increase of 1.5 °C,

• 51 years for a 66 % chance of avoiding a temperature increase of 2 °C, and

• 65 years for a 50 % chance of avoiding a temperature increase of 2°C

Each row of Figures 4 –7 shows the simulated trajectories of prices, quantities, profitability, and growth rates for all industries (the full list of industries is given in Table 7 in Appendix C). Each column corresponds to a different industry under decarbonization, where the carbon-intensive and the green processes are highlighted with less alpha transparency and larger width in red and green than the other industries, respectively. The vertical dashed line indicates the moment where the output of the green process overtakes the output of the carbon-intensive process. As they share the same price, the trajectory of the price of the carbon-intensive process is identical to the green one. However, it is the two bottom rows, showing the industry trajectories of profitability and growth rates, which are the most informative, as they highlight the differentials between the carbon and green processes regulated by fiscal policy that make decarbonization feasible: while the profitability differential is regulated by the cost advantage parameter θ (on which carbon pricing operates), the tax–subsidy scheme operates over the growth rate differential, progressively increasing it from scenario 2 to scenario 4.

Figure 4:

Scenario simulation of the low-carbon transition: no policy. Driven by lower production costs alone, profitability and growth differentials are not large enough to decarbonize in a hundred years. The vertical dashed line indicates the moment where the green process takes over the carbon process, i.e., σ ^ = 1 .

Figure 5:

Scenario simulation of the low-carbon transition: only carbon pricing. Despite an extreme carbon price (θ = 0.1), carbon pricing alone is not able to successfully increase the profitability differential of the green process enough to speed up the substitution process even below 150 years. The vertical dashed line indicates the moment where the green process takes over the carbon process, i.e., σ ^ = 1 .

Figure 6:

Scenario simulation of the low-carbon transition: only percentage carbon tax. A percentage carbon tax directly impacts growth (rather than profitability), speeding up decarbonization closer to, but still far from, UN IPCC targets. The vertical dashed line indicates the moment where the green process takes over the carbon process, i.e., σ ^ = 1 .

Figure 7:

Scenario simulation of the low-carbon transition: tax–subsidy mix. Only a combination of carbon taxes and green subsidies can accelerate decarbonization to reach UN IPCC targets in time. The vertical dashed line indicates the moment where the green process takes over the carbon process, i.e., σ ^ = 1 .

The specific scenarios are the following:

1. No policy (Figure 4), with a very high initial investment ratio of 25 % and very high cost advantage for the green process (θ = 0.7). Although this is a very favorable scenario for a fast technical substitution in absence of policy, profitability and growth differentials are not large enough as induced by lower production costs alone so that the fastest green phase-in takes place in around 250 years, for German mining and quarrying. For three industries, decarbonization time exceeds the maximum time of 500 years.

2. Only carbon pricing (Figure 5), with carbon pricing inducing an extremely high cost advantage for the green process (θ = 0.1), an initial investment ratio in line with the current empirical evidence. Even with an extreme carbon price, carbon pricing alone is not able to successfully increase the profitability differential of the green process enough to speed up the substitution process even below 150 years for any of the studied industries and below 500 years for Italian Mining and Quarrying and Japanese Electricity Production.

3. Only percentage carbon tax (Figure 6), with a 2.5 % percentage tax on carbon-intensive output, an initial investment ratio in line with the current empirical evidence, and similar cost advantage θ = 0.9. By directly impacting growth rather than profitability, the substitution process substantially speeds up to bring decarbonization in around 100 years, which is still considerably beyond the Paris Agreement targets.

4. Tax–subsidy mix (Figure 7), with a 2.5 % percentage tax on carbon-intensive output and 50 % subsidy rate, an initial investment ratio in line with the current empirical evidence, and similar cost advantage θ = 0.9. By using only half of the tax revenue raised from taxing carbon-intensive output (ρ = 0.5), decarbonization time finally falls within the 2018 UN IPCC targets, showing the multiplicative effect of green investment to accelerate decarbonization.

4.2 General Role of Fiscal Policy Parameters in Accelerating the Low-Carbon Transition

While the simulation results of the former section already highlight that it is more convenient to use fiscal policy to impact on growth directly rather than profitability, a more systematic computational analysis of the policy space can be conducted using the empirically calibrated growth model to complement the comparative-statics analysis of Section 3. In this section, 14,250 simulations are computed for specific ranges of the four policy parameters at stake in order to investigate how the duration of decarbonization t* depends on them: the range for relative cost advantage θ is (0.7, 1.1), the range for the percentage tax rate τ is (0, 0.24) (i.e., the share of carbon-intensive volume that is taxed), the range for the green subsidy rate ρ is (0, 1), and the range for the initial output ratio is (0.1, 0.25). The industries studied are “Manufacture of coke and refined petroleum products” (C19) for Germany, France, Italy, and Netherlands, and “Electricity, gas, steam and air conditioning supply” (D) for Japan. Table 3 shows the regression results of the dependence of decarbonization speed on the four policy parameters as regressors, which is highly significant for all of them and with a very high R ² value.

In this context, the regression slopes for each policy parameter correspond to the magnitude of their impact on decarbonization speed: the percentage tax on carbon-intensive output bears the largest impact on speed, around 10 times higher than the subsidy rate (which is expected, as ρ multiplies τ in the model) and the initial output ratio. Since the latter have very different upper thresholds, respectively, 1 and 0.25, green subsidies are thus expected to bear a four-time impact on the speed of decarbonization compared to the initial output ratio. Most strikingly, the impact of cost advantage θ is almost negligible, casting severe doubt on the feasibility of carbon pricing alone as an effective tool to decarbonize in time. Following these results, the most effective fiscal strategy would dispense with making firms internalize carbon prices in production and focus instead on a high percentage tax rate on real carbon-intensive output (as in sales), accompanied by substantial green investment, both in the form of a single large initial investment σ ₀ and green subsidies ρ. Keynesian dual dynamics addressing mark-up pricing can be used to properly assess to what extent firms pass on the carbon tax to customers.

For the illustrative sake of clarity, Figure 8 shows the dependence of decarbonization speed 1/t* (i.e., the inverse of the duration of decarbonization time) for a specific sector (C19, Germany) with respect to each parameter, for different values of the tax rate. In agreement with the regression results, the figure shows a very robust linear dependence of decarbonization speed on the tax rate τ. The relationship between speed and subsidy rate ρ is more complex, indicating a multiplicative interaction with the tax rate: linear at low tax rates and logarithmic at high tax rates, showing that green subsidies are most effective when carbon taxes are the highest. In contrast, cost advantage θ and initial investment ratios have negligible impacts on decarbonization speed within their respective domains. The impact of carbon pricing on decarbonization speed is relevant only if carbon prices are so high that θ is closer to zero.

Figure 8:

Impact of parameters on decarbonization speed for sector C19, Germany, for many values of the tax rate. When the tax rate is the dependent variable, the different lines correspond to different values of the subsidy rate. Horizontal dashed lines correspond to the necessary speed to meet the IPCC targets.

For instance, the top-left panel shows that, for a cost advantage θ = 0.8 and an initial output ratio of σ ₀ = 0.05, a very high tax rate of τ = 0.22 = 22 % decarbonizes sector C19 within an IPCC target of 23 years for any subsidy rate, even with ρ = 0. If all carbon tax revenues are instead re-invested as green subsidies (ρ = 1), then decarbonization within the same 23 years can already be achieved with half of the tax rate, τ = 0.11. For any subsidy rate from 0 to 100 %, a tax rate of τ = 0.05 decarbonizes sector C19 within a substantially higher IPCC target time of 51 years. Many current state policy guidelines aim at decarbonizing by 2050; this would require a tax rate on real output (as in sales) between 0.06 and 0.16, depending on the subsidy rate.