Don’t Forget About Technology!

(1)

Don’t Forget About Technology!

The role of technology inducing climate policy

Olav Abrahamsen

Thesis submitted for the degree of M.Phil in Economics

Department of Economics Faculty of Social Science

Submitted 11.2020

(2)

Olav Abrahamsen, 2020c

Print production: Reprosentralen, University of Oslo Supervisor: Christian Traeger

(3)

Abstract

I extend an analytical integrated assessment model of climate change to include endogenous technical change in the economy’s energy-producing sector. The technological change is induced both by direct investment in research and by learning-by- doing. The dynamic optimisation problem is solved by transforming the model into being linear in state-space. I find closed-form solutions for the social cost of carbon and the optimal allocation of funds to promote technological change in the energy sector. A simple policy simulation finds that the optimally controlled economy will curtail temperatures below 3^◦ Celsius above pre-industrial levels. Furthermore, I find that current subsidies in the energy sector are misallocated.

(4)

Preface

It can be somewhat daunting to contemplate how dependant the global economy is on fossil fuels knowing the damages they are projected to cause. Fossil fuel usage has, to some extent, been the foundation that modern society was built on top.

Even with the global outcry in fear for climate change, the fact remains that in 2019 84.3% of global primary energy was generated by either coal, oil, or natural gas. Almost all the goods and services that we enjoy today have most likely been transported by, powered by, or produced with fossil fuel energy. The challenge we have in front of us now is to put in place a new technological foundation, preferably without tearing down the house we have built on top.

Acknowledgements

Given my lifelong struggles with grammar, it is highly probable that the reader will encounter improper punctuation, misspellings, and strange sentence structure throughout this thesis. I take full responsibility for any grammatical errors left in this text, and I am more than willing to clarify any confusion they might create.

I would like to thank my supervisor Christian Traeger for his help, his patience, and his support during this project. His guidance has been essential to my ability to complete this thesis. I would also like to thank my friends and classmates that provided new ideas, and helped me improve the text.

(5)

List of Tables

1 Estimates of SCC in 2020 for different discount rates, denoted in 2010 US$

(Nordhaus, 2017) . . . 4

2 Production and damage parameters . . . 26

List of Figures

1 Outline of the model . . . 10

2 Response of the damage function to different temperatures . . . 28

3 Growth of GDP over 100 years . . . 29

4 Plots energy input usage . . . 31

5 Temperatures over time . . . 31

6 Damage caused by climate change as percentage of GDP . . . 32

7 Price of CO2 emissions . . . 33

8 Impact of emission pricing on emissions . . . 34

9 Price of CO2 emissions for different discount factors . . . 34

10 Evolution of the knowledge stocks . . . 35

11 Changed emission paths, and their effect on knowledge stocks in the fossil fuel sector . . . 36

(8)

1 Introduction

Economic historians argue that the availability of coal and the high wages for labor were among the reasons why the industrial revolution first took place in Great Britain (Allen (2009), Broadberry and Gupta (2009)). These factors contributed to an environment that induced innovations of technologies that could substitute human labor with energy from coal. More than 250 years after James Watt’s first patent on his steam engine, we find ourselves trying to reduce the dependency on fossil fuels that developed under the industrial revolution and turn to zero-emission technology. To speed up this transition, policymakers need to produce legislation that serves two distinct functions: disincentivize the use of fossil fuels and expedite the development of renewable energy technologies that will be effective substitutes. In this thesis, I create a model that allows the policymaker to construct a policy platform of both emission pricing to disincentivize fossil fuel usage and investment in research in the energy sector to ensure that the energy sector can function properly during the transition to a zero-emission economy.

During the last 30 years, economists have integrated the impacts of climate change into economic growth models. These integrated assessment models(IAMS) have enabled the calibration of economic policy that makes private actors account for damages caused by their emissions when deciding what inputs to use in production and what technologies to research, develop and deploy. Historically, IAMS have been solved with numerical methods that suffer from a lack of transparency, but recent developments have enabled their key results to be derived analytically. Except for the model of Hassler et al. (2019), analytical integrated assessment models (AIAMS) have treated technological change as an exogenous variable, limiting their ability to discuss the relationship between technological change and climate change.

In this paper, I attempt to correct these shortcomings by creating an (almost) analytical integrated assessment model where technological change in the energy sector is determined endogenously. The model is designed to accommodate transparent assumptions about how emissions impact our economy and how technological capabilities change over time. It

(9)

allows the social planner to implement a policy platform consisting of research investment and emission pricing. I find that an optimal policy response should consist of both tools, where the current optimal price of emissions is slightly under 20 USD per ton CO₂, and the optimal subsidy to knowledge promoting activities in the energy sector is slightly under 1% of GDP. The optimal social cost of carbon is found to be much higher than the current observed rates. In contrast, the total amount of spending on knowledge promotion appears to be close to the optimal levels found in the model, but the funds are found to be wrongly distributed. An additional benefit of implementing emission pricing is identified, as the reduction of present fossil fuel usage allows benefits from learning-by-doing to persist for longer.

Section two of this thesis reviews the relevant economics literature of integrated assessment models, technical change, and the interaction between them. These two strains of economics thought happened to share the 2018 Nobel Memorial Prize in Economic Sci- ences. The two recipients were William Nordhaus “for integrating climate change into long-run macroeconomic analysis”, and Paul Romer “for integrating technological innovations into long-run macroeconomic analysis”. In section three, I present the equations and assumptions that constitute the model. Section four solves the model’s social planner problem and derives analytical solutions to key policy rates. Section five calibrates the values for the simulation, and section six reviews the results of the simulation. Lastly, section seven summarizes the paper’s main findings, and I provide my concluding remarks.

2 Literature review

2.1 Economics of climate change

Early research by Nordhaus (1991) on the relationship between climate change and the economy extended the workhorse Ramsey growth model with a set of equations that detail how economic activity affects the atmospheric concentration of greenhouse gasses (GHS), radiative forcing, temperature changes, and subsequent damage. The optimal controls

(10)

are found under the assumption that the economy is in a resource steady-state. Meaning that in the equilibrium, all physical (per-capita) flows in the economy remain constant, including consumption, emissions, and concentrations of GHS in the atmosphere. As Nordhaus himself later wrote, this led to an unsatisfactory treatment of the dynamics of climate change¹. Building on his previous work, Nordhaus created the Dynamic Integrated Climate-Economy model(DICE) (1993). Instead of relying on an assumption of a resource steady-state, the model finds solutions through numerical optimisation. This allowed for a richer model of the climate that captured the novel dynamics of climate change. These changes enabled the estimation of the social cost of carbon(SCC). A value that represents

“[...] the change in the discounted value of consumption denominated in terms of current consumption per unit of additional current emissions” (Nordhaus, 2019).

The existence of externalities caused by climate change leads to a divergence between the social- and private net benefits. By logic dating back to the origins of welfare economics (Pigou, 1932)[first edition published 1920], we know that the value of the social cost of carbon corresponds to the optimal price of emissions. If implemented, an additional price that captures the marginal damage of emissions will make the social and private net benefit functions equate each-other, leading to an optimal allocation of resources. Economists have, in large part, favored the use of such price-based policies like carbon taxes and trad- able emission quotas, as opposed to command and control type policies such as technology standards and emission caps when addressing the externally of climate change. This is due to their ability to continually provide incentives to lower emissions and ensure that we lower emissions where it is most cost-effective to do so (Stiglitz et al., 2017). How- ever, implementing effective emission pricing schemes has proven to be difficult because of political reasons (Carattini et al., 2018). Currently, 22% of the world’s emissions have some additional price attached, but many of these schemes suffer from a lack of coverage and low prices. As a result, the average price of emissions is estimated to be around 1$

(World Bank, 2020).

1By “dynamics” he is referring to two distinct effects: That the greenhouse gasses we emit today will remain in the atmosphere for a long time and the thermal inertia caused by factors like oceanic cooling that creates lags between emissions and damages.

(11)

The optimal monetary value of carbon’s social cost is highly dependent upon the assumptions used in its calculation. Table 1 showcases estimates of the social cost of carbon from four different models (DICE-2013, DICE-2016R, PAGE09 & FUND 2013), under three different discount rates on consumption. The PAGE-model emphasizes the uncertainty associated with climate change-induced damages and includes valuations of non-economic values such as environmental degradation and social unrest caused by temperature changes (Hope, 2011), which leads to its high estimated SCC. Along with co-authors, Richard Tol argues that the human species’ ability to adapt to climate change is not accounted for in some IAMS (De Bruin et al., 2009). His specifications regarding adaptation, and the damage function in his FUND-model (Tol, 2013), results in substantially lowers estimated SCC compared to other IAMS. The change in estimated SCC between DICE-2013R and the DICE-2016R models can be explained by changes in assumptions regarding carbon cycles and the economic activity (Nordhaus, 2017).

Model r = 0.05 r = 0.03 r = 0.025

DICE-2013R 15 50 74

DICE-2016 23 84 140

PAGE 23 74 105

FUND 3 22 37

Table 1: Estimates of SCC in 2020 for different discount rates, denoted in 2010 US$

(Nordhaus, 2017)

Furthermore, discount rates impact the social cost of carbon substantially. Merely chang- ing discount rates from 3% to 2.5% increases the social cost of carbon by almost 50% in all models. This dependency on the discount factor has made the way discount rates are determined into a topic of discussion in environmental economics. A task-force headed by the economist, Sir. Nicholas Stern (2006) produced a rapport which used a discount rate on future consumption of 1.4%, leading to unusually high estimates of the social cost of carbon. These high policy rates were derived by setting the pure rate of social time preference parameter in Ramsey’s social discount rate equation to almost zero² based on

2The reason for not setting this parameter to exactly zero was to account for the possibility that

(12)

a partially utilitarian view. His findings were criticized by Nordhaus (2007), who argues that though ethically, he finds high discount rates imperfect, discount rates in economic models should more closely be linked with how human beings value future consumption rather than philosophical conviction.

The drawback of models that are solved numerically is their lack of transparency. Models such as DICE, FUND, MERGE, and PAGE rely on sensitivity analysis to infer how solutions are affected by differing assumptions. As a result, they have garnered criticism for being “black boxes” (Pindyck, 2017). Given the impactfulness of changes in assumptions and choices of parameters on their key results, this lack of transparency becomes a problem. It becomes hard to communicate what determines the social cost of carbon precisely.

However, recently models that allow for closed-form solutions for the price of carbon have been created. The first significant contribution here was Golosov et al. (2014), who finds that the optimal social cost of carbon should be proportional to GDP in a model that solves almost entirely in close form. Their model provides analytical solutions because it transforms into a linear-in-state model. This insight was used to create the ACE-model (Traeger, 2018), which provides the framework the model in this paper extends.

2.2 Technological change

One of the most famous macroeconomic theory results is that in an economy with close to steady-state levels of capital, growth in real wages is mostly a consequence of technological change (Solow, 1956). The Solow model depicts technology as hicks-neutral measure of total factor productivity³ that grows at an exogenously given rate. The main aim of the model was to explore the relationship between capital accumulation and growth. However, leaving the driving force behind real wage improvements for nations in steady-state as

“manna from heaven” turned out to be unsatisfactory for some.

A notable early attempt to develop a more tangible expression of technological change

humans will go extinct.

3See (Hulten, 2001) for a discussion of TFP

(13)

was made by Arrow (1962, 1971). Motivated by the empirical findings that the marginal cost of production decreased in cumulative output (Wright, 1936), Arrow constructed a model that allowed for increased productivity through learning-by-doing(LBD). In his model, investment and usage of capital goods make labor more efficient in making future capital goods. He uses the learning curve specification found in the paper by Wright and constructs a framework to discuss the implications of introducing LBD for wages and profits. Lucas extends LBD to a model where sector-specific investment leads to improved human capital as a whole, but with different learning rates (1988). He argues that countries that specialize in high skill production gain an extra advantage because higher-skilled production results in faster learning. This paper opens up the possibility of differences in learning speed, but knowledge only accumulates in sector-specific terms.

Technology in Arrow’s and Lucas’s models is considered a pure public good that is improved mainly as a byproduct of capital investment or production. The problem with this depiction of technology is that it fails to explain private investment directly in research and development⁴ (Romer, 1994). Paul Romer (1990) used developments in the monopolistic competition literature⁵ to create a model where research firms receive rents for discoveries of new intermediate good patents(patents on machines). These rents are given as wages to human capital allocated in the research sector. The price of interme- diates and the productivity of human capital in both the research and final good sectors determine the optimal allocation of human capital, determining the growth rate in the balanced growth path. It is assumed that new patents create more “giant shoulders” to stand on for future researchers so that research’s productivity gains are increasing in the amount of previous research. The social benefits from these spillovers are not captured in the private returns to research. Hence, some research subsidy is warranted.

4Consider a perfectly competitive market where knowledge is a pure public good. If rivals immediately gain access to patents stemming from costly research, they will press the price down to the marginal cost of production while innovative firms still need to pay for the research. This will make profits negative for the firm that did the research.

5The expanding variety framework developed by Dixit and Stiglitz (1977), and its subsequent extension to production (Ethier, 1982) were vital in the development of the Romer model

(14)

The Romer model allowed for the endogenous determination of the growth rate and aggregate technological change through active investment. However, this paper is not concerned with aggregate technological change, but rather factor specific technological change. Daron Acemoglu expands the framework the Romer model was built upon to construct a model where technological change can differ between factors (Acemoglu, 2002).

He finds that technological change can be biased⁶ towards factors that are expensive, and factors that have a large market. The determining factor for which of these forces is the strongest is the elasticity of substitution between the production inputs.

Representing technological change as an exogenously given scalar for the production inputs we use can, in many cases, be a useful abstraction. However, it is a paradigm that makes it difficult to discuss the determining factors of technological change. Technology in economics tends to be some residual term that captures all factors that impact production output after controlling for the inputs. Better machines, better layouts of factories, and better-trained workers can lead to better technology, though their causes may be very different. Direct investment in research will usually improve our knowledge in some codified form(patents, rapports, academic papers). In contrast, the knowledge gained by a worker learning how to use inputs more efficiently will often be tacit (for a discussion on tacit and codified knowledge, see (Cowan et al., 2000)). Leaving out either of the two paths for technological change will limit the model’s capabilities in discussing how policies impact emission and technological change. Therefore, this paper follows in the footsteps of Young (1993) by including both direct investment and learning but modified to fulfill the requirements of linear-in-state models. However, the introduction of both codified and tacit knowledge comes at a high cost of measurably.

6Considered the production function f(A, L, Z), where Lis labor inputs, Z is capital inputs, andA is the technology index. The Ais consideredLaugmenting if the production function can be written as f(AL, Z), and L biased if ^∂(

∂f /∂L

∂f /∂Z)

∂A >0.

(15)

2.3 Technological change in the economics of climate change

Integrated assessment models have provided insight into how market-based policies can change prices to account for future damages caused by emissions so that private actors face better incentives when deciding the number of inputs to use in production. The literature concerning endogenous technological change argues that changes in prices and production input usage can affect both the speed and direction of technological change.

There is mounting empirical evidence that emission pricing induces innovation in renewable technologies. Costantini et al. (2017) find that public policy can induce more environmentally friendly innovation when looking at patent data. They find that demand pull-policies (e.g., emission pricing) have a more significant effect than technology-push policy (e.g., research investment subsidies). However, a balanced approach consisting of demand-pull and technology-push policies led to a larger impact. By using matching between European firms, Calel and Dechezleprˆetre (2016) find that being regulated by the European Union Emissions Trading System (EU ETS) led to a 10% increase in low- emission patents. By looking at data from the car-manufacturing industry, Aghion et al.

(2016) find evidence that higher emission prices lead to innovation shifting from dirty to clean innovation, which they use as an argument for the importance of emission pricing.

Evidence from the oil and gas industry suggests the presence of learning-by-doing in the energy sector, particularly if the relationship between the oil company and the contractors manning the drills remains stable over time (Kellogg, 2011). Nemet (2012) provides evidence of productivity increases from the cumulative output in the wind-power sector in California but finds that the returns to learning are rapidly diminishing. Tang (2018) investigates learning at the multiple stages of wind energy production, from learning in research to learning in operation. She finds evidence that cumulative experience from wind farm operation does have a positive effect on output. For a more comprehensive review of the empirical literature on technological change in the energy sector, see (Popp, 2019).

It is tempting to think that we can innovate our way out of the climate crisis by funding renewable energy research alone. However, the context in which the investments are

(16)

made is important. In Acemoglu et al. (2012), the authors augment the framework of Acemoglu (2002) into a model with clean and dirty inputs and a simple model of the climate. They find that either temporary or permanent subsidies in renewable technology research are warranted and that investment in research and emission pricing is needed as an optimal policy response. Their results depend on how backward the renewable sector is and the elasticity of substitution between clean and dirty inputs. A paper by David Popp (2004) extends the DICE-model to include endogenous technical change, similar to what this paper tries to do with the ACE-model. In Popps’s ENTICE-model, technological change is caused by direct spending on research in the energy sector, which improves factor specific knowledge stocks. He finds that research spending should only play a supporting role to emission pricing in climate policy. Only relying on research spending in his model surprisingly leads to an increased concentration of GHS in the atmosphere.

This is because the improved renewable energy technologies lower the price of energy, leading to more energy in total being demanded.

3 The model

The present model presents a dynamic general equilibrium framework. It derives analytical results for the associated social planner problem by transforming into a linear-in- state-space system and identifying the conditions where an affine trial solution holds. The broad outline of the model is depicted in figure 1.

The model admits a representative consumer with an utility function that is logarithmic in consumption C_t. The representative consumer is assumed to discount future utility by the discount factor β.

U =

∞

X

t=0

β^tln(C_t) (1)

(17)

Figure 1: Outline of the model

3.1 Final good sector

A representative final good Y_t is produced with capital K_t, energy E_t, and exogenously given labour inputs L_t. Technology in the final goods sector is assumed to be hicks- neutral⁷, scaled with a measure of total factor productivity A_t which grows at an exogenously given rate.

Y_t =A_tK_t^κE_t^νL^1−κ−ν_t (2)

It is assumed that the initial stock of capital K0 is given and that the stock of capital is fully depreciating. Therefore, the stock of capital in any period is entirely dependent on the amount set aside from the previous periods’ production to capital investment.

7This implies that all technological changes in the final good sector come throughA, while the share parameters remain constant.

(18)

Kt+1 = ˆYt−Ct (3) The term ˆY_t represents the total production of the final good net of damages caused by climate change, investment in renewable energy inputs G_t, and research investments in renewable energy technology i_G,t and fossil fuel technology i_B,t.

Yˆ_t=Y_t[1−D(T_t)]h(G_t, i_G,t, i_B,t) (4) Here D(T_t) captures the damages from increased temperature T_t on the final good production. The costs of G_t, i_G,t and i_B,t are represented by the homogeneous function h.

What is left after we have accounted for damages and costs, will either be consumed, or invested in capital for the next period.

3.2 Energy sector

The energy sector’s production is represented by a Cobb-Douglas production function with green renewable energy inputs and brown fossil energy inputs. An unfortunate requirement of this Cobb-Douglas specification is that the share parameter α needs to remain constant. As the income share of renewable energy is projected to be increasing over time, this is an unrealistic assumption and makes calibrating its value difficult. As the production function is multiplicative, it also assumes that we will never completely stop using fossil fuels.

E_t= (A_G,tG_t)^α(A_B,tB_t)^1−α (5) The green energy inputs G_t represents the metal, concrete, and electronics required to operate renewable energy sources. The stock ofG_tis assumed to be fully depreciating and can be imagined as being rented. The brown fossil fuel inputs B_t represents the usage of coal, oil, and natural gas in energy production and is measured in tons of emitted carbon⁸. The fossil fuels are assumed to be costlessly extracted from a finite stock of fossil fuels R_t. The initial size of the stock of fossil fuels R₀ is assumed to be given.

R_t+1 =R_t−B_t (6)

8e.g if we use fossil fuels that emit 200 tons of carbon,B_t= 200

(19)

How effective the inputs are in producing power is dependant on the technological levels AG,t and AB,t. This model will consider technologies Gt augmenting (see footnote 6) if they only directly affect the efficiency of some weighted sum of the tonnage of solar energy inputs, concrete and water trapped by hydroelectric dams, metals used in windmills, and thermal energy pipes and pumps. Furthermore, electrical storage technologies and smart grid technologies that reduce the intermittency issues associated with wind and solar are also considered G_t augmenting. Technologies will be considered B_t augmenting if they only affect the carbon intensity to energy output ratio from the fossil fuel sector. Hence, the technological advances in Hydraulic fracturing (“fracking”) are considered to be B_t augmenting as it lowered the carbon intensity of the US (Acemoglu et al., 2019). Carbon capture technology will also be considered B_t augmenting here, as it will allow for more energy output while not increasing emissions. I have chosen to omit nuclear power from energy production, even though nuclear power constitutes 4.3% of the global energy mix.

Therefore, nuclear energy generation will be split into the technology terms AGt and AB,t. Downstream energy technologies that will affect both the efficiency of B_t and G_t (e.g., more energy-efficient machines) are not addressed in either AG or AB, and end up in the residual term A_t in the final good production function.

3.3 The climate

Using fossil fuels to produce energy will emit greenhouse gasses into the atmosphere. In subsequent periods, some of the gasses emitted to the atmosphere will cycle through to other reservoirs. How many tons of carbon there are in the different reservoirs is captured in the vectorMMM_t, where the individual elementsMMM_q,t, q∈ {1, m} represent the individual reservoirs. In this paper,MMM_q,t consists of three reservoirs, where elementM_1,t is the stock of pollution in the atmosphere, element M_2,t is the stock of carbon in the biosphere and shallow ocean, and M_3,t represents carbon storage in the deep oceans. The transition matrix Φ captures how the pollutants move between the different reservoirs⁹. Since the

9Say that we have three reservoirs: 1,2 & 3, whereMMM2,trepresents the biosphere. Let us say that 80%

of the pollutant in reservoir 2 stays put, 20% of the pollution in reservoir 1 moves to reservoir 2, along

(20)

atoms in the different reservoirs never really disappear, each column’s sum in the matrix needs to be equal to one. The m×1 vector eee1 = (1,0,0, ...)^T moves the emitted gasses from this periods energy production into the atmosphere in the next period.

MMMt+1 = ΦMMMt+eee1Bt (7) Increasing the stock of greenhouse gasses in the atmosphere increases the net influx of energy to the earth through the greenhouse effect. The measure of this net energy balance is known as radiative forcing or climate forcing. If the net energy balance changes, so do the long-run equilibrium temperature of the earth. We construct a variable F_t as the long-run equilibrium temperature given this period’s level of radiative forcing. Its value is determined by comparing the current amount of carbon in the atmosphere to industrial-era levels.

F_t=sln(_M^M^1,t

1,pre)

ln(2) (8)

The parameter s is a measure of how sensitive the long-run equilibrium temperature is to a doubling of carbon in the atmosphere over the pre-industrial level. It is a highly uncertain parameter since its value depends a lot on secondary effects and the size and sign of those secondary effects. A meta-analysis of academic papers found that the value of s likely lies between 1.5 and 4.5^◦C(Knutti et al., 2017), which is the same interval used in the IPCC AR5 rapport. In this paper, s is set to the mean value of 3^◦C. A doubling of greenhouse gasses in this model occurs when the current carbon reservoir M_1,t contains twice as much carbon compared to the pre-industrial levels M_1,pre. The logarithmic functional form implies that the marginal increase in Ft is decreasing in the atmosphere’s current stock of pollutants.

Though higher levels of forcing correspond to more energy being trapped, forces like oceanic cooling create inertia in surface-level temperature T_t. This inertia is modeled by

with 10% of the pollutant from layer 3. We can then writeMMM_2,t+1= 0.2MMM_1,t+ 0.8MMM_2,t+ 0.1MMM_3,t. The second row vector in this Φ matrix will then be (0.2,0.8,0.1).

(21)

expressing the surface level temperature next period as an average between the current temperature and long-run equilibrium temperature for the current level of forcingFt. The parameter σ will determine the speed of convergence, and the parameter ξ₁ determines the non-linearity of this change. As the value of Ft is zero if M1,t =M1,pre, surface-level temperature T_t is measured in degrees Celsius above pre-industrial levels.

Tt+1 = 1

ξ₁ ln((1−σ) exp[ξ1Tt] +σexp[ξ1Ft] (9) Increases in temperatures will cause damage to our planet (see IPCC rapport (Field, 2014)). The damages that impede our final good production are related to the economy through the damage function D. A parameter ξ₀ is included, which scales damages for any given temperature.

D(T_t) = 1−exp[−ξ₀exp(ξ₁T_t) +ξ₀] (10)

3.4 The technology sector

The productivity variables A_G,t and A_B,t in the energy producing sector are determined endogenously. Given their initial states {AG,0, AB,0}, they follow the equation of motion seen in equation 11.

A_j,t+1=A_j,texp(I_t), j ∈ {G, B} (11)

The growth rate of productivity in both sectors is determined by the state variable I_j,t ∈ [0,1], which represents a sector-specific stock of knowledge. Though not all-inclusive, this variable can be thought of as a measure of relevant sector-specific blueprints, patents, academic papers, construction rapports, organizational ability, and tacit knowledge at any point in time. When this stock is high, productivity in the sector will grow at a faster rate.

(22)

Ij,t+1 =ρIj,t+ηjij,t+ωjln(jt) (12) A fraction of the stock (1−ρ), ρ ∈ (0,1), will either be forgotten or rendered irrelevant in the subsequent period. There is evidence of such knowledge depreciation in the energy sector, though it is claimed that it is mostly tacit knowledge that depreciates (Grubler and Wilson, 2014). The variable i_j,t is a fraction of the final good that is set aside to research activity in sectorj ∈ {G, B}. The parameterη_j measures how effective spending on research is in sector j. As I assume thatη_j is constant throughout time, it is implicitly assumed that there is decreasing returns in research since a percentage of investment in a larger economy will lead to the same change in I_j,t+1 as the same percentage of investments in a smaller economy. The j denotation of η_j implies that the effectiveness of research investment might vary depending on which sector one does research. By looking at patent citations, Dechezleprˆetre et al. (2013) finds an indication that clean energy research generates more knowledge spillovers, making investments in green technology more effective.

LBD in the model is represented by the term ω_jln(j_t), where ω_j determines the speed of learning. Including LBD helps to emulate the market size effect of Acemoglu (2002), as it will ensure that the relative usage of factors helps determine relative productivity growth rates. The functional form implies that learning is linear in current logarithmic factor usage. It should be noted that this is an unconventional specification of LBD. A more typical depiction of learning-by-doing is a function of cumulative factor usage. However, this is impossible to implement in the mathematical framework this model applies to find analytical solutions, as it requires the optimal controls to be independent of all state variables. Nevertheless, given the decreasing marginal learning returns to current factor usage caused by the logarithmic function, combined with the linear decay of ρI_j,t, we still find that LBD becomes less effective if the stock of knowledge is large.

(23)

4 Solving the social planner problem

4.1 Initial problem and Bellman equation

The social planner problem maximises an infinite stream of utility from consumption for the representative consumer, given initial states and the equation of motion for those states. To transform this system into a linear-in-state model, we will be using consumption rates rather than total consumption x_t= ^C_ˆ^t

Yt, such that ln(C_t) = ln(x_t) + ln( ˆY_t). We also define a_t:= ln(A_t) and k_t := ln(K_t). Lastly we will need to use generalised temperature τ_t = exp (ξ₁T_t), and set ξ₁ = ^ln(2)_s . Let us now construct the Bellman-equation of this problem.

V(k_t, τ_t, MMM_t, R_t,ln(A_G,t),ln(A_B,t), I_G,t, I_B,t, t) =

xt,Gt,Bmaxt,iG,t,iB,t

n

ln(x_t) + ln( ˆY_t)

+βV(k_t+1, τ_t+1, MMM_t+1, R_t+1,ln(A_G,t+1),ln(A_B,t+1), I_G,t+1, I_B,t+1, t+ 1)o

(13)

4.2 Trial solution

The solution to the Bellman-equation will be a value function V. By transforming our model to satisfy the requirements of a linear-in-state model, we can propose an affine trial solution. The affine trial solution needs to be linear in the state variables, and all optimal controls in the system needs to be independent of the state variables. All state variables will have an associatedϕwhich acts as the shadow values of the stocks. In equation (14) we can see the proposed solution for the value function.

V(k_t, τ_t, MMM_t, R_t,ln(A_G,t),ln(A_B,t), I_G,t, I_B,t, t) = ϕ_kk_t+ϕ_ττ_t+ϕϕϕ^T_MMMM_t+ϕ_RR_t

+ϕ_A_Gln(A_G,t) +ϕ_A_B ln(A_B,t) +ϕ_I_G,tI_G,t+ϕ_I_BI_B,t+ϕ_t

(14)

(24)

By inserting forV_tandV_t+1in (13), and use the equations of motion for the state variables in Vt+1, we get the following expression as the Bellman equation.

ϕ_kk_t+ϕ_ττ_t+ϕϕϕ^T_MMMM_t+ϕ_R_tR_t

+ϕA_Gln(AG,t) +ϕA_B ln(AB,t) +ϕI_G,tIG,t+ϕI_BIB,t+ϕt=

xt,Gt,Bmaxt,iG,t,iB,t

(

ln(x_t) +a_t+κk_t+ναln(A_G,t) +ναln(G_t) +ν(1−α) ln(A_B,t) +ν(1−α) ln(B_t) + (1−κ−ν) ln(L_t)−ξ₀τ_t+ξ₀+ ln(h(G_t, i_G,t, i_B,t)) +βh

ϕ_k

ln(1−x_t) +a_t+κk_t+ναln(A_G,t) +ναln(G_t) +ν(1−α) ln(A_B,t)+

ν(1−α) ln(B_t) + (1−κ−ν) ln(L_t)−ξ₀τ_t+ξ₀+ ln(h(G_t, i_G,t, i_B,t)) +ϕϕϕ^T_M

ΦMΦMΦM_t+eee₁B_t

+ϕ_τ

(1−σ)τ_t+σ M_1,t M1,pre

+ϕ_R_t+1(R_t−B_t) +ϕ_A_G

ln(A_G,t) +I_G,t

+ϕ_A_B

ln(A_B,t) +I_B,t +ϕ_I_G

ρI_G,t+η_Gi_G,t+ω_Gln(G) +ϕ_I_B

ρI_B,t+η_Bi_B,t+ω_Bln(B)

+ϕ_t+1i )

(15)

In the following calculations it is assumed thath(G_t, i_G,t, i_B,t) = (1−γ_tG_t)(1−i_G,t)(1−i_B,t).

The parameter γt transforms the amount of green capital used to cost in terms of a percentage of final output. As the economy will be growing over time,γ_t will need to vary with time. The cost function is multiplicative; hence, timing becomes important. There needs to be a set order in which funds are removed. After damages are accounted for in production, spending on Gt inputs is accounted for, followed by investment in iG,t, and lastly investment in i_B,t. What is left is divided between consumption and investment in capital for the next period. Henceforth, γ∗Gt is now a measure of spending of the final good after accounting for damages. i_G,t is the percentage of final good spent in renewable energy research after accounting for damages and spending on Gt, and iB,t is a measure of spending on the research of non-renewable technologies after that.

(25)

4.3 First order equations

We derive the first-order equations by taking the partial derivative of the right-hand side(R.H.S) of equation 15 with respect to the set of optimal controls{x_t, G_t, B_t, i_G,t, i_B,t}, and set them equal to zero.

∂R.H.S

∂x_t = 1

x_t − βϕ_k (1−x_t) = 0

⇒x^∗_t = 1 (1 +βϕ_k)

(16)

Unsurprisingly, if the shadow value of (log) capital goes up, implying that capital becomes more valuable to us, we will consume less (x_t goes down). More myopic tendencies (β goes down) leads to higher current consumption levels, leaving less capital for the next period.

∂R.H.S

∂Bt

= ν(1−α) Bt

+βϕ_kν(1−α) Bt

+βϕϕϕ^T_Me₁−βϕ_R_t+1 +βϕ_I_Bω_B Bt

= 0

⇒B_t^∗ = ν(1−α)(1 +βϕ_k) +βω_Bϕ_I_B β(ϕ_R_t+1 −ϕ_M₁)

(17)

Determining the optimal use of “brown” inputs is more complex. The share parameters ν,(1−α) positively affect the use of brown inputs. If the shadow value of capital increases, more capital will be used, resulting in more energy demand. The term βωBϕI_B captures the additional benefit of using B_t due to learning-by-doing. A higher value of ω_B implies that we learn a lot from the use of brown inputs, and a higher shadow value of knowledge stock ϕ_I_B captures that we value knowledge more. Both terms will lead to more use of brown inputs. Even though we assumed that brown energy inputs are extracted and used costlessly, we can consider the denominator to represent the “cost” of using fossil fuels.

It captures both the opportunity cost of limiting future use of the non-renewable and how much the usage will hurt our economy in the future. The term ϕ_R_t+1 captures the Hotelling rent of the brown resource¹⁰. The term ϕ_M₁ acts as the shadow value of the stock of carbon in the atmosphere. Since this ϕ_M₁ term reflects “shadow damages”, its value will be negative. Higher absolute values of ϕ_M₁ indicates that emissions are more

10It can be thought of as a measure of how scared are we that this resource will run out soon. If we are terrified of it running out soon and demand rent to extract, the extraction rate will be lower

(26)

damaging to the economy.

∂R.H.S

∂G_t = να

G_t− γ

(1−γG_t) +βϕk

να

G_t − γ (1−γG_t)

+βϕ_I_Gω_G G_t = 0

⇒G^∗_t = να(1 +βϕ_k) +βω_Gϕ_I_G γ_t (1 +να)(1 +βϕ_k) +βω_Gϕ_I_G

(18)

The numerator in equation 18 has a similar interpretation to the numerator of the FOC w.r.t B. The share parameters, the shadow value of capital, and the value of learning associated with G_t all positively impact green energy usage. Since green energy inputs are assumed to be rented, there are costs associated with their use. The denominator here relates the benefits of using green inputs to the costs of using the inputs. A higher γ_twill represent higher levels of cost and will result in less use of renewable energy. It should be noted that if we insert forG^∗_t into the cost equationh, the gammas cancel each other out.

Hence the rest of the terms in (18) directly determine the optimal percentage of the final good that should be invested in renewable energy inputs. As both the β and γ values are less than unity, the total effect of a higher valuation of knowledge of renewable energy usage or faster learning will have a positive impact on the use of green energy inputs.

∂R.H.S

∂i_G,t =− 1

(1−i_G,t) − βϕ_k

(1−i_G,t)+βϕ_I_Gη_G = 0

⇒i^∗_G,t = 1− (1 +βϕk) βϕ_I_Gη_G

(19)

As iG is the percentage of final goods used on green research, we find it to be given as a 1−f raction expression. The numerator of the fraction captures the opportunity cost of not saving for more capital in the next period. The denominator consists of how valuable improved knowledge stock in the green sector is to us (ϕ_I_G) and how effective research spending is (ηG). A smaller value of the fraction will lead to higher levels of research of renewable technologies, so if we value knowledge in the green sector more, or spending on research becomes more effective, we invest more in green energy research. This logic is the same for the optimal investment in non-renewable energy technologies.

∂R.H.S

∂i_B,t =− 1

(1−i_B,t)− βϕ_k

(1−i_B,t)+βϕI_BηB,t = 0

⇒i^∗_B,t = 1−(1 +βϕ_k) βϕ_I_Bη_B

(20)

(27)

4.4 Verifying the trial solution

We can insert for the optimal controls in our Bellman-equation. By isolating all terms related to the stocks on the equation’s left-hand side, we obtain the following.

ϕ_k−κ−βϕ_kκ k_t+

ϕ_τ +ξ₀+βϕ_kξ₀−βϕ_τ(1−σ) τ_t +

ϕ

ϕϕ^T_M −βϕϕϕ^T_MΦΦΦ−βϕ_τσ eee₁ M1,pre

M M M_t+

ϕ_R_t −βϕ_R_t+1 R_t +

ϕ_A_G−να−βϕ_kνα−βϕ_A_G

ln(A_G,t) +

ϕA_B −ν(1−α)−βϕkν(1−α)−βϕA_B

ln(AB,t) +

ϕ_I_G,t −βϕ_A_G−βϕ_I_Gρ

I_G,t+

ϕ_I_B −βϕ_A_B −βϕ_I_Bρ

I_B,t+ϕ_t

=

ln(x^∗_t) +a_t+ναln(G^∗_t) +ν(1−α) ln(B_t^∗) +(1−κ−ν) ln(L_t) +ξ₀+ ln(h(G^∗_t, i^∗_G,t, i^∗_B,t)) +βh

ϕ_k

ln(1−x^∗_t) +a_t+ +ναln(G^∗_t) +ν(1−α) ln(B_t^∗) +(1−κ−ν) ln(L_t) +ξ₀+ ln(h(G^∗_t, i^∗_G,t, i^∗_B,t))

+ϕϕϕ^T_M eee₁B_t^∗

+ϕ_R_t+1(−B_t^∗) +ϕIG

ηGi^∗_G,t+ωGln(G^∗_t)

+ϕIB

ηBi^∗_B,t+ωBln(B_t^∗)

+ϕt+1

i

(21)

For the equality in equation 21 to hold, the coefficients in front of state variables on the left-hand side must equate to zero. This is used to pick out the shadow values.

ϕ_k−κ−βϕ_kκ= 0 −→ϕ_k = κ

1−βκ (22)

We find that the shadow value of capital capital depends on the share parameter of capital, and how we value future consumption.

ϕ_τ +ξ₀+βϕ_kξ₀−βϕ_τ(1−σ) = 0−→ϕ_τ =− (1 +βϕ_k)ξ₀

1−β(1−σ) (23) The shadow value of generalized temperature is found to be negative. As this value captures the damage of increasing temperature, the negative value is intuitive. A larger discount factor of utility makes us care more about future welfare, hence increases the absolute value of ϕ_τ, along with the measure of how damaging any level of temperature

(28)

is ξ₀. Furthermore, if the next periods’ temperature is highly dependant on the current temperature, implying that (1−σ) is high, the shadow value of temperature grows in absolute value¹¹.

ϕϕϕ^T_M −βϕϕϕ^T_MΦΦΦ−βϕ_τσ eee₁

M_1,pre = 0 −→ϕϕϕ_MM_M^T =βϕ_τσ 1

M_1,preeee^T₁(111−βΦΦΦ)⁻¹ (24) The vector ϕϕϕ_M_M_M^T contains the shadow values of all the different reservoirs. If we extract the top most element in the vector, we obtain the shadow value of the stock of pollutant in the atmosphere.

ϕ_M₁ =βϕ_τσ 1

M_1,pre[(111−βΦΦΦ)⁻¹]_1,1 (25) As we can see, the shadow value of the atmospheric emission stock is connected withϕ_τ. Because ϕ_τ is negative, so is ϕ_M₁. A higher absolute value of ϕ_τ, implying that increases in temperature are very damaging, increases the absolute value of ϕ_M₁. A high value of σ implies that temperatures converge rapidly to the long-run equilibrium temperature, leading to a higher absolute value for the emission stock’s shadow value. The vector eee^T₁ multiplied with the row vector extracted from the ΦΦΦ, pics out the element with indices 1,1 from the matrix (111−βΦΦΦ)⁻¹. This element contains information on how much of the current atmospheric pollution stock is left in the next period. If much of the pollutants remain in the atmosphere, the absolute value of ϕ_M₁ increases.

ϕ_R_t −βϕ_R_t+1 = 0−→ϕ_R_t =β^−tϕ_R₀ (26) The shadow value of the stock of fossil fuels is needed to be at levels which makes the rate of extraction comply with the Hotelling rule. In the calibration, this value is pinned down by the assumption of total resource depletion.

ϕ_A_G−να−βϕ_kνα−βϕ_A_G = 0−→ϕ_A_G = να

(1−β)(1−βκ) (27) ϕA_B −ν(1−α)−βϕkν(1−α)−βϕA_B = 0 −→ϕA_B = ν(1−α)

(1−β)(1−βκ) (28) Both ϕA_G and ϕA_B are very similarly determined. Both are positively related to their respective share parameter and the discount factor. If the share parameter of capital

11A high value of (1−σ) would imply ”sticky temperatures”, so that an increase in temperature persist for longer

(29)

becomes larger so that more capital is demanded, more energy is required, leading to higher shadow values of productivity in the energy sector. These parameters similarly influence their respective knowledge stocksϕ_I_G andϕ_I_B but weighted by the ρparameter, which indicated how fast knowledge is forgotten. If we lose knowledge more slowly, the value of the knowledge stock grows. It should be noted that neither ofϕ_A_G, ϕ_A_B, ϕ_I_G, ϕ_I_B is dependant on how damaging emissions are to the economy. This is a major shortcoming of the present model, as intuitively it should be more important to develop technologies in the energy sector if emissions are more damaging. The issue arises because of the linear separability required by the linear-in-state framework.

ϕIG−βϕAG−βϕIGρ= 0 −→ϕIG,t = βνα

(1−β)(1−βκ)(1−βρ) (29) ϕ_I_B −βϕ_A_B −βϕ_I_Bρ= 0 −→ϕ_I_B,t = βν(1−α)

(1−β)(1−βκ)(1−βρ) (30) An issue with the shadow values ϕIG and ϕIB is that they measure the change in utility per unit change in I_j, j ∈ {G, B}. As the stock of knowledge determines the growth rate of productivity for energy inputs, a one unit increase leads to an increase in the growth rate by 100%. A more apt measurement when finding policy rates would be the change in utility per percentage point change in Ij. Let us define a value ϕ^1%_I_j as the shadow value of a one percent change in I_j,t.

ϕ^1%_I_j = ϕ_I_j

100 (31)

With these shadow values, the left-hand side of equation 21 mostly disappears, and the remaining expression can be found in equation 32.

ϕ_t−βϕ_t+1=

ln(x^∗_t) +a_t+ναln(G^∗_t) +ν(1−α) ln(B_t^∗) +(1−κ−ν) ln(L_t) +ξ₀+ ln(h(G^∗_t, i^∗_G,t, i^∗_B,t)) +βh

ϕ_k

ln(1−x^∗_t) +a_t+ +ναln(G^∗_t) +ν(1−α) ln(B_t^∗) +(1−κ−ν) ln(L_t) +ξ₀+ ln(h(G^∗_t, i^∗_G,t, i^∗_B,t)) +ϕϕϕ^T_M

eee1B_t^∗

+ϕRt+1(−B_t^∗) +ϕI_G

ηGi^∗_G,t+ωGln(G^∗_t)

+ϕ_I_B

η_Bi^∗_B,t+ω_Bln(B_t^∗)i

(32)

(30)

For this equation to hold, we need to pick a sequence{ϕ₀, ϕ₁, ϕ₂, ϕ₃, ..}. The initial value is found under the condition limt→∞β^tV(kt, τt, MMMt, Rt,ln(AG,t),ln(AB,t), IG,t, IB,t, t) = 0 ⇒ limt→∞β^tϕ_t= 0.

We can now insert for the shadow values in the optimal controls and derive expressions that only depend on exogenously given parameters.

x^∗_t = 1

(1−β_1+βκ^κ ) = 1−βκ (33)

B_t^∗ = ν(1−α) _1−βκ¹

1 + (1−β)(1−βρ)^β²^ω^B

β(β^−tϕ_R₀ −βϕ_τσeee₁_M¹

1,pre(1−βΦΦΦ)⁻¹) (34)

G^∗_t = να 1 + (1−β)(1−βρ)^β²^ω^G

γt

1 +να 1 + (1−β)(1−βρ)^β²^ω^G

(35) i^∗_G,t = 1− (1−β)(1−βρ)

β²η_Gνα (36)

i^∗_B,t = 1−(1−β)(1−βρ)

β²η_Bν(1−α) (37)

As we can see, the fraction of the final good delegated to consumption after damages and costs are accounted for, x^∗_t, only depend on β and κ. This percentage will remain constant over time. G^∗_t will become larger if the cost parameterγ_tgoes down, representing that any unit of renewable energy inputs becomes a smaller fraction of the final good.

As previously discussed, the share of final output after accounting for damages devoted to spending on renewable energy inputs will not depend on the cost parameter. Hence, spending onG^∗_t will be a constant fraction of output over time. Two factors reduce the use of fossil fuels. The first is that as we extract, the stock of fossil fuels becomes more scarce, leading to increases in β^−tϕ_R₀. The second is that emissions become more damaging to the economy. The results of i^∗_B,t and i^∗_G,t tells us that a constant share of GDP should be devoted to research spending. This share is increasing how efficient the research is and how slow knowledge is forgotten.

(31)

4.5 The social cost of carbon

The shadow value ϕ_M₁ represents the utility loss of emitting one more ton of carbon. As seen in section 2.1, the social cost of carbon is denoted in consumption units, not utility.

We can transform the change in utility to a change in consumption units by finding the the marginal value of consumption u(C) = ln(C).

du=u⁰(C)dC = 1 x^∗_tYˆt

dC −→x^∗_tYˆtdu =dC (38) We know that x^∗_t = 1−βκ, and per definition the change in utility caused by an extra unit of emission in the atmosphere du is equal to ϕ_M₁. Since we want to represent the optimal price of emissions as a positive number, andϕ_M₁ <0, we need to multiply by−1.

By inserting for ϕ_τ and ϕ_k we obtain the following result for the social cost of carbon.

SCC_t=−(1−βκ) ˆY_tβϕ_τσ 1

M_1,pre[(111−βΦΦΦ)⁻¹]_1,1

⇒SCC_t= βYˆ_t M1,pre

ξ₀σ

1−β(1−σ)[(111−βΦΦΦ)⁻¹]_1,1

(39)

The expression only depends on known values and constant parameters, hence it is a fully analytical expression for the social cost of carbon. With dependency on the discount factor in both the denominator and numerator, both of which leads to a higher SCC, the term can vary widely given how we value future consumption. The social cost of carbon is positively related to the amount of pollutant that remain in the atmosphere, the damage parameter ξ₀ and the speed of convergence parameter σ. As most people think of emissions in terms of CO₂, we can convert the social cost of carbon into the social cost of carbon dioxide by dividing SCC_t with a conversion rate of 3.67, as there is approximately one ton of carbon in 3.67 tons of CO2.

4.6 Optimal spending on technological advancement

Just as we found found the optimal price of carbon by transforming the shadow value of emissions into consumption equivalents, we can find the optimal subsidies for research promoting activities S_j,t, j ∈ {G, B} by transforming the shadow value of the knowledge

Don’t Forget About Technology!