Optimal Extraction of Fossil Fuels when Emissions cause Climate
Change
Jørgen Larsen
Master thesis – Department of Economics UNIVERSITY OF OSLO
13 May 2019
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Optimal Extraction of Fossil Fuels when Emissions Cause Climate Change
Jørgen Larsen
Master thesis – Department of Economics
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Abstract
This thesis connects climate change to a classical resource extraction model. It employs a similarity between resource extraction under climate change and resource extraction under stock dependent extraction costs, building on a recent finding that temperature change is approximately proportional to cumulative historic emissions. After discussing the general model, the thesis develops a new closed-form solution under a flexible damage function and a general intertemporal elasticity of substitution. This generality in consumption and damage specification comes at the cost of abstracting from an explicit production sector. The
consumption of fossil fuels follows a “flatter” trajectory where fossil fuel consumption starts lower and falls slower as compared to the case without climate change. The model derives the optimal amount of fossil fuel reserves that should never be extracted.
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Acknowledgements
My supervisor, Christian Traeger, has been an invaluable support throughout this thesis. His expertise in resource and environmental economics has provided crucial guidance to my reasoning and writing. Christian has demanded excellent quality out of every sentence. This strictness has elevated the thesis profoundly, and improved my skills in writing. Nils Christian Framstad has provided mathematical guidance which has been crucial in the development of the model. Liselotte Seljom has proofread the thesis and found errors which I embarrassingly would have overlooked.
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Table of contents
1 Introduction ... 1
2 The general model ... 3
2.1 The model ... 3
2.2 Deriving the Euler equation ... 6
3 A novel closed form solution ... 9
3.1 The model ... 9
3.2 Proof of Proposition 1... 15
3.3 Proof of Proposition 2... 19
4 Applying the model ... 21
4.1 Fitting the model to an arbitrary planning period beginning ... 21
4.2 Calibration ... 22
5 Conclusion ... 27
Bibliography ... 29
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1 Introduction
Fossil fuels are an important resource in a modern economy. 80.1% of global energy consumption is accounted for by fossil fuels (The World Bank 2014).They are an essential factor of production in air and maritime transport services, and they are not easily substituted for in transport services on land. Fossil fuels are also the dominant factor of production in electrical power production, accounting for 65.2% of global power production (The World Bank 2014).
It has been a long standing research question in economics how to optimally manage the existing and scarce reserves of fossil fuels. Hotelling’s (1931) classic paper on extraction of exhaustible resources, finds that the scarcity rent of the optimally managed exhaustible resource has to grow at the rate of interest. This trajectory for the scarcity rent is since known as the “Hotelling’s rule” and is necessary to avoid intertemporal arbitrage; if the trajectory of the scarcity rent were different than that of Hotelling’s rule, it would be profitable to extract the entirety of the stock today as the present value of the stock would be lower than that of the assets the owner could buy with the revenue generated by extraction. In the opposite case, the owner would never extract, as the present value of the stock would exceed that of the revenue generated by extraction. Hence, in a competitive market for fossil fuels, the price net of extraction costs should increase by the rate of interest, making the owner indifferent to when to extract the marginal unit fuel.
Combusting fossil fuels, e.g. for energy production or transport, releases carbon dioxide (CO2) into the atmosphere. Some of the emitted CO2 is absorbed by natural carbon sinks, such as oceans or by plants through photosynthesis. The CO2 which is not absorbed remains in the atmosphere and enhances radiative forcing, causing an increase in average atmospheric temperature. When temperature increases ice sheets and glaciers melt which increases the sea level, and extreme weather such as droughts, heat waves and flooding become more frequent (IPCC 2013). These effects have negative consequences for economic activity. Depreciation of physical capital such as infrastructure and buildings increases as they become more costly to maintain. Areas at sea level become uninhabitable or require construction of dykes, and farmlands become unfertile.
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The relationship between emissions and economic harm from temperature has been studied by Integrated Assessment Models (IAM) for climate change. Classically, IAMs have been
numeric models built for policy advice. More recently, IAMs with closed form solutions have been developed to provide structural insights. Golosov et al. (2014) developed an IAM for climate change with closed form solutions using log-utility. Hambel, Kraft and Schwartz (2018) and Traeger (2018) developed IAMs for climate change with Epstein-Zin preferences, using unit elasticity of intertemporal substitution and general isoelastic relative risk aversion.
Recently, there has been an increasingly accepted finding that the increase in temperature is approximately proportional to cumulative historic emissions. This relationship was proposed by Matthews et al. (2009) and has been further established by Gillett and Arora (2013). A recent overview of the performance of this relationship is found in the 2013 IPCC report (IPCC 2013). When temperature is a function of accumulated emissions, temperature effectively acts as a stock-dependent cost of extraction. Schultze (1974) develops a general model for a resource market with a stock-dependant externality, and finds that such “[…]
externalities support efforts to conserve exhaustible resources to a greater extent than do private markets”. A closed-form solution to this problem using linear demand and quadratic stock dependant extraction costs was developed by Tahvonen (1995) to analyse strategic behaviour between countries which are net importers of fossil fuels and countries which are net exporters. Karp (1991) analysed a game between importers and exporters of fossil fuels where importers exercise monopsony power, and where marginal extraction costs were linear in remaining reserves, also with linear demand.
In this thesis, I build a simple stylised model relating climate change and optimal fossil fuel extraction, exploiting the aforementioned linear relation between cumulative emissions and temperature change. Chapter 2 presents the general model structure. Chapter 3 develops a new closed-form solution for utility with general isoelastic intertemporal elasticity of substitution. Chapter 4 calibrates the model with data on contemporary fossil fuel reserves and the increase in temperature that has already occurred since the industrial revolution.
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2 The general model
This chapter introduces a model of optimal fossil fuel extraction in the presence of climate change. First, I introduce the general model structure. It makes use of insight that temperature increase since preindustrial times has been and will remain approximately constant in
cumulative emissions. Second, I discuss the intertemporal optimality conditions (Euler equations) of this general problem.
2.1 The model
The consumption or flow of fossil fuels measured in units of carbon is denoted 𝑥𝑡. Combustion of fossil fuels causes CO2 emissions. CO2 emissions then cause atmospheric temperate to rise due to the greenhouse effect. The atmospheric temperature is denoted 𝑧𝑡, and is measured as the warming since preindustrial time in degree Celsius. Non-combustive use of fossil fuels will be neglected in this model, and the model does not distinguish between consuming and burning fossil fuels. Moreover, we abstract from an explicit production sector and the representative agent derives consumption directly from the flow of fossil fuels. Let 𝑢(𝑥𝑡, 𝑧𝑡) be the instantaneous utility function of a representative global consumer whose utility increases in fossil fuel use and decreases in temperature increase. A social planner maximises the future sum of utility by regulating the carbon flow:
(1)
𝑊 = max
𝑥𝑡,𝑧𝑡,𝑡∈[0,𝑇){∫ 𝑢(𝑥𝑡, 𝑧𝑡)𝑒−𝜌𝑡𝑑𝑡
𝑇 0
},
𝑇 ≥ 0, 𝑥𝑡 ≥ 0 ∀ 𝑡 ∈ [0, 𝑇)
where 𝜌 ≥ 0 is the rate of interest. t denotes a point in time, where 𝑡 = 0 is preindustrial time.
T is the duration of the planning period, and is determined endogenously. The global reserves of fossil fuels at a given point in time are 𝑦𝑡 = ∑𝑛𝑖=1𝜓𝑖𝑦𝑖𝑡. The subscript i denotes a type of fossil fuel, like crude oil of a certain quality. 𝜓𝑖 is a parameter that converts the quantity of type i reserves to units of carbon. 𝑦0 is then the untapped preindustrial level of reserves, and is considered a constant. Extraction of fossil fuel reduces the level of the reserves:
(2) 𝑦̇𝑡= −𝑥𝑡
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where the condition 𝑦𝑡 ≥ 0 ∀ 𝑡 ∈ [0, 𝑇) imposes the fact that reserves must always remain non-negative. Let utility from consumption have the properties
(3) 𝜕𝑢
𝜕𝑥𝑡∗ > 0
(4) 𝜕2𝑢
𝜕𝑥𝑡2 < 0.
Property (3) says utility from consumption is positive on the margin at any time throughout the planning period, when optimally controlled. Property (3) thus allows there to be
consumption levels where consuming an additional unit fuel causes more harm than benefit to the representative agent, but such consumption levels are never realised in optimum. Property (4) says utility from consumption is diminishing in the intratemporal consumption level.
Furthermore, let the impact of 𝑧𝑡 on 𝑢(𝑥𝑡, 𝑧𝑡) have the properties
(5) 𝜕𝑢
𝜕𝑧𝑡 < 0 ∀ 𝑧𝑡> 0
(6) 𝜕2𝑢
𝜕𝑧𝑡2 ≤ 0 ∀ 𝑧𝑡 > 0.
Property (5) says any increase in atmospheric temperature above preindustrial level impacts utility negatively. The statement captures that a warmer climate harms economic activity, e.g.
through destruction of physical capital when sea levels rise, or mass migration when human habitations are rendered uninhabitable by droughts or flooding. Property (6) says the harmful effects of a unit increase in temperature is either linear or becomes larger the higher
temperature is.
Temperature is assumed to be linearly increasing in accumulated emissions. This relationship is attributed to Matthews et al. (2009), who simulated the carbon-climate response when subject to exponentially increasing carbon emissions, and to instantaneous emission pulses.
They find that “after an initial adjustment of period about a decade, the [carbon-climate response] remained almost constant at approximately 1.7 °C [per thousand GtC (Gigatonne carbon) emitted]”. According to their simulations, the saturation of natural carbon sinks approximately offsets the saturation of radiative forcing, making the temperature response of the marginal unit carbon emitted independent of the units already emitted. Additionally, they
5 find that temperature per unit atmospheric carbon is increasing over time, but this effect is approximately offset by the rate at which carbon is absorbed by carbon sinks, making the temperature response of a unit carbon emitted constant over time. The relationship between temperature and emissions is then
(7) 𝑧𝑡 = 𝜅 ∫ 𝑥𝜏𝑑𝜏
𝑡 0
where 𝜅 is the carbon-climate response parameter. Taking the time derivative of equation (7) yields the equation of motion for temperature:
(8) 𝑧̇𝑡= 𝜅𝑥𝑡.
Let 𝜆𝑦,𝑡 be the discounted scarcity rent on fossil fuel reserves for the point in time t. The discounted scarcity rent is the opportunity cost of consuming a unit fossil fuel today, and contains two elements: that the unit consumed today cannot be consumed at another point in time, and that consuming the unit deteriorates utility for future consumption through
increased temperature. Then
(9) −𝜆𝑧,𝑡 = 1
𝜅𝜆𝑦,𝑡
is the discounted shadow value of temperature. It carries the same interpretation as 𝜆𝑦,𝑡, but it is measured in units of temperature, instead of consumption.
(10) 𝜇𝑗,𝑡≡ 𝑒𝜌𝑡𝜆𝑗,𝑡, 𝑗 = {𝑦, 𝑧}
are then, respectively, the current value scarcity rent of reserves and current shadow value of temperature, which I will from now on simply refer to as the “scarcity rent” and “shadow value of temperature”. While 𝜆𝑗,𝑡 is the opportunity cost measured in present value utility, 𝜇𝑗,𝑡 is the amount of utility that actually is foregone when the point in time t arrives.
Using the equation of motion for temperature (8) and the definition of the shadow value of temperature (9) and (10), the current value Hamiltonian is
(11) ℋ𝑡𝑐 = 𝑢(𝑥𝑡, 𝑧𝑡) + 𝜇𝑧,𝑡𝜅𝑥𝑡
where 𝑥𝑡 is the control variable and 𝑧𝑡 is the state variable. The Hamiltonian measures the instantaneous contribution to welfare. It takes into account how consumption and temperature
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at the given point in time generates utility, and how that consumption affects future utility through reserve depletion and increased temperature. When denoted in “current value”, the Hamiltonian measures the instantaneous contribution to welfare in units of utility at the point in time t.
2.2 Deriving the Euler equation
The first order conditions of the Hamiltonian are
(12) 𝜕ℋ𝑡𝑐
𝜕𝑥𝑡 = 𝜕𝑢
𝜕𝑥𝑡+ 𝜇𝑧,𝑡𝜅 =! 0
(13) 𝜕ℋ𝑡𝑐
𝜕𝑧𝑡 = 𝜕𝑢
𝜕𝑧𝑡=!− 𝜆̇𝑧,𝑡 = 𝜌𝜇𝑧,𝑡− 𝜇̇𝑧,𝑡 Taking the time derivative of equation (11) yields
(14) 𝜇̇𝑧,𝑡 = −1 𝜅
𝜕2𝑢
𝜕𝑥𝑡2𝑥̇𝑡−1 𝜅
𝜕2𝑢
𝜕𝑥𝑡𝜕𝑧𝑡𝑧̇𝑡 = −1 𝜅
𝜕2𝑢
𝜕𝑥𝑡2𝑥̇𝑡− 𝜕2𝑢
𝜕𝑥𝑡𝜕𝑧𝑡𝑥𝑡. Substituting equation (12) and (14) into equation (13) returns the Euler equation:
(15) −
−𝜕2𝑢
𝜕𝑥𝑡2
𝜕𝑢
𝜕𝑥𝑡 𝑥̇𝑡
⏟
𝑎1<0
+
− 𝜕𝑢
𝜕𝑧𝑡
𝜕𝑢
𝜕𝑥𝑡 𝜅
⏞
𝑐1>0
+
𝜕2𝑢
𝜕𝑥𝑡𝜕𝑧𝑡
𝜕𝑢
𝜕𝑥𝑡 𝜅𝑥𝑡
⏞
𝑑1
⏟
𝑏1
= 𝜌
The term labelled 𝑎1 captures the change of marginal utility as a consequence of consumption growth. As 𝑎1 < 0, consumption must decrease over time, as long as 𝑏1 < 𝜌. 𝑏1 captures that a consumed unit fossil fuel has a negative impact on utility because temperature increases by κ unit Celsius. 𝑏1 consists of two terms: 𝑐1 is the marginal rate of substitution between
consumption and emissions, and states the least amount of consumption needed to tolerate the increase in temperature caused by a unit carbon emitted. 𝑐1 is thus a consumption
denominated measure of the harm caused when temperature increases by 𝜅 unit Celsius. The term 𝑑1 captures how increases in temperature affects the marginal utility of consumption.
7 If there were no relationship between consumption and temperature, the term 𝑏1 would be 0, and the Euler equation would read −𝑎1 = 𝜌. This would replicate Hotelling’s rule, which says the scarcity rent of an optimally managed exhaustible resource grows at the rate of interest (Hotelling 1931). Hotelling discussed the competitive market equilibrium, but by the First Theorem of Welfare, his result applies to the social optimum as well (Strøm and Vislie 2007). In Hotelling’s model, the only opportunity cost is consumption at other points in time.
Thus the rule is equivalent to asserting that consumption should decrease over time as long as the interest rate is positive. If Hotelling’s rule is not adhered to, one could re-allocate
consumption between points in time, and achieve a higher level of discounted utility at the point in time which consumption is allocated to than the decrease suffered at the point in time consumption is allocated from. Hence, by Hotelling’s rule, the agent should be indifferent to which point in time to allocate the marginal unit of fuel.
However, fossil fuel consumption causes a temperature increase that reduces welfare, as captured by 𝑏1.The contribution 𝑏1 is typically positive. For a larger 𝑏1the Euler equation requires a less negative growth rate of consumption, i.e., the time path of consumption becomes flatter the more harmful the consequences of higher temperature are. The presence of harmful emissions accordingly dampens Hotelling’s rule. If 𝑏1 > 𝜌, the Euler equation says consumption should increase over time because harm from temperature is so severe that the representative agent must be compensated in the future by getting more consumption than today. However, for any finite amount of reserves, it is not feasible to continuously increase consumption as the constraint 𝑦𝑡 ≥ 0 becomes binding. Hence, 𝑏1 > 𝜌 implies the first order conditions cannot be satisfied, and the corner solution with no consumption at any time is optimal. If 𝑏1 = 𝜌, the Euler equation is satisfied and asserts that consumption should be constant over time. But a constant consumption trajectory is feasible only when consumption is zero. Consequently, for any consumption to be optimal, the condition 𝑏1 > 𝜌 must be satisfied.
The term 𝑑1 in the Euler equation contributes or dampens the effect of 𝑏1 on the consumption trajectory, depending on the sign of 𝜕
2𝑢
𝜕𝑥𝑡𝜕𝑧𝑡. When 𝜕
2𝑢
𝜕𝑥𝑡𝜕𝑧𝑡> 0, it implies that increased temperature has a positive effect on the marginal utility of consumption, e.g. through increased need for refrigeration or fuel intensive replacement investments. In this case, 𝑑1contributes to 𝑏1 because a warmer climate makes fossil fuels even more important for the
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points in time to come. The social planner thus wants to conserve the remaining reserves. In the opposite case, when 𝜕
2𝑢
𝜕𝑥𝑡𝜕𝑧𝑡< 0, a warmer climate makes fossil fuels less important, so the planner does not need to be as considerate with them, and the effect of 𝑏1 on the consumption trajectory is dampened. If the magnitude of − 𝜕2𝑢
𝜕𝑥𝑡𝜕𝑧𝑡 is sufficiently large, the effect of 𝑑1 dwarfs that of 𝑐1, making 𝑏1 negative, and the effect of temperature is steepening on the consumption trajectory.
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3 A novel closed form solution
This chapter introduces an explicit utility function that carries the utility function properties (3), (4), (5) and (6), and where the optimal trajectories for consumption, fossil fuel reserves and temperature have closed form solutions. First, I introduce the utility function and state the optimal trajectories. Second, I prove that the stated trajectories satisfy the first order
conditions of the problem. Third, I prove that the trajectories satisfy Arrow’s sufficient condition for optimality.
3.1 The model
The instantaneous utility function is
(16) 𝑢(𝑥𝑡, 𝑧𝑡) = { 𝜎
𝜎 − 1(𝑥𝑡(1 − 𝛾𝑧𝑡)𝜃)
𝜎−1
𝜎 for 𝜎 ≠ 1 ln 𝑥𝑡+ 𝜃 ln(1 − 𝛾𝑧𝑡) for 𝜎 = 1
where 𝜎 > 0 is the elasticity of intertemporal substitution. The utility function states the agent’s utility is isoelastic with elasticity of intertemporal substitution in the net consumption flow,
(17) 𝑐𝑡 ≡ 𝑥𝑡(1 − 𝛾𝑧𝑡)𝜃.
Net consumption is proportional1 to fossil fuel extraction and reduced by temperature
increase. This feature captures that the higher temperature is the more fuel has to be diverted to replacement investments, like building dykes and maintenance of infrastructure. 𝑐𝑡 is accordingly the amount of fuel left for utility generating consumption after damages from temperature are taken into account. The share of consumption that is damaged is
(18) 𝐷𝑡 ≡ 1 − (1 − 𝛾𝑧𝑡)𝜃.
In 𝐷𝑡, units of temperature are transformed to shares of consumption units by the parameters 𝛾 and 𝜃. As marginal utility becomes negative if 𝑧𝑡 > 1
𝛾, 𝛾 captures the highest temperature tolerated, which is 1
𝛾. {𝜃 ≥ 0: 𝜎 > 1 ⇒ 𝜃 ≤ 𝜎
𝜎−1} determines the curvature of the deterioration
1 The proportionality constant is normalised to unity as a proportional constant can be pulled out of the integral and, thus, does not affect the optimal trajectories.
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of net consumption by temperature, where the parameter restriction ensures the validity of property (6). For a given value of 𝛾, 𝜃 determines the rate of convergence to the limiting value of the damage share, lim𝑡→𝑇𝐷𝑡. For 𝜃 = 1, the damage fraction is linear in temperature, so a degree Celsius increase in temperature implies the fraction of consumption lost to
damages is 𝛾. For 𝜃 ∈ (0,1) the damage fraction is strictly convex, so the second degree Celsius of temperature increase implies a higher toll on economic activity than the first. The case 𝜃 = 0 or 𝛾 = 0 reduces the model to a standard Hotelling problem with isoelastic preferences. Additionally, the sign of the cross derivative is entangled with the intertemporal elasticity of substitution, such that
𝜕2𝑢
𝜕𝑥𝑡𝜕𝑧𝑡 > 0 ⇔ 𝜎 > 1, 𝜕2𝑢
𝜕𝑥𝑡𝜕𝑧𝑡= 0 ⇔ 𝜎 = 1,
𝜕2𝑢
𝜕𝑥𝑡𝜕𝑧𝑡 < 0 ⇔ 𝜎 ∈ (0,1).
Proposition 1: The trajectories of 𝑥𝑡, 𝑦𝑡, 𝑧𝑡 and 𝜇𝑦,𝑡 that satisfy the Euler equation and are, thus, necessary conditions for an optimum are
(i) 𝑥𝑡∗ = 𝜌𝜎
1 + 𝜃 𝜙
⏟ 𝛾𝜅
𝑥0
𝑒−𝜌𝜎𝑡 [1 − 𝜙(1 − 𝑒−𝜌𝜎𝑡 )]1+𝜃−𝜃
(ii) 𝑦𝑡∗ = 𝑦0− 1
𝛾𝜅+ 1
𝛾𝜅[1 − 𝜙(1 − 𝑒−𝜌𝜎𝑡)]1+𝜃1
(iii) 𝑧𝑡∗ =1
𝛾−1
𝛾[1 − 𝜙(1 − 𝑒−𝜌𝜎𝑡)]1+𝜃1
(iv) 𝜇𝑦,𝑡∗ = (1 + 𝜃
𝜌𝜎 𝛾𝜅
𝜙)
1
𝜎𝑒𝜌𝑡[1 − 𝜙(1 − 𝑒−𝜌𝜎𝑡)]1+𝜃𝜃
where
𝜙 = {
1 if 𝛾 > 1 𝜅𝑦0
1 − (1 − 𝛾𝜅𝑦0)1+𝜃 if 𝛾 ≤ 1 𝜅𝑦0.
The planning period duration approaches infinity in optimum, i.e. 𝑇∗ → ∞.
𝜙 ∈ [0,1] is a parameter that determines how abundant fossil fuel reserves are, of which 𝜙 = 1 implies there is no fossil scarcity at all. The larger the initial reserve 𝑦0, scarcity is
11 lower. But it is also lower the higher severity of harm from temperature, 𝛾, is. The reason reserve scarcity is determined by 𝛾, is because harm from temperature deteriorates utility of consumption. The higher the damage parameter 𝛾 the more damaging is fossil fuel
consumption. If 𝛾 > 1
𝜅𝑦0, the temperature effect is so detrimental that there is a point in time where consuming an additional unit of fossil fuel contributes negatively to utility. In this case, 𝜙 = 1, and the consumption trajectory becomes independent of reserves. Consequently, when 𝛾 > 1
𝜅𝑦0, fossil fuels are no longer a scarce resource, as consuming all of the reserves is more harmful than beneficial. Instead, it becomes optimal to leave
(19) lim
𝑡→∞𝑦𝑡∗ = 𝑦0− 1 𝛾𝜅
units of the reserves in the ground. Equivalently, the total amount of fuel consumed throughout the planning period is 𝑋 ≡ ∫ 𝑥0∞ 𝑡𝑑𝑡= 1
𝛾𝜅. Temperature then increases over time to its tolerance level, lim𝑡→∞𝑧𝑡∗ = 1
𝛾. 𝜙 ∈ [0,1) implies 𝛾 < 1
𝜅𝑦0, and the temperature effect is sufficiently mild that the reserve constraint, 𝑦0, is binding, and fossil fuels remain scarce. In this case, 𝜕𝑢
𝜕𝑥𝑡 > 0 ∀ 𝑦𝑡 ∈ [0, 𝑦0], so the representative agent would have liked to consume even more fuel after the reserves are fully depleted. In the special case where 𝛾 = 1
𝜅𝑦0, the point in time where damages from temperature turn marginal utility non-positive is precisely when the reserves are fully depleted. In the two latter cases, the total amount of fuel consumed throughout the planning period is naturally 𝑋 = 𝑦0.
Using equation (i), the growth rate for the consumption trajectory is
(20) 𝑔 ≡𝑥̇𝑡∗
𝑥𝑡∗ = −𝜌𝜎 (1 − 𝜙 𝜃 1 + 𝜃
1
𝑒𝜌𝜎𝑡− 𝜙(𝑒𝜌𝜎𝑡 − 1))
⏟
𝑎2
.
When there is no reserve scarcity, the factor labelled 𝑎2 reduces to 𝑎2 = 1
1+𝜃, and the consumption growth rate remains constant over time at 𝑔𝜙=1= − 𝜌𝜎
1+𝜃= 𝑔𝐻
1+𝜃, where 𝑔𝐻≡
−𝜌𝜎 is the consumption growth rate implied by Hotelling’s rule. In this case, the
consumption trajectory becomes flatter the higher 𝜃 is. This is because a higher 𝜃 means
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relatively more consumption must be allocated to the future to counteract that consumption today causes more harm to future utility.
Whenever 𝜙 ∈ (0,1), i.e. fossil fuel reserves are scarce, the growth rate is time dependent. At 𝑡 = 0 the growth rate is at its highest (the least negative), at 𝑔𝜙∈(0,1),𝑡=0 = −𝜌𝜎1+𝜃(1−𝜙)
1+𝜃 . As seen in this expression, the growth rate at 𝑡 = 0 is higher and closer to the no-scarcity growth rate the less scarce the reserves are. The more scarce reserves are, the closer to 𝑔𝐻 the growth rate at 𝑡 = 0 is. Furthermore, when 𝜙 ∈ (0,1), the factor 𝑎2 has the properties 𝑑𝑎2
𝑑𝑡 > 0 and lim𝑡→∞𝑎2 = 1. These properties mean the consumption growth rate becomes lower (more negative) over time, and gradually converges to the steady state level 𝑔𝜙∈(0,1)𝑠𝑠 = 𝑔𝐻. The rate of convergence towards Hotelling’s growth rate is faster the more scarce reserves are, i.e. the lower 𝜙 is.
There are consequently two forces driving the consumption growth rate. On the one hand, the intertemporal allocation of a scarce exhaustible resource which “wants” to follow Hotelling’s rule. On the other hand, the harm caused by temperature increase “wants” a flatter
consumption trajectory, as more consumption must be allocated to the future to compensate for the increase in temperature caused by today’s consumption. The less scarce reserves are, the more emphasis the social planner puts on harm from temperature, and the consumption trajectory becomes flatter. The scarcer the reserves are, the less important the temperature aspect, and the consumption trajectory becomes more similar to that of the standard Hotelling case. Additionally, as 𝑎2 can never be larger than unity, the consumption trajectory cannot be steeper than that of the standard Hotelling case. This is in contrast to the general model of chapter 2.2, where there could be a scenario where the effect of temperature on marginal utility is negative and of a sufficiently high magnitude that the consumption growth rate becomes even lower (more negative) than in the standard Hotelling case.
The initial consumption level is
(21) 𝑥0 = 𝜌𝜎
1 + 𝜃 𝜙 𝛾𝜅.
It is lower the higher is 𝜃 because the planner wants relatively more consumption to be allocated to the future. And the only way to achieve a higher future consumption level with a finite stock of reserves and an unaffected planning period duration, is to decrease
13 consumption today. In the case of 𝜃 = 0, temperature causes no harm, and 𝑥0 is reduced to the initial consumption level of the standard Hotelling case, 𝑥0𝐻 ≡ 𝜌𝜎𝑦0. 𝑥0 is also decreasing in 𝛾𝜅, which is harm from temperature measured in carbon units. In the case where 𝜙 = 1, a higher 𝛾𝜅 means more of the reserves is to be left in the ground, so the initial condition must be lower to accompany that. When 𝜙 ∈ (0,1), all of the reserves are consumed, but the initial condition is nevertheless decreasing in 𝛾𝜅 as the consumption trajectory is flatter when temperature is more harmful, so relatively more consumption is allocated to the future. The effect on the initial condition of an increase of the reserve constraint is
(22) 𝜕𝑥0
𝜕𝑦0 = { 𝜌𝜎 1 + 𝜃
1 𝛾𝜅
𝜕𝜙
𝜕𝑦0 > 0 if 𝜙 ∈ [0,1) 0 if 𝜙 = 1.
The initial consumption level trajectory is accordingly higher the more fossil fuel reserves there are to begin with, but only as long as fossil fuels remain scarce. This is intuitive as when fuel is scarce, more reserves means more consumption is possible so the initial condition should increase. And when 𝑦0 > 1
𝛾𝜅, accumulated consumption remains at 𝑋 = 1
𝛾𝜅 no matter the magnitude of 𝑦0, so the initial condition should remain constant.
Inserting for the temperature trajectory (iii) in the definition of the damage share (18), the optimally controlled damage share becomes
(23) 𝐷𝑡∗ = 1 − [1 − 𝜙(1 − 𝑒−𝜌𝜎𝑡)]1+𝜃𝜃 .
The damage share increases over time as past consumption has accumulated carbon in the atmosphere which deteriorates utility of contemporaneous consumption. If fuel reserves are scarce, the damage share converges to lim𝑡→∞𝐷𝑡∗ = 1 − [1 − 𝜙]1+𝜃𝜃 < 1, as it does not exist enough fuel reserves to deteriorate all marginal utility of consumption. If fuel reserves are sufficiently large, the damage share converges to unity, and it becomes optimal to leave some reserves in the ground.
Substituting the initial value of the consumption trajectory (21) and the damage share trajectory (23) in the optimally controlled scarcity rent (iv) yield
(24) 𝜆𝑦,𝑡∗ = 𝑥0−
1
𝜎(1 − 𝐷𝑡∗)
14
where the scarcity rent is discounted by the definition of the scarcity rent (10). When 𝜃 = 0, i.e. temperature causes no harm to utility, the discounted scarcity rent of the Hotelling case is replicated: 𝜆𝑦,𝑡𝐻 ≡ (𝑥0𝐻)−𝜎1. When 𝜃 > 0, the scarcity rent is changed by two effects. On the one hand, the scarcity rent becomes larger because the consumption trajectory is shifted downwards to compensate for future damages of temperature. As less consumption is
awarded, marginal utility of consumption is higher, which makes the opportunity to consume a unit fuel more valuable. On the other hand, the higher the damage share the less utility a unit fuel generates, which makes the opportunity to consume a unit less valuable.
Overall, the consumption trajectory effect is larger than the damage effect in the beginning of the planning period, as damages are small. But as time passes, the damage effect becomes more dominant. At one point, the damage effect dwarfs the consumption trajectory effect, and the scarcity rent becomes smaller than that of the Hotelling case. Consequently, when
increased temperature causes harm the agent prefers to consume the marginal unit fuel sooner rather than later. This is in contrast to the Hotelling case, where the time independence of the discounted scarcity rent implies the agent is indifferent to when to allocate the marginal fuel unit.
Inserting for the trajectories of consumption (i) and temperature (iii) in the definition of net consumption (17) yield
(25) 𝑐𝑡∗ = 𝑥0𝑒−𝜌𝜎𝑡.
Equation (25) states that net consumption decreases over time at the same rate as implied by Hotelling’s rule. As there are no damages from temperature at 𝑡 = 0, the initial net
consumption level and the initial consumption level are both 𝑥0. Equation (25) implies Hotelling’s rule holds true in the utility sense: In optimum, damages evolve in such a manner that one cannot shift utility between points in time and obtain more utility at the point which utility is shifted to than utility lost at the point which utility is shifted from. In contrast to the standard Hotelling case, the initial consumption level, 𝑥0, is lower, so the utility trajectory is shifted downwards. This is because some of the reserves must be used to compensate for the damages, whereas in the Hotelling case, all the reserves can be allocated to utility generating consumption.
15 Proposition 2: When 𝜎 > 1, the parameter restriction 𝜃 ∈ [0, 1
𝜎−1] is a sufficient condition for the trajectories to be optimal. When 𝜎 ∈ (0,1], no additional parameter restriction is needed and the candidate trajectories of Proposition 1 are optimal.
If 𝜃 ∈ ( 1
𝜎−1, 𝜎
𝜎−1] ∧ 𝜎 > 1, the Hamiltonian is strictly convex, as long as 𝛾 > 0. In this this case, harm from temperature is so severe that even an infinitesimal amount of consumption causes more harm than benefit. Hence, the corner solution, where there is no consumption at any time, is optimal. Furthermore, the corner solution is optimal when 𝜌 = 0 for any allowed values of 𝜎,𝛾 and 𝜃. In this case, consumption is suboptimal because any reduction of the reserves leaves too little for the infinitely many and equally important future points in time.
3.2 Proof of Proposition 1
Let temperature be transformed to (26)
𝜔𝑡= 1 − 𝛾𝑧𝑡
Using the equation of motion for temperature (8), the equation of motion for 𝜔𝑡 is
(27) 𝜔̇𝑡 = −𝛾𝜅𝑥𝑡.
Substituting equation (26) and (27) and the explicit utility function (16) into the current value Hamiltonian (11) yield
(28)
ℋ𝑡𝑐 = {
𝜎
𝜎 − 1(𝑥𝑡𝜔𝑡𝜃)
𝜎−1
𝜎 − 𝜇𝜔,𝑡𝛾𝜅𝑥𝑡 for 𝜎 ≠ 1 ln 𝑥𝑡+ 𝜃 ln 𝜔𝑡− 𝜇𝜔,𝑡𝛾𝜅𝑥𝑡 for 𝜎 = 1.
𝑥𝑡 remains the control variable and 𝜔𝑡 is now the state variable.
(29) 𝜇𝜔,𝑡 = −𝛾𝜇𝑧,𝑡
is then the current value co-state variable to 𝜔𝑡. The first order conditions are
(30) 𝜕ℋ𝑡𝑐
𝜕𝑥𝑡 = 𝑥𝑡−
1 𝜎𝜔𝑡
𝜎−1 𝜎 𝜃
− 𝜇𝜔,𝑡𝛾𝜅 =! 0
16
(31) 𝜕ℋ𝑡𝑐
𝜕𝜔𝑡 = 𝜃𝑥𝑡
𝜎−1 𝜎 𝜔𝑡
𝜎−1 𝜎 𝜃−1
=! 𝜌𝜇𝜔,𝑡− 𝜇̇𝜔,𝑡.
The time derivative of the shadow value is (32) 𝜇̇𝜔,𝑡 = − 1
𝛾𝜅 1 𝜎𝑥𝑡−
1 𝜎−1
𝜔𝑡
𝜎−1 𝜎 𝜃
𝑥̇𝑡+ 1 𝛾𝜅
𝜎 − 1 𝜎 𝜃𝑥𝑡−
1 𝜎𝜔𝑡
𝜎−1 𝜎 𝜃−1
𝜔̇𝑡.
Inserting for equation (30) and (32) in equation (31) returns the Euler equation:
𝜃𝑥𝑡
𝜎−1 𝜎 𝜔𝑡
𝜎−1 𝜎 𝜃−1
= 𝜌 1 𝛾𝜅𝑥𝑡−
1 𝜎𝜔𝑡
𝜎−1 𝜎 𝜃
+ 1 𝛾𝜅
1 𝜎𝑥𝑡−
1 𝜎−1
𝜔𝑡
𝜎−1 𝜎 𝜃
𝑥̇𝑡− 1 𝛾𝜅
𝜎 − 1 𝜎 𝜃𝑥𝑡−
1 𝜎𝜔𝑡
𝜎−1 𝜎 𝜃−1
𝜔̇𝑡.
Multiplying by 𝜎𝛾𝜅𝑥𝑡1+
1 𝜎𝜔𝑡−
𝜎−1 𝜎 𝜃+1
gives
(33) 𝜎𝛾𝜅𝜃𝑥𝑡2 = 𝜌𝜎𝑥𝑡𝜔𝑡+ 𝜔𝑡𝑥̇𝑡− (𝜎 − 1)𝜃𝑥𝑡𝜔̇𝑡. Taking the time derivative of the equation of motion (27) yields
(34) 𝜔̈𝑡 = −𝛾𝜅𝑥̇𝑡.
Substituting equation (34) and the equation of motion (24) into equation (33) expresses the Euler equation as a second order differential equation of 𝜔𝑡:
(35) 𝜃𝜔̇𝑡2+ 𝜔𝑡𝜔̈𝑡= −𝜌𝜎𝜔𝑡𝜔̇𝑡.
Let 𝑣𝑡 = 𝜔𝑡1+𝜃. Then the first and second order time derivatives of 𝑣𝑡 are respectively
(36) 𝑣̇𝑡 = (1 + 𝜃)𝜔𝑡𝜃𝜔̇𝑡
(37) 𝑣̈𝑡 = (1 + 𝜃)𝜔𝑡𝜃−1(𝜃𝜔̇𝑡2+ 𝜔𝑡𝜔̈𝑡).
Substituting equation (35) into equation (37) yields
𝑣̈𝑡 = −𝜌𝜎(1 + 𝜃)𝜔𝑡𝜃𝜔̇ = −𝜌𝜎𝑣̇𝑡
⇒ 𝑣̇𝑡 = 𝐶1𝑒−𝜌𝜎𝑡
⇒ 𝑣𝑡 = 𝐶2𝑒−𝜌𝜎𝑡+ 𝐶3, 𝐶2 = −𝐶1 𝜌𝜎
⇒ 𝜔𝑡 = [𝐶2𝑒−𝜌𝜎𝑡+ 𝐶3]1+𝜃1
17
(38) ⇒ 𝑦𝑡 = 𝑦0− 1
𝛾𝜅+ [𝐶4𝑒−𝜌𝜎𝑡+ 𝐶5]1+𝜃1
where in equation (38), fossil fuel reserves are substituted in for using definition (26) and the carbon-climate response (7). The constants are 𝐶4 = 𝐶2
(𝛾𝜅)1+𝜃 and 𝐶5 = 𝐶3
(𝛾𝜅)1+𝜃.
To represent the two degrees of freedom in terms of 𝑦0 and 𝑥0, instead of integration constants, we impose 𝑡 = 0 in equation (38), and find
(39) 𝐶5 = 1
(𝛾𝜅)1+𝜃− 𝐶4.
Inserting equation (39) in (38) and taking the derivative of 𝑦𝑡 with respect to time yield
(40) −𝑦̇𝑡 = 𝑥𝑡 = 𝜌𝜎
1 + 𝜃𝐶4𝑒−𝜌𝜎𝑡[𝐶4𝑒−𝜌𝜎𝑡+ 1
(𝛾𝜅)1+𝜃− 𝐶4]
− 𝜃 1+𝜃.
Imposing 𝑡 = 0 in equation (40) gives us
(41) 𝐶4 = 1 + 𝜃
𝜌𝜎 1 (𝛾𝜅)𝜃𝑥0.
Substituting equation (39) and (41) into equation (38) and (40) give respectively
(42) 𝑦𝑡 = 𝑦0− 1
𝛾𝜅+ [−1 + 𝜃 𝜌𝜎
1
(𝛾𝜅)𝜃𝑥0(1 − 𝑒−𝜌𝜎𝑡) + 1 (𝛾𝜅)1+𝜃]
1 1+𝜃
(43) 𝑥𝑡 = 1
(𝛾𝜅)𝜃𝑥0𝑒−𝜌𝜎𝑡[−1 + 𝜃 𝜌𝜎
1
(𝛾𝜅)𝜃𝑥0(1 − 𝑒−𝜌𝜎𝑡) + 1 (𝛾𝜅)1+𝜃]
−𝜃 1+𝜃.
Using the first order condition for consumption (30) and the shadow value transformation relationships (9), (10) and (29), the discounted scarcity rent becomes
(44) 𝜆𝑦,𝑡= 𝑥0−
1
𝜎(𝛾𝜅)𝜃[−1 + 𝜃 𝜌𝜎
1
(𝛾𝜅)𝜃𝑥0(1 − 𝑒−𝜌𝜎𝑡) + 1 (𝛾𝜅)1+𝜃]
𝜃 1+𝜃.
Highfill and McAsey (1991) proved that for any control problem, the optimal duration of the planning period is finite only when the derivative of the objective function with respect to the control variable is finite as the control variable approaches 0. For all 𝑥𝑡 yielding non-negative marginal utility (property (3)), we have
18
(45) lim
𝑥𝑡→0
𝜕𝑢
𝜕𝑥𝑡= lim
𝑥𝑡→0𝑥𝑡−
1
𝜎(1 − 𝛾𝑧𝑡)𝜎−1𝜎 𝜃 → ∞.
Accordingly, the duration of the planning period approaches infinity in optimum, i.e. 𝑇∗ → ∞.
The initial value for the consumption trajectory, 𝑥0, is determined by the transversality condition
(46) lim
𝑡→∞𝜆𝑦,𝑡∗ (𝑦𝑡− 𝑦𝑡∗) ≥ 0 ∀ 𝑦𝑡 ∈ [0, 𝑦0]
where condition (46) is a generalised transversality condition that holds true for all control problems with infinite horizon (Sydsæter et al. 2002). Equation (46) establishes that at the end of the planning period there shall be no more utility to be obtained from the reserves. There are two potential causes. Either the resource loses its value, i.e. 𝜆𝑦,𝑇∗ = 0, or the reserves become fully exhausted, 𝑦𝑇∗ = 0. In the case where 𝜆𝑦,𝑇∗ = 0 and 𝑦𝑇∗ > 0, the discounted scarcity rent decreases to zero over time because the harm from temperature is so severe that there is a point in time where consuming an additional unit fossil fuel contributes negatively to utility. In this case, it is optimal to leave some of the fossil fuel reserves in the ground. In the case where 𝜆𝑦,𝑇∗ > 0 and 𝑦𝑇∗ = 0, harm from temperature is sufficiently mild that even when reserves are fully depleted, marginal utility of consumption remains positive, and the representative agent would have liked to consume more if reserves were not exhausted. In the special case where 𝜆𝑦,𝑇∗ = 𝑦𝑇∗ = 0, both causes coincide as the point in time where
temperature deteriorates all utility from consumption is precisely when the reserves are fully depleted. The limits of 𝑦𝑡 and 𝜆𝑦,𝑡 yield
(47) lim
𝑡→∞𝜆𝑦,𝑡=! 0 ⇒ 𝑥0 = 𝜌𝜎 1 + 𝜃
1 𝛾𝜅
(48) lim
𝑡→∞𝑦𝑡 =! 0 ⇒ 𝑥0 = 𝜌𝜎 1 + 𝜃
1
𝛾𝜅(1 − (1 − 𝛾𝜅𝑦⏟ 0)1+𝜃)
≡𝜙
where 𝜙 ∈ [0,1] is a parameter that determines how abundant fossil fuel reserves are.
Condition (47) and (48) coincide when 𝛾 = 1
𝜅𝑦0. Hence, when 𝛾 > 1
𝜅𝑦0, harm from temperature is so severe that the reserves are in abundancy. When 𝛾 < 1
𝜅𝑦0, harm from temperature is sufficiently mild that fossil fuel reserves become a scarce resource. Finally, 𝛾 = 1
𝜅𝑦0 is the special case where marginal utility of consumption is fully deteriorated precisely when the
19 stock is fully depleted. Summarising condition (47) and (48) give the general expression for the initial value of the consumption trajectory:
(49) 𝑥0 = 𝜌𝜎
1 + 𝜃 1
𝛾𝜅𝜙, 𝜙 = {
1 ⇔ 𝛾 > 1 𝜅𝑦0
1 − (1 − 𝛾𝜅𝑦0)1+𝜃⇔ 𝛾 ≤ 1 𝜅𝑦0.
Substituting (49) into equation (42) returns the optimally controlled trajectory for the resource reserves:
(50) 𝑦𝑡∗ = 𝑦0− 1
𝛾𝜅+ 1
𝛾𝜅[1 − 𝜙(1 − 𝑒−𝜌𝜎𝑡)]1+𝜃1 .
Taking the time derivative of solution (50) returns the optimally controlled consumption trajectory:
(51) −𝑦̇𝑡= 𝑥𝑡∗ = 𝜌𝜎 1 + 𝜃
𝜙
𝛾𝜅𝑒−𝜌𝜎𝑡 [1 − 𝜙(1 − 𝑒−𝜌𝜎𝑡 )]1+𝜃−𝜃.
Inserting solution (50) into the carbon-climate response (7) returns the optimally controlled trajectory for temperature:
(52) 𝑧𝑡∗ =1
𝛾−1
𝛾[1 − 𝜙(1 − 𝑒−𝜌𝜎𝑡)]1+𝜃1 .
Using the definition of 𝜔𝑡 (26), and inserting for the solutions (51) and (52) in first order condition (30), the optimally controlled scarcity rent becomes
(53) 𝜇𝑦,𝑡∗ = (1 + 𝜃 𝜌𝜎
𝛾𝜅 𝜙)
1
𝜎𝑒𝜌𝑡[1 − 𝜙(1 − 𝑒−𝜌𝜎𝑡)]1+𝜃𝜃 .
3.3 Proof of Proposition 2
Define ℋ̂𝑡𝑐(𝜎) ≡ max𝑥𝑡ℋ𝑡𝑐(𝑥𝑡, 𝜔𝑡, 𝜇𝜔,𝑡). Then
(54) ℋ̂𝑡𝑐(𝜎 ≠ 1) = 1
𝜎 − 1( 𝜔𝑡𝜃 𝜇𝜔,𝑡𝛾𝜅)
𝜎−1
.
20
By Arrow’s sufficient condition, the trajectories for the control variable, the state variable and the co-state variable which satisfy the first order conditions of the Hamiltonian are optimal when the following condition is satisfied:
(55) 𝜕2ℋ̂𝑡𝑐(𝜎 ≠ 1)
𝜕𝜔𝑡2 = 𝜃(𝜃(𝜎 − 1) − 1) 𝜔𝑡𝜃(𝜎−1)−2
(𝜇𝜔,𝑡𝛾𝜅)𝜎−1 ≤ 0 ⇒ 𝜃(𝜎 − 1) ≤ 1.
Condition (55) states that the trajectories implied by the first order conditions are optimal only when 𝑢(𝑥𝑡, 𝜔𝑡) has a degree of homogeneity equal to or less than unity, where 𝑢(𝑥𝑡, 𝜔𝑡) is utility written in terms of consumption and 𝜔𝑡. For the case of logarithmic preferences, we have
(56) ℋ̂𝑡𝑐(𝜎 = 1) = − ln(𝜇𝜔,𝑡𝛾𝜅) + 𝜃 ln 𝜔𝑡− 1.
Arrow’s sufficient condition demands
(57) 𝜕2ℋ̂𝑡𝑐(𝜎 = 1)
𝜕𝑦𝑡2 = − 𝜃 𝜔𝑡2 ≤ 0.
Summarised, the trajectories implied by the first order conditions are optimal when
(58) 𝜃 ∈ [0, 1
𝜎 − 1] ∧ 𝜎 > 1, 𝜃 ∈ ℝ0+∧ 𝜎 ∈ (0,1]
where condition (58) uses condition (55) and (57) while taking into account the domain of 𝜃.
If condition (58) is violated, the Hamiltonian is strictly convex, and the trajectories implied by the first order conditions return the minimum of the Hamiltonian. In this case, it is optimal to never consume any fuel at any time, i.e. the corner solution is optimal.
21
4 Applying the model
The trajectories derived in chapter 3 were the optimal policy as of preindustrial time. Fossil fuels have been consumed at a non-negligible amount for the last 250 years, and temperature has increased accordingly. This chapter calibrates the model to fit an arbitrary point in time to be the beginning of the planning period, where an increase in temperature net of preindustrial time has already occurred. Then data on contemporary fossil fuel reserves and temperature are used to look at the optimal trajectories from today and onwards.
4.1 Fitting the model to an arbitrary planning period beginning
Let 𝑡 = 𝑠 be the beginning of the planning period, and 𝑋𝑠 ≡ ∫ 𝑥0𝑠 𝑡𝑑𝑡= 𝑦0− 𝑦𝑠 be the amount of fuel that has been consumed between 𝑡 = 0 and 𝑡 = 𝑠. Define 𝑧̃ to be the exogenously given increase in temperature that has occurred at 𝑡 < 𝑠. Using the temperature trajectory (iii) of Proposition 1, the point in time s that corresponds to 𝑧̃ is
𝑧̃ = 1 𝛾−1
𝛾[1 − 𝜙𝑠(1 − 𝑒−𝜌𝜎𝑠)]1+𝜃1
(59) ⇒ 𝑠 = − 1
𝜌𝜎ln (𝜙𝑠 − 1 + (1 − 𝛾𝑧̃)1+𝜃
𝜙𝑠 )
where
(60) 𝜙𝑠 =
{
1 if 𝛾 > 1 𝜅(𝑦𝑠+ 𝑋𝑠)
1 − (1 − 𝛾𝜅(𝑦𝑠+ 𝑋𝑠))1+𝜃 if 𝛾 ≤ 1 𝜅(𝑦𝑠+ 𝑋𝑠).
Expression (59) is a normalisation of the initial point of the planning period that allows us to use the model to look at the optimal trajectories when increases in temperature have occurred before the initial point of the planning period. Inserting for 𝑋𝑠 and 𝜙𝑠 into the optimal
trajectories for consumption (i), fuel reserves (ii), temperature (iii) and the damage share (23), the trajectories calibrated for 𝑡 = 𝑠 are
22
(61) 𝑥𝑡𝑠 = 𝜌𝜎
1 + 𝜃 𝜙𝑠
𝛾𝜅𝑒−𝜌𝜎𝑡 [1 − 𝜙𝑠(1 − 𝑒−𝜌𝜎𝑡 )]1+𝜃−𝜃
(62) 𝑦𝑡𝑠 = 𝑦𝑠+ 𝑋𝑠− 1 𝛾𝜅+ 1
𝛾𝜅[1 − 𝜙𝑠(1 − 𝑒−𝜌𝜎𝑡)]1+𝜃1 .
(63) 𝑧𝑡𝑠 = 1
𝛾−1
𝛾[1 − 𝜙𝑠(1 − 𝑒−𝜌𝜎𝑡)]1+𝜃1 (64) 𝐷𝑡𝑠 = 1 − [1 − 𝜙𝑠(1 − 𝑒−𝜌𝜎𝑡)]1+𝜃𝜃 .
4.2 Calibration
I now calibrate these trajectories based on global fossil fuel reserves, the quantitative carbon- climate response and the temperature tolerance stipulated by the Paris Agreement.
Table 1: Proven global reserves of fossil fuels at the end of 2017
Crude oil Natural gas Coal Total
1.69 trillion barrels 193.45 trillion m3 1.06 trillion tonnes
= 10 001.36 exajoules = 7 737,55 exajoules = 19 836.56 exajoules 37 575.45 exajoules
= 200.03 GtC = 118.38 GtC = 525.67 GtC = 844.08 GtC
Sources: BP (2018), IEA (2017) and Norsk Petroleum (2019).
Table 1 lists global reserves of fossil fuel reserves as of the end of 2017. “Proven reserves”
are an industry measure on reserves that captures both geological data and economic
feasibility of extraction (Investopedia 2018). Proven reserves are accordingly a measure that varies with the development of extraction technology, the resource market price and new discoveries. Proven reserves increase as extraction becomes more productive and new
discoveries are made. The development of substitutes to fossil fuels decreases proven reserves as lower demand for fossil fuels reduces their prices. Using proven reserves as a proxy for reserve level, the estimated value for the initial reserve level, 𝑦𝑠, is 𝑦̂𝑠 = 844.08 GtC.
Matthews et al. (2009) estimated 𝜅 to be 𝜅̂ = 0.0017 °C per GtC emitted, with a corresponding 95 % confidence interval of 0.001-0.0021 °C. In 2010, temperature had increased by 0.8 °C since 1880 (National Research Council 2010). Assuming 0.8 °C is an appropriate estimate for the contemporaneous temperature net of preindustrial time, total fuel consumption between today and the industrial revolution is estimated to be
23
(65) 𝑋̂𝑠 = 𝑧̃
𝜅̂= 470.59
unit GtC. The Paris Agreement (United Nations 2015) stipulated that humanity should hold
“[…] the increase in the global average temperature to well below 2 °C above pre-industrial levels and to pursue efforts to limit the temperature increase to 1.5 °C above pre-industrial levels”. Using 2 °C as the maximum tolerated temperature, 𝛾 is assessed to be
(66) 𝛾̂ =1
2. As 𝛾̂ =1
2 > 1
𝜅
̂(𝑦̂𝑠+𝑋̂𝑠)≈ 0.45, 𝜙̂ = 1 and global proven reserves of fossil fuels are too large for it to be optimal to consume it all. On the contrary, it is optimal to leave
(67) lim
𝑡→∞𝑦̂𝑡𝑠 = 𝑦̂𝑠+ 𝑋̂𝑠− 1
𝛾̂𝜅̂≈ 138.2
GtC worth of fossil fuels in the ground, or 16.4% of proven reserves. 705.8 GtC is then what is left for consumption, and must be rationed between all future points in time according to the consumption trajectory equation (61). Temperature is thus the binding constraint, and the optimal consumption trajectory implies a gradual convergence of the damage share to unity, at which temperature is 2 °C above pre-industrial level. The estimates rely however heavily on the estimated value of the carbon-climate response. If 𝜅 = 0.001, i.e. the lower bound on the confidence interval, all proven fuel reserves can be consumed without temperature increasing to more than 1.64 °C. If 𝜅 = 0.0021, the upper bound of the confidence interval, 42.9% of the reserves must be left in the ground.
Using Matthews et al.’s (2009) mean estimate for the carbon-climate response, 0.0017 °C/GtC, and a rate of interest of 2%, Figure 1, 2, 3 and 4 plot the optimal trajectories for consumption, fuel reserves, temperature and the damage share, respectively.
The trajectories are plotted for 6 scenarios, as tabulated in Table 2. The chosen parameter values
capture convexity of the damage share in temperature (𝜃 ∈ (0,1)) and concavity (𝜃 > 1) for 3 regimes of 𝜎; when the income effect dominates the substitution effect (𝜎 ∈ (0,1)), log-utility (𝜎 = 1) and when the substitution effect dominates the income effect (𝜎 > 1). In many
Table 2: Parameter values for different scenarios plotted
Scenario Value of 𝝈 Value of 𝜽
Scenario 1 0.3 0.5
Scenario 2 0.3 1.5
Scenario 3 1 0.5
Scenario 4 1 1.5
Scenario 5 1.5 0.5
Scenario 6 1.5 1.5
