# Battle for Climate and Scarcity Rents: Beyond the Linear-Quadratic Case

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## Abstract

Industria imports oil, produces final goods and wishes to mitigate global warming. Oilrabia exports oil and buys final goods from the other country. Industria uses the carbon tax to impose an import tariff on oil and steal some of Oilrabia’s scarcity rent. Conversely, Oilrabia has monopoly power and sets the oil price to steal some of Industria’s climate rent. We analyze the relative speeds of oil extraction and carbon accumulation under these strategic interactions for various production function specifications and compare these with the efficient and competitive outcomes. We prove that for the class of HARA production functions, the oil price is initially higher and subsequently lower in the open-loop Nash equilibrium than in the efficient outcome. The oil extraction rate is thus initially too low and in later stages too high. The HARA class includes linear, loglinear and semi-loglinear demand functions as special cases. For non-HARA production functions, Oilrabia may in the open-loop Nash equilibrium initially price oil lower than the efficient level, thus resulting in more oil extraction and climate damages. We also contrast the open-loop Nash and efficient outcomes numerically with the feedback Nash outcomes. We find that the optimal carbon tax path in the feedback Nash equilibrium is flatter than in the open-loop Nash equilibrium. It turns out that for certain demand functions using the carbon tax as an import tariff may hurt consumers’ welfare as the resulting user cost of oil is so high that the fall in welfare wipes out the gain from higher tariff revenues.

## Keywords

Exhaustible resources Hotelling rule Efficiency Carbon tax Climate rent Differential game Open-loop Nash equilibrium Subgame-perfect Nash equilibrium HARA production functions## JEL Classification

C73 H30 Q32 Q37 Q54## 1 Introduction

An oil-exporting cartel such as the OPEC can exert monopoly power on the world market, especially if the price elasticity of oil demand is not too high. This monopoly power can result in significantly different oil extraction patterns relative to a perfectly competitive market. A key factor that determines the effect of the exporter’s monopoly power is the nature of the importer’s oil demand, since the price elasticity of oil demand is for most demand functions not constant. However, many studies when modeling oil extractions either choose iso-elastic demand or choose a particular specification of the demand function (typically, linear demand) for reasons of convenience/simplicity to make the models tractable. A key result in the literature on oil extraction is that with zero extraction costs and iso-elastic demand, the monopolist oil extraction is efficient and coincides with what would prevail in a competitive market [18]. In that case, the oil price increases according to the Hotelling rule at a rate equal to the market rate of interest [10]. For non-iso-elastic demand, the oil price path can be steeper or flatter, depending on the demand’s functional form. It is thus important to analyze the robustness of results to the chosen specific functional forms for the production function and oil demand. This allows us to break down the way various demand specifications affect the bias of the exporter’s monopoly power on the oil extraction rate and thus on carbon emissions.

The monopolist’s oil extraction rate also plays a prominent role in the context of climate policies. As there is no global agreement on battling climate change, increasingly the developed countries are beginning to implement their own carbon emission reduction policies, while the oil-exporting nations are opposed to such measures. There is a range of studies modeling strategic interaction between a monopolist oil exporter who sets the oil price and an importer who combats climate change by setting a carbon tax [17, 19, 20, 24] and surveyed in [15]. In equilibrium, the oil-importing country uses the carbon tax as an import tariff to capture some of the monopolist’s rents, while the oil-exporting country marks up his price so that he can capture a part of the carbon tax revenue collected by the oil-exporting country.

Liski and Tahvonen [14] explicitly analyze such strategic interactions. Their main results are twofold: First, the subgame-perfect oil extraction path is flatter than in the efficient outcome; second, this path is also flatter than in the pure cartel outcome where no carbon tax is levied. Furthermore, they find that the level of damages due to global warming has a significant effect on equilibrium dynamics. The level of damages determines whether the import tariff or the Pigouvian environmental tax is the dominant component of the importer’s carbon tax, and hence, whether the carbon tax decreases or increases over time. The authors even find that for very high damages, the tariff component may be negative—a subsidy. However, Liski and Tahvonen [14] restrict their analysis to linear demand functions and parameterizations that lead to interior solutions and thus keep some oil left in situ. As we will see, these assumptions do end up affecting their result.

Our contribution is to investigate the generality of these results to a variety of specifications of production functions and oil demand. Our framework of analysis is partial equilibrium, since we abstract from saving, investment and capital accumulation and take the interest rate as given.^{1} Although we offer formal propositions where it is feasible, we also have to resort to numerical simulations to gain insights. For this purpose, we use an illustrative calibration of our model to real-world values. We obtain the following findings.

First, we recall that unlike the case of zero extraction cost, a constant price elasticity of oil demand and no climate concerns [5, 18], if oil extraction costs are nonzero and increase as remaining oil reserves diminish, monopolistic extraction is never efficient. We establish the nature of the inefficiency for the open-loop Nash equilibrium too.

Second, we prove that for the class of HARA production functions and associated oil demand functions, the oil price is initially higher and subsequently lower in the open-loop Nash equilibrium than in the efficient outcome. The oil extraction rate and carbon emissions are thus initially too low and in later stages too high relative to the efficient levels. The HARA class includes linear [14], loglinear and semi-loglinear demand functions as special cases. This proof also holds for the case of a cartel.

Third, we show that on the other hand, a shifted loglinear demand function can lead Oilrabia to initially price oil lower in open-loop Nash equilibrium than in the efficient level, thus resulting in more oil extraction and climate damages. This establishes that there exist non-HARA demand functions that yield opposite results without violating any arbitrage conditions. Lewis et al. [13] have demonstrated a similar result in a context without carbon taxation: If the price elasticity of demand increases in oil consumption, monopolistic extraction is initially faster than the efficient rate. Our study thus differs in that we allow for carbon pollution and global warming and the consequent need for the oil-importing country to impose a carbon tax.^{2}

Fourth, our simulation results indicate that the feedback Nash equilibrium leads initially to flatter oil price paths than the efficient outcome. We establish numerically that there is also a downward initial bias in oil extraction rates and carbon emissions for the feedback Nash equilibrium. The consumer oil price increases slower in the feedback Nash equilibrium than in the open-loop Nash equilibrium, and our simulations also indicate that the optimal carbon tax path will always be higher in the feedback Nash equilibrium than in the open-loop Nash equilibrium. This occurs because in the feedback Nash equilibrium the oil-importing region increases the carbon tax in order to capture the scarcity rents of the oil-exporting country.

Fifth, we find that for oil demand specifications which asymptotically lead to full oil exhaustion, the level of carbon damages does not influence the carbon tax dynamics.^{3} For such oil demand functions, the monopolist exporter earns significant amounts of scarcity rent, and thus, the carbon tax is used primarily as a tariff so that the importer can capture some of these rents back. However, as Oilrabia and Industria compete to capture each other’s rents, they raise the combined user cost of oil so high that the resulting welfare loss to the consumers is larger than the extra-gain from stolen scarcity rents.

Finally, our simulations indicate that for a reasonable calibration of the model, no matter how high the intensity of global warming damages, the oil-importing country will always impose a positive import tariff on top of the carbon tax in the feedback Nash equilibrium. Thus, the result of Liski and Tahvonen [14] that in some cases, the importer may have an incentive to subsidize oil consumption in order to ease the burden caused by the high prices set by the oil-exporting countries does not seem to be relevant.

We must make two important provisos to our findings. The first one is that we abstract from political economy issues. For example, oil exporters such as Saudi Arabia may have a desire to charge a higher export price of oil if their populations are sizeable and need to be pacified with transfers. Saudi Arabia would then be hurt if the oil-importing countries levy carbon taxes, especially if a lot of these are shifted to Saudi Arabia. Also, oil-importing countries may hesitate to fully internalize the social cost of carbon if they fear the income distributions of regressive carbon taxes, but we abstract from such political economy effects too. The second proviso is that we focus at open-loop and feedback Nash equilibrium outcomes and abstract from open-loop and feedback Stackelberg equilibrium outcomes for a supply side consisting of a cartel of monopolistic oil exporters and a competitive fringe of smaller oil exporters.^{4}

Although our analysis has antecedents in the economics of natural resources (e.g., [5, 13, 14, 17, 18, 19, 20, 24]), it is important to realize that our analysis also has roots in the industrial organization and international trade literature. For example, Brander and Spencer [3] examine how a tariff on imports produced by a foreign Cournot–Nash oligopoly is affected by linear and other plausible demand specifications and Brander and Spencer [4] also analyze how export subsidies might affect the optimal import tariff and how in a context of three countries export subsidies might affect the optimal import tariff or the nature of the Nash equilibrium in tariffs and subsidies between the three countries. Our analysis builds on this literature by explicitly taking account of the scarcity of fossil fuel and allow for both the Pigouvian and the import tariff motive for carbon taxation. Our work is also related to Mrázová and Neary [16] and Xie [21] who identify which demand function features, especially whether demand is super- or sub-modular indicated by the sign of the so-called super-elasticity, affect results in industrial organization and international trade. These insights are more relevant in Stackelberg than in Nash setups, so are of less relevance for our present analysis of the monopolist’s price markup and the importer’s carbon tax.

The outline of the paper is as follows. Section 2 sets up the model of the oil-importing and oil-exporting blocks of countries and discusses the differential game that is played between them. Section 3 derives the open-loop Nash equilibrium outcome, compares this with the efficient and competitive outcomes and proves that for the class of HARA production functions, the resulting oil price path is flatter and the oil extraction rate path is steeper in open-loop Nash equilibrium than in the efficient outcome. Section 4 derives the subgame-perfect Nash equilibrium outcome (also known as the feedback Nash equilibrium). Section 5 offers some illustrative simulations that confirm our proposition on HARA production functions, provides an example of demand for which the monopolist initially extracts less than the efficient rate, and more generally highlights the differences between the open-loop and subgame-perfect equilibrium and the efficient and competitive outcomes. Section 6 concludes with a summary of results and suggestions for further research.

## 2 The Model

^{5}The government of Oilrabia has a given initial stock of oil \(S_0\), which it manages optimally to maximize the present discounted value of its consumption stream which corresponds to the present value of its oil profits. Industria also has utilitarian preferences and is concerned with the present value of its consumption stream. It also levies a specific carbon tax \(\tau \) to limit the damages from global warming. We are concerned with strategic interactions between Industria and Oilrabia, where the former sets the carbon tax and the latter sets the oil price.

### 2.1 Demand for Oil and Private Consumption in Industria

### 2.2 The Government of Industria

^{6}Although all oil extraction takes place outside of Industria’s borders, Industria’s government is able to observe the remaining oil stock \(S\). The dynamics of oil extraction follow from:

### 2.3 The Government of Oilrabia

## 3 Efficient and Open-Loop Nash Equilibria

We first consider the efficient equilibrium where a social planner maximizes the welfare sum of Oilrabia and Industria. We then compare it to the open-loop Nash equilibrium where each of the countries takes the time path of the other country’s policy as given when setting its own policy. This requires commitment to announced policies and is distinct from the feedback Nash equilibrium where the countries cannot pre-commit to a time path, but instead base their strategies on the remaining oil stock \(S\) and pollution stock \(E\). We will derive the feedback Nash equilibrium in Sect. 4.

### 3.1 Efficient Equilibrium

In the efficient equilibrium, the social planner sets the user cost of oil, which is defined as \(q\equiv p+\tau \), such that it maximizes total global consumption less the environmental damages, subject to the resource extraction constraint in (2). The Hamiltonian for the social planner is then \(H\equiv F(R(q))-G(S)R(q)-D(E)-\mu R(q)\), where the co-state variable \(\mu \) corresponds to the sum of the social cost of carbon and the Hotelling rent. We assume \(F^{\prime }(0)>G(S_0 )+D^{\prime }(E_0 )/\rho \) in order to avoid the possibility of zero extraction throughout. Indeed, if the inequality would not hold, even the marginal revenues of low initial extraction would not cover the lowest possible sum of marginal oil extraction cost and the social cost of carbon.

### **Proposition 1**

It is optimal to have nonzero oil extraction rates forever in the efficient outcome.

### *Proof*

It is optimal to have an interior solution \(R(t)>0\) for all \(t\ge 0\), because if \(R(t)=0\) for all \(t\ge T\) for some \(T\ge 0\), then this \(T\) is the optimal stopping time and the Hamiltonian should equal zero at \(T\) so that \(F(0)=D^{\prime }(E_0 +S_0 -S(T))=0\). This is ruled out by the assumption \(D^{\prime }(E)>0\) for all \(E>0\). Moreover, there cannot be positive extraction followed by zero extraction and then positive extraction again, which follows from strictly positive discounting and the fact that in the zero extraction phase, the state of the system is not changing. \(\square \)

### **Proposition 2**

The optimal oil extraction rate is monotonically decreasing over time and vanishes only asymptotically in the efficient outcome. The corresponding stock of oil reserves gradually falls to zero if \(F^{\prime }(0)-G(0)>\frac{D^{\prime }(E_0 +S_0 )}{\rho }\) (full exhaustion). Otherwise, it diminishes to a strictly positive steady-state level (partial exhaustion).

### *Proof*

The two cases described in Proposition 2 are relevant not only just for the efficient equilibrium but also for the open-loop and the feedback Nash equilibrium outcomes. In fact, as Liski and Tahvonen [14] demonstrate, in the long run, both the feedback and the open-loop Nash equilibrium outcomes converge to the efficient stock of oil reserves. Thus, the amount of oil left in situ remains the same across the different equilibrium outcomes. Which of the above two cases occurs depends on the nature of oil demand and the parameterization.

### 3.2 Open-Loop Nash Equilibrium: Carbon Taxation in Industria

### 3.3 Open-Loop Nash Equilibrium: Oil Pricing by Oilrabia

### 3.4 Open-Loop Nash Equilibrium Outcome

### 3.5 Comparing the Open-Loop Nash and Efficient Equilibria

We now compare the results under the various outcomes described by (13), (13′), (14) and (15). To provide some initial insights, we first consider in Sect. 3.5.1, the case of iso-elastic demand as this produces the well-known result that monopolistic extraction is efficient [18] and thus provides a useful benchmark. If oil demand is not iso-elastic, we see from (13) that the solution becomes less trivial and depends on the sign of the super-elasticity and thus on the degree the consumers can substitute away from oil as the price rises, and thus how much scarcity rent the monopolist can capture. We spend the remainder determining the way in which various production function specifications affect equilibrium dynamics. In Sect. 3.5.2, we generalize the benchmark case of iso-elastic demand to the much more general class of HARA production functions, which captures most of the commonly used demand functions, and prove that for this class initial oil prices in the open-loop equilibrium outcome are too low and later on are too high relative to the efficient outcome. Section 3.5.3 discusses an example of a non-HARA production function, which is useful as this can give the opposite to our main result on HARA production functions.

#### 3.5.1 Case: Iso-elastic Oil Demand

#### 3.5.2 HARA Class Demand

### **Proposition 3**

For the class of HARA production functions with elastic demand \((\varepsilon >1)\), initial oil extraction along the open-loop Nash equilibrium is less than that in the efficient equilibrium.

### *Proof*

This proposition establishes that for all members of the HARA class of production functions, the extraction rates in the open-loop Nash equilibrium are initially too low and later on are too high compared with the efficient outcome. Of course, oil prices are then initially too high and later on too low in the open-loop Nash equilibrium. This result also holds for the monopolist. Indeed, if this were not the case, it would be unprofitable for the monopolist to enter the market. In that sense, monopolistic oil barons are the conservationist’s best friend.

The class of HARA production functions has super-elasticity \(\varTheta (q)=\frac{\chi }{1-\varphi }\left[ (q/\psi )^{\frac{1}{\varphi -1}}-\chi \right] ^{-1}.\) If \(\chi >0,\) then \(\varTheta (q)>0\) as positive marginal demand requires \(\left[ {(q/\psi )^{\frac{1}{\varphi -1}}-\chi } \right] (1-\varphi )>0.\) If \(\chi =0,\) then \(\varTheta (q)=0.\) Thus, the members of the HARA class of production functions have a positive (or zero for the case of iso-elastic demand) super-elasticity, which tends to flatten the price path (13) relative to the case of iso-elastic oil demand. Proposition 3 proves that the extraction rate in open-loop Nash equilibrium is in fact slower than the efficient rate.

It is important to realize that the class of HARA production functions is quite general, since it includes many familiar production functions as special cases. Table 1 presents five special cases of the HARA production functions: quadratic, exponential, Cobb–Douglas, power and logarithmic production functions. These correspond to, respectively, linear, semi-loglinear, iso-elastic, shifted loglinear and shifted unit-elastic oil demand specifications. For each of the HARA production functions, the associated demand function, the elasticity and the parameter restrictions that must hold are given in Table 1 too.

Various HARA production functions

Production function | Demand function | Demand elasticity | Parameter restrictions |
---|---|---|---|

\(F(R)=-\frac{1}{2}\left( {\chi -\psi R} \right) ^{2}+\frac{1}{2}\chi ^{2}\) | \(R=(\chi \psi -q)/\psi ^{2}\) | \(\varepsilon =\frac{q}{\chi \psi -q}\) | \(\varphi =2, \chi >0, q<\chi \psi \) |

(quadratic) | (linear) | ||

\(F(R)=1-e^{-\psi R}\) | \(R=-\frac{1}{\psi }\ln \left( {\frac{q}{\psi }} \right) \) | \(\varepsilon =-\frac{1}{\ln \left( {\frac{q}{\psi }} \right) }\) | \(\varphi \rightarrow -\infty , \chi =1\) |

(exponential) | (semi-loglinear) | ||

\(F(R)=\frac{1-\varphi }{\varphi }\left( {\frac{\psi R}{1-\varphi }} \right) ^{\varphi }\) | \(R=\frac{1-\varphi }{\psi }\left( {\frac{q}{\psi }} \right) ^{\frac{1}{\varphi -1}}\) | \(\varepsilon =\frac{1}{1-\varphi }\) | \(\chi =0, \varphi >0\) |

(Cobb–Douglas) | (loglinear/iso-elastic) | ||

\(F(R)=\frac{1-\varphi }{\varphi }\left[ {(R+\chi )^{\varphi }-\chi ^{\varphi }} \right] \) | \(R=(\frac{q}{1-\varphi })^{\frac{1}{\varphi -1}}-\chi \) | \(\varepsilon =\frac{(\frac{q}{1-\varphi })^{\frac{1}{\varphi -1}}}{(1-\varphi )\left[ {(\frac{q}{1-\varphi })^{\frac{1}{\varphi -1}}-\chi } \right] }\) | \(0<\varphi <1, \chi >0, \psi =1-\varphi \) |

(power) | (loglinear with shift in oil demand) | ||

\(F(R)=\log (R+\chi )-\log (\chi )\) | \(R=\frac{1}{q}-\chi \) | \(\varepsilon =\frac{1}{1-\chi q}\) | \(\varphi \rightarrow 0, \psi =1, \chi q<1\) |

(logarithmic) | (shifted unit-elastic) |

#### 3.5.3 Non-HARA Class Demand

In fact, our simulations establish that the open-loop Nash equilibrium outcome can lead to a faster initially extraction rate than the efficient rate. We cannot make any definitive conclusions, because we can also find other examples of shifted loglinear demand function where in the open-loop equilibrium, the monopolist is initially more conservative than in the efficient case.

## 4 Feedback Nash Equilibrium

We assume that both Industria and Oilrabia have perfect information on both remaining oil reserves and the atmospheric carbon stock at each instant of time.^{7} Hence, the optimal carbon tax imposed by Industria and the world market price of oil determined by Oilrabia can both be represented as a function of merely remaining oil reserves, since the stock of atmospheric carbon is the sum of the initial carbon stock and all oil burnt so far. We denote the equilibrium price and the equilibrium tax by \(p(S)\) and \(\tau (S)\) and their derivatives by \(p_S (S)\) and \(\tau _S (S)\).

### 4.1 The Feedback Rule for the Optimal Carbon Tax

### 4.2 The Optimal Feedback Rule for the Oil Price

### 4.3 Equilibrium

### 4.4 Time-Domain Representation and Comparison with the Open-Loop Nash Equilibrium Outcome

We begin by demonstrating that we can express \(p(S)\) and \(\tau (S)\) as optimal paths \(p(t)\) and \(\tau (t)\), without any information loss. Formally, the feedback Nash equilibrium implies that the players do not pre-commit to optimal policy paths and instead the policies are expressed as functions of the state variable. However, an interior solution is characterized by positive oil demand, i.e., \(R(t)>0\) for all \(t\ge 0.\) This implies that the oil stock \(S\) is a monotonically decreasing function of time \(\dot{S}=-R<0\). Therefore, for every instant of time \(t\), there is a corresponding unique value of \(S(t)\in (S_0 ,0)\). Thus, the solutions to the feedback equilibrium \(p(S(t))\) and \(\tau (S(t))\) can also be expressed simply as optimal equilibrium paths: \(p(t)\) and \(\tau (t)\).

We collect some of our results on the feedback Nash equilibrium in the following proposition.

### **Proposition 4**

For the feedback Nash equilibrium, the optimal carbon tax consists of the sum of a Pigouvian component and an import tariff on oil from Oilrabia. The monopoly markup on the sum of extraction cost and the scarcity rent increases if oil demand is less elastic. Oilrabia captures part of Industria’s climate rent, and more so if it has more monopoly power on the world oil market.

The feedback Nash equilibrium solution can now be obtained by numerically solving (5) and (31) for the time paths of \(S\) and \(q\). From these, one can calculate the time paths for the carbon stock \(E=E_0 +S_0 -S,\) the carbon tax \(\tau =[ {\dot{q}+D^{\prime }(E)} ]/\rho \) (from 31), and the market price of oil \(p=q-\tau \). We can calculate the pure Pigouvian component of the carbon tax as \(\tau ^{P}(t)\equiv \int _t^\infty {e^{-\rho (s-t)}D^{\prime }\left( {E(s)} \right) \mathrm{d}s} \), upon which the import tariff component can be found as \(\tau (t)-\tau ^{P}(t)\). The export markup for oil follows from \(p-G(S)-\lambda ^{**}\), where the scarcity rent is calculated as \(\lambda ^{**}(t)=-\int _t^\infty {e^{-\rho (s-t)}G^{\prime }\left( {S(s)} \right) R(s)\mathrm{d}s}\). These price and tax components can then be used to demonstrate how the demand specification influences the amount of monopolist rent that can be stolen by the importer and vice versa.

Solving this feedback Nash equilibrium problem is relatively straightforward, due to the fact that our model collapses to only one state variable (the oil stock). If instead two or more state variables were required to describe the state of the world, we would have to explicitly solve for optimal strategies as functions of the states. For example, if we had included atmospheric decay of carbon in our model, the atmospheric carbon stock would no longer be the complement of the oil stock, and the two countries would base their strategies on both stocks. Similarly, if we would have allowed for capital accumulation in the model, the optimal strategies would also be a function of the capital stock.

## 5 Illustrative Numerical Simulations

We supplement our analytic results by illustrating the efficient, the open-loop and the feedback Nash equilibrium outcomes with some numerical solutions. For these purposes, we use the functional forms and calibration reported in Table 2. All our emission variables are expressed in 1000 Giga tons of carbon (GtC), and prices and costs are expressed in $1000 USD per ton carbon (tC).

We adopt a linear extraction cost function with an initial stock of oil reserves of 10 GtC. Extraction costs rise from an initial value of $300 to $1150 per tC when reserves are exhausted corresponding to $30 and $115 per barrel of oil, and following the [11] long-term cost curve. The initial carbon stock is set to its pre-industrial level of 800 GtC. Global warming damages are given by a quadratic function, with \(\kappa >0\) as the damage intensity parameter. We consider quadratic, exponential and Cobb–Douglas production functions, which are all members of the HARA class (see Table 1), as well as a non-HARA production function. They lead to, respectively, linear, semi-loglinear, iso-elastic and shifted loglinear oil demand functions. Further calibration details are reported in “Appendix 1.”

Calibration

Functional form | Parameter values | |
---|---|---|

Extraction cost | \(G(S)=\gamma _1 +\gamma _2 (S_0 -S)\) | \(\gamma _1 =0.3,\gamma _2 =0.085,S_0 =10\) |

Damages | \(D(E)=\kappa E^{2}\) | \(E_0 =8,\kappa =0.00004\) |

Quadratic production | \(F(R)=\chi \psi R-\psi ^{2}R^{2}/2\) | \(\psi =0.855,\beta =2.61\) |

(linear demand) | ||

Exponential production | \(F(R)=\frac{A}{\psi }(1-e^{-\psi R})\) | \(\psi =3.672,A=.7430\) |

(semi-loglinear oil demand) | ||

Cobb–Douglas production | \(F(R)=\frac{1-\varphi }{\varphi }\left( {\frac{\psi R}{1-\varphi }} \right) ^{\varphi }\) | \(\varphi =0.35,\psi =13.94\) |

(iso-elastic demand) | ||

Non-HARA production | \(F(R)=\chi R+\frac{R^{1-1/\phi }}{1-1/\phi }\). | \(\phi =2.7, \chi =1\) |

(shifted loglinear demand) |

We solve the model for various levels of the damage intensity parameter, \(\kappa \), demonstrating that given a reasonable calibration, the damage parameter has a relatively minor effect on the resource extraction rate. Section 5.3 presents the results for the benchmark case of a Cobb–Douglas production function with loglinear or iso-elastic oil demand. Here the government of Industria in the desire to capture the exporter’s scarcity rent raises the tariff, to which Oilrabia’s government responds by raising the price even higher, causing ever more damage to the consumer. Section 5.4 discusses the example of the non-HARA production function with shifted loglinear oil demand discussed in Sect. 3.5.3 in which the monopolist initially extracts more oil than the efficient rate.

### 5.1 Quadratic Production Function and Linear Oil Demand

We first consider the case of a quadratic production function and linear oil demand, which is the functional form used by Liski and Tahvonen [14]. Figure 1 compares time paths for oil reserves and the rate of oil extraction for the various outcomes. It confirms that in the open-loop Nash equilibrium, oil is initially extracted more slowly than in the efficient outcome, but faster than in the feedback Nash equilibrium. However, in the long run, the stock of oil reserves converges to the same level for these three outcomes. The user cost of oil in the open-loop Nash equilibrium outcome is initially higher and later on lower than the efficient outcome reflecting the slower depletion of oil reserves, thus confirming proposition 3 and the results of Liski and Tahvonen [14]. This bias in the extraction rate and slowing down of the depletion of the stock of oil reserves is bigger in the feedback Nash equilibrium than in the open-loop Nash equilibrium, thus confirming the insight of proposition 4 that the price path in the feedback Nash equilibrium is flatter than in the open-loop Nash equilibrium.

Welfare in the various outcomes

Oil demand | Industria’s welfare | Oilrabia’s profit | ||||
---|---|---|---|---|---|---|

Efficient | Open-loop | Feedback | Efficient | Open-loop | Feedback | |

Linear | \(-\)0.188 | \(-\)0.242 | \(-\)0.195 | 0.276 | 0.309 | 0.250 |

Semi-loglinear | \(-\)0.183 | \(-\)0.237 | \(-\)0.189 | 0.279 | 0.313 | 0.252 |

Iso-elastic | 10.887 | 8.797 | 8.075 | 2.802 | 3.552 | 2.938 |

Figure 2 decomposes the feedback Nash equilibrium paths for the optimal carbon tax set by Industria into its Pigouvian tax and its import tariff component and similarly decomposes the optimal oil price set by Oilrabia into the scarcity rent and the export price markup. The first panel of Fig. 2 shows that the Pigouvian tax component rises over time as the stock of atmospheric carbon rises and marginal climate damages go up. The second panel of Fig. 2 also indicates that the import tariff component falls over time as the potential oil rents that can be captured decline as oil reserves fall. Effectively, Industria uses its monopsony market power to extract rents from Oilrabia with a high initial import tariff component that decreases over time. Since the calibration we use assumes fairly mild damages, the import tariff component of the carbon tax dominates the Pigouvian tax components.

The fourth and third panels of Fig. 2 indicate that the scarcity rent \(\lambda \) falls over time and that the export price markup falls over time too, since the extraction cost rises and the cartel loses its monopoly power.

However, we could not find a damage intensity that is high enough to ensure that the optimal feedback Nash carbon tax becomes smaller than the Pigouvian tax. Instead, for a very high climate damage intensity, oil extraction becomes too harmful, and Industria sets a prohibitive carbon tax such that no oil is extracted. This result differs from [14]. Recall that in that study, for very high values of climate damage intensity, Industria sets the carbon tax below the optimal Pigouvian level, thus imposing an import subsidy (i.e., a negative tariff). This contradiction is mainly driven by our calibration. Thus, for our calibration, we can conclude that no matter how high the climate damages, the importer will always add an import tariff on top of the Pigouvian carbon tax.

### 5.2 Exponential Production Function with Semi-loglinear Oil Demand

Time at which 95 % of oil is extracted and untapped oil reserves at that time

Oil demand | 95 % Extraction time | Untapped oil reserves | ||
---|---|---|---|---|

Efficient | Open-loop | Feedback | ||

Linear | 192.3 | 300.4 | 345.8 | 6.295 |

Semi-loglinear | 204.3 | 321.2 | 370.2 | 6.179 |

Iso-elastic | 116.3 | 258.6 | 489.7 | 0 |

Non-HARA: Shifted unit-elastic | 114.5 | 116.8 | 144.6 | 0 |

As we mentioned earlier, the long-run stock of untapped oil reserves is only approached asymptotically. For all equilibrium outcomes, some extraction will take place. As can be seen from Table 4, with semi-loglinear oil demand, Oilrabia leaves more oil in situ than with linear demand.

Table 3 indicates that with semi-loglinear demand, Industria’s welfare is still worse in the feedback Nash equilibrium than in the open-loop Nash equilibrium and a fortiori worse than in the efficient outcome. Oilrabia’s welfare is again worse than the efficient outcome in the feedback Nash equilibrium, but better than the efficient outcome in the open-loop Nash equilibrium.

### 5.3 Cobb–Douglas Production Function and Iso-elastic Oil Demand

The implications for Industria’s welfare, however, are ambiguous. As can be seen from Table 3, with unit-elastic oil demand Industria’s welfare in the feedback Nash equilibrium is lower than in the open-loop Nash equilibrium in contrast to the case of linear and semi-loglinear oil demand. The intuition is that the unit-elastic oil demand curve does not set any upper bound on the reservation price. Thus, as Oilrabia and Industria compete to maximize their rent, they set the price too high and end up harming Industria’s consumers, thereby decreasing Industria’s welfare. This implies that the use of the carbon tax to steal rents from the importer is not as beneficial as has hitherto been suggested in the literature as for certain demand specifications it can harm the consumers in the oil-importing nation.

### 5.4 Non-HARA Production Function: Shifted Unit-Elastic Oil Demand

As Figs. 1 and 5 indicate, with HARA oil demand, the open-loop Nash equilibrium path crosses the efficient path only once. With the shifted semi-loglinear demand example of non-HARA demand, we see that from the second panel of Fig. 7 that there are multiple crossing points. The open-loop Nash equilibrium path starts out steeper than the efficient one and then flattens out, crossing the efficient path again. Thus, initially, the open-loop Nash equilibrium price is lower than the efficient price. This result may seem counterintuitive: Why would the oil price in open-loop Nash equilibrium be lower than in the competitive market outcome? The intuition lies in the behavior of the price elasticity. Imagine that Industria’s population consists of two groups. The first has a fairly low reservation price for oil and can substitute away easily, while the second has a higher reservation price and cannot substitute away from oil. For example, Industria might consist of city dwellers that can switch to public transport and country dwellers that have to drive to work. Thus, for lower oil prices, demand is fairly elastic (city dwellers can stop using oil and switch to public transport) but beyond a certain threshold, the elasticity of demand decreases. The monopolist then initially sets the price slightly lower to capture some of the elastic demand (while the extraction cost is low), and then once all the cheap oil is extracted, it increases the price to serve exclusively the more inelastic portion of demand. As a result of these strategic dynamics, more oil is extracted initially than in the efficient case. Ergo, in the short run, the monopolistic power of Oilrabia is bad for the environment. In the long run, however, the open-loop Nash equilibrium user cost of oil path is flatter than the efficient one just as is the case with HARA preferences.

## 6 Concluding Remarks

Curbing fossil fuel use is essential to prevent further global warming. This goal is complicated by the fact that reserves of fossil fuels are highly concentrated in a few non-western oil-and gas-rich nations with considerable monopolistic power. Previous studies [14, 17, 19, 20, 24] have argued that western oil importers might nevertheless successfully limit fossil fuel consumption via a carbon tax. However, monopolistic exporters are likely to respond to the carbon tax by increasing the oil price more than they would have done in a competitive oil market, which would result in even fewer emissions. In this way, oil exporters can try to capture some of the climate rent. Additionally, the carbon tax can be used as an import tariff to extract scarcity rents from monopolistic oil exporters. This might further benefit the oil-importing nation, but the ability of the importer to capture the monopolist’s scarcity rents depends on the intensity of carbon damages. However, the aforementioned studies limit their attention to linear oil demand in order to keep the analysis tractable. The nature of demand for oil can of course have an important effect on how consumers respond to an increase in the carbon tax. Our paper’s aim was to examine how the specific nature of oil demand affects the oil importer’s efforts to prevent climate change and capture the monopolist’s rents.

We analyze the problem in the framework of a dynamic game between an oil-exporting country with monopoly power on global oil markets and an oil-importing country which is concerned with combating climate change. We find equilibrium conditions for a variety of production functions and their corresponding demand specifications. We prove that for all HARA class production functions, the open-loop Nash equilibrium extraction will initially be slower and later on faster than the efficient rate. As most commonly used demand specifications fall under the HARA class, this implies that in most cases, the monopolist is the conservationist’s best friend. However, we also find a numerical example of a shifted loglinear demand function for which the open-loop equilibrium extraction rate is initially too high. Thus, there are some non-HARA production functions for which in the open-loop equilibrium the oil-exporting country hinders the oil-importing country’s effort to battle climate change. We also solve the model for the feedback Nash equilibrium. Although we do not get any general analytical results, our numerical illustrations indicate that the initial feedback equilibrium consumer price is always higher than in the open-loop Nash equilibrium, which leads to a delay in oil extraction and carbon emissions, and hence lower damages from global warming. This initial consumer price increase is caused by the government of the oil-importing country using the carbon tax as a tariff to steal the oil exporter’s scarcity rents on oil, while the oil-exporting country responds by raising the oil price to steal back some of the climate rents of the oil-importing country. For the iso-elastic production function, the resulting increase in the consumer price of oil leads to a significant welfare loss which outweighs the gain of the captured scarcity rent.

Our main conclusion is that the demand structure plays a significant role when determining the optimal carbon tax or import tariff for foreign oil. Given the number of papers which choose their demand function based on computational convenience, these results serve as a cautionary tale: Demand specifications chosen out of simplicity rather than reflecting reality may make a carbon tax seem more or less beneficial in the context of strategic interactions in regard to exhaustible resources.

While this study generalizes the results of previous studies by examining a range of demand specifications, it remains limited in its scope. For instance, the model we examine excludes the possibility of saving and capital accumulation and a less-stylized carbon cycle dynamics. Without the capital stock, the economy collapses after oil runs out, a highly unrealistic outcome. Meanwhile, without a more sophisticated model of the carbon cycle, all \(\hbox {CO}_2\) that is emitted into the atmosphere stays there forever. Thus, we end up over-estimating the amount of climate change. To remedy these problems, we would have to construct a model with additional state variables (for example, using the two reservoirs that [7] uses to model the carbon cycle). The player’s strategies would then be based on the full state space, the two \(\hbox {CO}_2\) reservoirs and the oil stock. Finding the feedback equilibrium in this more complex setup would not only make our conclusions more realistic but also provide a contribution to the dynamic game literature through solving for a subgame-perfect Nash equilibrium with multiple state variables, one of which represents a fully exhaustible resource. Another interesting extension would be to introduce multiple oil exporters into the model, similar to [1, 2, 8, 9]. Adding a fringe in addition to the cartel exporter is a better model of reality and would lead to interesting dynamics with the cartel and the fringe responding differently to the importer’s carbon tax. We leave these extensions of our work for further research.

## Footnotes

- 1.
van Wijnbergen [23] and van der Meijden et al. [22] analyze two-country general equilibrium models with capital accumulation and international capital flows, but do not look at the strategic interactions of policies in a differential game framework. Jaakkola [12] does do this, but focuses at whether taxation of foreign interest income on oil exporters’ assets helps to overcome the Green Paradox.

- 2.
Wirl [25] demonstrates that decreasing marginal elasticity of utility can boost initial emissions in an international pollution control game, but this study has no exhaustible resource, and both players are affected by climate change.

- 3.
In most cases, studies restrict their attention to interior solutions to make them tractable, so we provide a numerical algorithm which can be used to solve such problems.

- 4.
- 5.
While utilitarian preferences may imply a variety of utility functions with different elasticities of intertemporal substitution, for computational simplicity, we assume linear utility functions.

- 6.
The social welfare function is linear in the individual utilities.

- 7.
Other informational assumptions are possible (e.g., Industria does not have knowledge of Oilrabia’s stock of remaining oil reserves), but we will abstract from these here.

## Notes

### Acknowledgments

We are grateful to the ERC Advanced Grant ‘Political Economy of Green Paradoxes’ for financial support (FP-IDEAS-ERC Grant 269788) and to Gérard Gaudet, Matti Liski, Peter Neary and Florian Wagener for useful feedback on an earlier version.

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