1 Introduction

Long [17] was the first to show that the adverse effects of a risk of future expropriation of the profits of an owner of natural resources under anticipated nationalisation leads to more rapacious extraction.Footnote 1 The insights are akin to those for over-exploitation of the common. Here we demonstrate how this pans out in a political context with two rival political factions, which alternatively hold political power. We thus depart from the notion of a homogenous government. Rather than analysing the risk of expropriation by a homogenous government as in Long [17], we thus examine the effects of a government that may be removed from office on the dynamics of natural resource depletion. We first show how rent sharing might attenuate the adverse effects of the risk of expropriation or the risk of a future 1-sided regime shift. We then show how the framework can be extended to analyse 2-sided regime switches and never-ending conflict cycles using the theory of differential games. Finally, we indicate how the framework can be used to analyse resource wars with hazard rates that are endogenous and depend on relative fighting efforts also using the theory of differential games.Footnote 2

We assume iso-elastic demand for natural resources and zero variable extraction costs. With no risk of being removed from office, monopolistic resource extraction is efficient [8, 29]. We thus focus on the inefficiencies that follow from expropriation risk, perennial political cycles, and dynamic resource wars. In addition, (Table 1) we assume that the incumbent incurs upfront costs of exploration investment to make natural resource reserves endogenous (cf., [11]. Our paper is also related to Leonard and Long [16], which analyses the endogenous evolution of a society’s property rights regime and regime switches albeit in an overlapping generation framework.Footnote 3

Table 1 Sensitivity of speed of oil extraction and fighting efforts
Full size table

2 Rent Sharing and the Risk of a Future Regime Shift

Consider a political system with two rival political parties. There is an incumbent political party A that has a one-off risk of being removed from office by the rival opposition political party B at some unknown future date \(T.\) Our model is specified in continuous time. This instantaneous risk is h > 0, so that the expected time that A stays in office is 1/h. To keep matters tractable, we make three simplifying assumptions. First, demand for oil (from now on a shorthand for natural resources) is iso-elastic, and marginal revenue is positive. The price elasticity, ε, is thus assumed to be constant and greater than one. Second, oil extraction is costless. Third, political parties are risk neutral. Hence, utility is given by \(U(R)={R}^{1-1/\varepsilon }/(1-1/\varepsilon ),\) where \(R>0\) denotes the rate of oil depletion. Since welfare is the area under the demand curve, the inverse demand function is given by \(p=U{\prime}(R)={R}^{-1/\varepsilon },\text{\hspace{1em}}\varepsilon >1.\) As marginal revenue is always finite and positive, reserves are exhausted asymptotically. Demand for oil, \(R={p}^{-\epsilon }\), is iso-elastic.

As a result of selling oil, the incumbent receives at time t oil rents \(p\left(t\right)R(t)\) given that extraction costs are zero. We define \({\tau }^{B}\) to be the share of oil rents that B, when in office, gives to A, and, similarly, \({\tau }^{A}\) to be the share of oil rents that A, when in office, gives to B. With a fully cohesive constitution the incumbent government must divide up oil rents equally in which case \({\tau }^{A}={\tau }^{B}=\tau =0.5\). With a poor constitution, \(0<{\tau }^{A}={\tau }^{B}=\tau <0.5\) and oil rents are less equally shared (cf. [3,4,5].

A special case of our model corresponds to Long [17], which considers an international, typically foreign-owned oil company that owns the stock of natural resources and is subject to the risk of confiscation by the government. This case emerges if we assume that the international oil company keeps all of the oil rents, i.e. \({\tau }^{A}=0\), and h is the risk of confiscation. Once the government has confiscated the oil company, it keeps all the rents so that \({\tau }^{B}=0\). Another possibility is that the government at an unexpected future time increase the profit tax on this company. We will come back to this case when we discuss our results in Sect. 2.3.

We study the effect of constitutional cohesiveness (i.e. full rent sharing), but also analyse two departures from the constitution. The first one is when the incumbent adheres to the constitution and shares rents according to \(\tau^{A} = \tau ,\) but faces a threat of being removed from office by a rogue rival faction that offers less than what is stipulated by the constitution, i.e. \(\tau^{B} < \tau^{A} = \tau .\) The second one is where the incumbent is the rogue administration that faces a threat of being replaced by an administration that shares rents more fairly. This corresponds to \(\tau^{A} < \tau^{B} = \tau .\) We show that both types of departures make oil extraction more voracious and depress exploration investment.

We use the Principle of Dynamic Programming and work backwards in time by first analysing what happens after the change in government at some unknown future date T and then analysing what happens in the preceding period when the incumbent A is still in office.

2.1 Actions of the New Incumbent Starting at Some Future Date

Working backwards in time in accordance with the principle of optimality of dynamic programming, the political party B once in office maximises its rent net of cohesive payments,

$$\underset{R,I}{\text{Max}}\text{\hspace{0.33em}}{\int }_{T}^{\infty }(1-{\tau }^{B})p(t)R(t){e}^{-r(t-T)}\mathrm{d}t,$$
(1)

subject to the inverse demand function and the oil depletion equations,

$$\dot{S}\left(t\right)=-R\left(t\right),\hspace{0.33em}\forall t\ge T, {\int }_{T}^{\infty }R\left(t\right)dt\le S\left(T\right), \mathrm{and }S\left(0\right)=\Omega \left(I\right),$$
(2)

where \(I\) denotes investment in initial oil exploration, \(S\) denotes the stock of oil reserves, and \(r\) is the exogenous and constant market interest rate. Here \({\tau }^{B}\) denotes the government take of oil revenues when B is in office given to the opposition in the form of cohesiveness payments (rent sharing). We discuss the setting of optimal oil exploration investment in Sect. 2.2.Footnote 4

The optimal policy requires that marginal oil revenue net of the government take must equal the scarcity rent, \(\lambda \), which according to the Hotelling rule must rise at a rate equal to the market interest rate, r:

$$\left(1-{\tau }^{B}\right)\left(1-\frac{1}{\varepsilon }\right)R{\left(t\right)}^{-\frac{1}{\varepsilon }}=\lambda \left(t\right),\dot{\lambda }(t)/\lambda (t)=r, \, t\ge T.$$
(3)

It follows from (2) and (3) that under the regime of B the price and oil depletion paths are efficient from time T onwards despite the country being a monopolist on world markets. This can be seen from the price of oil following a competitive price path with prices rising at a rate equal to the interest rate according to the Hotelling rule:

$$\frac{\dot{p}\left(t\right)}{p\left(t\right)}=r>0,\frac{\text{\hspace{0.05em}} \, \dot{R}\left(t\right)}{R\left(t\right)}=-\varepsilon r<0, \, t\ge T,$$
(4)

Hence, oil producers are indifferent between extracting now and in a future point of time. The efficiency of oil extraction from time T onwards stems from the assumptions of costless extraction and iso-elastic demand for oil.

Using (4) in the oil depletion Eqs. (2), we obtain the policy function for oil, \(R(t) = \varepsilon rS(t)\), and thus obtain

$$\begin{array}{c}R(t)=\varepsilon r{e}^{-\varepsilon r(t-T)}S(T), \, \, \, \, S(t)={e}^{-\varepsilon r(t-T)}S(T)\le S(T),\\ p(t)={e}^{r(t-T)}{\left(\varepsilon rS(T)\right)}^{-1/\varepsilon }, \, \, \, \forall t>T.\end{array}$$
(5)

The optimal depletion and price paths follow Hotelling paths and are constrained efficient (conditional on \(S(T)\)). They do not depend on the government take, \({\tau }^{B},\) because this take corresponds to a lump-sum tax. Substituting (5) in (1), we get B’s welfare to go at time \(T:\)

$$ V^{B} \left( {S\left( T \right),\tau^{B} } \right) \equiv \mathop \smallint \limits_{T}^{\infty } \left( {1 - \tau^{B} } \right)R\left( t \right)^{1 - 1/\varepsilon } e^{{ - r\left( {t - T} \right)}} dt = \left( {1 - \tau^{B} } \right)\left( {\varepsilon r} \right)^{ - 1/\varepsilon } S\left( T \right)^{1 - 1/\varepsilon } . $$
(6)

where \({V}^{B}(.)\) denotes B’s (undiscounted) value function. Welfare to go for B increases in the stock of oil it inherits and decreases in the interest rate. Welfare to go for A from the moment it has been removed from office consists of cohesiveness payments it receives from B and equals

$${V}^{A}\left(S(T),{\tau }^{B}\right)\equiv {\tau }^{B}{(\varepsilon r)}^{-1/\varepsilon }S{(T)}^{1-1/\varepsilon }, \, \forall t\ge T.$$
(7)

2.2 Actions of the Incumbent Before Being Removed from Office

Given its future welfare to go (7), A must choose initial exploration investment, \(I,\) and the extraction path to maximise expected welfare to go with uncertainty regarding the end of its regime T. With an exponential density function for T, i.e. \(h\mathrm{exp}(-hT)\) with h the constant hazard rate, the problem for A from time zero is

$$ \begin{array}{*{20}c} {\mathop {{\text{Max}}}\limits_{R,I} {\text{E}}\left[ {\mathop \smallint \limits_{0}^{T} \left( {1 - \tau^{A} } \right)p\left( t \right)R\left( t \right)e^{ - rt} {\text{d}}t + e^{ - rT} V^{A} \left( {S\left( T \right),\tau^{B} } \right)} \right] - qI = } \\ {\left( {\mathop \smallint \limits_{0}^{\infty } h{\text{exp}}\left( { - hT} \right)\left( {\mathop \smallint \limits_{0}^{T} \left( {1 - \tau^{A} } \right)p\left( t \right)R\left( t \right)e^{ - rs} dt + e^{ - rT} V^{A} \left( {S\left( T \right),\tau^{B} } \right)} \right){\text{d}}T} \right)} \\ \end{array} $$
(8)

subject to the oil demand function, the oil depletion equations,

$$\dot{S}(t)=-R(t),\text{\hspace{0.33em}}\forall 0\le t\le T, \, (0)={S}_{0}>0, \, {\int }_{0}^{T}R(t)dt\le {S}_{0}-S(T),$$
(9)

and the oil exploration investment schedule,

$${S}_{0}=\Omega (I)={\omega }_{0}{I}^{\omega }, \, \Omega {\prime}>0,\text{\hspace{0.33em}}\Omega "<0, \, {\omega }_{0}>0,\text{\hspace{0.33em}\hspace{0.33em}}0<\omega <1,$$
(10)

where \(q\) is the exogenous price of oil exploration investment. All exploration investments have to be done up front, so ongoing exploration investments are ruled out for simplicity (cf. [11]). Concavity of Ω(.) ensures decreasing returns to exploration investment. The cumulative probability that the incumbent is removed from office by time t is \(\mathrm{Pr}\left(T\le t\right)=1-\mathrm{exp}\left(-ht\right),\text{\hspace{0.33em}}\forall t\ge 0, h\ge 0,\) and the probability that A is still in office by time t is \(\mathrm{Pr}\left(T>t\right)=\mathrm{exp}\left(-ht\right)\). The conditional probability that A is still in office during a small interval is independent of time. Expected duration and the standard deviation of A’s term of office are \(1/h\). The Hamilton–Jacobi–Bellman (HJB) equation for the incumbent A is

$$rV(S,{\tau }^{A},{\tau }^{B})=\underset{R}{\text{Max}} \, \left[(1-{\tau }^{A})U{\prime}(R)R-V{\prime}(S,{\tau }^{A},{\tau }^{B})R\right]-h\left[V(S,{\tau }^{A},{\tau }^{B})-{V}^{A}(S,{\tau }^{B})\right],$$
(11)

where \(p=U{\prime}(R)\) is the inverse oil demand function, \(V(S,{\tau }^{A},{\tau }^{B})\) the value function excluding exploration investment for A when is still in office, and \({V}^{A}(S,{\tau }^{B})\) is the value function for A after it has lost office. The optimality condition for oil extraction sets marginal revenue of oil extraction equals to the marginal value of oil reserves and gives optimal oil extraction,

$$R(t)={\left(\frac{V^{\prime}\left((S(t),{\tau }^{A},{\tau }^{B}\right)}{(1-{\tau }^{A})(1-1/\varepsilon )}\right)}^{-\varepsilon }, \, 0\le t\le T.$$
(12)

Upon substituting (12) into (11) and using the method of undetermined coefficients, we obtain \(V(S,{\tau }^{A},{\tau }^{B})=K{S}^{1-1/\varepsilon },\) where the unique value for K follows from the algebraic equation:

$$\frac{{(1-{\tau }^{A})}^{\varepsilon }}{\varepsilon }{K}^{1-\varepsilon }+h{\tau }^{B}{(\varepsilon r)}^{-1/\varepsilon }=(r+h)K.$$
(13)

The oil price and depletion rate when A is in office are

$$ p(t) = KS(t)^{ - 1/\varepsilon } /(1 - \tau^{A} )\quad {\text{and}}\quad \,R(t) = (1 - \tau^{A} )^{\varepsilon } K^{ - \varepsilon } S(t),\;0 \le t < T. $$
(14)

Defining \(L \equiv \left( {(1 - \tau^{A} )/K} \right)^{\varepsilon }\) and solving for the time paths from (14) and (2), we obtain

$$p(t)={e}^{Lt/\varepsilon }{(L{S}_{0})}^{-1/\varepsilon }, \, R(t)=L{e}^{-Lt}{S}_{0}, \, S(t)={e}^{-Lt}{S}_{0},\text{\hspace{1em}\hspace{0.33em}}0\le t<T.$$
(15)

Result 1: After the government switch, the oil price rises at the rate r and the oil depletion rate and reserves decline at the rate εr. Before the switch, the oil depletion rate and reserves decline at the rate L > εr and the oil price rises at the rate L/ε > r. More voracious oil depletion (higher L) and lower welfare to go for A occur with a higher risk of a change in government (higher h), and less rent sharing (smaller \({\tau }^{A}\) or \({\tau }^{B}\)).

A more cohesive constitution with more rent sharing (higher \({\tau }^{A}={\tau }^{B}\)) gives less voracious depletion and fewer inefficiencies. Whenever the incumbent or the rival faction shares rents less fairly, oil depletion will be more voracious, and A’s welfare will be lower. The incumbent faction thus has an incentive to share more equally. Oil extraction of the incumbent will be less voracious if it can count on cohesiveness payments when out of office from the new government B. If both factions share oil rents equally, we have \(L = \varepsilon r,\) so that the rate of extraction is efficient, and there are no political distortions whatsoever. This is also the case if there is no political risk, h = 0.

Returning to a special case inspired by Long [17], i.e. \({\tau }^{A}={\tau }^{B}=0\), Eq. (13) and the definition of L yields \(L={K}^{-\varepsilon }=\epsilon (r+h)\) and thus the oil price charged by the international oil company before its oil rents are confiscated rises at a rate larger than the market rate of interest (i.e. \(\frac{L}{\epsilon }=r+h>r\)). Long [17] also studies time-varying probabilities of confiscation and rising trends in the effective profit rate.

2.3 Effects of a Regime Switch on the Time Paths for Oil Depletion

Since oil depletion of the incumbent A is excessively fast, initially the path for the oil depletion rate is above the efficient path and the oil price path is below the Hotelling path. Along a Hotelling path, the market is indifferent between extraction in future and today so that today’s price of oil must equal the present discounted value of tomorrow’s price of oil. Along a Hotelling path, the price of oil thus grows at the market rate of interest, r. However, in a situation with a risk of a regime switch, the price of oil grows at the rate \(L/\epsilon \), which exceeds the rate of interest. It thus follows that the rate of oil extraction is faster than that under a Hotelling path too.

If the realised date of the change of government is far enough in future, the oil depletion rate before the switch can fall below and the oil price path can rise above the efficient path. Once A is removed, the oil depletion rate jumps down, and the price jumps up by a discrete amount. From then on depletion and reserves follow Hotelling paths, but they are inefficient as they start out from fewer oil reserves than without political risk. Oil prices rise at the interest rate but start from a higher level than without political uncertainty as voracious extraction during A’s incumbency has made oil scarcer when B enters office.

Figure 1 reports the results from setting ε = 2, r = 0.04, S0 = 100, h = 0.1, \(\tau^{A} = 0\) and \(\tau^{B} = 0.4.\) The expected political take-over thus occurs at time 10 while K = 2.46 and L = 0.165. The crossing time is 8.5. The reserves to production ratios before and after the switch are 6.1 and 12.5, respectively. Dotted lines indicate the efficient outcomes in case h = 0. Solid lines indicate the inefficient outcomes for if the switch occurs after 5 units of time. Since T = 5 < 8.5, oil depletion rates are always higher and oil prices are always lower than the efficient ones. Dashed lines correspond to a later change of government with T = 15 > 8.5, so the oil depletion and price paths cross the efficient paths before A leaves office. The simulations confirm that political uncertainty boosts oil depletion rates and depresses oil prices in the period before the change of government. After that, oil depletion jumps down while oil prices jump up and then continue at their less aggressive Hotelling rates.

Fig. 1
figure 1

Time paths of oil extraction before and after a regime switch

Full size image

2.4 Exploration Investment and the Hold-up Problem

Optimal exploration investment follows from setting its marginal value to its cost:

$$(1-1/\varepsilon )K\Omega {(I)}^{-1/\varepsilon }\Omega^{\prime}(I)=q\text{\hspace{1em}\hspace{1em}}\Rightarrow \text{\hspace{1em}\hspace{1em}}I=I(h,{\tau }^{A},{\tau }^{B},q),\text{\hspace{1em}}{I}_{h,}{I}_{{\tau }^{A}},{I}_{q}<0,\text{\hspace{0.33em}}{I}_{{\tau }^{B}}>0.$$
(16)

Higher political risk and more rent sharing (higher value of h, τA, and τB) make it less attractive to undertake exploration investment so that the discovered stock of oil reserves is less. This hold-up problem exacerbates the inefficiencies highlighted in Result 1.

3 Ongoing Conflict Cycles with 2-Sided Regime Switches

Consider ongoing switches in government. Assume that both factions share rents equally, \({\tau }^{A}={\tau }^{B}\equiv \tau ,\) and let the probability of a switch of government, h, be the same whoever is in office. With symmetric, asymptotic, Nash equilibrium outcomes, we do not need to distinguish the value functions for the two factions separately. Denote the in-office value function by \(V(S)\) and the out-of-office value function by \({V}^{*}(S).\) The HJB equation for the incumbent is

$$\underset{R}{\text{Max}}\text{\hspace{0.33em}}\left[(1-\tau )U^{\prime}(R)R-V^{\prime}(S)R\right]-h\left[V(S)-{V}^{*}(S)\right]=rV(S).$$
(17)

The value of being out of office is the present value of expected cohesiveness payments received when out of office plus the expected net rents upon gaining office, so that the HJB equation is

$$r{V}^{*}(S)=\tau U^{\prime}(R)R-{V}^{*{\prime}}(S)R+h\left[V(S)-{V}^{*}(S)\right].$$
(18)

From (17) marginal revenue equals the shadow value of oil, so \((1-1/\varepsilon )(1-\tau )U^{\prime}(R)=V^{\prime}(S).\)

Result 2: With ongoing political cycles, exhaustible resource depletion is rapacious and faster than predicted by the competitive Hotelling rule (L > εr), especially with weak constitutional cohesiveness (low τ), and frequent changes of government (high h).

Political uncertainty and less cohesive constitutions in a situation with ongoing political conflict (i.e. perennial two-sided regime shift) lead to more voracious extraction of the exhaustible resource. Furthermore, a partisan in-office bias also makes resource extraction more voracious.

4 Resource Wars

To study dynamic interactions between exhaustible resource extraction and resource wars, we let the hazard rates of a regime switch depend on fighting by the two factions. By fighting and diverting labour from productive activities, faction try to hold on to office or get into power. We thus study resource wars with repeated switches of government regime.Footnote 5

If A is in office, A and B fight \({f}^{A}\) and \({f}^{B*}\) units of time, respectively, and have \(N-{f}^{A}\) and \(N-{f}^{B*}\) units of time left for work with N exogenous labour supply of each faction. If B is in office, A and B fight, respectively, \({f}^{A*}\) and \({f}^{B}\) units of time and work \(N-{f}^{A*}\) and \(N-{f}^{B}\) units of time. The opportunity cost of fighting is the exogenous wage W. We let the hazard rates of A being replaced by B and of B by A depend on fighting be equal to

$${h}^{A}=\frac{2H{f}^{B*}}{{f}^{A}+{f}^{B*}},\text{\hspace{1em}}{h}^{B}=\frac{2H{f}^{A*}}{{f}^{A*} +{f}^{B}},\text{\hspace{1em}\hspace{1em}}H>0$$
(19)

(cf. [12, 31]. By fighting more, a faction improves its chances of staying in or gaining office and gaining control of oil. A rebel faction that does not fight, never gains office, \({h}^{A}=0\). If both the incumbent and the rebel faction fight with the same intensity, the hazard of being removed from office is \({h}^{A}={h}^{B}=H\). If the incumbent does not fend off rebels, its hazard of being removed from office is twice as high, \({h}^{A}=2H\), and B once in office will stay in forever, \({h}^{B}=0\). H stands for how fast elections take place or for core political instability while \(0<\tau \le 0.5\) is the cohesiveness of the constitution or share of oil rents the incumbent gives to the rebel faction.

A’s HJB equations for the non-cooperative subgame-perfect Nash equilibrium outcome are:

$$\underset{{f}^{A},{R}^{A}}{\text{Max}}\text{\hspace{0.33em}}\left\{(1-\tau )U^{\prime}({R}^{A}){R}^{A}-{V}_{S}^{A}(S){R}^{A}+W(N-{f}^{A})\right.\left.-{h}^{A}\left[{V}^{A}(S)-{V}^{A*}(S)\right]\right\}=r{V}^{A}(S),$$
(20)
$$\underset{{f}^{A*}}{\text{Max}}\text{\hspace{0.33em}}\left\{\tau U^{\prime}({R}^{B}){R}^{B}-{V}_{S}^{A*}(S){R}^{B}+W(N-{f}^{A*})\right.\left.+{h}^{B}\left[{V}^{A}(S)-{V}^{A*}(S)\right]\right\}=r{V}^{A*}\left(S\right).$$
(21)

There are two similar HJB equations for B in \(V^{B} (S)\) and \(V^{B*} (S).\) Equation (20) states that the A’s maximum oil rents (net of the share of oil rents given to B and net of the shadow cost of oil) plus income from productive activities minus the expected loss in value terms of losing office must equal the return from investing oil proceeds at the market rate of interest. Equation (21) states that the contender’s cohesiveness transfers plus wage income plus the expected gain of entering office must equal the market rate of return. Asymptotically, the effect of which faction started in office withers away and the in- and out-office value functions for the two factions converge. We will thus focus on the asymptotic subgame-perfect Nash equilibrium outcome.

4.1 Non-cooperative Outcomes

The non-cooperative outcome assumes that, if A is in office, it takes as given rebel fighting efforts \({f}^{B*}\) when choosing its optimal fighting efforts \({f}^{A}\) and oil depletion rate \({R}^{A}\). If A is out of office, it takes \({f}^{B},\) as given when deciding \({f}^{A*}.\) A’s marginal expected gain from, respectively, fighting in and out of office is set to the wage:

$$\left(\frac{2H{f}^{B*}}{{\left[{f}^{A}+{f}^{B*}\right]}^{2}}\right)\left[{V}^{A}(S)-{V}^{A*}(S)\right]=\left(\frac{2H{f}^{B}}{{\left[{f}^{A}+{f}^{B*}\right]}^{2}}\right)\left[{V}^{A}(S)-{V}^{A*}(S)\right]=W$$
(22)

and similarly for B. This gives two reaction functions for when A is in and out of office indicating that A will fight more if B fights more (both if A is in and out of office). If the intersection with the reaction functions for B exists, we obtain the non-cooperative symmetric Markov-perfect Nash equilibrium. Since hazard rates are given by symmetric contest functions, this equilibrium is symmetric and gives the fighting intensities:

$${f}^{A}={f}^{A*}={f}^{B}={f}^{B*}\text{=}\frac{\phi H}{2W} \left[V(S)-{V}^{*}(S)\right].$$
(23)

Fighting is thus more intense if the expected gain from office is high relative to the opportunity cost of fighting (W) and if it is easier to remove the incumbent from office (higher H). There is no direct effect of cohesiveness on fighting efforts, only via the value functions. In equilibrium \({h}^{A}={h}^{B}=H.\) The HJB equations for when in and out of office become, respectively:

$$ \mathop {{\text{Max}}}\limits_{{R^{A} }} \left\{ {({\text{1 - }}\tau ){{U}}^{\prime } ({{R}}){{R - V}}^{\prime } ({{S}}){{R + WN - 1}}.{{5H}}\left[ {{{V}}({{S}}){{ - V}}^{{{*}}} ({{S}})} \right]} \right\}{{ = rV}}({{S}}), $$
(24)
$$ \tau U^{\prime } (R)R - V^{{*\prime }} (S)R + WN + 0.5H\left[ {V(S) - V^{*} (S)} \right] = rV^{*} (S). $$
(25)

The incumbent sets marginal oil revenue net of cohesiveness payments to the marginal social cost of oil, \((1-\tau )(1-1/\varepsilon )p=V{\prime}(S).\) This implies that the oil price is low and the rate of oil depletion high if oil is abundant (high S and low \(V^{\prime}(S)\)), and cohesiveness is weak (low τ). We conjecture \({V}^{A}(S)={V}^{B}(S)= K{S}^{1-1/\varepsilon }+WN/r\) and \({V}^{A*}(S)={V}^{B*}(S)={K}^{*}{S}^{1-1/\varepsilon }+WN/r\) with K and K* to be found, so we have the optimal oil price and oil depletion rates:

$$p=\frac{K}{\beta (1-\tau )}{S}^{-1/\varepsilon },\text{\hspace{1em}}R=LS,\text{\hspace{1em}\hspace{1em}}L\equiv {\left[\beta (1-\tau )\right]}^{\varepsilon }{K}^{-\varepsilon }.$$
(26)

Using (26) in (24) and (25) and equating coefficients on \(S^{1 - 1/\varepsilon } ,\) we get:

$$\frac{1}{\varepsilon }{\left[(1-\tau )\right]}^{\varepsilon }{K}^{1-\varepsilon }-1.5H(K-{K}^{*})=rK,$$
(27)
$$\tau {\left[(1-\tau )\right]}^{\varepsilon -1}{K}^{1-\varepsilon }-(1-1/\varepsilon ){\left[\left(1-\tau \right)\right]}^{\varepsilon }{K}^{-\varepsilon }{K}^{*}+0.5H(K-{K}^{*})=r{K}^{*}.$$
(28)

which can be solved for K and K*. We thus find from (23) the fighting efforts:

$$f={f}^{*}=\frac{H}{2W}(K-{K}^{*}){S}^{1-1/\varepsilon }.$$
(29)

If all oil rents must be shared equally (\(\tau = 0.5\)), we have \(K={K}^{*}=0.5{(\varepsilon r)}^{-1/\varepsilon },\text{\hspace{0.33em}}L=\varepsilon r\) and thus \(R = \varepsilon rS\) and\(f=0\). Hence, a perfectly cohesive constitution ensures efficiency no armed conflict. If factions cannot be removed from office,\(H=0\), there is no fighting while \(K = \beta (1 - \tau )(\varepsilon r)^{ - 1/\varepsilon }\) gives \(p={(\varepsilon rS)}^{-1/\varepsilon }\) and \(R=\varepsilon rS.\) The outcome is also efficient independent of the degree of rent sharing,\(\tau \).

Result 3: Resource wars are more intense if reserves are high, and workers are paid poorly. Oil depletion is less rapid if a big part of oil rents is shared (big τ) and government stability is high (low H).

4.2 Some Simulations

Figure 2 shows the effects of rent sharing, τ, on oil depletion and fighting and Fig. 2 plots the values to go with baseline parameters set to ε = 2, r = 0.04, S0 = 100, H = 0.1, N = 0.2 and W = 8. We see that more rent sharing (higher \(\tau )\) makes oil depletion less rapacious and curbs armed conflict (cf. Result 3). Rent sharing also raises the value to go for the rebels and the joint value to go, but the incumbent’s value to go first decreases slightly and then increases slightly with the degree of rent sharing.

Fig. 2
figure 2

Oil depletion and fighting (f = f.*) versus rent sharing (τ)

Full size image

The short-dashed lines indicate that more political instability (H = 0.2 > 0.1) leads to more aggressive oil depletion and more fighting, especially if the political system is less cohesive and there is less rent sharing. If the chance of losing office is more imminent, the incumbent fears losing control of the oil stake and the rebels fight more. Political instability depresses value to go for the incumbent and the rebel factions, especially if there is little rent sharing. The model predicts that oil-rich countries with regular, hotly contested elections have more conflict.

Finally, it is easy to demonstrate that if the oil stake (proxied by oil reserves, \(S\)) is high or the opportunity cost of fighting (proxied by the wage, \(W\)), conflict is more intense even though the speed of oil depletion is unaffected. Also, armed conflict is less severe if rebels are more patient as then oil depletion is less rapid and payoffs to both the ruling and the rebel faction are higher.

4.3 Effects on Oil Exploration Investment and Initial Reserves

Since \(I=I\left(K(\tau ,H)\right), {I}_{H}<0,\) political instability depresses oil exploration investment and initial oil reserves. This hold-up problem depresses welfare further. Rent sharing has a non-monotonic effect on value to go for the incumbent and on oil exploration and initial reserves.

4.4 Empirical Evidence

Countries with high oil exports have more armed conflict especially in sub-Saharan Africa (e.g. [7, 10, 26]. This type of evidence also suggests that natural resources boost conflict, especially in ethnically polarised societies (e.g. [21, 25]). Giant oil discoveries increase armed conflict especially for countries that experienced armed conflict or coups in the previous decade [15]. Within-country evidence also points in this direction (e.g. [1] and finds that a higher price of capital-intensive oil induces guerrilla and paramilitary attacks [9].

5 Conclusion

We have shown that Long’s [17] analysis of expropriation risk can be extended to study the effects of (i) rent sharing and a one-off change in government, (ii) fighting to stay in or gain political power and of rent sharing on the voracity of oil extraction and initial oil exploration investments. We have thus analysed the effects of one-off and ongoing regime shifts with both exogenous and endogenous hazard rates. We have discussed how political uncertainty induces rapacious oil depletion and holds back exploration investment, especially if constitutional cohesiveness and rent sharing is weak. We have also shown why fighting for the control of oil is more intense with large oil reserves and a low wage, especially if there is little rent sharing.

It will take too much space to do justice to all the innovative contributions of Ngo Van Long, but it is worthwhile mentioning a few here that are related to the matters that have been discussed. First, Ngo Van Long is together with Hans-Werner Sinn the founding father of the extensive literature on the so-called green paradox, first exposited in Long and Sinn [19] and then much later worked out in more detail in Sinn [28]. As the review in van der Ploeg and Withagen [24] indicates, these crucial insights from almost forty years ago have spawned an extensive and ongoing literature about why a future surprise increase in the price due to an increase in future carbon tax increases resource extraction and accelerates global warming today. The intertemporal mechanisms are not that different from those underlying Long’s [17] theory of expropriation. The green paradox effect is particularly strong if the price elasticity of resource reserves is small.

Second, Long [18] shows in his masterful review of dynamic (differential and difference) games in the economics of natural economics how useful this type of analysis is to improve our understanding of this part of economics. The survey discusses both exhaustible and renewable resources, discusses the tragedy of the commons, considers different market structures varying from a monopoly, an oligopoly, a cartel with a competitive fringe, and open access, gives applications to carbon taxation, and pays attention to technology and strategic aspects of investments in R&D. A prominent example of the wide applicability of these tools is Benchekroun, Gaudet and Long [2]. They analyse why a non-renewable resource cartel that anticipates being forced, at some date in future, to break up into an oligopolistic market will produce more over the same interval of time than it would if there were no threat of dissolution. Furthermore, they show that the rate of extraction decreases with the life of the cartel and that the depletion rate may increase during the cartel phase.

Third, Long [20] is an authoritative survey on how to manage, induce, or prevent regime shifts. It deals with regime shifts, thresholds, and tipping points and pays careful attention to uncertainty including the role ambiguity aversion. It discusses applications to political economy, where it distinguishes between gradual and big-push political regime shifts including issues such as repression, redistribution, and gradual democratisation. It also discusses games of resource exploitation and games in industrial organisation with regime shifts (e.g. R&D contests). This survey illustrates the broad width of Long’s scholarship and achievements in the areas of optimal control theory and differential games and his genuine interests to apply to them to important and policy relevant problems in economic theory.

The political economy and dynamic resource wars games discussed in this paper owe much to the many insights offered over many decades by my dear friend and colleague Ngo Van Long. He will be sorely missed by young and old alike for his kindness, curiosity, enormous intellect, and unbounded generosity.