On Political Regime Changes in Arab Countries

Raouf Boucekkine, Fabien Prieur, Klarizze Puzon

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Working Papers / Documents de travail

WP 2014 - Nr 01

On Political Regime Changes in Arab Countries

Raouf BoucekkineFabien Prieur

Klarizze Puzon

On political regime changes in Arab countries∗

Raouf Boucekkine†, Fabien Prieur,‡and Klarizze Puzon§

January 22, 2014

Abstract

We develop a dynamic game to provide with a theory of Arab spring-type events. We consider two interacting groups, the elite vs. the citizens,two political regimes, dictatorship vs. a freer regime, the possibility toswitch from the �rst to the second regime as a consequence of a revolution,and �nally the opportunity, for the elite, to a�ect the citizens' decisionthrough concession and/or repression strategies. In this framework, weprovide a full characterization of the equilibrium of the political regimeswitching game. First, we emphasize the role of the direct switchingcost of a revolution (for the citizens) and of the elite's self-preservationoptions. Under the concession strategy, when the switching cost is low, theelite can't avoid the political regime change. She optimally adapts to theoverthrow of their political power by setting the rate of redistribution tothe highest possible level, thereby extending the period during which shehas full control on resources. This surprising result actually illustrates therole of the timing of events in these situations of interaction between theruling elite and the people. When the direct switching is high, the elitecan ultimately select the equilibrium outcome and adopts the oppositestrategy, i.e. she chooses the lowest level of redistribution that allows herto stay in power forever. The same kind of results are obtained whenthe elite relies on repression to keep the citizens under control. Next, theequilibrium properties under a mix of repression and redistribution areanalyzed. It is shown that in situations where neither repression (only)nor redistribution (only) protect the elite against the uprising of citizens, asubtle mixture of the two instruments is su�cient to make the dictatorshippermanent. Based on our theoretical results, we �nally examine the reasonfor such a large variety of decisions and outcomes during the Arab Springevents.

Keywords: political transitions, revolution, natural resources, opti-mal timing, regime switching, dynamic game.

JEL Classi�cation: C61, D74, Q34.

∗We thank Rabah Arezki, Gustav Feichtinger, Miguel Sanchez-Romero and participants inworkshops in Paris and Vienna for useful suggestions. The usual disclaimer applies.†Aix-Marseille University (Aix-Marseille School of Economics), EHESS and CNRS. E-mail:

raouf.boucekkine@univ-amu.fr‡LAMETA, Universite Montpellier I and INRA. E-mail: prieur@supagro.inra.fr§LAMETA, University of Montpellier I. E-mail: puzon@lameta.univ-montp1.fr

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1 Introduction

The Arab spring, which started in Tunisia in late 2010, is arguably the mostimportant event of the beginning of this century, certainly comparable to thefall of Berlin's Wall in 1989. While the political and geostrategic consequencesof this event are not yet settled, it's enough to look a few months or yearsbackward to realize how it is important and somehow striking: who could expectin early 2011 that �erce dictators and regimes as those of Benali or Mubarakwill be ousted after quite a few weeks of heavy protests and demonstrations?Who could expect the cascade of events which have a�ected the whole Arabworld after the Tunisian Jasmine revolution? After so many decades of brutal(and even insane) dictatorship, many analysts did lose hope in any institutionalevolution (not speaking about democratization!) in the Arab world before theamazing sequence initialized in Tunisia. It's therefore fair to state that the Arabspring came as a surprise, not only to political leaders around the world, butalso to the traditional media of the Arab World.1

i) A �rst puzzling aspect of the Arab spring is henceforth the timing. Whydid Arab populations wait 4 or 5 decades (after independence) beforerevolting? All these countries have in common the fact that they have beenruled for decades by elites controlling �ercely the rents deriving either fromthe exploitation of natural resources (that's oil and gas) or from economicliberalization (like in Tunisia, celebrated as one of the most open Arabcountries by the World Bank's Doing Business successive surveys priorto the Jasmine Revolution).2 What are the mechanisms leading fromsuch a regime with an omnipotent elite controlling the whole resource-dependent economy and deciding about everything to a regime changetowards democracy?

ii) A second striking feature is the large variety of outcomes observed sofar. In some cases, the ruling parties and dynasties were eventually ousted(as in Tunisia or Egypt) after weeks of political and social turmoil reg-ularly punctuated by massive and sometimes deadly demonstrations. Inothers, the same massive demonstrations have not been so successful andthe incumbents remained in o�ce as in Bahrein: it is likely that the Shiitedemonstrators didn't expect the bloody intervention of Saudi Arabia (forthe Sunnite minority to keep power in Bahrein) when they started the up-rising. Another case of external intervention is Libya, which resulted in theopposite outcome: the ruling dictator and associated �elite� were violentlyexpelled by the opposition backed by NATO massive bombings. A much

1A funny example of this unawarness is this French minister spending her 2010 Christmasholidays in Tunisia, and showing little concern about the situation even after the �rst registereddeaths. The whole French government led by Sarkozy was apparently sure that the �rm ally,Benali, will be able to control the situation and stay in power at the end of the day!

2Egypt was ranked in the Top 10 of the most reforming countries in the world by Doing

Business in 2008/2009, and Tunisia was ranked 55th out of 183 in 2011 by Doing Business

ranking.

2

less bloody case is Algeria: with a mixture of soft repression and massiveredistribution, the ruling elite has been able to retain the power (see Achy,2011). Indeed, the January 2011 protests have led the Algerian govern-ment to markedly enlarge the scope of its staple food subsidy programand its youth employment support packages leading to a durable destruc-turing of the national budget. The Regulation Fund, created in 2000 forthe stabilization of public expenditures thanks to the oils and gas exports,has been used to �nance the ongoing budget de�cits (about 50% of the2011 budget de�cit was �nanced by this Fund, according to Boucekkineand Bouklia-Hassane, 2011), which endangers the public �nance systemsustainability of this country. Morocco is another case where the elite (inother words, the King) has prevented a major uprising so far by conductingearly institutional changes towards a constitutional monarchy (see Traub,2012), signaling more redistribution of economic and political rights inthe future. This variety of outcomes calls for a deep theoretical researchon the use of redistribution/repression by ruling elites and the result-ing economico-political equilibria. Two questions arise quite naturally:Couldn't the ruling elite, with all the powers in hands, systematicallyprevent the occurrence of such moves by taking the adequate decisions,either �erce/soft repression and/or massive/moderate redistribution of therents, in the right time? For a given repression/redistribution policy ofthe elites, what does determine the decision of the opposition (that's thevast majority of the population) to go for a revolution and the inherenttiming?

The two striking aspects described above suggest that the underlined phe-nomena cannot be properly addressed without two key ingredients: in �rst place,the dynamic aspects to be able to fully understand the timing of the events, andin second place, the strategic aspects with ruling elites acting as strategic leadersin the initial regime. Accordingly, we shall develop a full-�edged dynamic gameframework to provide with a comprehensive theory of Arab spring-type phenom-ena as captured by the two features i) and ii). In this framework, we introducehierarchy: elites do act as strategic leaders. Moreover, the considered games ex-plicitly distinguish between the pre-revolution regime (dictatorship of the elite)and the post-revolution regime (say, common access to resources). Finally andmore importantly, the determination of (Markov perfect) equilibria includes thetiming of the transition (if it occurs) from the �rst to the second regime. Thisultimately leads to a methodological innovation as the latter requires mergingdynamic games with multi-stage optimal control (see Boucekkine et al., 2013afor the use of multi-stage optimal control in a non-strategic model of technologyadoption).3

In the �rst regime, the elite has full control on the stock of resources (to betaken in a broad sense, it could either come from an � unmodelled � extraction

3Two related papers are Long et al., 2013, and Boucekkine et al., 2011. But, among othernotable di�erences, the former does not have a strategic leader structure, and the latter isonly concerned with open-loop equilibria.

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sector or from economic resources derived from economic globalization, e.g.from FDIs or import licenses for example) and decides about: i) how muchresources to transfer to population (redistribution or concession), ii) how muchto use for own consumption, iii) how much resources to employ for repression(modelled as a �ow) and iv) how much to invest. The production function is AKto have a chance to extract (partial) analytical results. Of course, redistributionand/or repression are the instruments the elites may use (simultaneously or not,we examine both cases) to retain the power (self-preservation) in exchange forown consumption and investment. From the point of view of the elites, neitherredistribution nor repression make sense in the absence of a revolution threat.Given the elites' policy, the opposition has to decide whether she rebels (or not)against the elites and when (if she does so). A key aspect in the oppositiontradeo� is the direct (political regime) switching costs (DSC hereafter) faced.These costs depend either on the repression exerted by the elites and/or theirallies (possibly foreign countries) or on the coordination costs inherent in anycollective action. Knowing the resulting reaction function of the opposition, theelites as strategic leaders will then choose their optimal redistribution and/orrepression policies. We shall address all the research questions listed abovewithin this far nontrivial framework.

In our analysis, we cannot explicitly model important aspects of the Arabspring in order to come with a comprehensive enough analytical characteriza-tion of the economico-political equilibria. As one can already guess from thedescription just above, our simpli�ed setting is already heavily sophisticatedfrom the algebraic point of view. In most of the cases, we are able however toidentify shortcuts allowing to partially account for the missing pieces such likethe education level or external military interventions. This is possible thanksto our broad interpretation of the DSC. For example, concerning the level ofeducation, it's important to notice that the �rst country of the Arab spring,Tunisia, is the one which has invested the most in human capital in the recentdecades in % of the national budgets (for example, about 23% of the Tunisiantotal public expenditures was devoted to education in 2008, according to theWorld Bank's Edstats).4 If education increases the perception of grievance inthe face of unequal distribution of rents, then of course it should play a rolein the Arab spring story. Though we don't integrally consider such a channelin our model, we might view it partially through exogenous moves in the DSC.One might think that as the average education level of the countries becomeshigh enough, it is de�nitely easier to coordinate on a collective action,5 thuspushing down the DSC. The contrary occurs in the case of an exogenous in-crease in repression due to external pro-elite intervention. A key aspect for theexistence and nature of the economico-political equilibria is the controllabilityof the DSC by the elites. In reality, the DSC can be only partially controlledby the elites. Control obviously results from repression and is also linked to

4The Gulf monarchies and Jordan invest also a lot in public education, although slightlybelow 20% in the 2008 Edstats report.

5While this is a common idea in labour economics, see Becker and Murphy (1992), it seemsrelevant in any context involving collective actions.

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education, given the importance of public education in these countries (see theTunisian case above). A signi�cant part of the DSC is however out of the con-trol of the elites. Uncontrolled factors a�ecting the cost of revolting against theelites include opposite external military interventions (Libya or Syria), most ofthe elements linked to globalization (e.g. the role of worldwide communicationnetworks), and demography. In this paper, we shall study the two polar cases:the case where the DSC are fully controllable vs fully exogenous. The mixedcase is discussed in the last section before the conclusion.

Last but not least, we shall omit uncertainty. In the context of the Arabspring, an important source of uncertainty originates in the size of the rents(because of random resource prices in international markets for example). Thisuncertainty is likely to a�ect the elites' behaviour: in the Algerian case describedby Boucekkine and Bouklia-Hassane (2011), it is shown that the ruling nomen-klatura is much more prone to political and economic liberalization in periodswhere the oil prices are persistently low. Boucekkine et al. (2013b) have alreadyprovided with a stochastic game modelling of the latter aspect but they omittedthe two fundamental components of our framework: elites as strategic leadersand a multi-regime setting. Indeed, incorporating the latter components in aproper stochastic dynamic game structure sounds as a daunting task.

Our research can be related to two distinct streams of the economic liter-ature, one in the line of democratization games à la Acemoglu and Robinson(2006, 2008) and the other one (see for instance, Torvik, 2002) viewing natu-ral resources as a source of con�icts. It's de�nitely more closely related to theformer where a key question is the strategic behaviour of the elites as a centralactor of institutional change. While the dynamic aspect is not always absent (seeAcemoglu and Robinson, 2008), no dynamic games with explicit state variableand timing decisions are considered. We are not aware of any other paper alongthis line dealing with the fundamental features i) and ii) outlined above. Animportant aspect of the Arab spring and other democratization processes whichis well accounted for in Acemoglu and Robinson (2008) and which is missing inour paper is the distinction between political power and economic power, andtherefore between political and economic liberalization. As mentioned aboveabout Tunisia and Egypt, what led to the disgrace of their respective dictatorsis the lack of political liberalization while these economies are close to fully liber-alized. Non-simultaneity of both liberalizations has been more than problematicin these countries as analysed in Dunne and Revkin (2011). Concerning the sec-ond stream, several authors have recently pointed out that resource abundanceinduces political instability if competing factions try to obtain control over theassociated rents. Resources cause rent seeking and �ghting activities betweenrival groups, which weaken property rights and lead to a curse (see Torvik, 2002,Hodler, 2006, Mehlum et al., 2006, or Gonzalez, 2007). Cross-country evidencesuggests a positive correlation between rents from natural resources and civilwars. Perhaps the major contribution along this line is the greed and grievancestory told by Collier and Hoe�er (2004). In this seminal paper, the causes ofcivil war are studied using a new data set of wars during 1960-99. The authors

5

outline that rebellion may be explained by atypically severe grievances, such ashigh inequality, a lack of political rights, or ethnic and religious divisions in so-ciety. Accordingly, most subsequent studies have considered con�icts as a resultof grievance and greed between rival groups. Lower transfers to the oppositionmight induce grief as they sense injustice, and con�ict over resource rents arefurther exacerbated when society is fractionalized by competing interest groups(Caselli and Coleman, 2006). Perhaps the most comprehensive theories on re-source wars so far are due to Acemoglu et al. (2012) and van der Ploeg andRohner (2012). In the latter for example, some mechanisms leading to this typeof wars are singled out in a quite comprehensive setting where rebels �ght agovernment, and �ghting, armament, and extraction method, speed and invest-ment, are all endogenous. Interestingly enough, rapacious resource exploitationcan be preferred in this context to balanced depletion due to lowered incentivesfor future rebel attacks. Acemoglu et al. (2012) build their resource war modelin the framework of international trade with unequal endowment of resourcesacross countries. While in both cases the stories told are quite far from the Arabspring, one can hardly build on their respective conceptual settings to infer atheory for the latter. Indeed, a major ingredient of the Arab spring is lacking:a strong hierarchical relationship from the strategic point of view between theruling elites and population. That's in the Arab spring story, the elites play therole of strategic leaders. As correctly pointed out by Caselli and Cunnigham(2009) in an excellent theoretical re�ection on the �channels through which re-source rents will alter the incentives of a political leader,...these mechanismscannot be fully understood without simultaneously studying leader behaviour�.We eagerly adopt this point of view and build dynamic games theories for theArab spring with elites acting as strategic leaders. Pioneering works using dy-namic games in contexts of strategic exploitation of resources by rival groupsare due to Tornell (1996) and Lane and Tornell (1996). We depart from thesecontributions in two major ways. First, we do introduce hierarchy as mentionedjust above, and elites act as strategic leaders. Second, the considered games ex-plicitly distinguish between the pre-revolution regime (dictatorship of the elite)and the post-revolution regime (say, common access to resources). More impor-tantly, the determination of (Markov perfect) equilibria includes the timing ofthe transition (if it occurs) from the �rst to the second regime.

On theoretical grounds, we provide a comprehensive analysis of the equilib-rium of a political regime switching game when the elites adopt a redistributionstrategy and/or a repression policy. First, we emphasize the role of the di-rect switching cost of the citizens and of the elite's self-preservation options byfocusing on the particular cases where the elites use a single instrument. Con-cession made by the elites a�ects the citizens' uprising decision indirectly andmay not be su�cient to avoid a revolution. The direct switching cost is crucialin understanding the equilibrium outcome. When this cost is low, a politicalregime switching always occurs. The crucial point is then to understand howthe elites optimally adapt to the overthrow of their political power. Contraryto conventional wisdom, we show that the optimal strategy for the elites is to

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redistribute as many resources as possible to the people in the �rst dictatorialregime. This result clearly di�ers from the rapacious resource exploitation em-phasized by van der Ploeg and Rohner (2012). It highlights the importance ofthe �rst striking aspect underlined above, i.e. the timing of events of our rev-olution game. Once this temporal dimension is taken into account, it comes atno surprise that making high redistribution levels is worthwhile since it allowsthe elites to lengthen the period of political and economic control. When thedirect switching is high, the elites are able to provide su�cient transfers to theopposing citizens and choose the lowest level of redistribution compatible with apermanent dictatorship. The same kind of results are obtained when the elitesdecide to keep the people under control through repressive means. Repressionmakes it possible to directly change the switching cost and generally implies thatthere is more room for self-preservation. In situations where neither repression(only) nor redistribution (only) prevent the elites from being removed, a policy-mix between these two instruments succeeds in maintaining the non-democraticregime. Based on the criterion of retaining power, there is a temptation toconclude that this policy-mix is the best strategy for the elites. However, itis not always true that the payo�s associated with an equilibrium featuring apermanent dictatorship are higher than those corresponding to a solution witha regime change. Finally, we address the second important point raised in thebeginning of the introduction, which is related to the large variety of outcomesthat has characterized the Arab Spring. A discussion on the determinants ofthe cost of revolution (here, switching cost) is conducted. We emphasize howthe results of our theoretical model may contribute to the analysis of the ArabSpring events by explaining why some countries have experienced a transition to(more) democratic political regimes while others are still stuck in dictatorships.

The article is organized as follows. Section 2 presents the details of our dy-namic setup. Section 3 then explains the benchmark case when a revolutionarythreat is absent and presents our approach. We formally discuss our politicalcon�ict game when the elite adopts a strategy based on concessions for givenDSC in Section 4. Section 5 then analyzes the other polar case where the eliteuses repression in order to keep the citizens under control, and where repressionis the unique determinant of the DSC. Section 6 provides a synthesis of ouranalysis by considering the equilibrium under a policy-mix where. Section 7discusses the implications of our analysis for the Arab spring. Finally, Section8 concludes.

2 The setup

We consider a modi�ed AK growth model where the representative agent isreplaced by two in�nitely lived groups. We note them as the following players:the incumbent elite (E) and the opposition (P , for the poor citizens). Thesegroups comprise a fractionalized society with a resource-dependent economy.We abstract from any assumption regarding the size of the population and eachrival group. Emulating the framework of Lane and Tornell (1996), let K be the

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stock of resources and A the rate of return on this asset. Resources are de�nedin a broad sense and may refer to economic resources, or wealth and windfallsfrom natural resources.

There are two political or institutional regimes that describe the ways inwhich these groups interact: dictatorship (regime 1) and a freer regime withcommon access to the resources (regime 2), the switch from regime 1 to regime2 being the result of a successful revolution by the citizens. The economy initiallybelongs to regime 1. In this regime, the elite has the full control over the econ-omy and the economic resources and has to divide them between three di�erentuses: self-preservation, consumption, and investment. In order to lengthen thedictatorial system, the elite may choose either to make concessions to the citi-zens or to rely on repression. Self-preservation through concessionary spendingtakes the form a transfer of a share 1 − uE of the output AK to the opposi-tion. More formally, the concession strategy of the elite is modeled as a choiceuE ∈ [uE , uE ], where uE , the share of resources accruing to the elite, also repre-sents the inequality in the access to economic resources. Alternatively or jointly,the elite may decide to opt for a repression strategy. The repression strategyconsists of a �ow of military expenditures rE ∈ [rE , rE ], that can be interpretedas bribes paid to those who repress (or a periodic �xed cost to pay to bene�tfrom the protection to get some military support). The two last decisions of theelite involve the classical consumption-investment intertemporal tradeo�. LetCE be the elite consumption at period t.6 Then, the dynamics of the stock ofeconomic resources are:

K = uEAK − CE − rE , (1)

with K(0) = K0, given.Citizens' only source of wealth is from the elite's transfers, used for con-

sumption: CP = (1−uE)AK.7 Concessions and repression a�ect the revolutionstrategy of the citizens in two di�erent ways. From the expression of the citi-zens' consumption, one can easily see that concessions indirectly shape citizens'decisions by modifying the opportunity cost of the revolution. Other thingsequal, the higher the transfer (the lower uE), the lower the incentive to revolt.By contrast, repression directly impinges upon the opposition. It is worth clari-fying this point by describing the sequence of events leading to a regime changeand its consequences. A switch from regime 1 to regime 2 results from a revo-lution by the opposition. A revolution (if any) succeeds with probability one.A revolt is associated with a global cost χ > 0. This is the continuous timeanalog of Acemoglu et al. (2012) who assume that a �xed amount of resourceis destroyed when violent uprisings occur:

K(T+) = K(T )− χ, (2)

6When there is no risk of confusion, the time index is omitted.7Leaving aside self-preservation, regime 1 is the one wherein the elite consumes and invests,

while the opposition only consume. This structure shares similarities with the literature onthe interaction between capitalists and workers (Lancaster, 1973, Hoel, 1978), except that inour framework citizens are completely passive and subject to the control of the elite.

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which means that the state variable experiences a downward jump at the rightof the switching time T , provided that T is �nite. T represents the switchingstrategy of the citizens.

In addition, when conducting a revolt, the opposition incurs a direct switch-ing cost (DSC), ψ > 0. This cost may be due to e�orts from collective action,i.e. the opposing citizens need to coordinate when trying to instigate con�ict.Nonetheless, the magnitude of the DSC can also be interpreted as a measureof regime contestability. The higher is this cost, the more di�cult it is for theopposition to lead a popular uprising against the ruling elite. The importantpoint to note is that repression by the elite makes the cost of revolting rela-tively higher.8 As explained in the introduction section, the DSC are thereforepartially controllable by the elites through rE . In this paper, we will studythe polar cases where the DSC are fully exogenous (Section 4) vs fully control-lable (Sections 5 and 6) via rE . Hereafter, we make some assumptions on thedependence of ψ on rE when it is postulated:

Assumption 1 ψ(rE) from [rE , rE ] to [ψ,ψ], with 0 < ψ < ψ <∞, ψ′(rE) > 0and ψ′′(rE) ≤ 0.

After a revolution, the system switches to regime 2 where common access toeconomic resources prevails. The dynamics of the stock K simply becomes:

K = AK − CE − CP . (3)

Our view is that this second regime is characterized by more (political)freedom but may not be a well functioning democracy yet. Put di�erently,our groups engage in a political rent-seeking competition: Both groups try toextract transfers from resource wealth in a non-cooperative manner. The elite isno longer a leader but survives to the revolution, i.e. continues to take decisionseven after she loses the control of the economy. This assumption di�ers fromwhat is usually done in the literature (Acemoglu and Robinson, 2006). Wedo believe however that it is relevant to describe the situations of some Arabcountries where the Arab spring events have been successful in overthrowing theruling elite.

The preferences of the two groups are the same and invariant with the regime.A logarithmic function is utilized, e.g. U(Cji ) = ln(Cji ) with i = E,P andj = 1, 2. The rate of pure time preference is δ and the time horizon is in�nite.

In the next section, we �rst present the benchmark situation where the eliteis not subject to the revolution threat. Then, we examine the di�erential gamecorresponding to the second regime. This preliminary discussion will serve as abasis for the original part of the paper, devoted to the analysis of the politicalregime change game under concessions and repression.

8Alternatively, we may assume that a revolution does not succeed with probability one andmake the probability of success dependent on the level of repression. This would not changethe general message delivered by the subsequent analysis.

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3 Perpetual dictatorship, common pool game

3.1 Perpetual dictatorship of the elite

Our problem may generate two possible political modes: a perpetual dictatorialregime, and a regime switching from elite rule to common access. It is �rstworth studying the benchmark case with perpetual dictatorship, i.e. a regimein which one player (the incumbent elite) has permanent control of the politicalsystem. It boils down to considering that the ruling elite faces no threat ofrevolution. For the time being, assume that both uE and rE are given. Then,the elite solves the following optimization program:

max{CE}

∫ ∞0

ln(CE)e−δtdt

subject to (1) and K(0) = K0 given.Straightforward manipulation of the necessary optimality conditions yield

the time path of K, CE and CP :K(t) = (K0 − rE

uEA)e(uEA−δ)t + rE

uEA

CE(t) = δ(K0 − rE

uEA

)e(uEA−δ)t

CP (t) = (1− uE)A[(K0 − rE

uEA

)e(uEA−δ)t + rE

uEA

] (4)

The resulting present value, for the elite, is given by:

VE(uE , rE) =1

δ

[ln(δ) + ln

(K0 −

rEuEA

)+uEA

δ− 1

]. (5)

Several remarks can be made. First, the growth rate of the elite's consumption isequal to g = uEA− δ. Therefore, modifying the sharing rule is accompanied bya positive growth e�ect. By increasing uE , the elite has more resources availablefor consumption and investment. This stimulates growth. In addition, there isalso a positive scale e�ect. Given the level of repression, an increase in uE alsoimplies that the elite has more resources left for consumption. In the absence ofthreat of con�ict, the optimal choice of the elite is to set the sharing rule to thelargest possible value, i.e. to uE . Second, the repression decision only involvesa negative scale e�ect. So, it is also clear that when the elite does not have anypolitical challenger, there is no incentive to repress and the optimal repressionlevel is rE . For simplicity, hereafter, we will take rE = 0. Third, the presentvalue earned by the citizens, in this case, is:

VP (uE , 0) =1

δ[ln((1− uE)A) + ln(K0) +

uEA

δ− 1], (6)

while they also bene�t from the growth e�ect associated with an increase in uE ,they incur a negative rent capturing e�ect. Increasing uE means that the elitegrabs more resources at the expense of the opposition.

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Fourth, when a revolutionary threat is present, one logically expects that theelite will no longer be able to set her instruments to the levels uE and rE = 0.In the subsequent analysis, we will envision two extreme cases:•Case 1. Concessions, no repression: In this �rst case, the elite will respond

to the threat of revolution by making concessions only, which allow her to modifythe opportunity cost of revolting for the citizens. Therefore, the strategy willbe to choose uE ∈ [uE , uE ], taking rE = 0.• Case 2. Repression, no concessions: In contrast, here we consider that

the elite sets the sharing rule to her most preferred level uE and deals with thethreat of revolution through military expenditures. Thus, the choice is to pickup a rE ∈ [0, rE ].

Let's now discuss the critical boundaries for the domain of de�nition of ourstrategies, in the two cases. The basic idea we want to convey is that wheneverthe elite can maintain the citizens under control by appropriately choosing herinstrument, she should bene�t from her resulting leadership. Put di�erently, itseems reasonable to disregard situations where player E controls the economyand can't take advantage of this power. This would mean that she obtains avalue that is less than the value of player P , who is subject to her domination. Ina permanent dictatorial regime, we rather want to have: VE(uE , 0) ≥ VP (uE , 0)for all uE ∈ [uE , uE ] (case 1) and VE(uE , rE) ≥ VP (uE , rE) for all rE ∈ [0, rE ](case 2).9 This consideration leads us to impose the following restrictions, sum-marized in two di�erent assumptions:

Assumption 2 Under the concession strategy, the lower bound of the domainof de�nition of uE is de�ned such that VP (uE) = VE(uE)⇔ uE = 1− δ

A .

Assumption 3 Under the repression strategy, the upper bound of the domain

is de�ned by: rE = uEAK0(δ−(1−uE)A)δ , which implies that VE(rE) > VP (rE) for

all rE ≤ rE.

All the proofs are relegated in the appendices (see Appendix A.1).In order to simplify the analysis even further we will consider simple once-

and-for-all strategies for concessions and repression. In addition, note that inboth cases, the model exhibits a leader-follower structure regarding decisions uEand/or rE and T . It thus encompasses a two-stage optimal control/di�erentialgame problem that has to be solved backward. In the next section, we brie�yexamine the game played by our two players following a revolution.

3.2 After the regime switch: common access

After the revolution, the two groups interact in a common-pool resource di�er-ential game. Indeed, regime 2 is characterized by the lack of strong institutions.It can be seen as reduced form model where each group has its own resource

9To save notations, we will not make the dependance of the value function with respect tothe second �xed instrument explicit. For instance, in the �rst case, we will denote the valuefunction of player i as Vi(uE), i = E,P .

11

stock, with ability to appropriate part of the other's wealth (Lane and Tornell,1996). Given the structure of the model described above, this game is sym-metric. Using Markov-perfect equilibrium (MPE) as the solution concept andguessing linear (proportional) feedback strategy for players, which implies thatCj = aj + bjK with aj , bj two constants, each player solves:

max{Ci}

∫ ∞T

ln(Ci)e−δtdt

subject to (3), given K(T+) = K − χ, where K is the level of the capital stockat the instant of the revolution, and the guess formulated above. Direct manip-ulations of the necessary optimality conditions yield the MPE (the superscript2 refers to the second regime){

K2(t) = (K − χ)e(A−2δ)(t−T ),

C2i (t) = δ(K − χ)e(A−2δ)(t−T ),

(7)

and the value corresponding to this problem is, for each player i = E,P :

V 2i (K − χ) =

1

δ

[ln(δ) + ln(K − χ) +

A

δ− 2

].

This value yield the continuation payo� as seen from regime 1.We conclude this section by discussing growth prospects in the di�erent

regimes. A restriction on the parameters is imposed, which is necessary andsu�cient to ensure that the regime following a revolution is compatible withnon-negative growth:

Assumption 4 The productivity parameter is high enough compared to the dis-count rate: A− 2δ ≥ 0.

This assumption is the most relevant in our AK model where the analysisis conducted in terms of balanced growth path.10 Given the lower bound onthe sharing rule (see Assumption 2), this condition also ensures that elite'sconsumption grows at a positive rate in � the non permanent dictatorial �regime 1, whatever the case under scrutiny.

We are now equipped to analyze the political regime change problem. Thenext two sections alternatively conduct the equilibrium analysis in the �rst andthe second case considered. Mixed strategies, involving the simultaneous use ofredistribution and repression, are examined in section 6.

4 Equilibrium under the concession strategy

The purpose of the analysis below is twofold. First, we want to discuss the con-ditions under which a revolution might occur or, on the contrary the elite might

10Allowing for negative post-revolution growth, as it is observed in Tunisia or Lybia followingthe Arab Spring revolutions, would only make sense in the transitional dynamics, i.e. in theshort run, that are absent here.

12

stay in power forever. Second in case of unavoidable uprising, we wonder whatis the best concession strategy for the elite. If a revolution takes place in �nitetime, then the optimization problem can be decomposed into two subproblems,one for each regime. This is solved backward starting from the di�erential gamestudied above.

4.1 Before the regime switch: Elite rule

Let us proceed with the analysis of the problem faced by the elite given thepotential occurrence of a revolution at some date T . Under the once and forall choice of the resource sharing by the incumbent elite,11 �rst we solve theoptimal control problem for any uE . Then, we examine the elite's choice of thesharing rule. The optimization program of the elite is given by:

maxCE

∫ T

0

ln(CE)e−δtdt+ e−δTV 2E(K − χ)

subject to (1), K(0) = K0, and given that T 5∞. The level K(T ) = K is freeif T <∞.

Simple computations give the general solution, valid in the dictatorial regime,indexed by 1, for any uE :

K1(t) = φeuEAt + (K0 − φ)e(uEA−δ)t

C1E(t) = δ(K0 − φ)e(uEA−δ)t

C1P (t) = (1− uE)A[φeuEAt + (K0 − φ)e(uEA−δ)t]

(8)

with φ an unknown. If T <∞, then K1(T ) = K is free and the correspondingtransversality condition is:12

e−δTλ1E(T ) = e−δT∂V 2

E(.)

∂K. (10)

The crucial point is that the elite does not directly choose whether T 5∞.However, in some circumstances, he will be able to in�uence the choice of thecitizens to revolt or not. The purpose of the next section is to discuss thecondition(s) under which either a solution with T <∞, or T =∞ exists.

11To keep the forthcoming solutions tractable, we assume that there is no hold-up problem.The commitment mechanism of the elite is credible.

12Otherwise (T =∞), we have the usual standard transversality condition (and we get backto the permanent dictatorship studied in section 3.1):

limt→∞

e−δtλ1E(t)K1(t) = 0. (9)

13

4.2 Timing of the revolt

If the opposition �nds it optimal to challenge political control by the elite, thenhe earns the following present value:13

VP (K0, T ) =

∫ T

0

ln(C1P )e−δtdt+ e−δT [V 2

P (K − χ)− ψ]. (11)

The optimal condition for switching results from the maximization of (11) w.r.tT , which yields:

ln(C1P (T ))− V 2

P (K − χ) = −δψ.

If there exists an optimal T for switching then the marginal bene�t from delayingthe switch (LHS) must be equal to the marginal direct switching cost at thisinstant (RHS). Using (2) and (8), this condition can be rewritten as:

ln

(K

K − χ

)=A

δ− 2 + ln

(δ

[1− uE ]A

)− δψ. (12)

The LHS is a measure of the extent of the switching cost. The �rst term inthe RHS represents the growth rate achieved under the second regime. Thesecond term is a ratio of the proportion of resource consumed by the citizens inthe second and �rst regimes. The last entity represents the discounted privateswitching cost. Denote the RHS of (12) as ω(uE) and de�ne the critical thresholdfor the DSC, ψu, as follows:

ψu =1

δ(A

δ− 2) (13)

Then, it can be established that:

Lemma 1 A necessary and su�cient condition for the existence of a solutionto (12) is: ω(uE) > 0.

1. If ψ < ψu (low DSC) then, ω(uE) > 0 for all uE ∈ [uE , uE ]: There

always exists a unique K(uE) for switching with,

K(uE) =χeω(uE)

eω(uE) − 1. (14)

2. Else, ψ ≥ ψu (high DSC), there exists a critical threshold uE such thatω(uE) > 0⇔ uE > uE with,

uE = 1− δ

Aeδ(ψu−ψ), (15)

this threshold is admissible i.e. uE > uE.

13Under the concession strategy, the DSC is equal to the lowest possible value ψ, i.e. theelite can't change the DSC directly.

14

Irrespective of the size of the global switching cost (GSC), χ, the occurrenceof a political regime change is more likely when the DSC is low enough. Fora high enough DSC, the decision to undertake a revolution will be bound tothe sharing of resource �xed by the elite. So, the actual DSC and its posi-tion compared to the threshold ψu is of crucial importance to understand theoptions available to the elite. Clearly, when the DSC is low, the elite can'tavoid the revolution through concessions and the question is what is her beststrategy given that the regime change is inevitable. By contrast, when the costof switching regime is high enough, the elite seems to have the choice betweenmaking sizeable concessions in order to stay in power forever or grabbing a lotof economic resources in the �rst regime till a switch to regime 2 occurs.

Based on this discussion, hereafter a distinction will be made between twoscenarios, depending on the magnitude of the opposing citizens' switching cost.

4.3 Magnitude of the switching costs and nature of the

equilibrium

4.3.1 Low DSC scenario

In the case where the DSC of the citizens is low, ψ < ψu, a series of questionsnaturally arise: Does a solution with a revolution always exist? What doesthe incumbent do in its anticipation of a future regime change? Despite theexistence of an interior solution, is the alternative (permanent elite dictatorship)still possible?

Part of the answer to the �rst question is provided by the proposition belowwhere we characterize, for uE given, the solution to the switching problem.

Proposition 1 The optimal switching time is implicitly given by

K0e(uEaA−δ)T = χ

(1

eω(uEa) − 1+ e−δT

). (16)

This equation has a unique solution, denoted by T (uE), if and only if

K(uE) > K0 ⇔χeω(uE)

eω(uE) − 1> K0. (17)

where K(uE) is de�ned in (14).If K(uE) > K0, and furthermore χ > K0, then this solution exists for all

uE ∈ [uE , uE ].

Proof. See the Appendix A.2.Proposition 1 states that the switching level must be higher than the initial

stock. This implies that the opposition allows the resource to accumulate �rst.This ensures that the remaining �cake� after the switch is high enough to com-pensate for the loss incurred during the regime change. This also guaranteesthat what is competed for in the common access regime is abundant. More

15

importantly, Proposition 1 indicates that the existence condition is dependenton the GSC. The existence of a solution, for a given uE , surprisingly requiresthe GSC be high compared to the initial stock, K0. The cost of switching forsociety must be high enough for the citizens to revolt against the elite. Indeed,the explanation relies on a remaining cake size e�ect. In order for the secondregime to be valuable to the citizens, the amount of resource available at thebeginning of the common pool game should be su�ciently high. In other words,the remaining cake K(uE) − χ should be large enough for the revolution tooccur. Now, this critical level is increasing in χ.

Assuming that a regime change will occur in �nite time, we now tackle thechoice of the sharing rule by the elite. Before proceeding to the analysis, it isuseful to assess the features of the solution in regime 1:

K1(t) = χeuEA(t−T )(

1 + e−δ(t−T )

eω(uE)−1

),

C1E(t) = δχ

eω(uE)−1e(uEA−δ)(t−T ),

C1P (t) = (1− uE)AK1(t).

(18)

Let us start the discussion with a comparative statics exercise. From (14)and (16), it appears that:{

∂K∂uE

< 0; ∂K∂χ > 0; ∂K

∂ψ > 0;∂T∂uE

< 0; ∂T∂K0

< 0; ∂T∂χ > 0; ∂T

∂ψ > 0.

As expected the desired switching level, for the citizens, is decreasing in uE :The higher uE , the lower the opportunity cost of switching. A high uE impliesthat the �rst regime is painful for the citizens. In contrast, the higher the GSCor the DSC, the higher the desired switching level of resource. This is due to thecake size e�ect. Regarding the switching date, the revolution will occur morerapidly if the elite chooses an unequal sharing rule during the �rst regime. Theswitching date decreases with the initial endowment too. The more abundantis the initial stock of natural resource in the economy, the more rapidly willcon�ict occur. With a higher initial stock, the resource level that triggers therevolt is achieved earlier. This observation is consistent with resource curseliterature related to civil wars (see Hodler, 2006; Ploeg and Rohner, 2012 fordetailed examples). Finally, the impact of a change in the switching costs onT is positive. To make the revolt valuable, the citizens must accept a longerphase of resource accumulation during the �rst regime in order to reach a largerK(uE) that will compensate for the cost.

Also central to the elite decision is the impact of a change in uE on herconsumption level during the �rst regime.14 The elite's consumption can be

14Another interesting feature of the �rst regime concerns the evolution of the ratio betweenthe consumption levels of the two groups, which can be seen as a measure of the level of

inequalities in the society. From (18), it appears that the ratioC1P (t)

C1E(t)

=(1−uE)A

δ[1+(eω(uE)−

1)eδ(t−T )] increases over time. As time passes by, the di�erence between consumption levels

16

rewritten as:

C1E(t;uE) = δK0(uE)e(uEA−δ)t with K0(uE) = K0 − χe−uEAT (uE) (19)

Comparing this expression with the one in (4), with rE = 0, it turns out that thegrowth rate of consumption is similar. The striking di�erence is the existenceof a scale e�ect. Elite consumption is lower at the equilibrium with a regimechange, uE being given. The derivative of elite consumption with respect to uEis

∂C1E(t;uE)

∂uE= δ

[K ′0(uE) +AtK0(uE)

]e(uEA−δ)t, (20)

with,K ′0(uE) = χAT (uE)e−uEAT (uE)[1 + σ(T )],

where σ(T ) = uET′(uE)

T (uE) is the elasticity of the switching date with respect to

the sharing rule uE .The RHS of (20) can be decomposed into two terms. The second term is a

positive growth e�ect. A larger uE implies greater investment. This eventuallytranslates into a higher consumption growth rate. The �rst term is a combi-nation of two opposing forces: the wealth e�ect due to rent-capturing, and theregime instability e�ect. When uE increases, indeed, the elite is able to consumemore. However, increasing uE always decreases the waiting time T (uE) when arevolution will occur. A higher uE makes elite rule less cohesive.

If |σ(T )| ≤ 1, the wealth e�ect dominates the instability e�ect. Consumptionin the �rst regime increases with uE . On the other hand, if the impact of uEon T is strong enough (|σ(T )| > 1), then the instability e�ect dominates. Inthis case, when investment uE increases, elite consumption takes the oppositedirection. In anticipation of con�ict, the elite is willing to sacri�ce her ownconsumption. From now on, we assume |σ(T )| ≤ 1, i.e. the switching timechosen by the citizens is not too sensitive to the sharing rule. This is a relevantcharacteristic of resource-dependent economies with mediocre levels of socialcapital, e.g. awareness towards collective action.

Let us now look at the incumbent elite's choice of uE , under the constraintthat uE ∈ [uE , uE ]. The value obtained by the elite in the equilibrium with aninterior regime switching is:

VE(uE) =

∫ T (uE)

0

ln[C1E(t;uE)]e−δtdt+ e−δT (uE)V 2

E [K(uE)− χ]. (21)

decreases. However, the consumption level of the elite at any date up to T is larger than theone of the citizens because C1

E(T ) > C1P (T ). This feature may appear to be at odd compared

to what is observed in reality. It however re�ects the optimal reaction of the elite to thefuture occurrence of a political regime change. In anticipation to the coming events, the eliteoptimally decides to smooth her consumption during the transition between the two regimes.This is ultimately accompanied by a decrease of the consumption ratio. Note however thatwhat may seem to be a drawback of our stylized model is not robust to the removal of thenormalization of the lower bound on the repression strategy to nought. With a strictly positiverE , this by-product of the analysis will not be observed.

17

The derivative of VE w.r.t. uE is:

∂VE(uE)

∂uE=

∫ T (uE)

0

∂C1E(t;uE)

∂uE

C1E(t;uE)

e−δtdt

+T ′(uE)e−δT (uE){ln[C1E(T (uE))]− δV 2

E [K(uE)− χ]}+e−δT (uE) ∂V

2E [K(uE)−χ]

∂K K ′(uE)

(22)

The �rst term represents the cumulative impact of a change in the redis-tribution rate on consumption. As shown above, this is positive as long as|σ(T )| ≤ 1. The second positive term states a comparison between the cost andbene�t at the switching time T (uE). The term between the brace brackets isthe marginal gain of delaying the switch. It is weighted by the derivate of theswitching date w.r.t uE , which is negative. It compares the discounted value ofwhat the elite gains in the second regime, and her loss in utility (from consump-tion) during the switch. The third term exhibits the remaining cake size e�ect.This is always negative as K ′(uE). Simply put, this term indicates the loss dueto resource destruction at the start of the second regime.15 Careful analysis ofthis derivative leads to:

Proposition 2 If |σ(T )| ≤ 1 for all uE ∈ [uE , uE ], then under the conditionsof Proposition 1, there exists an equilibrium with political regime switching in�nite time in which the elite sets the sharing of resource to the level u∗E = uE.

Proof. See the Appendix A.3.Given that a revolt will occur in the future, one may expect that the elite's

best response is to set the sharing rule to the highest possible value uE as shetakes advantage of her period of control. This is the kind of message deliveredby the related literature (see for instance Ploeg and Rohner, 2012). It turns outthat the optimal choice of the elite is the exact opposite, i.e. she chooses anuE equal to the lower bound uE . This quite surprising result emphasizes therole of the timing, which is typically ignored by the literature. However, oncewe account for the timing issue, it comes at no surprise that setting uE is goodto the elite. It allows her to delay the regime switching and to lengthen theduration in o�ce.

The last question is whether there exist multiple equilibria in the low DSCscenario. The other possible solution corresponds to the situation where theopposing citizens never �nd it optimal to undertake a revolt. This means thatwe have a corner solution for the regime switching problem. Assuming that theinitial regime is dictatorship and immediate switching is not allowed, the cornersolution of interest is of the never switching type. If there is no T < ∞ for

15An alternative reading of the expression in (22) is that the impact of a change in uEcan be divided between three di�erent e�ects, depending on the period during! which thechange is felt: The marginal impact of choosing uE before the regime change (�rst term), themarginal impact at this instant of the regime change (second term) and the marginal impactof varying uE after the political switch (last term).

18

switching, then it must hold that:

limT→∞

∂VP (.)

∂T≥ 0⇔ lim

T→∞e−δT

[ln

(K

K − χ

)− ω(uE)

]≥ 0 (23)

with VP (.) de�ned in (11). This condition implies that postponing the switchis associated with a marginal gain that is not lower than the marginal loss offoregoing for an instant the bene�t from switching to the second regime.

Given that a general solution with perpetual elite dictatorship is character-ized by (4), we have:

Proposition 3 For a given uE, a necessary condition for the citizens' problemto have a never switching solution is ω(uE) ≤ 0. A su�cient condition for theconverse to be true is ω(uE) > 0.

Proof. See the Appendix A.4.To sum up, the analysis conducted in the �rst scenario can be summarized

in the following corollary.

Corollary 1 In the low DSC scenario, there exists a unique equilibrium wherethe citizens challenge political control by the elite in �nite time and the elite setsthe sharing rule to the lower bound uE.

According to Lemma 1, in the low DSC scenario, ω(uE) > 0 for all admissibleuE . Despite her leadership and ability to manipulate the sharing rule, theincumbent elite is unable to avoid the revolution by the opposition.

4.3.2 High DSC scenario

We now pay attention to the second scenario characterized by a high DSC,ψ ≥ ψu. This is an interesting case because the elite now has the capacity toindirectly in�uence the choice of the citizens. When the elite is willing to makesu�cient concessions by sharing the resource in such a way that citizens arenot too harmed, then the latter may not instigate con�ict. Hence, there mightbe conditions wherein the opposition prefer staying under permanent elite rule.In what follows, we conduct the analysis of the high DSC scenario using thematerial presented in the previous subsection.

As stated in Lemma 1, when the DSC is high (and such that ω(uE) < 0), theelite has two options available. Either, she chooses a sharing rule uE ∈ [uE , uE ],which implies that ω(uE) ≤ 0. In this case, from Lemma 1 and Proposition 3,the unique equilibrium must be of the never switching type. Or, the elite can �xthe sharing rule to a level uE ∈ (uE , uE ]. In this case, the equilibrium candidateexhibits a regime change, characterized in Lemma 1 Proposition 1.

Nonetheless, even if there are two possible solutions, we cannot have multipleequilibria, i.e. one equilibrium of each type for the same uE . In the end, theelite seems to have the power to select the particular outcome that is best forher. The reason is that in this second scenario, he can strategically a�ect the

19

switching decision of the opposition. Consequently, the elite has a choice tomake between these two options. Obviously, she will select the option yieldingthe highest value.

To better understand this decision, we further examine the optimal sharingrule corresponding to each of these options. In the �rst - never switching -case, based on the �ndings for a perpetual dictatorship with no revolutionaryoption (see Section 3.1), the elite should choose u∗E = uE : Her value, de�ned in(5), with rE put equal to zero, is increasing in uE , which implies that she setsthe sharing rule to the highest level compatible with a permanent dictatorship.The conditions are rather di�erent under the second option as it now considers arevolt by the citizens. The elite must decide on the sharing rule in (uE , uE ] withthe aim of maximizing her value, which is now given by (21). If we want to beconsistent with the condition imposed in Proposition 2 (regarding the elasticityof the switching date with respect to uE), then the answer is trivial. Given thatthe value of the elite is decreasing in uE , her optimal decision would be to setuE to the lowest possible level. However, this critical redistribution rate, uE , isnot achievable because the interval corresponding to a solution with a regimechange is open on the left. Since there is no solution to the elite's problem,there is no equilibrium featuring a transition between the two political regimes.This analysis is summarized in:

Corollary 2 In the high DSC scenario, there exists a unique equilibrium wherethe elite stays in power forever and sets the sharing rule to the intermediatelevel uE, de�ned in (15).

5 Equilibrium under the repression strategy

The analysis of the second case where the elite adopts the repression strategyfor given redistribution is similar to the one conducted in Section 4. We shalltherefore brie�y summarize our main �ndings. Suppose that the share uE is�xed at the largest possible value, uE .

From the resolution of the timing problem of the citizens and the de�nitionof:

ω(rE) =A

δ− 2 + ln

(δ

[1− uE ]A

)− δψ(rE),

and,

ψr =1

δ

(ln(δ)− ln[(1− uE)A] +

A

δ− 2

),

the second critical threshold for the DSC in this case,16 we obtain

Lemma 2 A necessary and su�cient condition for the existence of a solutionto the switching problem of the citizens is: ω(rE) > 0.

16Quite naturally we have ψr > ψu: Other things equal, when the elite doesn't provide anytransfers to the citizens, the latter group is willing to revolt for a relatively lower switchingcost. The main di�erence with the �rst case is that now the elite is able to modify directlythis switching cost.

20

1. If ψ ≥ ψr then dictatorship is necessarily permanent.

2. If ψ < ψr (low returns to repression, RR) then the revolution is

unavoidable and occurs for a switching level K(rE) = χeω(rE)

eω(rE)−1 .

3. If ψ ≥ ψr > ψ (high RR) then there exists a critical repression level

rE = ψ−1(ψr) such that: Any rE < rE will trigger a revolution in �nitetime whereas choosing rE ≥ rE is a means to avoid the revolution.

Proof. See the appendix A.5.The solution to the �rst case is trivial. From the discussion conducted in

Section 3.1, the optimal solution is r∗E = 0. By analogy with Section 4, twointeresting cases remain. On the one hand, when the RR are low, the elite isnot able to keep the citizens under control and the political regime change isinevitable. On the other, the elite may avoid the revolution provided that therepression technology is e�cient enough and by investing at least rE in militaryexpenditures. Put di�erently, a permanent policy that consists in devoting aconstant level of resources rE to the military budget protects the elite from anuprising of the citizens.

Next, we can establish that:

Proposition 4 Suppose rE is given. If χ > K0 then the switching time T (rE)is uniquely and implicitly de�ned by:(

K0 −rEuEA

)e(uEA−δ)T = K(rE)− χ+

(χ− rE

uEA

)e−δT ,

with K(rE) de�ned in Lemma 2. The comparative statics are:

∂T

∂rE> 0,

∂K

∂rE> 0;

∂T

∂uE< 0,

∂K

∂uE< 0;

∂T

∂χ> 0,

∂K

∂χ> 0 and

∂T

∂K0< 0.

Proof. See the Appendix A.5.The same condition as in Proposition 1, for the concession strategy, is also

su�cient for the existence of a solution to the switching problem. As far asthe impact of repression on this solution is concerned, intuitively we obtain thatincreasing repression expenditures is a means to delay the revolution, which willoccur for a larger stock of resource. Indeed, the larger rE , the higher the costof switching and the larger the compensation must be for the citizens. Thiscompensation takes the form of the achievement of the second regime, whosepro�tability is determined by the di�erence K(rE) − χ (the remaining cakesize). Regarding the other parameters, the larger uE , i.e. the more unequal thecountry is in the absence of concessions by the elite, the lower the opportunitycost of switching and the sooner the political regime change. In addition, witha large uE , the citizens accept to start the second regime with a lower amountof economic resources. Finally, a larger initial wealth of stock of resources tendsto expedite the decision to revolt.

21

Suppose that T (rE) exists. Then, the solution corresponding to regime 1can be written as:

K1(t; rE) =(χ− rE

uEA

)euEA(t−T (rE)) + (K(rE)− χ)e(uEA−δ)(t−T (rE)) + rE

uEA

C1E(t; rE) = δ(K(rE)− χ)e(uEA−δ)(t−T (rE))

C1P (t; rE) = (1− uE)AK1(t; rE).

(24)The last proposition summarizes our results in the two possible scenarios,

low vs. high RR. Let us �rst de�ne σ(K) = rEK′(rE)

K(rE)> 0. Then,

Proposition 5 If σ(K) < − rEψ′′(rE)

ψ′(rE) for all rE ∈ [0, rE ] and∂C1

E(t;rE)∂rE

|rE=rE >

0 then, ∂VE(rE)∂rE

> 0 for all admissible rE.

• In the low RR scenario, there is a unique equilibrium where the revolutionoccurs in �nite time and the elite sets r∗E = rE.

• In the high RR scenario, the unique equilibrium features permanent dic-tatorship and r∗E = rE.

Proof. See the Appendix A.5.Under the repression strategy, the trick consists once again in determining

the sign of the derivative of the value function of the elite with respect to therE . This boils down to determining how the consumption of the elite, in the�rst regime, responds to a change in the repression level. It appears that anincrease in rE has two opposing (scale) e�ects on the elite consumption. Thee�ects are more easily seen when rewriting consumption as:

C1E(t; rE) = δ

[K0 −

rEuEA

−(χ− rE

uEA

)e−uEAT (rE)

]e(uEA−δ)t

Both the initial condition and the GSC are reduced by an amount rEuEA

. Theresulting di�erences de�ne the true values, or the values that matter to the elitewhen deciding how much to repress. Then, the analysis runs as follows. As inthe �rst case (concession strategy, see (19)). The constant term in the expres-sion above is a rescaling of the true initial condition obtained by subtractingthe true GSC incurred at the date of the revolution, T (rE), discounted from theinitial period at the autonomous growth rate of the stock of economic resources.Other things equal, more repression means that revolution will occur later andconsequently this loss is felt less acutely, which stimulates consumption. How-ever, at the same time, an increase in rE implies that less resources are left forconsumption and investment at every date in regime 1. This in turn tends tolower the elite consumption. Now, it turns out that if the elasticity of the desiredswitching level, K(rE), with respect to repression is low enough and lower thanthe sensitivity of the marginal switching with respect to this strategy, then theoverall e�ect is positive: regime 1 consumption is higher the larger rE .

17 Under

17The second su�cient condition is Proposition 5 is a technical boundary condition, see theAppendix A.5.

22

this condition we �nally obtain that the value function of the elite is strictlyincreasing in rE when the political regime switch is inevitable. Therefore, inthe low RR scenario, the elite will choose the highest level of repression in orderto lengthen the �rst dictatorial regime. By contrast, when the RR are high, apermanent dictatorship is the only possible equilibrium. In this case, and giventhat her present value is now decreasing with the level of repression, the elitesets the repression level to the lowest possible value, rE , allowing for the absenceof regime switch.

6 Policy mix

In the two preceding sections, the analysis has revealed under which conditionsa revolution might occur in an economy ruled by an elite having two options tocontrol the citizens. Let us now go back to the other important question raisedby the paper: What is the best strategy for the elite? The answer to this ques-tion is more complicated. At �rst glance, one may reply that the best strategyis the one that allows the elite to stay in power forever, if possible. A quick in-spection of the ordering between the critical switching costs gives some insightsinto this crucial point. Assume that ψ < ψu(< ψr). In this situation, adopt-ing the concession strategy doesn't succeed in avoiding the revolution whereasrepression may be su�cient to keep the citizens under control provided that itrenders the switching cost prohibitive, which requires ψr ≤ ψ. Thus based onthis criterion, the elite should implement a repression policy. The conclusion isreversed however, when the ordering is ψu ≤ ψ < ψ < ψr. Here repression isnot e�cient enough to prevent political challenge by the citizens. In this case,an elite mostly interested in holding power should rather provide citizens withlarge transfers so that regime 1 is good enough to them and the opportunitycost of revolting is too high compared to the resulting bene�ts associated witha freer political system.

A last question arises ultimately: could a strategy mix perform better thanthe single-instrument strategies studied so far? The answer to this question isfar nontrivial. Let us see under which conditions the strategy mix is better forthe elite in the sense that it allows him to stay in power when a single strategyfails to do so.

From lemmas 1 and 2, we know that the revolution is unavoidable underconcession (resp. repression) when ψ < ψu (resp. ψ < ψr). Let's assume that

these two conditions hold and the overall ordering is: ψ < ψu < ψ < ψr.18 In

that case, using a strategy mix can be a means to maintain the dictatorship.The proof is as follows: From now on, we allow rE and uE to vary in their

respective domains of de�nition, i.e. [0, rE ] and [uE , uE ]. Adopting the samemethodology as before, we can show that the switching problem of the citizens

18The following reasoning can easily be extended to the other possible cases.

23

de�nes a unique K(rE , uE) if and only if ω(rE , uE) > 0 with:

ω(rE , uE) =A

δ− 2 + ln

(δ

[1− uE ]A

)− δψ(rE).

Solving the equation ω(rE , uE) = 0 boils down to solving ψ(rE) = ϕ(uE). Thefunction ϕ(uE) = ψr − 1

δ [ln(1 − uE) − ln(1 − uE)](> 0 for uE ≥ uE) satis�es

ϕ′(uE) > 0, ϕ(uE) = ψu and ϕ(uE) = ψr. Given our ordering, it's clear thatψ(rE) = ϕ(uE) has a solution because we have ψ < ϕ(uE) < ψ < ϕ(uE). More

precisely, we can de�ne uE ∈ (uE , uE) such that ϕ(uE) = ψ and rE ∈ (0, rE)such that ψ(rE) = ψu. Given that ψ is invertible, for any uE ∈ [uE , uE ],the equation ψ(rE) = ϕ(uE) can rewritten as a relationship rE = κ(uE)(=ψ−1′(ϕ(uE))) with κ′(uE) > 0, κ(uE) = rE and κ(uE) = rE . By de�nition,any pair (rE , uE) belonging to this locus is such that ω(rE , uE) = 0. Then, letζ be the set such that for any pair taken in this set a revolution cannot occur(dictatorship is permanent) because ω(rE , uE) ≤ 0:

ζ = {(rE , uE) ∈ [uE , uE ]× [rE , rE ]/rE ≥ κ(uE)} ,

the complementary set being denoted by ζ. For any pair in ζ, there might existan equilibrium with a revolution in �nite time.

The second step consists in determining the optimal combination (rE , uE) ∈ζ. Under permanent dictatorship, the value of the elite is given by (5), whichis decreasing in rE and increasing in uE . Thus, for any, uE , the elite choosesthe lowest level of repression compatible with dictatorship. This means that theoptimal combination necessarily lies in the frontier given by rE = κ(uE). Then,the value can be rewritten as:

VE(uE , κ(uE)) =1

δ

[ln(δ) + ln

(K0 −

κ(uE)

uEA

)+uEA

δ− 1

].

Taking the derivative w.r.t uE , one obtains:

∂VE∂uE

=κ(uE)

u2EA(K0 − κ(uE)uEA

)(1− σ(κ)) +

A

δ2with σ(κ) =

uEκ′(uE)

κ(uE),

and the condition σ(κ) < 1, together with K0 − κ(uE)uEA

> 0, is su�cient to have∂VE∂uE

> 0. The conclusion is that the optimal combination is (uE , rE), whichyields a present value:

V PE (uE , rE) =1

δ

[ln(δ) + ln

(K0 −

rEuEA

)+uEA

δ− 1

]. (25)

The third step shows that permanent dictatorship is indeed the unique equi-librium when the ordering is ψ < ψu < ψ < ψr. Here our aim is to generalizethe analyses of the previous sections and to prove that under the conditionsused in propositions 1 and 5, very few is needed to obtain the result.

24

Let's assume that the elite picks up a policy mix from ζ. This implies thatthe switching problem has a solution (K(rE , uE), T (rE , uE)), which is similar tothe one characterized in lemma 2, proposition 4 and (24) when one replaces uEwith any uE provided that (rE , uE) ∈ ζ.19 To determine under which conditionsthe general properties derived in the two preceding sections are still valid, let'ssee how the �rst period elite's consumption, C1

E(t; rE , uE) responds to changesin rE and uE given that:

C1E(t; rE , uE) = δ(K(rE , uE)− χ)e(uEA−δ)(t−T (rE ,uE)),

with K(rE) = χeω(rE,uE)

eω(rE,uE)−1 .

Partial derivative w.r.t rE for any uE ∈ [uE , uE ]:20

∂C1E(t; rE , uE)

∂rE= C1

E(t; rE , uE)

(KrE

K(rE , uE)− χ− (uEA− δ)TrE

),

We know that under the conditions of Proposition 5, this derivative is positive

at uE = uE . Given thatKrE

K(rE ,uE)−χ = δψ′(rE)K(rE ,uE)χ is decreasing in uE

(because KuE < 0), we haveKrE

K(rE ,uE)−χ |uE≤uE >KrE

K(rE ,uE)−χ |uE=uE . Note also

that uEA − δ < uEA − δ for all uE ∈ [uE , uE ]. Then the condition TrEuE > 0

for all (rE , uE) ∈ ζ is su�cient to conclude that∂C1

E(t;rE ,uE)∂rE

> 0 providedthat the conditions of Proposition 5, adapted to the case studied, hold. The

�rst condition is σ(K) < − rEψ′′(rE)

ψ′(rE) where the RHS is independent of uE and

one can easily check that the LHS is decreasing in uE . Thus, this condition

must be strengthened to require that σ(K)|uE=uE< − rEψ

′′(rE)ψ′(rE) . The second

technical condition is satis�ed as well. The new condition, that involves thecross derivative TrEuE > 0, basically means that when the elite makes lessconcessions (uE increases), the switching time chosen by the citizens becomesmore sensitive to the level of repression.

By analogy with section 4, consumption can be rewritten as: C1E(t;uE , rE) =

δK0(uE , rE)e(uEA−δ)t with

K0(uE , rE) = K0 −rEAuE

− (χ− rEAuE

)e−uEAT (uE ,rE).

The partial derivative w.r.t uE for any rE ∈ [rE , rE ]:

∂C1E(t;uE)

∂uE= δ

[K0,uE +AtK0(uE)

]e(uEA−δ)t,

with,

K0,uE =rEAu2E

(1− e−uEAT (rE ,uE)

)+

(χ− rE

uEA

)AT (rE , uE)(1 + σ(T )),

19Of course, the argument uE becomes apparent in the functions K(.) and T (.).20With a slight abuse of notation, we will denote the derivate w.r.t rE for K and T as KrE

and TrE .

25

where σ(T ) =uETuET (rE ,uE) is increasing in rE . Then, extending the condition of

Proposition 2 to |σ(T )|rE=rE ≤ 1 is su�cient to conclude that∂C1

E(t;uE)∂uE

> 0.Finally, under the conditions stated above, it's straightforward to show that

the present value of the elite at a solution featuring a regime change is decreasingin uE and increasing in rE . Thus, there is no equilibrium with a political regimeswitching because one cannot �nd a pair (rE , uE) that maximizes the elite value.For instance, the elite would like to choose the highest concession level uE butthen it doesn't exist a repression level such that the pair of instruments belongsto ζ. In sum, when the ordering is ψ < ψu < ψ < ψr, there exists a uniqueequilibrium with permanent dictatorship, which produces a payo� equal to (25).

In concluding this discussion, we want to emphasize a crucial point. Atthe beginning of this section we argued that a success in keeping the politicalcontrol is a good criterion to evaluate the best strategy. However, this criterionmay not be so clear and forceful. Indeed, it may well be that the elite, byusing an instrument only and accepting the regime change, is better o� thanby implementing a policy mix of the type discussed above. Put di�erently, thepresent value associated with the former strategy may likely be higher than theone yielded by the latter. Therefore, according to the criterion of (maximizing)the present value, accepting the revolution and adapting to this event mayconstitute the optimal strategy of the elite. Unfortunately, any attempt to godeeper into this discussion is vain. The comparison between the values providedby the di�erent strategies appears to be a di�cult exercise and doesn't allow usto identify the set of conditions under which this provocative result holds.21 Asan illustration of the various con�icting (growth and scale) e�ects at stake, let'shave a quick look at the comparison between the concession strategy and thepolicy mix. In this particular case, it can easily be shown that growth prospectsare higher at the policy mix solution. But, how the scale e�ect exactly playis unclear. If the elite redistributes much more resources at the solution withconcession, she doesn't spend a penny to repress the citizens whereas, with thepolicy mix, she devotes a lot of resources to her military budget. In general,it is possible neither to know which solution the scale e�ect bene�t to, norto conclude which e�ect prevails (when the scale e�ect pushes in the oppositedirection as the growth e�ect).

21From the results of sections 4 and 5, we know that the degenerated combination of instru-ments corresponding to these two cases are respectively given by (0, uE) (concessions) and(rE , uE) (repression). The values produced by these extreme strategies are:

V CE (0, uE) = 1δ

(ln(δ) + ln(K(0, uE)− χ)− (uEA− δ)T (0, uE) +

uEA−δδ

),

V RE (rE , uE) = 1δ

[e−δT (rE ,uE)

(A(1−uE)−δ

δ

)+ ln(δ) + ln(K(rE , uE)− χ)− (uEA− δ)T (rE , uE) + uEA−δ

δ

],

and one cannot �nd simple conditions to �nd the ordering between these values and the oneproduced by the policy mix (25).

26

7 Implications for the Arab spring

As mentioned in the introduction, the DSC are neither fully controllable as inSections 5 and 6 nor totally exogenous as in Section 4. Recall that the DSCre�ect either the cost of a collective action (or the cost of coordination) or theextent to which the regime set by the elites is contestable, that's in particularthe extent of their repressive forces. Because the DSC are a�ected in real lifeby such diverse factors as national demography, international geopolitics andconnectability of the countries to global markets and media, our theoreticalanalysis (of the polar cases) need to be quali�ed. A few points are worth tomake in the Arab spring context.

a) A major �nding of the theoretical analysis is that the equilibrium essen-tially depends on the level of the DSC. An interesting example is Bahrein.Since his arrival in o�ce in 1999, Sheikh Hamad has started a sequenceof democratization and political liberalization steps leading many inter-national organizations to believe that the country has de�nitely improvedregarding protection of human rights. This low level of repression washowever associated with what the religious majority of the country (theShias) felt as an unfair redistribution of resources. In this context, our the-ory would predict a revolution and a regime change at �nite time (Section4). Indeed, the Shias revolt took place in 2011 but the massive externalintervention of Saudi Arabia to back the Khalifa dynasty changed the out-come. A large enough exogenous increase in the DSC through externalintervention as in Bahrein would change the equilibrium to permanentdictatorship.22 We might also interpret the Libyan case symmetrically:Initially repression was strong in this country (and so was the militarythreat of the former dictator) and redistribution quite sizeable, leading topermanent dictatorship according to our theory if no external interventionoccurs. The NATO bombings implied a rapid drop in the DSC faced bythe opposition, opening the doors wide open for an alternative equilibriumwith a revolution and regime change at �nite time.

b) As referred to in the introduction, as the DSC can also be interpretedas coordinations costs, it might help explaining the role of human capitallevel in the genesis of the Arab spring. In particular, if one admits that thelarger the education level, the lower the coordination costs in collectiveactions (so the lower the DSC), the Tunisian lead sounds more naturalgiven the weight of public education in the budget of this country andthe impressive literacy performances registered in the relevant populationage classes.23 While this single criterion cannot explain the whole picture,it is a relevant part of the story. The institutional stability of the Gulf

22In a straightforward extension of our model with random revolution success such as theprobability of success is a decreasing function of total repression forces, we would get that theShias would have not revolted if they would have anticipated the Saudi intervention.

23According to the World Bank Edstats, the literacy rate in the 15-24 years old populationis Tunisia was about 97% in 2008, among the very best in the MENA region.

27

monarchies, which are also known to have high education performances,does not go at odds with this view: Just like Algeria, these countries havespent a substantial part of their petrodollars to buy time. In none of thelatter cases, the status-quo seems a stable economico-political equilibria.In particular, the Gulf monarchies' stability is undermined by deep contra-dictions, the discord between religious and secular schooling, and betweenwomen's education and their actual role in public life being only two ex-amples of these contradictions among others (see the illuminating analysisby Bahgat, 1998, many years before the Arab spring).24 The repressionforce of the Big Brother of this region, Saudi Arabia, has been so far,if not persuasive, certainly implacable (as it transpires from the Bahreincase). Much of the future of this region depends on how this monarchywill tackle its contradictions. It is far unclear whether the recent stepstaken (for example in favor of women's rights) by the King are necessaryand su�cient for that.

c) The interpretation of the DSC as a coordination cost is also interesting tounderstand the cascade of events following the Jasmine revolution. Onecan perfectly attribute to the striking success of this early revolution a sig-ni�cant psychological e�ect on the Arab populations, a kind of awarenessshock easing their rallying towards a common objective. In this sense,the Jasmine revolution has supposed a signi�cant drop in the DSC facedby these populations, and therefore a larger propensity to revolt. It isbecause the rulers of the Arab world are precisely aware of this drivingforce that they have launched so many initiatives after the Tunisian revo-lution towards more concessionary policies both with respect to economicand political rights. This concerns countries of the Maghreb (Algeria andMarocco as mentioned in the introduction) but also Gulf monarchies start-ing with Saudi Arabia.

d) Last but not least, the demographic aspects, and in particular the relativedemographic size of the elites, are worth accounting for (see the modellingin Acemoglu and Robinson, 2008). Just like human capital accumula-tion, demographic change involves slow dynamics and might be a goodcandidate for explaining the timing of the revolts in the Arab countries.But the Arab world demographics are tricky to model for several rea-sons (see for example Rashad, 2000, about the demographic transition inArab countries), and a minimally realistic (endogenous) speci�cation ofthese demographics would have made our analytical work quite uncom-fortable. In our theoretical setting, population growth has a priori anambiguous e�ect on the DSC: on one hand, it lowers them because repres-sion is potentially less e�cient when the size of the opposition increases;but on ther other hand, for a given �xed education level, coordinationcosts across a bigger population are larger leading to a DSC increment.

24See also the excellent recent book by Christopher Davidson (2013) on the sustainabilityof Gulf monarchies.

28

Moreover, demographic growth has also an impact on the redistributionas it leads to a mechanical increase of the redistribution bill. Accordingly,a straightforward extension of our model including exogenous growth ofthe opposition size would deliver a threshold value for this growth rateabove which a regime change occurs.

8 Conclusion

In this paper, we develop a dynamic game to provide with a comprehensivetheory of Arab spring-type events with the following main ingredients. We havetwo interacting groups, the elite vs. the citizens, two political regimes, dicta-torship vs. a freer regime, the possibility to switch from the �rst to the secondregime as a consequence of a revolution by the citizens and �nally the oppor-tunity, for the elite, to a�ect the citizens' decision through concession and/orrepression strategies. In this framework, we provide a full characterization ofthe equilibrium of our political regime switching game. First, we emphasize therole of the direct switching cost of the citizens and of the elite's self-preservationoptions. Under the concession strategy, when the switching cost is low, the elitecan't avoid the political regime change. She optimally adapts to the overthrowof their political power by setting the rate of redistribution to the highest pos-sible level, thereby extending the period during which she has full control onresources. This surprising result actually illustrates the role of the timing ofevents in these situations of interaction between the ruling elite and the people.When the direct switching is high, the elite can ultimately select the equilibriumoutcome and adopt the opposite strategy, that consists in choosing the lowestlevel of redistribution that allows her to stay in power forever. The same kind ofresults are obtained when the elites rely on repression to keep the citizens undercontrol. Next, the equilibrium properties under a mix of repression and redis-tribution are analyzed. It is shown that in situations where neither repression(only) nor redistribution (only) protect the elite against the uprising of citizens,a subtle mixture of the two instruments is su�cient to make the dictatorshippermanent. Based on our theoretical results, we �nally examine the reason forsuch a large variety of decisions (taken by the elites) and outcomes during theArab Spring events.

Our analysis can be extended in several ways. First, we may allow the elite torevise her repression and/or redistribution strategy in face of the threat of rev-olution. This can be done by formulating the choice of uE and rE as a regimeswitching problem (and applying the methodology developed by Long et al.,2013). Second, we should account for additional aspects of the mechanics of theArab Spring. In particular the role of human capital and education (throughtheir impact on citizens' claims for freedom) or the importance of the demo-graphic structure seem to be crucial for understanding the timing and successof the revolts in Arab countries. Among the other promising developments ofour work are the introduction of (endogenous) probabilities of (successful) rev-olution, capital evasion (when the elite can divert resources to its own bene�t)

29

and secure property rights in the second regime.

A Appendix

A.1 Assumption 2 & 3

The elite in the �rst case (resp. the second case) is obtained by replacing rE = 0(resp. uE = uE) in (5). The value of the citizens in the �rst case is derivedfrom (6) by substituting uE with any uE .

First case: One can directly check that VE(uE) ≥ VP (uE)⇔ B ≥ ln[ (1−uE)Aδ ]

or uE ≥ 1− δA . This leads to the Assumption 2.

Second case: The citizens' value cannot be computed so easily. So we resortto the condition that the elite's consumption is always higher than the citizens'consumption under permanent dictatorship. A su�cient condition for this to

hold is CE(0) > CP (0) ⇔ rE ≤ uEAK0(δ−(1−uE)A)δ . This in turn ensures that

VE(rE) > VP (rE). All this information is summarized in Assumption 3.

A.2 Proof of proposition 1

First, K is continuous everywhere except at T , where it's only left-continuous.The left-continuity of the capital stock at the switching date T , the generalexpression of K1(t) for all t ≤ T being given in (8), implies

K1(T ) = K(uE)⇔ φeuEAT + (K0 − φ)e(uEA−δ)T = K(uE). (26)

Second, we use the fact that the switching decision is taken by citizens butcitizens don't have any in�uence on the dynamics of the �rst regime i.e. thevalue of the state at which the switch occurs is controlled by the elite. So,condition (10) must hold, which together with (8) yields:

e−δTλ1E(T ) = e−δT∂V 2

E(.)

∂K⇔ (K0 − φ)e(uEA−δ)T = K(uE)− χ. (27)

In sum, we are left with a system of two conditions (26)-(27) in the tworemaining unknowns φ and T . Substituting the value of φ given by (26) in (27),one obtains

K0e(uEA−δ)T = χ

(1

eω(uE) − 1+ e−δT

),

which must be studied to show the existence of a strictly positive and �-nite switching date. Noticing that under Assumption 1., necessarily we haveuEA − δ > 0 for all uE , then we obtain that there exists a unique 0 <T (uE) <∞ i� K(uE) > K0. Finally, given that ω′(uE) = 1

1−uE and K ′(uE) =

30

−−χω′(uE)eω(uE)

(eω(uE)−1)2 < 0, K(uE) > K0 is su�cient to ensure existence for all

uE ∈ [uE , uE ]. 25

A.3 Proof of proposition 2

After direct manipulations, the derivative in (22) can be rewritten as

∂VE(uE)

∂uE=

1

δ

{1

1−uE

((1−uE)A

δ − eω(uE)

eω(uE)−1

)−AT (uE)[1 + σ(uE)] + δT ′(uE)

−e−δT (uE){T ′(uE) [(1− uE)A− δ] + A

δ

}.

(28)Under Assumption 2, it's easy to check that if uE ≥ uE = 1− δ

A and |σ(uE)| ≤ 1

then ∂VE(uE)∂uE

< 0 for all uE ∈ [uE , uE ]. Hence, the optimal choice is uE = uE(if the second order optimality condition is satis�ed).

A.4 Proof of proposition 3

Under Assumptions 2 and 4, we have uE > δA for all admissible uE . This

implies that limt→∞K(t) =∞ under a permanent regime 1 and we know that

limK→∞ ln[

KK−χ

]= 0. Then, (23) is equivalent to ln

[(1−uE)A

δ

]− A−2δ

δ ≥ 0⇔ω(uE) ≤ 0.

A.5 Proof of lemma 2 and Propositions 4 & 5

• Citizens' switching problem (Lemma 2): The opposition solves

maxT

VP (K0, T ) =

∫ T

0

ln(C1P )e−δtdt+ e−δT [V 2

P (K − χ)− ψ(rE)],

The necessary optimal condition for switching is (the su�cient optimality con-dition is satis�ed):

ln

(K

K − χ

)=A

δ− 2 + ln

(δ

[1− uE ]A

)− δψ(rE). (29)

Denote the RHS of (29) as ω(rE). As before, there exists a unique K(rE) =χeω(rE)

eω(rE)−1 that solves (29) i� ω(rE) > 0. In addition, we can de�ne a second

critical threshold for the DSC:

ψr =1

δ

(ln(δ)− ln[(1− uE)A] +

A

δ− 2

).

This is su�cient to state Lemma 2.

25One may be concerned with the meaning of this su�cient condition because till now wehaven't discussed the level of the upper bound uE . For the sake of simplicity, given that theshare uE is bounded from above by 1, we may simply replace K(uE) > K0 with the strongerrequirement: K(1) > K0.

31

• Switching date (Proposition 4): From the transversality condition of regime1 and the left continuity of the resource stock we obtain

φ =

(K(rE)− rE

uEA−(K0 −

rEuEA

))e−uEAT

(1− e−δT

)−1And T is implicitly given by:(

K0 −rEuEA

)e(uEA−δ)T = K(rE)− χ+

(χ− rE

uEA

)e−δT (30)

The condition χ > K0 is su�cient for the existence of a unique solutionT (rE) to this equation (for a given rE). This is the same condition as the onederived in the draft for the concession case.

Suppose that T (rE) exists. Then, the solution corresponding to regime 1reads as follows:{

K1(t; rE) =(χ− rE

uEA

)euEA(t−T (rE)) + (K(rE)− χ)e(uEA−δ)(t−T (rE)) + rE

uEA

C1E(t; rE) = δ(K(rE)− χ)e(uEA−δ)(t−T (rE))

• Proof of Proposition 5:Impact of a change in rE on consumption (other comparative statics can

easily be derived from the de�nition of K(rE) and (??)):

∂C1E(t; rE)

∂rE= C1

E(t; rE)

(K ′(rE)

K(rE)− χ− (uEA− δ)T ′(rE)

),

From (30),

T ′(rE) =K ′(rE) + 1

uEA

(e(uEA−δ)T (rE) − e−δT (rE)

)K(rE)− χ+ uEA

uEA−δ

(χ− rE

uEA

)e−δT (rE)

Then,∂C1

E(t;rE)∂rE

> 0 ⇔ K′(rE)

K(rE)−χ > (uEA − δ)T ′(rE), which after straight-

forward manipulations, is equivalent to:

uEA

uEA− δK ′(rE)

K(rE)− χ

(χ− rE

uEA

)>

1

uEA

(euEAT (rE) − 1

). (31)

Denote the RHS of the inequality by G(rE). This function is increasing inrE . Let's de�ne

σ(K) =rEK

′(rE)

K(rE)> 0 and σ(ψ′) = −rEψ

′′(rE)

ψ′(rE)> 0

If σ(ψ′) > σ(K) for all rE ∈ [0, rE ] then, the LHS of the inequality aboveis decreasing in rE . Let F (rE) be this LHS. Now, imposing F (rE) > G(rE) issu�cient to conclude that C1

E(t; rE) is increasing in rE for all t ≥ 0.

32

Optimal choice of the repression level rE ∈ [0, rE ]: The present value of theelite when she uses repression but the revolution occurs in �nite time is:

VE(rE) =

∫ T (rE)

0

ln(C1E(t; rE))e−δtdt+ e−δT (rE)V 2

E(K(rE)− χ),

Taking the derivative w.r.t rE , one obtains:

∂VE(rE)

∂rE=

∫ T (rE)

0

∂C1E(t;rE)

∂rE

C1E(t;rE)

e−δtdt

+(

ln(C1E(T (rE); rE)− δV 2

E(K(rE)− χ))T ′(rE)e−δT (rE)

+∂V 2

E

∂K K ′(rE)e−δT (rE).

This can be decomposed into 3 di�erent e�ects: the marginal impact ofchoosing rE before the regime change (�rst term); the marginal impact at thisinstant of the regime change (second term) and the marginal impact of varyingrE after the switch (last term). Using the expression of C1

E , this derivative canbe rewritten as:

∂VE(rE)

∂rE=

1

δ

(K ′(rE)

K(rE)− χ−(uEA− δ + (A(1− uE)− δ)e−δT (rE)

)T ′(rE)

),

which is positive under the same two conditions that yield a consumption in-creasing in rE .

33

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