Finally, it is interesting to reflect on the role that (6.7) plays in Proposition 6.2. Repression is attractive to the elites when democracy threatens to enact policies that are very pro-citizen. However, if policies are insufficiently majoritarian, it is unlikely that (6.7) will hold; thus, the elites will be forced to repress when µ < µ* because democracy will not avoid revolution.
7. A Dynamic Model of Democratization
We now develop an infinite horizon model of democratization, the main motivation of which is that it allows us to model the issue of commitment to future policy in a more satisfactory way. The citizens demand democracy and changes in the structure of political institutions precisely because of the fact that such changes influence the allocation of political power in the future. Thus, the problems we are considering are inherently dynamic and intertemporal. In the static model, we had to model this by introducing a rather arbitrary assumption that the elites might be able to reoptimize after they had initially chosen their policy. We now show that results similar to those derived with this crude assumption flow naturally from the time structure of a repeated game.
The model is a direct extension of the one developed in Chapter 5, Section 6, and the one in the previous section. We adopt the same notation and again refer to the infinite horizon discounted repeated game as G∞(β). There is again a total population of 1 with rich elites and poor citizens as before, with fractions δ and 1 - δ. Initially, there is a nondemocracy, but the citizens can contest power through collective action, and in a democracy, the median voter will be a poor citizen. The structure of de facto power is exactly as in Chapter 5, Section 6, so that the cost of revolution is µt, where µt ∈ {µL, µH} and Pr(µt= µH) = q irrespective of whether µt-1 = µH or µL. We again normalize so that µL = 1 and use the notation µH = µ.
The timing of the stage game is similar. In each period, the elites can decide whether to create democracy and whether to repress. If democracy is created, the median voter - a poor citizen - sets the tax rate. We assume that if democracy is created, it cannot be rescinded, so the society always remains a democracy. As before, we assume that if repression is chosen, revolution cannot be undertaken and the stage game is over for that period, with agents getting the repression payoffs.
As a result, utilities are now given by Ui =where, as in the previous section, incomes are given by (6.8) and as in Chapter 5, Ui applies only when there is no revolution in equilibrium.
The timing of moves in the stage game is now as follows:
(1) The state µt ∈ {µL, µH} is revealed.
(2) The elites decide whether to use repression, ω ∈ {0, 1}. If ω = 1, the poor cannot undertake revolution and the stage game ends.
(3) If ω = 0, the elites decide whether to democratize, φ ∈ {0, 1}. If they decide not to democratize, they set the tax rate τN.
(4) The citizens decide whether to initiate revolution, ρ ∈ {0, 1}. If ρ = 1, they share the remaining income forever. If p = 0 and φ = 1, the tax rate τD is set by the median voter (a poor citizen). If p = 0 and φ = 0, the tax rate is τN.
We initially characterize Markov perfect equilibria of this game in which players are restricted to playing Markov strategies that are functions only of the current state of the game. Although the focus on Markovian equilibria is natural in this setting, for completeness in the next section, we drop the restriction to Markov strategies and discuss non-Markovian subgame perfect equilibria. As in Chapter 5, we show that this does not change the qualitative nature of our general results.
The state of the game consists of the current opportunity for revolution, represented by either µL or µH, and the political state P, which is either N (nondemocracy) or D (democracy). More formally, let σr= {ω(·), φ(·), τN(·)} be the notation for the actions taken by the elites, and σp = {ρ(·), τD} be the actions of the poor. The notation σr consists of a decision to repress ω : {µL, µH} → {0, 1}, or to create democracy φ : {µL, µH} → {0, 1} when P = N, and a tax rate τN : {µL, µH} → [0, 1] when φ = 0 (i.e., when democracy is not extended). Clearly, if φ = 0, P remains at N, and if φ = 1, P switches to D forever; thus, we do not make these strategies explicit functions of the political state. The actions of the citizens consist of a decision to initiate revolution, p : {µL, µH} × {0, 1}2 × [0, 1] → {0, 1} and possibly a tax rate τD ∈ [0, 1] when the political state is P = D. Here, ρ(µ, ω, φ, τN) is the revolution decision of the citizens that is conditioned on the current actions of the elites, as well as on the state, because the elites move before the citizens in the stage game according to the timing of the previous events. Then, a Markov perfect equilibrium is a strategy combination {r,p}, such thatp andr are best responses to each other for all µt and P.
We can characterize the equilibria of this game by writing the appropriate Bellman equations. Define Vp(R, µs) as the return to the citizens if there is revolution starting in state µS ∈ {µL, µH}. This value is naturally given by:
(6.11)
which is the per-period return from revolution for the infinite future discounted to the present. Also, because the elites lose everything, Vr(R, µS) = 0 whatever is the value of µS. Moreover, recall that we have assumed µL = 1, so Vp(R, µL) = 0, and the citizens would never attempt revolution when µt = µL.
In the state (N, µL), the elites are in power and there is no threat of revolution; therefore, in any Markov perfect equilibrium, φ = ω = 0 and τN = τr = 0. This says simply that when the elites are in power and the citizens cannot threaten them, the elites do not repress and set their preferred tax rate, which is zero. Therefore, the values of the citizens and the elites, i = p or r, are given by:
(6.12)
Now, (6.12) says that the value to an agent of type i in a nondemocracy when there is no threat of revolution is equal to a payoff of yi today, plus the expected continuation value discounted back to today (which is why it is multiplied by β). The payoff today is yi because taxes are set at zero and everyone simply consumes their income. The continuation value consists of two terms; the second, (1 — q) Vi (N, µL), is the probability that µL arises tomorrow times the value of being in that state V’(N, µL). In this case, tomorrow is the same as today, which is why the same value “recurs.” The first term, q Vi(N, µH), is the probability that µH arises tomorrow multiplied by the value of that state Vi(N, µH). This value is different because now there is a potential threat to the regime. To see how this plays out, we need to understand what the value Vi(N, µH) looks like.
Consider the state (N, µH), where there is a nondemocracy but it is relatively attractive to mount a revolution. Suppose that the elites play φ = ω = 0 and τN= τr; that is, they neither create democracy nor repress nor redistribute to the citizens. Then, we would have:
The revolution constraint is equivalent to Vp(R,µH) > Vp(N, µH), so that without any redistribution or democratization, the citizens prefer to initiate revolution when µt = µH. This is equivalent to θ > µ, which is identical to (6.3) in the previous section, and says that revolution becomes attractive when θ is sufficiently high (i.e., when inequality is sufficiently high).
Because revolution is the worst outcome for the elites, they will try to prevent it. They can do this in three different ways. First, the elites can choose to maintain political power, φ = 0, but redistribute through taxation. In this case, the poor obtain Vp(N, µH, τN) where τN is the specific value of the tax rate chosen by the elites. Second, the elites can create democracy. Third, the elites can use repression. Let Vi(O, µ |κ) be the value function of agent i = p, r in state µ when the elites pursue the strategy of repression and the cost of repression is K. We condition these values explicitly on K to emphasize the importance of the cost of repression and to simplify notation when we later define threshold values.
If the elites create democracy or attempt to stay in power by redistributing, the citizens may still prefer revolution; thus:
where Vp(D) is the return to the citizens in democracy. (Note here how the va
lue of the citizens depends on the decision variables ω and φ of the elites). If ω = 1, the elites choose to repress; the citizens cannot revolt and get the continuation value Vp(O, µH |κ). If ω = 0, then what the citizens compare Vp (R, µH) to depends on the decision by the elites about creating democracy. If φ = 1, then they choose between revolution and democracy. If φ = 0, they choose between revolution and accepting the promise of redistribution at the tax rate τN.
We first focus on the trade-off for the elites between redistribution and democratization and then integrate repression into the analysis. The return to the citizens when the elites choose the redistribution strategy is:
(6.13)
The elites redistribute to the citizens, taxing all income at the rate τN. The citizens, therefore, receive their income yp from their own earnings and a net transfer of τN(- yp) - C(τN). If in the next period we are still in state µt+1 = µH, redistribution continues. But, if the state switches to µt+1 = µL, redistribution stops and the citizens receive Vp(N, µL). This captures our intuitive ideas that the elites cannot commit to future redistribution unless the future also poses an effective revolution threat.
The second strategy to prevent revolution is to democratize, φ = 1. Because 1 - δ > 1/2, in a democracy the median voter is a citizen and the equilibrium tax rate is τp and T = (τp - C(τp)). The returns to the citizens and the elites in democracy are, therefore:
(6.14)
These expressions follow because in this chapter we are assuming that once created, democracy consolidates and there are never any coups.
Will democratization prevent revolution? The answer is not obvious. It might be that revolution in the state µt = µH is so attractive that even democratization is not sufficient to prevent it. It is obvious that the condition for democratization to prevent revolution is Vp(D) > Vp(R, µH), which is exactly the condition we derived in Section 5 (i.e., (6.7)).
To determine whether the elites can prevent revolution with the redistribution strategy, let Vp(N, µH, τN = τp) be the maximum utility that can be given to the citizens without democratizing. This maximum utility is achieved by setting τN = τp in (6.13). Therefore, combining (6.12) and (6.13), we obtain:
(6.15)
(6.15) has a nice interpretation. It says that Vp(N, µH, τN = τp) is equal to the present discounted value of yp, the pretax income of citizens, plus the expected present value of net redistribution from the elites to the citizens. Net redistribution is given by the expression (τp(- yp) - C(τp)) but this only occurs today, and a proportion q of the time in the future when the state is µH. (The reason this leads to the expression (1 - β(1 - q))/(1 - β) is exactly the same as the one discussed after (5.28) in Chapter 5.)
If Vp(N, µH,τN= τp) < Vp(R, µH), then the maximum transfer that can be made when µt = µH is not sufficient to prevent revolution. As long as (6.7) holds, Vp(D) ≥ VP(R, µH). It is clear that Vp(N, µH = 1, τN = τp) > VP(R, µH = 1) because revolution generates a zero payoff to the citizens forever. This implies that when µH = 1, it must be the case that the value to the citizens of accepting redistribution at the rate τp in state uH is greater than the value of revolution. Also note that:
(6.16)
so that the payoff from revolution must be greater when µH = 0. Because Vp(R, µH) is monotonically increasing and continuous in µ, by the intermediate value theorem there exists a unique µ* ∈(0, 1), such that when µH = µ*:
(6.17)
When µ < µ*, concessions do not work so the elites are forced to either democratize or repress. When µ ≥ µ*, they can prevent revolution by temporary redistribution, which is always preferable to them when the alternative is democratization (because with democratization, redistribution is not temporary but rather permanent). In this case, the tax that the elites set, which we denote by, will be set exactly to leave the citizens indifferent between revolution and accepting concessions under a nondemocratic regime - that is,satisfies the equation Vp(N, µH, τN= ) = Vp(R, µH).
To determine equilibrium actions, we need to compare the payoffs to the elites from staying in power using redistribution and from democracy to the cost of repression. Without loss of generality, we limit attention to situations in which the elites play a strategy of always repressing rather than more complicated strategies of repressing sometimes and using redistribution other times (this also is without generality because of the “one-shot deviation” principle, discussed in greater detail in the next chapter; see also Fudenberg and Tirole 1991, pp. 108-10). By standard arguments, these values satisfy the Bellman equations:
(6.18)
which takes into account that the cost of repression will only be incurred in the state where the revolution threat is active - that is, when µt = µH.
Together with the definition for Δ, the Bellman equations can be solved simultaneously to derive the values to the elites and the citizens from repression:
(6.19)
The value function Vr(O, µH | κ) has a clear interpretation: the payoff to the elites from a strategy of repression is the discounted sum of their income, yr /(1 - β) minus the expected cost of repressing. The net present value of the cost of repressing is (1 - β(1 - q))κ yr/(1 - β) for the elites because they pay this cost today and a fraction q of the time in the future.
To understand when repression occurs, we need to compare Vr (O, µHκ) to Vr(D) when µ < µ* and to Vr(N, µH, τN =) when µ ≥ µ*. As in the extensive-form game of the previous section, we now determine two threshold values for the cost of repression - this time called κ* and- such that the elites are indifferent between their various options at these threshold levels. More specifically, let K* be such that the elites are indifferent between promising redistribution at the tax rate τN =and repression Vr (O, µH|κ*) = Vr(N, µH, τN =). This equality implies that:
(6.20)
Similarly, letbe such that at this cost of repression, the elites are indifferent between democratization and repression - that is, Vr (O,µH|) = Vr(D), which implies that:
(6.21)
It is immediate that> κ*; that is, if the elites prefer repression to redistribution, then they also prefer repression to democratization. Therefore, the elites prefer repression when µ ≥ µ* and κ < κ* and also when µ < µ* and κ <.
Given our previous analysis, the strategies that constitute equilibria in different parts of the parameter space can easily be constructed. Therefore, we have (as in Propositions 6.1 and 6.2, stated without specifying the full set of strategies):
Proposition 6.3: There is a unique Markov perfect equilibrium {r, p} in the game G∞(β), and it is such that:
• If θ ≤ µ, then the revolution constraint does not bind and the elites can stay in power without repressing, redistributing, or democratizing.
• If θ > µ, then the revolution constraint binds. In addition, let µ* be defined by (6.17) and κ* and be defined by (6.20) and (6.21). Then:
(1) If µ ≥ µ* and κ ≥ κ*, repression is relatively costly and the elites redistribute income in state µHto avoid a revolution.
(2) If < µ* and κ < , or κ ≥ and (6.7) does not hold, or if µ ≥ µ* and K < κ*, the elites use repression in state µH.
(3) If µ < µ*, (6.7) holds, and κ ≥, concessions are insufficient to avoid a revolution and repression is relatively costly. In this case, in state µHthe elites democratize.
Democracy arises only when µ < µ*, repression is relatively costly (i.e., κ ≥), and (6.7) holds. This critical threshold for the cost of repression,, is increasing in inequality (increasing in θ); more specifically, we can again show by an argument identical to the one used in the last section that:
Intuitively, when inequality is higher, democracy is more redistributive (i.e., τp is higher) and hence more costly to the rich elites, who are therefore more willing to use repression.
As also shown by the static model in the previous section, democracy emerges as an equilibrium outcome only in societies with intermediate levels of inequality. In ver
y equal or very unequal societies, democracy does not arise as an equilibrium phenomenon. In very equal societies, there is little incentive for the disenfranchised to contest power and the elites do not have to make concessions, neither do they have to democratize. In very unequal societies, the elites cannot use redistribution to hang onto power; however, because in such a society democracy is very bad for the elites, they use repression rather than having to relinquish power. It therefore tends to be in societies with intermediate levels of inequality that democracy emerges. Here, inequality is sufficiently high for challenges to the political status quo to emerge, but not high enough that the elites find repression attractive. Thus, the intuition behind Corollary 6.2 applies in this model directly.
We show in the next section that even without the restriction to Markov perfect equilibria, similar results obtain: revolution can be stopped with temporary redistribution when µ ≥** where** < µ* - hence, for a larger range of parameters- but if µ <**, the elites cannot use concessions to avoid revolution.
Economic Origins of Dictatorship and Democracy Page 29
