Economic Origins of Dictatorship and Democracy
Page 25
This book does not enter into a lengthy discussion of the theory of repeated games, so the analysis is brief (see Fudenberg and Tirole 1991, Chapter 5, for more on repeated games; and Powell 2004 for an analysis of the circumstances under which punishment strategies can solve problems of commitment in a class of games close to those we study herein). What we want to show here is that this type of promise can go some way towards resolving commitment problems, but the underlying commitment problem will remain. It will still be the case that the elites cannot credibly promise arbitrarily large amounts of redistribution in the state where the revolution threat is not present and, as a result, the spirit of Proposition 5.4 applies even with non-Markovian strategies.
We now take a situation in which, in terms of Proposition 5.4, θ > µ and µ < µ*, so with the restriction to Markov perfect equilibria, the unique equilibrium involves a revolution. Let us see whether the elites can avert a revolution by using incentive-compatible promises supported by future punishments. To do this, we first find the maximum value that the elites can give to the citizens, once we consider potential punishment strategies. Because in general, repeated games have many subgame perfect equilibria, we focus on the subgame perfect equilibrium that is best for the elites. This subgame perfect equilibrium will prevent a revolution for the largest possible set of parameter values; however, there are other subgame perfect equilibria that also prevent a revolution for the same set of parameter values but give the citizens more. Nevertheless, this analysis of the specific equilibrium gives the flavor of what types of outcomes can be supported in non-Markovian equilibria.
Suppose also that we start when the state is µL. We first calculate the value to the elites if they redistribute at the rate τN = τH ≤ τp in the state µt = µH and at the rate τN = τL≤ τp in the state µt = µL (because we are no longer looking at Markovian strategies, τL > 0 is now possible). We also suppose for now that the citizens will not undertake a revolution (later, we impose this as a constraint on the tax vector). By the same arguments as in the previous section, this value is given by
(5.32)
We are now using a different notation, Vr (N, µL, [τL, τH]), rather than Vr (N, µL) as we did in the previous section. This is because, while in the Markov Perfect Equilibrium, the elites always set τN = 0 when τt= µL; this is no longer true. In particular, we are looking at situations in which the elites make credible promises of a tax rate of τ L when µt = µL and set a tax rate of τ H when µt, = µH. The new notation captures this. The term µL refers to the fact that we are in state µt = µL, and [τL, τH] is the vector of promised taxes starting with the tax rate in the state µt= µL.
The intuition for (5.32) is straightforward: the first term, yr + (τL(- yr) - C (τL) y), is again the current return to the elites, given that there is taxation at the rate τL. The second term is the continuation value, considering the fact that taxation changes to τH if the state switches to µH. By the same token, we also have:
as the value starting in the state µH. Combining these two expressions, we obtain:
(5.33)
as the value that the elites will receive if they adhere to their “promised” behavior summarized by the tax vector [τL, τH]. The key is whether this behavior is “incentive-compatible” for them - that is, whether they wish to deviate from it now or in the future.
What happens if they deviate? Clearly, the answer depends on how the citizens react. We want to see whether we can make the promise by the elites to redistribute at the tax rate τL > 0 in state µL credible. It is more likely to be credible when deviation from it is less profitable or when deviation from this prescribed behavior is met by a severe punishment. The most severe punishment is that of revolution by the citizens when the opportunity occurs again (it is never profitable for the citizens to undertake revolution in the state µt= µL, because µL = 1, so the threat to undertake such revolution in the state µt, = µL is not credible and, therefore, never part of a subgame perfect equilibrium). Consequently, the best way to ensure that the elites do not deviate from their promises is to threaten them (credibly) with as severe a punishment as possible - that is, revolution as soon as the state switches to µt= µH. So, there will be revolution the first time the state is µt = µH. What will happen until then? The elites are now deviating from their promised behavior so, in the meantime, they adopt the best policy for themselves, so τN = τr = 0. Thus, what we have is a value(N, µL) µL) for the elites, in which the subscript d denotes that they have deviated from their prescribed behavior. This value is given by the following recursion:
where we know that Vr (R, µH) = 0. Using this fact, we have that:
(5.34)
This analysis immediately establishes that only redistribution at the rate τL in the state µt = µL, such that:
(5.35)
is credible. If the inequality were reversed, the elites would prefer to deviate and give the citizens no redistribution in the state µL, suffering the consequences rather than tax themselves at the rate τL now (and at the rate τH when the state becomes high). Therefore, (5.35) is necessary for redistribution at the tax rate τL to be “incentive-compatible” for the elites and thus a credible promise to the citizens. The tax rate τH ≤ τp in the state µt = µH is automatically credible because we are looking at the part of the parameter space where µ < µ *; therefore, any deviation by the elites from their promised actions in the high state can be immediately punished.
The subgame perfect equilibrium that is best for the elites, starting in the state µL, can be characterized as the solution to the following maximization problem:
(5.36)
subject to (5.35) and
(5.37)
where Vp (N, µH, [τL,τH]) is the value to the citizens starting in the state µH from the tax vector [τL, τH], and Vp (R, µH) , as usual, is the value to the citizens from revolution in the state µH given by (5.21) in the previous section.
Whereas the first constraint ensures that the elites do not wish to renege on their promises, the second constraint requires that the citizens do not wish to undertake revolution in the high state.
The value Vp(N, µH, [τL, τH]) is obtained analogously to the values for the elites. In particular, we have the following value functions for the citizens. In the low state:
and in the high state:
Combining the two expressions, we obtain:
(5.38)
Before providing a full solution to this maximization problem, it is straightforward to characterize the minimum value of µH such that a revolution can be averted. We denote this threshold by µ** using an analogy with the threshold µ* in the previous section. Formally, this threshold corresponds to the minimum value of µH such that the constraint set of the optimization problem is nonempty. When the constraint set is empty, this implies that there is no tax vector [τL, τH ] that is simultaneously credible and can convince the citizens not to undertake revolution, so there has to be an equilibrium revolution in the state µH.
To calculate this threshold, note that the largest value that τH can take is τp. Intuitively, in the high state, the elites are willing to give the maximum redistribution to avoid revolution. What about τL? Once τH= τp, τL is then given by treating the incentive-compatibility constraint of the elites, (5.35), as an equality. Therefore, the largest amount of redistribution that can credibly be promised is that which stems from levying the tax ratein the state µt= µL such that eitherVr(N, µL, [’,τp]) and‘ ≤ τp, or’ = τp. More specifically let” be such that
Substituting for the definition of yr and simplifying terms, we obtain:
(5.39)
Then the maximum credible tax rate is’ = min{“, τp}.
This tax rate,’, can be shown to be an increasing function of β; the more valuable the future, the less attractive it is for the elites to deviate from the promised behavior, so the higher is the maximum tax rate they can promise. This is intuitive and, in fact, a fundamental principle of analyses of r
epeated games; for players not to take the action in their immediate interest, the benefits from this action need to be counterbalanced by some other future considerations. Here, if they take these actions, they will be punished in the future. The more players discount the future or the less severe is the expected punishment, the more difficult it will be to convince them to adhere to these promises.
The important point highlighted by (5.39) is that the elites do not have unrestricted powers to make promises: they have a limited capability, supported by the threat of future punishments. Any promises they make will be credible only if it is in their interests to carry out this promise at the time. Here, some positive redistribution even without the threat of revolution might be in their interests because otherwise they know they will have to tolerate revolution later. Nevertheless, this threat of future punishments can support only a limited amount of redistribution (i.e., the elites cannot credibly promise a tax rate greater than’ in the low state).
This analysis then implies that the question of whether revolution can be averted boils down to whether the value to the citizens from redistribution at the tax rate’ in the state µt = µL and at the tax rate τp in the state µt = µH, starting in the state µt = µH, is better than revolution for the citizens. Or, stated differently, this is equivalent to whether the tax vector [’, τp] is in the constraint set of the maximization problem given by inequalities (5.35) and (5.37).
By analogy to the analysis in the previous section, we can see that the tax vector [‘, τp] is in the constraint set for all µ ≥ µ**, where µ** is such that Vp(N, µH, [’, τp]) = Vp(R, µH) when µH = µ**. More explicitly, the threshold µ** is the solution to:
(5.40)
where’ is given by (5.39). It can be verified that µ** > 0.
Recall that, using the notation in this section, µ* is defined by Vp (N, µH, [0,τp])=Vp(R,µH), so for all’ > 0, we have:
which is clear from formulas (5.30) and (5.40).
This implies that once we allow for the use of punishment strategies, there will be situations in which a revolution can be averted by incentive compatible promises but could not have been otherwise. This is true when µ ∈ [µ**, µ*). Nevertheless, since µ** > 0, there will still be situations (i.e., when µ < µ**) in which the best that the elites can promise is not enough to avert a revolution.
This discussion leads to the main result of this section, which we informally state as the following:
Result: When we allow non-Markovian strategies, a revolution can be averted for all µ ≥ µ**. Here, µ** < µ*, which means that greater redistribution is now possible, but µ** > 0, which means that a revolution can happen if µ is sufficiently small.
To state the results of this section more carefully and to complete the characterization of the equilibrium, we must define what a strategy is in this game. The main difference with the previous section is that we have dropped the restriction to Markov strategies and now a strategy can depend not just on the state at any date t but also on the history of play up to that date. Let Ht-1 denote the set of all possible histories of play up to t - 1 with a particular history being denoted ht-1 ∈ Ht-1. The actions of the elites and the citizens are now denoted by σr = {τN(·, ·)} and σp = {p (·, ·,·)}, where τN(µt, ht-1) is the tax rate set by the elites at date t when the current state is µt = µH or µL and the observed history is ht-1. Hence, τN : {µL,µH} × Ht-1→ [0, 1]. Similarly, ρ(µt, τN, ht-1) is the decision by the citizens to initiate a revolution conditional on the current state, the current actions of the elites, and the history. We have that ρ : {µL, µH} × [0, 1] x Ht-1 → {0, 1}. Then, a subgame perfect equilibrium is a strategy combination, {r,p}, such thatr andp are best responses to each other for all possible histories ht-1 ∈ Ht-1 and prior actions taken within the same stage game.
When µ < µ**, the following strategy profile is the unique subgame perfect equilibrium: τN(µt, ht-1) = 0 for µt ∈ {µL, µH} and any ht-1, ρ(µL, ·, ht-1) = 0 and ρ(µH, ·, ht-1) = 1 for any ht-1. For this set of parameter values, a revolution is sufficiently attractive that concessions will not work; the first time µH arises there will be a revolution whatever the previous history of play or the current tax rate. Because the elites know this, they simply set zero taxes when µL occurs.
To understand the nature of the subgame perfect equilibrium when µ ≥ µ**, it is also useful to note that in this case there is an additional motive for the elites: “tax-smoothing.” Intuitively, the elites want to deliver a given amount of redistribution to the citizens at the minimum cost to themselves. Because the cost of taxation given by the function C(·) is convex, this implies that taxes should exhibit as little variability as possible - in other words, they should be smooth.16 This idea was first suggested by Barro (1979) in the context of optimal fiscal policy, but it applies equally here. Such tax-smoothing was not possible before because the elites could never promise to redistribute in the state µL. Now that this type of redistribution is possible, tax-smoothing also emerges as a possibility.
The tax-smoothing argument makes it clear that the cheapest way to the elite of providing utility of Vp(R, µH) is to set a constant tax rate, τs, such that:
(5.41)
or, more explicitly, τs is given by:
(5.42)
Therefore, redistributing at this rate is the best possible strategy for the elites. The question is whether this tax vector is incentive-compatible - that is, whether it satisfies (5.35). The same arguments immediately imply that the vector [τs, τs] will be incentive-compatible as long as τs ≤s wheres is given by:
(5.43)
which is similar to (5.39) with the vector [s,]s] replacing [’, τp].
Then the question of whether perfect tax-smoothing can be achieved simply boils down to whether any tax rate τs ≤s satisfies (5.41). Again, similar arguments immediately establish that there exists a level of µH, here denotedS and given by:
(5.44)
such that when µ≥S, a perfectly smooth credible tax policy will prevent revolution.
Clearly,s > µ**; on the other hand,S can be greater than or less than µ* . When µ >S, the best possible subgame perfect equilibrium for the elites is a strategy combination that corresponds to the tax vector [τs, τs] (which, by construction, prevents revolution at the lowest possible cost). More explicitly, let us define the historyt such that ht =t if for all s ≤ t, τN(µL, hs) = τs, where τs is given by (5.41). Then, the subgame perfect equilibrium is given by the following strategy combination. For the elites:
(5.45)
for µt, ∈ {µL, µH}, and for the citizens: ρ (µL, ·, ht-1) = 0, and:
In this case, as before, strategies specify how a player will play even off the equilibrium path, which now includes all possible histories up to that point. In particular, here ht-1 denotes the equilibrium path. Then, as long as play is on this path, the elites set τs in both states and the citizens never revolt. However, if the elites ever set a tax rate less than τ s, we will move along some history ht-1≠t-1 and the strategies say that the first time the state is µt = µH, the citizens undertake revolution. How do we know that in such a situation it will actually be credible for the citizens to undertake revolution? This comes from (5.45), which states that if the elites find themselves setting the tax rate after some history different fromt-1, they set the tax rate to zero. Thus, the poor understand that if they do not undertake revolution following a deviation from the prescribed behavior, they will never get any redistribution from that point on in the game. Therefore, as long as the revolution constraint θ > µ holds, it is optimal to undertake revolution following a deviation by the elites.
Finally, when µ ∈ [µ **,S), revolution can be averted, but perfect-tax smoothing is no longer possible. In this case, it can be seen that the best subgame perfect equilibrium for the elites is a tax vector [L, H], which is the solution to (5.36) and satisfies:
(5.46)
and:
(5.47)