(5.22)
These value functions have a form that recurs throughout the dynamic analysis in this book, so it is important to understand the reasoning behind them. We focus on the elites for concreteness.
The value functions in (5.22) say that the value to a member of the elite in a nondemocracy and in the state µt= µL consists of two terms: (1) what happens today, the first term yr; and (2) what is expected to happen tomorrow, or the continuation value, represented by the second term, β [q Vr (N, µH) + (1 - q) Vr (N, µL)]. Today, given the decision τN = τr, there is no redistribution, and a member of the elite obtains yr, which is the first term. The second term is multiplied by β because it starts tomorrow and, therefore, is discounted back to today by the discount factor β. Tomorrow, there is a new draw from the distribution of µ, and with probability 1 - q, the state µL recurs, so we have µt+1 = µL. In this case, exactly the same reasoning as today implies that the value to an elite agent from that point onward is Vr(N, µL); hence, this term is multiplied by 1 - q and included as part of the future value. The value Vr(N, µL) recurs because the world looking forward into the infinite future from state µt = µL looks identical to the world looking forward into the infinite future from state µt+1= µL (recall equation (5.20)). With the remaining probability, q, there is a change in the state, and we have µt+1 = µH; in this case, we have a different value for a member of the elite tomorrow, denoted by Vr(N, µH).
The same argument also applies for citizens and gives the corresponding expression for Vp(N, µL), again consisting of two terms: what they receive today, yP, and what they will receive tomorrow, β [q Vp(N, µH) + (1 - q) Vp (N, µL)].
The nice thing about the value functions in (5.22) is their “recursive” structure. Basically, the future is much like today, so the same value that applies today in the state µL also applies tomorrow if the state happens to be µL.
Naturally, (5.22) is not sufficient to characterize the equilibrium because we do not know what happens in the state µt = µH or, in other words, we do not know what is Vr(N, µH) and, similarly, what is Vp(N, µH). In this state, there may be an effective threat of revolution. So, we must first check whether the revolution constraint is binding. To do so, we define Vr(N) and Vp(N) as the payoffs that would apply if society remains in nondemocracy all the time (i.e., no revolution) and the elites never redistribute to the citizens (i.e., τN = τr). We clearly have:
because the elites always receive the income yr as there is no taxation, and this future income stream is discounted to the present at the discount factor β. Similarly:
(5.23)
We say that the revolution constraint binds if the poor citizens prefer revolution in the state µt = µH rather than to live in nondemocracy without any redistribution; that is, if:
where Vp(R, µH) is given by (5.21). Using the definitions in (4.7), the revolution constraint is equivalent to:
(5.24)
In other words, inequality needs to be sufficiently high (i.e., θ sufficiently high) for the revolution constraint to bind. If inequality is not that high so that we have θ ≤ µ, there is no threat of revolution even in the state µt = µH, even with no redistribution ever. In this case, the elites always set their unconstrained best tax rate, τN = τr, and we have no revolution along the equilibrium path.
It is useful to recall the analysis of our “static” model in the previous section. The formula for the revolution constraint in the dynamic model (5.24) is identical to that in the static model (5.4). In both cases, they simply link inequality to the cost of mounting revolution. This is the basis of the parallel we draw between the static and dynamic models.
The more interesting case is the one in which the revolution constraint (5.24) binds. If, in this case, the elites set τN = τr in the threat state µt= µH, there will be revolution. So, the elites make some concessions by setting a tax rate r N => 0. We denote the values to the elites and the citizens in the state µt, = µH when the elites set a tax rateand are expected to do so in the future, and there is no revolution, by Vr (N, µH, τN= ) and Vp (N, µH, τN=). At this tax rate, an agent of type i has net income of (1 -)yi, plus he receives a lump-sum transfer of. From the government budget constraint, this lump-sum transfer is T = (- C ())whereis total tax revenue and C () y is the cost of taxation.
By the same argument as before, we have the value functions Vr(N, µH,τN =) and Vp(N, µH,τN=) given by:
(5.25)
For the purpose of illustration, we focus on the value function for a member of the elite. The first term is now yr + ((y - yr) - C()y), which is his or her net income after taxation at the rate. The second term is again the continuation value, β [q Vr (N, µH,τN= ) + (1- q) Vr (N, µL)]. With probability q, the state µH arises again tomorrow and, in this case, the rich continue to set τN =and receive Vr(N, µH,τN =). With probability 1 - q, the state switches to µL, and the corresponding value is Vr (N, µL, τN =). The entire term is multiplied by β to discount it to the present.
A similar argument underlies the expression for Vp(N, µH, τN =). A citizen receives a relatively high income today because there is redistribution at the rate. But, what happens in the future is uncertain. If the state remains at µH, redistribution continues. However, there is no guarantee and, in fact, the threat state could switch to µL where the threat of revolution disappears. As we saw previously, now irrespective of what they promise, the elites will stop redistributing and set τN = τr. Therefore, the expression for Vp (N, µH, τN =) already incorporates the potential “noncredibility” of the promise of future redistribution made today. Today’s redistribution arises because the citizens have de facto political power: they have a relatively effective revolution threat and, if the elites do not make some concessions in the form of redistribution, they can overthrow the system. Political power, therefore, gets them additional income. This redistribution might cease tomorrow, however, if what gives political power to the citizens - the revolution threat - disappears. This is the essence of the problem of commitment in this society.
Note also at this point the similarity of the reasoning to that used in the simple game of the previous section. There, the elites made a promise to redistribute at the tax rate, but after the threat of revolution disappeared, nature decided whether they could reset the tax. Here, the elites can successfully redistribute to the citizens today, but what the citizens care about is not only redistribution today but also tomorrow, the day after tomorrow, and so on. Today’s redistribution is supported by the citizens’ political power: the threat of revolution. The elites might like to promise redistribution tomorrow, but when nature decides that the revolution threat disappears tomorrow (i.e., the state switches to µL with probability 1 - q), they no longer keep their promise and cut taxes down to 0, τN = τr. Therefore, as claimed there, the simple game of the previous section was a reduced-form way of capturing the dynamic commitment problems being more carefully modeled here.
Returning to the analysis of the current game, we still need to determine the action of the citizens after the elites decide to redistribute at the tax ratein the state µH. Clearly, they have a choice between no revolution, p = 0, and revolution, p = 1. If they decide to undertake revolution, then once the game reaches this point, the value functions for revolution, Vr(R, µH) and Vp(R, µH), will apply. Otherwise, we have Vr(N, µH, τN= ) and Vp(N, µH,τN= ). Moreover, clearly, a citizen will choose p depending on whether Vp(N, µH, τN =) or Vp(R, µH) is greater. Hence, we can write:
(5.26)
This decision calculus is the same for all citizens. In other words, a citizen takes part in revolution if he or she gets a higher return with revolution than with redistribution at the ratetoday, which again can be thought of as a “semicredible promise of redistribution by the elites” - there will be redistribution today at the tax rateand there might be tomorrow if nature determines that there is an effective threat of revolution tomorrow. We proceed by assuming in (5.26) that if Vp(R, µH) = Vp (N, µH,τN=), then ρ
= 0 so that indifference is broken by not undertaking revolution.
With p given by (5.26), we also have that:
(5.27)
As we know, the elites would like to prevent revolution if they can; the question is whether they will be able to do so. To answer this question, we need to see what is the maximum value that the elites can promise to the citizens. Clearly, this is when they set the tax most preferred by the citizens, τp, given by (4.11). Hence, the relevant comparison is between Vp(R, µH) and Vp(N, µH, τN = τp). If Vp(N, µH, τN= τp) ≥ Vp(R, µH), then a revolution can be averted but not otherwise.
As one would expect, the value function Vp(N, µH, τN= τp) crucially depends on q, the probability that the state will be µH in the future, because this is the extent to which redistribution recurs in the future (i.e., in one sense, how much future redistribution the rich elites can credibly promise). To derive an expression for Vp(N, µH, τN = τp), we substitute Vp(N, µH, τN = τp) = VP(N, µH) in (5.22) and note that (5.22) and (5.25) are two linear equations in two unknowns, the value functions Vp(N, µH, τN = τp) and Vp(N, µL). Solving these two equations, we find:
(5.28)
Equation (5.28) has a straightforward interpretation: Vp(N, µH, τN = τp) is equal to the present discounted value of yp, the pretax income of a citizen, plus the expected present value of net redistribution. Net redistribution is τp(y - yp) - C(τp) y, but this only occurs when the state is µH, something that happens a proportion q of the time. However, in (5.28), (τp(y - yp) - C(τp)) is multiplied by (1 - β (1 - q)), not by q. This reflects the fact that today we start in the state µH and, given that today is more important than the future because of discounting (i.e., because β < 1), the state µL, where there will be no redistribution, gets the weight β(1 - q), not (1 - q). As a result, the state µH received the remaining weight, 1 - β(1-q). (Expressed differently, because we start in the high state, the citizens receive transfers today and a fraction q of the time in the future, so the net present discounted value of the transfer is multiplied by 1 + βq/(1 - β) = (1 - β(1 - q))/(1 - β).) Notice also that as β → 1 (i.e., as discounting disappears), the weight of the state µH indeed converges to q.
Given this value function, we can see that revolution can be averted if Vp(N, µH, τN = τp) ≥ Vp(R, µH), or if:
which can be simplified to:
(5.29)
If this condition does not hold, even the maximum credible transfer to a citizen is not enough, and there will be revolution along the equilibrium path. We can now use (5.29) to define a critical value of µH, again denoted µ* such that Vp(N, µH,τN = τp) = Vp(R, µH), when µH = µ* or:
(5.30)
where µ* < θ. Naturally, we have that when µ ≥ µ*, Vp (N, µH, τN = τp) ≥ Vp (R, µH), and revolution is averted. Whereas when µ < µ*, Vp (N, µH, τN = τp) < Vp(R, µH), future transfers are expected to be sufficiently rare that even at the best possible tax rate for the citizens, there is not enough redistribution in the future, and the citizens prefer revolution rather than living under nondemocracy with political power in the hands of the elites.
It is also useful to point out that the expression in (5.30) is identical to that in (5.15) from the static model with p = 1 - β(1 - q), again emphasizing the similarity between the two models.
As in the static model, when µ > µ*, the elites can avert revolution by setting a tax rate< τp. This tax rate is such that Vp (N, µH, τN =) = Vp (R, µH); that is, it just makes the citizens indifferent between revolution and living under nondemocracy with redistribution only during revolutionary periods. Using (5.21) and (5.28), we have thatis given by:
(5.31)
Putting all these pieces together, we have the key proposition of this section, which - although more complicated - in many ways mirrors Proposition 5.3. This is also a common feature of many of the games analyzed in this book. We start with the simpler reduced-form (static) model and then, most of the time, show that our results hold in a more satisfactory dynamic model.
To state the main result of this section more formally, we can appeal directly to the notation we used to specify the strategies before Proposition 5.3. There, actions were conditioned on whether µ was high or low, and now this is the crucial state variable. This implies that a Markov strategy in the repeated game under consideration has exactly the same form as the equilibrium strategies in the game whose equilibria were analyzed in Proposition 5.3. This enables us to state:
Proposition 5.4: There, in a unique Markov perfect equilibrium {r,p} of the game G∞(β). Let µ* and be given by (5.30) and (5.31). Then, in this equilibrium:
• If θ ≤ µ, the elites never redistribute and the citizens never undertake a revolution.
• If θ > µ, then we have:
(1) If µ < µ*, promises by the elites are insufficiently credible to avoid a revolution. In the low state, the elites do not redistribute and there is no revolution, but in the high state, a revolution occurs whatever tax rate the elites set.
(2) If µ ≥ µ*, the elites do not redistribute in the low state and set the tax rate in the high-threat state, just sufficient to stop a revolution. The citizens never revolt.
Here, we used the intuitive alternative form for stating the proposition. The differences between Propositions 5.3 and 5.4 are the formula for µ* and the fact that the strategies are now Markov strategies in a repeated game, not strategies in an extensive-form game.
It is interesting to focus on the cases where θ > µ. Starting with the elites in power, if µ < µ*, then they set a zero tax rate when µt = µL; however, when the state transits to µH, they are swept away by revolution. The problem is that although the elites would like to stay in power by offering the citizens redistribution, they cannot offer today enough to make the present value of nondemocracy to the citizens as great as the present value of revolution. To avoid revolution, they would have to redistribute not just now but also in the future. Unfortunately, however, they cannot credibly promise to redistribute enough in the future and, as a result, the citizens find it optimal to revolt. In contrast, when µ > µ*, the elites can prevent a revolution by redistributing. So, in the state µt = µL, they set τN = 0, and when µt = µH, they set a tax rate, τN =, just high enough to prevent a revolution.
This proposition, therefore, shows how in a dynamic setting the ability of the elites to transfer resources to the citizens - in other words, the “credibility” of their promises - depends on the future allocation of political power. When q is very low, the citizens may have de facto political power today because of an effective revolution threat, but they are unlikely to have it again in the future. In this case, any promises made by the elites are not credible, and the citizens prefer to use their political power to transform society toward one that is more beneficial for themselves. It is only when q is high, so that the de facto political power of the citizens is likely to recur in the future, that the promises made by the elites are sufficiently credible that a revolution can be averted.
There is an interesting paradox here. When q is high, so that the de facto political power of the citizens is more permanent, it is easier to avoid a revolution. This follows from the fact that µ* defined by (5.30) is decreasing in q in the same way as µ* defined by (5.15) is decreasing in p. This is because when the power of the citizens is not transitory, it is easier for the elites to make credible promises of redistribution in the future. This is somewhat counterintuitive because a simple intuition might have been that when the citizens were better organized and more powerful, revolution would have been more of a threat. This is not the case because the future threat of a revolution also enables more credible promises by the elites to stave off a revolution. Once we introduce democracy into the model, this feature of the equilibrium allows us to provide an interesting interpretation to some historical facts about the incidence of democracy (see Chapter 7).
Also, as in the last section, the critical threshold µ* depends on the extent of inequality in society. In partic
ular, the more unequal is society (i.e., the higher is θ), the higher is µ* and the more likely are revolutions. The reason is simple: with greater inequality, revolution is more attractive, and a greater amount of credible redistribution is necessary to avert a revolution.
7. Incentive-Compatible Promises
The analysis in the previous section focused on Markov perfect equilibria, and showed how a revolution may arise as an equilibrium outcome. Because the political power of the citizens in the future was limited, any promise made by the elites when they keep political power in their own hands is imperfectly credible, and the citizens may prefer to take power today by revolution. An important ingredient of this scenario was the commitment problem: the elites find it optimal to revert back to their most preferred tax rate as soon as the threat of a revolution disappears. This was a consequence of our restricting attention to Markovian strategies because we imposed that, once the threat of revolution subsides, the elites would always choose the strategy that is in their immediate interests.
It is possible, however, that the elites can make certain other promises - for example, they might promise to redistribute in the future even if it is not in their immediate interests. They can support this promise by the implicit understanding that if they deviate from it, when the threat of revolution recurs again, the citizens would undertake a revolution, giving the elites a very low payoff. In other words, these promises could be supported by the threat of future punishments or by “repeated-game” strategies. Punishments correspond to actions that the citizens will take in the future (i.e., revolution), once the elites deviate from their prescribed behavior (i.e., renege on their promises), that will hurt the elites. When we allow players to play non-Markovian strategies, the result is the survival of nondemocracy for a larger set of parameter values. The important difference between Markovian and non-Markovian strategies is that the latter allow players to condition their actions at date t, not only on the state at that date but also on the previous history of play until that date.
Economic Origins of Dictatorship and Democracy Page 24
