Dynamic Stability and Reform of Political Institutions

Transcription

1 Dynamic Stability and Reform of Political Institutions Roger Lagunoff October 10, 2004 Preliminary Abstract This paper introduces a framework designed to study dynamic, endogenous reform and stability of political institutions. Dynamic political games (DPGs) are dynamic games in which institutional choice is both recursive and instrumental. That is, future political aggregation rules are decided under current ones, and the resulting institutional choices do not affect payoffs or technology directly. We examine properties of the Markovian equilibria of DPGs. In any Markovian equilibrium, a political rule is stable if the subsequent rule is chosen to be the same as the current one. A rule admits institutional reform if the subsequent political rule is chosen to be different. Which environments tend toward institutional stability? Which tend toward reform? When private sector decisions are inessential, any feasible social planner s continuation payoff can achieved by public sector decisions alone. In that case, reform never occurs: all political rules are stable. More generally, we identify sufficient conditions for stability and reform in terms of recursive self selection, an incentive compatibility concept that treats rules as players in all feasible sequences of play with private individuals. We also address the political fixed point problem in which a current political rule (e.g., majority voting) admits a solution only if all political rules feasible in the future admit solutions in all states. If the class of political rules is dynamically consistent, then DPGs are shown to admit political fixed points. This result is used to prove two equilibrium existence theorems, one of which implies that all decision rules are smooth functions of the economic state. JEL Codes: C73,D72, D74 Key Words and Phrases: Recursive, dynamic political games, institutional reform, institutional stability, political fixed points, dynamically consistent rules, inessential. Department of Economics, Georgetown University, Washington DC USA lagunofr@georgetown.edu. This project began while the author visited Johns Hopkins University and the W. Allen Wallis Institute of Political Economy at the University of Rochester. I am grateful for their hospitality. I also thank Luca Anderlini, William Jack, Jim Jordan, Mark Stegeman, and seminar participants at Georgetown, Penn State, VPI, and the Latin American Meetings of the Econometric Society for their helpful comments. 1

2 1 Introduction Reforms of political institutions are common throughout history. They come in many varieties. In some cases, the reforms correspond to changes in the voting franchise. Periodic expansions of voting rights occurred in governments of ancient Athens ( BC), the Roman Republic (509BC-25AD), and most of Western Europe in the 19th and early 20th centuries, to name just a few examples. 1 In other cases, modifications were made to the voting procedure itself. Medieval Venice ( ), for instance, gradually lowered the required voting threshold from unanimity to a simple majority in its Citizens Council. Nineteenth century Prussia, where votes were initially weighted by one s wealth, eventually equalized the weights across all citizens. The U.S. changed its rules under the 17th Amendment to require direct election of senators. In still other instances, the scope of a government s authority changed. For example, England and France privatized common land during the 16th and 17th century enclosure movement thus reducing scope for rules governing the commons. 2 The U.S. government, on the other hand, increased its scope under the 16th Amendment by legalizing federal income tax in Numerous historical examples suggest that institutional change is often gradual and incremental. Consider the progress of reforms in the Roman Republic. In 509BC, the Senate and Assembly were founded; in 494 BC the Patricians conceded the right of the plebs (the commoners ) to participate in the election of magistrates; in 336 BC one of the consulships became available for election by plebians; and in 287 BC Hortensian Law was introduced which gave resolutions in the plebian council the force of law. Gradual reform also characterized expansion of voting rights in 19th century England. In 1830, the voting franchise restricted to 2% of the population. In 1832, the First Reform Act extended the franchise to 3.5% of population. The Second Reform Act of 1867 extended it to some 7.7%. By 1884 it had been extended to 15% of population. Universal suffrage only passed in 1928 (see Finer (1997). The present paper has two related objectives. First, we seek a coherent framework to understand how gradual, endogenous, institutional change works. To build such a framework, one natural consideration is that the design process be recursive; parameters of future political institutions are a part of the explicit decision made under the current institution. Another consideration is that the institutional decisions should be instrumental. That is, they do not affect payoffs or technology directly. Hence, society modifies its existing institutions not because the details of political procedures enter into the utility functions. Rather, institutions are modified because these changes modify substantive private and public sector decisions in the future. 1 See Fine (1983), Finer (1997), and Fleck and Hanssen (2003). 2 See MacFarlane (1978) and Dahlman (1980). 1

3 As a second objective, we seek to understand why institutional reform occurs. Most basically, which environments tend toward institutional stability? Which environments admit change? What are the relevant forces that drive this change? To address these objectives, we introduce the class of dynamic political games (DPG) in which the rules for choosing public decisions are themselves part of the decision process. A dynamic political game (DPG) is an infinite horizon stochastic game in which at each date t, private and public sector decisions jointly determine the date (t+1) distribution of economic and political states of the world. Economic states are substantive parameters that affect preferences and technology directly. Political states are procedural parameters that characterize the explicit political rule for determining public sector decisions. If, for example, the current political rule is a simple majority rule, then the feasible set of outcomes of majority rule is the set of Condorcet Winners the choices which cannot be defeated by any alternative choice in a majority vote. We study the Markovian equilibria of dynamic political games. A Markovian equilibrium in a DPG is a collection of state-contingent private and public sector decision rules such that (a) the private sector rule for each individual is optimal for him in each period and in each state, and (b) public sector decisions are consistent with the prevailing political rule in each period and in each state. The choice of restricting to Markovian equilibria follows the general rule of thumb: in an economy-wide context, the participants are less apt to coordinate on non payoff-relevant history than they would if they were in a small group. DPGs satisfy the objective that the process be both recursive and instrumental. Parameters of next period s political rule are a part of the explicit decision made under the current political rule. These parameters constitute the political state each period. Moreover, political states are not directly payoff relevant. There is a modest literature on dynamic, endogenous political institutions. 3 Informal discussions in North (1981) and Ostrom (1990) both hint at recursivity in the process of institutional change. In more formal work, Messner and Polborn (2002) examine an OLG model of endogenous changes to future voting rules under current ones. Lagunoff (2001) studies a dynamic recursive model of endogenously chosen civil liberties. Greif and Laitin (2004) model institutions as equilibrium outcomes in a repeated game. An influential model of Acemoglu and Robinson (2000, 2001) examines endogenous voting rights in a dynamic game. In their model an enfranchised player representing the elite can choose in any period whether to make a once-and-for-all extension of voting rights to the other player representing the masses. The extension of rights is motivated by the elite s desire to head off social unrest. Models in which endogenous extensions of voting rights are gradual and incremental 3 In focusing attention on dynamic models, I neglect a larger literature on static models of endogenous political institutions such as, for example, Lizzeri and Persico (2002) and Aghion, Alesina, and Trebbi (2002). 2

4 are proposed by Justman and Gradstein (1999), Roberts (1998, 1999), Barbera, Maschler, and Shalev (2001), Jack and Lagunoff (2003), and Gradstein (2003). In certain respects, the model of Jack and Lagunoff (2003) is a prototype for the present framework. The dynamics of institutional choices in that model are fully recursive and instrumental. The current paper presents new results and extends their framework to institutional choices other than voting rights. In addition, fairly arbitrary types of agent heterogeneity and general functional forms for payoffs and transition technologies are considered here. Given these features, the framework is used to address two questions. First, when (i.e., in what class of DPGs) do Markovian equilibria exist? Second, in a given equilibrium, when do institutional reforms occur, or alternatively, when are rules stable over time? The paper is divided into two halves, each addressing one of these questions. The first half of the paper addresses the second question first. By definition, a political rule admits reform occurs whenever next period s political rule is chosen to be different than the present one. A rule is stable whenever no reform occurs. For tractability we restrict attention to games in which all rules are dynamically consistent. A political rule is dynamically consistent if it is rationalized by a time separable social welfare function. When all rules are dynamically consistent, our findings suggest that institutional reform (alternatively, institutional stability) depends on how useful the private sector is relative to public sector. More precisely, the private sector is said to be inessential in the current political state if for any private sector decision, any social continuation that is feasible tomorrow can always be reached or exceeded today by a public sector decision. Hence, private decisions are inessential if their effects on social welfare can always be replaced by those of the public sector. Clearly, a private sector can be inessential only in restrictive circumstances. Nevertheless, it provides an important benchmark with telling consequences for institutional reform. We prove that institutional reform never occurs if private sector decisions are inessential. Consequently, the existing political rule is always stable. The intuition for the result may be seen in the special case of endogenous voting rights. The Acemoglu-Robinson (2000) argument asserts that the elites of 19th century Europe chose to extend the voting franchise because policy concessions alone could not buy off the external threat of an uprising. In other words, franchise extension could only occur because the private sector was essential. Of course, this condition provides only a necessary, not a sufficient condition for a reform such as a franchise expansion. Jack and Lagunoff (2003) display Acemoglu-Robinson argument as a special case in a recursive model in which gradual extension of the voting franchise is possible. The result therefore helps to make sense of their logic in a larger context; it identifies some necessary features of an environment in order for institutional change to occur. 3

5 More generally, by fixing a continuation value and an economic state in each period, one can interpret play of the dynamic political game in that period as a distinct normal form game. A rule is recursively self-selected if, in each induced game, the current rule never chooses to delegate decision authority to another rule. A rule is recursively self-denied if, in each induced game, it chooses some other rule. Recursive self-selection and self-denial are institutional incentive constraints that treat each distinct rule as an institutional player in the induced normal form game with private individuals. Recursive self-selection bears some relation to self-selected rules in the static social choice models of Koray (2000) and Barbera and Jackson (2000), and also to the infinite regress model of choice of rules by Lagunoff (1992). These all posit social orderings on the rules themselves based on the outcomes that these rules prescribe. Rules that select themselves do so on basis of selecting the same outcome as original rule. The present model has two differences. First, institutional choice occurs in real time next period s rule is chosen by the present one. This makes possible an analysis of explicit dynamics of change. Second, the present model is more concrete; the trade-offs are explicitly derived from the interaction between public and private sector decisions. We show first that recursively self-selected rules are stable. Second, if a rule is either (a) recursively self-denied by another recursively self selected rule, or (b) is recursively self-denied by every other rule, then the rule admits institutional reform. Ideally, one eventually hopes to obtain a complete set of necessary and sufficient conditions that distinguish environments that admit reform from those that do not. Our results provide only sufficient conditions for both, and somewhat restrictive ones at that. Nevertheless, the notions of inessentiality and self-selection/denial seem more representative of general ideas than their application suggests. The second half of the paper deals with issues of equilibrium existence. A number of interesting issues arise that make it more than a technical exercise. Any recursive political model incurs the following political fixed point problem. Political rules are defined here on average discounted payoffs of the form (1 δ)u t + δv t+1 where u t is the current stage payoff, δ is the discount factor, and V t+1 is the continuation. This payoff is endogenous: the continuation V t+1 is the result of the application of a political rule in period t + 1, which, in turn, is defined on payoffs that depend on the application of a political rule in period t+2, etc... This recursive consistency condition requires that the implied map from future (state-contingent) decisions rules to current ones has a political fixed point. But if the political rule is a voting rule, then voting cycles may lead to nonexistence of political fixed points. Standard fixes such as single peakedness do not work in DPGs because public decisions are inherently multi-dimensional: both the current policy and the future political rule are chosen each period. 4

6 Our results emphasize the importance of dynamic consistency for resolving this problem. We show that if rules are dynamically consistent then the DPG admits political fixed points. An important special case is the class of all voting rules. If stage game payoffs admit an affine representation then each voting rule is rationalized by the preferences of the median voter in each state. This affine preference representation is similar to (and somewhat more restrictive than) the class of Intermediate Preferences introduced by Grandmont (1978) to prove a multi-dimensional Median Voter Theorem. Since these median voter preferences are dynamically consistent, political fixed points exist under voting rules and affine stage payoffs. We use these results to establish two equilibrium existence theorems. The first is a more standard result asserting that equilibria exist when state and action spaces are finite. The second theorem adapts an elegant result of Horst (2003) to find sufficient conditions under which equilibria exist that are smooth functions of the economic state. The smooth existence result, in particular, is useful for practical applications. It allows for economically interesting dynamic environments with unbounded state space (e.g., capital stocks). Smooth existence is also required for computational techniques such as those developed by Klein, Krusell, and Rios-Rull (2002) for politico-economic models. 4 The paper is organized as follows. In Section 2, we present a less formal version of the model in order to highlight some issues and problems of recursive institutional choice. The general model is described in Section 3. There, we introduce the class of political rules. Political rules are the natural objects of choice in a recursive model of endogenous institutions. A dynamic political game combines both the public and private sectors of the environment. An equilibrium combines standard Markov Perfection in private decisions with an implementations requirement for public decisions. Section 4 examines the issue of institutional reform. Section 5 examines the political fixed point problem and describes the existence results more generally. Section 6 concludes with a discussion of dynamic consistency requirements of the political rules. Proofs of the main results are in the Appendix, Section 7. 2 An Introductory Pure Public Sector Model A less formal, stripped down version of the model is presented in this Section to highlight some basic issues in recursive institutional choice. Consider initially a simple model of period-by-period majority voting. This is a frequently studied model, and one in which the institutional environment is fixed. At each date t =1, 2,..., a set I = {1,...,n} of individuals must vote to decide a policy p t at date t. To fix ideas, one can think of p t as an income tax schedule from a set, P, of feasible 4 Their approach computes optimal policy functions resulting from median voter and other political rules using generalized Euler equations that are higher order differential equations of the policy function. 5

7 schedules. The current state is ω t drawn from a set Ω. One can interpret ω t as the distribution of incomes across individuals. Since current tax rates may affect savings behavior, the tax rate affects both one s current payoff and the future state. For now, omit (notationally) private sector behavior such as individuals savings decisions. 5 A policy rule ψ is a Markov strategy specifying the policy p t = ψ(ω t ) as a function of state ω t at date t. Given state ω t, individual i s (i =1,...,n) average discounted dynamic payoff over policies is expressed in a recursive form by U i (ω t ; ψ)(p t ) (1 δ)u i (ω t,p t )+δ V i (ω t+1 ; ψ)dq(ω t+1 ω t,p t ) (1) where δ is the discount factor, u i is the stage game payoff received in each period, q denotes the stochastic transition function mapping current states and policies into probability distributions over future states, and V i is i s continuation payoff given policy rule ψ. 6 By construction, U i (ω t ; ψ)(ψ(ω t )) = V i (ω t ; ψ). The political fixed point problem. If p t is chosen each period by a simple majority vote, then pairwise comparisons of policy are evaluated by each individual using his payoff function U i (ω t ; ψ)( ) in (1). The profile of payoff functions is U(ω t ; ψ) = (U i (ω t ; ψ)) n i=1. For each such profile, the outcome of majority voting is typically represented by the set of Condorcet Winners outcomes that survive all pairwise comparisons in a majority vote. Denote this set by C(U(ω t ; ψ)). For the policy rule ψ to be consistent with C(U(ω t ; ψ)), it must satisfy the political fixed point problem ψ(ω t ) C(U(ω t ; ψ) ), ω t (2) Notice that (2) is both a fixed point and an implementation property. If voting is cycle proof, then Condorcet Winners exist. But since voting takes place each period, the continuation payoff V i already encodes future voting outcomes, and so the voting rule at date t is cycle-proof only if Condorcet Winners exist in all future dates. In certain cases, the problem is resolved by assuming that p t is uni-dimensional and then showing that single peaked stage game preferences generate single peaked recursive preferences. However, single peakedness will not suffice if both the policy and the political institution itself are part of the decision problem. Consequently, let θ t denote a parameter that determines the political institution. θ t is chosen from a finite index set Θ. For example, suppose Θ = {θ 1,θ 2 } where θ t = θ 1 means that the tax schedule is determined by a majority vote. By contrast, if θ t = θ 2 then the tax schedule is imposed by a dictator, whom we assume to be individual i = 1. The political state θ t is then distinct from the payoff-relevant economic state ω t. The political state summarizes 5 Private sector behavior will be introduced in the next Section. 6 The transition q is assumed to satisfy the standard measurability assumptions. 6

8 the political process by which public decisions are made. In particular, the political state θ t determines the political rule, in this case either majority rule or dictatorship, for choosing both the policy p t and the subsequent political state, θ t+1. As before, ψ determines policy p t. Again, omit the private sector. Now, the public sector decision includes the choice of institution for the following period. An institutional decision rule µ is a mapping that determines the future political state θ t+1 = µ(ω t,θ t ) given the current economic and political state. The institutional decision rule describes a recursive process of institutional change. The rule in period t produces the new rule for period t + 1. The public sector decision each period is a pair (p t,θ t+1 ). To save on notation, let s t =(ω t,θ t ) be the composite state. An individual s payoff now is U i (s t ; ψ,µ)(p t,θ t+1 ) (1 δ)u i (ω t,p t )+δ V i (s t+1 ; ψ,µ)dq(ω t+1 ω t,p t ) (3) If s t =(ω t,θ 1 ), then the set C(U(s t ; ψ,µ),s t ) describes the set of Condorcet Winning public decisions as before. However, if s t = (ω t,θ 2 ), then C(U(s t ; ψ,µ),s t ) describes the public sector decisions that maximize dictator s payoff function U 1 (s t ; ψ,µ)( ). The political fixed point problem may be restated as (ψ(s t ),µ(s t )) C(U(s t ; ψ,µ),s t ), s t (4) The fixed point problems (2) and (4) are distinct in several respects. In (2), the map admits a fixed point in the space of policy rules. Since the institution majority voting was fixed, the recursive payoff profiles were required to admit Condorcet Winners for each economic state ω. This is a nontrivial problem by itself. However, the mapping in (4) varies by institutional state, θ t, as well as by economic state ω t, and so we further require that (4) admits solutions for all such institutions in Θ. Moreover, since the decision problem is multidimensional, the simplest Median Voter Theorems are not useful for solving (4). Finally, if private sector decisions are considered, then (4) must hold for individuals private decision rules that best respond to public sector decisions and to each other in all states. Stability vs Reform. If, indeed, a satisfactory solution to the political fixed point problem is found, this model can determine when and if institutional change takes place. For instance, when (i.e., for what values of the economic state ω t ) is it true that dictators relinquish power: µ(ω t,θ 2 )=θ 1? When is it true that democracies turn over power to dictators: µ(ω t,θ 1 )= θ 2? The answer will depend crucially on the interaction between public and private sector decisions. In particular, without any private sector, we have: Proposition Let (ψ,µ) be a political fixed point (a pair that satisfies (4) in each state s t ) with the property that the Condorcet Winning choice in state θ 1 is the most preferred decision 7

9 of one of the voters (e.g., a median voter). Then each political state θ Θ is everywhere politically stable in the sense that for every ω, µ(ω, θ) =θ. The proof is omitted since the result is a special case of Theorem 1 later in the paper. 7 The intuition is: by maintaining the current political state, the current pivotal decision maker (either the dictator in θ 2 or the pivotal voter in θ 1 ) holds on to power. By doing so, the decision problem reduces to a single agent dynamic programming problem. It is well known in such problems that the resulting sequence of decisions is optimal from the decision maker s point of view. No other individuals make choices to offset the pivotal decision maker s choices. A private sector typically fills that role. Significantly, the absence of inalienable rights to make private decisions is the defining feature of a totalitarian government. This is the traditional definition, according to which democracies can be totalitarian if all choices are filtered through the voting mechanism. Consequently, both of these political rules are stable in a totalitarian state! Since most governments are not totalitarian to this extreme, we introduce a private sector in the general model that follows. 3 The General Model 3.1 Political Rules In general, the set Θ can define a larger set of political institutions than merely dictatorships and democracies. Following standard conventions, we will find it useful to drop time subscripts, and adopt instead the use of primes, e.g., θ to denote subsequent period s variables, θ t+1, and so on. Let denote the cardinality of a set. Let S =Ω Θ denote the composite state space. Let v i denote an arbitrary function expressing the payoff v i (p, θ ) over current policy p and next period s political state, θ. In the dynamic model of the previous section, v i is a notational shorthand for v i (p, θ )=U i (s; ψ,µ)(p, θ ). Let V denote the set of all profiles, v =(v 1,...,v n ), of such payoff functions. A class of political rules corresponds to a (possibly empty valued) correspondence C : V S P Θ that associates to each profile v and to each state s, a set C(v, s) of public decisions. If (p, θ ) C(v, s), then (p, θ ) is a feasible public decision under C. Each particular political rule in the class C is given by C(,s). The class of rules C is single valued if for each v V and for each s, there exists a (p, θ ) pair such that C(v, s) ={(p, θ )}. Fix the state s =(ω, θ). To get a better sense of how broadly the framework describes various political institutions, we sketch a few examples below. 7 A special case of this result also appears in Jack and Lagunoff (2003). 8

10 I. Voting over the Voting Rule. The political state identifies the fraction, θ 1/2 of individuals required to pass a public decision. A super-majority voting rule therefore determines which supermajority rule is used in the future: formally Θ (.5, 1] and let (p, θ ) C(v, s) if for all (ˆp, ˆθ ) {i I : v i (ˆp, ˆθ ) >v i (p, θ )} θn II. Voting over the Voting Franchise. The political state θ identifies the subset of individuals who currently possess the right to vote (the voting franchise). The chosen public decision is the one that is majority preferred within this restricted group. Each restricted voting franchise today uses a majority vote to determine what group of individuals have the right to vote tomorrow: formally, Θ 2 I, and let (p, θ ) C(v, s) if for all (ˆp, ˆθ ), {i θ : v i (ˆp, ˆθ ) >v i (p, θ )} 1 2 θ In this model, the current voting franchise decides on a new voting franchise in the following period. An interesting subclass of these rules is the class of Delegated Dictatorship rules. 8 Under these rules, θ varies only over the singletons {i}, i =1,...,n. In each such state, the current dictator chooses his most preferred policy and then delegates the decision to possibly a new dictator in the future. If the date t describes the length of a generation, then delegated dictator rule might be useful for describing a particular process of dynastic succession in which the king anoints his own successor. III. Voting over the Scope of Government. The political state identifies the domain of public decisions. Let θ P so that θ denotes the set of feasible policies. Let (p, θ ) C(v, s) ifp θ and for all (ˆp, ˆθ ) satisfying ˆp θ, {i I : v i (ˆp, ˆθ ) >v i (p, θ )} 1 2 n Numerous hybrids can be derived from these (e.g., delegated dictator from a limited oligarchy). In many cases, the political rules of interest are those that can rationalized by some social welfare criterion. Formally, a class of rules C is (partially) rationalized by a social welfare function F :IR n S IR if F is weakly increasing in each dimension of IR n and if C(v, s) =( )arg max F (v(p, θ ),s) p,θ Clearly the Delegated Dictator Rule is rationalized by v θ where θ identifies the dictator. When the political rule C(,s) is some type of voting rule (e.g., Examples I and II above), then there are two well known conditions under which C is rationalized by the preferences of a Median Voter. The first is the standard restriction to single peaked preferences. 9 The second is the order restriction property of Rothstein (1990). Similar results can be found in application of single crossing properties by Roberts (1977) and by Gans and Smart (1996). 8 These were introduced and studied by Jack and Lagunoff (2003). 9 See Arrow (1951) and Black (1958). 9

11 3.2 Dynamic Political Games Recall that I = {1,...,n} is the set of individuals in this society. P is the set of feasible policies in each period, and S = Ω Θ is the composite state space. We now introduce private sector decisions. Let e it denote i s private decision at date t, chosen from a feasible set E. A profile of private decisions is e t =(e 1t,...,e nt ). These decisions may capture any number of activities, including labor effort, savings, or investment activities. They may also include non-economic activities such as religious worship or one s participation in a social revolt. To express the dependence of payoffs and technology in private sector decisions, let u i (ω t,e t,p t ) denote i s stage payoff and let q(b ω t,e t,p t ) denote the probability that ω t+1 belongs to the (Borel measurable) subset B Ω, both given the economic state ω t, the private decision profile e t, and the policy p t. Given that economic states evolve according to q, each individual s dynamic objective is to maximize average discounted payoff, δ t (1 δ) u i (ω t,e t,p t ) (5) t=0 Denote the initial state by s 0 =(ω 0,θ 0 ) and define a Dynamic Political Game (DPG) to be the collection economic structure {}}{ G (u i ) i I,q,E,P,Ω, political structure {}}{ Θ,C, initial state {}}{ s 0 The class of dynamic political games (DPGs) constitutes a broad set of problems in which institutional changes occur endogenously and incrementally. While DPGs include all the exogenous elements of the game, the relevant payoff inputs in C are endogenous recursive payoffs that depend on strategies to be defined in the next section. For tractability, we restrict attention to dynamic political games that satisfy one of the following two exclusive sets of assumptions. (A1) Θ is a finite set. P and E are compact, convex subsets of Euclidian spaces, and Ω is a convex subset of a Euclidian space; the payoff function u i for each i is continuous and uniformly bounded above by some K>0; for each (ω, e, p), q( ω, e, p) admits a norm continuous, conditional density f( ω, e, p) with respect to a probability measure η. 10 (A1 ) E, P and Ω are all finite sets. 10 These assumptions are fairly standard for proving existence of Markovian equilibria in the dynamic games. However, they are not harmless. In particular, dynamic payoffs are bounded by K and norm continuity of a density f precludes deterministic transitions. See Dutta and Sundaram (1994) for a cogent discussion of the role of these assumptions. 10

12 Unless otherwise stated, all the subsequent results assume that either (A1) or (A1 ) holds. 3.3 Strategies and Equilibrium To make the theory tractable, we restrict attention to Markov strategies. Such strategies only encode the payoff-relevant states of the game. Consequently, individuals are not required to coordinate on the history of past play. Recall that ψ : S P and µ : S Θ describe the policy and institutional rules, respectively. Together they constitute the public sector rules. A private sector rule for individual i is a function σ i : S E i that prescribes private action e it = σ i (s t ) in state s t. Let σ =(σ 1,...,σ n ). The strategy profile is therefore summarized by the triple private sector rule }{{} policy rule }{{} institutional rule }{{} π ( σ, ψ, µ ) An individual deviation from π is denoted by, for example, π\σ i. For any s t =(ω t,θ t ) the payoff to citizen i in profile π at date t is defined recursively by: V i (s t ; π) = (1 δ)u i (ω t,σ(s t ),ψ(s t )) + δ V i (ω t+1,µ(s t ); π)dq(ω t+1 ω t,σ(s t ),ψ(s t ) ) (6) The function V depends on and varies with arbitrary Markov strategy profiles π =(σ,ψ,µ). Along an equilibrium path (defined below), the function V i defines a Bellman s equation for citizen i. Given any strategy π, and any state s t at date t, an individual s public payoff function U i (s t,π):p Θ IR is defined by U i (s t,π)(p t,θ t+1 ) (1 δ)u i (ω t,σ(s t ),p t )+δ V i (ω t+1,θ t+1 ; π)dq(ω t+1 ω t,σ(s t ),p t ) (7) Let U(s t,π)=(u i (s t,π)) i I. We now drop time subscripts and define an equilibrium for any dynamic political game. Definition 1 An Equilibrium of a dynamic political game, G, is a profile π =(σ,ψ,µ)of Markov strategies such that for all states s =(ω, θ), (a) Private decision rationality: For each citizen i, and each private decision rule, ˆσ i, V i (s; π) V i (s; π\ˆσ i ) (8) 11

13 (b) Public decision implementation: political fixed point problem The public decision pair (ψ(s),µ(s) ) satisfies the (ψ(s),µ(s) ) C (U(s, π), s) (9) According to (a), in each state, s, private sector actions are individually optimal. Part (b) asserts the existence of political fixed points. Public sector decisions are required to be consistent with political rules in the class C. In keeping with the standard definition of a stochastic game, both types of decisions are simultaneous. Therefore, an equilibrium of a DPG requires both Markov Perfection from individuals private sector choices and recursive consistency of public sector choices with a political rule. The latter requirement must hold in each state s, and so the consistency condition also satisfies kind of perfection constraint. 4 Institutional Reform vs Institutional Stability In this Section, statements beginning with for all ω and there exists ω are to be interpreted as for all ω excluding possibly a set of Lebesgue measure zero and there exists a set of ω with positive measure..., respectively. All functions defined on Ω will be assumed to be measurable. The term institutional reform refers simply to the idea that institutions are deliberately modified. Given an equilibrium π =(σ,ψ,µ), a political state θ is said to admit institutional reform in π if there exists a (set of positive measure) ω such that µ(ω, θ) θ. Alternatively, a political state θ is institutionally stable in π if it does not admit reform, i.e, if µ(ω, θ) =θ for (almost) every ω. When do political states admit reform? When are they stable? Minimally, the answer should depend only on exogenous features of the dynamic political game. In this Section, we investigate properties of dynamic political games that fit this requirement. 4.1 Dynamically Consistent Rules Recall that most of the common political rules of interest are those that are rationalized or partially rationalized by social welfare functions. Among these, the most obvious ones satisfy a dynamic consistency requirement. Most dynamic models of government policy, for example, presume a dynamically consistent decision maker. 11 The discussion of institutional reform will, for now, limit attention to dynamically consistent rules as defined as follows. 11 A notable exception is Krusell, Kuruscu, and Smith (2002). 12

14 Consider any additively separable public sector payoffs: payoffs function of the time separable form, v(p t,θ t+1 )=(1 δ)v 1 (p t )+δv 2 (θ t+1 ). (Clearly, dynamic payoffs in the present model satisfy this requirement). For our purposes, a class of political rules, C, will be said to be dynamically consistent if it is partially rationalized by a continuous social welfare function F that satisfies in every state s t =(ω t,θ t ), F ( (1 δ)v 1 (p t )+δv 2 (θ t+1 ),s t ) =(1 δ)f ( v 1 (p t ),θ t ) + δf ( v 2 (θ t+1 ),θ t ) This definition embodies two critical properties. First, F is time separable and linearly homogeneous in the discount weights (1 δ, δ). 12 Second, F does not vary directly with the economic state. Section 5 shows that dynamically consistent rules admit solutions to the political fixed point problem. In the concluding section, we discuss some potential implications of dynamically inconsistent aggregation. The remainder the Section assumes that C is rationalized by a single valued, dynamically consistent social welfare function, F. 4.2 An Inessential Private Sector In the Pure Public Sector Model of Section 2, it was shown that equilibria never admit institutional reform. It seems clear from that model that the private sector matters somehow. Here, we try to make clear what it means for the private sector to matter. First, the following notation is required. For any initial economic state ω, afeasible continuation payoff profile is a measurable function x :Ω [0,K) n for which there exists an arbitrary pair of (measurable) Markov functions σ and ψ defined only on Ω such that for each i =1,...,n, [ x i (ω) =E (1 δ)δ t u i (ω t,σ(ω t ),ψ(ω t )) ] ω = ω0, t=0 where E is the expectation on {ω t } induced by the transition law q. Feasible continuations are those that could be reached by any pair of Markov strategies, equilibrium or otherwise. Now given any feasible continuation x, define for each i, H i (ω,e,p,x) (1 δ)u i (ω, e, p)+δ x i (ω )dq(ω ω, e, p). Given e, p, and x, the vector H(, e, p, x) =(H 1 (, e, p, x),...,h n (, e, p, x) ) is a profile of current payoffs for which x is the feasible continuation. Notice that H(, e, p, x) is itself a feasible continuation. 12 Broader definitions of dynamic consistency do not require the linear homogeneity in discount factor requirement. The results here may also be amenable to a more general definition, but at a cost in transparency. 13

15 Recall that a dynamically consistent rule C is rationalized by a social welfare function F that does not vary across economic states, i.e., F (,s) = F (,θ), and which satisfies: F (H(ω,e,p,x), θ) = (1 δ)f (u(ω, e, p), θ)+δ F (x(ω ),θ) dq(ω ω, e, p) We refer to F (x(ω ),θ) as a feasible social continuation in state s =(ω,θ). Definition 2 Fix the political state θ. Formally, we will say that the private sector decisions are inessential in θ if the following holds for any economic state ω, and any feasible continuation profile, x: for every private sector profile e there exists a public sector decision p such that F (H(ω,e,p,x), θ) F (x(ω), θ) (10) In words, private sector decisions are inessential in θ if in any economic state and for any private sector decision, any social continuation that is feasible tomorrow can always be reached or exceeded today by a public sector decision. Informally, private decisions are inessential if their effects on social welfare can always be replaced by those of the public sector. The definition utilizes the recursive idea that a given social payoff available today, is always available tomorrow if the political aggregation rule (indexed by θ) is still available. An extreme case was examined in Section 2. When there is no private sector (the totalitarian state), then private decisions are, by definition, inessential. A less extreme example is one where u, F, and θ satisfy: ω, e, max p F (u(ω, e, p), θ) = maxf (u(ω, e, p), θ) (11) p,e In other words, the social welfare function, evaluated at the stage game payoffs, can achieve its optimum in each state by varying p alone. In this case, the absence of the private sector need only occur at the optimum. It is straightforward to verify that any dynamic political game satisfying (11) has an inessential private sector. As the examples suggest, a private sector is inessential only in restrictive circumstances. Private sector actions must be redundant from political type s point of view. 13 Despite its severity, the definition provides a baseline environment for evaluating institutional reform. 13 This suggests that inessentiality is a nongeneric property. However, mathematical genericity is not really relevant here since redundancy, in this context, is partly an artifact of property rights arrangements. The delineation between private and public actions may be built into the political process itself. Hence, it is endogenous. 14

16 Theorem 1 Consider a dynamic political game in which C is rationalized by a single valued, dynamically consistent class of political rules. Fix a political state θ. If private decisions are inessential in state θ, then θ does not admit institutional reform in any equilibrium. The conclusion of the Theorem is that in any equilibrium, the type θ is institutionally stable, i.e., µ(ω, θ) = θ. According to the Theorem, an essential private sector is necessary for reform. The Proposition in a previous Section is a special case. The proof is in the Appendix, but the logic of the argument is the following. Interpret the parameter θ as a player the social planner whose payoff is F (,θ). We refer to this player as the institutional type. In any state s =(ω, θ), the institutional type θ chooses the public policy p and designates a subsequent decision maker, type θ, the following period. A standard result of dynamic programming is that if type θ faces a single agent decision problem, it need never designate the decision authority over future policies to another player. Its own choice of policies {p t } would be optimal in each realized state. To put it another way, Type θ is willing to relinquish is authority over future decisions only if its designated choice can induce a more favorable response from actions of others. Consequently, inessentiality of the private sector collapses the problem to a pure policy hence single agent decision problem. The planner of type θ can unilaterally reach any alternative social payoff using policies alone. Hence, this type need never relinquish decision making authority to another type ˆθ θ. In such a case, the policy-path is optimal for type θ if that same type makes decisions each period. The institutional type θ is therefore stable. 4.3 Recursively Self Selected Institutions Inessentiality of the private sector is a sufficient condition for institutional stability. Hence, its opposite essentiality is a necessary condition for change. It is clearly not sufficient. For even if the private sector is essential, change need not occur if the private sector response matters in a way that favors the current political institution. Moreover, it may be the case that change may or may not occur because the equilibrium selects good equilibria in certain states and bad equilibria in others. This Section addresses some of these issues. Fix a continuation profile, x, and a state s =(ω, θ). Let N(s, x) denote the set of Nash equilibria of the (n + 1)-player normal form game defined by the following payoffs. The first n players are the same as individuals i =1,...,nin the original game. Each Player i =1,...,n has payoff H i (ω,e,p,x). Player n + 1 has payoff F (H(ω,e,p,x),θ). Player n + 1 is the social planner of type θ who maximizes welfare function F (, θ). A standard result establishes that under assumption (A1) (or (A1 ) ), N(s, x) in any state and any continuation. 15

17 Given payoffs described above, define, for each s =(ω, θ), each x, and each θ, F (s, x, θ ) = max {F (H(ω,e,p,x), θ ) : (e, p) N(s, x)} The value F (s, x, θ ) is the largest social welfare for institutional type θ over all possible Nash equilibria of the stage game with payoff profile (H 1 (ω,e,p,x),...,h n (ω,e,p,x),f(h(ω,e,p,x), θ). (Note that if θ θ, then type θ, and not θ, is the institutional player in the stage game.) The reason for this notation is as follows. Recall that the choice of next period s type θ is viewed as a strategic delegation decision. For a fixed ω and x, one can view this choice as a choice between equilibria from the sets N(ω,x,θ) and N(ω, x, ˆθ) for any pair θ and ˆθ. However, multiplicity of equilibrium potentially muddles the comparison. In order to make a consistent comparison across types, the preferred equilibria in N(ω,x,θ) and N(ω, x, ˆθ) are compared by the current type θ making the delegation decision. 14 The following definition and Lemma will prove useful for subsequent results. Definition 3 An equilibrium π is institutionally optimal for θ if for any other equilibrium ˆπ and for each s =(ω, θ), F (V (ω, µ(s); π), θ ) F (V (ω, ˆµ(s); ˆπ), θ ) (12) The institutionally optimal equilibrium is the one that generates the most preferred social welfare for type θ in each state among all equilibria. The idea behind institutional optimality extends the best-case comparison to the full dynamic model. A straightforward application of the definitions establishes the following. Lemma 1 If π is institutionally optimal, then for any state s =(ω, θ) and for any θ, then F (s, V (,µ(s); π), θ ) = F (V (s; π), θ ) (13) According to the Lemma, when the equilibrium profile V (,µ(s); π) is the continuation, an institutionally optimal equilibrium is the most preferred Nash equilibria for type θ among the set of stage game Nash equilibria N(s, V (,µ(s); π). Clearly, if an equilibrium exists, then it is clear from the definition that an institutionally optimal one exists as well. 14 There are obviously other ways to make a consistent comparison. For example one could always select the most preferred equilibria for the designated type next period that plays the game. This type of comparison is avoided in order to find and isolate reasons other than simple coordination failure as the basis for reform. 16

18 Definition 4 Type θ is recursively self-selected (RSS) if for all ω, all ˆθ, and all x F (ω,θ,x, θ) F (ω, ˆθ, x, θ) Recursive self-selection describes an implicit incentive constraint on the institutional types. A recursively self-selected θ is an institutional type that would never delegate decision authority in the public sector to another type, regardless of the realized state and continuation payoff. Theorem 2 Consider a dynamic political game (DPG) in which C is rationalized by a single valued, dynamically consistent class of political rules. Suppose that the DPG admits at least one equilibrium. Then, if a political state θ is recursively self-selected, then there exists an equilibrium in which θ is institutionally stable. As with the inessentiality condition, recursive self selection is clearly sufficient, but is too restrictive to constitute a necessary condition for institutional stability. Unfortunately, necessary conditions are elusive. Consider, for example, the following restrictive condition. Definition 5 Type θ is recursively self-denied (RSD) if there exists a ˆθ θ such that for all ω and all x, F (ω, ˆθ, x, θ) > F (ω,θ,x, θ) In this instance, θ is said to be recursively denied by ˆθ. A recursively self-denied θ is a type that never delegates decision authority to itself in the subsequent period s game in any state and for any continuation. Surprisingly, a recursively self-denied type, θ, need not admit reform. Consider, for example, a dynamic political game with three types: θ 1,θ 2,θ 3. Suppose that θ 1 is recursively self-denied by type θ 2, which, in turn, is recursively self-denied by type θ 3. It may, nevertheless, be the case that F (ω, θ 1,x, θ 1 ) > F (ω, θ 3,x, θ 1 ), ω x Suppose in some institutionally optimal equilibrium, that µ(ω, θ 2 )=θ 3 for all ω. Ifδ is close enough to one, then it can easily be shown that type θ 1 is stable. Basically, type θ 1 does not delegate to θ 2 because θ 2 is expected to delegate to θ 3. The failure to induce reform is due the fact that recursive self-denial is not transitive. Type θ 1 would delegate to θ 2 if it were certain that θ 2 were stable. Generally, θ 2 cannot commit not to delegate further. The lack of commitment leads to stability of θ 1. 17

19 In a related paper, Jack and Lagunoff (2003) construct parametric examples with this time consistency problem: an elite chooses very conservatively to expand the voting franchise precisely because a larger franchise subsequently expands rights farther than is desirable for this elite. In light of these problems, the following result gives sufficient (though less than satisfactory) conditions for reform. Theorem 3 Consider a dynamic political game (DPG) in which C is rationalized by a single valued, dynamically consistent class of political rules. Suppose that the DPG admits at least one equilibrium. The following hold. (i) If a political state θ is recursively self-denied by another type ˆθ which is itself recursively-self selected, then there exists an equilibrium in which θ admits institutional reform. (ii) If a political state θ is recursively self-denied by a every other type, then again there exists an equilibrium in which θ admits institutional reform. The result provides two scenarios in which reforms can occur. First, a type admits reform if it is self-denied by a stable type. In that case, the aforementioned commitment problem does not arise. Second, a type admits reform if it is self-denied by all other types. In this case, the public sector public sector is completely ineffective. So much so, that any form of delegation is preferred by type θ. In both scenarios, future private sector decisions compensate for the loss of control by type θ in the policy arena. The best outcome from type θ s point of view is one where it delegates decision authority to another type. In order to induce the preferred alternative, the social planner must buy-off the private sector by delegating future decision authority to another type ˆθ θ. The delegation represents a strategic commitment in which a type cedes control over public decisions to gain more favorable treatment from decisions over which it is has no direct control. Hence, Player θ designates a new player, ˆθ, in order to elicit the desired response from the private sector. Theorems 2 and 3 unify a number of results in the literature on progressive expansion of voting rights. Under the external conflict explanation of Acemoglu and Robinson (2000), the voting franchise historically was extended by an elite to head off social unrest. In their model there are only two types: a restricted voting franchise of the elite and a full, universal manhood suffrage. Public decisions in the restricted franchise may be undercut by the threat of revolt. Consequently, the political state of the restricted franchise was self-denied by the stable state of the universal franchise. According to the internal conflict explanation of Lizzeri and Persico (2002), rights are extended to gain support when there is ideological or class conflict among the elite. Jack and Lagunoff (2003) construct an example of this in which taxes sustain investment in public literacy. Conflict between the median voter within the elite, and the population median individual s private investment in literacy leads to an expansion of voting rights. An excessively 18