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Multidimensional voting model in describes stability of coalitions structure, equiliberium, probablistic voting models, public finance and market equiliberium.
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A Thesis submitted to the Department of Economics in fulfillment of the requirements for the degree of Doctor of Philosophy (PhD)
Declaration
I, Valerio Dotti confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis.
Valerio Dotti
Firstly I would like to thank my PhD supervisors, Ian Preston and Aureo de Paula for their constant guidance and advice throughout my PhD. Antonio Cabrales and John Roemer have provided insightful and helpful guidance and advice at various points throughout my PhD, for which I would like to state my appreciation. I would also like to thank Marco Bassetto, V. Bhaskar, Christian Dustmann, Antonio Guarino, Johannes H¨orner, Tobias K¨onig, Suehyun Kwon, Guy Laroque, Gilat Levy, Konrad Mierendorff, Nikita Roketskiy and Larry Samuelson for their helpful comments and suggestions. I gratefully acknowledge financial support from ESRC and UCL. This PhD thesis could not have been accomplished without the support and encouragement of my parents, so I would like to thank them. Lastly, I would like to thank Sanghmitra for her constant support, patience and encouragement.
Chapter 1 Introduction
dramatic in the early Social Choice literature. Specifically, Arrow’s General Impossibility Theorem (Arrow 1950) implies that one cannot ensure the existence of a stable social preference ordering that satisfies some minimum properties for unrestricted individual preference orderings. Moreover, the so-called Gibbard-Satterthwaite Theorem (Gibbard 1973, Satterthwaite 1975) states that any non-dictatorial deterministic voting rule that allows all potential alternatives to be chosen is prone to tactical voting. This means that a voter with full knowledge of how the other voters are to vote and of the rule being used has an incentive to vote in a way that does not reflect his or her preferences. This result implies that such voting rule is manipulable, meaning that the incentive compatibility of a voter behavior that reflects the true voter preferences cannot be ensured. The traditional literature - which I extensively survey in the next chapter - has tackled these problems by imposing restrictions on voters’ preference orderings. Specifically, the early contributions by Black (1948, 1958) and Downs (1957) have shown that, if voters preferences over electoral outcomes satisfy some ordinal conditions such as Single-Peakedness, then a Condorcet winner exists and corresponds to the platform that is preferred by the median individual in the population. Thus, in a simple deterministic model of electoral competition - usually referred as Downsian model - it is possible to derive a simple and clear characterization of the policy outcome that prevails in equilibrium. This result, known as Median Voter Theorem, proved extremely popular in the Political Economy literature. The main reason is that it can be adopted to derive testable implications about the relationship between some characteristics of the voting population and the policy outcome, abstracting from other features of the political process. The shortcoming of this approach is that the conditions under which the Median Voter Theorem holds are extremely restrictive in some cases, because a Condorcet winner may not exist. Specifically, two of these cases have been extensively studied in the literature because of their relevance in several economic applications. This thesis focuses primarily on these two cases. The first is the (i) analysis of electoral competition over a multidimensional policy choice domain. A review of the literature about this topic is provided in chapter 2. The second case concerns problems in which (ii) voter preferences exhibit non-convexities, and it is extensively described chapter 5. In particular, I study collective choices over policies consisting of the public provision of certain goods and services, for which private alternatives are also available to voters on the market. A typical example of this kind of policy is public intervention in education, which is the object of the analysis in chapter 5. The contribution of this thesis to the literature in Political Economy is twofold. The first contribution is methodological. Specifically, I provide a set of theoretical tools to analyse questions in which either the multidimensionality of the policy space, or the existence of non-convexities in voters’ preferences (or both) play a crucial role in shaping the economic trade-offs. Moreover, the proposed theoretical tools preserve - at least to some extent - the strong predictive power that the Downsian framework exhibits in presence of single-peaked preferences.
Introduction
The second contribution is the application of such theoretical tools to analyse some of the most important policy decisions that democratic countries face in this decade, which correspond to some of the most popular topic of research in the literature. In detail, the thesis is structured as follows. In chapter 2 I propose a new model of electoral competition that possesses very useful properties to tackle problems of type (i), and I compare strength and weaknesses of this new theoretical framework with the ones that characterize the alternatives in the literature. In the following chapters I present the applications of the theoretical analysis. First, in chapter 3 I study the relationship between income inequality and size of the government. Specifically, I extent the analysis by Meltzer and Richard (1981) to the case in which more than one public spending policy is available to voters. Second, in chapter 4 I analyse the effects of population ageing on collective choices over immigration policies and fiscal policies. A different approach underpins the analysis in chapter 5, in which I study the relationship between income inequality and public investment in education. One has to tackle both issues of type (i) and of type (ii) to analyse such relationship, which implies that the theoretical framework proposed in chapter 2 cannot be successfully employed in this case. Thus, I adopt a more traditional model of electoral competition, namely a Probabilistic Voting Model. This approach proves useful to tackle this specific question, but it can only deliver a limited set of analytical results, because of reasons that I describe in section 2.5.5. A common feature of chapters 3, 4 and 5 is to show that the extension of the analysis proposed by existing studies to multiple endogenous policy dimensions can radically change the trade-offs faced by voters. Such exercise allows me to identify the relevant economic channels that drive voters’ choices and to shed light on some patterns observed in the data regarding each specific question. Lastly, in chapter 6 I highlight some of the achievements and of the shortcoming of the analysis presented in the previous chapters, and I identify some directions of future research. The theoretical tools proposed in this thesis to address an heterogeneous set of questions are examples of a large and growing literature that explores possible alternatives to the traditional Downsian framework. An extensive review of this literature is provided in chapter 2 of this thesis. An important contribution of this thesis is to show that - in order to capture the correct trade-offs that underpin several important questions in the Political Economy literature - it is crucial to relax some of the restriction that are commonly imposed in studies that employ the Downsian model, in particular the assumption of a unidimensional of policy space. This is shown to be the case for the three questions that I analyse in chapters 3, 4 and 5 of this thesis. Most papers in the literature - because of the technical issues mentioned above - address these and other questions abstracting from the role played by the interaction among multiple endogenous policy dimensions. As a result, such studies often overlook some crucial trade-offs that underpin the choices made by voters and obtain empirically controversial predictions. In this thesis I show that, if one possesses a tool to tackle the problem of multidimensionality, then the
I investigate the equilibrium properties of a deterministic voting model in which the policy space is multidimensional and politicians have limited ability to commit to platforms. Specifically, a politician running alone can only offer his ideal policy. Voters can form coalitions to increase the commitment ability of politicians. Coalition structures are required to be stable in any equilibrium. This analysis is useful to answer a large class of Political Economy questions in which the multidimensional nature of the policy is crucial to model voters’ trade-offs. I show that, under suitable restrictions on voter preferences, a Median Voter Theorem holds. The main result consist of two monotone comparative statics results for the equilibrium policy outcome. Moreover, I characterize the types of coalitions that can be stable in an equilibrium. Lastly, I show how this model relates to popular alternatives in the literature, and that the main result is robust to a variety of different assumptions about the notion of stability.
JEL classification: D72, C71.
Keywords: Multidimensional policy space, Coalitions, Median Voter.
The model of electoral competition proposed by Downs (1957) is a simple and useful tool that has proved to be extremely successful in the Political Economy literature. Under suitable assumptions on voter preferences the model delivers a very sharp result known as the Median Voter Theorem. Such result states that in the unique equilibrium the median voter is a Condorcet Winner, i.e. he cannot be defeated by any other individual candidate, and that the policy that is chosen in equilibrium is the one that is most preferred by the median voter. The theorem implies that the levels and the comparative statics of the equilibrium policy outcome reduces to the ones of a single pivotal individual. As a consequence, the predictions of the model regarding the comparative statics of the equilibrium policy outcome are typically easy to derive and interpret. These desirable features made this framework very popular in the literature in Political Economy, such that the median voter result has been applied to an incredible
Chapter 2 Generalized Comparative Statics for Voting Models
variety of questions. Examples are the analysis of the relationship between income inequality and size of governmental intervention in redistributive policies (Meltzer and Richard, 1981), the study of the determinants of immigration policies (Razin and Sadka, 1999), of the extent of taxation on different tyes of income (Bassetto and Benhabib, 2006), and many more. Unfortunately the conditions under which this useful result of the Downsian model holds become extremely restrictive if the policy space is multidimensional. Specifically, the preference restrictions required in order to ensure the existence of a Condorcet Winner over a multidimensional choice domain are so demanding that the adoption of such framework to any relevant economic question becomes almost impossible (see section 2). The aim of this paper is to provide a tool that shares the desirable features of the Downsian model regarding the characterization of the equilibrium policy outcome, but that can be easily applied to problems in which the policy space is multidimensional. Following a successful stream of literature (Roemer 1999, Levy 2004, 2005), the approach adopted in this paper in order to achieve the target is to recognize that the political interaction in democratic countries involves a number of actors (voters, politicians) and institutions (e.g. political parties) that interact strategically, and that such interaction is affected by commitment issues. Specifically, I assume that individual politicians have limited ability to commit to specific policy platforms, because they cannot write binding contracts with their electors. As a result, a single citizen-candidate can only credibly propose his own ideal policy. Nevertheless, the ability of political agents to commit to platforms other than their own ideal points is enhanced by the existence of institutions, such as coalitions or political parties, that can ensure the credibility of commitment through internal self-enforcing agreements. In line with most of the recent literature (e.g. Levy 2004, 2005, Roemer 1999), I assume that a coalition can propose any policy in the Pareto set of its members. I do not explicitly model the process of coalition formation, but I require coalitions to be stable in equilibrium. The notion of stability adopted - that I define formally in section 3 - admits a simple intuitive description. Specifically, a coalition A is stable if and only if it can credibly commit to a policy platform x such that, given the policy proposed by other coalitions, there is no other platform x′^ in the policy space that possesses the following features: (i) x′^ makes each member of a subcoalition A ′^ ⊆ A strictly better off with respect to x; (ii) x′^ is in the Pareto set of the subcoalition A ′; (iii) there is no policy x′′^ in the Pareto set of the complementary subcoalition A \A ′^ that can defeat x′^ by majority voting. This informal description suggests that this is a realtively weak concept of stability. The intuition is that a coalition is not stable if, for any policy platform that this coalition can put forward, there is a subcoalition that can deviate and credibly propose another platform that makes all its member strictly better off, and such that the remaining members of the original coalition do not have access to any alternative that can discourage such deviation. The main results of the paper are robust to stronger stability requirements, such as coalitions being merger-proof, or less extreme assumptions about
