Parliament Shapes and Sizes: Parliament Shapes and Sizes...PARLIAMENT SHAPES AND SIZES RAPHAEL...

PARLIAMENT SHAPES AND SIZES

RAPHAEL GODEFROY and NICOLAS KLEIN∗

This paper proposes a model of Parliamentary institutions in which a society makesthree decisions behind the veil of ignorance: whether a Parliament should compriseone or two chambers, what the relative bargaining power of each chamber should be ifthe Parliament is bicameral, and how many legislators should sit in each chamber. Wedocument empirical regularities across countries that are consistent with the predictionsof our model. (JEL D71, D72)

I. INTRODUCTION

Parliamentary institutions vary widelyacross countries. For instance, the IndianLower Chamber—Lok Sabha—has 543 to 545seats, and the Indian Upper Chamber—RajyaSabha—has 250. The U.S. Congress has 535seats: 435 in the House of Representatives and100 in the Senate. The Luxembourg Parliamenthas 60 legislators, in a single chamber.1 As of2012, there were 58 bicameral and 110 unicam-eral systems recorded in the Database of PoliticalInstitutions (DPI; Beck et al. 2001 [updatedin 2012]).

The main questions that arise when design-ing Parliamentary institutions are fundamentallyquantitative: should there be one or two cham-bers, what should be their respective bargainingpower, what should be the size of Parliament, andso on. James Madison, in the “Federalist No. 10,”postulates a concave and increasing relationship

∗The authors gratefully acknowledge financial supportfrom the Fonds de Recherche du Québec—Société et Culture.The second author also gratefully acknowledges financialsupport from the Social Sciences and Humanities ResearchCouncil of Canada.Godefroy: Assistant Professor, Department of Economics,

Université de Montréal and CIREQ, Montréal, QuebecH3T1N8, Canada. Phone 514-343-2397, Fax 514-343-7221, E-mail [email protected]

Klein: Assistant Professor, Department of Economics,Université de Montréal and CIREQ, Montréal, Que-bec H3T1N8, Canada. Phone 514-343-7908, Fax514-343-7221, E-mail [email protected]

1. Throughout the paper, we use the term “legislator” torefer to a member of Parliament. If a Parliament is unicameral,we refer to its chamber as the House or the Lower Chamber,indifferently. If there is a second chamber, we refer to it as theSenate or Upper Chamber, indifferently.

between the population and the number of repre-sentatives.2 In “Federalist No. 62,” he argues infavor of an Upper Chamber as a safeguard againstthe errors of a large Lower Chamber.3

However, little is known on the institutionalregularities of Parliaments across countries.Indeed, only two stylized facts have been docu-mented: a linear relationship between the log ofthe size of the population and the log of the sizeof Parliament (Stigler 1976) and an increasingrelationship between the population size andthe probability to have a bicameral Parliament(Massicotte 2001).

The purpose of this paper is twofold: to pro-pose a model of Parliament design and to doc-ument empirical institutional regularities acrosscountries, following the lead of the predictions ofthe model. One of our contributions is that oursimple model generates predictions concerningvariables that have an unambiguous equivalent inthe data we observe.

2. “In the first place, it is to be remarked that, howeversmall the republic may be, the representatives must be raisedto a certain number, in order to guard against the cabals of afew; and that, however large it may be, they must be limitedto a certain number, in order to guard against the confusion ofa multitude. Hence, the number of representatives in the twocases [will be] proportionally greater in the small republic”(Madison 1788a).

3. “The necessity of a senate is not less indicated by thepropensity of all single and numerous assemblies, to yieldto the impulse of sudden and violent passions, and to beseduced by factious leaders into intemperate and perniciousresolutions” (Madison 1788d).

ABBREVIATIONS

DPI: Database of Political Institutionsi.i.d.: Identically and Independently Distributed

1

Economic Inquiry(ISSN 0095-2583)

doi:10.1111/ecin.12584© 2018 Western Economic Association International

2 ECONOMIC INQUIRY

There are two stages in the model: a Constitu-tional stage and a Legislative stage. At the Con-stitutional stage, society, which is made up of adiscrete population of individuals, designs a Con-stitution by writing three decisions into the socialcontract: whether a Parliament should compriseone or two chambers, what the relative bargain-ing power of each chamber should be if the Par-liament is bicameral, and how many legislatorsshould sit in each chamber. These decisions aremade behind a veil of ignorance, that is, beforethe members of society know their own prefer-ences. As everyone is identical behind the veil ofignorance, they will unanimously agree that thegoal should be to maximize the expected utilityof the representative agent. They know that eachindividual’s utility is the opposite of the quadraticdifference between the policy adopted by the Par-liament and the individual’s bliss point, whichare both real numbers, minus the costs associatedwith the functioning of Parliament.

We assume that the population is partitionedinto parties and that two types of issues may arise:partisan and nonpartisan. For nonpartisan issues,individuals’ bliss points are identically and inde-pendently distributed (i.i.d.), whereas for partisanissues, all members of a party share the samebliss point, the party’s bliss point. Parties’ blisspoints are i.i.d. across parties. Individuals do notknow their partisan affiliations at the Constitu-tional stage. Nor do they know which type ofissue may arise, the distribution of shares of par-ties’ members in the population, or the realizeddistribution of bliss points in the population oramong legislators.

At the Legislative stage, the Parliamentdecides on a policy. To formalize society’s infor-mation about Parliament’s decision process, weassume that only one issue arises, and that thepolicy adopted for that issue is the weighted aver-age of the policies that maximize the aggregateutility of each chamber, subject to some error.4

We consider two allocation systems of Parlia-mentary seats: a proportional representation anda nonproportional representation system. In theproportional representation system, the distribu-tion of seat shares across parties is equal to thedistribution of shares of partisans in the popula-tion. In the nonproportional system, these distri-butions may differ. In an extension (Appendix C),we show that the distribution of legislators in theproportional representation system is obtained

4. In a unicameral system, the policy adopted is the policyproposed by the unique chamber.

from a two-stage game. In the first stage, par-ties may form coalitions to maximize the share ofseats they obtain in a proportional single-districtelection. In the second stage, each individualcasts one vote, either for a party or for a coalitionof parties, to maximize his expected utility, whichis a function of the policy adopted, under per-fect information about legislators’ partisan affili-ations, but imperfect information about the exactvalue of their bliss points.5,6

The problem at the Constitutional stageinvolves two trade-offs. On one hand, there is atrade-off between the unit cost of a member ofParliament and the marginal benefit of having alarger Parliament to lower the risk that the policywill be decided by legislators whose preferencesare far from those of the population at large.One could see this trade-off in terms of externalcosts and internal or decision-making costs.(External costs are the costs that individuals haveto bear as a result of others’ decisions wheneveran action is chosen collectively. Internal costsstem from an individual’s participation in anorganized activity, such as legislative bargaining,see Buchanan and Tullock 1962, Chapter 15.)The former decrease in expectation as the sizeof Parliament increases; indeed, the larger theParliament the more precisely it will estimate theaverage bliss point of the population. The lattercosts, by contrast, will increase as Parliamentbecomes larger; indeed, the more numerous theassembly, the costlier it will be to ensure itsproper functioning. In our model, we assume thatthe marginal internal cost of parliamentarians isconstant. On the other hand, there is a trade-offbetween the unit cost of a chamber and thebenefit of instituting a Senate to mitigate thenegative impact of the error term in the adoptedpolicy. That error term formalizes in the simplestway possible the common point in defensesof bicameralism, such as Madison’s, which isthat members of the Lower Chamber might not

5. Any game using a nonproportional voting systemrequires many more assumptions than the game in which vot-ing is at-large and proportional. For the latter, we would needto specify the number of districts, the size of each district(which may not be uniform in practice), the distribution ofpartisans in each district, the possibility for parties to formcoalitions within and across districts, and so on. We do notpropose such a model; instead, we make an assumption on thesystem of allocation of seats that is consistent with an empir-ical analysis of nonproportional voting systems.

6. The conclusions of the model would be the same ifwe considered an electorate whose votes are determined bypartisan loyalty, which is the main determinant of votingbehavior according to Achen and Bartels (2016), rather thanpolicy preferences.

GODEFROY & KLEIN: PARLIAMENT SHAPES AND SIZES 3

adopt the best policy according to some measureof welfare.

The model provides several predictions. First,the log of the size of Parliament increases linearlywith the log of the size of the population, whichis consistent with one of the two stylized factsmentioned above.

Second, the number of legislators ought not todepend on the unicameral or bicameral structureof the legislature.

Third, the relative bargaining power of achamber in a bicameral system should optimallybe equal to the share of legislators belonging tothis chamber.

Fourth, countries with a larger population aremore likely to have a bicameral legislature, theother stylized fact mentioned above. Fifth, themodel predicts no impact of the factors that affectthe size of Parliament, except the size of the pop-ulation, on whether a country has a unicameral orbicameral Parliament.

We find that the number of legislators dependson whether the system is proportional or not. Oursixth prediction is that in a proportional votingsystem, the makeup of the partisan structure ofa country does not affect the number of legisla-tors. Our seventh prediction is that, in nonpro-portional systems, by contrast, we predict that asthe partisan structure becomes more dispersed, asmeasured by the Herfindahl–Hirschman index,the number of legislators ought to rise. To derivetestable predictions, we compute the optimalnumber of seats for a specific nonproportionalsystem in which a fixed number of seats aregranted to the party with the biggest share ofpartisans in the population, and the remainingseats are randomly distributed across all parties,including the largest party, according to theirshare of partisans in the population.

Empirically, we find that all these predic-tions are consistent with the results of a seriesof estimations that we conduct on a sample of75 nonautocratic countries for which we havedetailed political information over the period1975 to 2012. For the average bargaining powerof Upper Houses across bicameral countries, weuse the estimation provided in Bradbury andCrain (2001).

Our model also provides specific predictions,which are borne out by the data, concerning thecoefficient of the Herfindahl–Hirschman indexof partisan fractionalization as well as that ofother terms.

Finally, we provide two tests of the assump-tion we use to compute the size of Parliament

in nonproportional representation systems. First,we estimate that, on average across observationsat the country/party/election year level, the partyranked first by decreasing share of votes obtainsaround 10% of seats in the Lower Chamber. Sec-ond, we find that the rest of the seats are allo-cated across all parties, including the party rankedfirst, so that the share of seats and the share ofvotes of a party are the same, regardless of theparty’s rank.

The rest of the paper is set up as follows:Section II reviews related literature; Section IIIpresents the model, which is used to derive thepredictions exhibited in Section IV; Section Vpresents the empirical analysis; Section VI con-cludes. Descriptive statistics are in Appendix A.Appendix B presents tests of the assumptionsused in nonproportional representation systems.Appendix C presents a two-stage voting gamethat substantiates the reduced-form analysis ofproportional systems.

II. LITERATURE REVIEW

Our contribution relates to the literature onendogenous political institutions.7 The distinc-tion between a Constitutional stage, in whichdecision procedures are created, and a subsequentlegislative stage, in which a policy is chosen, wasstudied in Romer and Rosenthal (1983). WhileRomer and Rosenthal (1983) find a certain una-nimity rule to be optimal in their setting, Aghionand Bolton (2003) find that it is optimal to requirean interior majority threshold for a change in thestatus quo policy.8

The seminal theory concerning the size of leg-islatures is the cube root formula proposed byTaagepera (1972), which holds that the num-ber of legislators should equal the cube rootof the population of a country. This formula isdesigned to prevent excessive disproportionalityin representation—see Lijphart (2012). Theoret-ically, it results from the minimization of thenumber of communication channels for assemblymembers, who need to communicate with their

7. Stigler (1992) claimed a role for economists inthe study of legal institutions, writing: “Understanding thesource, structure, and evolution of a legal system is the kind ofproject that requires skills that are possessed but not monopo-lized by economists, for it is in good part an empirical projectaddressed to rational social policy.”

8. The crucial difference is that Aghion and Bolton(2003) assume that there is a deadweight loss associated withimplementing monetary transfers designed to compensate thelosers of the reform being proposed.

4 ECONOMIC INQUIRY

constituents while also communicating with theirfellow assembly members. Empirically, this for-mula underpredicts the sizes of Parliaments.

Auriol and Gary-Bobo (2012) adopt amechanism-design approach to the design ofParliamentary institutions. In their model, theprincipal (the “Founding Fathers”) does notknow the distribution of preferences over aone-dimensional policy choice to be made, andknows that no agent in society will know them.Rather, the “Founding Fathers” have a diffuseprior over the set of possible distributions ofpreferences. Furthermore, there is an execu-tive branch, made up of a randomly chosensingle agent who is the residual claimant ofall decision rights not specifically delegated tothe legislative branch. The legislative branch,by contrast, is made up of n randomly chosenagents, whose role consists of revealing theirpreferences truthfully. Truthful revelation iseffected by a Vickrey-Clarke-Groves mechanismapplied to the agents making up the legislativebranch. In our model, by contrast, there is noexecutive branch, and the legislators’ salaries areexogenously given. This simpler model allows usto derive additional predictions concerning thestructure of Parliaments, such as their bicameralor unicameral nature, the impact of differentvoting systems, and the effect of preferences thatare homogeneous within given groups.

We do not explicitly model the decisionprocess in Parliament, which is the focus ofthe legislative bargaining literature, starting withBaron and Ferejohn (1989), Baron and Diermeier(2001), and Diermeier and Merlo (2000).

As the predictions of these bargaining modelsoften depend on the fine details of the bargainingprotocol, which varies widely across countries,we instead use a spatial model of policy pref-erences and assume that each chamber adoptsas its policy the average bliss point of its mem-bers. This policy maximizes the sum of leg-islators’ utilities, and thus corresponds to thebehavior they would optimally adopt in a set-ting in which utilities are transferrable.9 In con-trast to, for instance, the analysis in Tsebelis andMoney (1997), our legislators do not anticipatethe impact of their decisions on the bargainingprocess with the other chamber in a bicam-eral system.

The literature on bicameralism has devotedsome attention to the impact of a second chamber

9. See also our discussion in Section III.B.

on the process of legislative bargaining and sub-sequent policy choices. Ansolabehere, Snyder,and Ting (2003) analyze a situation where theHouse has one member per district while the Sen-ate has one member per state. As in Baron andFerejohn (1989), the object of legislative bargain-ing is to divide a fixed budget of resources. Housedistricts are equal in population size, while statesmay encompass several House districts. Legis-lators in both chambers are responsive to theirrespective median voter. Both chambers have toagree to a proposed division of the resources, andboth vote by majority rule, but only representa-tives can propose a bill. They show that smallerstates, which are over-represented in the Senate,do not get a higher per capita share of spend-ing. The reason is that a minimum winning coali-tion in the House carries a majority in the Senateas well. However, small state bias reappears ifthere are supermajority rules in the Senate, Sen-ators have proposal power, or goods are lumpy(i.e., they cannot be targeted toward a single dis-trict). Kalandrakis (2004) analyzes a variant ofthis model where resources can only be targetedat the state level, and Senators can be recognizedas proposers as well. He finds that supermajoritiesmay occur in equilibrium, but they only ever doin one chamber at a time. Parameswaran (2018)adopts this model by letting the goods be targetedat the House-district level. He finds that smallstates may actually fare worse with a Senate inwhich they are over-represented than in a uni-cameral system without malapportionment. Thereason is that, if a Senator from a big state is rec-ognized as the proposer, he can at no cost buyoff all the House members from his state. Thus,he needs to include fewer small state legislatorsin his winning coalition, which in turn makes itless likely that representatives from a small statewill be included in the winning coalition. Knight(2008) investigates empirically the role of repre-sentation for the division of funds in the UnitedStates. He finds that small states receive a largershare of appropriations originating in the Senatethan of those originating in the House. He dis-tinguishes between the proposer-power channelresulting from a state’s representation on the rele-vant committees and the vote-cost channel result-ing from the proposer’s need to build a winningcoalition. Snyder, Ting, and Ansolabehere (2005)show that when votes are weighted in a unicam-eral setting, a legislator’s ex ante expected shareof the resources equals his voting weight. Thus,the biases introduced by a malapportioned sec-ond chamber could in principle be replicated by


a unicameral system with weighted voting. Vespa(2016) runs a laboratory experiment testing theimpact of weighted voting in a unicameral settingas compared to bicameralism with a malappor-tioned second chamber, confirming the theoreti-cal predictions that there will be a small state biaswith weighted voting. In a bicameral system, thisbias appears if and only if Senators have proposalpower, as predicted by Ansolabehere, Snyder, andTing (2003).10

Facchini and Testa (2005) study the inter-action of legislators with lobbying groups in abicameral setting. Rogers (1998) also formalizesan informational justification for bicameralism.He assumes that larger chambers have lower costsfor information acquisition, giving them a largerfirst-mover advantage.

Our paper complements these papers. Insteadof looking in detail at a model that tries to formal-ize the U.S. Congressional system in a CollectiveBargaining framework, we use a reduced formmodel to formalize as simply as possible the mostuniversal motivation in favor of a Senate, which isto reduce the probability of legislative errors. Wedo not attempt to explain where such error comesfrom. Instead, we formalize in greater detail whatthe Parliament designers’ objectives may be. Forthe same reasons, our empirical analysis aims lessat explaining the policies adopted by a specificParliament, such as the U.S. Congress, than atexplaining what may have motivated the specificdesign features (number of seats and number ofChambers) of Parliaments across countries in thefirst place.

We also do not consider issues pertaining tothe acquisition or transmission of informationwithin the legislature, all legislators being per-fectly informed of their respective bliss points.By contrast, an early investigation of legislators’incentives to acquire information is provided byGersbach (1992). Austen-Smith and Riker (1987)analyze legislators’ incentives to reveal or con-ceal private information they may have.

Our paper is also related to the literature onelectoral systems. The argument most often citedin favor of proportional systems is that they leadto a more faithful representation of a popula-tion’s opinions in Parliament, whereas pluralitysystems are more likely to obviate the need formultiparty coalitions, thus leading to greater

10. See also Buchanan and Tullock (1962, Chapter 16),Riker (1992), Diermeier and Myerson (1994), Diermeier andMyerson (1999), Rogers (1998), Bradbury and Crain (2001),Tsebelis and Money (1997), and the references in thesepapers.

stability (Blais 1991; Grofman and Lijphart1986). Aghion, Alesina, and Trebbi (2004)show that countries with greater ethno-linguisticfractionalization are more likely to have plu-rality voting systems, which are interpreted asa means of achieving greater insulation of thepolitical leadership. While also minimizing therole of strategic behavior in the establishment ofpolitical parties, Lijphart (1990) finds that thevoting system has little impact on the number ofparties. Finally, a few studies link voting systemsto voter turnout (Herrera, Morelli, and Nunnari2016; Herrera, Morelli, and Palfrey 2014). Ourpaper contributes to this literature by relating thestructure of Parliament to the voting system andto partisan fractionalization.

III. MODEL

A. Setup

We consider a population of M individuals.At the Constitutional stage, they want to maxi-mize the representative agent’s expected utilitythrough their choice of the setup of Parliament.They choose whether Parliament will be unicam-eral or bicameral, and set the number of mem-bers of the House nH . If they choose a bicameralParliament, they also set the number of mem-bers of the Senate nS, and the bargaining powerof the House and the Senate, denoted 𝛼 and1− 𝛼, respectively.

Preferences. An individual i’s utility is:

ui (x) = −(x − xi

)2 − ((NC + nc) ∕M)

where x∈ℝ is the policy to be adopted by the Par-liament, xi ∈ℝ is i’s bliss point, N ∈ {1, 2} is thenumber of chambers, C the unit cost of a cham-ber, n the number of members of Parliament, andc the unit cost of a member of Parliament.11

The first term represents the payoff obtainedfrom the policy adopted, and the second termrepresents the per capita contribution to the fund-ing of Parliament.12

Distribution of Bliss Points. The population ispartitioned into partisan groups or parties, and

11. The unit cost of a chamber may comprise its main-tenance cost and the potential rent of the building where itsmembers meet. The unit cost of a member of Parliament maycomprise his salary.

12. While we make the assumption of quadratic util-ity for the purpose of tractability, one could interpret thequadratic utility function as a second-order Taylor approx-imation of a more general utility function. Indeed let ui(x)be agent i’s smooth utility function depending on the policy

6 ECONOMIC INQUIRY

two types of political issues may arise: partisanand nonpartisan. For partisan issues, all membersi of a partisan group share the same bliss point;bliss points are drawn independently across par-ties. For nonpartisan issues, bliss points are drawnindependently across individuals.

We assume that the distribution from whichbliss points are drawn, either parties’ bliss pointsfor partisan issues or individual bliss points fornonpartisan issues, is continuous and has a vari-ance 𝜎2 ∈ (0,∞).

Policy Adopted at the Legislative Stage. Legisla-tors have the same type of preferences as the restof the population. Once in Parliament, legislatorsadopt a policy for the unique issue that has arisen.With probability q ∈ [0, 1], the issue that arisesis partisan.

If Parliament members are numbered 1through nH + nS, with members of the Housenumbered first and members of the Senate num-bered second, we assume that the policy adoptedfor this issue is:

x∗ = 𝛼

(1

nH

nH∑k=1

xk + Z̃H

)(1)

+ (1 − 𝛼)

(1nS

nH+nS∑k=nH+1

xk + Z̃S

)where xk is legislator k’s bliss point for theissue discussed, 𝛼 represents the bargainingpower of the House, and Z̃X is an error term forchamber X.

The policy adopted is the weighted averageof the policies that maximize the sum of the

decision x. By taking the Taylor expansion of ui at agent i’spreferred policy xi we can write

ui (x) = ui

(xi

)+(x − xi

)u′i(xi

)+(x − xi

)2

2u′′i(xi

)+ o((

x − xi

)2).

If the population taken into account at the Constitutionalstage is homogeneous so that x will be close enough to xi, theterms o((x − xi)

2) will be small enough to be neglected. As,by definition, ui assumes its maximum at xi, u′i (xi) = 0 ≥ u′′i(xi). If individual i is risk averse, u′′i (xi) < 0, and maximiz-

ing 𝔼[ui

]will be tantamount to minimizing 𝔼

[−(x − xi

)2].

Anecdotal historical evidence suggests that Parliamentaryinstitutions have often been ushered in by and for groups oflike-minded people, which is consistent with the fact that thedegree of population homogeneity is a critical determinant ofthe size of a nation (Alesina and Spolaore 2005).

utilities of the members of each chamber, subjectto some error, where the weight is the bargainingpower conferred to each chamber at the Constitu-tional stage.

We assume that the error term Z̃X is indepen-dently drawn from a distribution of mean zeroand square mean vX > 0. The error term cap-tures any difference between the policy adoptedby a chamber and the cooperatively optimal pol-icy for its members. It may be interpreted asthe result of imperfections in the deliberationprocess, which may manifest themselves in var-ious ways: it could be the probability that asubgroup of members imposes its favorite pol-icy, or that negotiations among members fail,and so on.

Information. At the time of the design of Par-liament, it is not known which issue will arise,or what the actual distribution of bliss pointsacross individuals or across legislators will be. Itis known, however, that there will be G parties.Society also has a prior belief at the Constitu-tional stage regarding the shares of these groupsin the overall population, 𝛄 =

(𝛾g

)G

g=1∈ (0, 1)G

(with∑G

g=1 𝛾g = 1). They know too how the leg-islators’ preferences map into the adopted policy(Equation (1)).

Representation in Parliament. We consider twosystems of representation: a system of propor-tional representation in which the partisan distri-bution of shares of seats in Parliament is equalto the distribution of shares of partisans in thepopulation, and a system of nonproportional rep-resentation. Legislators’ bliss points for nonpar-tisan issues are i.i.d.

B. Discussion of Assumptions

Society Sets Up a Parliament. Why would soci-ety set up a Parliament in the first place? Thisis a question broached by Buchanan and Tul-lock (1962, Chapter 15). On one hand, dele-gating decision-making powers to a Parliamentincreases the external costs, as compared to directdemocracy, because there will always be somechance that Parliamentary representation mightlead to a biased sample of population preferences.Yet, as Buchanan and Tullock (1962, paragraph3.15.6) argue, the risk of bias has to be tradedoff against the decrease in decision-making costs.Representative (as opposed to direct) democracyfacilitates collective action. Equations (2) and (3)can be interpreted as the formalization of this


very trade-off13; the benefit part in the equationscorresponds to a decrease in external costs, whilethe cost part could be interpreted as an increasein decision-making costs, as the size of Parlia-ment increases. Madison also discusses the opti-mal size of the House of Representatives, interalia, in the “Federalist No. 55” and “FederalistNo. 58,”14 see the discussion of the size princi-ple in Ostrom (2008, Chapter 5). Ostrom (2008)points out that the larger a legislative assemblybecomes, the less time individual representativeswill have to make their arguments, as only onerepresentative can speak at a time. Thus, largeassemblies will tend to be dominated by an oli-garchic leadership, as is arguably the case in theHouse of Commons in the United Kingdom. Atthe same time, a certain minimum number of rep-resentatives is needed “to secure the benefits offree consultation and discussion, and to guardagainst too easy a combination for improper pur-poses” (Madison 1788b).

While it is necessary for society to know thevariance in the distribution of bliss points at theConstitutional stage already, none of our calcu-lations depend on society’s already knowing themean or any other characteristics of this distri-bution. If the dispersion of this mean is largeenough, it will not be practical for society to seta policy at the Constitutional stage already.

Wages c Are Exogenous. We could endogenizewages by assuming that higher wages will leadto better legislators being selected, that is, vX isdecreasing in c. We refrain from doing so herebecause we do not observe vX in the data and thushave no way of ascertaining the functional formof the dependency of vX on c.

The Population Considered at the ConstitutionalStage May Differ from the Actual Population.Such a case may arise for two reasons: a fixed sizeof Parliament may still apply many years afterit was set, when the size of the population haschanged, or the preferences of parts of the popu-lation, for example, a disenfranchised population,may not be included in the Constitution Design-ers’ objective functions.

We need not interpret M as the actual popula-tion size; in fact, our predictions would continueto hold if M were a fraction or a multiple of theactual population size.

13. We are indebted to an anonymous referee for pointingout this connection to us.

14. See Madison (1788b, 1788c).

The Electoral System Is Exogenously Given. Inour model, society should always prefer a pro-portional voting system over a nonproportionalsystem. This is because we do not integrate anyoffsetting benefits of nonproportional systems,which may for instance lead to greater stabil-ity of Parliamentary majorities and may favorstronger ties between legislators and their con-stituents, thus leading to greater accountability. Inparticular, we assume that there is no relationshipbetween the voting system and the cost of Par-liament or with the error terms that arise in theadopted policy of either chamber. This assump-tion implies that the voting system is not signif-icantly correlated with the probability to have asecond chamber; we examine this implication inSection V.

In a system of proportional representation, aparty’s share of seats corresponds to its share ofthe votes. Thus, we assume that voters vote fortheir respective parties. We show in AppendixC that this is the unique subgame-perfect Nashequilibrium outcome if voters only know theirpartisan affiliations at the time of the vote. In par-ticular, it is assumed that they do not know therealizations of the various parties’ bliss pointsat the time of voting. This assumption capturesthe idea that voters will be uncertain about thedetails of the question that will arise beforethe Legislature.

Achen and Bartels (2016) argue that vot-ers choose parties and candidates on the basisof social identities and partisan loyalties, evenadjusting their policy views to match these loy-alties. Such behavior would be perfectly in linewith our assumptions.15 The observation thatthe link between voters’ partisan loyalties andtheir policy preferences was tenuous would cor-respond to a low q in our model.

The Role of the Second Chamber. Whileone rationale that is often used to justify theexistence of second chambers is to give some(over-)representation to certain minorities, ourmodel makes an implicit assumption that eitherchamber equally represents any citizen. Indeed, ifthe goal were to (over-)represent some minority,this could also be achieved by quotas or weightedvoting in a unicameral system (see Snyder, Ting,and Ansolabehere 2005 for a Baron and Ferejohn1989 bargaining type model). By the same token,

15. Of course, Achen and Bartels (2016) focus theiranalysis on the United States, which does not use proportionalrepresentation.

8 ECONOMIC INQUIRY

if the role of a second chamber were merelyto make it more difficult to pass legislation,the same could be achieved in a unicameralsystem, for example, by imposing supermajorityrules. Indeed, Cutrone and McCarty (2006) con-clude: “Regardless of the theoretical frameworkor the collective action problem to be solved,we find that both the positive and normativearguments in favor of bicameralism tend to beweak and underdeveloped. Most of the effects ofbicameralism are due primarily to quite distinctinstitutional choices, such as malapportionmentand super-majoritarianism, which correlateempirically with bicameralism. There seems tobe no logical reason why the benefits of theseinstitutions—like the protection of minorityrights and the preservation of federalism—couldnot be obtained by a suitably engineered uni-cameral legislature or through vigorous judicialreview.”

By contrast, we see the goal of bicameral-ism, which cannot easily be reproduced withina unicameral system, as providing for a second,independent, deliberative process in the law-making procedure.16 Indeed, we view represen-tative democracy as a sampling of populationpreferences, which are subsequently aggregated.Each Parliamentary chamber corresponds to suchan aggregation process. As aggregation processesare subject to error, it can be useful to have a sec-ond, independent aggregation process. Consis-tent with this view, we abstract from any designdifferences in the two legislative chambers, as itis not clear why certain peculiarities of secondchambers, for instance those designed to protecta minority or the federal structure of a country,could not be replicated in a unicameral setting.The analysis of the impact of a malapportionedsecond chamber, or malapportionment in general,which motivates much of the literature on bar-gaining in bicameral legislatures (see our discus-sion in Section II), is thus left outside the scopeof this paper.

The Legislative Bargaining Process and theDetermination of the Adopted Policy. We assumethat, at the Constitutional stage, the Constitutiondesigners anticipate that legislators within eitherChamber act cooperatively when choosing thepolicy to be adopted (see Equation (1)). While

16. As we shall see below, more heterogeneous countriesare not significantly more likely to have a bicameral Parlia-ment. One could interpret this finding as suggesting that theprimary role of a second chamber was not to enhance the rep-resentation of minorities in heterogeneous countries.

this is no doubt a somewhat naïve view of thelegislative process, there is some evidence thatlegislators will often trade their votes intertem-porally across issues (so-called log-rolling, see,e.g., Stratmann 1992). If one takes for grantedthat the utility legislators derive from the enact-ment of particular policies is transferable inthis way, legislators should always adopt thepolicy that maximizes the sum of their utilities,as we assume in Equation (1). Any failure to doso would amount to an error on their part, asit reduces the surplus to be distributed amongthem. In our opinion, this admittedly idealizedview of the legislative bargaining process has themerit of not depending on the fine details of theParliamentary procedures governing coalitionformation within and across legislative chambers(e.g., the choice of proposer, the navette systembetween chambers, etc.), which vary widelyacross countries.

IV. RESULTS

We first analyze the case of proportional vot-ing, before turning to nonproportional systems.

We neglect integer problems throughoutour analysis.

A. Proportional Representation

For Proportional Representation, we assumethat the shares of a party’s members in the popu-lation and in Parliament are equal.

Unicameral Parliament. If a nonpartisan issuearises, which happens with probability (1− q) ∈(0, 1), the policy choice is

x∗ =∑n

d=1 xd

n+ Z̃H ,

where n is the number of legislators, and(xd

)nd=1

are their bliss points. These are i.i.d. draws froma distribution with variance 𝜎2.

If a partisan issue arises, the policy choice is:x∗ =

∑Gg=1 𝛾gxg + Z̃H .

At the Constitutional stage, society maxi-mizes:

𝔼[ui

]= −𝔼

[(x∗ − xi

)2]−

C + nHc

M(2)

= −𝜎2

{q

(1 − 𝔼

[G∑

g=1

𝛾2g

])

+ (1 − q)(

1 + 1nH

)}− vH −

C + nHc

M.


Thus, when all parties are guaranteed propor-tional representation in Parliament regardless ofits size, only the nonpartisan issues matter forthe optimal size of Parliament. Indeed, the onlyrole of Parliament in our model consists in thesampling of population preferences. In particular,the size of Parliament will be independent of thepartisan fractionalization of society. The optimalnumber of members of Parliament is given by:

n∗ = n∗H = 𝜎

√(1 − q) M

c.

Bicameral Parliament. With two chambers, therepresentative agent’s ex ante expected utility isgiven by:

𝔼[ui

]= −𝜎2

{q

(1 − 𝔼

[G∑

g=1

𝛾2g

])(3)

+ (1 − q)(

1 + 𝛼2

nH+ (1 − 𝛼)2

nS

)}−(𝛼2vH + (1 − 𝛼)2 vS

)−

2C +(nH + nS

)c

M.

It thus follows that, in the unique can-didate for an interior optimum, the relativebargaining power of a chamber equals itsshare of seats: 𝛼∗ =

(n∗H∕(n∗H + n∗S

)). More-

over, n∗H = 𝜎vS

vH+vS

√(1 − q) M

cand n∗S =

𝜎vH

vH+vS

√(1 − q) M

c, so that 𝛼* = (vS/(vH + vS)),

and the overall number of members of Parliamentis equal to the number of members of Parliamentin a unicameral system,

n∗ = 𝜎

√(1 − q) M

c.

Finally, the difference in welfare between abicameral and unicameral system is:(

v2H∕(vH + vS

))− (C∕M).

If vH , vS, and C are uncorrelated with the pop-ulation of a country, this difference decreases asthe size of the population increases. This impliesthat more populous countries are more likely tohave a Senate.

B. Nonproportional Representation

In a nonproportional representation system,the shares of a party’s members in the populationand among legislators may differ.

Unicameral Parliament. First, fix a realizationof partisan shares 𝜸. Let kg be the number oflegislators who belong to party g. Given that arepresentative agent i’s utility conditional on anonpartisan issue arising is given by the sameexpression regardless of the electoral system, wehere compute his utility conditional on a partisanissue arising. We call this event 𝒬. This expectedutility is given by

𝔼[−(x∗ − xi

)2 |𝒬, 𝛄]= −𝔼

⎡⎢⎢⎣G∑

g=1

𝛾g

(xg −

G∑g′=1

kg′

nxg′

)2 |𝛄⎤⎥⎥⎦ − vH

= −G∑

g=1

𝛾g𝔼⎡⎢⎢⎣(

xg −G∑

g′=1

kg′

nxg′

)2 |𝛄⎤⎥⎥⎦ − vH

= −𝜎2G∑

g=1

𝛾g𝔼

[1−2

kg

n+

G∑g′=1

(kg′

n

)2 |𝛄] − vH ,

where we have used that the xg are i.i.d. drawsfrom a distribution with a variance of 𝜎2. We have

𝔼[−(x∗ − xi

)2 |𝒬, 𝛄]= −𝜎2

{1 − 2

n

G∑g=1

𝛾g𝔼[kg|𝛄]

+ 1n2

G∑g=1

𝔼[k2

g|𝛄]}

− vH .

We can write kg =∑n

d=1 1d,g, where 1d, g is anindicator equal to 1 if and only if the legislatorassigned to seat d (d ∈{1,· · ·, n}), is of party g,and 0 otherwise.

Thus,

𝔼[kg|𝛄] = n∑

d=1

𝔼[1d,g|𝛄]

=n∑

d=1

Pr(1d,g = 1|𝛄) = nλg (𝛄) ,

where λg (𝛄) ∶=1n

n∑d=1

Pr(1d,g = 1|𝛄) is the aver-

age probability (over Parliamentary seats) that amember of party g becomes a legislator.

By the same token,

𝔼[k2

g|𝛄] = Var(kg|𝛄) + (𝔼 [kg|𝛄])2

= Var(kg|𝛄) + n2λ2

g (𝛄) .

10 ECONOMIC INQUIRY

Now,

Var(kg|𝛄) = Var

(n∑

d=1

1d,g|𝛄)(4)

=∗n∑

d=1

Var(1d,g|𝛄)

=n∑

d=1

[Pr(1d,g = 1|𝛄) (1 − Pr(1d,g = 1|𝛄))]

= nςg (𝛄) ,

where ςg (𝛄) ∶=1n

∑nd=1

[Pr(1d,g = 1|𝛄) (1 − Pr

(1d,g = 1|𝛄))] = 1n

∑nd=1 Var

(1d,g|𝛄) is the arith-

metic mean over seats d of the variance of 1d, g,and where we have used the assumption that thepartisan affiliations of legislators are indepen-dently drawn across seats for the equality markedby *.

Thus, we have

𝔼[−(x∗ − xi

)2 |𝒬, 𝛄]=−𝜎2

{1−2

G∑g=1

λg (𝛄) 𝛾g

+G∑

g=1

λ2g (𝛄) +

1n

G∑g=1

ςg (𝛄)}

− vH .

Therefore, the size of Parliament matters forthe representative agent’s ex ante expected utilityunless Var(kg|𝜸)= 0 for all g, that is, unless thecomposition of Parliament is deterministic, as ina proportional voting system.

In order to reduce the dimensionality of theproblem of estimating Pr(1d, g = 1| 𝜸) for all (d, g)∈{1, · · ·, n} × {1, · · ·, G} (which would be neces-sary to compute the (ς1, · · ·, ςG)), further assump-tions on the seat-allocation process are needed.We shall discuss two simple sets of assumptionsbelow: (1) the case of an i.i.d. allocation of seatsand (2) a bonus for the biggest party.

(1) i.i.d. Allocation of Seats

In this case, Pr(1d, g = 1| 𝜸)=λg(𝜸) for all seatsd ∈{1,· · ·, n}. We shall furthermore assume thatthe allocation of seats is fair in the sense thatλg(𝜸)= 𝛾g for all g ∈{1,· · ·, G}. In this case,society’s objective at the Constitutional stage isgiven by

−𝜎2

{q

[1 − 𝔼

[G∑

g=1

𝛾2g

]+ 1

n

(1−𝔼

[G∑

g=1

𝛾2g

])]+ (1 − q)

(1 + 1

n

)}− vH − C + nc

M.

Thus, the optimal number of members of Par-liament is given by

n∗ = 𝜎

√Mc

√q (1 − 𝔼 [ℋ ]) + 1 − q,

where we write ℋ ∶=∑G

g=1 𝛾2g for the

Herfindahl–Hirschman index of parti-san fractionalization.

(2) Bonus to the Largest Group

Without loss of generality, the group g= 1,also denoted Party 1, refers to the party with thelargest number of members in the population atthe time of the allocation of seats. If the allo-cation of each seat were determined by the out-come of an election in a unique district attachedto that seat, if there were no coalitions formed,if individuals voted sincerely to elect membersof Parliament and if the distribution of parti-san affiliations were homogeneous across dis-tricts, Party 1 would win all seats in Parliament.Although all these assumptions may fail in coun-tries that use nonproportional voting systems,the party that represents the largest share of thepopulation may still have some advantage overother parties.

To formalize that advantage, we assume thatparty g= 1 automatically wins a proportion 1− 𝜉of seats. In the remaining proportion 𝜉 of the nseats, we make the assumption that the seat allo-cations are i.i.d., with the probability that a seatis allocated to party g equal to its share of par-tisans in the overall population. This assumptionis motivated by the fact that, in practice, at thetime of the design of Parliamentary institutions,the Parliament Designers may not know perfectlyhow individuals will migrate from one districtto another, or may not even know how districtboundaries will be defined, etc. We examine thisassumption empirically in Appendix B.

Thus, we have λ1(𝜸)= 1−𝜉(1−𝛾1), λg(𝜸)=𝜉𝛾g for g≠ 1, ς1(𝜸)= 𝜉(1−𝛾1)(1−𝜉(1−𝛾1)), andςg(𝜸)= 𝜉𝛾g(1−𝜉𝛾g) for g≠ 1. Using this in our

expression for 𝔼[−(x∗ − xi

)2 |𝒬, 𝛄], we find

𝔼[−(x∗ − xi

)2 |𝒬, 𝛄] =(5)

−𝜎2

{1 + (1 − 𝜉)2

(1 − 2𝛾1

)− 𝜉 (2 − 𝜉)

G∑g=1

𝛾2g

+𝜉

n

[2(1 − 𝛾1 (1 − 𝜉)

)−𝜉

(1+

G∑g=1

𝛾2g

)]}− vH .


Thus, society’s objective at the Constitutionalstage is given by

−𝜎2

{q

[1 + (1 − 𝜉)2

(1 − 2𝔼

[𝛾1

])− 𝜉 (2 − 𝜉)

(6)

𝔼

[G∑

g=1

𝛾2g

]+𝜉

n

[2(1 − 𝔼

[𝛾1

](1 − 𝜉)

)−𝜉

(1 + 𝔼

[G∑

g=1

𝛾2g

])] ]+ (1 − q)

(1 + 1

nH

)}− vH − C + nc

M.

Optimizing over n gives us the optimal size ofParliament,

n∗ = 𝜎

√Mc√

1−q + q𝜉[2(1−𝔼

[𝛾1

](1−𝜉)

)−𝜉 (1+𝔼 [ℋ ])

].

We note that the case 𝜉 = 1 corresponds to ourprevious i.i.d. case.

For 𝜉 = 0 and given 𝜸, there is no uncertaintyconcerning the partisan composition of the leg-islature, as party g= 1 will capture all the seats.Thus, as in the case of proportional voting, thenumber of Parliamentary seats matters only whenit comes to nonpartisan issues. It is thereforeno surprise that, in this case, the optimal sizeof Parliament corresponds to that under propor-tional representation.

Bicameral Parliament. We now examine the caseof a bicameral system in a nonproportional sys-tem. The representative agent i’s ex ante expectedutility, conditional on a partisan issue arising, isgiven by

𝔼[−(x∗ − xi

)2 |𝒬, 𝛄](7)

= −G∑

g=1

𝛾g𝔼

[(xg − 𝛼

G∑g′=1

kHg′

nHxg′

− (1 − 𝛼)G∑

g′=1

kSg′

nSxg′

)2 |𝛄⎤⎥⎥⎦−𝛼2vH − (1 − 𝛼)2 vS,

where kXg is the number of legislators of party g

in chamber X.

One shows by calculations similar to thoseabove that

𝔼[−(x∗ − xi

)2 |𝒬, 𝛄] = −𝜎2

{1 − 2

G∑g=1

𝛾g

(8)

(𝛼

nH𝔼[kH

g |𝛄] + 1 − 𝛼

nS𝔼[kS

g|𝛄])+

G∑g=1

𝔼

[(𝛼

nHkH

g + 1 − 𝛼

nSkS

g

)2 |𝛄]}−𝛼2vH − (1 − 𝛼)2 vS

= −𝜎2

{1 − 2

G∑g=1

𝛾g

(𝛼λH

g (𝛄) + (1 − 𝛼) λSg (𝛄))

+G∑

g=1

(𝛼λH

g (𝛄) + (1 − 𝛼) λSg (𝛄))2

+G∑

g=1

(𝛼2

nHςH

g (𝛄) + (1 − 𝛼)2

nSςS

g (𝛄))}

−𝛼2vH − (1 − 𝛼)2 vS,

where λXg (𝛄) denotes the average probability

over districts that a candidate of party g iselected to chamber X, and ςX

g (𝛄) is the arith-metic mean (over districts) of the variance ofthe random variable 1X

d,g conditional on 𝜸. Therandom variable 1X

d,g is 1 if a candidate of partyg is elected to chamber X for the seat d, and0 otherwise.

To make further predictions, we again analyzethe i.i.d. case and the case of a bonus to the largestparty, as above.

(1) i.i.d. Allocation of Seats

In this case, (8) simplifies to

𝔼[−(x∗ − xi

)2 |𝒬, 𝛄]= −𝜎2

(1 + 𝛼2

nH+ (1 − 𝛼)2

nS

)(1 −ℋ ) − 𝛼2vH − (1 − 𝛼)2 vS,

where ℋ ∶=∑G

g=1 𝛾2g again denotes the

Herfindahl–Hirschman index of partisan frac-tionalization. At the Constitutional stage, society

12 ECONOMIC INQUIRY

maximizes

− 𝜎2

(1 + 𝛼2

nH+ (1 − 𝛼)2

nS

)[q (1 − 𝔼 [ℋ ])

+1 − q]− 𝛼2vH − (1 − 𝛼)2 vS

−2C +

(nH + nS

)c

M,

and the optimum is given by

𝛼∗ =(vS∕(vH + vS

)),

n∗H = 𝜎𝛼∗√

McH

√q (1 − 𝔼 [ℋ ]) + 1 − q,

n∗S = 𝜎(1 − 𝛼∗

)√McS

√q (1 − 𝔼 [ℋ ]) + 1 − q,

implying

(9) n∗ = 𝜎√

q (1 − 𝔼 [ℋ ]) + 1 − q

√Mc,

as in the unicameral case.

(2) Bonus to the Largest Group

Straightforward calculations show

𝔼[−(x∗ − xi

)2 |𝒬, 𝛄] =(10)

−𝜎2

{1 + (1 − 𝜉)2

(1 − 2𝛾1

)− 𝜉 (2 − 𝜉)

G∑g=1

𝛾2g

+𝜉(𝛼2

nH+ (1 − 𝛼)2

nS

)[2(1 − 𝛾1 (1 − 𝜉)

)−𝜉

(1 +

G∑g=1

𝛾2g

) ]}− 𝛼2vH − (1 − 𝛼)2 vS.

Thus, society’s objective at the Constitutionalstage is given by

−𝜎2 {q[1 + (1 − 𝜉)2

(1 − 2𝔼

[𝛾1

])(11)

−𝜉 (2 − 𝜉)𝔼

[G∑

g=1

𝛾2g

]]}

+𝜉(𝛼2

nH+ (1 − 𝛼)2

nS

)(2(1 − 𝔼

[𝛾1

](1 − 𝜉)

)−𝜉

(1 + 𝔼

[G∑

g=1

𝛾2g

]))]

+ (1 − q)(

1 + 𝛼2

nH+ (1 − 𝛼)2

nS

)}− 𝛼2vH

− (1 − 𝛼)2 vS −2C +

(nH + nS

)c

M.

Optimizing, we again find

𝛼∗ =(vS∕(vS + vH

)),

and for the optimal total number of legislators

n∗ = 𝜎

√Mc

(12)√1−q+q𝜉

[2(1−𝔼

[𝛾1

](1−𝜉)

)−𝜉 (1 + 𝔼 [ℋ ])

],

as in the unicameral setting. As before, 𝛼 =n∗H∕n∗, or n∗H= 𝛼*n* and n∗S = (1−𝛼*)n*.

Both models (1) and (2) predict that the trade-off between a unicameral and a bicameral systemis the same as under a proportional voting system,implying that a bicameral system is better if andonly if (

v2H∕(vH + vS

))≥ (C∕M) .

Thus, a country’s choice of bicameralismdepends on the size of its population, but doesnot depend on its voting system or its level ofpartisan fractionalization.

C. Predictions

This analysis yields the following predictions.Prediction 1. The log of the number of legis-

lators is linearly increasing in the log of the sizeof the population, with a coefficient close to 0.5.

Prediction 2. The number of legislators isindependent of whether a Parliament is unicam-eral or bicameral.

Prediction 3. In bicameral systems, the rela-tive bargaining power of a given chamber is equalto the share of legislators sitting in this chamber.

Prediction 4. More populous countries aremore likely to have a bicameral Parliament.

Prediction 5. The factors that impact the sizeof Parliament in the model (except the size of thepopulation) have no impact on the probability thata country has a bicameral Parliament.

Prediction 6. In proportional systems, the levelof partisan fractionalization has no impact onthe size of Parliament. The log of the size ofParliament is:

log n = log 𝜎 + 0.5 log M − 0.5 log c(13)

+ 0.5 log (1 − q) .


Prediction 7. In nonproportional voting sys-tems, for a given q and 𝜉, the log-linearization ofEquation (12) implies that the log of the size ofParliament is:


+0.5 log(1 − q + q𝜉

[2(1 − 𝔼

[𝛾1

](1 − 𝜉)

)−𝜉 (1 + 𝔼 [ℋ ])]) .

Approximating the last term, we have:


+0.5 log (1 − q) − 0.5q

1 − q𝜉2𝔼 [ℋ ]

−0.5q

1 − q

(2𝜉 (1 − 𝜉)𝔼

[𝛾1

]+ 𝜉2 − 2𝜉

).

V. EMPIRICAL ANALYSIS

A. Methodology

This section documents a certain number ofempirical regularities of Parliamentary institu-tions across countries. The size of the sam-ple, small by nature, and the lack of exogenoussources of variation in the explanatory variableslimit causal inference. With this caveat in mind,we formulate the identification assumptions of allestimations, but do not discuss their plausibility.Instead, we discuss the results of the estimationsin the context of the predictions of the model.

Data. The estimations use data from the DPI2012 (Beck et al. 2001 [last update in 2012]).These data contain information, for almost everycountry and every year between 1975 and 2012,concerning the number of chambers, the numberof members of either chamber, the voting sys-tem (proportional/nonproportional), and electoraloutcomes in election years, namely vote sharesand seat shares across parties, for the LowerChamber.17 Population by country comes fromthe World Bank Database. To assess the “degreeof democratization,” we use the Polity IV scorefrom Polity data.18 These data indicate the PolityIV score of most countries for every year between1800 and 2015. The Polity IV score ranges from−10 to 10, and is used to partition regimes into

17. There is no information on Upper Chamber elections.In fact, in many countries (e.g., in the United Kingdom or inCanada), members of the Upper Chamber are not elected.

18. The data are available on the Polity IV website http://www.systemicpeace.org/inscrdata.html.

“Autocracies” (score between −10 and −6), andother regimes.

Sample of Observations. The longest period cov-ered by all the data sources spans 1975 to 2012(with some missing information). Many countrieshave been governed by an autocratic regime forat least some years of that period. Since we con-sider that our model does not explain the institu-tional features of such regimes, we aim to excludethese observations from our sample. To do so,we include only “Nonautocratic” regimes in thePolity IV classification (i.e., we exclude countrieswith a score between −10 and −6 in 2012).

In addition, we exclude countries that have nothad at least two legislative elections for whichno party obtained all the votes and there were noreports of substantial fraud (the DPI contains abinary variable that indicates whether an electionwas marred by fraud).19

Since observations in a country across yearsusually cannot be considered independent—forinstance, the size of the U.S. Congress overthe whole period is 535, due to a rule set in1911—we run all regressions on a sample of datawith a unique observation by country.20

Definition of Variables. To assess the value ofinstitutional variables unrelated to electoral out-comes (the number of chambers of Parliament,the size of each chamber of Parliament) of a coun-try, which are all nonrandom, we use the obser-vation for 2012. To assess the size of the popu-lation of that country, we use the observation atthe time of the most recent change in those insti-tutional variables before 2012.21 To assess thevalues of variables related to electoral outcomes,which may be random, we compute their empir-ical means over all the elections (with no fraudreported and with more than one party) that tookplace between 1975 and 2012.22

19. This restriction excludes countries that may have hadmore than one election, but for which there is no electoralinformation.

20. Including many years of observations would artifi-cially increase the significance of the impact of any variablethat changes little between 1975 and 2012, such as the size ofthe Parliament, the population, the number of chambers, andso on.

21. In most countries, Parliament features change “regu-larly.” Only a few countries have not changed the number ofParliamentary seats since 1975.

22. All results are similar if we use 2012 population levelsfor all countries instead.

http://www.systemicpeace.org/inscrdata.html

http://www.systemicpeace.org/inscrdata.html

14 ECONOMIC INQUIRY

As in the model, Party 1 refers to the party that,in a given election, obtained the largest share ofthe total number of votes in the country.23

The variables related to electoral outcomesare the Herfindahl–Hirschman index of partisanfractionalization ℋ =

∑g 𝛾

2g , the share of total

votes obtained by Party 1 𝛾1, and 𝜉. The actualvalue of ℋ and 𝛾1 for any democratic electionthat took place between 1975 and 2012 can becomputed directly from the data, and we usetheir empirical means over democratic electionsas proxy variables for their expected values.24

Conversely, the actual value of 𝜉 cannot beobtained directly from any variable in the data.In fact 1−𝜉, which we informally refer to as“the bonus to the largest party,” may dependon many institutional features—such as redis-tricting rules—that cannot be easily quantified.Instead, to assess the value of 𝜉, we use the factthat, by definition, the expected proportion ofseats won by Party 1, 𝔼

[λ1

], is 𝔼

[1 − 𝜉 + 𝜉𝛾1

].

The proxy variable for 𝜉 is then1−𝔼[λ1]1−𝔼[𝛾1] , where

we use averages across elections to estimate theexpectations in the equation.25,26

Appendix A reports the descriptive statisticsof the variables used in the empirical analysis.

Specification. The basic specification for the esti-mations is:

log nk = β0 + β1 log Mk + β2𝜉2k𝔼 [ℋ ]k + β3

(16)

(2𝜉k

(1 − 𝜉k

)𝔼[𝛾1

]k+ 𝜉2

k − 2𝜉k

)+ εk

with one observation by country k, and where Mkis the size of the population,𝔼 [ℋ ]k is the averageHerfindahl–Hirschman index across democraticelections in country k, 𝔼

[𝛾1

]k

is the average share

23. The identity of Party 1 may of course change fromone election to the next.

24. In certain cases, the vote shares of some par-ties may be either missing or not consistent. Since pre-cise information on vote shares is crucial to compute theHerfindahl–Hirschman index, we include only observationssuch that the sum of the known shares of the parties is between0.90 and 1.10.

25. The model assumes that ξ is nonrandom. If it wererandom, the relationship 𝔼

[ξγ1 + 1 − ξ

]= 𝔼[λ1

]still holds,

but cannot be used to derive 𝔼[ξ]

in general if ξ and γ1 are notindependent. In fact, there would be no general proxy variablefor the expected values of the polynomials of ξ and the shareof votes that appear in the model if we did not assume that ξand the vote share γ1 were independent.

26. We note that the maximum value of ξ is larger than1, which is due to the fact that, in a few cases, the partythat obtained the largest share of votes got fewer seats inParliament.

of votes obtained by Party 1 in country k acrossdemocratic elections, 𝜉k is the ratio of one minusthe average share of seats obtained by Party 1across democratic elections over one minus theaverage share of votes obtained by Party 1 acrossdemocratic elections in country k. The dependentvariable log nk is the log of the total number ofmembers of Parliament in country k.

Equation (16) does not include variables forwhich there is no information (c) or that have noobvious empirical proxy (q and 𝜎).

Predictions 6 and 7 imply that:

• Under any voting system, the model pre-dicts β1 = 0.5.

• Under a proportional voting system, 𝜉 isequal to 1, β2 = 0. Term (3) and the interceptwould be collinear in theory, yet in practice, if𝜉k = 1+ uk where uk is some random error ofmean 0—due for instance to integer problems inthe attribution of seats—we may have β3 = 0.

• Under a nonproportional voting systemwith an i.i.d. distribution of seats across parties,𝜉 is equal to 1, β2 = −0.5 q

1−q, which implies

that β2 < 0. Term (3) and the intercept would becollinear in theory, yet in practice, as under aproportional voting system, we may have β3 = 0.

• Under a nonproportional voting system thatgives a bonus of seats to Party 1, β2 = −0.5 q

1−qand β3 = −0.5 q

1−q, which implies that β2 < 0,

β3 < 0, and β2 =β3.

B. Results of the Estimations

Table 1 reports the estimation of the coeffi-cients of Equation (16) for the sample of nonau-tocratic regimes. The identification assumptionsare: (1) The unobservable variables that alsoaffect the size of Parliament of a country, that is,the variance of bliss points 𝜎2 and the salariesof members of Parliament, are not correlatedwith the size of the population, the probabilityof bicameralism, or the terms specific to partisanissues. (2) The total number of seats in Parliamenthas no impact on the size of the population, theprobability of bicameralism, or the terms specificto partisan issues.

Column 1 shows that the log-linear relation-ship from the model holds, with a coefficientclose to 0.5, which is consistent with Prediction 1.

In column 2, we include as a covariate a binaryvariable equal to 1 if and only if country k has abicameral system. This variable has no significantimpact on the log of the size of Parliament, whichis consistent with Prediction 2.


TAB

LE

1Si

zeof

Parl

iam

ents

inN

onau

tocr

atic

Reg

imes

Log

Size

ofP

arlia

men

t

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Log

popu

latio

n0.

417**

*0.

407**

*0.

410**

*0.

403**

*0.

355**

*0.

353**

*0.

427**

*0.

412**

*

(0.0

23)

(0.0

23)

(0.0

22)

(0.0

23)

(0.0

40)

(0.0

40)

(0.0

29)

(0.0

30)

Her

finda

hl–

Hir

schm

an𝔼[ℋ

]−

0.80

1**−

0.34

5−

1.04

9*

(0.3

67)

(0.5

88)

(0.5

25)

𝜉2×𝔼[ℋ

]−

0.76

7**−

0.39

9−

1.04

9**

(0.3

20)

(0.6

00)

(0.4

63)

2×𝜉×(1

−𝜉)×

𝔼[𝛾

1 ]+𝜉

2−

2×𝜉

−0.

847**

0.19

4−

1.13

2**

(0.3

33)

(1.0

01)

(0.4

82)

Bic

amer

alPa

rlia

men

t0.

147

(0.0

89)

Voti

ngsy

stem

Any

Any

Any

Any

Pro

port

iona

lP

ropo

rtio

nal

Non

prop

orti

onal

Non

prop

orti

onal

#O

bser

vatio

ns74

7474

7432

3242

42R

20.

825

0.83

10.

836

0.84

00.

736

0.73

90.

874

0.87

9

Wal

dp

valu

eH

0:β2

=β3

0.5

18

Not

es:

Thi

sta

ble

repo

rts

the

estim

atio

nof

the

regr

essi

onof

the

log

ofth

eto

taln

umbe

rof

mem

bers

ofPa

rlia

men

ton

the

log

ofpo

pula

tion

and

othe

rco

vari

ates

.The

reis

one

obse

rvat

ion

byco

untr

y.T

hesa

mpl

eco

mpr

ises

allc

ount

ries

that

wer

eno

taut

ocra

ticin

2012

inth

ePo

lity

IVcl

assi

ficat

ion

and

that

have

had

atle

astt

wo

elec

tions

with

nopa

rty

getti

ngal

lthe

vote

san

dw

ithou

trep

orte

dfr

aud

betw

een

1975

and

2012

.See

Sect

ion

Vfo

rde

tails

.Sta

ndar

der

rors

inpa

rent

hese

s.*p

<.0

5,**

p<

.01,

***p

<.0

01.

Sour

ces:

DPI

;Pol

ity;W

orld

Ban

k.

16 ECONOMIC INQUIRY

TABLE 2Probability to Have a Bicameral Parliament

Bicameral Parliament

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Log population 0.067** 0.062** 0.073** 0.080** 0.076** 0.090*** 0.074** 0.077** 0.068** 0.066**

(0.030) (0.030) (0.030) (0.031) (0.030) (0.032) (0.030) (0.030) (0.031) (0.032)Proportional voting −0.138

(0.116)Herfindahl–Hirschman 𝔼 [ℋ ] 0.582

(0.492)𝜉2 × 𝔼 [ℋ ] 0.506

(0.430)2 × 𝜉 × (1 − 𝜉) × 𝔼 [𝛾1] + 𝜉2 − 2 × 𝜉 0.686

(0.448)Ethnic F [1] 0.062

(0.243)Linguistic F [1] −0.006

(0.231)Religious F [1] 0.370

(0.241)Linguistic F [2] 0.019

(0.202)Ethnic P [3] 0.335

(0.229)Religious P [3] −0.089

(0.185)

# Observations 74 74 74 74 73 70 73 73 66 66R2 0.066 0.084 0.084 0.102 0.085 0.107 0.114 0.085 0.091 0.064

Notes: This table reports the estimation of the regression of a binary variable equal to 1 if and only if the country has abicameral Parliament, on the log of population and other covariates. There is one observation by country. The sample comprisesall countries that were not autocratic in 2012 in the Polity IV classification and that have had at least two elections with noparty getting all the votes and without reported fraud between 1975 and 2012. F [1], F [2], P [3] refer respectively to theFractionalization indices of Alesina et al. (2003), the Fractionalization index of Desmet, Ortuno-Ortin, and Wacziarg (2012), andthe Polarization indices of Montalvo and Reynal-Querol (2005a, 2005b). See Section V for details. Standard errors in parentheses.

*p < .05; **p < .01; ***p < .001.Sources: DPI; Polity; World Bank.

Columns 3 and 4 of Table 1 show that thecoefficients of the Herfindahl–Hirschman index(column 3), or of the second and third terms ofEquation (16) (column 4) have the sign predictedby the model for a nonproportional voting sys-tem. These results are consistent with the factthat the coefficients are each some average ofthe coefficient in the proportional voting system(which is null) and in the nonproportional system(which is negative).

In fact, if we restrict the sample to countrieswith a Proportional voting system (columns 5 and6), we find no statistically significant impact ofthe Herfindahl–Hirschman index, or of the coef-ficients of the second and third terms of Equation(16), which is consistent with Prediction 6.

If we then restrict the sample to countrieswith a nonproportional voting system (columns7 and 8), we find a negative and statisticallysignificant impact of the Herfindahl–Hirschman

index, as well as of the coefficients of the secondterm of Equation (16), which is consistent withPrediction 7.

Column 8 also shows that the coefficient of thethird term is negative, statistically significant, andclose to the value of the coefficient of the secondterm. The p value of a Wald test of the hypothesisH0: β2 =β3, reported at the bottom of column 8,shows that we cannot reject the possibility thatthese coefficients are equal at the 0.05 (or 0.1)level of significance. These results are consistentwith Prediction 7 in the case of a “bonus to thelargest group.”

Remark. These results may be used to assessthe probability q that a partisan issue arises, aparameter that has no obvious measurable equiv-alent. We have here that 0.5 q

1−qis around 1, so

that q is around 67%.To assess Prediction 3, we rely on Bradbury

and Crain (2001) who provide the only estimation


of the ratio in the bargaining powers of the Lowerand the Upper Chamber across countries, thatis, 𝛼*/(1− 𝛼*) with the previous notations. Theyfind that the ratio of bargaining powers is 3.5on average. This means that 𝛼* is around 0.78on average across countries. With our data onthe sizes of Lower and Upper Chambers, weestimate that the ratio of the number of membersof the Lower Chamber over the total numberof members of a Parliament is equal to 0.73on average among nonautocratic regimes. Thisaverage is very close to the estimate of Bradburyand Crain (2001).27

Table 2 examines the determinants of anUpper Chamber. The identification assumptionsare: (1) The unobservable variables that alsoaffect the probability of bicameralism, that is,square means of the error terms and the fixed costof an extra Chamber, are not correlated with thesize of the population, the voting system in place,or the terms specific to partisan issues. (2) Theprobability of bicameralism has no causal impacton the size of the population, the voting systemin place, or the terms specific to partisan issues.

In column 1, we regress a binary variable equalto 1 if and only if the country has a BicameralParliament on the log of the population of a coun-try. We find that larger countries are significantlymore likely to have a Senate, which is consistentwith Prediction 4.

In column 2, we include a covariate equal to1 if and only if the country has a Proportionalvoting system. This variable is not significantlycorrelated with the dependent variable, which isconsistent with Prediction 5.

In columns 3 and 4, we find no impact ofthe other terms considered before. In particular,countries with greater partisan fractionaliza-tion are not significantly more likely to havea Senate. This finding might be interpreted assomewhat disputing the view that the role ofsecond chambers is primarily to afford represen-tation to otherwise under-represented minorities.To further examine this point, we also presentadditional estimations that use measures of eth-nic, religious, and linguistic fractionalization (F)and polarization (P) presented in Alesina et al.(2003) ([1] in the table), Desmet, Ortuno-Ortin,and Wacziarg (2012) ([2] in the table), andMontalvo and Reynal-Querol (2005a, 2005b)([3] in the table). No indicator has any effect on

27. If we restrict the sample to the sample of countriesused in Bradbury and Crain’s (2001) estimations, the share is0.74.

the coefficient of the log of the population on theprobability to have a Senate.28

VI. CONCLUSION

We have analyzed a simple model of Parlia-mentary institutions, testing the consistency ofits predictions with cross-country data. We haveseen that the log of the size of a country’s Par-liament increases in a linear manner with thelog of the size of its population. The size of acountry’s Parliament does not depend on whetherit is unicameral or bicameral. In bicameral sys-tems, the relative weight of a chamber shouldcorrespond to the share of legislators sitting inthis chamber. Furthermore, the only observablevariable impacting the probability that a givencountry has a bicameral Parliament is the size ofits population.

A second set of results pertains to the impactof partisan fractionalization and voting systemson Parliament size. We find that the mode of elec-tion has no impact on the probability that a givencountry has a second chamber, and that greaterfractionalization increases the size of a country’sParliament if and only if it has a nonproportionalvoting system. In proportional systems, fraction-alization has no impact.

Our model could be extended in several ways.For instance, we do not model the bargainingprocess among legislators, assuming instead thatlegislators within a given chamber act cooper-atively and do not take the bargaining processwith the other chamber into account. Whether itwould be possible to account for the huge cross-country differences in Parliamentary proceduresin a richer model, which would focus on a smallersubset of countries, and whether such a modelwould preserve, or possibly even increase, thepredictive power of our model, is an interestingquestion for future research.

Furthermore, we have made the assumptionthat, in a bicameral system, errors are uncorre-lated across chambers. This is clearly a strongassumption, made for the purpose of tractability,as one could well imagine both chambers fallingunder the sway of the same lobbying effortsor similar mood swings in published opinion,making for positively correlated errors. On the

28. This result does not mean that ethnic, linguistic, orreligious diversity has no impact on the setup of a country’sParliament. In fact, diversity might affect the number ofmembers of Parliament if, for instance, it had an impact onq or σ.

18 ECONOMIC INQUIRY

other hand, in our simple model, the ConstitutionDesigners would endeavor to ensure that theerrors are as negatively correlated as possible.Indeed, in reality, many countries use differentmodes of selection for the two chambers oftheir Parliaments, which one could arguablyinterpret as one way of effecting a negativecorrelation between the errors they are proneto. Furthermore, in many bicameral systems,there are important design differences acrosschambers, from which we have abstracted in

our analysis. Indeed, sometimes, the secondchamber is malapportioned in such a way asto over-represent some minority, as is the case,for example, with smaller states in the UnitedStates. One could view the fact that often onlyone chamber is malapportioned in this way as anattempt by the Constitution Designers to inducenegatively correlated errors between chambers.We recommend for future research a moredetailed investigation of differences betweenchambers in bicameral systems.

APPENDIX A: DESCRIPTIVE STATISTICS AND GRAPH

TABLE A1Descriptive Statistics

Variable Mean Std. Dev. Min. Max. N

Population in million 41.33 137.48 0.1 1,127.14 74Size of Parliament 253.14 220.97 28 955 74Size of Lower Chamber/House 211.84 171.63 15 650 74Size of Upper Chamber/Senate 87.31 82.25 11 326 35Bicameral Parliament 0.47 0.5 0 1 74Proportional voting system 0.43 0.5 0 1 74Share of legislators in Lower Chamber 0.73 0.09 0.54 0.91 35Herfindahl–Hirschman index ℋ 0.36 0.12 0.17 0.83 74𝜉 0.93 0.2 0.48 2 74𝜉2 ×ℋ 0.33 0.37 0.12 3.31 742 × 𝜉 × (1 − 𝜉) × 𝛾1 + 𝜉2 –2 × 𝜉 −0.94 0.36 −3.63 −0.39 74

Notes: This table reports descriptive statistics for all nonautocratic countries in 2012. The size of the population is measuredat the time of the most recent change in Parliament features before 2012. The electoral outcomes terms ℋ , 𝜉, 𝜉2 ×ℋ and2 × 𝜉 × (1 − 𝜉) × 𝛾1 + 𝜉2 –2 × 𝜉 are the empirical averages of these terms over legislative elections that took place between1975 and 2012. The number of members in a chamber may differ from the number of seats in that chamber if some seats are nottaken up.

Sources: DPI; Polity; World Bank.


FIGURE A1

Population and Parliament Size

alb

arg

ausaus

bahbar

bel

bel

bol

bot

bra

bul

bur

cab

can

chi

col

cos

cro

cyp

den

dom

el

est

fin

fra

gam

ger

gha

gre

gre

hon

hun

ice

ind

ind

ire

isr

ita

jam

jap

lux

mac

madmal

mal

mal

mex

nam

net

new

nor

pan

par per

pol

por

sen

slo

sou

sou

spa

sri

st.

swe

swi

tri

tun

tur

uk

uru

usa

van

ven

34

56

7

Log

# M

embe

rs o

f Par

liam

ent

12 14 16 18 20

Log Population

Notes: Each country is represented by the first three letters of its name. The line is the linear interpolation of the relationshipbetween the log of population and the log of Parliament size.

Sources: DPI; Polity; World Bank.

APPENDIX B: EMPIRICAL EXAMINATION OF THEBONUS TO THE LARGEST GROUP ASSUMPTION

In this section, we present a series of estimations that usedata of the DPI at the election and party level, for any electionthat took place between 1975 and 2012.

To do so, we use the data’s detailed information on sharesof seats and votes obtained by the largest parties acrosselections to test the main implication of this assumption,namely that there exists 𝜉 such that:

(A1) 𝔼[λg

]= 𝜉 × 𝔼

[𝛾g

]+ (1 − 𝜉) × 1g=1

with the notations of the model, and where 1g= 1 isa binary variable equal to 1 if and only if g= 1, and0 otherwise.

As mentioned in Section V, we do not observe 𝜉 directly,but infer its value from solving Equation (A1) with g= 1.Empirically, we find that, in nonproportional voting systems,𝜉 is equal to 0.89 on average (whereas it is equal to 0.98 onaverage in proportional systems). This estimation indicatesthat countries that use nonproportional voting systems indeed

give an advantage to the party with the largest share of votes inthe country, which amounts to 11% of House seats on average.

Our estimation method for 𝜉 implies that Equation (A1)is trivially satisfied for Party 1 empirically. It is also triviallysatisfied empirically in elections with only two parties. Wetherefore focus on estimating the relation between the sharesof seats and votes for parties g> 1 only in elections with threeparties or more.

The basic specification we use here is:

Seats Sharek,g,t = δ0 + δ1 𝜉k × Votes Sharek,g,t

(A2)

+G−1∑r=2

δr Party rank = rk,g,t + εk,g,t

where Seats Sharek, g, t is the share of seats and VotesSharek, g, t is the share of total votes obtained by party gin country k for the election that took place in year t, 𝜉kis defined as before for country k, and Party rank= rk, g, tis an indicator variable equal to 1 if and only if party g is

20 ECONOMIC INQUIRY

TABLE A2Distribution of Seat Shares in Nonproportional Voting Systems

Seats Share

(1) (2) (3) (4) (5) (6) (7)

𝜉 ×Votes Share 0.817*** 0.943*** 1.095*** 0.936*** 1.284*** 1.172*** 0.982***

(0.147) (0.146) (0.104) (0.176) (0.236) (0.071) (0.069)Party rank= 2 0.086 0.047 0.018 0.033 −0.014 −0.012 0.036

(0.065) (0.043) (0.030) (0.061) (0.038) (0.017) (0.023)Party rank= 3 −0.022 −0.034* −0.003 −0.026 −0.030 −0.016

(0.013) (0.016) (0.013) (0.022) (0.019) (0.010)Party rank= 4 0.001 −0.007 −0.043* −0.022 −0.003

(0.009) (0.009) (0.021) (0.015) (0.008)Party rank= 5 0.009 −0.027 −0.020* −0.002

(0.013) (0.016) (0.010) (0.007)Party rank= 6 −0.013 −0.006 −0.002

(0.008) (0.007) (0.006)Party rank= 7 −0.005** −0.001

(0.002) (0.005)

# Parties 3 4 5 6 7 8 3 to 8

# Observations 72 84 208 100 84 70 666R2 0.926 0.900 0.898 0.865 0.933 0.962 0.923

Wald p valueH0: δ = 1 .238 .702 .371 .723 .263 .061 .795

Notes: This table reports the estimation of the regression 34. The dependent variable is the share of seats in the House that partyg in country k obtained after the election in year t. There is one observation by country, party, and election year. All estimationsinclude a fixed effect for any country and election-year pair. The sample comprises all countries that were not autocratic in 2012in the Polity IV classification and that have had at least two elections with no party getting all the votes and without reportedfraud between 1975 and 2012. See Section V for details. Standard errors clustered at the country level are in parentheses.

*p< .05; **p< .01; ***p< .001.Sources: DPI; Polity; World Bank.

ranked rth in decreasing order of vote shares in country kfor the election that took place in year t. We also include ascovariates the fixed effects of any country/election year. (TheParty rank=G variable is excluded to avoid collinearity withthe constant term. The party with the smallest share of votesis thus the reference group.) Standard errors are clustered bycountry to account for the fact that error terms are correlatedwithin countries.

Table A2 reports the estimation of this regression sep-arately for all elections with a given number of partiesrunning—from three to eight parties—and for all elec-tions.29,30

29. The data do not give information on more than nineparties, so that they may aggregate both votes and seatsfor elections with more than eight parties. By definition,including elections with more than eight parties would createa measurement error that would bias the estimations of thecoefficients.

30. The electoral system of a country may give an advan-tage or a disadvantage to the smallest party instead of a partyof some given rank r. Reporting estimations on subsamplesof observations partitioned by number of parties allows tocheck whether this is indeed the case. There are not enoughobservations by country to run the regression separately forevery country, which would provide an even stronger test ofthe assumption.

These estimations show first that the coefficient of theterm 𝜉k ×Votes Sharek, g, t is significantly different from 0,and close to 1. In fact, the p value of a Wald test indicatesthat we cannot reject the hypothesis that the coefficient of𝜉k ×Votes Sharek, g, t is equal to 1. This result is consistentwith Equation (A1) above.

The estimations also show that most indicator variablesParty rank= rk, g, t have no significant effect at the 5% level,and none when all observations are included (column 7). Thisresult means that a party that obtains a share of votes suchthat it would be of rank r has no significant advantage ordisadvantage in terms of Parliamentary seats (with respect tothe party with the largest rank, which is the reference groupin these estimations). On average across countries, we thuscannot reject at the 5% level the hypothesis that the shareof votes and the share of seats (deflated by the factor 𝜉), arethe same.

APPENDIX C: A TWO-STAGE VOTING GAME

In a proportional voting system, the allocation of seats thatwe assume here is the outcome of the subgame-perfect Nashequilibrium that is coalition proof in the following sense.

A coalition is defined by a vectorω= (θ1, η1, … , θG, ηG)∈ ({0, 1}× [0, 1])G, such that theshare of seats obtained by the coalition that are attributed to


party g is θgηg, with∑G

g θgηg = 1.31 We say that g belongsto the coalition ω if θg = 1. In other words, a coalition isdefined by its members and by a contract (η1, ... , ηG) thatspecifies how the seats won in the chamber will be sharedamong them.

Let Ω denote the set of all possible coalitions, and Ωg theset of all possible coalitions that g belongs to.

We consider the following two-stage game.Stage 1. Suppose each party is headed by a party leader

who may agree on behalf of the party to form a coalitionwith one or several other parties. The party leader wants tomaximize the share of seats of his party.32 In the first stage,every party g’s leader chooses a unique ω ∈ Ωg. Coalition ωis running in the election if and only if all parties that belongto ω have chosen it. Otherwise, we impose that all parties thatpicked ω run independently. Let Ω

(ω1, … ,ωG

)∈ 2Ω be the

set of coalitions or parties that run in the election for a givenvector (ω1, ..., ωG).

Stage 2. In the second stage, every individual observesparties’ choices, and casts a vote for a coalition (or a party)that is actually running. An individual i’s strategy is definedby a function ΩG →Ω which sets, for any vector of choices(ω1, ..., ωG), which coalition (or party) ω ∈ Ω individual iwill vote for, and by the constraint that ω ∈ Ω

(ω1, … ,ωG

),

that is, that i cannot vote for a coalition that is not running inthe election.

Information. At the time of the election, individuals’ par-tisan affiliations are publicly known, but the type of issue toarise and individual bliss points are unknown. This assump-tion aims to account for citizens’ imperfect informationon the details of the issue that will come before the leg-islators to whom they delegate decision power. Over thecourse of a term, legislators may have to address unexpectedissues, for instance, a domestic or international crisis, orto acquire more information on an issue before decidingon a policy.

After the election, the type of issue is realized, individ-ual bliss points are known, and the policy adopted is as inEquation (1) of Section III.

To simplify notations, we assume, in this section only,that the issue is partisan, that error terms ZX are null, thatthe Parliament is unicameral, and that C = c= 0. The resultsare the same if we relax all these assumptions, and use theassumptions of the general model instead.

LEMMA 1. In any subgame-perfect Nash equilibrium ofthis game, any party necessarily obtains a share of seats equalto its share of partisans in the population.

Proof. Consider the second stage first. Let i be an individualwho belongs to party g, but who is not a legislator. For apartisan issue, individual i’s expected utility is:

−𝔼⎡⎢⎢⎣(

xg −G∑

g′=1

λg′xg′

)2⎤⎥⎥⎦31. If the coalition obtains a share of seats s, and hence a

number of seats sn in the chamber, party g will obtain snη ofthese sn seats if θg = 1, and none of them if θg = 0.

32. This assumption would hold in particular if the partyleader is a partisan who does not sit in Parliament, whichseems to be often the case empirically.

where λg′ is the share of seats obtained by party g′

in Parlia-ment. This term can be rewritten:

−𝜎2 [1 − λg

]2 − 𝜎2G∑

g′=1,g′≠g

(λg′)2

.

This term is increasing in λg ∈ [0, 1] and decreasing in λg′

∈ [0, 1] for g′≠ g.

Let λ̃g′ ≥ 0 be the share of seats obtained by party g′ ∈{1,

... , G} without individual i’s vote. i’s vote counts for 1/Mof the total share of votes, so, by definition of a proportional

voting system,G∑

g′=1λ̃g′ = 1 − (1∕M).

If party g runs independently and i votes for it, i’s payoffis:

A = −𝜎2

{(1 − λ̃g − (1∕M)

)2+

G∑g′=1,g′≠g

λ̃2g′

}.

If party g runs independently and i votes for a coalitionor for a party other than g, his vote will add some εg′ to λg′ ,where 0 ≤ εg′ ≤ (1∕M) and

∑g′∈G

εg′ = (1∕M).33 Individual i’s

payoff is then:

B = −𝜎2

{(1 − λ̃g

)2+

G∑g′=1,g′≠g

(̃λg′ + εg′

)2}

.

The monotonicity in λg ∈ [0, 1] and λg’ ∈ [0, 1] impliesthat A>B. Therefore, any party that runs independently willobtain a share of seats that is at least equal to its share ofpartisans in the population.

In the first stage, no party will choose a coalition unlessit can obtain at least that share of seats by doing so. Sincethe sum of these shares is equal to 1 no party can obtain ashare of seats strictly larger than its share of partisans in anyequilibrium. ◾

LEMMA 2. A situation in which all parties choose to runindependently in the first stage and each individual votes forhis own party in the second stage is a subgame-perfect Nashequilibrium.

Proof. In such a situation, a deviation by any one party will notchange the set of coalitions running, since no coalition can beformed after a unique deviation. ◾

REFERENCES

Achen, C. H., and L. M. Bartels. Democracy for Realists:Why Elections Do Not Produce Responsive Govern-ment. Princeton, NJ: Princeton University Press, 2016.

Aghion, P., A. Alesina, and F. Trebbi. “Endogenous PoliticalInstitutions.” Quarterly Journal of Economics, 119(2),2004, 565–611.

Aghion, P., and P. Bolton. “Incomplete Social Contracts.”Journal of the European Economic Association, 1(1),2003, 38–67.

Alesina, A., A. Devleeschauwer, W. Easterly, S. Kurlat, andR. Wacziarg. “Fractionalization.” Journal of EconomicGrowth, 8(2), 2003, 155–94.

33. If i votes for a coalition, εg′ may depend on thecontract that defines that coalition.

22 ECONOMIC INQUIRY

Alesina, A., and E. Spolaore. The Size of Nations. Cambridge,MA: MIT Press, 2005.

Ansolabehere, S., J. M. Snyder, and M. M. Ting. “Bargain-ing in Bicameral Legislatures: When and Why DoesMalapportionment Matter?” American Political ScienceReview, 97(3), 2003, 471–81.

Auriol, E., and R. J. Gary-Bobo. “On the Optimal Numberof Representatives.” Public Choice, 153(3–4), 2012,419–45.

Austen-Smith, D., and W. H. Riker. “Asymmetric Informationand the Coherence of Legislation.” American PoliticalScience Review, 81(03), 1987, 897–918.

Baron, D. P., and J. A. Ferejohn. “Bargaining in Legis-latures.” American Political Science Review, 83(04),1989, 1181–206.

Baron, D., and D. Diermeier. “Elections, Governments, andParliaments under Proportional Representation.” Quar-terly Journal of Economics, 116(3), 2001, 933–67.

Beck, T., G. Clarke, A. Groff, P. Keefer, and P. Walsh. “NewTools in Comparative Political Economy: The Databaseof Political Institutions.” The World Bank EconomicReview, 15(1), 2001, 165–76.

Blais, A. “The Debate over Electoral Systems.” InternationalPolitical Science Review, 12(3), 1991, 239–60.

Bradbury, J. C., and W. M. Crain. “Legislative Organizationand Government Spending: Cross-Country Evidence.”Journal of Public Economics, 82(3), 2001, 309–25.

Buchanan, J. M., and G. Tullock. The Calculus of Consent.Ann Arbor: University of Michigan Press, 1962.

Cutrone, M., and N. McCarty. “Does Bicameralism Matter?”in The Oxford Handbook of Political Economy, editedby B. R. Weingast and D. A. Wittman. Oxford: OxfordUniversity Press, 2006, 180–95.

Desmet, K., I. Ortuno-Ortin, and R. Wacziarg. “The PoliticalEconomy of Linguistic Cleavages.” Journal of Devel-opment Economics, 97(2), 2012, 322–38.

Diermeier, D., and A. Merlo. “Government Turnover in Par-liamentary Democracies.” Journal of Economic Theory,94(1), 2000, 46–79.

Diermeier, D., and R. B. Myerson. “Bargaining, Veto Power,and Legislative Committees.” Discussion Paper No.1089. Center for Mathematical Studies in Economicsand Management Science, Northwestern University,1994.

. “Bicameralism and Its Consequences for the Inter-nal Organization of Legislatures.” American EconomicReview, 89(5), 1999, 1182–96.

Facchini, G., and Testa, C. “A Theory of Bicameralism.”Working Paper, 2005. Accessed April 11, 2018.https://ssrn.com/abstract=681223.

Gersbach, H. “Collective Acquisition of Information by Com-mittees.” Journal of Institutional and Theoretical Eco-nomics (JITE)/Zeitschrift für die gesamte Staatswis-senschaft, 148(3), 1992, 460–7.

Grofman, B., and A. Lijphart. Electoral Laws and Their Polit-ical Consequences, Vol. 1. New York: Algora Publish-ing, 1986.

Herrera, H., M. Morelli, and T. Palfrey. “Turnout and PowerSharing.” The Economic Journal, 124(574), 2014,F131–62.

Herrera, H., M. Morelli, and S. Nunnari. “Turnout acrossDemocracies.” American Journal of Political Science,60(3), 2016, 607–24.

Kalandrakis, T. “Bicameral Winning Coalitions and Equilib-rium Federal Legislatures.” Legislative Studies Quar-terly, 29(1), 2004, 49–79.

Knight, B. “Legislative Representation, Bargaining Powerand the Distribution of Federal Funds: Evidence from

the US Congress.” The Economic Journal, 118(532),2008, 1785–803.

Lijphart, A. “The Political Consequences of Electoral Laws,1945–85.” American Political Science Review, 84(02),1990, 481–96.

. Patterns of Democracy: Government Forms and Per-formance in Thirty-Six Countries. New Haven, CT: YaleUniversity Press, 2012.

Madison, J. “The Federalist No. 10,” in The Federalist: ACollection of Essays, Written in Favour of the New Con-stitution, as Agreed upon by the Federal Convention,September 17, 1787. Originally published in New Yorkby J. and A. McLean, 1788a.

. “The Federalist No. 55,” in The Federalist: A Col-lection of Essays, Written in Favour of the New Con-stitution, as Agreed upon by the Federal Convention,September 17, 1787. Originally published in New Yorkby J. and A. McLean, 1788b.

. “The Federalist No. 58,” in The Federalist: A Col-lection of Essays, Written in Favour of the New Con-stitution, as Agreed upon by the Federal Convention,September 17, 1787. Originally published in New Yorkby J. and A. McLean, 1788c.

. “The Federalist No. 62,” in The Federalist: A Col-lection of Essays, Written in Favour of the New Con-stitution, as Agreed upon by the Federal Convention,September 17, 1787. Originally published in New Yorkby J. and A. McLean, 1788d.

Massicotte, L. “Legislative Unicameralism: A Global Surveyand a Few Case Studies.” Journal of Legislative Studies,7(1), 2001, 151–70.

Montalvo, J. G., and M. Reynal-Querol. “Ethnic Diversityand Economic Development.” Journal of DevelopmentEconomics, 76(2), 2005a, 293–323.

. “Ethnic Polarization, Potential Conflict, and CivilWars.” American Economic Review, 95(3), 2005b,796–816.

Ostrom, V. The Political Theory of a Compound Republic:Designing the American Experiment. Lanham, MD:Lexington Books, 2008.

Parameswaran, G. “Bargaining and Bicameralism.” Legisla-tive Studies Quarterly, 43(1), 2018, 101–39.

Riker, W. H. “The Justification of Bicameralism.” Inter-national Political Science Review, 13(1), 1992,101–16.

Rogers, J. R. “Bicameral Sequence: Theory and State Legisla-tive Evidence.” American Journal of Political Science,42(4), 1998, 1025–60.

Romer, T., and H. Rosenthal. “A Constitution for SolvingAsymmetric Externality Games.” American Journal ofPolitical Science, 27(1), 1983, 1–26.

Snyder, J. M., M. M. Ting, and S. Ansolabehere. “Legisla-tive Bargaining under Weighted Voting.” American Eco-nomic Review, 95(4), 2005, 981–1004.

Stigler, G. J. “Sizes of Legislatures.” Journal of LegislativeStudies, 5, 1976, 17.

. “Law or Economics?” The Journal of Law & Eco-nomics, 35(2), 1992, 455–68.

Stratmann, T. “The Effects of Logrolling on CongressionalVoting.” American Economic Review, 82(5), 1992,1162–76.

Taagepera, R. “The Size of National Assemblies.” SocialScience Research, 1(4), 1972, 385–401.

Tsebelis, G., and J. Money. Bicameralism. Cambridge: Cam-bridge University Press, 1997.

Vespa, E. I. “Malapportionment and Multilateral Bargaining:An Experiment.” Journal of Public Economics, 133,2016, 64–74.

Parliament Shapes and Sizes: Parliament Shapes and Sizes...PARLIAMENT SHAPES AND SIZES RAPHAEL...

Documents

Transcript of Parliament Shapes and Sizes: Parliament Shapes and Sizes...PARLIAMENT SHAPES AND SIZES RAPHAEL...