Socially Optimal Districting: An Empirical Investigation

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1 Preliminary Draft September 2005 Socially Optimal Districting: An Empirical Investigation Abstract This paper provides an empirical exploration of the potential gains from socially optimal districting. Districtings, or the allocation of voters of differing ideologies across legislative districts, matter for welfare because they influence the partisan composition of the legislature and hence policy outcomes. Under a socially optimal districting, the ideological composition of the legislature is welfare maximizing for any realization of voter preferences. We develop a methodology for analyzing how actual districting plans differ from optimal districting plans and for measuring the potential welfare gains that would emerge from implementing optimal districting. We apply this method to U.S. state legislative elections using data from the 1990s. We find that the districtings used by the states in our data set lead to electoral systems that are overly responsive to changes in voters preferences. While there is significant variation across states, the potential welfare gains from optimal districting are on average small relative to the overall surplus generated by legislatures. This appears to be because state districting plans are reasonably close to optimal rather than because there are little aggregate gains to be had from varying districting plans. Interestingly, implementing proportional representation would lead to welfare gains in all states and would produce welfare levels quite close to those achieved with socially optimal districting. Stephen Coate Department of Economics Cornell University Ithaca NY sc163@cornell.edu Brian Knight Department of Economics Brown University Providence RI brian knight@brown.edu We are greatly indebted to Jim Snyder for providing the data on state legislative elections used in this study.

2 1 Introduction The decennial redrawing of district lines used in electing candidates to federal, state, and local legislatures in the United States has generated intense interest among voters, politicians, and parties alike. The intensity of this interest in redistricting, as witnessed through both political and legal battles, should come as no surprise. Democrats and Republicans are often sharply divided in their preferences over the choice of public policies. These policies are determined in part by the representation of political parties in legislatures, which in turn depend upon the allocation of voters across legislative districts. This public interest in redistricting has helped to spurn an academic literature analyzing redistricting plans and the redistricting process. In this literature, redistricting plans are typically characterized by the implied relationship between seats in the legislature and support for parties among voters. In particular, the seat-vote curve relates the fraction of seats won by the Democratic party to its support among voters across all districts. Commonly-studied properties of seat-vote curves include responsiveness, defined as the increase in seats for the Democrats associated with an increase in voter support across all districts for the Democrats, and partisan bias, defined as the advantage or disadvantage afforded the Democratic party in terms of seats, relative to the Republican party, when voters are equally divided between the two parties. Researchers have developed statistical methods to estimate seat-vote curves and to measure the associated responsiveness and partisan bias parameters. Key questions are then how responsiveness and bias are altered following the redrawing of district lines and also how different institutional arrangements for redistricting influence these changes. While this literature is certainly interesting from a positive perspective, the normative lessons to be drawn from its findings are unclear. How should voters be allocated across districts? Is partisan bias necessarily a bad thing from the perspective of voters? What is the optimal degree of responsiveness? In Coate and Knight (2005), a companion paper, we develop a welfare economics approach to address these questions from a theoretical perspective. Following the literature, in our framework districting matters because it determines the equilibrium seat-vote curve. The fraction of seats held by Democrats determines the ideological make up of the legislature and thereby policy outcomes and citizen welfare. To understand optimal districting, we first derive the optimal seatvote curve, which describes the welfare-maximizing relationship between Democratic seats and 1

3 votes. This optimal seat-vote curve is linear with a slope, or responsiveness, that depends on the degree of variation in the preferences of Independent voters. Furthermore, it is biased in favor of the party with the largest base. We then show that, under seemingly reasonable conditions, there exists districtings under which the equilibrium and optimal seat-vote curves coincide. These districtings are socially optimal districtings because they generate the ideal relationship between seats and votes. These theoretical findings raise several natural empirical questions. How close are the equilibrium seat-vote curves associated with real world legislatures to the optimal seat-vote curves identified via this social welfare approach? Are the conditions under which there exist districtings that make equilibrium and optimal seat-vote curves coincide satisfied in practice? If so, how large are the potential welfare gains associated with implementing optimal seat-vote curves? What are the sources of these welfare gains? In this companion empirical paper, we begin the development of a methodology that can be used to answer these and related questions. We first develop an empirical methodology for using district-specific electoral returns in order to estimate equilibrium seat-vote curves. This approach, while allowing for a general functional form for the equilibrium seat-vote curve, has micro foundations, and the parameters of the equilibrium seat-vote curve can be related to the parameters of the theoretical model. Turning to the normative questions, we then develop expressions for the optimal seat-vote curve and its associated optimal bias and responsiveness parameters; this optimal seat-vote curve can also be identified using data on electoral returns along with information regarding the fraction of Independent voters across all districts. We then address whether or not the optimal seat-vote curve can be implemented in practice by developing a sufficient condition that depends upon the fraction of Independent voters and showing that this condition can be verified empirically. Finally, we develop expressions for the welfare gains from socially optimal districting and show that these depend on squared distance between the equilibrium and optimal seat-vote curves. We apply this method to U.S. state legislative elections using data from the 1990s. One key finding is that equilibrium seat-vote curves tend to be overly responsive, relative to optimal seatvote curves, suggesting that, if anything, elections in the U.S. are overly competitive and that there are too few safe seats from a social welfare perspective. In addition, we find that optimal seatvote curve can be implemented in practice in the sense that there exists districtings that make the 2

4 equilibrium and optimal seat-vote curves coincide. In general, while there is significant variation across states, our measures of the potential welfare gains from socially optimal districting are quite small, at least relative to the overall surplus generated by state legislatures. We argue that this is because state districting plans are reasonably close to optimal rather than because there are little aggregate gains to be had from varying districting plans. We also find that implementing proportional representation would lead to welfare gains in all states and would produce welfare levels quite close to those achieved with socially optimal districting. The paper proceeds as follows. Section 2 describes the formal model underlying the analysis and reviews the key theoretical results. Section 3 presents our empirical methodology for comparing equilibrium and optimal seat-vote curves and evaluating the potential gains from socially optimal districting. In Section 4, this empirical methodology is applied to data from U.S. state legislative elections during the 1990s. Section 5 compares our approach and results with the existing districting literature, and Section 6 concludes. 2 Theoretical background 1 We consider a state divided into n equally sized districts indexed by i {1/n, 2/n,..., 1}. Policymaking is done by a legislature consisting of a representative from each district. Each district chooses its representative in an election. The policy outcomes chosen by the legislature depend upon the average ideology of the elected representatives, where ideology is measured on a 0 to 1 scale. In terms of ideologies, citizens can be divided into three groups - Democrats, Republicans, and Independents. Democrats and Republicans have ideologies 0 and 1, respectively. Independents have ideologies that are uniformly distributed on the interval [m τ,m + τ]. Reflecting the fluid nature of these voters attitudes, the ideology of the median Independent is ex-ante uncertain. Specifically, m is the realization of a random variable uniformly distributed on [1/2 ε, 1/2 +ε], where ε < τ and ε + τ 1/2. The former assumption guarantees that both candidates receive support from Independents, while the latter assumption guarantees that the ideologies of the Independents are always between zero and one. The fraction of voters in district i who are Democrats, Republicans, and Independents are, respectively, π i D, πi R and πi I.Letπ D, π R and π I 1 This section will be brief, and readers are referred to Coate and Knight (2005) for additional interpretation and discussion of alternative modeling assumptions. 3

5 denote, respectively, the fraction of voters in the whole state who are Democrats, Republicans, and Independents. Each district must elect a representative. Candidates are put forward by two political parties: the Democrats and Republicans. Following the citizen-candidate approach, candidates are citizens and are characterized by their ideologies. Each party must select from the ranks of its membership, so that the Democrat Party always selects a Democrat and the Republican Party a Republican. Elections are held simultaneously in each of the n districts and the candidate with the most votes wins. The surplus a citizen with ideology x obtains from having a legislature with average ideology x 0 is given by β γ(x x 0 ) 2. The parameter β is the surplus a citizen would obtain from having a legislature that is perfectly congruent with his own ideology. The parameter γ measures the rate at which this surplus is dissipated as the ideology of the legislature diverges from the ideology of the citizen. The ratio γ/β will play an important role in the welfare analysis and can be interpreted as the fraction of the surplus a partisan (i.e., a Democrat or Republican) obtains from having a perfectly congruent legislature that is dissipated by having a legislature composed entirely of the opposition party. We assume throughout that this ratio is bounded between zero and one (γ/β [0, 1]). In each district, every citizen votes sincerely for the representative whose ideology is closest to his own. If the median Independent has ideology m, the fraction of voters in district i voting for the Democrat candidate is given by: V i (m) =π i D + π i I[ 1/2 (m τ) ]. (1) 2τ This consists of all the Democrats and the Independents whose ideologies are less than 1/2. Similarly, the statewide fraction of voters voting for Democrats is V (m) =π D + π I [ 1/2 (m τ) ]. (2) 2τ Let V and V denote, respectively, the average maximum and minimum Democratic vote shares; i.e., V = V (1/2 ε) andv = V (1/2+ε). 4

6 2.1 The equilibrium seat-vote curve We can use the model to derive the equilibrium relationship between seats and average votes. First, for all V [V, V ], let m(v ) denote the ideology of the median Independent that would generate the vote share V ; i.e., m(v )=V 1 (V ). It is straightforward to verify that m(v )= τ[π I +2π D 2V ]. (3) π I Substituting this into the district-specific support,weobtain V i (m(v )) = πd i + πi[ i V π D ]. (4) π I Thus, district i elects a Democrat if and only if V i (m(v )) 1/2, or, equivalently, if and only if: V V i Here Vi i is a safe Democrat (safe Republican) seatifvi Democrat or Republican seat is competitive. = π D + π I [ 1/2 πi D πi i ]. (5) is the average vote threshold above which district i elects a Democrat candidate. District V (V i V ). A seat which is neither a safe Relabelling as necessary, order the districts so that V1/n V 2/n... V 1. Then, the fraction of seats the Democrats receive when they have aggregate vote share V is: S(V )=max{i : V i V }. (6) This is the equilibrium seat-vote curve. Its range is the interval [V, V ] and its shape is determined by the pattern of threshold vote levels (Vi ). These threshold vote levels are in turn determined by the districting; that is, the allocation of citizen types across districts. 2.2 The optimal seat-vote curve Consider now the problem of a planner deciding on the number of seats S that should be allocated to the Democrats when their vote share is V. Aggregate surplus when the median Independent has ideology m and the Democrats have seat share S is given by: Z m+τ W (S; m) =β γ[π D (1 S) 2 + π R S 2 + π I (1 S x) 2 dx ]. (7) m τ 2τ To avoid tedious integer concerns, assume that the number of districts is very large, so that we can interpret S asthefractionofseatsheldbythedemocratsandtreatthechoicesetin 5

7 the optimization problem as the unit interval [0, 1]. It is then straightforward to establish the following result: Proposition 1: The optimal seat-vote curve S o :[V, V ] [0, 1] is given by S o (V )=1/2+(π D π R )(1/2 τ)+2τ(v 1/2). (8) Proof: See Appendix. Thus, the optimal seat-vote curve is linear. Its partisan bias -defined as the advantage or disadvantage accorded to the Democrats when the two parties split the vote - is (π D π R )(1/2 τ). The system is therefore optimally biased towards the party with the largest partisan base. This optimal bias reflects the fact that when the legislature is equally divided between Democrats and Republicans, a marginal change in the composition of the legislature has a greater impact on partisans surplus than on Independents. This in turn reflects the assumption that the loss experienced by citizens from having a legislature that diverges from their own ideology is a convex (indeed, quadratic) function of the extent of this divergence. The responsiveness of the optimal seat vote curve - defined as the rate at which Democrat seats increase as their vote share increases - is 2τ. Thus, as the policy preferences of Independent voters become more diverse, the Democrats seat share becomes more responsive to changes in their aggregate vote. Notice, however, that since τ<1/2, optimal responsiveness is always less than 1 meaning that if the Democratic vote share goes up by V, the Democrats share of seats in the legislature should optimally go up by less than V. To understand this result, note first that the Democratic seat share will optimally be such as to make average legislator ideology equal to average ideology in the population at large. Then observe that changes in vote share always over-estimate changes in average population ideology; that is, if the Democratic vote share goes up by V, the decrease in average population ideology will be less than V. This is because changes in vote share are driven by changes in the voting behavior of Independents and such changes can arise from fairly small shifts in preferences. To illustrate, consider the case in which τ is very small so that Independents all have basically the same ideology - namely m. Now suppose that m changes from 0.51 to 0.49 so that all the Independents switch from voting Republican to voting Democrat. Then there is a massive change in the Democratic vote share but virtually no change in the average ideology of the population and hence virtually no change in the optimal share of Democratic seats in the legislature. 6

8 2.3 The conditions for implementation The optimal seat-vote curve describes the ideal relationship between seats and votes. It is by no means obvious that there exist districtings that would actually make the equilibrium and optimal seat-vote curves coincide. When there do, we say that the optimal seat-vote curve is implementable and refer to the underlying districtings as socially optimal districtings. Whether the optimal seat-vote curve is implementable depends upon the fraction of Independents. Two special cases provide intuition for this. First, if there were no Independents, then the optimal seat-vote curve would be a single point [S o (V )=π D ] and could be implemented by creating a fraction π R of majority Republican districts and a fraction π D of majority Democrat districts. On the other hand, if the entire population were Independents, then all districts would necessarily be identical and the equilibrium seat-vote curve would jump from zero to one at V =1/2. The optimal seat-vote curve, by contrast, is linear in this case [S o (V )=1/2+2τ(V 1/2)] and is thus clearly not implementable. In the companion theoretical paper, we establish: Proposition 2: The optimal seat-vote curve is implementable if and only if Proof: See Coate and Knight (2005). π I [ ε 2τ + ε (τ + ε)ln(1+ ε τ )] min(π D,π R ). (9) Thus, the fraction of independents must be sufficiently small. The condition does not appear particularly restrictive since it is necessarily satisfied whenever π I 2min(π D,π R ). Moreover, as shown in Coate and Knight (2005), when the condition is satisfied, there will typically exist a large class of districtings under which the equilibrium coincides with the optimal seat-vote curve. This class of optimal districtings typically includes districtings that look quite straightforward in the sense that they do not require patterns of concentration of voter types that would seem likely to infeasible when account is taken of the geographic constraints that real world districters face. This nurtures the hope that the optimal seat-vote curve may be an attainable benchmark. 2.4 The welfare gains to socially optimal districting Assuming that the optimal seat-vote curve is implementable, what would be the welfare gains from moving to a socially optimal districting? We address this question next. While aggregate surplus has previously been expressed conditional on the aggregate vote share V, which can be 7

9 interpreted as welfare at a single point on the seat-vote curve, we require a measure of expected welfare over the entire seat-vote curve in order to rank alternative curves from a social welfare perspective. This expected welfare is defined by: E[W (S(V ))] = β γ Z V Z m(v )+τ [π D (1 S(V )) 2 +π R S(V ) 2 +π I (1 S(V ) x) 2 dx 2τ ]( dv ). (10) V V V m(v ) τ The welfare gain from socially optimal districting is therefore given by G = E[W (S o (V ))] E[W (S(V ))]. (11) This welfare gain turns out to be proportional to the squared distance between the optimal and equilibrium seat-vote curves. To see this, note first that expected citizen welfare can be expressed as a function of the equilibrium and optimal seat-vote curves: Lemma: It is the case that E[W (S(V ))] = β γ{c + E[S(V ) 2 ] 2E[S(V )S o (V )]} (12) where c is a constant given by c = π D + π I [1/4+ε 2 /3+τ 2 /3]. Proof: See Appendix. Using this formula to compute the welfare gain immediately establishes: Proposition 3: The welfare gain from socially optimal districting can be written as G = γe[(s o (V ) S(V )) 2 ]. (13) Intuitively, the smaller the degree of similarity between the optimal and equilibrium seat-vote curves, the larger are the welfare gains associated with socially optimal districting. 3 Empirical methodology We can now describe our methodology for comparing actual and optimal districting plans and evaluating the welfare gains from socially optimal districting. We develop empirical counterparts to each of the four theoretical concepts described above: the equilibrium seat-vote curve, the optimal seat-vote curve, the conditions for implementing the optimal seat-vote curve, and the welfare gains to socially optimal districting. In developing this methodology, we consider a researcher with 8

10 access to data on district-specific electoral returns, as well as the state-wide fraction of voters who identify as Independents Estimating the equilibrium seat-vote curve As explained in Section 2.1, the equilibrium seat-vote curve is determined by the range of possible aggregate Democratic vote shares [V, V ] and the pattern of district-specific threshold vote levels (Vi ). We will show that both the range of vote levels and the vote thresholds can be expressed as a function of the first two moments of the district-specific and statewide voting distributions. This then permits the estimation of the equilibrium seat-vote curve by simply estimating these moments. The first step in establishing this is to provide expressions for the moments of the voting distribution. Beginning with the district-specific moments, recall from the theoretical model that the fraction of votes received by the Democrat candidate in district i is given by: V i = πd i + πi[ i 1/2 (m τ) ]. (14) 2τ Given that m is uniformly distributed on [1/2 ε, 1/2+ε], the mean and standard deviation of votes in district i are µ i = E(V i )=πd i πi I = 1 2 [1 + πi D πi R ] σ i = p q πi Var(V i )= i2var(m) 4τ = πi 2 I ε 2 3τ where the second expression uses the fact that var(m) = ε 2 /3. 3 (15) Observe that the district expected vote share (µ i )reflects a partisan advantage for the Democratic candidate (µ i > 1/2) if πd i >πi R but for the Republican candidate (µ i < 1/2) if πd i <πi R. The standard deviation of vote shares is proportional to the fraction of Independents. It is increasing in the degree to which the median Independent shifts support between the two candidates from election to election (ε) but is decreasing in the diversity of preferences among Independents (τ). This latter property reflects the fact that a small swing in preferences of the median Independent towards the Democrats 2 Data on voter identification are typically available from a variety of sources, such as survey data or, in some states, voter registration rolls. 3 More generally, a random variable X distributed uniformly on the interval [a, b] has a mean equal to (a + b)/2 and a variance equal to (b a) 2 /12. 9

11 translates into a large shift in support for the candidate if the distribution of the preferences of Independents is tight but a small shift if this distribution is disperse. Turning to the statewide moments, we take cross-district averages of the district-specific means and standard deviations to obtain: µ = E(V )=π D π I = 1 2 [1 + π D π R ] σ = p q. (16) πi Var(V )= 2var(m) 4τ = πiε 2 2 3τ The statewide expected Democratic vote is independent of the fraction of Independents, reflecting the fact that these voters favor neither party ex-ante. The variance is again proportional to the fraction of Independents, reflecting the fact that there is no uncertainty in how partisans vote. Using the state-wide moments and equation (2), we can now write the maximum and minimum aggregate Democratic vote shares as V = V (1/2+ε) =µ 3σ (17) and V = V (1/2 ε) =µ + 3σ. (18) Moreover, using the district-specific and statewide moments and the definition of Vi in equation (5), we can write the vote threshold for electing a Democratic candidate in district i as follows: V i = µ + σ σ i (1/2 µ i ). (19) These results permit the researcher to estimate the equilibrium seat-vote curve by estimating the means and standard deviations of the Democratic vote share in each district and the associated statewide mean and standard deviation. Suppose that these estimates are given by (bµ i, bσ i ) 1 i=1/n and (bµ, bσ). Relabelling the districts so that bµ + bσ (1/2 bµ 1/n ) bµ + bσ (1/2 bµ 2/n )... bµ + bσ (1/2 bµ 1 ), bσ 1/n bσ 2/n bσ 1 the estimated equilibrium seat-vote curve is the function defined on the interval [bµ 3bσ, bµ+ 3bσ] given by: S(V )=max{i : bµ + bσ bσ i (1/2 bµ i ) V }. (20) 10

12 3.2 Estimating the optimal seat-vote curve Turning to the normative component of the empirical analysis, we next develop measures of the optimal seat-vote curve, thereby allowing for comparison with the equilibrium seat-vote curve. Using the fact that π D π R =2µ 1, we can re-write the optimal seat-vote curve described in Proposition 1 as follows: S o (V )=1/2+(2µ 1)(1/2 τ)+2τ(v 1/2). (21) Observe that the optimal seat-vote curve and, in particular, its responsiveness and partisan bias parameters depend critically upon the diversity of preferences among Independents (τ). Unfortunately, this preference parameter is not directly identified in the voting data. To see this, recall that the statewide standard deviation of votes for the Democratic candidate can be written as follows: σ = π Iε 2 3τ. (22) Thus, with only information on σ, we cannot separately identify the key underlying parameters π I, ε, andτ. Intuitively, large swings in the Democratic vote share within a state could be due to a large fraction of Independents (π I ), large swings in the preferences of the median Independent (ε), or a tight distribution of ideology among Independents (τ), in which case relatively small swings in the preferences of the median Independent translate into relatively large swings in voting outcomes. While we cannot separately identify these preference parameters, it is possible to identify their ratio (α = ε/τ), with data on the statewide fraction of Independents (π I ) and the statewide standard deviation of the voting distribution (σ) as follows: α = ε τ = 2 3σ. (23) π I Further, using the theoretical restriction on the sum of these preference parameters + τ 1/2, we can place an upper bound on the dispersion in the preferences of Independents: 1 τ 2(1 + α), (24) and, we can in turn place an upper bound on the optimal degree of responsiveness [r o =2τ< 1/(1 + α)]. In the baseline analysis of the empirical application to follow, we assume that the 11

13 optimal degree of responsiveness equals this upper bound. In addition, as a robustness check, we allow the optimal responsiveness to fall in a range below this upper bound. 3.3 Verifying the conditions for implementation The necessary and sufficient condition for implementability presented in Proposition 2 can not be verified directly without information on the underlying preference parameters (ε, τ). As just noted however, with outside information on the fraction of Independents, we can identify the ratio α = ε/τ. We can use information on this ratio to place an upper bound on the coefficient associated with the implementability of the optimal seat-vote curve. In particular, we can show that ( ε 2τ + ε (τ + ε)ln(1+ ε )) α/2, (25) τ andthisimpliesthatasufficient condition for implementability is: π I 2min(π D,π R ). (26) α Clearly as the ratio α approaches unity, this sufficient condition converges to π I 2min(π D,π R ), which is just that given in Section 2.3. For any given α<1, however, this ratio provides additional information to the researcher attempting to verify the conditions for implementation. Substituting in the expression for α from equation (23) and using the fact that µ = π D + π I /2, this sufficient condition can be re-written as π I 2min(µ 3σ, 1 µ 3σ). (27) Thus, the fraction of Independents must be below a critical value, the calculation of which only requires information on the statewide mean and standard deviation of the Democratic vote share. 3.4 Estimating the welfare gains from socially optimal districting Given that we only observe voting outcomes, which do not reveal the intensity of voter preferences for one party over another, the parameters of the surplus expression (β,γ) in equation (12) are clearly not identified in the empirical analysis. We can, however, use the theoretical restriction on the ratio of these parameters in order to calculatetherangeofproportionatewelfaregains. To this end, first note that the percentage increase in aggregate welfare from socially optimal districting can be written as follows: 12

14 G = E[W (So (V ))] E[W (S(V ))] (28) E[W (S(V ))] Then, using the expressions from equations (12) and (13) and dividing through by β, wehave that: G( γ γ β )= β E[(So (V ) S(V )) 2 ] 1 γ β {c + E[S(V )2 ] 2E[S(V )S o (29) (V )]} Recall from the theoretical section that the ratio γ/β is the fraction of the surplus a partisan obtains from having a perfectly congruent legislature that is dissipated by having a legislature composed entirely of the opposition party. When parties are not that polarized in terms of their underlying ideologies or when the legislature is responsible for choosing only policies on which there is little disagreement across ideologies (for example, spending on public safety and highway maintenance) this ratio may be close to zero. When parties are polarized and are choosing policies on which there is strong ideological disagreement (such as the level of transfer payments for the poor or the regulation of abortion), this ratio may be close to one. In the former case, districting is not very important, while in the latter case it is crucial to citizen welfare. Using the restriction that γ/β [0, 1], we can thus bound these proportionate welfare gains as follows: 0 G( γ ) G(1). (30) β Because the upper bound will only be relevant for legislatures in states in which parties are polarized and which choose policies on which there is strong disagreement, we will provide welfare calculations for different values of this key ratio (γ/β) in the empirical application to follow. We next turn to the measurement of this upper bound. Inserting equation (21) into equation (29), we have that G( γ β )=( γ β ){(ξo ) 2 +2ξ o r o µ +(r o ) 2 (µ 2 + σ 2 ) 2E[(ξ o + r o V )S(V )] + E[S(V ) 2 ]} 1 ( γ β ){c + E[S(V )2 ] 2ξ o E[S(V )] 2r o E[VS(V )]} (31) where r o = 2τ represents optimal responsiveness and ξ o = µ(1 2τ) represents the vertical intercept of the optimal seat-vote curve. Notice that we can express the constant as c = µ + 2 3τσ ε [ε 2 /3+τ 2 /3 1/4], and hence, given a particular value of the ratio γ/β and the parameter τ, this expression can be evaluated by computing three moments of the equilibrium seat-vote curve: E[S(V )], E[S(V ) 2 ], and E[VS(V )]. 13

15 We next show that these three expectations can be expressed as functions of the district-specific vote thresholds. To this end, let the associated range of average votes be [V, V ] and the districtspecific votethresholdsbe(vi ), and let i be the smallest district such that Vi V and i be the largest district such that Vi V. 4 Then, using straightforward area calculations, we can convert the three key expectations into functions of the district-specific vote threshholds as follows: and where V i = V i E[S(V )] = E[S(V ) 2 ]= E[VS(V )] = i+1/n X (i 1/n) V i V V, (32) i=i i+1/n X (i 1/n) 2 V i V V, (33) i=i i+1/n X i=i Vi (i 1/n) V V V, V = V V,and V i+1/n i i = Vi+1/n V i for i = i +1/n,..., i. 4 Application to U.S. state legislatures To illustrate the approach, we study redistricting plans for elections to U.S. state legislatures. We prefer this setting over the U.S. House because, in federal redistricting, state officials control the redistricting process and each redistricting plan thus only partially contributes to the resulting allocation of national seats across parties in Congress. Redistricting plans for state legislatures are also controlled by state officials, and redistricting plans thus perfectly correspond to changes in seats in state legislatures. In the empirical application, we examine state districting plans put in place during the early 1990s. For consistency with the theoretical framework, we focus on states with single-member districts; many states elect multiple members from each district to state legislatures. In addition, given the bicameral nature of state legislatures, we follow the existing empirical literature on redistricting and focus on the elections to the lower house. As shown in Table 1, we have complete data for 28 states that elect representatives from single-member districts, and most of these states adopted redistricting plans in 1992 and then again in 2002; for these states, we thus examine state (34) 4 As pointed out in Coate and Knight (2005), any seat-vote curve S(V ) may be equivalently described by a triple {i, i, (Vi )i i=i } - the so-called inverse seat-vote curve. The interpretation is that districts i =1/n,.., i 1/n are safe Democrat; i = i +1/n,.., 1 are safe Republican and Vi is the vote threshold for competitive district i {i,..,i}. 14

16 legislative elections held in 1992, 1994, 1996, 1998, and States deviating from this pattern of elections include Virginia, which has elections in odd years and adopted redistricting plans in 1991 and 2001, and Colorado, whose district lines were redrawn in 1998 following litigation over therepresentationofminoritygroups in the state legislature. 4.1 Data sources Recall from the empirical methodology section that our approach requires data on district-specific voting returns and the statewide fraction of voters identifying as Independents. We use data from Ansolabehere and Snyder (2002) on state legislative election returns, focusing only on returns for the two major party candidates and computing support for the Democratic candidate as the fraction of the two-party vote. We also use estimates of the fraction of Independent voters in each state. These data are derived from annual New York Times surveys in which voters are asked to self-identify as a Republican, Democrat, or Independent. 5 In order to compute the time-invariant fraction of Independents for each state, we take averages across the years listed in Table Estimating the moments of the voting distributions Consider some legislative district i in a particular state. Suppose that the state has legislative elections in 1992, 1994, 1996, 1998, and Assume that all these elections are contested in district i and let the vote share received by the Democratic candidate in year t be V it. Then, we form our estimates of µ i and σ i by just taking the mean and standard deviation of these vote 5 These data were downloaded from the website wright1/cbs7603 pct.zip. 15

17 shares; that is, 6 bµi = P V it t 5 bσ i = q P t (V it bµ i ) 2 5. (35) A problem that arises with this procedure is that many elections are in fact uncontested. In our data, roughly 30 percent of state legislative elections are uncontested, and the support for the Democratic candidate is thus censored for these elections. One potential solution to this problem would be to simply assume that uncontested Democratic victories correspond to safe Democratic seats and that uncontested Republican victories correspond to safe Republican seats. After all, it is only the vote thresholds for competitive districts that matter for the seat-vote curve. The primary drawback of this approach is that both the equilibrium and optimal seat-vote curves depend upon the statewide moments (µ, σ), which require explicit measures of the district-specific moments (µ i,σ i ) for both contested and uncontested districts. To get around this problem, for districts with at least two contested races, we simply estimate these two moments using voting data from these contested elections; that is, we use the formulas in equation (35), but with a smaller number of elections. For the 25 percent of districts that have either zero or only one contested race, we predict the district-specific moments by first regressing the mean and standard deviation of vote shares in contested districts on voter characteristics taken from Census data. These data, which are published in Barone et al (1998) are available for state legislative districts and include the fraction of residents living in urban areas, the fraction living in suburban areas, household income, percent of residents with a college degree, percent over age 65, percent African American, and percent Hispanic. To implement this prediction approach, we assume that the mean Democratic vote share in 6 An obvious limitation of this approach is that because of the limited time dimension of the panel, our estimates of the district-specific moments will be noisy. As an alternative approach, which we leave for future work, one could estimate statistical models in which the moments of the voting distribution are parameterized in terms of observed and unobserved district characteristics; the number of parameters to be estimated is smaller and hence estimates of district-specific moments should entail less noise in this case. For example, using the definitions of the district-specific moments, we can re-write the district-specific voteasv i = µ i + σ i w,where w = 3 (1/2 m) and is distributed uniform with mean 0 and variance 1. Adding a time dimension (t) to the model, we have that V it = µ i + σ i w t. Then, we parameterize the mean (µ i = Xi 0β + η i)andvariance(σi 2 =exp(xi 0γ + υ i)) of the voting distribution, where (η i,υ i ) are district-specific randomeffects, which are normally distributed with variances equal to ση 2 and p σν, 2 respectively. Substituting in these parameterizations, we have that V it = Xi 0β + η i + u it, where u it = exp(xi 0γ + υ i)w t. Using standard panel data estimation techniques, the researcher could then estimate the parameters of the model (β, γ,ση 2 and σν). 2 In this framework, the district-specific moments,and hence the equilibrium seat-vote curve, are random from the econometrician s point of view. Simulating the model by drawing random effects from the normal distribution, the researcher could then make statements about features of the distribution of the seat-vote curve and the associated welfare gains from socially optimal districting. 16

18 district i takes the following form, which restricts the value to be between zero and one: µ i = exp(x0 i β) 1+exp(Xi 0β) (36) where X i are voter characteristics and β represents parameters to be estimated. For estimation purposes, this equation can be written in log-odds form: ln(µ i /1 µ i )=X 0 iβ (37) Similarly, we parameterize the standard deviation of the voting distribution as follows: ln(σ i )=X 0 iγ (38) In order to increase the power associated with this prediction approach, we run this regression across all contested districts in the United States, implicitly assuming that the mapping from observed characteristics into partisan affiliation is homogeneous across states. While this assumption is somewhat restrictive, we do include a set of state dummy variables, thereby allowing two districts with identical observable characteristics but in different states to have different fractions of voter types. Table 2 provides the results from these two prediction equations. As shown in column 1, the partisan advantage for the Democratic party (µ i ) in contested districts is increasing in the percent urban and suburban (both of these are relative to the omitted category - percent rural), percent with a college degree, percent over age 65, percent African American, and percent Hispanic but is decreasing household income. Interestingly, the coefficients from the regression of these covariates on the standard deviation of the voting distribution, which is proportional to fraction of Independent votes (µ i ), are mostly reversed. As shown in column 2, the likelihood of being an Independent voter is decreasing in all of the observable measures. These coefficients and state-specific constant terms, which are not reported in Table 2, are then used in order to predict these two moments in uncontested districts. With our measured district-specific moments in contested districts and predicted moments in uncontested districts, we can now estimate the statewide moments in a straightforward manner: bµ = 1 n 17 nx bµ i (39) i=1

19 bσ = 1 n nx bσ i With estimates of the district-specific and statewide voting moments in hand, we proceed to estimation of the equilibrium seat-vote curve. 4.3 Equilibrium seat-vote curves Figure 1 plots the equilibrium seat-vote curve as estimated for each state in the sample. As shown, the range of possible statewide support for the Democratic Party seems reasonable. For example, in New York, typically considered a heavily Democratic state, the party receives support on the range [0.57,0.72], while in Utah, typically considered a heavily Republican state, the Democrats receive support on the range [0.34,0.51]. Notice also that the equilibrium seat-vote curve is close to linear in some states, such as Mississippi, New York, and Ohio, while it has important nonlinearities in other states, such as Delaware, Florida, and Montana. These differences in the degree of non-linearity across states highlight the importance of allowing for a general functional form in estimating equilibrium seat-vote curves. Column1ofTable3reportsameasureoftheresponsiveness of each of these equilibrium seatvote curves. This measure is obtained by computing the slope of the linear seat-vote curve that best approximates the equilibrium seat-vote curve. 7 Notice that in every state, responsiveness exceeds 1, and, in the vast majority of states, it exceeds 2. Column 3 of Table 3 reports the partisan bias associated with the equilibrium seat-vote curves. This is measured by S(1/2) 1/2 andisthedifference between the fraction of seats that the Democrats hold when they get half the statewide vote and 1/2. Notice that in the four states for which V =1/2 doesnotlieintherangeofpossibledemocraticvoteshares,thismeasure cannot be computed. The interesting thing to note is that in all but three states, the equilibrium seat-vote curve is biased towards the Republicans. The average bias of 4.27 percent implies that, when voters are equally split, Republicans would secure of the seats on average, relative to percent for Democrats, and would thus hold a significant advantage of 8.54 percent in the legislature. i=1 7 To be more precise, define the linearized equilibrium seat-vote curve to be S l (V )=1/2+b +r(v 1/2), where b and r are chosen in order to minimize the expected square distance between the equilibrium and the linearized equilibrium seat-vote curves. That is: r =cov[v,s(v )]/var(v )andb = E[S(V )] 1/2 r[e(v ) 1/2]. Then our measure of responsiveness is r. 18

20 4.4 Optimal seat-vote curves For comparison purposes, Figure 1 also includes the optimal seat-vote curves. These are plotted under the assumption that responsiveness, 2τ, is at its maximal level 1/(1 + α) ineachstate (see equation (24)). It is apparent from the Figures, that the equilibrium seat-vote curve is more responsiveness than the optimal seat-vote curve in all cases, suggesting that, if anything, U.S. state legislative elections are too competitive and that there are too few safe seats from a social welfare perspective! Column 2 of Table 3 reports the responsiveness of the optimal seat-vote curve. As noted in the theoretical section, optimal responsiveness is always below one, while the responsiveness associated with the equilibrium seat-vote curves substantially exceeds one in all cases. Column 4 of Table 3 reports the partisan bias associated with the optimal seat-vote curve, which is defined as S o (1/2) 1/2. It is notable that in all but two states the bias associated with the equilibrium seat-vote curve is less than that associated with the optimal seat-vote curve. Thus, the partisan bias towards the Republicans exhibited in the equilibrium seat-vote curves, cannot be justified as optimal partisan bias. Indeed, in twelve states, the equilibrium seat-vote curve is biased towards the Republicans, when it should optimally be biased towards the Democrats! A reasonable objection to this comparison of partisan bias under the optimal and equilibrium seat-vote curves is that it just tells us about the properties of the curves at the vote share V = 1/2. For a more global comparison, we computed the expected Democratic seat share (E(S(V ))) under the equilibrium and optimal seat-vote curves (E(S(V )) and E(S o (V ))). Column 6 of Table 3reportsthedifference in this expected seat-share under the equilibrium and optimal curves. When this difference is positive, the expected Democratic seat share is higher than optimal. The interesting thing to note is that, in this expected seat sense, there appears to be no obvious bias in one or the other direction. In almost half the states, the expected Democratic seat share is higher than optimal and the average difference is very close to zero. This makes us hesitant to draw any strong conclusions concerning the general direction of bias in U.S. state legislative elections. 4.5 Verifying the conditions for implementation As shown in Table 4, the conditions for implementation, the restriction that the fraction of Independents must be below a maximal level is satisfied in every state. The state closest to not satisfying the requirements is Rhode Island, which is reported to have 51% Independents, just 19

21 slightly below the maximal level of 55%. The next closest state is Connecticut, which is reported to have 42% Independents, relative to the maximal level of 65%. In summary, these results demonstrate that the conditions for implementability of the optimal seat-vote curve are indeed permissive, being satisfied in all of 28 states included in our analysis and by a large margin in all cases except for the state of Rhode Island Welfare gains Given that the optimal seat-vote curve is implementable in all states, we next turn to measuring the welfare gains associated with socially optimal districting. To begin with, we compute the percentage welfare gains under the assumptions that the ratio γ/β is at its maximal level (i.e., 1) and that optimal responsiveness is at its maximal level in each state (i.e., 1/(1 + α)). As shown in Table 5, the percentage welfare gains to socially optimal districting vary from just 0.31% in Pennsylvania to 6.5% in Rhode Island. A visual comparison of the equilibrium and optimal seatvote curves in Figure 1 supports these welfare calculations as the optimal and equilibrium seat-vote curves are similar in states with low potential welfare gains but quite different in states with large potential welfare gains. 9 On average, states could increase welfare by roughly 2%, suggesting that the gains from socially optimal districting are typically small relative to the surplus that is generated by state legislatures. Of course, if this surplus is itself very large, then these gains could be quite large in monetary terms. But without further assumptions on the underlying welfare parameters β and γ, these percentage gains in welfare cannot be converted into monetary terms. For a sense of how these results depend on the specific assumptions about the parameters γ/β and 2τ, Table 6 reports the average welfare gains associated with socially optimal districting 8 Note that in evaluating the conditions for implementation, we have not used external information on the statewide fraction of Democrats and Republican. It is comforting to note, however, that the implied fraction of partisans from voting behavior are highly correlated with the fraction of voters self-reporting as partisans. Using the fact that the implied statewide fraction of Democrats and Republicans are given by π D = µ π I /2and π R =1 µ π I /2, we calculate correlation between the implied and reported fraction of voters of roughly 0.8 for both Republicans and Democrats. 9 While a welfare comparison of the optimal seat-vote curve, which is continuous, and the measured seat-vote curve, which is a discrete step-function, is somewhat artificial, we do so in order to apply the results from the companion theoretical paper, which assumed a continuous seat-vote curve, in order to verify that the optimal seatvote curve is implementable. To provide a sense of the error associated with this approximation, we have derived a discrete optimal seat-vote curve, which is a step-function approximation of the continuous optimal seat-vote curve. Welfare associated with this discrete optimal seat-vote curve is very similar to welfare under the continuous optimal seat-vote curve, and the approximation error associated with the use of a continuous optimal seat-vote curve is thus small in practice. The small size of this error should not be surprising given that the discrete and continuous optimal seat-vote curve converge as the number of districts grows large. As shown in table 1, states have a large number of districts, averaging over 100 districts and ranging from 41 in Delaware to 203 in Pennsylvania. 20

22 arising under different parameter values. We allow the ratio γ/β to vary from 0.25 to 1 and allow 2τ to vary from 0.25/(1 + α) to1/(1 + α) in each state. The notation η refers to the numerator in the ratio η/(1 + α), so that the case in which 2τ is set equal to 0.25/(1 + α) ineachstate corresponds to η =0.25; the case in which 2τ is set equal to 0.5/(1 + α) ineachstatecorresponds to η = 0.5; etc. As shown, holding the ratio γ/β constant, the welfare gains to socially optimal districting, averaged across all states, are uniformly increasing as the parameter η is reduced. This pattern reflects the fact that reductions in the parameter η are associated with reductions in optimal responsiveness, which, as shown in Table 3, was already below the responsiveness associated with the equilibrium seat-vote curve in the baseline analysis. Holding the parameter η constant and reducing the ratio γ/β, we have that welfare is uniformly decreasing. As explained above, districting matters less for welfare as this ratio decreases, reflecting the fact that the conflict between parties over the available policy choices is less severe. Perhaps the most noteworthy lesson to be drawn from Tables 5 and 6 is that the welfare gains from socially optimal districting are relatively small as a proportion of the total surplus generated by state legislatures. In principle, there may be two reasons for this. The first is that the districting plans that states actually implement are relatively close to optimal plans. The second is that, because of the diverse ideological make up of the U.S. states, the aggregate gains that can be obtained by tinkering with districting plans are relatively small. To get a feel for which of these views is correct, we have computed the proportionate welfare gains that would arise from implementing the optimal seat-vote curve over the welfare minimizing seat-vote curve. We take the latter to be the seat-vote curve that makes the seat share of the party with the smallest partisan base equal to Theideaisthatifthesegainsaresmall,thenthesecond view is correct. Table 7 reports the results under the same parametric assumptions that underlie Table 5. They strongly suggest that it is the first view that is correct: the states seem to be doing districting in a way that is generating seat-vote curves that are relatively close to optimal. A further interesting benchmark of comparison is the seat-vote curve that would be generated by a proportional representation electoral system. Under such a system, seats are allocated in strict proportion to statewide votes and hence the implied PR seat-vote curve is S(V ) = V. As shown in Column 1 of Table 8, all states would experience a welfare gain by implementing 10 An important objection to this, is that such a seat-vote curve may not be implementable. We agree with this objection, but time constraints have prevented the computation of the welfare minimizing implementable seat-vote curve. 21

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