ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY

ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY

NATHAN CANEN, MATTHEW O. JACKSON, AND FRANCESCO TREBBI

Abstract. We develop a model of endogenous network formation as well asstrategic interactions that take place on the resulting network, and use it tomeasure social complementarities in the legislative process. Our model allowsfor partisan bias and homophily in the formation of relationships, which thenimpact legislative output. We identify and structurally estimate our modelusing data on social and legislative efforts of members for each of the 105th-110th U.S. Congresses (1997-2009). We find large network effects in the formof complementarities between the efforts of politicians, both within and acrossparties. Although partisanship and preference differences between parties aresignificant drivers of socializing in Congress, our empirical evidence paintsa less polarized picture of the informal connections of members of Congressthan typically emerges from congressional votes alone. Finally, we show thatour formulation is useful for developing relevant counterfactuals, including theeffect of political polarization on legislative activity (and how this effect can bereversed), and the impacts of networks in the congressional emergency responseto the 2008-09 financial crisis.

Date: February, 2020

Canen: Department of Economics, University of Houston.Mailing Address: 3623 Cullen Boulevard, Office 221-C, Houston, TX, 77204, USA.E-mail: [email protected].

Jackson: Department of Economics, Stanford University, CIFAR Fellow, and External Faculty of Santa FeInstitute.Mailing Address: 579 Serra Mall, Stanford, CA, 94305, USA.E-mail: [email protected].

Trebbi: Vancouver School of Economics University of British Columbia, and CIFAR Fellow.Mailing Address: 6000 Iona Drive, Vancouver, BC, V6T 1L4, Canada.E-mail: [email protected].

Kareem Carr and Juan Felipe Riano provided excellent assistance in the data collection. The authorswould like to thank Matilde Bombardini, Gabriel Lopez-Moctezuma, and seminar participants at variousinstitutions for their comments and suggestions. Particularly, we thank Antonio Cabrales and Yves Zenoufor fruitful discussion of their model. Jackson gratefully acknowledges support by the Canadian InstituteFor Advanced Research and the NSF under grant SES-1629446. Trebbi gratefully acknowledges support bythe Canadian Institute For Advanced Research and the Social Sciences and Humanities Research Councilof Canada. A previous version of this work circulated under the title “Endogenous Network Formation inCongress”.

ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 1

1. Introduction

Deliberative bodies, especially large ones, rely on informal interactions in order to function

productively. Individuals form relationships with each other to craft and pass legislation.

Because of the salient role of interpersonal ties in the legislative process, its study dates at

least to the 1930s (Routt, 1938), but only in the last fifteen years has this area of research

come to more prominence (Lazer, 2011).

The challenge of simultaneously modeling network formation and political decision-making

is one reason for this delayed uptake. Interpersonal ties are not drawn at random – legislators

strategically choose their connections (e.g., with whom to cooperate and collaborate). In

turn, then benefits are strategic as well – having key allies can enable a politician to craft

and pass legislation that would otherwise not be possible. Finally, these decisions are made

in an environment rife with identity-based (party) affiliation with an immense number of

possible connections, making empirical analysis challenging. The construction and analysis

of the model presented here addresses this challenge.

Our model simultaneously captures endogenous network formation, strategic decisions

made on this network, and homophily: group identity matters in payoffs and in link forma-

tion. We prove statistical identification of the parameters driving each of these features.1

In particular, we generalize the tractable and powerful framework of Cabrales, Calvo-

Armengol, and Zenou (2011) in several important directions. As in Cabrales et al. (2011), our

model has two strategic choices: legislators choose both how much socializing to do with other

politicians as well as how much effort to exert crafting and passing legislation. Socializing

efforts result in formed relationships that increase the success of legislative efforts, and so

social and legislative efforts are complements. Importantly, social and legislative efforts are

also complementary to those of the other politicians with whom a given politician has ties,

both within and outside of his/her party. The two main generalizations in our model are as

follows. First, while in Cabrales et al. (2011) relationships form completely at random, our

model admits homophily and allows social ties to form at a different rate within compared to

across groups – so that, for instance, legislators can collaborate with members of their own

party at a different rate than with members of the opposition. Second, we allow the returns

to social and legislative efforts to be party-specific. This captures important institutional or

time-specific differences across parties, including who holds a majority, which can make a

big difference in the returns to effort.

We structurally estimate our model employing data on cosponsorship and legislative efforts

of members of House of Representatives from the 105th-110th U.S. Congresses.

A first empirical finding is that the complementarities among politicians are significant

and stable across our sample period. The estimated social marginal multiplier on legislative

effort is between a tenth and a fourth of the direct incentive for legislative effort, with larger

1This model is the first to capture all of these features, which should be useful beyond the application tolegislative production. In Section 2.6, we compare our model to those in the literature.

2 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY

values for Democrats.2 This means that a nontrivial fraction of incentives for efforts of

politicians appears to be driven by what other politicians are doing.

We then examine differences between Democrats and Republicans. We find that the two

parties have different base payoffs from passing legislation, both in terms of average and

variance (both higher for Democrats). These differences lead to higher levels of social and

legislative efforts by Democrats, all else held equal.

Further, we find evidence that partisan bias is an empirically relevant feature and the

model with biased interactions fits the data significantly better than a model with no bias.

We also show that our model that imputes endogenous network formation has better in-

sample properties (model fit) than models of social networks in Congress based on exogenous

or predetermined graphs, including those based solely on cosponsorships, alumni connections

and committee membership. As such, legislative behavior can be better explained by ex-

plicitly modeling the strategic decisions that drive these underlying networks, than using a

proxy for them.

We also show, however, that social interaction in the U.S. Congress is far from being an

exclusively partisan affair. The data are more nuanced than the common narrative of a

balkanized Congress, segregated along party lines, that has emerged from recent literature

mostly based on post-1980 congressional roll call evidence (Fiorina, 2017). We find that

intermediate levels of partisanship – a partisan bias is in the range of 5-15 percent – fit the

data significantly better than a fully partisan model where 100 percent of interactions are

exclusively within party. It is hard to reconcile the thousands of bipartisan cosponsorships

in recent data with a hypothesis of unmitigated polarization between parties. The stark

posturing and divisive language (Gentzkow and Shapiro, 2015; Gentzkow, 2017), and some

metrics of formal political activity, miss bipartisan interaction that more informally takes

place among legislators – especially with respect to the bulk of less controversial bills that

constitute day-to-day law and budget making.

We then assess the specific equilibrium within each Congress among the multiple equilibria

in this class of games. We show that the estimated equilibria are interior and stable, and

that social effort is (inefficiently) under-provided in all Congresses. There are externalities

in efforts, given the complementarities, that are not internalized by the agents.

Our model and structural approach provide comparative statics, which enable us to per-

form counterfactuals based on our estimated parameters. In a first illustration, we estimate

the effects of political polarization, such as a drifting to the right of Republican types, on

effort levels. We find that an increase of GOP ideological extremism of about 20% leads

to an average decrease of 8.3% in social effort for Democrats and a decrease of 7.5% for

Republicans. Further, we show that this decrease can be reversed by an improved selection

of legislators even by one party alone, as the higher legislative activity by that party can

2As it will be clear in the analysis that follows, the parameter multiplying the full product of social andlegislative efforts ranges from 0.03 to 0.05, which when multiplied by other legislators’ efforts of 4 to 6, andsocial efforts around 1, leads to a multiplier of 0.12 to 0.3. This is compared to direct incentives for legislativeactivity ranging from 1.0 to 1.4.


spill over to the other. We also find that increases of bipartisanship (i.e., cross-party interac-

tions) do not imply unambiguous increases in governmental activity and legislative success.

A party can benefit from being less exposed to less-engaged low types in the opposition.

In another exercise, we examine how the amount of legislation that was generated in the

emergency response to the 2008-09 financial crisis would have changed if the Democrats had

not taken over the House in 2006. This emerged as a narrative for the evolution of the anti-

government intervention and Tea Party movement of 2010 (Mayer, 2016). Here we find a

quantitatively small change (between 1-2 percent lower likelihood of success) in the amount

of emergency legislation that would have passed had the legislature had remained identical

to the Republican 109th Congress, with virtually no difference in final outcomes.

1.1. Relation to the Literature. From the theoretical perspective, this paper contributes

to a literature that examines peer-influenced behavior when accounting for the endogeneity

of networks.3 Our model allows for homophily, so that people interact more within groups

than across groups, and also allows the value of social interaction to differ across groups.

This meaningfully generalizes the model of Cabrales et al. (2011) to have group membership

impact the value of social interaction and the rate at which that interaction happens within

compared to across groups. Both factors matter significantly in our empirical application.

Introducing a theoretically tractable and econometrically feasible form of asymmetry in the

process of socializing of members of Congress within this framework should be a valuable

base for other applications that involve multiple groups.4

This paper also contributes to the growing literature showing that social networks matter

in legislative environments. For instance, Fowler (2006) used a connectedness measure based

on cosponsorship to show that more connected members of Congress are able to get more

amendments approved and have more success on roll call votes on their sponsored bills.5 Also

using cosponsorship links, Cho and Fowler (2010) show that Congress appears subdivided in

multiple dense parts tied together by some intermediaries. These network features correlate

with legislative productivity over time (number of important laws passed, as defined by

Mayhew, 2005).6

3See Bramoulle, Djebbari, and Fortin (2009), Mauleon et al. (2010), Goldsmith-Pinkham and Imbens (2013a),Goldsmith-Pinkham and Imbens (2013b), Manski (2013), Jackson (2013), Badev (2017), Jackson (2019),Hsieh and Lee (2016), Mele (2017), Baumann (2017) (and see Jackson (2005), Jackson (2008), Jackson andZenou (2015) for surveys of the network formation and games on networks literatures).4Homophily in peer group formation is also theoretically explored in Baccara and Yariv, 2013, who furtherexplore group stability.5A similar study is Zhang et al. (2008).6There is other work on cosponsorhip. For example, Aleman and Calvo (2013), Koger (2003), and Brattonand Rouse (2011) study the incentives for cosponsoring in different settings (focusing on ideological similar-ity, tenure, etc.). Beyond their role in social networks, Wilson and Young (1997) study the signaling contentof cosponsorships, noting that cosponsorship is a cheap way of signaling to the median voter about one’s con-gressional activity. They identify three different explanations for cosponsorships and their possible signalingimpact: (i) bandwagoning (signaling strong support for the bill), (ii) ideology, and (iii) expertise. They finda null to moderate effect of cosponsorship on bill success, as measured as successive progress of the billsthrough Congress hurdles. Kessler and Krehbiel (1996) instead point out that the timing of cosponsorshipswould indicate that it is not as much a signaling to voters, as to other politicians (for example, they showthat extremists seem to cosponsor earlier). Still in the context of bill sponsorships, Anderson et al. (2003)


The network analysis of legislation is growing, and provides increasing evidence that social

relationships matter substantially and are causal in nature. For example, Kirkland (2011)

shows a correlation between bill survival and weak ties of the sponsor for eight state legisla-

tures and for the US House of Representatives. Cohen and Malloy (2014) employ identifica-

tion restrictions aimed at ascertaining causal effects of networks on voting behavior (using

the quasi-at-random seating arrangements of Freshman Senators).7 Rogowski and Sinclair

(2012) also use random spatial arrangements to estimate the causal effect of interactions

on legislative voting and cosponsorship in the House. In particular, the authors use lottery

office assignment affecting certain classes of members of the House of Representatives, not

finding a significant affect of office proximity on co-behavior.8 Harmon et al. (2019) studies

the role of exogenously shifted social connections within the European Parliament also using

seating arrangements.

Importantly, none of those papers model interaction as a choice variable. In an important

theoretical contribution, Squintani (2018) studies endogenous legislative networks, and the

role of ideological positions on information transmission. Our focus is on legislative produc-

tion rather than information transmission, and our analysis enables us to see how incentives

to socialize differ across parties, relate to legislative productivity, and have changed over time.

The role of political networks in connection to special interest politics is studied in Groll and

Prummer (2016) and Battaglini et al. (2018). In particular, Battaglini et al. (2018) focus

on legislative effectiveness of legislators based on their Bonacich centrality - taking it as ex-

ogenous, but employing a Heckman two-step procedure based on alumni networks to correct

for network endogeneity in the empirical analysis.9 In a complementary effort, we present a

tightly connected theoretical and empirical structure of endogenous network formation and

how it contributes to legislators’ decisions. This allows us to estimate socialization efforts

and how they affect legislation, and enables us to do comparative static/counterfactual ex-

ercises based on the estimated model. In subsequent work, Battaglini et al. (2019) study

a game of network formation with strategic decisions made on the graph in Congress. The

authors focus on the choice of directed links using a setting akin to general equilibrium:

while politicians can choose who to link to directly, in equilibrium the solution is charac-

terized by “prices” that clear the market for connections. We differ by modeling strategic

interactions within a game-theoretic framework and by proving identification (the focus in

find correlations with legislative productivity (i.e. the bill passing through different stages in Congress) forCongress member who sponsor more bills and use more floor time (albeit at a declining marginal rate).7For a similar approach see also Masket (2008).8They do not interpret these results as an absence of peer effects in Congress, but rather that office proximity- and the exogenous changes in connections caused by it - do not significantly explain congressional behavior.In fact, the most important network effects might be those from endogenously formed connections. However,these are hard to identify in reduced form and cannot be captured in their study. As they conclude, “Payingattention not only to the structure of networks but also to how that structure came to be can help remedymany of the difficulties in providing causal evidence for network effects.” (p. 327). Our structural model fillsthis gap.9For a similar reduced-form identification strategy, we exploit differential characteristics of networks withinthe same bill across Senate and House. The results are available from the authors upon request.


the cited paper is on Bayesian estimation). We also conduct multiple policy counterfactuals,

which are precluded in that alternative framework.

2. The Model

2.1. Legislators, Parties, and Partisanship. The legislature, henceforth referred to as

“Congress”, is composed of a set N = 1, 2, ..., n of members. For simplicity, we focus

on one chamber (e.g. the House), and clearly although we refer to Congress, the model

applies to a variety of deliberative bodies, legislatures, committees, and organizations more

generally.

The set of politicians N is partitioned into parties, with a generic party denoted P`.

Each party P` has a level of partisanship p`. This can be thought of as a structural

form of homophily. In particular, members of party ` spend a fraction of their interaction,

p`, at exclusively party ` events, so only mixing and meeting with own-party events, and

the remainder, 1 − p`, at events in which they mix with members of all parties. This

can include party and caucus meetings, joint sessions, fund-raising events, committee works,

social gatherings and formal events, etc. For our period of interest, examples of party-specific

events are closed sessions called Party Conferences for Republicans and Party Caucuses for

Democrats (their respective chairs represent the number 3 position of official party leadership

rankings).

Politician i is from party P (i), and p(i) denotes the level of partisanship of politician i’s

party.

In our empirical analysis, there are two parties, 1, 2, and then we index the n politicians

so that the first q of them belong to P1 = 1; ...; q and the remainder to P2 = q + 1; ...;n.Let q ≥ n/2, so that party 1 is the majority party.

2.1.1. Socializing. Each politician chooses an effort level of how much he or she socializes

(i.e., interacting with other politicians), denoted si ∈ <+. It is via this socializing that he

or she forms connections with other politicians.

The network G = gi,ji,j∈N that arises from the vector of social efforts, s, is described

by10:

gij(s) = sisjmij(s),

where if j ∈ P (i) then

mij(s) = p(i)p(j)∑

k∈P (i),k 6=i p(k)sk+ (1− p(i)) (1− p(j))∑

k 6=i(1− p(k))sk,

10This is for the case in which sj > 0 for at least two people in each party. If other agents are not putting insocial effort, then there can be nobody to match with, and then some of these equations do not apply (theydivide by 0, and if all s are 0). In those cases the matching is described as follows. If at most one sj > 0,then set mij = 0 for all ij and the entire network equal to 0. If there are at least two people with sj > 0,but also at least one party with sj > 0 for no more than one agent, then set mij = gi,j = 0 for all membersof a party that has nor more than one sj > 0, but use the remaining above specified equations for gi,j ’s forany other combinations.


and if j /∈ P (i) then

mij(s) = (1− p(i)) (1− p(j))∑k 6=i(1− p(k))sk

.

So, politicians meet their own party members in two ways: at their own events and at

bipartisan events. They meet members of the other party only at the bipartisan events.

Politicians are met with the relative frequency with which they are present at events.

This specification captures key features of how politicians socialize in practice, while main-

taining tractability of the model. First, members of the same party meet more often and,

hence, are more likely to connect with one another.11 Second, more social members (those

with higher si, which we will show to be the higher types in equilibrium) are more likely to

connect to members in their own party.12 Third, socialization is not deterministic: observ-

able decisions do not fully determine social connections. In this model, two socially active

members with equal characteristics will not necessarily link. As in any activity, we allow for

randomness to be a component of social relationships, although driven by efforts and social-

ization characteristics. In practice, personalities, unobserved characteristics and preferences

play a large role in such connections. Furthermore, the events in which legislators interact

are often unobserved by researchers and the public.13

Our specification with a biased random matching protocol allows us to capture both these

realistic socialization features, as well as maintaining a tractable framework in which we

can characterize equilibria, obtain theoretical predictions, and test them empirically. As an

alternative one could consider a model of directed links gij, where politician i identifies a

specific partner j deterministically and based on j’s other choices. This appears less tractable

and unrealistic. Not only is the action space gigantic (with 2435 potential directed links just

in the network formation stage), but these links need to endogenously anticipate all pairs’

strategic decisions over the entire network. Our framework instead allows for closed-form

solutions and for a realistic modicum of stochasticity in the formation of certain social ties.

In Appendix A we show that: ∑j 6=i

sisjmij(s) = si,

so that the total number of connections that i makes is proportional relative to si.

When p` = 0 for each `, then this simplifies to coincide with the model of Cabrales et al.

(2011). When p` = 1 for each `, instead, each party is completely cut off from the other.

Then, within each party again Cabrales et al. (2011) applies.14

11As previously described, examples of single party events include fund-raisers and party caucuses.12We will explore these patterns empirically in a later section, showing that these higher types coincide withthose in more influential committees, for example.13These include being at the gym at the same time, going to particular restaurants or attending certain cul-tural events. A journalistic description for the case of the Senate can be found in Roll Call’s piece, https://www.rollcall.com/news/behind_the_doors_of_the_senate_gym-222790-1.html, accessed January 21,2019.14One could also consider endogenizing partisanship. Having three action variables for each party/agentwould render the model intractable analytically. Here we focus on the two that seem most important to

https://www.rollcall.com/news/behind_the_doors_of_the_senate_gym-222790-1.html

https://www.rollcall.com/news/behind_the_doors_of_the_senate_gym-222790-1.html


2.2. Legislative effort and Preferences. The other choice of politicians is their legislative

effort xi ∈ <+. The benefits from legislative efforts are described by:

αixi + φi∑j 6=i

sisjmij(s)xixj.

As in a large class of models, of which Cabrales et al. (2011) is a salient instance, there

is a direct benefit from private effort, with idiosyncratic weight αi. In addition, there are

complementarities in legislative efforts between politicians who have formed connections:

the more effort they both expend, the more likely their legislation is to pass. The size of

this interaction effect is governed by a party-specific parameter φi, whose quantification is

a relevant goal in the empirical analysis that follows. Note that φi = φP (i), but we keep the

first notation for simplicity.

Both forms of effort are costly for a politician. The cost of legislative effort is given byc2x2i , with c > 0, and the total cost of socializing is given by 1

2s2i . The parameter c governs

the relative cost of legislative effort to social effort.

Taken together, the politician’s preferences are the amount of legislation that he or she

produces less the costs of legislative and social efforts. This is given by:

(2.1) ui(xi, x−i, si, s−i) = αixi + φi∑j 6=i

sisjmij(s)xixj −c

2x2i −

1

2s2i .

If G was exogenous and known, equation (2.1) would collapse to the set-up in Ballester

et al. (2006). As a result, equilibrium legislative efforts would be proportional to a weighted

measure of the politician’s centrality. If that were the case, we could base an empirical

specification on this foundation, as done by Acemoglu et al. (2015) in the context of public

good provision on a geographical network. However, in this paper, we assume that the

politician network is endogenous and unknown. As a result, we must model the choice of

social effort that generates the underlying network in a way that can be identified in the

data. The current set-up accomplishes this through equilibrium restrictions and additional

data in an appropriate way, as we show in the next sections.

2.3. A Micro-Foundation Built Upon Reelection Preferences. There are many dif-

ferent ways in which we could justify the preferences in (2.1), as the natural presumption is

that politicians care to maximize the legislation that they pass. Here, we posit a realistic

microfoundation, which we will bring directly to the data.

Politicians care about being reelected and can affect the probability of being reelected

by exerting effort in Congress and by building connections instrumental to having specific

legislation passed (e.g. policy favorable to the politician’s constituents).

Each politician anticipates these effects on his/her reelection chances. More specifically,

each congressional cycle has two periods, 1 and 2, where the second period provides the

reelection incentives that drive activity in the first period. Politicians are career motivated

and exert costly efforts with the aim of increasing their chances of being reelected.

endogenize, and then estimate the third. At the end of Appendix B, we discuss some potential approachesto, and challenges with, endogenizing partisanship that could be addressed in future research.


In period 1, each Congress member can present a policy proposal, which for brevity we

refer to as a “bill”. The bill consists of a policy goal the Congress member intends to

fulfill, for instance passing a statute targeted to his or her constituency, landing a subsidy,

or obtaining an earmark beneficial to firms in the home district. We describe below how

getting i’s policy goal fulfilled maps into an increase in i’s chances of being reelected.

Suppose a politician’s utility is given by:

(2.2) ui = Pr(reelected)− c

2x2i −

1

2s2i .

The choice of xi, the level of legislative activity exerted by i, affects the support for i’s

legislation, Yi, through a function:

Yi = εixi

(∑j∈N

gi,j(s)xj

).

Both i’s own legislative effort, xi, and that of his or her connections in the network,∑

j∈N gi,j(s)xj,

matter for the ultimate support received by i’s bill.

Yi is stochastic and depends also on a random shock εi, assumed to be standard Pareto

distributed with scale parameter γP (i) > 0 and i.i.d. across politicians. We allow γ to be

party-specific, reflecting different average electoral returns by party. We assume that εi is

realized after the choice of x, the vector of xj across all politicians j ∈ N . Because εi is a

shock following the realized legislative support, i must take expectations over its value when

choosing (xi, si).15

The bill is approved if Yi > m, where m > 0 is a generic institutional threshold.16 The

probability of having the bill approved is thus given by:

Pr(Yi > m) = Pr

εi > m

xi

(∑j∈N gi,j(s)xj

)(2.3)

=(γP (i)

m

)(∑j∈N

gi,j(s)xj

)xi,

where we use the distributional assumption on ε.17 Actual passage of the bill sponsored by

i is represented by the indicator function I[Yi>m].

We interpret xi as the observable legislative effort by i, instrumental to the approval of i’s

bill, and we postulate that voters prefer politicians exerting higher legislative effort to lower

effort. We also allow for voters to care about whether in fact the bill passes conditional on

15Also notice that each link between politician i and j is an endogenous function of the social efforts ofeverybody else, hence the dependency gi,j(s) on s, the vector of sj efforts across all politicians j ∈ N .16Naturally, m can be function of a simple majority requirement or even supermajority restrictions.17More generally, one can take Yi to represent the average approval rate of i’s multiple bills. In this case,each b is a separate bill by a politician i. The conditions for our model are unchanged, as long as bills arenot strategically introduced (i.e., the shocks εb are still i.i.d. within i).


effort. That is, we allow for the political principals (the voters) to reward their agent i for

effort xi, networking si, and ultimately luck εi.

To get reelected, the politician must have an approval rate in his/her electoral district that

is sufficiently large. Similarly to Bartels (1993), the electoral approval rate of i is modeled

as a variable Vi:

Vi = ρVi,0 + ζP (i)I[Yi>m] + αixi + ηi

where ηi is assumed to be a mean zero electoral shock, uniformly distributed on [−0.5, 0.5],

and where Vi,0 ≥ 0 stands for the baseline approval rate before the start of the term (i.e.

before period 1 in the model). Hence, this set-up allows for approval rates to be persistent,

but also to react when a politician is capable of getting a bill approved I[Yi>m] or when i

exerts high legislative effort xi. The parameter ζ, which could be equal to zero empirically,

governs the relative importance of a bill actually passing vis-a-vis legislative effort. The

direct effect of xi is captured by αi, which is i-specific and may depend on party affiliation,

majority status, congressional delegation, etc. Finally, notice that, while there is no direct

value to the voters of the politician having more socializing, the value of si matters implicitly,

being instrumental in getting legislation approved.

In period 2, i is reelected if his/her electoral approval level, Vi is larger than an electoral

threshold w < 1. So, the probability of being reelected is given by:

Pr(ρVi,0 + ζP (i)I(Yi>m) + αixi + ηi > w) = min(1 + 0.5− w) + ρVi,0 + ζP (i)I(Yi>m) + αixi, 1,

where we have used the distributional assumption on η.18 We proceed with the empirically

relevant case of when reelection is uncertain (i.e. a politician does not know for sure whether

(s)he will lose or win).19

Note that, in period 1 when making his effort decisions, i does not know the value of εi.

So taking the expectation over ε of the above implies an expected probability of reelection,

when choosing (si, xi), given by:

Pr(reelected) = EεPr(ρVi,0 + ζP (i)I(Yi>m) + αixi + ηi > w)(2.4)

= (1.5− w) + ρVi,0 + ζP (i)

γP (i)

m

∑j∈N

gi,j(s)xixj + αixi

where EεI(Yi>m) = Pr(Yi > m), as given by (2.3).

Let φi = ζP (i)γP (i)

m. Replacing (2.4) into the utility function (2.2) yields:

ui(xi, x−i) = (1.5− w) + ρVi,0 + φi∑j∈N

gi,j(s)xixj + αixi −c

2x2i −

1

2s2i

18The above expression is non negative, since its terms are non negative.19When the probability of reelection is either 0 or 1, we have that si = xi = 0. This can be seen fromequation (2.2). If a politician i cannot influence his reelection prospects, he will not undertake costly effort.This is contrary to what is observed in the data, as described in Section 3, as well as theories of legislativebehavior (e.g. Mayhew (1974)).


Since the terms (1.5− w) and ρVi,0 do not affect the maximization problem, (2.3) can be

rewritten as the specification given in (2.1).

2.4. Solving For Equilibrium. We examine the pure strategy Nash equilibria of the game

in which all politicians simultaneously choose si and xi.

The first order conditions with respect to si and xi that characterize the best response of

politician i imply that interior equilibrium levels of (s∗i , x∗i ) must satisfy:20

(2.5) s∗i = φi∑j 6=i

s∗jmij(s∗)x∗ix

∗j

and

(2.6) cx∗i = αi + φi∑j 6=i

s∗i s∗jmij(s

∗)x∗j .

We rewrite (2.5) as

(2.7)s∗ix∗i

= φi∑j 6=i

s∗jmij(s∗)x∗j ,

To fully characterize equilibria, we work with the same approximation as in Cabrales et al.

(2011). We operate “at the limit”, when the number of politicians grows.21 In particular,

we solve for equilibrium under the assumption that∑

j 6=i s∗jmij(s

∗)x∗j is the same for all i of

the same party.

This implies thats∗ix∗i

is the same for all agents within a party. Using (2.7) in (2.6) yields:

cx∗i = αi + s∗iφi∑j 6=i

s∗jmij(s∗)x∗j

= αi +s∗2ix∗i.

Dividing through by x∗i implies that

(2.8) c =αix∗i

+s∗2ix∗2i

.

Sinces∗ix∗i

is the same for all agents within a party provided that φi is the same for all agents

within a party, (2.8) implies that αix∗i

is the same for all agents within a party. This further

implies that:

x∗i = αiXP (i),

for some XP (i). In addition, the fact thats∗ix∗i

is the same for all agents within a party,

implies that

s∗i = αiSP (i),

20Note that second derivatives are everywhere negative.21Alternatively, this could be justified via a continuum of politicians of each type, or by examining an epsilonequilibrium with a large n.


in equilibrium for some SP (i).22

To get explicit expressions for our empirical analysis of Congress, we now specialize the

analysis to the case of two parties.

For each party j = 1, 2 define

Aj =∑i∈Pj

αi,

Bj =∑i∈Pj

α2i .

Proposition 2.1. The (interior) Nash equilibria of the limit game of this model are positive

solutions to the system given by:

x∗i = αiXP (i), and(2.9)

s∗i = αiSP (i),(2.10)

where

(2.11)S1

X1

= φ1

(p1B1X1

A1

+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2S2X2

(1− p1)A1S1 + (1− p2)A2S2

),

(2.12)S2

X2

= φ2

(p2B2X2

A2

+(1− p2)2B2S2X2 + (1− p1)(1− p2)B1S1X1

(1− p1)A1S1 + (1− p2)A2S2

),

(2.13) cX21 = X1 + S2

1 , cX22 = X2 + S2

2 .

All proofs appear in Appendix A.

If p1 = 1 or p2 = 1, then things reduce to the case of two separate parties with no

interaction across them. That is, they are two copies of the model in Cabrales et al. (2011).

Similarly, if p1 = p2 = 0 then there is no impact of party affiliation, and again the model

simplifies to that of Cabrales et al. (2011). The novel case is when at least one partisanship

level is positive, yet both levels are below 1. This biases the interaction of at least one party,

leaving room for interaction across parties. In this case there will be both social mixing

across different parties and partisanship in socializing.

Generally, there are multiple equilibria. For instance, there is always an (unstable) equi-

librium in which si = 0 for all i. In that case, since no other politician provides effort, a

given politician’s efforts results in no connections and so the best response is also to provide

no effort.

A sufficient condition for existence of an interior equilibrium is as follows.

22In contrast to results in network games with exogenous networks, equilibrium actions in our model are notexpressed as only being proportional to a centrality measure of the network (e.g., Katz-Bonacich centrality).This results from the equilibrium interactions between social and legislative efforts. The legislative effortsstill have to satsify a version of the usual characterization on the margin. Therefore, an empirical approachusing only centrality measures for estimation instead of our structural equations would only capture onedimension of the model’s predictions.


Proposition 2.2. A sufficient condition for the existence of an interior equilibrium is

2c3/2

3√

3≥ max [φ1, φ2] max

[B1

A1

,B2

A2

].

In this setting with two parties and nontrivial partisanship, there will generally be either

two or four interior equilibria (except at a degenerate set of values where the system switches

from two to four equilibria).23

2.5. Pareto Efficient Efforts. Before proceeding with the empirical analysis, we comment

on the Pareto inefficiency of the equilibrium outcomes of the model. This is relevant for a

welfare analysis that checks whether there is over-provision or under-provision of social and

legislative effort in the strategic setting.

Generally, the fact that there are positive externalities in efforts – in particular in legislative

efforts – implies that there is under-provision of effort. In particular, the Pareto optimal social

and legislative effort levels are unbounded: any finite level of efforts are Pareto dominated

by some higher levels.24 Hence, all equilibria are characterized by an “under-provision” of

efforts.

To see this, we first note that the interaction term in equation (2.3) multiplies three

variables together: si, xi and xj. This has a cubic function property on efforts: doubling all

efforts produces an eight times higher interaction effect. Meanwhile, the costs on the social

effort (si) and legislative effort (xi) are quadratic. Hence, doubling those only quadruples

costs. It is then direct to check that the gains from increasing effort grow faster than their

costs, which implies the following result.

Proposition 2.3. Every finite profile of efforts is Pareto dominated by some larger level of

efforts.

This implies that, although higher payoffs are possible, the selfish attention to individual

costs limits the amount of effort that is produced in any equilibrium.25 If we were to cap

effort levels at some high level, then there would exist Pareto optimal efforts bound by the

caps.

23These equilibria correspond to when both parties exert high levels or low levels of social efforts, and thenfor some parameters there are also two additional equilibria in which one party does medium-high and theother does medium-low socializing.24The Pareto analysis in Cabrales et al. (2011) only applies if actions are bounded at some small enoughfinite level. The second derivatives in their proof flip signs if actions are large enough. Thus, there is alocal maximum of a weighted sum of utilities that is interior (which is the one identified in their analysis offirst-best actions), but the global maximum is actually unbounded. With a strict enough bound on efforts,there would exist an interior maximizer. Effectively, once the efforts are large enough, then the interactioneffects dominate the costs. One needs to constrain efforts to be below that level in order to get an Paretooptimal effort solutions. Note however that even with bounds, the equilibrium efforts tend to be inefficient,given the externalities.25It should be noted, that with high enough effort levels, then the best responses increase without bound inresponse to increases in others’ efforts. There is no equilibrium, because of the unbounded feedback. Again,if one imposed a cap, then there would be an equilibrium at those capped levels in the model.


The message here is that generally, given the complementarities and positive externalities,

there is underprovision of effort. A political party, or a government, could help overcome

some of the inefficiencies, for instance, by subsidizing meetings and interactions.

2.6. Discussion of the Model.

The model is designed to include five key features simultaneously: (i) strategic decisions on

a network - agents choose behaviors and their payoffs from those behaviors depend on their

neighbors’ behaviors, (ii) endogenous network formation - agents have discretion over whom

they interact with, and base those choices over the anticipated payoffs from their network

position accounting for the ensuing behaviors and externalities, (iii) group memberships and

homophily - both the payoffs and meeting rates of agents can be biased to privilege group

identity, (iv) statistical identification of the three previously discussed different features,

and (v) practical estimation of this model for a moderately sized network from a single

observation of a network and associated behaviors.26

These five features are all very important for our context and we anticipate they would

be in many other applications. This list of desirable features of the model requires some

stylizing of the model on various dimensions, trying to hit an appropriate point in the

tradeoff between tractability and richness. The model has to be rich enough to provide a

good fit and explain much of the variation in the data along several simultaneous dimensions

(more on this below), and yet be tractable enough to solve and estimate.

There are models that combine one or more of the various features mentioned above,

but none that allow for all of them. The growing literature on games on networks (e.g.,

see Jackson and Zenou (2015) for a survey) addresses peer interactions on networks and

how those vary as a function of the network. The linear-quadratic framework here, first

explored in Ballester, Calvo-Armengol, and Zenou (2006) has become a standard approach

in that literature given its tractability. Here this is combined with network formation and

homophily.

The extensive literature on network formation, starting from its early incarnation in Jack-

son and Wolinsky (1996); Dutta and Mutuswami (1997); Bala and Goyal (2000); Currarini

and Morelli (2000); Jackson and Watts (2002); Jackson (2005); Herings, Mauleon, and Van-

netelbosch (2009) provides insight into how networks form, when inefficient networks form,

and how that depends on the setting. More recently, the literature has also begun to de-

velop models that incorporate some heterogeneity and are still tractable enough to allow for

fitting the models to data, as in Leung (2015); Sheng (2016); Chandrasekhar and Jackson

(2016); Mele (2017); Graham (2017); de Paula, Richards-Shubik, and Tamer (2018); Leung

(2019); and some of that literature also allows for homophily, such as Currarini, Jackson,

and Pin (2009, 2010); Banerjee, Chandrasekhar, Duflo, and Jackson (2018). The models

that are tractable enough to fit to data require a structure that limits the multiplicity of

stable (equilibrium) networks, and such that those can be estimated with a practical number

of calculations.

26Congressional data allow for a time series, but the agents and their preferences change over time, and sowe need a model that can be fit from a snapshot of the network and behaviors.


The breadth of that class of models is still unknown and we only have a handful of such

estimable models that involve non-trivial network effects (payoffs that are not simply inde-

pendently determined link-by-link); and generally those models are stylized in some way.

For instance, as shown in Sheng (2016), the choice of the specific model can be important

as some models with indirect network effects (utility from friends-of-friends) lead to (i) a

lack of identification (many configurations of parameters leading to the same outcomes) and,

(ii) computational intractability with as few as 20 players, due to a curse of dimensional-

ity (p. 14). To make progress, she restricts the model to one with endogenous links that

have “dependence [that] has a particular structure such that conditional on some network

heterogeneity and individual heterogeneity, the links become independent.” An alternative

approach is that of Mele (2017). He proposes an empirical model of network formation that

allows for homophily in network formation. Again, he shows that there is a curse of dimen-

sionality in using standard estimation methods unless some strong asymptotic independence

conditions are satisfied. Other approaches are to have certain subgraphs generate value and

then model the formation of those subgraphs directly (Chandrasekhar and Jackson, 2016),

or to have payoffs based on combinations of individual characteristics, geography, or assor-

tativity (e.g., Currarini et al. (2009); Leung (2015); Graham (2017); Leung (2019)). Here

we want a model in which the value to a given pairing depends on their subsequent mutual

(legislative) efforts, and so we need a model in which expected values of links can be easily

calculated conditional upon future efforts, and those efforts can also be characterized easily

as a function of the pairings. Using random meeting probabilities to derive link formation

does exactly this by reducing the dimension of choices while allowing for interdependencies

and still yielding a clean characterization of both types of efforts.

In summary, one has to be judicious in modeling network formation to obtain a formulation

that is both well-identified and estimable, and to have heterogeneity. Meanwhile, existing

empirical models of games on networks that are well-identified (e.g., de Paula et al. (2019),

advancing the work of Bramoulle et al. (2009)) do not allow for endogenous networks - they

assume that the network is fixed and exogenous, and require a different data set-up than

ours.27

Thus one can see why models that incorporate both behavior and network formation

are few: Cabrales, Calvo-Armengol, and Zenou (2011); Konig, Tessone, and Zenou (2009);

Goldsmith-Pinkham and Imbens (2013b); Hiller (2017); Badev (2017). These models neces-

sarily sacrifice some richness in order to incorporate both network formation and endogenous

behaviors and to allow for an interaction between them. Nonetheless, they can still be quite

rich and, as we show here, can still fit data well. Of this class, in order to work with a

tractable model that we can extend to have yet a third dimension of group identity and

27For example, to recover an unobserved exogenous network as they set out to, de Paula et al. (2019) assumes(i) an exogenous network that is sufficiently sparse, (ii) the network does not change over time, (iii) a paneldata structure, with large enough time-series dimension, and (iv) a linear in means model. Our set-up anddata structure do not have any of these 4 properties, as alluded to previously. Furthermore, the estimationof this set-up must involve shrinkage estimators and their resulting bias due to the size of the parameterspace (N2 parameters to recover from just the network itself).


homophily, and still take to (static) data, we build upon the model of Cabrales, Calvo-

Armengol, and Zenou (2011). This introduces another dimension to the estimation, of group

interaction rates, and thus requires that the model be tractable enough to still solve with a

third dimension of endogeneity. Finally, a close look at the fit of the data provides support

for our modeling choices.28 In Section 5.1, we show that the predicted links from the model

are highly correlated with measures that use disaggregate data between congressmembers

(e.g. Fowler (2006)), even if we do not use the latter in estimation. Secondly, strategic

outcomes on this network, such as the probability of bill approval, are fit well within and

across parties/politicians. Third, homophily is quantitatively important: a model that with

homophily fits the data significantly better than one without it. Fourth, our model is shown

to outperform alternative approaches to the characterization of G in terms of in-sample mean

squared error. Finally, this model allows us to investigate behavior beyond what is observed:

it allows us to perform meaningful counterfactuals, shown later in the paper. Such exercises

include the effects and reversal of polarization, and the impacts of networks on the passage

of key bills in the Great Recession.

2.7. Preliminaries to Estimation. Generally, effort levels s∗i , x∗i are not measured exactly

and are observed with noise. For instance, the bill cosponsorships often used as the basis for

the construction of political networks are end products that miss other forms of socializing

(e.g. close-doors meetings, fund-raisers, and so on). Similarly, although we can partially

observe legislative effort through standard proxies (e.g. times the Congress member was

present on the floor for speeches, presence in roll call voting, or number of bills written29),

these are imperfect proxies for the legislative efforts that politicians exert. Thus, we account

for measurement error in our analysis.

LetNτ denote the politicians comprising Congress τ and note that this is a set which varies

across different τ .30 Introducing classical measurement error, for politician i in Congress τ ,

we observe:

si,τ = s∗i,τe−εi,τ(2.14)

xi,τ = x∗i,τe−vi,τ .(2.15)

s∗i denotes what is chosen, but it is hit with independent noise and si is observed (and

similarly for xi). The measurement error, conditional on this observation, is mean zero, and

independent of all the other measurement errors across individuals and time. We do not need

to impose that the measurement errors in both types of effort have the same distribution.

28Most notably, we see the value of (i) a (biased) random socialization protocol, (ii) choices made on effortlevels, and (iii) mean-zero i.i.d. measurement errors on our observed proxies.29Both highlighted as important for legislative success in Anderson et al. (2003).30The data is observed for multiple Congresses and we provide identification results for parameters specificto each Congress. This means we allow our parameters to differ across different Congresses and we canconstruct time-series estimates of the parameters.


From Proposition 2.1, (S1, S2, X1, X2) are completely determined by the parameters that

govern the system. Then, all individual choices are functions of parameters and of the set of

types αjj∈Nτ .Let:

(2.16) αi,τ = ez′i,τβP (i),τ

where zi,τ indicates a vector of individual observables31 (e.g. ideology, tenure, committee

membership), and βP (i),τ are party-specific and Congress-specific parameters that will be

estimated.32

Measurement error’s independence with respect to individual politician covariates is not

a particularly stringent assumption in our context and is needed to allow the possibility of

a partial mismatch between data and model effort predictions. Covariates capture much of

the individual-level heterogeneity in benefits from legislative effort and common behavior:

in the model, through αi, and in the data, as we allow αi to vary according to ideology,

tenure, strength of committee positions held (this is verified in the model fit section). As

a result, our i.i.d. assumption is on the (mis)measurement of equilibrium actions, not the

actions themselves. It implies that our mismeasurement on average is not worse for certain

politicians than others, conditional on their characteristics. While we make this assumption

explicitly in this paper, it is commonly (implicitly) used for inference in most empirical

models, whether reduced form (e.g. using instrumental variables as in Battaglini et al.,

2018) or structural for the validity of normal approximations of test statistics/asymptotic

distributions.

In addition, our specification is not oblivious to common shocks driving social and leg-

islative effort and legislative success. Quite the contrary, Proposition 2.1 shows how our

structural approach in fact operates under the theoretical result of dependence of individual

efforts from party and time specific common XP (i) and SP (i) factors through Equations (2.9)-

(2.10). The model’s system of equations makes explicit how common and possibly correlated

shifts affecting all members of a specific party in a Congress affect individual equilibrium

choices. In fact, XP (i) and SP (i) work as sufficient statistics that capture all common be-

haviors across the network - so only individual characteristics and those parameters must

be specified to fully capture the endogenety of network formation. In contrast, an approach

based on Ballester et al. (2006) would have to specify a full, observable and exogenous,

network and deal with its nonlinearities. We revisit this after the Results section, when

discussing model fit.

31Unobservables are already present when we introduce measurement errors. Note that if we had αi =ez

′iβ+ηi , we could rewrite equation (2.14), using the equilibrium results, simply as:

sieεi−ηi = ez

′iβSP (i),

and a redefinition of the measurement error to εi − ηi (still mean-zero and i.i.d.) would suffice in returningto the model presented in the main text, as long as A1, A2, B1, B2 were not functions of ηi.32Identification of the model does not rely on the parametrization of α, as we prove in Appendix I. However,this is useful for estimation purposes. A nonparametric α would require us to estimate a parameter αi foreach politician in each Congress, when we only observe one set of (si, xi) per period.


The information we employ in the analysis is the following. Let yi,τ = I(Yi,τ>m) indicate

whether each bill was approved or not, where i ∈ Nτ and τ is a given Congressional cycle.33

si,τ indicates the (log of hundreds of) cosponsorship decisions per politician i ∈ Nτ . This

is our proxy for the equilibrium social efforts∗i,τ

. The use of logs and rescaling allows us

to keep this effort proxy in the same scale as our proxy variable for legislative effort. xi,τindicates a vector of observable proxies for legislative effort

x∗i,τ

. As discussed in more

detail in the following section, this is constructed using data on floor speeches (word counts

per politician during a term) and roll call presence/votes. We employ a procedure (Non-

Negative Matrix Factorization, see Trebbi and Weese, 2019), to reduce the dimensionality

of this set of proxies to a single dimension.34

As we perform our analysis within a Congress, we suppress the notation τ . We assume

that a single pure strategy Nash equilibrium, as defined in Proposition 2.1, is played in each

Congress. We do not impose, however, that the same equilibrium is played across different

Congresses, rather we characterize the equilibrium played empirically in Section 6.

Given yi, si, xi, zii∈N , we estimate the parameters (c, φ1, φ2, ζ1, ζ2, γ, p1, p2, β1, β2). For

identification, we set m = 1, so that the random variable εi is scaled in terms of the institu-

tional threshold. The basis for identification is Proposition 2.1 and the systems of equations

that it provides.

For identification of the parameters of our model it is not necessary to identify the full set

of equilibria, but instead just to use the implications that we are observing some (interior)

equilibrium. More precisely, we show that, given the observed data, one can uniquely pin

down the equilibrium that is played, as long as only one is played during each Congressional

term. Formal identification of our model is demonstrated in Appendix D.

3. Data

We use the cosponsorship data from Fowler (2006), compiled from the Library of Congress,

covering the 105th to the 110th United States Congress (from 1997 to 2009). This data

contains cosponsorship decisions by politician, and within that data, who sponsors and

who cosponsors each bill. It also contains information on whether each bill was approved in

Congress or not (we focus on passage in the House of Representatives). Figure 1a shows that

measures of inter-connectedness of Congress, for example the total number of cosponsorship

links in legislative acts across members of the House (Fowler (2006)), have been steadily

increasing. Figure 1b then breaks down how cosponsorships vary within and across parties.

Per Congressional cycle, we compute the log of how many hundred bills each politician

cosponsors, which is the variable Cosponsorships. This function of cosponsorships acts as an

empirical proxy for the social effort s∗i,τi∈Nτ .We note here that cosponsorship differs from bill sponsorship. Sponsoring a bill refers to

the introduction of a bill for consideration (and can be done by multiple legislators drafting

the bill, the “sponsors”). These sponsors are the authors of the bill. Instead, cosponsorships

33A complete data description section follows below.34Sponsorship of bills is already included, as we use the separate bills independently. Further details on this,the data and procedure to lead to the effort proxy are in the next section.


Figure 1. Number of Cosponsorships per Congressional Cycle

(a) Total Number of Cosponsorships

(b) Number of Cosponsorships Within and Across Parties

The figure shows the evolution of the total number of (unique) cosponsorships during a congressionalcycle (i.e. anytime a politician has cosponsored another in a directed way) over time. The first figureshows the total number of cosponsorships, while the second decomposes it by party.

refer to the decision of adding one’s name as a supporter of the bill (becoming a “cosponsor”

of the bill). In contrast to sponsorship of a bill, the decision to cosponsor does not involve any

writing of legislation. Instead, cosponsorships serve as a signal of support to that current bill

(or potentially, to its authors), without ownership of the legislation itself. Cosponsorships

are prevalent in Congress, as can be seen in Table 1, and the presence of cosponsorship

across party lines is still quite common, notwithstanding the trends in polarization discussed

in Fiorina (2017) or Canen et al. (2020), as evident from the time series in Figure 1b.


The individual bill success outcome (i.e. if the bill passes or not) maps into yi,τi∈N .

We then use the sponsorship information to link the outcome of the bill to the network

characteristics and individual decisions.

To compute our proxies for legislative effort, x∗i,τi∈Nτ , we first collect data on Roll Call

voting and floor speeches in Congress. Data for Roll Call voting comes from VoteView.

We compute an index, for each politician and for each term in Congress, as the times the

Congress member voted as a proportion of total Roll Call votes. This measure, which we

call Roll Call Effort, is defined as 1− (number of times i was “Not Voting”/ total Number

of Roll Call votes in a Congress).

Following Anderson et al. (2003), we also use data on floor speeches as a measure of

individual legislative effort. To do so, we compile the amount of words that each Congress

member used in his/her floor speeches across the duration of one term. Our Floor Speeches

variable is constructed as log(1 +Wordsi,τ/100). We log and rescale this variable to a scale

comparable to other legislative activities.35 Data on floor speeches comes from Gentzkow

and Shapiro (2015), available on ICPSR.36

That these measures of social interaction and legislative activity may be germane to one

another is evident from the significant and positive raw correlation of link formation and

proxies of legislative activity and effort, for instance floor speeches in Figure 2. This com-

plementarity between effort choices is fully consistent with our theoretical setup.

We proceed to construct xi,τi∈Nτ , by using both Roll Call Effort and Floor Speeches. An

appropriate combination of these variables can be obtained through dimensionality reduc-

tion methods. Since effort should be non-negative, we employ a procedure that guarantees

positive values (i.e. we do not use methodologies like principal components analysis that

involve a centering of data and negative values).37 We employ Non-Negative Matrix Fac-

torization (NNMF), a dimensionality reduction procedure which imposes constraints so that

the resulting elements are all non-negative.38

We also use observable characteristics, namely ideology (measured by DWNominate from

VoteView), tenure (how many terms a politician has served in Congress, with data coming

from the Library of Congress), and committee memberships.

35Dividing the number of words by 100, reflects an appropriate scale to compare cosponsorships to thesespeeches. It is a reasonable scale as House rules explicitly limit one minute speeches, a useful tool forpoliticians (Schneider (2015)), to 300 words.36As there are changes in the composition of Congress within a term, for instance due to death or resignationamong other reasons, we have some observations whose cosponsorship numbers and word counts do notcorrespond to a full term. To mend this, we scale up values proportionally to the recorded behavior whilein Congress. In other words, if a politician leaves halfway through his term, we double the values of theseobservations.37Our qualitative results about the parameters still hold if we use either of these variables individually.However, the magnitudes of the estimates change due to the different scales of Roll Call Effort (between 0and 1) and the floor speech data (in hundreds of words).38Wang and Zhang (2013) provide a discussion of this methodology. NNMF works by factorizing a matrix, callit A, into two positive matrices W,H, under a quadratic loss function. The product WH is an approximationto A of smaller dimension, as there are less columns in W than rows in A. We then use the main factor inW as our proxy.


Figure 2. Correlation between the raw data of log(1+Words) in FloorSpeeches and Cosponsorship decisions.

The figure shows the positive correlation between proxies for socializing (log number of cosponsor-ships/100) and legislative effort (log number of words in floor speeches/100). The graph presents thevariables in raw form, without rescaling or removal of members with low cosponsorship. The rawcorrelation is 0.171. In red, we present a LOWESS (locally weighted scatterplot smoothing) fit, withbandwidth (span) equal to 0.9, fitting the relationship between the variables. We do remove, as de-scribed in the Data section, observations that have total words equal to zero, which are mostly due todeath/resignations in that term.

Data on committee memberships comes from the work of Stewart and Woon (2016). To

quantify the value of the committees a politician is in, we use the Grosewart measure

(Groseclose and Stewart, 1998). Groseclose and Stewart (1998) and Stewart (2012) estimate

a cardinal value of how much an assignment to a given committee is valuable to politicians.

Such estimates are based upon data on how often politicians accept transfers from one

committee to another. The more desirable committees are those that politicians accept to

be transferred to often, but rarely accept to be transferred away from. The Grosewart

measure sums up the values of the committees in which a politician is present. We use the

estimates given in Stewart (2012) for our study, since they are the updated values for the

period we study.39

Summary statistics for all our variables of interest can be found for reference in Table 1.

39Below, we also consider an alternative measure for committee memberships. There, we construct dummyvariables for whether a politician has been assigned to a given committee during that congressional term.We then focus on the main committees for parsimony: Appropriations, Energy and Commerce, Oversightand Government Reform, Rules, Transportation and Infrastructure, and Ways and Means. We also includea variable Leadership of whether the politician was the Speaker, the Majority or Minority Leader, or theMajority or Minority Whip.


Table 1. Summary Statistics

Congress105 106 107 108 109 110

Cosponsorships

Mean 185.74 234.57 229.79 226.75 230.74 269.65Standard Deviation 85.79 102.91 127.03 124.08 119.48 135.90

Floor Speeches (Words)


Roll Call Effort


Ideology (DWNominate)


Tenure


Grosewart


Approval of House Bills


Number of Politicians N 442 435 440 439 438 445Number of Bills 4874 5681 5767 5431 6436 7340

The table presents summary statistics for the variables used in the structural estimation, across Con-gresses.40 Roll Call Effort is defined as the proportion of Roll Call votes that the politician does notappear as “Not Voting”. Number of words said in floor speeches aggregates the number of words saidby a politician across all his speeches in a term. Cosponsorships and number of words are scaled tofull term length (i.e. if a politician leaves mid-office and is replaced mid-office; then both him and thereplacement have those variables multiplied by 2.). For estimation, we remove the observations (billsand politicians) we do not have or cannot match to identifying numbers, and those with less than 3Cosponsorships (see the Data Section). These are mostly Congressmen who substitute others mid-term.Data used for bills is House bills (H.R.).


We restrict the data to Congresses 105th-110th for multiple reasons. First, the data

we employ to compute effort from floor speeches is only available from the 104th Congress

onwards. Second, the 104th Congress (corresponding to the Republican Revolution) provides

a structural break in the analysis of Congressional behavior. With multiple changes to

Congressional composition and structure during the 104th, it becomes hard to compare the

costs and socializing of this specific Congress to others, preceding or following, without

having to further delve into the exceptionality of this particular congressional cycle, which

is not the aim of this work.41

4. Estimation

4.1. Moment Equations. Let zi = [1, Ii∈P2 , z′i, z′iIi∈P2 ], where Ii∈P2 denotes a dummy

variable of whether politician i is in Party 2. Further define: βs = [log(S1), log(S2) −log(S1), β1, β2 − β1], βx = [log(X1), log(X2)− log(X1), β1, β2 − β1].

Appendix E shows that the moment conditions necessary to identify and estimate the

model’s parameters are:

Ezi(log(si)− z′iβs) = 0(4.1)

Ezi(log(xi)− z′iβx) = 0(4.2)

E(

2(log(si)− log(xi))− log(c− 1

X1

)−(log

(c− 1

X2

)− log

(c− 1

X1

))Ii∈P2

)= 0

(4.3)

(ElogP (yi = 1)− log(

1

ζ1

)− log(s2

i ))Ii∈P1 = 0.(4.4)

(ElogP (yi = 1)− log(

1

ζ2

)− log(s2

i ))Ii∈P2 = 0.(4.5)

S1 = φ1X1 (B1S1X1m11 +B2S2X2m12)(4.6)

S2 = φ2X2 (B2S2X2m22 +B1S1X1m12) .(4.7)

These moment conditions are based on rewriting the equations of Proposition 2.1 using

our parameterization for α and measurement errors, given in equations (2.14) and (2.16).

They allow us to identify (c, ζ1, ζ2, S1, S2, X1, X2, β1, β2) and set identify the φ parameters.

41In addition, without ad-hoc modifications to the estimating model specifically designed to accommodatethe idiosyncrasies of the 104th Congress, this lack of stability would also likely undermine any effort ofstructural estimation.We also perform a final, additional, trimming of the data across all Congresses. We drop a set of 19observations (out of 2636), that have the number of words in Floor Speeches set to 0 in the data of Gentzkowand Shapiro (2015). These observations relate almost exclusively to a politician who either resigned or diedduring that term (e.g. Representatives Jo Ann Davis in the 110th Congress, Sony Bono in the 105th, orresignations as Representative Bobby Jindal in the 110th). Since the data is zero, the rescaling above doesnot prove to be adequate, so we drop these observations. We also drop one observation in which politiciansthat have cosponsorship figures less than 3 bills over a full term, since identification relies on the existenceof cosponsorship and most cosponsor in the hundreds, so scaling is also inappropriate. The results do notdepend on this cutoff.


Specifically, lack of point identification of (φ1, φ2) is the result of lack of point identification

of p1 and p2. The parameters p1 and p2 enter nonlinearly in equations (4.6) and (4.7) through

mij(s), identifying a ridge of (p1, p2) pairs satisfying (4.6) and (4.7).

In Appendix H, we demonstrate how to obtain point identification of the φs when we

impose additional restrictions on the proxy variables (si, xi). The restrictions there are on

the second moments of the effort proxies, similarly in spirit to random coefficient models.

Such restrictions are justified under the assumption that more partisanship may result in

noisier measurement of social interactions and legislative effort. This alternative approach

is heavier in terms of assumptions and for this reason we do not adopt it for the derivation

of our main results in Section 5. Instead, we report estimated values for all parameters

(including p1, p2) under these additional assumptions and specification in Appendix H.42

For our main empirical exercise, we let Party 1 denote the Democratic Party (with its vari-

ables denoted by the subscript Dem) and Party 2 denote the Republican Party (analogously

denoted with a Rep subscript).

We carry out the estimation process using a two step procedure. In the first step, we

compute estimates for the parameters (c, ζDem, ζRep, SDem, SRep, XDem, XRep, βDem, βRep) from

the moment equations (4.1) - (4.5) above, via GMM.

In the second step, we use the first-step estimates to derive a set estimate for (φDem, φRep).

This is done by using equations (4.6) and (4.7). We grid all pairs (pDem, pRep) ∈ [0, 1]× [0, 1],

and, employing the estimated ADem, ARep, SDem, SRep from the first step, we calculate the

values of mij for each pair (pDem, pRep). The set estimate for (φDem, φRep) are all the values

that satisfy equations (4.6) and (4.7) for any pair (pDem, pRep).

Concerning the information of whether a bill passed or not yi,τi∈N , the model is agnostic

on how many bills a politician proposes. Because a good fraction of members of Congress

sponsor multiple bills, however, we work with L > N bills in the actual data. This is easily

accommodated in the estimation. Recall that εb are i.i.d. across time and bills. For each

politician i, all i’s bills have the same associated network gi,j, as it comes from the same

politician and his same network and effort choices (as well as those of his network). The

different ε realizations, however, represent different bill qualities or institutional arrange-

ments within politician, meaning that the same politician may have one bill approved and

not another. The dimensionality of the problem can be decreased by simply averaging out

each bill’s success by politician. This is made possible by the fact that equation (4.4) holds

for all bills, implying that it must hold for all politicians as well. Hence, we use the average

pass rate of bills for politician i as its estimate of the probability of bill approval.

4.2. Estimation via GMM. To estimate the model, we replace equations (4.1)-(4.4) by

their empirical counterparts and stack them into a vector of the form 1n

∑ni=1 g(si, xi, yi, zi; θ).

Since all moments have expectations taken over εi, vi, which are i.i.d. and mean zero for all

politicians, the empirical counterpart replaces the expectation operator by the mean over

42As second moments restrictions allow us to obtain point estimates of p1 and p2, we use such estimates inthe last part of Section 5 for assessing additional implications of our model and validation.


i.43 Furthermore, we average over the approval rates for bills for each politician to get the

estimated probability of approval at the politician level.

We then minimize the quadratic form:

(4.12)

(1

n

n∑i=1

g(si, xi, yi, zi; θ)

)′W

(1

n

n∑i=1

g(si, xi, yi, zi; θ)

),

where 1n

∑ni=1 g(si, xi, yi, zi; θ) is given by stacking up the empirical counterparts of equations

(4.1)-(4.4), for a total of 2k + 2 equations (k being the dimensionality of zi). W is the

weighting matrix, which can be taken as the identity matrix (an inefficient choice), or the

optimal weighting matrix for the GMM.

Given these first stage estimates, we then estimate the set of feasible values for (φ1, φ2)

as previously described. Further details about the empirical implementation are discussed

in Appendix F.

5. Results

Table 2 presents our parameter estimates, which delineate a series of intuitive relationships

emerging from the data. Table 3 shows the distributions of the estimated direct benefits

from passing policy αi, over time and by party. These distributions appear stable across

Congresses.

Splitting the samples by party, we observe important differences in the estimated distribu-

tions of Republicans and Democrats. Democrats have a higher average and dispersion, while

Republicans have tighter distributions. This implies different social effort patterns across

parties, as Democrats socialize more and (by our social meeting function) more often with

other Democrats.

43That is, the expectation operator has one observation for each politician, and averages across all politicians.For example, the empirical counterparts to (4.1)-(4.2) are:

1

n

n∑i=1

zi(log(si)− z′iβs) = 0,(4.8)

1

n

n∑i=1

zi(log(xi)− z′iβx) = 0,(4.9)

or in matrix form:

Z ′(log(s)− Zβs)n

= 0,(4.10)

Z ′(log(x)− Zβx)

n= 0,(4.11)

where Z stacks up zi, and log(s), log(x) stack-up log(si), log(xi) respectively.


Table 2. Main Results, Specification 1

Congress

105 106 107 108 109 110c 0.266 0.246 0.268 0.275 0.266 0.256

(0.008) (0.005) (0.007) (0.008) (0.006) (0.006)

φDem [0.037,0.037] [0.042,0.043] [0.048,0.048] [0.045,0.046] [0.045,0.045] [0.045,0.045]

φRep [0.031,0.031] [0.032,0.032] [0.034,0.034] [0.033,0.034] [0.034,0.034] [0.035,0.036]

ζDem 25.287 61.911 39.332 38.862 47.861 16.266(2.758) (9.762) (5.256) (5.516) (6.835) (1.213)

ζRep 8.911 8.332 9.129 7.584 9.678 20.176(0.676) (0.758) (0.786) (0.614) (0.760) (1.763)

SDem 0.897 1.085 1.046 1.000 1.041 1.185(0.034) (0.034) (0.033) (0.033) (0.029) (0.028)

SRep 0.711 0.979 0.922 0.847 0.884 0.982(0.036) (0.036) (0.044) (0.038) (0.041) (0.053)

XDem 4.392 4.594 4.129 4.064 4.261 4.761(0.161) (0.139) (0.131) (0.148) (0.125) (0.101)

XRep 4.224 5.479 5.182 4.791 4.885 5.017(0.201) (0.196) (0.235) (0.21) (0.225) (0.287)

Ideology -0.709 -0.426 -0.616 -0.692 -0.601 -0.428(0.096) (0.073) (0.074) (0.087) (0.074) (0.059)

Tenure -0.008 0.005 0.004 0.004 0.002 -0.001(0.002) (0.002) (0.002) (0.003) (0.002) (0.002)

Grosewart -0.01 -0.015 -0.026 -0.013 -0.01 -0.003(0.009) (0.008) (0.008) (0.011) (0.009) (0.007)

Ideology ×Rep 0.318 -0.011 -0.039 0.1 0.122 0.14(0.083) (0.06) (0.075) (0.066) (0.068) (0.073)

Tenure×Rep 0.019 0.006 0.005 0.007 0.002 0.001(0.003) (0.002) (0.003) (0.003) (0.003) (0.003)

Grosewart×Rep 0.02 -0.006 -0.019 -0.021 -0.029 -0.032(0.01) (0.009) (0.01) (0.01) (0.009) (0.011)

N 436 433 435 437 433 437

Notes: Standard errors in parentheses. The table presents the GMM estimates using the OptimalWeighting Matrix for the parameters of interest, as described in the Estimation section. Standard Errorsare computed from estimates of the variance for a GMM estimator with the Optimal Weighting Matrix.Details are in Appendix F. The estimate of φ1 and φ2 are their estimated sets. Rep represents the dummyvariable of whether a politician was in the Republican Party. Hence, a variable Tenure × Republicanrepresents the additional estimate of the Tenure variable for the Republican Party, as compared to theDemocratic one.


Table 3. Differences in the Distributions of αi Across Parties

Congress 105 106 107 108 109 110

Democrats:Mean αi 1.245 1.199 1.277 1.329 1.278 1.162Standard Deviation of αi 0.128 0.079 0.114 0.129 0.106 0.072

Republicans:Mean αi 1.269 1.025 1.000 1.095 1.079 1.089Standard Deviation of αi 0.098 0.022 0.024 0.036 0.039 0.045

We show the mean and the standard deviation of the (estimated) distributions of αi for each party,highlighting the differences in those distributions. The estimates presented are those from Table 2.

From the main results, we see that both φDem and φRep have estimated sets that exclude

0. For Democrats, the estimates range from [0.037, 0.045], while for Republicans, the range

is [0.031, 0.036]. Both are slightly increasing over time. To put this into context, note that

the marginal utility of an increase in xi is

αi + φisi∑j 6=i

xjsjmij(s)− cxi.

The direct benefit αi ranges from 1 to 1.4, while the network benefit φisi∑

j 6=i xjsjmij(s)

ranges from about 0.1 to 0.3. So, the social incentive is somewhere between a tenth to a

fourth of the direct incentives, a substantial consideration in politician’s choices.

We note that the value for socializing, φDem, for Democrats is higher than that for Re-

publicans, even as we control for the difference in types, partisan bias and preferences across

both parties. This difference appears historical, not changing with majority status. It is

also quantitatively large, approximately 17% of the returns to social effort (in the order of

0.006/0.036 across Congresses). Furthermore, it is of a significant magnitude relative to c,

which ranges in [0.25, 0.28]) and does not change much over time, on average.

The relative cost of legislative effort to social effort, c, is stable over time. As a result,

interactions between politicians are more valuable, as there is an increasing return of social-

izing against a stable cost. This may be consistent with an increase in the complexity of

extant statutes, making it more difficult to approve legislation. This is for example evident

from an average number of pages per statute of 3.6 in 1965-66 to 18.8 in 2015-1644, making

interactions between politicians in drafting and drumming up support for legislations on the

Capitol more important. Although this would not change the costs to social effort, it would

appear to change the returns from it.

The estimates of ζDem and ζRep are also significant and large in magnitude. This indicates

that politicians see positive gains from having bills approved. The larger magnitude of ζ

44Vital Statistics of Congress 2017, Chapter 6, available at www.brookings.edu


relative to the other parameters is needed for the model to be internally consistent. This is

because ζ also operates as a normalizer that guarantees that the probability of bill approval

(in equation (4.4)) is between 0 and 1. However, we can see large differences across parties.

In particular, the Democrats have the largest ζ between Congresses 105-109, when they are

in the minority. In Congress 110, this is reversed, with the minority Republicans having

the largest ζ. This intuitively suggests that the returns from getting bills approved for a

politician depend on their party’s majority status in the House. A high ζ for the minority

party indicates that voters reward legislators more when it is harder for them to pass bills.

This is consistent, for instance, with voters learning from legislative successes about the

quality of their representatives.

Using the estimated values of ζP (i) for both parties, we can also calculate the probability

of bill approvals for each politician. We show these in Figures 3a for Democrats and 3b for

Republicans.

By comparing Figures 3a - 3b with the average bill passage rates in the summary statistics

(Table 1), we can see that the model can generate a good match of the mean bill success rates

(which we observe). Our structural assumptions allow us to represent the whole distribution

of expected probabilities of having a bill approved across different politicians. These indicate

some variation over time. Later Congresses (108th and 110th) show a higher predicted

approval rate for most politicians. Furthermore, approval rates are highly correlated with

majority party status. Democrats have a much higher rate as a majority in Congress 110,

while Republicans have higher ones as a majority in 105-109.

We can also discuss the statistical significance of different covariates in explaining direct

benefits, αi. With our baseline specification that uses Ideology, Tenure, Grosewart for zi in

Table 2, we see that ideology is statistically significant (especially in later Congresses). The

estimates suggest that those on the left of the ideological spectrum (extremist Democrats,

moderate Republicans), have higher direct returns of exerting legislative effort. Meanwhile,

the Grosewart variable, capturing the impacts of committee assignments, appears to be

noisy.

We also consider another specification where we replace the Grosewart variable by dummy

variables of committee assignments to each of Congress main committees. This is shown in

Table C.3 in Appendix. We can see that the results from our main specification are robust.45

Finally, we investigate the relationship between estimated party types (AP (i) =∑

i∈P (i) αi)

and partisan bias in social interactions, pi.46 Figure 4 reports the results for Democrats and

Republicans separately.

45It is noteworthy that in this specification, the estimate of being in the Rules Committee is positive andsignificant. The Rules Committee is the committee in charge of determining the rules that allow each billto come to the floor, fundamental for the progress of legislation. It seems consistent that politicians in thatcommittee are rewarded for effort in it, even conditional on having the same ideology, party, and tenure.46Absent an estimate for p1 and p2 from the baseline model where these parameters are not identified, weemploy the estimates generated by a specification imposing second moment restrictions, reported in AppendixH. As discussed in the next section and in Appendix, these estimates appear to fit empirical patterns in thedata better than corner solutions.


Figure 3. Estimated Probability of Approval

(a) Democrats

(b) Republicans

We find a clear negative correlation between types and partisan bias for Republicans, even

with our small sample of Congresses, while the relationship is absent for Democrats. Our

theory does not provide a microfoundation of the partisan bias of each party, yet an intuitive

explanation for these two relationships may arise from the model’s internal mechanism.

Recall that social externalities are stronger for the higher types, who have larger direct

benefits from legislative activity and a larger social network to benefit from. Republicans

benefit from socializing with Democrats who are higher types on average. As Republican

types increase, then they will choose higher social effort and may benefit even more from

bipartisan connections with the Democrats. This would encourage a lower partisan bias for

GOP as the average type of its politicians increases. At the same time, the same incentives

may not necessarily hold for Democrats and this mechanism can reverse. Democrats have


Figure 4. Relationship between Partisan Bias and Estimated Types

(a) Democrats (b) Republicans

The figure shows the relationship between estimated party types (the sum of each party’s estimatedα’s from Table 2, defined as A1, A2 in the model) and the partisan biases (estimated as described inAppendix H).

the highest types on average, so the largest externalities are when their legislators interact

only within their own party. With more segregation across party lines (higher partisanship

pi), Democrats may deal with higher types relative to when also mixing with Republicans.

This suggests a possible opposing incentive in selecting a low pi for Democrats as their direct

incentives grow and it can rationalize the mixed evidence in the Democrats’ panel of Figure

4. The reader can find additional discussion of the problem of endogenous partisanship in

Appendix.

5.1. Fit and Further Discussion. To conclude the section, we conduct three exercises. In

the first, we consider an out-of-sample validation of our approach, by assessing the fit of our

model of moments of the socialization patterns not used in estimation. In the second, we

look at an in-sample validation comparing the fit of empirical approval rates of bills to the

estimated values. In the third, we compare the prediction of legislative behavior according

to our model generated network to other salient examples of political networks used in the

literature, including cosponsorship networks, alumni-connections networks, and those based

on committee memberships.

Although our analysis employs i’s total cosponsorships to proxy for his/her social effort

s∗i,τ , the more fine-grained data on pairwise cosponsorship information between i, j politician

pairs is not used in estimation. In this first exercise, we predict the i, j links for each

pair of politicians based on what predicted by our theoretical gi,j(s) function based on the

estimated parameters for each Congress in our sample and then compare them to the actual

cosponsorship pairs. The goal is to show that our estimated network model fits such proxies

for social ties, commonly employed in the literature following Fowler (2006), even without

explicitly modeling pairwise socialization decisions.


The correlations between the estimated gi,j(s) and any i, j pairwise cosponsorships are

reported in Table 4. The Table illustrates correlations for two possible definitions of links

based on actual i, j cosponsorship in the data. In the top panel cosponsorships are consid-

ered directed from i to j and in the second panel cosponsorships are considered a-directional.

In the two cases the correlations with the model-implied gi,j(s) are 0.375 and 0.456 respec-

tively and statistically significant. Thus, the model appears able to capture disaggregated

socializing proxies not directly targeted in estimation, reassuring on the plausibility of our

socialization structure.

We can draw an additional relevant conclusion from this exercise. Results of Fisher’s

z-transformation tests also suggest that our model with pDem > 0, pRep > 0 is better at

capturing the relationships from the pairwise cosponsorship data than alternative models

with either full partisanship (at least one of pDem = 1 or pRep = 1) or without partisanship

(pDem = pRep = 0). These comparisons are possible as different gi,j(s) can be generated

using different values for pDem, pRep.47

Although recent political economy research highlights a hollowing out of the moderate

middle ground in congressional voting (Fiorina, 2017; McCarty et al., 2006; Canen et al.,

2020), our model with pDem, pRep around 0.1 (values in this range for pDem, pRep are obtained

in Appendix H) produces a substantially better fit of the cosponsorship data than a model

with complete polarization pDem = pRep = 1, which is statistically dominated. Also, while

the exact point estimates of pDem, pRep rely on our assumptions on second moments, we

believe that the rejection of pDem = pRep = 1 has to be considered more general. The raw

data in Figure 1b display a sufficient degree of cross-party cosponsorship to cast doubt on an

hypothesis of “full sorting” among party members in the House. A model with full polariza-

tion would predict 0 relationships emerging across different parties, that is obviously not the

case. Meanwhile, a model with no polarization would predict a much higher degree of social

connection across party lines than observed (it predicts approximately 60% of Democrat

connections to other Democrats, compared to around 70% for our model and the data; and

fewer connections among Republicans alone, relative to Republicans and Democrats, when

compared to our model and the data).

Possibly, reconciling a world of more polarized legislators and the thousands of cospon-

sorships across party lines reported in Figure 1b, may come from noting that, as ideology

may diverge, engagement across party lines becomes more important for getting legislation

to the floor and passed. Our model appears to capture such phenomena.

In a second exercise, we fit the empirical success rate yi for each politician in each Congress

to the estimated one, given by equations (4.4)-(4.5).

For this exercise, we focus on politicians who have sponsored a sufficiently large amount

of bills (over 10 House Bills sponsored), so to have a reasonable empirical approximation to

their average success rate. The results are shown in Figure 5. We find a significant positive

correlation of 0.265 between estimated and empirical bill success rates. Thus our model

47As before, we use the estimates of p1 and p2 generated by a specification imposing second moment restric-tions, reported in Appendix H. This is because these parameters are not identified in the baseline model.Nevertheless, this test shows that our model captures part of the empirically observed relationship.


Table 4. Model Fit: Correlation of Estimated Network of the Model to theCosponsorship Networks in the Data

Congress Correlation Fisher’s z-statistic

Data from Directed Cosponsorships:

Model: pDem > 0, pRep > 0 0.375 -

Model: pDem = 0, pRep = 0 0.365 9.427***

Model: pDem = 1 0.349 22.725***

Data from “Combined” Cosponsorships:

Model: pDem > 0, pRep > 0 0.456 -

Model: pDem = 0, pRep = 0 0.443 12.713***

Model: pDem = 1 0.426 27.891***

We compare the performance of the partisan model (with pDem > 0, pRep > 0) to the performance of themodel without partisanship (pDem = pRep = 0) and complete partisanship (pDem = 1), in explainingthe observed cosponsorships in the data. In the first panel, cosponsorships are measured by the directednumber: how many times i cosponsors j. In the second panel, “combined cosponsorships” are measuredby the number of times i cosponsors j and j cosponsors i, creating a symmetric undirected graph.To calculate the statistics for each model, we first generate the links using the theoretical definitiongij(s) = sisjmij(s) under our estimated parameters. That is done for the 3 cases. We then show thecorrelations of the model links to the values of the cosponsorships in the data. The estimated valuesfor pDem > 0, pRep > 0 come from the estimates using second moments of (si, xi), from Table H.1in Appendix H. We present Fisher’s z-transformation statistic, for the test that the correlation of theadjancency matrix of the Model with pDem > 0, pRep > 0 with the data is equal to the correlation ofthe alternative model (without partisanship/complete partisanship) with the data. Since our modelgenerates a symmetric adjacency matrix by construction, we consider the correlations of the lowertriangular adjacency matrices. ∗∗∗ represents that the null hypothesis of equal correlations can berejected at 1% significance level, ∗∗ at 5%. Note that, when estimating the model, we did not usecosponsorships at the ij level. We aggregate all Congresses in the analysis above.

captures part of the empirically observed relationship. Reassuringly, as we use more precise

measures of the empirical success rate of politicians, this correlation tends to increase.48

Finally, we conduct a horse race comparing the in-sample properties of our endogenously

estimated network to exogenous alternative G’s in the literature. We compare these networks

in terms of their predictive power/model fit of legislative behavior using the outcome from

48The correlation is stronger as we increase the threshold of 10 sponsorships. It is 0.283 if we look atpoliticians with over 20 sponsorships, 0.362 for those over 30, and 0.373 for those over 50.


Figure 5. Model Fit: Estimated and Empirical Approval Rate of Bills

The figure shows the correlation between estimated and empirical approval rates for politicians. Weconsider politicians with over 10 sponsorships, so that a meaningful estimate of the average approvalrate can be obtained. The correlation is 0.265 in this case, and is stronger when we look at politicianswith a higher number of sponsorships.

our main specification. To do so, assume that gij(s) is known, as we will feed it from the

data. Then, the game defined by (2.1) collapses to that in Ballester et al. (2006), with the

Nash equilibrium given by:

(5.1) (I − φG)x∗ = α,

where α = (αi)i. Using our measurement error assumption of x∗ in (2.14) and the

parametrization of α in (2.16), equation (5.1) can be rewritten as:

log(xi) = log((I − φG)−1ez′iβ) + vi.

≈ log((I + φG+ φ2G2 + φ3G3)ez′iβ) + vi,(5.2)

for small φ and G of (approximate) full rank. We can then compare the fit of equation

(5.2) across different networks G, as this model does not include any of the endogenous

network structure from our own model. We compare the predicted endogenous network from

our model to graphs based on directed cosponsorships (Fowler, 2006), alumni connections

(Battaglini and Patacchini, 2018; Battaglini et al., 2018), and committee membership (from

the main committees used in our estimation).49

49For illustration purposes, we present figures of these networks in Congress 110 in Figure C.1 in Appendix.From those graphs, one can check that the alumni and committee-based networks are much sparser thancosponsorship-based networks. For example, approximately 270 legislators are isolated in the alumni network,


To estimate φ, β in (5.2) across models, we first look at the subgraph of non-isolated

nodes for each case. We estimate those parameters by Non-Linear Least Squares using

the covariates of Ideology, Tenure and Grosewart, used in our main specification of Table 2.

Then, we use the estimated parameters to fit the Mean Squared Error across all observations

for each alternative network G. Note that covariates are essential in this specification so that

one can generate heterogeneity in αi.50 The results for this exercise are shown in Figure 6.

Figure 6. Comparison of Model Fit Across Our Model Network and Competitors

The figure compares the performance in terms of mean squared error (MSE) of different legislativenetworks used in the literature (our estimated model, one based only on directed cosponsorships, onebased on university alumni networks, and one based on committee memberships) in fitting legislativebehavior (our xi). To do so, we estimate the Nash equilibrium of the game in which the network G isgiven, which collapses to the problem in Ballester et al. (2006).

Our endogenous network outperforms all competitors in explaining legislative behavior in

terms of MSE. For example, our model fits legislative behavior much better than the sparser

alumni and committee networks. This is because it can capture correlated behavior across the

hundreds of nodes that those alternative models assume are isolated, but that in reality are

highly correlated, better captured by our endogenously formed network. While alumni and

with an average degree below 2, while no legislator is isolated in the cosponsorship case. The data for thecommittee and cosponsorship based networks is the same as used in estimation in the previous section.The committee network has edges defined by whether legislators sat on the same committees. For alumninetworks, we scrape the Congressional bioguide webpage and use fuzzy matching based on the politician’suniversity and date of graduation to generate the network. A link on this network exists if politicians wentto the same university within 8 years of one another (as in Battaglini and Patacchini, 2018).50Monte Carlo simulations have found that φ, β can be recovered reliably in this set-up. Nevertheless, thesesimulations also illustrate the empirical limits of a model based on (5.2). For example, discrete covariatesgenerate identification problems due to the lack of variation in the support of those variables. For this reason,we only keep the covariates from our main specification in this exercise.


committee networks can fit the behavior of very central and influential nodes, they miss out

on the majority of legislators who do not have such easily codified connections. In addition,

our model appears to outperform even a pure cosponsorship network. The cosponsorship

network can appear too dense, since observed cosponsorships might be only weakly related

to underlying relationships. Meanwhile, our model uses cosponsorship information more

efficiently to find underlying links, as it is able to explore how cosponsorships correlate

with other legislative behavior through equilibrium restrictions. We note that this is a fair

comparison across models - while we do use cosponsorships to inform our model, we only

use aggregate information about one’s cosponsorships. However, the directed cosponsorship

network in the alternative uses information about pairwise decisions to completely determine

G.

Altogether, this last exercise further illustrates the benefits of simultaneously modeling

simultaneous network formation and strategic interactions on that network.

6. Multiplicity of Equilibria and Counterfactuals

Our theory allows for multiplicity of equilibria. In this section, we first discuss what we

can say about the equilibrium being played in each Congress. In the second part of this

section, we present counterfactual exercises using the estimated model parameters.

6.1. Stability of Equilibrium and Other Equilibrium Properties. Let us first discuss

the stability of the equilibria that we find.

A preliminary technical consideration deals with the fact that we only infer noisy esti-

mates of social and legislative effort (SDem, SRep, XDem, XRep) from the data. Those values,

however, do not necessarily solve the original (exact) system defined in Proposition 2.1 under

our set of estimated parameters, they only approximate its solution. To find the equilib-

rium values (SDem, SRep, XDem, XRep) that are consistent with our estimated parameters for

(c, φDem, φRep, βDem, βRep), we solve the system in Proposition 2.1 exactly.51 Those solutions

are presented in the upper panel of Table C.4 in Appendix. They are used to compute the

model-consistent bill approval rates shown in Figures 3a - 3b, as well as the counterfactuals

presented below. Based on the results in Figures 3a - 3b and in the upper panel of Table

C.4, Congress is always at an interior equilibrium of our model.

As there can be multiple interior equilibria, we check that the estimated equilibria are

stable.52

51This procedure is described in more detail in Appendix G. For example, for φs, we use the median valuein their set.52Generally, when there are multiple interior equilibria, only some are stable. This is in contradiction withProposition 1 in Cabrales et al. (2011) which claims stability of all interior equilibria. In their model, contraryto the original proof, the largest equilibrium is unstable. In the proof of that proposition the matrix Π cannotbe approximated by setting off-diagonal terms to 0. In fact, the eigenvalue can change sign if the off-diagonalterms are included and are on the order of 1/n. This reverses their conclusion.


To perform this analysis, we use the best response dynamics.53 Starting at some vectors

s0, x0 and iterating through t, one obtains that the best response dynamics are described by:

sti = xt−1i φi

∑j 6=i

mij(st−1)st−1

j xt−1j ,

and

xti =αic

+ st−1i

φic

∑j 6=i

mij(st−1)st−1

j xt−1j .

Then we check whether perturbations away from equilibrium socialization and legislative

efforts converge back to the estimated equilibrium efforts through the best responses of the

players in the network. This is done by starting slightly below or above the values of the

low equilibrium and successively iterating the best response dynamics.54

In all Congresses 105th-110th, we find that best-response dynamics converge back to our

estimated equilibrium (from the upper panel in Table C.4) after few iterations.55

Second, we compare our estimated equilibrium effort levels to those from a possible equi-

librium with higher levels of effort in each Congress. To do so, we take the system defined

by Proposition 2.1 under the estimated parameters and numerically search for solutions of

this system at higher values of (SDem, SRep, XDem, XRep).

These results are reported in the lower panel of Table C.4 in Appendix. We find that

there exists a higher effort level equilibrium for every Congress considered in our analysis.

These other equilibria are distinct from the ones we estimate, with effort levels that are

approximately 20% larger in magnitude to the empirically assessed ones. We also verify that

these unobserved equilibria are unstable. To do so, we repeat the stability exercise, starting

below the values of the higher equilibria. Here, we find that the dynamics diverge away from

these higher equilibria. Given that this is the only equilibrium with higher effort levels in

our set-up (see footnote 23), we don’t have a stable equilibrium with higher effort levels with

our parameters.

From these exercises one deduces that Congress is generally in an interior, low-effort

equilibrium and, moreover, that all Congresses operate at effort levels lower than at an

unobserved, unstable, high-effort equilibrium. These equilibria Pareto dominate the observed

equilibria.56

53For details, we refer the reader to Appendix B. Here sti takes xt−1i as given, but one could also solve for

simultaneous best reply dynamics and get the same results.54This is not a full check of stability, as we are not verifying all perturbations. However, given the structureof equilibria, all best responses in a party are proportional to the same X,S and have similar dynamics.55Only an adjustment is needed for Congress 107, where the pointwise estimate for c generates a numericalinstability. Other values within its confidence interval confirm this result.56We conclude this discussion by briefly noting that there also always exists a “semi-corner” equilibrium.In this equilibrium legislative effort is exerted and is chosen at level x∗i = αi/c, but there is no socializing,i.e. s∗i = 0 for all i. This occurs since there is no return to social effort if no other politician is socializing.Each politician acts in autarky. Effort is still provided in the model, because there are direct incentives αifor legislative effort, but no law is passed. Such an outcome is not desirable for politicians or voters due toProposition 2.3. Furthermore, this semi-corner equilibrium is unstable, in the sense that, were any politicianto deviate to a positive social effort sj , so would all the other politicians. The semi-corner equilibrium is not


6.2. Counterfactuals. We use the model to assess four counterfactuals.

First, we investigate the topical issue of how social networks vary endogenously with

political polarization and its consequences for legislative activity. We consider an exercise

in which there is an increase in polarization stemming from either drifts in the ideology of

certain members of Congress or from partisanship in mixing with politicians from a different

party. We then discuss how those results can be overturned if the selection of politicians to

the House of Representatives changes.

Second, we continue with a counterfactual analysis of the Congressional emergency re-

sponse to the 2008-09 financial crisis by modifying the partisan composition of the 110th

House back to the 109th GOP majority. This counterfactual provides evidence against a

common claim during the Tea Party movement surge of 2010 that excessive government in-

tervention during the crisis was rooted in the Democratic party’s congressional control. We

show that a counterfactual GOP controlled House would have not had much lower likelihood

of passing the same legislation in the fall of 2008. The final two counterfactuals investigate

the nonlinear properties of the model. We briefly describe them at the end of the section

and present the full results in Appendix.

6.2.1. The Effects of Polarization on Effort Levels. Do social and legislative activity in Con-

gress decrease with political polarization? What are the changes in the likelihood of bill

passage and legislative activity when bipartisanship becomes more rare? And, if the effects

of polarization are welfare-reducing, can they be overturned? Our model provides guidance

in answering these questions.

In our setting, polarization can be interpreted in two distinct ways. First, polarization

can be the drifting apart of ideologies of individual legislators. Second, polarization can map

into partisanship: a reduction in the likelihood that members of two parties mix socially as

opposed to only within their own party.

We begin with the first approach in changing the relative ideologies. In our empirical

model, type αi correlates significantly with ideology and in fact the direct benefit from

legislative effort appears to decrease as ideologies drifts to the right (Table 2). This is the case

when Republicans become more extreme, for example, as it has been extensively documented

in a large literature on ideological polarization in the modern Congress (see McCarty et al.,

2006). Our first exercise looks at the effect on equilibrium activity level of Republicans’

ideological stances shifting to the right by 20% holding the ideology of Democrats constant.

This translates in lower direct benefits from legislative activity for Republicans by 10%.57

As we see from Table 5, an increase in polarization leads to an average decrease of 8.3%

and 7.5% in social effort, for Democrats and Republicans respectively, and of 3.4% and 1.9%

for legislative effort. Intuitively, Republicans now have lower benefits, so they provide less

observed, given that we observe positive socialization in Congress. Such a complete shutdown of socializationeffort would be unstable, and so should not be observed for any length of time.57An increase of approximately 20% in ideological stance of for Republicans correlates with a decrease of10% in their direct benefit from legislative activity. To see this, recall that αi = ez

′iβ , the magnitude of

zi = Ideology is of the order of 1 and we seek the coefficient ∆ such that 0.9 = e∆βIdeology . For a value ofβIdeology at approximately -0.5 (Table 2), we get that ∆ = −ln(0.9)/0.5 ≈ 20%.


Table 5. Effects of Polarization on Equilibrium Effort Levels (and How toOverturn it)

Congress 105 106 107 108 109 110

Baseline Equilibrium

SDem 0.922 1.506 1.443 1.403 1.412 1.484

SRep 0.724 0.916 0.808 0.831 0.881 0.988

XDem 4.478 5.680 5.111 5.058 5.206 5.484

XRep 4.231 4.773 4.196 4.235 4.426 4.722

Change in Effort Levelswhen ARep decreases by 10%

SDem -0.070 -0.130 -0.095 -0.109 -0.123 -0.152

SRep -0.056 -0.071 -0.047 -0.056 -0.067 -0.093

XDem -0.092 -0.214 -0.150 -0.170 -0.194 -0.246

XRep -0.063 -0.094 -0.057 -0.069 -0.086 -0.127

Change in Effort Levelswhen ARep decreases by 10%,but ADem is 10% higher than estimated

SDem 0.053 0.223 0.156 0.231 0.269 0.233

SRep 0.023 0.156 0.135 0.157 0.173 0.165

XDem 0.071 0.381 0.251 0.372 0.441 0.392

XRep 0.027 0.220 0.174 0.207 0.237 0.239

We compute the effects of polarization on effort levels in each Congress. In the second panel, we decreasethe types of Republicans by 10%, representing a drift to the right of their ideologies by about 20%. Weshow the changes to (SDem, SRep, XDem, XRep) relative to the values in the baseline (first panel). In thethird panel, we show the changes to equilibrium effort levels when Republican party types decrease by10%, but when the types of the Democrats (ADem =

∑i∈Dem αi) increases by 10% relative to estimated

levels. Although polarization decreases effort levels relative to the estimated values, it is sufficient forone party’s types to increase moderately to generate spillovers sufficient to increase effort levels. Thisis even the case when Democrats are the minority. Further results are in Appendix (Table C.5)

legislative and social effort in equilibrium. This also impacts Democrats, however, whose

socialization with Republicans involves dealing with even lower types, lower effort levels and

less dense social networks. There are fewer externalities from social connections, decreasing

the returns from social activity for all politicians. Since legislative effort is complementary,


it decreases for all legislators, but not equally. The impacts on legislative productivity are

even greater for Democrats, who are more active legislators and higher αi to begin with.

What can be done to reverse the perverse effects of polarization, described in Table 5? It

turns out that even small increases in the types of elected politicians within one party can

generate enough social activity to neutralize increased party bias. In Table 5, we show that

an increase in 10% in the average type of the politicians in the Democratic party is sufficient

to generate equilibrium effort levels above the baseline of no polarization (i.e. before the

initial change in GOP occurred).58 Note that this is true even though Democrats are the

minority party in most of the sample. This is driven by spillovers across parties from having

high αi types in Congress. If Democrats’ types increase, they interact with each other

and with Republicans more (under pDem < 1). This implies that there are higher returns

for Republicans to provide more social effort to connect with those active Democrats. All

legislators, in turn, increase their legislative effort. The equilibrium effects of this increase

in types are amplified in a large enough way to overturn the initial decrease in effort due to

polarization.

We note that social connections are the key mechanism by which the behavior of more

active members can permeate throughout Congress, leading to gains from one party reaching

all legislators. Other models that treat legislators as independent actors would miss such

externalities from strategic behavior.

A policy takeaway from this exercise is that unilateral improvements in the screening of

candidates by one party could neutralize the effects of polarization in the aggregate. These

improvements do not need to be coordinated, as decisions by one party can be sufficient. In

practice, such policies may include the adoption of different recruitment procedures for new

candidates, different primary election rules, or changes in party campaign financing. Finally,

even if some of these policies were to increase polarization (see Barber and McCarty, 2015 for

a review on the evidence of primaries driving polarization, for example), our results suggest

that the externalities of these policies might be large enough for a resulting positive effect

on legislative productivity.

A second interpretation of polarization within our model comes from changing partisan-

ship, via the pis. Polarization then corresponds to the hollowing out of bipartisanship,

captured by increases in (pDem, pRep). This is another form of polarization, as Democrats

and Republicans meet less frequently and connect less with one another to approve legisla-

tion. The incentives under this type of polarization differ from the case of ideological drift

and produces a somewhat more ambiguous interpretation of the effects of polarization on

legislative effort and productivity.

Numerical exercises are shown in Table C.5 in Appendix, where we increase pRep to 0.3

relative to baseline estimates of 0.05 − 0.15.59 The results suggest that polarization itself

is not necessarily effort reducing. In fact, polarization may lead to higher effort levels for

58We note that this argument holds for values below 10% as well, given that the gains of such a change arelarge enough (see the third panel in Table 5). We stick to 10% as a focal number.59Changing both parties’ biases to 0.3 instead of just the Republicans’ does not significantly change theresults.


one party than in the non-polarized equilibrium. As a result, that party also obtains higher

approval rates for legislation.

The intuition relies on having one party with types that are sufficiently higher than the

other (ADem >> ARep), in line with what is estimated in the data. This type of polariza-

tion makes legislators from the high type party interact mostly within their own party.60

This isolation generates large externalities from connections, as high type politicians con-

nect more often with other high types within their party. This outcome may be preferable

to dealing with members of the opposition (low types on average), who choose low effort

levels, do not have as many connections and, as a consequence, do not generate many

spillovers/externalities from interactions.

Table C.5 suggests that in this case reducing bipartisanship may be welfare enhancing, as

long as one party has sufficiently higher αi than the other, and it would benefit to work only

within their group to get legislation approved.61

6.2.2. Counterfactual of the Democratic Party Takeover in the 110th Congress and Tea Party

narrative. We now propose a counterfactual of congressional behavior during the 110th cycle.

Elected in November 2006, the House of Representative turned Democratic majority after

twelve years of consecutive Republican control. This revealed to be a particularly consequen-

tial election, as it was the 110th Congress that voted between the Summer and the Fall of

2008 a host of emergency economic measures in response to the 2008-09 financial crisis (for

an analysis of these Congressional votes, see Mian et al., 2010). Some of this legislative ac-

tivity happened to be extremely momentous, including the vote of the Emergency Economic

Stabilization Act of 2008 (EESA, also known as the “TARP” from the Troubled Asset Relief

Program), which initially failed passage in the House, inducing one of the largest intra-day

losses in NYSE’s history.

An important counterfactual is to assess how relevant the role of congressional networks

was in eventually guaranteeing a responsive legislative intervention to the financial crisis.

How different would legislative activity have looked absent the Democratic party takeover

in 2006?

Within our framework, this counterfactual corresponds to keeping the composition of the

109th Congress in the 110th Congress. That is, we keep the observed characteristics and

estimated αi for the members of the 109th Congress in an analysis of outcomes in the 110th

Congress. Meanwhile, the institutional setting of the 110th Congress remains the same (with

the cost and returns to social effort, as well as the other institutional parameters kept at

their estimated values for the 110th Congress).

Table 3 implies that the distribution of αi from the 109th Congress appears mostly mildly

to the left of what is estimated for the 110th Congress. However, even small differences in the

vector of α could potentially produce substantial effects through the network in terms of bill

likelihood of success. In Table 6 we inspect the magnitude of the counterfactual reductions in

the likelihood of bill passage for some of the most important emergency response legislation

60A particular example is the case of full polarization, also shown in Table C.5 in Appendix.61This suggests that parties would have incentives to affect bipartisanship. At the end of Appendix B wediscuss endogenous partisanship in this model.


during the Fall of 2008. This set includes in addition to the EESA, the Housing and Economic

Recovery Act of 2008 (aimed at foreclosure prevention, also studied by Mian et al., 2010),

the Economic Stimulus Act of 2008 and the Supplementary Appropriations Act, both large

bills precursor of the fiscal intervention of 2009. Table 6 reports the relative differences in

bill passage probabilities between the counterfactual and the estimated model, as well as the

baseline probabilities.62

It appears all differences are in the range of a 10−15 percent reduction in the likelihood of

success, a quantitatively small effect considering the baseline probability of approval. That

is, the social network composition of the House would have not changed in a sufficiently

different way to substantially affect final voting outcomes. This is counter to the claim that,

absent the Democratic takeover of the House in 2006, the financial crisis response would

have been substantially different, with a more restrained government intervention under

a Republican Congress, taking as given emergency legislation being set forth by Treasury

Secretary Hank Paulson and Federal Reserve Chairman Ben Bernanke. This is a relevant

consideration, as it speaks to one of the crucial claims behind the Tea Party movement surge

within the Republican party in 2010, whose primary targets were government intervention

during the crisis and the consequences of the emergency stabilization and stimulus programs

on U.S. national debt (Mayer, 2016).

In Appendix C, we provide two additional counterfactuals that assess basic properties of

the model, first by changing the relative cost of legislative effort c, and then by reducing

bill quality (but not the incentives to socialize - that is, changing γP (i) while keeping φiconstant). These two exercises further emphasize nonlinear effects at play within our model

- in the first, increases in c lead to large nonlinear decreases in the probability of bill approval.

The mechanism is through the complementarity between legislative and social effort: as the

first becomes more costly, it is reduced, which by complementarity induces decreases in the

equilibrium levels of the other form of effort. Through feedback, as all politicians incorporate

this, the new equilibrium has lower effort levels in both dimensions. Meanwhile, the second

exercise finds that simple changes to average bill quality without changing socialization

outcomes do not generate externalities in effort. From that, we conclude that reforms aimed

at increasing effort levels and bill approval rates should alter the incentives to socialize to

generate nonlinear externalities - through the distribution of types, the relative costs of

socialization c, or the relative benefits from it, φi.

62Some of these bills’ complex histories align with a low likelihood of success. The EESA of 2008 failedthe House. In addition, about H.R. 3221, Mian et al. (2010) write: “Roll call 301: “On Agreeing to theSenate Amendment with Amendment No. 1: H.R. 3221 Foreclosure Prevention Act of 2008. This vote isconsidered by many the first crucial roll call in the political economy of the crisis and was characterized bystrong opposition (and a veto threat) by the executive branch. The Wall Street Journal (May 9, 2008) refersto the vote as follows: “The House voted 266-154 in favor of the centerpiece of the legislation $300 billionin federal loan guarantees -despite a White House veto threat. In particular, “The heart of the legislationis a program to help struggling homeowners by providing them with new mortgages backed by the FederalHousing Administration. The guarantees would be provided if lenders agree to reduce the principal of aborrower’s existing mortgage. H.R. 3221 had also previously failed cloture in the Senate in February 2008.


Table 6. Counterfactuals in α: Looking at the Changes in (Ex-Ante) pre-dicted probability of Emergency Crisis bills in the 110th Congress, if the Re-publicans who lost their seats remained

Act Proportional Baseline ProbabilityChange of Success

Emergency Economic Stabilization Act of 2008. (H.R. 1424) -0.119 0.165Sponsor: Patrick Kennedy, Democrat - RIHousing and Economic Recovery Act of 2008 (H.R. 3221) -0.110 0.175Sponsor: Nancy Pelosi, Democrat - CAEconomic Stimulus Act of 2008 (H.R. 5140) -0.110 0.175Sponsor: Nancy Pelosi, Democrat - CASupplementary Appropriations Act, 2008 (H.R. 2642) -0.120 0.141Sponsor: Chet Edwards, Democrat - TX

The table presents the proportional change (Counterfactual - Model)/Model of the probability of eachbill passing under our counterfactual scenario. The counterfactual scenario is keeping the Republicanmajority and composition from the 109th Congress, in the 110th Congress. To do so, we keep thecharacteristics of all politicians from the 109th Congress with the estimated parameters of the 110thCongress. The only difference to characteristics is we add 1 for each politician’s Tenure variable, asthose politicians would have stayed 1 extra term in the counterfactual. We do not change the Committeecomposition or ideology. We then calculate the projected probability of bill approval using the estimatedparameters from the 110th Congress. The baseline (model) probability is shown in the second column,computed using pDem, pRep from Table H.1 in Appendix H.

7. Conclusions

We have developed and estimated a structural model of legislative activity for the U.S.

Congress in which endogenous, partisan social interactions play an important role in pro-

moting bill passage. We estimate that social effort matters substantially and significantly

for legislative activity.

By endogenizing both legislative and social efforts, we are able to accommodate comple-

mentarities in actions that appear to be strong. In particular, we find that complementarities

among politicians are quantitatively substantial (on the order of 0.1 to 0.25 of the direct in-

centives), and are fairly stable across our sample period.

We also find that the two parties have different base payoffs from passing bills, both in

terms of the average and variance across party members (both are higher for the Democrats).

Overall, we show how the process of informal social interaction among legislators may paint

a less extreme, although still partisan, picture of the internal operation of Congress.

Multiple equilibria arise naturally within our theoretical setting (as it is typical of models

of endogenous network formation). By careful consideration of the theoretical model and its

behavior around the estimated equilibrium, we are able to show that Congress appears in a

stable, low-socialization equilibrium, with effort levels lower than in a Pareto superior, but

unstable, equilibrium present in all Congresses.

Finally, our estimated model enables us to perform relevant counterfactuals. We show

how polarization can lead to decreased social and legislative activity in Congress, but that

can be overturned by a unilateral improvement in the selection of politicians. Politicians


who are more active can generate large enough externalities from their behavior that spills

over across parties. We also estimate that the response to the 2008-09 financial crisis would

not have changed in terms of levels of legislation if the Democrats had not taken over the

House and show substantial impacts from changing the relative cost of social effort in terms

of probability of bill passage.

From the methodological perspective, our tractable model of endogenous network for-

mation with biased socialization may be fruitful for further investigation of the behavior of

other political agencies beyond the U.S. context or in more general environments where both

endogenous socialization and homophily are relevant features.

References

Acemoglu, D., C. Garcıa-Jimeno, and J. A. Robinson (2015). State capacity and economic

development: A network approach. American Economic Review 105 (8), 2364–2409.

Aleman, E. and E. Calvo (2013). Explaining policy ties in presidential congresses: A network

analysis of bill initiation data. Political Studies 61 (2), 356–377.

Anderson, W. D., J. M. Box-Steffensmeier, and V. Sinclair-Chapman (2003). The keys to

legislative success in the us house of representatives. Legislative Studies Quarterly 28 (3),

357–386.

Baccara, M. and L. Yariv (2013). Homophily in peer groups. American Economic Journal:

Microeconomics 5 (3), 69–96.

Badev, A. (2017). Discrete games in endogenous networks: Theory and policy. Mimeo,

University of Pennsylvania.

Bala, V. and S. Goyal (2000). A noncooperative model of network formation. Economet-

rica 68 (5), 1181–1229.

Ballester, C., A. Calvo-Armengol, and Y. Zenou (2006). Who’s who in networks. wanted:

The key player. Econometrica 74 (5), 1403–1417.

Banerjee, A. V., A. G. Chandrasekhar, E. Duflo, and M. O. Jackson (2018). Changes in

social network structure in response to exposure to formal credit markets. SSRN paper

3245656 .

Barber, M. and N. McCarty (2015). Causes and consequences of polarization. Political

Negotiation: A Handbook 37, 39–43.

Bartels, L. M. (1993). Messages received: the political impact of media exposure. American

Political Science Review 87 (02), 267–285.

Battaglini, M. and E. Patacchini (2018). Influencing connected legislators. Journal of Po-

litical Economy 126 (6), 2277–2322.

Battaglini, M., E. Patacchini, and E. Rainone (2019). Endogenous social connections in

legislatures. Technical report, National Bureau of Economic Research.

Battaglini, M., V. L. Sciabolazza, and E. Patacchini (2018). Effectiveness of connected

legislators.

Baumann, L. (2017). A model of weighted network formation. SSRN Working Paper .

Bramoulle, Y., H. Djebbari, and B. Fortin (2009). Identification of peer effects through social

networks. Journal of Econometrics 150 (1), 41–55.


Bratton, K. A. and S. M. Rouse (2011). Networks in the legislative arena: How group

dynamics affect cosponsorship. Legislative Studies Quarterly 36 (3), 423–460.

Cabrales, A., A. Calvo-Armengol, and Y. Zenou (2011). Social interactions and spillovers.

Games and Economic Behavior 72 (2), 339–360.

Cameron, A. C., J. B. Gelbach, and D. L. Miller (2011). Robust inference with multiway

clustering. Journal of Business & Economic Statistics 29 (2), 238–249.

Canen, N., C. Kendall, and F. Trebbi (2020). Unbundling polarization. Econometrica

(forthcoming).

Chandrasekhar, A. G. and M. O. Jackson (2016). A network formation model based on

subgraphs. SSRN Working Paper .

Cho, W. K. T. and J. H. Fowler (2010). Legislative success in a small world: Social network

analysis and the dynamics of congressional legislation. The Journal of Politics 72 (01),

124–135.

Cohen, L. and C. J. Malloy (2014). Friends in high places. American Economic Journal:

Economic Policy 6 (3), 63–91.

Currarini, S., M. O. Jackson, and P. Pin (2009). An economic model of friendship: Ho-

mophily, minorities, and segregation. Econometrica 77 (4), 1003–1045.

Currarini, S., M. O. Jackson, and P. Pin (2010). Identifying the roles of race-based choice and

chance in high school friendship network formation. Proceedings of the National Academy

of Sciences 107 (11), 4857–4861.

Currarini, S. and M. Morelli (2000). Network formation with sequential demands. Review

of Economic Design 5, 229 – 250.

de Paula, A., I. Rasul, and P. Souza (2019). Recovering social networks from panel data:

identification, simulations and an application.

de Paula, A., S. Richards-Shubik, and E. Tamer (2018). Identifying preferences in networks

with bounded degree. Econometrica 86 (1), 263–288.

Dutta, B. and S. Mutuswami (1997). Stable networks. Journal of Economic Theory 76, 322

– 44.

Fiorina, M. P. (2017). The political parties have sorted. Hoover Institution Essay on Con-

temporary Politics, Series No. 3 .

Fowler, J. H. (2006). Connecting the congress: A study of cosponsorship networks. Political

Analysis 14 (4), 456–487.

Gentzkow, M. (2017). Polarization in 2016. Essay, Stanford University .

Gentzkow, M. and J. Shapiro (2015). Congressional record for 104th-110th congresses: Text

and phrase counts. ICPSR33501-v4.

Goldsmith-Pinkham, P. and G. Imbens (2013a). Rejoinder: Social networks and the identi-

fication of peer effects. Journal of Business and Economic Statistics 31:3, 279–281.

Goldsmith-Pinkham, P. and G. Imbens (2013b). Social networks and the identification of

peer effects. Journal of Business and Economic Statistics 31:3, 253–264.

Graham, B. S. (2017). An econometric model of network formation with degree heterogeneity.

Econometrica 85 (4), 1033–1063.


Groll, T. and A. Prummer (2016). Whom to lobby? targeting in political networks. Working

Paper 808, London.

Groseclose, T. and C. Stewart, III (1998). The value of committee seats in the house, 1947-91.

American Journal of Political Science 42 (2), 453–474.

Harmon, N., R. Fisman, and E. Kamenica (2019). Peer effects in legislative voting. American

Economic Journal: Applied Economics 11 (4), 156–80.

Herings, P. J.-J., A. Mauleon, and V. Vannetelbosch (2009). Farsightedly stable networks.

Games and Economic Behavior 67 (2), 526–541.

Hiller, T. (2017). Peer effects in endogenous networks. Games and Economic Behavior 105,

349–367.

Hsieh, C.-S. and L. F. Lee (2016). A social interactions model with endogenous friendship

formation and selectivity. Journal of Applied Econometrics 31 (2), 301–319.

Jackson, M. O. (2005). A survey of models of network formation: Stability and efficiency.

Group Formation in Economics; Networks, Clubs and Coalitions, ed. G Demange, M

Wooders. Cambridge, UK: Cambridge Univ. Press .

Jackson, M. O. (2008). Social and Economic Networks. Princeton, NJ, USA: Princeton

University Press.

Jackson, M. O. (2013). Comment: Unraveling peers and peer effects (com-

ments on goldsmith-pinkham and imbens’ “social networks and the identification of

peer effects”). Journal of Business and Economic Statistics 31:3, 270–273, DOI:

10.1080/07350015.2013.794095.

Jackson, M. O. (2019). The friendship paradox and systematic biases in perceptions and

social norms. Journal of Political Economy 127 (2), 777–818.

Jackson, M. O. and A. Watts (2002). The evolution of social and economic networks. Journal

of Economic Theory 106 (2), 265–295.

Jackson, M. O. and A. Wolinsky (1996). A strategic model of social and economic networks.

Journal of Economic Theory 71 (1), 44–74.

Jackson, M. O. and Y. Zenou (2015). Games on networks. Volume 4 of Handbook of Game

Theory with Economic Applications, pp. 95–163. Elsevier.

Kessler, D. and K. Krehbiel (1996). Dynamics of cosponsorship. American Political Science

Review 90 (03), 555–566.

Kirkland, J. H. (2011). The relational determinants of legislative outcomes: Strong and weak

ties between legislators. The Journal of Politics 73 (03), 887–898.

Koger, G. (2003). Position taking and cosponsorship in the us house. Legislative Studies

Quarterly 28 (2), 225–246.

Konig, M., C. J. Tessone, and Y. Zenou (2009). A dynamic model of network formation with

strategic interactions.

Lazer, D. (2011). Networks in political science: Back to the future. PS: Political Science &

Politics 44 (01), 61–68.

Leung, M. P. (2015). Two-step estimation of network-formation models with incomplete

information. Journal of Econometrics 188 (1), 182–195.

Leung, M. P. (2019). A weak law for moments of pairwise stable networks. Journal of

Econometrics .

Manski, C. F. (2013). Comment: Social networks and the identification of peer effects.

Journal of Business and Economic Statistics 31:3, 273–275.

Masket, S. E. (2008). Where you sit is where you stand: The impact of seating proximity

on legislative cue-taking. Quarterly Journal of Political Science 3, 301–311.

Mauleon, A., H. Song, and V. Vannetelbosch (2010). Networks of free trade agreements

among heterogeneous countries. Journal of Public Economic Theory 12 (3), 471–500.

Mayer, J. (2016). Dark Money: how a secretive group of billionaires is trying to buy political

control in the US. Scribe Publications.

Mayhew, D. R. (1974). Congress: The electoral connection. Yale University Press.

Mayhew, D. R. (2005). Divided we govern: Party control, lawmaking and investigations,

1946-2002. Yale University Press.

McCarty, N., K. Poole, and H. Rosenthal (2006). Polarized America. MIT Press.

Mele, A. (2017). A structural model of segregation in social networks. Econometrica 85 (3),

825–850.

Mian, A., A. Sufi, and F. Trebbi (2010). The political economy of the us mortgage default

crisis. The American Economic Review 100 (5), 1967–1998.

Rogowski, J. C. and B. Sinclair (2012). Estimating the causal effects of social interaction

with endogenous networks. Political Analysis 20 (3), 316–328.

Routt, G. C. (1938). Interpersonal relationships and the legislative process. The Annals of

the American Academy of Political and Social Science 195 (1), 129–136.

Schneider, J. (2015). One-minute speeches: Current house practices. CRS Report RL30135.

Sheng, S. (2016). A structural econometric analysis of network formation games. UCLA

mimeo.

Squintani, F. (2018). Networks and diverse preferences. Working Paper .

Stewart, III, C. (2012). The value of committee assignments in congress since

1994. MIT Political Science Department Research Paper 2012-7. Available at SSRN:

https://ssrn.com/abstract=2035632 (2), 453–474.

Stewart, III, C. and J. Woon (2016). Congressional committee assignments, 103rd to 114th

congresses, 1993–2017: House, august 12, 2016. Data Archive.

Trebbi, F. and E. Weese (2019). Insurgency and small wars: Estimation of unobserved

coalition structures. Econometrica 87 (2), 463–496.

Wang, Y. X. and Y. J. Zhang (2013). Nonnegative matrix factorization: A comprehensive

review. IEEE Transactions on Knowledge and Data Engineering 25 (6), 1336–1353.

Wilson, R. K. and C. D. Young (1997). Cosponsorship in the u. s. congress. Legislative

Studies Quarterly 22 (1), 25–43.

Zhang, Y., A. Friend, A. L. Traud, M. A. Porter, J. H. Fowler, and P. J. Mucha (2008).

Community structure in congressional cosponsorship networks. Physica A: Statistical Me-

chanics and its Applications 387 (7), 1705–1712.


ONLINE APPENDIX

Endogenous Networks and Legislative Activity

Nathan Canen, Matthew O. Jackson and Francesco Trebbi

Appendix A. Proofs

Lemma A.1. ∑j 6=i

sisjmij(s) = si.

Proof.∑j 6=i

sisjmij(s) =∑

j 6=i,j∈P (i)

sisjmij(s) +∑

j 6=i,j /∈P (i)

sisjmij(s)

=∑

j 6=i,j∈P (i)

sisj

(p(i)

p(j)∑k∈P (i),k 6=i p(k)sk

+ (1− p(i)) (1− p(j))∑k 6=i(1− p(k))sk

)+

+∑

j 6=i,j /∈P (i)

sisj(1− p(i))(1− p(j))∑

k 6=i(1− p(k))sk

= si

(p(i)

(∑j 6=i,j∈P (i) p(j)sj∑k 6=i,k∈P (i) p(k)sk

)+ (1− p(i))

∑j 6=i,j∈P (i)(1− p(j))sj∑

k 6=i(1− p(k))sk

)+

+si(1− p(i))

(∑j 6=i,j /∈P (i)(1− p(j))sj∑

k 6=i(1− p(k))sk

)

= si

(p(i) + (1− p(i))

(∑j 6=i,j∈P (i)

∑j 6=i,j /∈P (i)(1− p(j))sj∑

k 6=i(1− p(k))sk

))

= si

(p(i) + (1− p(i))

(∑j 6=i(1− p(j))sj∑k 6=i(1− p(k))sk

))= si.

Proposition 2.1: The limit equilibrium is defined by equations (2.11)-(2.13).

Proof of Proposition 2.1. Recall that we have from equations (2.8) and (2.7), from the First

Order Conditions, that:

(A.1) c =αix∗i

+s∗2ix∗2i

,

and

(A.2)s∗ix∗i

= φi∑j 6=i

s∗jmij(s∗)x∗j .


We also use that:

x∗i = αiXP (i)(A.3)

s∗i = αiSP (i),(A.4)

for some XP (i), SP (i), which comes from the fact thats∗ix∗i

andx∗iαi

are the same for all agents

within a party. Let P (i) ∈ 1, 2 be arbitrary.

Using (A.3) in (2.8) implies:

c =αix∗i

+s∗2ix∗2i

=αi

αiXP (i)

+α2iS

2P (i)

α2iX

2P (i)

=1

XP (i)

+S2P (i)

X2P (i)

.

Multiplying both sides by X2P (i) yields:

cX2P (i) = XP (i) + S2

P (i),(A.5)

which is (2.13).

Let us now substitute (A.3) in (2.7):

αiSP (i)

αiXP (i)

= φP (i)

∑j 6=i

αjSP (j)mij(s∗)αjXP (j)

SP (i)

XP (i)

= φP (i)

∑j 6=i

α2jXP (j)SP (j)mij(s

∗)

= φP (i)

∑j 6=i

α2jXP (j)SP (j)

(p(i)

p(j)∑k∈P (i),k 6=i p(k)s∗k

+ (1− p(i)) (1− p(j))∑k 6=i(1− p(k))s∗k

)Ij∈P (i)

+φP (i)

∑j 6=i

α2jXP (j)SP (j)

((1− p(i)) (1− p(j))∑

k 6=i(1− p(k))s∗k

)Ij /∈P (i).

Note that for the first two terms, p(i) = p(j) because they are only summed when j ∈ P (i).

For the last, p(i) 6= p(j) as it is summed when j /∈ P (i).

Rewriting the above with this implies:


SP (i)

XP (i)

= φP (i)

∑j 6=i

α2jXP (i)SP (i)

(p(i)

p(i)∑k∈P (i),k 6=i p(i)s

∗k

+ (1− p(i)) (1− p(i))∑k 6=i(1− p(k))s∗k

)Ij∈P (i)

+φP (i)

∑j 6=i

α2jXP (j)SP (j)

((1− p(i)) (1− p(j))∑

k 6=i(1− p(k))s∗k

)Ij /∈P (i).

Using that s∗k = αkSP (k) leads to:

SP (i)

XP (i)

= φP (i)

∑j 6=i

α2jXP (i)SP (i)

(p(i)2

p(i)∑

k∈P (i),k 6=i αkSP (k)

+(1− p(i))2∑

k 6=i(1− p(k))αkSP (k)

)Ij∈P (i)

+φP (i)

∑j 6=i

α2jXP (j)SP (j)

((1− p(i))(1− p(j))∑k 6=i(1− p(k))αkSP (k)

)Ij /∈P (i).

Let us focus on the case of P (i) = 1, as the other case is symmetric.

S1

X1

= φ1

∑j 6=i

α2jX1S1

(p1∑

k∈P (i),k 6=i αkS1

+(1− p1)2∑

k 6=i(1− p(k))αkSP (k)

)Ij∈P (i)

+φ1

∑j 6=i

α2jX2S2

((1− p1)(1− p2)∑k 6=i(1− p(k))αkSP (k)

)Ij /∈P (i).

Finally, we use that:

∑k 6=i

(1− p(k))αkSP (k) =∑

k 6=i,k∈P (i)

(1− p(k))αkSP (k) +∑

k 6=i,k/∈P (i)

(1− p(k))αkSP (k)

=∑

k 6=i,k∈P (i)

(1− p1)αkS1 +∑

k 6=i,k/∈P (i)

(1− p2)αkS2

= (1− p1)S1

∑k 6=i,k∈P (i)

αk + (1− p2)S2

∑k 6=i,k/∈P (i)

αk

= (1− p1)S1A1 + (1− p2)S2A2.

To finalize the calculations, we use the simplification above for the denominators of the

second and third terms.

Note that only αj is now a function of the summand j itself, in the main expression. We

also note that we can now use the indicators of j ∈ P (i) for the first two terms, and j /∈ P (i)

of the last term, within sums. These observations lead to the final equation:


S1

X1

= φ1X1S1

∑j 6=i

α2j

(p1

S1

∑k∈P (i),k 6=i αk

+(1− p1)2

(1− p1)S1A1 + (1− p2)S2A2

)Ij∈P (i)

+X2S2φ1

∑j 6=i

α2j

((1− p1)(1− p2)

(1− p1)S1A1 + (1− p2)S2A2

)Ij /∈P (i)

= φ1X1S1

∑j 6=i,j∈P (i)

α2j

(p1

S1A1

+(1− p1)2

(1− p1)S1A1 + (1− p2)S2A2

)

+X2S2φ1

∑j 6=i,j /∈P (i)

α2j

((1− p1)(1− p2)

(1− p1)S1A1 + (1− p2)S2A2

)

= φ1X1S1B1

(p1

S1A1

+(1− p1)2

(1− p1)S1A1 + (1− p2)S2A2

)+X2S2φ1B2

((1− p1)(1− p2)

(1− p1)S1A1 + (1− p2)S2A2

)= φ1

(X1B1p1

A1

+X1S1B1(1− p1)2

(1− p1)S1A1 + (1− p2)S2A2

+X2S2B2(1− p1)(1− p2)

(1− p1)S1A1 + (1− p2)S2A2

)= φ1

(p1X1B1

A1

+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2X2S2

(1− p1)A1S1 + (1− p2)A2S2

).

Proof of Proposition 2.2. Recall that an interior equilibrium is a solution to (2.11) to (2.13).

So, rewriting these:

(A.6) S1 = X1φ1

(p1B1X1

A1

+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2S2X2

(1− p1)A1S1 + (1− p2)A2S2

).

(A.7) S2 = X2φ2

(p2B2X2

A2

+(1− p2)2B2S2X2 + (1− p1)(1− p2)B1S1X1

(1− p1)A1S1 + (1− p2)A2S2

).

(A.8) cX21 = X1 + S2

1 , cX22 = X2 + S2

2 .

Substituting (A.6) into (A.8) leads to

cX21 = X1 +X2

1φ21

(p1B1X1

A1

+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2S2X2

(1− p1)A1S1 + (1− p2)A2S2

)2

.

or

(A.9) cX1 = 1 +X1φ21

(p1B1X1

A1

+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2S2X2

(1− p1)A1S1 + (1− p2)A2S2

)2

.

There is a similar expression for S2, X2. Note that the right hand side of (A.9) lies above

the left hand side as we approach X1 = 0 (same for X2). To have an interior solution, we

need the right hand side to sometimes fall at or below the left hand side for positive X1.

Suppose that the equilibrium (when it exists) is such that X1 ≥ X2, and the other case is

analogous just reversing subscripts everywhere. Then the right hand side is less than what


we get by replacing X2 by X1, and so we want

(A.10) cX1 ≥ 1 +X31φ

21

(p1B1

A1

+(1− p1)2B1S1 + (1− p1)(1− p2)B2S2

(1− p1)A1S1 + (1− p2)A2S2

)2

.

for some interior X1. Rewriting

(A.11) cX1 ≥ 1 +X31φ

21

(p1B1

A1

+(1− p1)2B1 + (1− p1)(1− p2)B2

S2

S1

(1− p1)A1 + (1− p2)A2S2

S1

)2

.

The right hand side is maximized either at S2

S1= 0 or S2

S1=∞, and so it is sufficient to have

(A.12) cX1 ≥ 1 +X31φ

21

(p1B1

A1

+ (1− p1) max

[B1

A1

,B2

A2

])2

.

Let

D1 = p1B1

A1

+ (1− p1) max

[B1

A1

,B2

A2

]Then (A.12) can be rewritten as

(A.13) cX1 ≥ 1 +X31φ

21D

21.

for some positive X1. Note that

D1 ≤ D = max

[B1

A1

,B2

A2

]So, it is sufficient to have

(A.14) cX1 ≥ 1 +X31φ

21D

2.

for some positive X1.

It is necessary and sufficient to check that the left hand side and right hand side are tangent

at the point at which the slope of the right hand side is c. This happens at X1 =√

c3φ21D

2

and then the corresponding sufficient condition becomes:

(A.15) c

(c

3φ21D

2

)1/2

≥ 1 +

(c

3φ21D

2

)3/2

φ21D

2,

or

(A.16)2c3/2

3√

3≥ φ1D.

Having this hold also for the other case, leads to the claimed expression.

Proof of Proposition 2.3. Assume by way of contradiction that there is sFBi , xFBi first best

that is finite. Consider the allocation s′i, x′i = λsFBi , λxFBi , with λ > 1. The later

increases all politician’s utility by a cubic rate (from equation (2.3)), while the costs increase

quadratically. Hence, s′i, x′i is feasible and yields a higher utility to all agents, which is a

contradiction.


Appendix B. Additional Aspects of the Theory

B.1. Best Response Dynamics. Best response dynamics are described as follows. Con-

sider starting at some vectors s0, x0. Then the best response dynamics are described by:

(B.1) sti = xt−1i φP (i)

∑j 6=i

mij(st−1)st−1

j xt−1j ,

and

(B.2) xti =αic

+ st−1i

φP (i)

c

∑j 6=i

mij(st−1)st−1

j xt−1j .

It follows that if s0 = 0, then mij(st−1) = 0 for all ij (recall Footnote 10) and we get

immediate convergence to sti = 0, xti = αic

for all t. Otherwise, st, xt will be positive for all t.

To see how these best response dynamics work for a special case, let us consider the

situation in which there is some S0, X0 such that s0i = αiS

0 and x0i = αiX

0 (which has to

eventually hold at any limit point).63

In that case, working with the limiting or continuum case, in which the matching function

is symmetric within a party, and presuming that St−1k > 0 for each party (which happens

after the first period if some s0j > 0 and otherwise the solution is already described above),

we end up with the following dynamics. For party k (letting k′ denote the other party):

(B.3) Stk = X t−1k φk

(mkk(S

t−1)BkSt−1k X t−1

k +mkk′(St−1)Bk′S

t−1k′ X

t−1k′

),

and

(B.4) X tk =

1

c+ St−1

k

φkc

(mkk(S

t−1)BkSt−1k X t−1

k +mkk′(St−1)Bk′S

t−1k′ X

t−1k′

).

where

mkk(St−1) =

pkSkAk

+(1− pk)2

(1− p1)S1A1 + (1− p2)S2A2

,

and

mkk′(St−1) =

(1− p1)(1− p2)

(1− p1)S1A1 + (1− p2)S2A2

.

B.2. Endogenous Partisanship. A natural extension of our model would be to endogenize

the pi’s. We comment here on potential directions and issues that arise.

First, it is easy to see that if one simply endogenized the pi’s within the current model

without introducing any costs of affecting pi, then the solutions would be corner solutions.

If a group can choose its pi without having any costs of selecting pi, then (generically in the

parameters) one of the two groups would want to be entirely partisan, since one of the two

groups would find interacting with itself more beneficial than interacting across the aisle.

Such a corner solution is clearly of little interest, and is incompatible with our empirical

estimates.

More generally, there are interactions, both within and across parties, that happen natu-

rally due to committee membership among other things and would be difficult to prevent,

and others that might be costly to encourage. This suggests that there would minimum

63This is also useful in determining the instability of equilibria.


and maximum levels of partisanship that could be attained and also that one would need to

model a nonlinear cost of partisanship. Once one provided a nonlinear cost to capture the

high cost of going to either extreme of pi = 0 or pi = 1, one would end up with an interior

equilibrium. A challenge would be that this could be dependent upon the cost formulation,

and so one would need to work with a flexible enough cost function to allow the model to fit

the data.

Having three endogenous choices for each of the two parties - partisanship, social effort, and

legislative effort - would then end up producing a model for which analytic characterizations

of the equilibrium would no longer be possible, and for which the multiplicity of equilibria

would more difficult to ascertain. There would be two approaches. One would be to work

entirely with numerical simulations. Since the interest in endogenizing partisanship levels

would presumably be to understand how they interact with other variables and change

incentives, this would require a very rich and complex set of simulations, especially as they

would be sensitive to the choice of the cost function.

Another approach, and perhaps the most fruitful, would be to fix one of the other effort

variables and return to a model in which there are just two different action variables that

agents/parties are making. Given the importance of partisanship on the endogeneity of the

network, a starting point might be to fix the xi’s and then work with the other variables. This

could be an interesting approach for further research. We chose to work with endogenizing

the network and legislative effort, holding partisanship constant, as these seem to be the

first-order questions, but understanding partisanship is also a very interesting topic.

Appendix C. Additional Tables, Figures and Counterfactuals

In this section we present additional results referenced in the main text. We first discuss

two additional counterfactuals that illustrate the model’s comparative statics in Appendix

C.1: how changes in c and changes in γP (i) affect equilibrium effort choices. Section C.2 has

additional Tables and Figures references in the main text and in Appendix C.1.

C.1. Additional Counterfactuals.

C.1.1. Change in c. We now examine counterfactuals with respect to socialization costs c.

Specifically, by increasing c we increase the cost of legislative effort relative to social effort.

Table C.1 reports what would happen to the model equilibrium activity levels if c were

higher. This exercise makes evident the nonlinear effects emerging from social interactions

in our environment, drivers of the results in the previous sections.

We find a negative effect of increasing c by 1 or 2% on the likelihood of bill passage.

Moreover, this effect is quantitatively large, substantially more than linear, consistent with

the evidence so far pointing towards a substantial role for social complementarities across

legislators. There is an increase of bill success probability of approximately 15 percent on

average for Democrats, and 10 percent for Republicans, across all Congresses, vis-a-vis a

decrease of 1 percent in c. Similarly large effects are present for a 2 percent cut. These

magnitudes appear consistent with the overall importance of social interactions in legislative

activity reported in Section 5.


Table C.1. Counterfactuals in c: Predicted (Proportional) Change in the(Mean) Probability of Bill Approval

Congress 105 106 107 108 109 110

Increase in 1% in c Democrats -0.0673 -0.156 -0.170 -0.149 -0.142 -0.156Republicans -0.0616 -0.118 -0.126 -0.114 -0.111 -0.123

Increase in 2% in c Democrats -0.128 -0.261 -0.278 -0.253 -0.243 -0.262Republicans -0.118 -0.204 -0.212 -0.198 -0.195 -0.212

The table presents the change in the average probability of bill approval under thecounterfactual, where the estimated cost c, is increased by either 1% or 2% (of theestimated value c). We do this by calculating the the implied optimal x∗i , s∗i fromProposition 2.1 under the appropriate value of c and calculate the probability of approvaldefined as 1

ζP (i)(s∗i )

2. We then find the percentual change over the predicted values under

Table 2.

The rationale for this effect is that an increase in c leads to a decrease in the choice of

legislative effort. Since the latter has a sufficiently strong complementarity with social effort,

socializing by the individual also decreases in this equilibrium (similarly to what happens

at the low effort equilibrium of Cabrales et al., 2011). As a result, returns to socializing

by others also decrease, who then further decrease their own legislative efforts until a new

equilibrium is reached. This feeds back and leads to a large (nonlinear) negative effect,

highlighting the importance of socializing in the approval of bills. These dynamics contrast

to an opposite positive feedback effect that can occur, for example, in the higher Pareto

equilibrium of Cabrales et al. (2011). In that case, an increase in c may lead to an increase

in bill approval. This is because, at the high equilibrium, a decrease in c makes socialization

more costly, which reduces the incentives for socialization. In turn, this reduces the returns

for xi due to the complementarity of effort choices.

C.1.2. Change in γ with φ constant. To conclude, let us recall that the shock to the bill

passage ε is assumed to be standard Pareto distributed with scale parameter γP (i) > 0,

hence a lower γ determines a lower median draw of the positive shock ε and a lower chance

of legislative success. We now investigate the effects of an institutional change in which γ

gets reduced. For each of the 105th-110th Congresses, Table C.2 shows what would happen

to equilibrium efforts if γ was reduced while keeping φ constant (noticing that ζ must adjust

inversely).

As the system of equations in Proposition 2.1 does not change - the equations depend

on φi directly - only the probability of approval is affected as per equation (2.3). Hence, γ

only changes the shape of the bill approval function. Quantitatively a decrease in γP (i) by

10 percent leads to a sizeable shift of the probability of approval curve to the left (which in

Table 1 are shown to vary from 9.57 to 12.85 percent). This is shown in Table C.2, which


Table C.2. Counterfactual in γ: Predicted Probability of Bill Approval

Congress 105 106 107 108 109 110

DemocratsDecrease in 10% in γDem 0.047 0.048 0.078 0.081 0.062 0.165

RepublicansDecrease in 10% in γRep 0.086 0.095 0.064 0.098 0.084 0.052

We calculate the average probability of bill approval by party, when γP (i) is decreasedby 10% (keeping both φDem, φRep constant). This is done by calculating the probabilityof bill approval, given by P (yi = 1) = 1

ζP (i)(s∗i )

2, with s∗i the solution to the non-

noisy equilibrium system from Proposition 2.1 (further details in Appendix G). We thendecrease γP (i) by 10%.

reports the average probabilities of bill approval under a smaller γP (i) for each party. As we

only change the values of γ in equation (4.4)-(4.5), the percentage change in the expected

probability of bill passage is linear, dropping by 10 percent as well.

C.2. Additional Tables and Figures. In this subsection, we include additional Tables

and Figures referenced in the main text and in Appendices A-C.


Figure C.1. Examples of Alternative Congress 110 Networks

(a) More than 3 Directed Cosponsorships

(b) Committee Network

(c) Alumni Network

We show illustrations of alternative networks used in the literature that we use to compare our modelagainst. A link in the committee network exists if two legislators sit in one of the 7 main committeestogether (see the Data section). A link in the alumni network exists if two legislators attended thesame university within 8 years of one another. While in the empirical specification of equation (5.2)the cosponsorship network is taken as the amount of directed cosponsorships, we illustrate it hereby plotting the upper triangular matrix of directed cosponsorships, with a link formed if a legislatorcosponsors more than 3 bills by another.


Table C.3. Main Results, Specification 2

Congress105 106 107 108 109 110

c 0.248 0.249 0.263 0.272 0.262 0.251(0.006) (0.006) (0.007) (0.007) (0.006) (0.005)

φDem [0.038,0.038] [0.043,0.043] [0.048,0.048] [0.045,0.0046] [0.045,0.045] [0.045,0.045]

φRep [0.031,0.031] [0.032,0.032] [0.034,0.034] [0.033,0.034] [0.034,0.034] [0.036,0.036]

ζDem 25.731 62.28 40.259 39.726 48.87 16.324(2.774) (9.675) (5.308) (5.561) (6.934) (1.174)

ζRep 8.885 8.208 9.134 7.612 9.704 20.274(0.661) (0.698) (0.769) (0.601) (0.751) (1.705)

SDem 0.908 1.069 1.067 1.005 1.051 1.176(0.03) (0.033) (0.034) (0.034) (0.029) (0.027)

SRep 0.872 0.989 0.999 0.875 0.919 1.071(0.039) (0.038) (0.05) (0.045) (0.047) (0.062)

XDem 4.415 4.488 4.188 4.082 4.296 4.726(0.141) (0.134) (0.135) (0.142) (0.124) (0.097)

XRep 5.186 5.547 5.604 4.942 5.065 5.426(0.23) (0.222) (0.276 (0.25 (0.258 (0.315

Ideology -0.492 -0.466 -0.591 -0.708 -0.561 -0.438(0.074) (0.072) (0.073) (0.083) (0.074) (0.062)

Tenure 0.003 0.004 0.001 0.003 0.003 0(0.003) (0.002) (0.002) (0.002) (0.002) (0.002)

Appropriations -0.041 -0.013 -0.071 -0.086 -0.079 -0.045(0.035) (0.029) (0.031) (0.036) (0.034) (0.025)

Energy and Commerce 0.018 0.025 -0.019 0.025 0.034 0.027(0.03) (0.028 (0.032) (0.035) (0.029) (0.02)

Oversight 0.034 0.041 0.053 0.027 0.045 0.002(0.025) (0.03) (0.032) (0.037) (0.039) (0.033)

Rules 0.151 0.138 0.125 0.14 0.181 0.12(0.022) (0.024) (0.029) (0.033) (0.031) (0.021)

Leadership 0.059 0.009 0.08 -0.114 -0.096 -0.751(0.066) (0.044) (0.082) (0.071) (0.045) (0.234)

Transportation -0.015 0.04 0.018 0.007 0 0.006(0.027) (0.028) (0.023) (0.029) (0.024) (0.023)

WaysAndMeans -0.061 -0.043 -0.023 -0.033 -0.023 -0.014(0.033) (0.027) (0.032) (0.037) (0.038) (0.03)

Ideology ×Rep 0.046 -0.039 -0.114 0.082 0.083 0.062(0.067) (0.06) (0.073) (0.067) (0.068) (0.076)

Tenure×Rep 0.009 0.006 0.002 0.006 0.002 0.001(0.003) (0.002) (0.003) (0.003) (0.003) (0.003)

Appropriations×Rep -0.008 -0.037 -0.139 -0.099 -0.107 -0.129(0.026) (0.024) (0.033) (0.034) (0.035) (0.04)

Energy and Commerce×Rep -0.067 -0.009 -0.039 -0.046 -0.027 -0.041(0.027) (0.024) (0.027) (0.028) (0.03) (0.034)

Oversight×Rep 0.069 0.091 0.081 0.055 0.068 0.03(0.029) (0.02) (0.027) (0.027) (0.031) (0.036)

Rules×Rep 0.098 0.082 0.055 0.037 0.119 0.096(0.035) (0.024) (0.025) (0.031) (0.036) (0.044)

Leadership×Rep -0.089 -0.06 -0.181 -0.213 -0.249 -0.04(0.063) (0.083) (0.149) (0.126) (0.148) (0.086)

Transportation×Rep -0.035 -0.031 -0.051 0 -0.019 -0.039(0.028) (0.024) (0.027) (0.028) (0.027) (0.031)

WaysAndMeans×Rep -0.037 -0.002 -0.01 -0.019 -0.07 -0.138(0.031) (0.035) (0.036) (0.032) (0.04) (0.044)

N 436 433 435 437 433 436

Notes: Standard errors in parentheses. The table presents the results from the GMM estimation underthe second specification. That is, we replace the Grosewart measure by dummy variables for the mostimportant committees. The variable Leadership represents a dummy of whether the politician was theSpeaker, the Majority or Minority Leader, or the Majority or Minority Whip. Rep is a dummy variablefor belonging to the Republican Party. The estimates of φDem and φRep are their estimated sets. Theoptimal weighting matrix is used, and standard errors are estimated as discussed in Appendix F. Allother notes follow those in Table 2.


Table C.4. Estimated and High Effort Equilibria

Congress 105 106 107 108 109 110

Effort Level in the Estimated Equilibrium:

SDem 0.922 1.506 1.443 1.403 1.412 1.484

SRep 0.724 0.916 0.808 0.831 0.881 0.988

XDem 4.478 5.680 5.111 5.058 5.206 5.484

XRep 4.231 4.773 4.196 4.235 4.426 4.722

Effort Level in the Higher Equilibrium:

SDem 3.972 2.296 2.058 2.196 2.302 2.285

SRep 2.075 1.200 1.009 1.110 1.217 1.324

XDem 9.816 7.081 6.138 6.389 6.729 6.882

XRep 6.326 5.188 4.460 4.613 4.903 5.226

We numerically assess whether the equilibrium we have estimated is the equilibrium with the highestvalues of social and legislative efforts (SDem, SRep, XDem, XRep). To do so, for each Congress we com-pute two (possibly) distinct solution to the system of equations in Proposition 2.1 under our estimatedparameters. First, we find the solutions under our estimated parameters, with starting values for effortlevels at those estimated in Table 2 (upper panel). Second, we search for a higher effort equilibrium(lower panel). This is done by searching for a solution to equations (2.11) - (2.13), while starting from avector with large values of effort relative to the estimated ones (namely, (SDem, SRep, XDem, XRep)+2).The table shows that there exists an equilibrium with higher levels of both social and legislative effortin all Congresses. These effort levels are higher than the equilibrium ones we estimated and that weobserve in the data. For Congress 107, the pointwise estimate of c has a numerical quirk which gener-ates instability in the algorithms to find zeros. Instead, we use a value within its confidence interval,c+0.007, presented above. This change does not significantly impact estimated equilibrium effort levels.


Table C.5. Effects of Polarization on Equilibrium Effort Levels (and How toOverturn it), Additional Scenarios

Congress 105 106 107 108 109 110

Change in Effort Levels when pRep = 0.3,Relative to the Baseline in Table 5

SDem 0.004 0.251 0.242 0.312 0.232 0.086

SRep -0.003 0.035 0.022 0.043 0.027 0.000

XDem 0.006 0.430 0.393 0.507 0.380 0.143

XRep -0.003 0.048 0.027 0.054 0.036 0.000

Change in Effort Levels when pRep = 0.3,and ADem is 10% higher than estimated

SDem 0.154 0.202 0.130 0.198 0.238 0.221

SRep 0.073 0.124 0.088 0.107 0.130 0.151

XDem 0.214 0.344 0.209 0.318 0.389 0.370

XRep 0.088 0.174 0.111 0.138 0.176 0.219

Change in Effort Levels with Full Polarization,pDem = 1, Relative to the Baseline Estimates

SDem 0.040 0.142 0.095 0.144 0.165 0.169

SRep -0.025 -0.269 -0.295 -0.265 -0.259 -0.227

XDem 0.053 0.241 0.152 0.230 0.268 0.282

XRep -0.028 -0.331 -0.318 -0.298 -0.307 -0.297

We compute the effects of polarization on effort levels in each Congress, as in Table 5. We consider 3different scenarios and compare effort levels to those in the estimated equilibrium in Table C.4. In thefirst, we increase the partisan bias for only Republicans to 0.3 in all Congresses in the second panel. Inthe second, we show the equilibrium effort levels when partisan bias for Republicans is 0.3, but whenthe types of the Democrats (ADem =

∑i∈Dem αi) increases by 10% relative to estimated levels. In

the third, we show a case in which there is full polarization. We can see that Democrats may stillincrease effort levels with polarization, and will do so especially with higher types. Therefore, increasedpolarization means there is increased interaction within the Democratic party and with higher types,leading to higher spillovers that can compensate the lack of interactions with the additional politiciansin the opposition.


Appendix D. Parametric Identification, with set identification of (φ1, φ2)

Recall from equation (2.16) that αi = ez′iβP (i) , and that we observe proxies for (s∗i , x

∗i ).

We also have that mij = (1−p1)(1−p2)(1−p1)A1S1+(1−p2)A2S2

if i, j are from opposing parties and mij =p(i)

AP (i)SP (i)+ (1−p(i))2

(1−p1)A1S1+(1−p2)A2S2if they are from the same. Let us denote mij = m12, if i, j

belong to different parties, mij = m11 if they both belong to party 1, and mij = m22 if they

both belong to party 2.

We now proceed with identification of the parameters of the model. Applying (2.14) and

(2.16) in (2.10), for an arbitrary politician i from party P (i) we obtain:

sieεi = ez

′iβP (i)SP (i), and(D.1)

log(si) = log(SP (i)) + z′iβP (i) − εi.(D.2)

Since Eεi = Eziεi = 0, we now have elementary moment conditions (like an OLS) to esti-

mate α.64 The moment conditions (D.1) for each party identify the respective party specific

parameters. To see this more clearly, one can use the moment equations just described to

get:

E(log(si)− log(SP (i))− z′iβP (i)

)= 0(D.3)

Ezi(log(si)− log(SP (i))− z′iβP (i)

)= 0.(D.4)

As long as E[1, zi][1, zi]′ is invertible, then βP (i) and log(SP (i)) are identified. βP (i)

is identified off the different zi for members of the same party. SP (i) is identified from the

average proxy for effort within a party (the constant in the regression within a party).65

Similarly for xi:

xievi = ez

′iβP (i)XP (i)

log(xi) = log(XP (i)) + z′iβP (i) − vi,(D.5)

which can be written as another moment condition in terms of the i.i.d. mean zero random

variable vi. XP (i) can be similarly identified for each party.

Since we know βP (i), SP (i), XP (i) for both parties, c is identified from equation (2.13),

where it is the only unknown. However, p1, p2 cannot be uniquely identified from the system

above. It is clear that p1, p2, φ1, φ2 show up only in the same 2 equations: (2.11) and (2.12).

64Identification with nonparametric α is proved in a following Appendix.65Implicitly, we require that zi does not include the constant, as that cannot be separately identified fromlog(SP (i)) without further assumptions. The average (log) socializing is party-specific.


To identify φ1, φ2, we pursue a set identification approach. To do so, let us first notice that

equations (2.11) - (2.12) can be rewritten as66:

S1 = φ1X1 (B1S1X1m11 +B2S2X2m12)(D.6)

S2 = φ2X2 (B2S2X2m22 +B1S1X1m12) .(D.7)

To proceed with set identification of (φ1, φ2), we calculate all triples m11,m12,m22 that

are consistent with some p1, p2 ∈ [0, 1] × [0, 1]. Hence, any pair (φ1, φ2) that satisfies the

above equations for some (p1, p2) ∈ [0, 1]× [0, 1] is identified.67

A few comments about mij(s∗) are in order. Even if mij(s

∗) was recovered uniquely, that

would not guarantee a unique identification of p1, p2. There might be multiple p1, p2 that can

yield the same meeting probabilities (usually a continuum of them, governed by a hyperbolic

function). Second, mij(s∗) is not a parameter of interest for us, since it is a normalization

on the gij function that governs the probability of linking.

Having identified all other parameters of the model, we can now prove the identification

of the parameters γP (i), ζP (i). To do so, we use the data on bill passage.

D.0.1. Identifying the components of φi: γP (i) and ζP (i). We recall that φi =ζP (i)γP (i)

m, where

γ was the scale parameter of the distribution of εi (the shock on bill approval, such that the

same politician could have different bills passing or not), m was the institutional threshold

for approval of a bill (i.e. the minimum amount of support needed for approval), and ζ was

the return to the politician from the voters of having the bill approved. We further assumed

that m = 1.

If we want to identify the components of γP (i) and ζP (i), we can use the probability of bill

approval equation (2.3):

P (yi = 1) =γP (i)

m

∑j 6=i

gij(s∗)x∗ix

∗j .

This, in turn, can be rewritten using the First Order Condition for si (equation (A.2)) as:

66This follows from an alternative rewriting of the first order conditions presented in the model:

SP (i)

XP (i)= φP (i)

∑j 6=i

α2jSP (j)XP (j)mij(s

∗)

= φP (i)

∑j 6=i,j∈P (i)


∗) +∑j /∈P (i)


∗)

= φP (i)

SP (i)XP (i)mij(s∗)Ii,j∈P (i)

∑j 6=i,j∈P (i)

α2j + SP (−i)XP (−i)m12(s∗)

∑j /∈P (i)

α2j

.

67In Appendix H, we also provide a solution to point identify p1, p2, but that relies on additional momentconditions, coming from the second moments of the error terms of the proxy variables (si, xi). However,such a solution requires additional structure outside of the theoretical model.


P (yi = 1) =γP (i)

m

∑j 6=i

gij(s∗)x∗ix

∗j

=γP (i)

ms∗ix∗i

∑j 6=i

s∗jx∗jmij(s

∗)

=γP (i)

ms∗ix∗i

(s∗iφix∗i

)=

γP (i)

mφis∗2i

=1

ζP (i)

s∗2i .(D.8)

where P (yi = 1) is the probability that a bill from politician i is approved.

Since s∗i = sieεi , we can rewrite (D.8) as:

P (yi = 1) =1

ζP (i)

s2i e

2εi .(D.9)

Taking logs implies that:

logP (yi = 1) = log

(1

ζP (i)

)+ log(s2

i ) + 2εi.(D.10)

Since Eεi = 0, taking expectations on both sides means the only unknown is ζP (i), which

is identified when looking at si and yi within each party.

ζP (i) must also satisfy the following restriction, due to γP (i) being the scale parameter of

the Pareto distributed εi68:

mφis∗2i

≥ γP (i), ∀i

ζP (i) ≥ miniα2iS

2P (i)

Since (φ1, φ2) had been previously (set) identified, and m = 1, then γP (i) = φiζP (i)

is (set)

identified. This completes the identification of the model. Finally, we note the importance

of the measurement errors for identification of this model.

D.1. The need for exponential measurement errors even when α is parametric.

The exponential form for the measurement errors is very useful for two main reasons. First,

we bypass the truncation issue (having to guarantee that si ≥ 0 for any ε). Second, it is

very tractable (as seen in equation (D.1)).

68This, in turn, implies that the support for εi within each party is [γP (i),∞). The restriction holds empiri-cally.


One could think that measurement errors would not have to show up on s∗i and/or x∗i .

However, measurement errors on s∗i , x∗i are needed even with α being parametric. To see

this, consider dividing equation (2.10) by (2.9):

(D.11)s∗ix∗i

=SP (i)

XP (i)

.

The α’s cancel out, and the model is rejected as (D.11) does not hold in the data (without

randomness). Hence, there must be an error term in both s∗i and x∗i .

Appendix E. Rewriting the Model in terms of Moment conditions over i

In this Section, we provide the derivation for transforming the model from the equations

in Proposition 2.1 to the moment equations described in Section 4. We first note that one

can stack equations (D.1) and (D.5) across both parties, to get:

log(si) = log(S1) + (log(S2)− log(S1))Ii∈P2 + z′iβ1 + z′iIi∈P2(β2 − β1)− εi,log(xi) = log(X1) + (log(X2)− log(X1))Ii∈P2 + z′iβ1 + z′iIi∈P2(β2 − β1)− vi,

where Ii∈P2 is an indicator of whether i is in party 2. We have simply introduced the

dummy variable for party 2 to stack up the equations. Just like in Ordinary Least Squares,

the parameter in front of the indicator recovers the difference of that variable across parties.

Note that the above equations can be rewritten as:

log(si) = ziβs − εi,(E.1)

log(xi) = ziβx − vi,(E.2)

where zi = [1, Ii∈P2 , z′i, z′iIi∈P2 ], βs = [log(S1), log(S2)−log(S1), β1, β2−β1], βx = [log(X1), log(X2)−

log(X1), β1, β2 − β1].

Recall that Eεi = Eziεi = 0, which is now rewritten together as Eziεi = 0. Similarly,

Evi = Ezivi = 0 is rewritten as Ezivi = 0.

We are now ready to rewrite the model in terms of moment conditions. Equations (E.1)

and (E.2) are rewritten in terms of moments as:

Ezi(log(si)− z′iβs) = 0,(E.3)

Ezi(log(xi)− z′iβx) = 0.(E.4)

Each of the above equations is k by 1, where k is the dimensionality of zi. Note that the

constant is included in zi.

Focusing on (2.13), for all i in Party 1, one way we can rewrite it is:


0 = c− 1

X1

− S21

X21

= c− 1

X1

− s2i

x2i

= c− 1

X1

− s2i e

2(εi−vi)

x2i

.

Simplifying and taking logs:

log(c− 1

X1

) = 2log

(eεi−vi

sixi

)= 2(log(si)− log(xi) + εi − vi).

where we used (2.9) and (2.10) and rewrote them with our proxies (si, xi).69

Since an analogous version holds for Party 2, one can rewrite the equation above across

parties as:

(E.5) E(

2(log(si)− log(xi))− log(c− 1

X1

)Ii∈P1 − log

(c− 1

X2

)Ii∈P2

)= 0,

where we have taken expectations on both sides over εi, vi, across all agents. This can be

rewritten as equation (4.3) by replacing Ii∈P1 = 1− Ii∈P2 . Finally, to rewrite equation (4.4),

we simply take expectations over both sides of equation (D.10).

Appendix F. Details on Estimation

We now provide further details on how the estimation procedure was implemented, in-

cluding the starting values for the numerical solution to the GMM optimizer and numerical

details on the computation of standard errors.

F.1. OLS and plug-in Approach as Starting Values for Optimization. For the start-

ing values for GMM optimization, we use a simple closed form estimate for the parameters of

interest. This new estimator is the OLS estimator for a subset of the parameters, and a plug-

in of those for the remaining parameters. Such an estimator is consistent, but inefficient.

Yet, it is a good starting value for the optimization procedure.

The inefficiency comes from an OLS approach to equations (4.8) and (4.9) neglecting

that βDem, βRep are the same across both equations. To note that we can find this OLS

estimator, (4.8) and (4.9) are the moment conditions associated with an OLS problem. The

OLS estimator is then given by:

βs = (Z ′Z)−1Z ′log(s),(F.1)

βx = (Z ′Z)−1Z ′log(x),(F.2)

69While we can replace X1 separately to obtain analogous equations, they will be linearly dependent, as weare already using (2.9) and (2.10) in the estimation and the previous equation has been introduced.


where Z is an N by k matrix of z′i, log(s) and log(x) are the vectors of log(si) and log(xi)

respectively.

By our definition of βs, βx, such an estimator is consistent for log(S1), log(X1), log(S2), log(X2)

(from the first two parameters in each of βs, βx). Similarly, we recover consistent estimates

for β1, β2 in each equation. One replaces them into either equation (2.13) to recover c (that

was not present in any of the other equations).

F.2. Computation of the Optimal Weighting Matrix and of Standard Errors.

F.2.1. The Optimal Weighting Matrix and Computation of Estimates for the Standard Er-

rors. To compute the standard errors for our GMM estimates, we use a two step procedure

that is common in the literature. First, we estimate the model using an inefficient weighting

matrix W = I, with I the identity matrix, with starting values from the OLS and plug-in

approach (described above). Given these inefficient estimates, we then compute the optimal

weighting matrix for GMM, based on its asymptotic formula.

The optimal weighting matrix for GMM is defined as W = Ω−1, where

Ω = E(g(si, xi, yi, zi, θ)g(si, xi, yi, zi, θ)′). As it is well known, the asymptotic variance matrix

(of√n times) our parameters of interest is then given by (Γ′Ω−1Γ)−1, where Γ = E∂g(si,xi,θ)

∂θ′.

We compute Γ analytically, by taking derivatives of each moment equation in relation to

each parameter. We then replace the expectation by its empirical counterpart (the mean

across all politicians).

F.2.2. Finite Sample Corrections for the Standard Errors. In finite samples, Ω can be close

to singular. This appears to be the case in some of the specifications based on the set-up in

Appendix H, using second order moment restrictions. We provide corrections that improve

the finite sample performance, used only for those specifications in Appendix.

We implement the correction used in Cameron et al. (2011). This involves increasing the

standard errors in Ω by adding a small perturbation to its eigenvalues. This perturbation is

sufficient to remove singularity.

Such a procedure uses the spectral decomposition of Ω = DΛD′, where Λ is a diagonal

matrix of eigenvalues. We then add a small δΩ > 0 to the diagonal of Λ, therefore increasing

the eigenvalues of Ω. Since this procedure increases standard errors, the new standard errors

are still valid for our parameters.

In practice, we pick δΩ = 0.0001, and use it on the eigenvalues that are smaller than 10−7.

This is typically 1 or 2 of the eigenvalues of our estimated Ω. This correction is used for

both the calculation of the optimal weighting matrix, as well as for the standard errors of

the parameters.

Appendix G. Computation of Comparative Statics

Our estimates of SDem, SRep, XDem, XRep, are useful for our comparative statics and fitting

our model. However, those estimates are not necessarily the values that solve the equilibrium

system in Proposition 2.1. They are estimates that solve the system in expected value (the

moment conditions), as we only observe proxies for social and legislative effort.


To calculate the values that are consistent with our model, we solve the system in Proposi-

tion 2.1 using those values as starting points, under the estimated values of (c, φ1, φ2, βDem, βRep).

The solution are values S∗Dem, S∗Rep, X

∗Dem, X

∗Rep that, under the estimated parameters, solve

the original game.

For example, when we are interested in the changes in the probability of bill approval, we

use S∗P (i), αi and equation (4.4), i.e. P (yi = 1) = 1ζP (i)

(αiS∗P (i))

2.

Appendix H. Identification and Estimation using second moments of the

proxies of (s∗i , x∗i )

In this section, we impose additional structure on the measurement errors of the proxies,

denoted εi, vi, such that we can recover point identification of p1, p2, φ1, φ2. In the following

subsection, we then estimate this version of the model.

As before, let us introduce the measurement errors εi, vi, such that we observe proxies of

(s∗i , x∗i ) of the following form:

si = s∗i e−εi(H.1)

xi = x∗i e−vi .(H.2)

However, we now assume that:

εi = ωip(i) + ui(H.3)

vi = ωip(i) + ηi,(H.4)

with ui, ωi, ηi being i.i.d. across i and we have the (standard) assumptions, for all i: E(ui |zi, ωi, p(i)) = 0,Eu2

i = σ2u,E(ηi | zi, ωi, p(i)) = 0,Eη2

i = σ2η,E(ωi | zi, p(i)) = 0,Eω2

i = σ2ω.

These conditions imply the restrictions E(εi | zi) = E(vi | zi) = 0 used in the main section

of the paper.

The structure allows the measurement errors to vary by individual and by party. An

intuition for this is that, when partisanship increases, it is harder to observe the true so-

cial/legislative effort of a politician. This implies we get noisier proxies for s∗i , x∗i with more

partisanship. When pi = 0, we only have classical measurement error in the proxies. With

no partisanship, there are no differences in how well we observe socializing across politicians

and parties.

The identification arguments presented in the previous approach still hold, up to the

identification of (φ1, φ2). Let us continue from there.

Under the assumptions stated above, note that:

Eε2i = E(ωip(i) + ui)2

= E(ω2i p(i)

2 + u2i + 2uiωip(i))

= p(i)2σ2ω + σ2

u.(H.5)


Similarly,

Ev2i = p(i)2σ2

ω + σ2η(H.6)

E(εivi) = p(i)2σ2ω.(H.7)

We make the following normalization, such that all variances are now written in terms of

the variance of ω: σω = 1.70 In that case, from (H.7) we get that:

(H.8) p(i)2 = E(εivi).

Since we can take expectations over i ∈ P1 or i ∈ P2, we get identification of both p1 and

p2. Plugging in p(i) into (H.5) identifies σ2u. Plugging p(i) into (H.6) identifies σ2

η.

Note that the moment conditions given by equations (H.5), (H.6), (H.7) (for each party)

are of very simple form to use. Given (p1, p2), we can identify φ1 immediately from equation

(4.6). Analogously, φ2 follows from plugging in the previously identiied parameters in (4.7).

H.1. Estimation with Second Moment Conditions on the Proxies of s∗i , x∗i . To

estimate this version of the model, we can retain all moment conditions presented in the

main part of the text, while adding the moment equations (H.5), (H.6), (H.7), together with

a moment equation derived for each of φ1, φ2. To derive the latter, equations (2.11) - (2.12)

are rewritten as:

E (log(si)− log(xi)− log(φ1)−Ψ0) Ii∈P1 = 0.(H.9)

and

E (log(si)− log(xi)− log(φ2)−Ψ1) Ii∈P2 = 0.(H.10)

where

Ψ0 = log

(p1B1X1

A1

+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2S2X2

(1− p1)A1S1 + (1− p2)A2S2

), and

Ψ1 = log

(p2B2X2

A2

+(1− p2)2B2S2X2 + (1− p1)(1− p2)B1S1X1

(1− p1)A1S1 + (1− p2)A2S2

).

Estimation then follows the routines described in the main part of the text, except we now

estimate φ1, φ2, p1, p2, σ2u, σ

2η.

H.2. Results under Restrictions on the Second Moments of the Proxies for (s∗i , x∗i ).

These are presented in Tables H.1 - H.2 below. We can also show that our results for the

counterfactuals by changing c and by changing α remain quantitatively and qualitatively

similar to those in Tables 6 and C.1. This shows the robustness of our results to different

parametrizations. However, due to space considerations, we do not present them here.

70p1, p2 are always multiplied by σω.


Table H.1. Results, Specification 1, second moments of the (s∗i , x∗i ) proxy

Congress105 106 107 108 109 110

c 0.249 0.224 0.236 0.275 0.266 0.243(0.006) (0.006) (0.008) (0.007) (0.006) (0.006)

φDem 0.038 0.042 0.048 0.046 0.045 0.046(0.001) (0.002) (0.002) (0.003) (0.002) (0.002)

φRep 0.031 0.032 0.034 0.033 0.034 0.036(0.001) (0.001) (0.001) (0.002) (0.002) (0.002)

ζDem 29.143 59.436 44.775 34.08 43.058 17.419(1.924) (6.06) (3.348) (2.434) (3.356) (1.228)

ζRep 9.134 9.414 9.436 8.043 10.448 21.379(0.692) (0.853) (0.812) (0.65) (0.818) (1.866)

pDem 0.066 0.091 0.043 0.097 0.051 0.045(0.02) (0.013) (0.03) (0.021) (0.026) (0.03)

pRep 0.076 0.13 0.126 0.093 0.103 0.145(0.017) (0.01) (0.014) (0.018) (0.018) (0.011)

σ2u 0.232 0.23 0.214 0.211 0.221 0.238

(0.005) (0.007) (0.007) (0.006) (0.007) (0.014)

σ2η 0.143 0.144 0.18 0.192 0.16 0.19

(0.003) (0.004) (0.003) (0.024) (0.01) (0.005)

SDem 0.902 1.146 1.162 0.999 1.042 1.266(0.03) (0.037) (0.042) (0.033) (0.029) (0.037)

SRep 0.854 1.389 1.41 0.847 0.884 1.037(0.034) (0.072) (0.096) (0.038) (0.041) (0.057)

XDem 4.41 4.855 4.585 4.059 4.258 5.076(0.144) (0.151) (0.172) (0.148) (0.125) (0.139)

XRep 5.077 7.791 7.931 4.797 4.886 5.304(0.194) (0.41) (0.543) (0.21) (0.225) (0.312)

Ideology -0.498 -0.045 -0.039 -0.697 -0.603 0.033(0.077) (0.075) (0.095) (0.087) (0.074) (0.08)

Tenure 0.003 0.017 0.017 0.004 0.002 0.013(0.002) (0.003) (0.003) (0.003) (0.002) (0.002)

Grosewart -0.013 -0.02 -0.027 -0.013 -0.01 -0.035(0.008) (0.009) (0.01) (0.011) (0.009) (0.01)

Ideology ×Rep 0.072 -0.506 -0.657 0.098 0.122 0.039(0.067) (0.084) (0.111) (0.066) (0.068) (0.075)

Tenure×Rep 0.01 -0.009 -0.009 0.007 0.002 0.005(0.003) (0.003) (0.004) (0.003) (0.003) (0.004)

Grosewart×Rep -0.016 -0.032 -0.029 -0.021 -0.029 -0.027(0.009) (0.01) (0.012) (0.01) (0.009) (0.011)

N 437 434 436 438 434 437Notes: Standard Errors in parentheses. The table presents the results from the GMM estimationunder second moment conditions on the proxies of (s∗i , x

∗i ). The optimal weighting matrix is used, and

standard errors are estimated as discussed in Appendix. All other notes follow those in Table 2.


Table H.2. Results, Specification 2, second moments of the (s∗i , x∗i ) proxy

Congress105 106 107 108 109 110

c 0.213 0.22 0.263 0.272 0.262 0.253(0.007) (0.006) (0.007) (0.007) (0.006) (0.005)

φDem 0.037 0.043 0.049 0.046 0.046 0.045

(0.001) (0.002) (0.002) (0.002) (0.002) (0.002)φRep 0.03 0.032 0.034 0.033 0.034 0.036

(0.001) (0.001) (0.001) (0.001) (0.001) (0.001)ζDem 29.518 61.179 44.847 34.1 42.763 17.513

(1.901) (6.074) (3.324) (2.427) (3.241) (1.179)

ζRep 9.078 9.158 9.454 8.101 10.505 21.623(0.675) (0.776) (0.795) (0.638) (0.811) (1.817)

p1 0.081 0.088 0.074 0.105 0.049 0.077

(0.02) (0.012) (0.015) (0.014) (0.026) (0.02)p2 0.158 0.125 0.101 0.092 0.095 0.074

(0.01) (0.009) (0.014) (0.016) (0.017) (0.026)σ2u 0.218 0.196 0.195 0.198 0.218 0.208

(0.005) (0.005) (0.005) (0.004) (0.005) (0.005)

σ2η 0.188 0.134 0.141 0.161 0.17 0.146

(0.003) (0.003) (0.003) (0.011) (0.009) (0.006)SDem 1.041 1.203 1.068 1.004 1.052 1.171

(0.045) (0.044) (0.034) (0.034) (0.029) (0.029)

SRep 1.181 1.27 1 0.876 0.921 1.07(0.061) (0.059) (0.05) (0.045) (0.047) (0.063)

XDem 5.07 5.051 4.185 4.077 4.293 4.689

(0.218) (0.181) (0.135) (0.142) (0.124) (0.105)XRep 7.044 7.137 5.606 4.952 5.074 5.428

(0.369) (0.345) (0.276) (0.251) (0.258) (0.319)Ideology 0.049 0.014 -0.592 -0.709 -0.558 -0.464

(0.093) (0.085) (0.073) (0.083) (0.075) (0.071)

Tenure 0.021 0.015 0.001 0.003 0.003 0(0.004) (0.003) (0.002) (0.002) (0.002) (0.002)

Appropriations -0.018 0.03 -0.071 -0.085 -0.08 -0.044

(0.043) (0.032) (0.031) (0.036) (0.035) (0.025)Energy and Commerce -0.16 -0.044 -0.02 0.024 0.033 0.027

(0.047) (0.033) (0.032) (0.035) (0.029) (0.02)

Oversight -0.008 0.065 0.057 0.032 0.046 0.001(0.039) (0.038) (0.032) (0.038) (0.039) (0.033)

Rules 0.035 0.074 0.125 0.14 0.183 0.106

(0.039) (0.025) (0.029) (0.033) (0.031) (0.032)Leadership 0.168 0.166 0.082 -0.113 -0.299 -1.102

(0.04) (0.054) (0.081) (0.071) (0.041) (0.336)Transportation -0.162 -0.032 0.018 0.006 -0.001 0.007

(0.038) (0.034) (0.023) (0.029) (0.025) (0.023)WaysAndMeans -0.131 -0.028 -0.023 -0.033 -0.024 -0.015

(0.039) (0.027) (0.032) (0.037) (0.038) (0.03)

Ideology ×Rep -0.353 -0.358 -0.114 0.079 0.082 0.063

(0.078) (0.073) (0.073) (0.067) (0.068) (0.077)Tenure×Rep -0.003 -0.006 0.002 0.006 0.002 0.001

(0.003) (0.003) (0.003) (0.003) (0.003) (0.003)

Appropriations×Rep -0.141 -0.071 -0.139 -0.1 -0.108 -0.133(0.029) (0.025) (0.033) (0.034) (0.035) (0.04)

EnergyandCommerce×Rep -0.134 -0.045 -0.039 -0.047 -0.03 -0.053

(0.03)2 (0.028) (0.027) (0.028) (0.03) (0.037)Oversight×Rep 0.077 0.065 0.081 0.055 0.068 0.031

(0.033) (0.022) (0.027) (0.027) (0.031) (0.036)Rules×Rep 0.092 0.036 0.055 0.037 0.118 0.104

(0.045) (0.033) (0.025) (0.031) (0.036) (0.052)

Leadership×Rep 0.179 0.048 -0.182 -0.213 -0.187 0.226(0.061) (0.069) (0.15) (0.126) (0.133) (0.119)

Transportation×Rep -0.101 -0.051 -0.051 0 -0.019 -0.04

(0.031) (0.028) (0.027) (0.028) (0.027) (0.031)WaysAndMeans×Rep -0.102 -0.029 -0.01 -0.019 -0.07 -0.139

(0.033) (0.035) (0.036) (0.032) (0.04) (0.044)

N 437 434 436 438 434 437

Notes: Standard Errors in parentheses. The table presents the results from the GMM estimationunder second moment conditions on the proxies of (s∗i , x

∗i ). The optimal weighting matrix is used, and

standard errors are estimated as discussed in Appendix. All other notes follow those in Table C.3.


Appendix I. Identification with Nonparametric α

In this section, we prove that our model is identified even with a nonparametric α, sug-

gesting that the parametrization of α is useful for data purposes, but not strictly needed.71

Let us maintain the same error structure as in the parametric approach in the main text,

given in equation (2.14). Let i1 be a normalizer (αi1 = 1). Without loss, let him/her

be from party 1. Then, we can rewrite (2.10) for i1 as: s∗i1 = αi1S1, which implies that:

log(si1) + εi1 = log(S1).

Taking expectations on both sides yields implies that: Elog(si1) = log(S1), which is now

identified. Note that Esi1 is known from the data, as it is a moment of the observed proxy.

Now using (2.10) for any i in party 1, leads to:

(I.1) Elog(si) = log(αi) + Elog(si1),

and αi is identified for any politician in party 1. Intuitively, once we normalize someone,

we know everyone else’s scale, since in equilibrium all politicians in party 1 choose social

efforts that are proportional to S1. It follows that A1 =∑

j∈P1αj, B1 =

∑j∈P1

α2j are

identified given knowledge of party affiliations.

For any politician in party 1, we must also have from (2.9) that:

log(x∗i ) = log(xi) + vi = log(αi) + log(X1)

which implies that Elog(xi) = log(αi) + log(X1) since vi is mean zero. It follows that X1

is identified. S1/X1 is then identified. Then, equation (2.13) for Party 1 gives us:

(I.2) c =1

X1

+

(S1

X1

)2

,

so that c is identified (since S1, X1 have been identified). Let us now use equations (2.9)

and (2.10) for a party member k from party 2. (2.10) divided by (2.9) yields:

s∗k = x∗kS2

X2

⇒ log(sk) + εk = (log(xk) + vk)log

(S2

X2

).

In turn, this leads to Elog(sk) = Elog(xk)S2

X2and, hence, S2/X2 is now identified.

Since c and S2/X2 are already known, we have that X2 is identified by equation (2.13) for

Party 2, i.e. c − S2

X2= 1

X2. It follows that S2 is identified. Using this with equation (2.10)

implies that αj is identified for all politicians in party 2. It follows that A2, B2 are known.

(Set) identification of (φ1, φ2) follows the parametric approach.

71A nonparametric α requires us to estimate αi for each politician in each Congress, but we have only oneset of observations per politician per term.

ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY

Documents

Transcript of ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY