Discussion Papers In Economics
And BusinessInformatization, Voter Turnout and Income Inequality
Ryo Arawatari
Discussion Paper 07-28

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInformatization, Voter Turnout and Income Inequality
Ryo Arawatari
Discussion Paper 07-28

July 2007

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInformatization, Voter Turnout and Income Inequality∗
Ryo Arawatari†
July 2007
Abstract
In recent years, voter turnout has been decreasing in most industrial countries, and about
40% of all electors abstain from voting. This may affect income inequality and the GDP growthrate through a redistribution policy determined by majority voting. In this paper, we explore
the reasons for this continuing decrease in voter turnout and assess its social costs. We conclude
that informatization lowers voter turnout by generating an information overload, and that adecrease in voter turnout lowers GDP growth by limiting income redistribution.

JEL classification: D31; O15; P16
Keywords: Income inequality; Information; Informatization; Voter turnout; Voting∗

I am very grateful to Kazuo Mino for his invaluable help and encouragement. I acknowledge the technical support by
Takeo Hori. I also thank anonymous referee and editors, Jacques Silber, Francisco H. G. Ferreira and Branko Milanovic,
for very constructive comments and suggestions. I also acknowledge the helpful comments by Francisco Ferreira, Koichi
Futagami, Ryo Horii, Nobuhiro Hosoe, Masako Ikefuji, Yutaka Imai, Kazuaki Kagami, Junji Kageyama, Phil Keefer,
Branko Milanovic, Akira Momota, Yasutomo Murasawa, Ryoji Ohdoi, Hirotaka Ohmura, Tetsuo Ono, Makoto Saito,
Kengo Soga, Hanako Takashima, Ryuichi Tanaka, Takuma Wakayama, Hideo Yunoue, and seminar participants at
Osaka University, Osaka Prefecture University, Meikai University, III World Bank Conference on Inequality and PET
06 Hanoi. Any remaining errors are mine. Financial aid from JSPS Research Fellowships for Young Scientists is
gratefully acknowledged.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan; E-mail:ege001ar@mail2.econ.osaka-u.ac.jp; Tel: +81-6-6835-5710.

1

Introduction
In recent years, voter turnout has been decreasing in most industrial countries, and about

40% of all electors abstain from voting. Moreover, a common feature of industrial countries isthat agents with low income, low human capital or limited information tend to abstain fromvoting. Figure 1 shows the movement of voter turnout in the US and UK from 1918 to 2005. It
shows that the evolution of voter turnout is described by an inverse U-shaped curve, and that
voter turnout has been decreasing in recent years. Nardulli et al. (1996) also point out that
voter turnout in the US is described by an inverse U-shaped curve; voter turnout in the US
increased from 1920 to 1960 and decreased from 1960 on.
[Figure 1 around here.]
The continuing decrease in voter turnout may affect income inequality and the GDP growthrate through a redistribution policy determined by majority voting. Although there have been
many studies of the relationship between the endogenous determination of the tax rate andincome inequality (Persson and Tabellini, 1994, Alesina and Rodrik, 1994, Li and Zou, 1998,
etc.), to our knowledge, there has been no study of the reasons for the continuing decrease invoter turnout and the social costs of this decline.
Many studies of voting behavior have their origins in Downs (1957). He develops the wellknown “Paradox of Voting”, which explains why rational voters invest their personal time andenergy in voting despite the minimal probability of their vote being a decisive in majority voting.
Many political scientists and economists have analyzed and presented solutions to the “Paradox
of Voting”.
Downs (1957) presents the simplest solution to this paradox. He introduces a term, D, torepresent the value of democracy continuing. He argue that, if everyone abstains from voting,democratic institutions would cease to exist. Therefore, some people may vote to perpetuate
democratic institutions. This idea is expanded by Riker and Ordeshook (1968), who develop thenotion of “citizen duty”. They argue that “citizen duty” represents the value of doing one’s duty
as a citizen, and voters obtain a value, D, from voting regardless of the outcome. Feddersen
and Pesendorfer (1996, 1999) proposes a game-theoretic voting model. They investigate how
asymmetries in information across voters affect voting behavior and the election result. In their1

model, voting is costless for all voters and, thus, abstention cannot be explained by differencesin the cost of voting. By contrast, Ghirardato and Katz (2002) focus not on the quantity, buton the quality of information. They explain how the quality of the information available tovoters affects their voting behavior. Their key result is that the voter who is averse to ambiguity
considers abstention strictly optimal when the candidates’ policy positions are both ambiguous
and they are “ambiguity complements”. Lohmann (1993, 1994) develops a signaling model ofmass political action. She shows that some self-interested voters have incentives to undertake
costly political action to provide private information to the political leader, whereas others
abstain from voting, hoping to benefit if the leader makes an uninformed decision.
These approaches all predict that agents with low income, low human capital or limited
information tend to abstain from voting (cross-sectional differences in voting behavior). Thisprediction is supported by many experimental studies. For example, using cross-section data overa 32-year period on 3100 US counties, Filer et al. (1993) find that better educated people havea greater tendency to vote. Using data on counties where many African Americans live, Smith(1984) shows that voter turnout is positively correlated with the average academic level. Usinga unique data set based on telephone interviews with Copenhagen voters, carried out in 2000,
Lassen (2005) find that being informed has a statistically significant effect on the propensity tovote.
Thus, the “Paradox of Voting” seems to have been resolved by these theoretical and experimental studies. However, the problem with these studies is that, because they analyze astatic model, they cannot explain the observed decrease in voter turnout (over-time differences
in voting behavior). Since average incomes, human capital and the degree of informatization
have increased in most developed countries, in a dynamic context, the above models suggestthat voter turnout increases with economic growth. However, this result is not consistent withthe empirical evidence that voter turnout has been decreasing in most developed countries. Thissuggests another paradox. In a static context, voters with low incomes, low human capital or
limited information tend to abstain from voting; this is supported by existing theoretical andempirical studies. However, from a dynamic context, a decreasing voter turnout has been accompanied by increases average incomes, human capital and the degree of informatization. We
refer to this as the ‘New Paradox of Voting’. Hence, to analyze the dynamic decline in voter

2

turnout, we need another model of the choice between voting and abstaining.
In this paper, we present a new dynamic model that provides a solution to the ‘New Paradox
of Voting’. First, we introduce the concepts of ‘informatization’ and the ‘cost of collecting information’ into a standard static voting model and show that our model is consistent with earlier
studies cited above. Second, we apply the static model to a discrete-time nonoverlapping generations model and analyze the dynamic evolution of voter turnout. In this way, we demonstratethat our model can explain the cross-sectional differences and the over-time differences in voting
behavior simultaneously; it can offer a solution to the ‘New Paradox of Voting’. Our model
also allows us to analyze: (1) the relationship between voter turnout and informatization; (2)the social costs of a decrease in voter turnout; and (3) the effect of informatization on income
inequality. We conclude that informatization generates an inverse U-shaped pattern in voter
turnout, and that a decrease in voter turnout reduces the GDP growth rate by limiting income
redistribution.1
This paper is organized as follows. The model in this paper consists of two parts: the voting
model and the economic model. In Section 2, we discuss the voting model. We explain therelationship between informatization and the cost of collecting information and show that votersand abstainers may coexist. In Section 3, we develop the economic model. We demonstratethat an increase in abstention lowers the GDP growth rate. In Section 4, we combine the voting
model and the economic model and derive a politico-economic equilibrium. In Section 5, we
present numerical examples and their implications. Section 6 concludes the paper.

2

The Voting ModelIn this section, we analyze a simple but novel static voting model. We introduce informati-

zation and the cost of collecting information into the standard spatial voting model. We argue
that our model is consistent with existing theoretical and experimental studies and that it offersa solution to the ‘New Paradox of Voting’.

1

We do not claim that ours is the only possible explanation of the observed facts. We aim to describe the significant
effects of informatization on voter turnout and present our findings as a possible explanation of the observed facts.

3

2.1

Multidimensional Policy and Voters’ Preferences

We consider the following spatial voting model. We denote a multidimensional policy by Pt =(τt , Zt ) ∈ [0, 1] × Rl+ (l ≥ 2), where t is the time index, τt is the tax rate and Zt is a vector ofpolicies except for taxation. For example, Zt includes foreign policy, environmental policy anddefense policy. We assume that τt and Zt are defined over the Euclidean space.
There is a continuum of voters i ∈ [0, 1] who decide between voting under the pure majority
rule2 and abstaining from voting. Let us assume that the utility function of a voter i is givenby:
Uti = u(cit , hit+1 |τt ) + V (It , hit ) + W (Zt ),(1)

where cit is consumption, hit and hit+1 are human capital at period t and t + 1, respectively, τt
and Zt are implemented policies at period t.3 It is the degree of informatization. This variable
represents the number of sources of information, including the circulation of newspapers, books
and magazines, the number of television channels and the number of websites. We assume that
It is defined over the Euclidean space.
Each voter’s utility comprises three components. First, u(cit , hit+1 |τt ) represents utility fromthe individual’s own consumption and the human capital of that person’s offspring (altruism).4
Second, V (It , hit ) represents utility from voting. This is important in explaining the long-run
decline in voter turnout. Third, W (Zt ) represents the utility obtained from policies other thantaxation. This is not the main focus of the analysis, but it is included to illustrate the importanceof informatization in the remainder of this section.

2

See Persson and Tabellini (2000) for details of the pure majority rule. Persson and Tabellini (1994), Alesina andRodrik (1994), and Li and Zou (1998) also assume this rule.
3
Implemented policies are determined by majority voting.
4
This is explained in detail in Section 3.

4

2.2

Citizen Duty and Information Cost

We specify V (It , hit ) in (1) as:

V (It , hit ) = D − IC(It , hi ) if vote,
t



0

(2)

if abstain,

where each variable is defined in what follows.
The first term, D, is the utility derived from voting per se. The idea that utility is obtained
from voting was initially developed by Downs (1957) and expanded by Riker and Ordeshook(1968). Using the term “citizen-duty”, Riker and Ordeshook (1968) argue that D represents
the value of doing one’s duty as a citizen, by, for example, expressing support for one’s country
and democratic institutions. Voters consider that voting is a citizen’s right and duty, and thus
they value voting per se whatever the outcome.5 We assume that the utility represented by D
is common to all voters and is time invariant.
We also assume that, when voters vote, they must have information about the policies, Pt =(τt , Zt ), and incur a cost, IC(It , hit ), of collecting information. We can interpret IC(It , hit ) as theopportunity cost of the leisure time used to collect information about policies, Pt = (τt , Zt ). Note
that, if there is only taxation policy, voters can easily evaluate the policy because they know the
optimal tax rate. However, to evaluate non taxation policies, Zt , voters need information about
policy content and objectives. For example, a voter considering what is the best foreign policy
must collect information about the current diplomatic situation and the expected outcome of
implementing a particular foreign policy. Then, the voter must forgo some leisure time to collectinformation. This opportunity cost is IC(It , hit ), which is henceforth termed ‘information cost’.
Note that we do not claim that information requirements for informed voting are high. As Lupia(1992), Lupia and McCubbins (1998) and Popkin (1994) argue, voters use little information
about political issues very effectively, therefore information requirements for informed voting
are actually low. However, this fact do not means that we can ignore the information cost. As
Aldrich (1993) points out, turnout is a low-cost, low-benefit decision-making problem. Smallchanges in costs and benefits alter the voting-abstention choice for the electorate. Therefore,
5

The term D is also used in Fiorina (1976), Crain and Deaton (1977), Hinich (1981), Aldrich (1993), Kanazawa(1998), among others.

5when we consider the voting-abstention choice of the electorate, we cannot ignore the information
cost. As already mentioned, It > 0 represents the degree of informatization: the larger is It ,the greater is informatization. Equation (2) shows that the degree of informatization affects theinformation cost. Note that It is an endogenous variable; we discuss It in detail in Subsection
2.5.
We specify the information cost as follows:

IC(It , hit ) =






χ
·
AIt−ψ +
| {z }(hit )ζ access cost






BItφ
,

|{z}


filtering cost(3)

where χ > 0, ζ > 0, φ > 1 and ψ > 0 are parameters. The information cost, IC(It , hit ), consistsof two components: the ‘access cost’ and the ‘filtering cost’. First, AIt−ψ is the cost of accessing
information, and is decreasing in the degree of informatization. This cost can be interpreted
as a measure of the time required to access information sources. An increase in the number of
information sources reduces the time taken to access information sources and, hence, lowers thecost of collecting information. We term this cost the access cost. Second, BItφ refers to the costof filtering the information, which increases with the degree of informatization. We can interpret
this cost as a measure of the time taken to filter out the required information. That is, the more
sources of information there are, the longer it takes to find the information source that has therequired information. Moreover, as the number of information sources increases, it is harder todecide what is true and what is false.6 A recent survey by the University of California at Berkleyreported that, globally, about two exabytes (a billion gigabytes, or 1018 bytes) of information is
produced annually; this amounts to about 250 megabytes per person. Arguably, such an excess
of information makes it difficult to filter the necessary information. Therefore, informatization
increases the cost of filtering information. We can interpret this as the effect of information
overload. We term this the filtering cost.7

6

Jinwon and Tang (2001) use five industry case studies to investigate the information overload problem and findthat the quality of information is negatively related to informatization and information overload.
7
Reis (2006a, 2006b) also claims that it is costly to acquire, absorb, and process information.

6

The information cost function has the following properties:∂IC(It , hit )
< 0,
∂hit

³ ´ 1

ψA ψ+φi∂IC(It , ht )  < 0 if It < φB=
³ ´ 1 .
∂Iti
 ≥ 0 if It ≥ ψA ψ+φ
φB

(4)

(5)

First, (4) shows that, when the degree of informatization is fixed, an increase in humancapital, hit , reduces the information cost. This means that voters who have more human capitalare better able to collect information. For example, voters with higher human capital are moreskilled in using personal computers (and so have a lower access cost) and in filtering out usefulwebsites (and so have a lower filtering cost). Hence these voters collect information easily.
Second, (5) means that, when the degree of informatization is low, an increase in the degree of
informatization lowers the information cost. However, when the degree of informatization is too
high, an increase in the degree of informatization raises the information cost (See Figure 2).
[Figure 2 around here.]
Figure 2 illustrates a U-shaped relationship between the degree of informatization and theinformation cost. When It is low, an increase in It lowers the information cost. That is because
the decrease in the access cost dominates the increase in the filtering cost. Suppose that there are
no newspapers, television, or the Internet. Then, the diffusion of newspapers would reduce theaccess cost considerably. However, when the degree of informatization is low, the increase in thefiltering cost is not excessive. Therefore, a decrease in the access cost dominates an increase in
the filtering cost and, hence, the information cost decreases. By contrast, when It is excessively
high, an increase in It increases the information cost. This is because the increase in the filtering
cost outweighs the decrease in the access cost. For example, although the Internet has lowered
access costs, it also generates excess information and considerably increases the filtering cost(through information overload).
The information cost is the key variable in our model. Most existing studies of the endogenous
determination of policy implicitly assume that all voters have perfect information and that thecost of collecting information is zero. In this paper, we introduce the information cost explicitly.

7

As we go on to show, this cost induces voter abstention and plays a key to role in explaining the‘New Paradox of Voting’.

2.3

Voting-Abstention Choice

We assume that voters vote when the utility of voting, D, is higher than the information cost,
IC(It , hit ), and abstain from voting otherwise. Note that we may interpret abstention behavior
under this assumption as Downs’s “rational ignorance”. Voters choose to abstain because theinformation cost is sufficiently high. They make a rational choice to be ignorant of politics andabstain from voting.
Hence, IC(It , ·) is continuous, monotonically decreasing in hit , and limhit →0 IC(It , hit ) = +∞,limhit →+∞ IC(It , hit ) = 0 by (3). Thus, there must exist an h∗t such that:o1
 n −ψ
ζχ AIt + BItφ ,h∗t = 
D

where



 hi ≥ h∗ ⇒ voting,tt hit < h∗t ⇒ abstention.

(6)[Figure 3 around here.]
Figure 3 illustrates the threshold level of human capital corresponding to the choice between
voting and abstaining.8 Figure 3 shows that our model is consistent with existing theoreticaland experimental studies: our model shows that voters with low income, low human capital or
limited information tend to abstain from voting (cross-sectional differences in voting behavior).

2.4

The Median VoterIn this subsection, we discuss the median voter, under the assumption that some voters abstain.9
We assume that there is a continuum of agents, i ∈ [0, 1], and that human capital is lognormally
distributed: ln hit ∼ N (mt , vt2 ), mt ∈ R, vt ∈ R+ ,10 where mt and vt2 are the mean and varianceof the log-human capital distribution at period t, respectively.11
8

We assume that when, D = IC(It , hit ), the electors choose to vote.
The method used in this subsection is also applied in Benabou (2000).
10
The lognormal distribution is an appropriate representation of the empirical data on income and human capital(Benabou, 2000). In addition, the lognormal approach facilitates dynamic analysis of income inequality.
11
Note that mt is the mean of log-human capital; hence, it lies between −∞ and +∞.
98By definition, when no one abstains, the median voter’s human capital, hmed
, is determinedt
as Pr(ln hit ≤ ln hmed
) = (1/2) = Ω(0), where Ω(·) is the cumulative distribution function oft¡¢− mt )/vt , if no one
a standard normal distribution. Because Pr(hti ≤ hmed
) = Ω (ln hmedttabstains, the median voter’s human capital is ln hmed
= mt . However, if some do abstain, wetmust identify the median voter among electors who actually vote. As already mentioned, agentswith less human capital than h∗t abstain from voting; hence, the larger is the number of voters
who abstain, the higher is the median voter’s human capital (see Figure 4).
[Figure 4 around here.]
When some voters abstain, the median voter’s human capital is hmed
, which must satisfytPr(hit ≤ hmed
|hit ≥ h∗t ) = (1/2). By using a standard normalization, we can redefine the mediant
voter and rewrite voter turnout as:µ¶
1 1
ln h∗t − mt
=
+ ·Ω
,
Ω
2 2
vt
¶
µ
ln h∗t − mt.
The voter turnout = 1 − Ωvtµln hmed
− mt
t
vt

¶

(7)
(8)

¢
¡
Given that Ω ((ln h∗t − mt )/vt ) ≥ 0, the left-hand side of (7) must satisfy Ω (ln hmed
− mt )/vt ≥t(1/2). This property unsures that, if some abstain, the median voter’s log-human capital exceeds
the mean log-human capital, mt . We define the following:
µ µ
¶¶
µ
¶
ln h∗t − mt
1 1
ln h∗t − mt
Ω λ
≡ + ·Ω
,
vt
2 2
vt

(9)

where λ(·) ∈ [0, +∞) and λ0 (·) ≥ 0. From this definition and from (7), we can represent themedian voter’s log-human capital as:µln hmed= mt + vt · λtln h∗t − mt
vt¶
≥ mt .

(10)

We can now state the following proposition.
1

Proposition 1. When It < (ψA/φB) ψ+φ , an increase in the degree of informatization raises
voter turnout and makes an individual with less human capital become the median voter. When9

1It > (ψA/φB) ψ+φ , an increase in the degree of informatization lowers voter turnout and makes
an individual with more human capital become the median voter.
Proof. See Appendix A.
This proposition is one of the main results of this paper. It tells us that informatization
has positive and negative effects on voter turnout. When the degree of informatization is low,
informatization reduces the information cost and raises voter turnout. However, when the degreeof informatization is high, informatization raises the information cost and lowers voter turnout(through information overload). Consequently, there is an inverse U-shaped relationship between
voter turnout and the degree of informatization.
Some empirical studies support this result. For example, using panel data on US counties
during the period in which radio expanded (from 1920 to 1930), Strömberg (2004) presents
evidence that the penetration of radio increased voter turnout. In the context of our model,
1

this development could be represented by the case where It < (ψA/φB) ψ+φ . By contrast, using
panel data on US cities from 1940 to 1970, Gentzkow (2005) shows that the introduction of
television significantly reduced voter turnout. In our model, this corresponds to the case where1

It > (ψA/φB) ψ+φ .
Note that, as we explained in Section 4 below, the policy preferred by the median voter isimplemented in the politico-economic equilibrium. Therefore, an increase in the median voter’shuman capital means that the implemented policy becomes more desirable for agents with higher
human capital.

2.5

Advancement of Informatization

We specify the advancement of informatization as:

It = σ · exp(mt ),(11)

where σ > 0.1213 Equation (11) means that the degree of informatization, It , is determined byaverage human capital. We can then interpret It as externalities associated with human capital.
12

We may interpret the degree of informatization as externalities of the average human capital, as Lucas (1988)
formulates.
13
As long as It is increasing with mt , the analysis that follows is not materially affected.

10For example, we may consider that an increase in average human capital reflects an increasein the literacy rate. An increase in the literacy rate increases the degree of informatization bymaking it easier for agents to exchange information through newspapers, books, magazines, forexample. We may also explain this externality as residual product of knowledge spillovers. When
average human capital increases, the knowledge and skills of each agent spread to other agents.
Then, in the process of spilling over, knowledge and skills are translated into information in theform of articles, books, videos and so on, which promote informatization.
Substituting (11) into (6), we can rewrite the threshold human capital, h∗t , as:ln h∗t = Ξ + g(mt ),

(12)

£
¤
where Ξ ≡ (1/ζ) · ln (χ/D) and g(mt ) ≡ (1/ζ) · ln Aσ −ψ exp(−ψmt ) + Bσ φ exp(φmt ) . By
definition, the first derivative of g(mt ) is given by:

³
´

ψA
1

−ψ
φ
<
0,
if
m
<
ln
t
−ψAσ exp(−ψmt ) + φBσ exp(φmt )ψ+φφBσ ψ+φg 0 (mt ) =
³
´ .
−ψ
φ
ζ {Aσ exp(−ψmt ) + Bσ exp(φmt )} ψA ≥ 0, if mt ≥ 1 ln
ψ+φφBσ ψ+φ(13)

Therefore, ln h∗t is also U-shaped with respect to mt . Equation (13) indicates the following. When
average human capital is low, the degree of informatization is also low. Hence, an increase in
the average of log-human capital, mt , lowers the information cost and increases the threshold,ln h∗t , by increasing in the degree of informatization, and vice versa. This explains the inverse
U-shaped relationship between voter turnout and average log-human capital, or GDP.
We can now state the following proposition.
Proposition 2. An increase in the mean of log-human capital, mt , ceteris paribus, raises voter£¤
turnout when mt < (1/(ψ + φ)) · ln (ψ + ζ)A/(φ − ζ)σ ψ+φ B , and vice versa. Moreover, an
increase in the variance of log-human capital, vt2 , ceteris paribus, raises (lowers) voter turnout
when the voter turnout is over (under) 50%.
Proof. See Appendix B.
This proposition suggests a solution to the ‘New Paradox of Voting’. When mt (or GDP) is
high enough, an increase in the mean of log-human capital, mt , has a negative effect on voter

11

turnout. This negative effect arises from the increase in the filtering cost (or from information
overload). Excesssive informatization increases the filtering cost considerably. Consequently,
the information cost becomes high for all voters, and voter turnout decreases. By contrast, an
increase in the variance of log-human capital, vt2 , has a negative (positive) effect when voter
turnout is over (under) 50%. These results are summarized in Figure 5.
[Figure 5 around here.]Suppose that mt is increasing with economic growth. Then, if the effect of an increase
in mt is high, relative to the effect of an increase in vt2 , voter turnout have an inverse Ushaped relationship with mt and economic growth. This means that voter turnout decreases
with economic growth when the economy is sufficiently developed. We have already shown that
our model is consistent with existing theoretical and empirical studies. Thus, our model offersa solution to the ‘New Paradox of Voting’ and it can explain the cross-sectional and over-timedifferences in voting behavior simultaneously.14
Note that this model also predicts that voter turnout decreases because agents with lowincome, low-human capital and limited information abstain from voting. Cavanagh (1981) supports this prediction. Cavanagh (1981) shows that the decrease in voter turnout from 1964 to1976 in the US was disproportionately among the poorest and least informed groups of citizens:
the steepest drops in voter turnout have been recorded among the poor and the least educatedcitizens.
Finally, note that from (10) the median voter’s human capital is given by:µln hmed= mt + v t · λtΞ + g(mt ) − mt
vt

¶
.

(14)

In this section, we have discussed the formulation of a model of the choice between voting
and abstaining and the determination of the identity of the median voter when some abstain.
Moreover, we have shown that informatization and information overload are the keys to solving
14

Note that this model does not claim that the cross-country association between GDP and voter turnout is inverse
U-shaped. Determining factors in the voter turnout is not only the level of GDP but also other economic and politicalfactors. For example, distribution of human capital, advancement of informatization, citizen’s preference for policies(these are representers by the parameters in this model) also affect to the voter turnout. Therefore, the countries withthe similar levels of GDPs may have different rates of voter turnout.

12the ‘New Paradox of Voting’. In the rest of this paper, we discuss the relationships between the
dynamics of the income distribution, the effect of informatization, and long-run decline in voter
turnout.

3

The Economic ModelIn this section, we discuss the second part of the model; that is, the economic model. We set

up a neoclassical dynamic macroeconomic model, which has also been used in Benabou (2000)
and Takii and Tanaka (2005). We describe the dynamic relationship between redistribution
policy and the distribution of human capital.

The economy is populated by nonoverlapping generations of families, i ∈ [0, 1], and there
is no population growth. We assume that the initial distribution of human capital between
families is ln hi0 ∼ N (m0 , v02 ). The sequence of events is as follows. Each agent’s human capitalis determined by their parents’ investment in education. Agents choose between voting andabstaining. The equilibrium tax rate in period t is determined on the basis of pure majority
voting. Agents learn the value of their idiosyncratic productivity shocks, zti and ηti . Agentsproduce goods by using their human capital. Gross income and disposable income are thus
determined. Agents then allocate their disposable incomes between their own consumption andeducation investment for their children. Thus, their offspring’s human capital is determined.
Agent i of generation t produces output by use of his or her human capital, hit , subject to anindependently and identically distributed idiosyncratic productivity shock, zti :yti = zti (hit )γ ,(15)

where yti is gross income and γ ∈ (0, 1) is a parameter. In this economy, agents have preferences defined over their own consumption, cit , and their offspring’s human capital, hit+1 (which
represents altruism). The taxes and transfers specified below transform gross income, yti , into
disposable income, ŷti . Agents allocate this disposable income to their own consumption, cit , andto education investment for their offspring, eit . For simplicity, we assume that their offspring’s
human capital is determined solely by education investment and the idiosyncratic shocks, ηti .

13This assumption means that children cannot inherit human capital from their parents. We
specify the production function of human capital as:
hit+1 = κηti (eit )β ,(16)

where κ > 0 and β ∈ (0, 1) are parameters, and ηti is an independently and identically distributed idiosyncratic productivity shock. As argued above, both shocks, zti and ηti , are unknown when people vote. Hence, redistribution policy acts as a hedge against risk.15 For
simplicity, both shocks are assumed to be lognormally distributed: ln zti ∼ N (−r2 /2, r2 ) andln ηti ∼ N (−q 2 /2, q 2 ), r, q > 0.
Taxes and transfers transform agent i’s gross income, yti , into disposable income, ŷti , asfollows:ŷti = (yti )1−τt (ỹt )τt = cit + eit ,(17)

where τt ∈ [0, 1] is the tax rate in period t, and ỹt is the break-even level of income. Therefore,
agents with gross incomes exceeding ỹt pay tax, and those with gross incomes lower than ỹtreceive subsidies. Note that, under this scheme, the ratio of tax payments (or subsidies) to gross¡¢τincome is (yti − ŷti )/yti = 1 − y˜t /yti t . Thus, there is a progressive taxation system.16 Equation(17) implies that, when τt = 1, all agents have the same disposable income. The break-evenlevel of income, y˜t , is determined by the government’s budget constraint, under which the sumof net transfers must be zero:
Z 10(yti )1−τt (ỹt )τt di = yt ,(18)

where yt is average income in period t: yt = E[yti ].
We specify u(cit , hit+1 |τt ) in (1) as:u(cit , hit+1 |τt ) = (1 − ρ) ln cit + ρ ln hit+1 .

15
16

(19)

Both productivity shocks, zti and ηti , also ensure that the variance of log-human capital, v 2 , is strictly positive.
See Benabou (2000) for details of this policy scheme.

14where ρ ∈ (0, 1) defines the relative weights assigned to agents’ felicity and altruistic motives.
We assume that u(cit , hit+1 |τt ) is the utility function in the remainder of this paper.17
Proposition 3. Given the tax rate τt , agents choose a common saving rate ξ:
eit = ξ ŷti ,cit = (1 − ξ)ŷti ,(20)

where ξ ≡ ρβ/(1 − ρ + ρβ) ∈ (0, 1).
Proof. See Appendix C.

We discuss the dynamics of human capital distribution and of GDP. Using Proposition 3,
and given (15) to (18), and given the properties of the lognormal distribution, E[zti ] = 1, and¡¢given that E[(hit )γ ] = exp γmt + γ 2 vt2 /2 (see Appendix D), it follows that human capital
accumulation is simply determined by:ln hit+1 = ln ηti + ln(κξ β ) + β(1 − τt ) ln zti + βγ(1 − τt ) ln hit+βγτt mt + β ·ª
γ 2 vt2βr2 ©
· τt (2 − τt ) +· (1 − τt ) − (1 − τt )2 .
2
2

(21)

Taking expectations and the variances of both sides of (21), we obtain the following difference
equations, which govern the evolution of the economy:

mt+1 = −q2
βγ 2 vt2
βr2+ ln(κξ β ) + τt (2 − τt ) ·− (1 − τt )2 ·+ βγmt ,
2
2
2

2
vt+1
= q 2 + β 2 (1 − τt )2 r2 + β 2 γ 2 (1 − τt )2 vt2 .

(22)
(23)

The effect of income redistribution on the dynamics of human capital (or income) inequality
is clear: the tax rate, τt , determines the persistence of family wealth inequality, β 2 γ 2 (1 − τt )2 .
Given (22) and (23), when the sequence of tax rates, {τt }∞t=0 , is determined, the dynamics of themean and variance of log-human capital follow simple first-order difference equations. Moreover,from equation (23), it follows that the productivity shocks, zti and ηti , affect the dynamics of
17

Note that, given (1), utility from voting, V (It , hit ), and utility from non-taxation policies, W (Zt ), do not affect the
optimization of cit and eit .

15the variance of log-human capital; that is, the dynamics of income inequality. This means that
productivity shocks increase income inequality.
Now consider the GDP growth rate. We have assumed that the economy is populated by
nonoverlapping generations of families, i ∈ [0, 1]; hence, average income can be interpreted asGDP. From (18), average income, yt , is:ln yt = γmt +γ 2 vt2.
2

Both mt and vt2 positively affect ln yt , despite the concavity of the production function.18
Now,we define the GDP growth rate as ln (yt+1 /yt ). Then, using (22) and (23), we obtain:
µ
ln

yt+1
yt

¶

q2
= −(1 − βγ) ln yt + γ(γ − 1) + γ ln(κξ β ) − Θt ·
2

¶
µ 2 2
γ vt + r 2,
2

where Θt ≡ βγ(1 − τt )2 (1 − βγ) ≥ 0. The first term on the right-hand side, −(1 − βγ) ln yt ,represents a standard convergence effect. Because we have assumed that β ∈ (0, 1) and γ ∈ (0, 1),¡¢
this term decreases as the ln yt increases.19 The term −Θt · (γ 2 vt2 + r2 )/2 represents the effectof the variance of log-human capital and the effect of the productivity shock, zti . Because Θt ≥ 0,
increases in the variance of human capital and variance of productivity shock lower GDP growth.
The reason for this is simple. Given that the saving rate is the same for all agents and giventhat there are diminishing returns to education investment, an increase in the variance of humancapital lowers average human capital and the income of the next generation. In addition, because
Θt is a decreasing function of τt , we can state next proposition.
Proposition 4. In this economy, the tax rate that maximizes the GDP growth rate in period t
is τt = 1.
This result is consistent with that of Alesina and Rodrik (1994).20

18

This feature depends on the assumption of a log-normal distribution. When ln hit is normally distributed, hit has
a right-skewed distribution function. This means that the mean of hit increases with the variance of ln hit . Hence, anincrease in the variance of ln hit has a positive effect on GDP.
19
If βγ > 1, this term increases with the growth rate, in which case, there is a possibility of endogenous growth.
20
When βγ > 1, the tax rate that maximizes the GDP growth rate in period t is τt = 0. This result is consistent
with that of Li and Zou (1998).

16

In this section, we have discussed the dynamics of log-human capital distribution and of GDP
growth. We found that τt = 1 maximizes GDP growth in period t. However, in this economy,
the tax rate is determined not by the government but by the pure majority voting. In the next
section, we discuss the equilibrium tax rate that is determined on the basis of the pure majority
voting and investigate the steady state of economy.

4

Politico-Economic Equilibrium
In this section, combining the voting model and the economic model discussed above, we

derive a politico-economic equilibrium. Moreover, we analyze the dynamics of voter turnout, theequilibrium policies determined on the basis of the pure majority voting, and changes in income
inequality and GDP. We also discuss the social costs of a fall in voter turnout in terms of theassociated negative effects on GDP growth.

Given (19) and Proposition 3, agent i’s expected utility before voting under a given tax rate,τt , is given by:21
r2
q2
−ρ·
2
2
γ 2 vt2r2+ρ ln κ + τt (1 − ρ + ρβ)γmt + τt (2 − τt )(1 − ρ + ρβ) ·− τt (τt − 1)(1 − ρ + ρβ) · .
22

E[u(cit , hit+1 |τt )] = ln[(1 − ξ)1−ρ ξ ρβ ] + (1 − ρ + ρβ)(1 − τt )γ ln hit − (1 − ρ + ρβ)(1 − τt ) ·(24)

Given (24), the optimal tax rate for agent i who has human capital of hit is:(τti )∗ =







1−






1
γ(ln hit −mt )γ 2 vt2 +r20

if ln hit ≤ mt ,if mt < ln hit < mt +if mt +

γ 2 vt2 +r2
,
γ

(25)

γ 2 vt2 +r2≤ ln hit ,γ

where (τti )∗ is the optimal tax rate for agent i in period t, and (25) must satisfy (τti )∗ ∈ [0, 1] for∀hit ≥ 0. An increase in the variance of the productivity shock, r2 , or an increase in the varianceof human capital, vt2 , increases the optimal tax rate, (τti )∗ . This is because the higher are these
21

Note that we assumed that the productivity shocks, zti and ηti , are unknown when agents vote.

17

factors, the greater is risk, in which case, people prefer greater redistribution for more of a hedge
against risk. Moreover, an increase in human capital, ln hit , lowers the optimal tax rate, (τti )∗ .
This is because the taxation system is progressive .
As argued in Section 2, when some voters abstain, the median voter’s human capital isln hmed= mt + vt · λ ((Ξ + g(mt ) − mt )/vt ). Therefore, by substituting (14) into (25), we obtaint
the tax rate that is preferred by the median voter, τtmed , which is:´ ³Ξ+g(mt )−mt
½
¾


γv
·
λ
med
t
vt
γ(ln ht − mt )τtmed = max 1 −
,
0
=
max
1
−
,
0
.


γ 2 vt2 + r2
γ 2 vt2 + r2

(26)

We can now state the next proposition.
Proposition 5. A fall in voter turnout lowers the GDP growth rate.
Proof. Given (8) and (12), voter turnout is 1−Ω ((ln h∗t − mt )/vt ) = 1−Ω ((Ξ + g(mt ) − mt )/vt ).
Then, given equation (9) and (10), a fall in voter turnout increases the median voter’s humancapital. From (26), it follows that:∂τtmed< 0.
∂ ln hmedtHence, a fall in voter turnout in period t lowers the equilibrium tax rate and lowers the degreeof income redistribution in period t. From Proposition 4, it follows that the GDP growth rate
in period t is maximized when τt = 1. Hence a decrease in voter turnout in period t lowers the
GDP growth rate in period t.
Proposition 5 reveals that a short-run social cost is associated with a fall in voter turnout.
This result is very intuitive. When agents with less human capital (income) have a greater
tendency to abstain from voting than agents with more human capital (income), the lower voter
turnout makes the equilibrium tax rate determined on the basis of the pure majority voting
more favorable to those with a higher level of human capital (income). Hence, under a progressive taxation system, a fall in voter turnout increases income inequality by promoting a lower
degree of income redistribution. Moreover, if the human capital production function exhibits
decreasing returns, the GDP growth rate in period t declines. This prediction is supported bysome empirical studies. For example, using cross-national data set spans from 1960 to 1990 on

18

“strong democratic”22 countries, Mueller and Stratmann (2003) show that citizen participation
increases government size or transfers, and these in turn reduce income inequality. Moreover,using cross-natilnal data set spans from 1960 to 1990 on democratic countries, they show that afall in voter turnout lowers the GDP growth rate.
Next, we characterize the steady state.
Proposition 6. The policy that is preferred by the median voter, Ptmed = (τtmed , Ztmed ), is theequilibrium policy in period t ∀t.
Proof. See Appendix E.
According to the median voter theorem, the equilibrium tax rate determined on the basisof the pure majority voting is τtmed . Hence, from (22) and (23), the dynamic behavior of theeconomy is represented by:

mt+1 = −βγ 2 vt2
βr2
q2+ ln(κξ β ) + τtmed (2 − τtmed ) ·− (1 − τtmed )2 ·+ βγmt ,
2
2
2

2
vt+1
= q 2 + β 2 (1 − τtmed )2 r2 + β 2 γ 2 (1 − τtmed )2 vt2 .

(27)
(28)

Given (26), the equilibrium tax rate that is preferred by the median voter depends only on
mt and vt2 . Hence, by substituting (26) into (27) and (28), we can rewrite these equations as
first-order nonlinear difference equations in mt and vt2 .

In this section, we have discussed the politico-economic equilibrium. In the next section, we
use a numerical example to demonstrate the features of the transition path and the steady stateof this economy.

5

Numerical Example
In Sections 2 and 3, we described the voting model and the economic model. In Section

4, we combined both models and derived the politico-economic equilibrium. We also described
the system that determines the long-run transition path of the economy. It is worth illustrating
22“Strong democratic” category contains the US, EU menbers, Canada and so on.

19the transition path by using a numerical example to explaining the relationships between thevariables.

Figure 6 displays the phase diagram of the system described by equations (26) to (28) with
χ = 0.07, A = 1, B = 2, ζ = 0.5 ψ = 0.3, φ = 2, D = 0.2, σ = 1, γ = 0.6, β = 0.8, κ = 1.5,
q = 1.5, r = 0.1, ρ = 0.7.
[Figure 6 around here.]In this figure, the solid line depicts the dmt = 0 locus and the dashed line depicts the
dvt2 = 0 locus. Note that, from (26), (27) and (28), the dmt = 0 locus and the dvt2 = 0 locus
are dominated on the basis that either τtmed > 0 or τtmed = 0. The dotted line depicts theτtmed = 0 locus. Above the τtmed = 0 locus, the dmt = 0 locus and the dvt2 = 0 locus are basedon τtmed > 0; below the τtmed = 0 locus, the dmt = 0 locus and the dvt2 = 0 locus are based on
τtmed = 0. Given (27) and (28), the dmt = 0 locus and the dvt2 = 0 locus can be rewritten as:
½ 2
¾
q
βr2
1
β
· − + ln(κξ ) −,mt =1 − βγ22vt2 =q2 + β 2 r2,
1 − β2γ2when τtmed = 0. Below the τtmed = 0 locus, the dmt = 0 locus is a vertical line and the dvt2 = 0
locus is a horizontal line. In Figure 6, the steady-state tax rate is zero.
Figure 7 displays the phase diagram based on χ = 0.04 with the other parameters taking thesame value as in Figure 6.
[Figure 7 around here.]
Given (3), a lower χ implies a lower information cost. From Figures 6 and 7, it is apparent
that a decrease in χ lowers the kink point of the dmt = 0 locus. In comparison with Figure 6, thesteady state shifts lower and to the right, which implies an increase in GDP and fall in income
inequality. Intuitively, a lower χ (that is, a lower information cost) increases voter turnout andraises tax rate. A rise in the tax rate reduces the degree of income inequality by increasing thedegree of income redistribution. Moreover, as shown in Proposition 4, the rise in the tax rate
also raises GDP.

20

Now we consider the transition path. Figure 8 displays the transition path in the case ofχ = 0.01, A = 1, B = 2, ζ = 0.5 ψ = 0.3, φ = 2, D = 0.2, σ = 12, γ = 0.6, β = 0.8, κ = 1.5,
q = 1.5, r = 0.1 and ρ = 0.7. This parameter set generates plausible sequences for voter turnout
and GDP.
[Figure 8 around here.]In this case, voter turnout exhibits an inverse U-shape. From Proposition 2, when mt <£¤(1/(ψ + φ)) · ln (ψ + ζ)A/(φ − ζ)σ ψ+φ B , an increase in mt raises voter turnout. When mt
and It are low, an increase in the degree of informatization lowers the access cost considerably,
but a rise in the filtering cost is limited. Hence, the information cost, which is the sum of theaccess cost and filtering cost, decreases, and voter turnout increases. By contrast, when mt >£¤(1/(ψ + φ)) · ln (ψ + ζ)A/(φ − ζ)σ ψ+φ B , an increase in mt reduces voter turnout. When mt
and It are high, an increase in the degree of informatization raises the filtering cost considerably,
but the decline in the access cost is small. Hence, the information cost rises, and voter turnout
falls. In Figure 8, the effect of the degree of informatization is different in period 2, and voter
turnout is represented by an inverse U-shaped curve. Thus, our model explains the observed
patterns described in the Introduction (see Figure 1). Nardulli et al. (1996) argue that voter
turnout in the US is described by an inverse U-shaped curve; it increased from 1920 to 1960
and decreased form 1960 on. In this numerical example, period 2 corresponds to the 1960 in the
US.23
Moreover, after period 2, voter turnout decreases as GDP, the average log-human capital andthe degree of informatization increase.24 In Section 2, we showed that voters with low income,low human capital or limited information tend to abstain from voting. Therefore, this transition
path shows that our model can explain the cross-sectional and over-time differences in voting
behavior simultaneously, and provide a solution to the ‘New Paradox of Voting’.
Figure 8 also illustrates the relationships between the variables. A decrease (increase) in
23

Nardulli et al. (1996) also argue that voter turnout in the US decreased from 1876 to 1920. Our model cannot
describe such a decline. Two reasons may explain this inconsistency. First, the advancement of informatization was
weak before the 1920s. For example, Strömberg (2004) argues that radio expanded from 1920 to 1930 in the US. Thus
we can interpret that full-scale informatization in the US began from 1920. Second, we have assumed that all agentsare eligible to vote, but women in the US were not enfranchised until 1920. Therefore, the movement in voter turnout
in the period from 1876 to 1920 in the US can be exempted from the analysis of this model.
24
Recall that the degree of informatization, It , is determined by average human capital, mt .

21

voter turnout means that an agent with a higher (lower) level of human capital becomes themedian voter (see Proposition 1). Given (26), an increase in voter turnout lowers (raises) theequilibrium tax rate determined on the basis of the pure majority voting. Consequently, a fall(rise) in the tax rate lowers (raises) the degree of income redistribution, and thereby increases(reduces) the variance of log-human capital, vt2 , which implies an increase (a fall) in income
inequality. Mueller and Stratmann (2003) support this prediction. Using the cross-national
data set spans from 1960 to 1990 on “strong democratic”25 countries, Mueller and Stratmann(2003) show that citizen participation raises the government size or transfers, which in turn
reduces income inequality.

In this section, we have obtained two implications of our model by using numerical examples
base on the analysis of previous sections. First, higher filtering cost (or information overload)
could explain the dynamic decrease in voter turnout, and thus provide a solution to the ‘New
Paradox of Voting’. Second, a lower voter turnout generates social costs in the form of lower
GDP and greater income inequality.

6

Conclusion
In this paper, we have proposed the ‘New Paradox of Voting’, which means that, from a static

point of view, voters with low incomes, low human capital or limited information tend to abstain
from voting (cross-sectional differences in voting behavior), however, from a dynamic point ofview, voter turnout decreases with increases in average incomes, human capital and the degreeof informatization (over-time differences in voting behavior). We have introduced the ‘accesscost’ and the ‘filtering cost’ into a neoclassical macroeconomic model and offered a solution tothe ‘New Paradox of Voting’. That is, increasing informatization raises the filtering cost by
generating information overload; then, the increase in the filtering cost lowers voter turnout.
We have used an illustrative numerical example, and have shown that our model can explain
the cross-sectional differences and the over-time differences in voting behavior simultaneously;
it can offer a solution to the ‘New Paradox of Voting’.
25“Strong democratic” category contains the US, EU menbers, Canada and so on.

22We have also discussed the social costs of a decrease in voter turnout. When the production
functions for goods and human capital are both strictly concave, a higher tax rate at time t
raises GDP growth at t. Then, the fall in voter turnout makes an individual with a higher levelof human capital become the median voter, which reduces the tax rate through majority voting.
Consequently, a fall in voter turnout lowers the GDP growth rate, hence raising social costs.
This model generates a policy implication. That is, a decrease in voter turnout generatessocial costs. Therefore, it would be desirable to halt the fall in voter turnout over time indeveloped countries. We also found that an increase in the filtering cost associated with greater
informatization may be the cause of the continuing decline in voter turnout. However, it isdifficult to control informatization and limit information overload. This suggests that halting thecontinuing decline in voter turnout by controlling informatization would be difficult. Therefore,
perhaps voting should be compulsory, as is the case in, for example, Italy, Australia, Belgium
and Singapore.26

Appendix A: Proof of Proposition 1.
Proof. We start with the proof relating to voter turnout. From (8):
∂
∂It

µ
µ
¶¶
µ
¶
ln h∗t − mt−ψAIt−ψ−1 + φBItφ−1∂ln h∗t − mt
n
o ·
1−Ω
=−
Ω
.
vt
∂ ln h∗tvt
ζ AI −ψ + BI φtt

Since Ω(·) is the standard normal distribution function, (∂/∂ ln h∗t )Ω ((ln h∗t − mt )/vt ) > 0.
1

Therefore, when It < (ψA/φB) ψ+φ , an increase in the degree of informatization, It , raises
voter turnout, and vice versa. Moreover, by (10):∂ ln hmed∂t= vt ··λ∂It∂ ln h∗tµln h∗t − mtvt¶

−ψAIt−ψ−1 + φBItφ−1no ,·ζ AIt−ψ + BItφ
1/∂It ) < 0 when It < (Aψ/Bφ) ψ+φ , and vice versa.
where λ0 (·) > 0. Therefore, (∂ ln hmedt26

Using cross-country data from 91 countries for the period 1960-2000, Chong and Olivera (2005) shows that compulsory voting, when enforced strictly, reduces income inequality, as measured by the Gini coefficient.

23

Appendix B: Proof of Proposition 2.
Proof. We can rewrite voter turnout as 1 − Ω ((ln h∗t − mt )/vt ) = 1 − Ω ((Ξ + g(mt ) − mt )/vt )by use of (12). Therefore:
∂
∂mt

µ
½
µ
¶¾
¶
Ξ + g(mt ) − mt
1 − g 0 (mt )0 Ξ + g(mt ) − mt
·Ω
1−Ω
=
.
vt
vt
vt

Note that Ω(·) is a standard normal distribution function, and hence Ω0 (·) > 0. Given thedefinition of g(mt ), we obtain:·¸
1(ψ + ζ)Ag (mt ) < 1 ⇔ mt <
· ln
.
ψ+φ
(φ − ζ)σ ψ+φ B
0£¤Hence, if mt < (1/(ψ + φ)) · ln (ψ + ζ)A/(φ − ζ)σ ψ+φ B , an increase in the mean of log-humancapital raises voter turnout; the opposite is also true. Moreover:
∂
∂vt2

¶¾
¶
½
µ
µ
ln h∗t − mt
Ξ + g(mt ) − mt
0 Ξ + g(mt ) − mt
=
.
1−Ω
·Ω
vt
vt
2vt3

Therefore, if ln h∗t < mt (that is, if voter turnout is over 50%), an increase in the variance of
log-human capital lowers voter turnout, and vice versa.

Appendix C: Proof of Proposition 3.
Proof. Let us define ωti ≡ (eit /ŷti ). Then, given (15), (16), (17) and (19), we obtain:£¤u(cit , hit+1 |τt ) = (1 − ρ) ln(1 − ωti ) + τt ln ỹt + (1 − τt )[ln zti + γ ln hit ]
£
©
£
¤
ª¤
+ ρ ln κ + ln ηti + β ln ωti + (1 − τt ) ln zti + γ ln hit + τt ln ỹt .
Given (C.1), we obtain (20).

24

(C.1)

Appendix D: Properties of the lognormal distribution.
First, let us assume that ln x ∼ N (µ, σ 2 ) and G(x) is cumulative distribution function of x.
Then, by definition, it follows that:
Z ln w
G(w) = Pr(x ≤ w) =
−∞

·
¸
1
(ln x − µ)2√ · exp −d ln x.
2σ 2
σ 2π

By applying Leibniz’s rule, we can express the probability density function of x as:·¸(ln w − µ)21
1g(w) = G (w) = · √ · exp −.
w σ 2π2σ 20If we define s ≡ (ln x − µ)/σ, E[(x)r ] is given by:·¸(ln x − µ)2
1√ · exp −
dx
2σ 2
xσ 2π
−∞
·
¸ Z ∞
·
¸
(rσ)2
1
(s − rσ)2√ · exp −= exp(rµ) · exp
·
ds
2
2
2π
−∞
µ
¶
r2 σ2
= exp rµ +.
2

E[(x)r ] =Z ∞(x)r ·¡¢¡¢
Therefore, when ln hit ∼ N mt , vt2 and ln zti ∼ N −q 2 /2, q 2 , we obtain:
E[(hit )γ ] = exp¶µ
γ 2 vt2,γmt +2

E[zti ] = exp

µ 2
¶
q
q2
− +
= 1.
2
2

Appendix E: Proof of Proposition 6.
Proof. Given (1) and (24), voter i’s pre-election utility, E[Uti ], is:
r2
q2
−ρ·
+ ρ ln κ22
γ 2 vt2r2+τt (1 − ρ + ρβ)γmt + τt (2 − τt )(1 − ρ + ρβ) ·− τt (τt − 1)(1 − ρ + ρβ) ·+ W (Zt ). (E.1)
2
2

E[Uti ] = ln[(1 − ξ)1−ρ ξ ρβ ] + (1 − ρ + ρβ)(1 − τt )γ ln hit − (1 − ρ + ρβ)(1 − τt ) ·25

Note, V (It , hit ) is already determined when voters vote and, hence, we can treat it as a sunkbenefit. Then, we can rewrite (E.1) as:
E[Uti ] = J(Pt ) + H(Pt ) · ln hit ,q2r2− (1 − ρ + ρβ)(1 − τt ) ·22
γ 2 vt2r2+τt (1 − ρ + ρβ)γmt + τt (2 − τt )(1 − ρ + ρβ) ·− τt (τt − 1)(1 − ρ + ρβ) ·+ W (Zt )2
2

J(Pt ) ≡ ln[(1 − ξ)1−ρ ξ ρβ ] + ρ ln κ − ρ ·H(Pt ) ≡ (1 − ρ + ρβ)(1 − τt )γ,
Since ln hit is a monotonic function, voters’ preferences are the intermediate preference.27 Thismeans that the policy preferred by the median voter is the equilibrium policy.

27

See Persson and Tabellini (2000) for details of intermediate preference.

26

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Voter Turnout (%)
50
60
70

USA

95% confidence interval
Fitted values

40

USA

1920

1940

1960
Year

1980

2000

Voter Turnout (%)
60 70 80 90UK

95% confidence interval
Fitted values

50

UK

1920

1940

1960
Year

1980

2000

Source: International Institute for Democracy and Electoral Assistance (IDEA); Interuniversity Consortium for Political and Social Research (ICPSR), study number 8611; House of Commons Research
Paper 04/61 UK Election Statistics 1918-2004.
Note 1: Fitted values are estimated by using quadratic regression.
Note 2: The graph for the US describes the presidential election turnout.
Note 3: The graph for the UK describes the parliamentary election turnout.

Figure 1: Movements in voter turnout.

31

information cost

0

³

´ 1

ÃA Ã+Á
ÁB

Figure 2: The information cost.

32It

D, IC(It ; hit )Dabstention

votinghit

h¤t
Figure 3: The citizen duty and information cost.

33

p.d.f.

abstention

0 ln h¤tln hmedmt

Figure 4: Abstention.

34

ln hitvoter turnout
100%

mt " ) voter turnout "mt " ) voter turnout #v2t " ) voter turnout #v2t " ) voter turnout #mt " ) voter turnout "mt " ) voter turnout #v2t " ) voter turnout "v2t " ) voter turnout "

50%

0%
h

(Ã+³)A1
Ã+Á ln (Á¡³)¾ Ã+ÁBi

Figure 5: The effect of increases in mt and vt2 .

35

mt

10

9

8

7

6

5

4

3

2

1

0
‑5

‑4

‑3

‑2

‑1

0

1

2

3

4

5

Figure 6: Phase diagram based on χ = 0.07, A = 1, B = 2, ζ = 0.5 ψ = 0.3, φ = 2, D = 0.2, σ = 1,
γ = 0.6, β = 0.8, κ = 1.5, q = 1.5, r = 0.1 and ρ = 0.7.

36

10

9

8

7

6

5

4

3

2

1

0
‑5

‑4

‑3

‑2

‑1

0

1

2

3

4

5

Figure 7: Phase diagram based on χ = 0.04, A = 1, B = 2, ζ = 0.5 ψ = 0.3, φ = 2, D = 0.2, σ = 1,
γ = 0.6, β = 0.8, κ = 1.5, q = 1.5, r = 0.1 and ρ = 0.7.

37

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Figure 8: Transition path of an economy based on χ = 0.01, A = 1, B = 2, ζ = 0.5 ψ = 0.3, φ = 2,
D = 0.2, σ = 12, γ = 0.6, β = 0.8, κ = 1.5, q = 1.5, r = 0.1 and ρ = 0.7.

38