1

Dynamic Electoral Competition and

Constitutional Design

Marco Battaglini Princeton University,

CEPR and NBER

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Introduction

• Consider the comparison between proportional vs. majoritarian electoral systems.

• A robust finding: PS lead to higher g and lower r than MS.

• In a static model, this comparison is done ceteris paribus: same underlying parameters, etc.

• In a dynamic model, some of these values are endogenous.

• Debt affects the performance of the voting rule; but the voting rule affects debt.

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• If in the steady state we have more public debt in a PS than in a MS, then even if politicians desire to have a larger g, they may be able to afford only a lower g.

• The main result of this paper is that this is indeed the case:• PS tend to be more dynamically inefficient, and so

accumulate more debt;• This may lead to a lower g and higher r in the steady

state, despite the fact that a lower fraction of citizens is “represented.”

• This phenomenon may reverse the received wisdom on welfare comparisons.

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Plan for the talk

I. The model

II. The political equilibria:

1. The Proportional System (PS);

2. The Majoritarian Systems: single district (SDS), multiple districts (SMS)

III. Comparing electoral systems

IV. Empirical implications

I. The modelI.1 The economy

• A continuum of infinitely-lived citizens live in n identical districts. The size of the population in each district is one.

• There are three goods - a public good g, private consumption z, and labor l.

• Each citizen's per period utility function is:

• We assume A evolves according to a Markov process.

5

1(1 )

1.

lz Ag

6

• Linear technology: z=wl and g=z/p.

• The discount factor is δ.

• There are markets for labor, the public good, and one period, risk free bonds.

• In a competitive equilibrium:

– price of the public good is p,

– the wage rate is w,

– and the interest rate is ρ=1/δ-1.

I.2 Politics and policies

• Public decisions are chosen in national elections.

• A policy choice is described by an n+3-tuple:

{r,g,x,s1,…,sn}

• The government faces 3 feasibility constraints:

‒

‒ Non negative transfers: si > 0.

‒ An upper bound on debt (no Ponzi schemes):

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( , , ; ) ( ) (1 ) iiR r xB r g x b pg b s

x x

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I.3 A model of electoral politics

• Electoral competition is modelled a' la Lindbeck and Weibull [1987] as in Persson and Tabellini [1999].

• Candidates L, R run for office, simultaneously and noncooperatively committing to

• Voters vote for the preferred party and the electoral rule determines the winning party.

• A key difference: election is embedded in a dynamic game. Debt creates a strategic linkage between electoral cycles.

1{ , , , ,...., }i i i i i inp r g x s s

I.3.1 Voters

•Voter l’s utility for policy p={r,g,x,s1,…,sn} in a state A is:

where v(x; A’) is the expected continuation value function.

•Voters care about the policy and about an intrinsic quality ofthe candidate. Voter l in district j will vote for L iff:

• κj is an ideological preference for party R in district j

• σl is idiosyncratic to voter l

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( ; ; ) ( ; , )L Rljl lW v A vp Ap W

( ; , ) ( , , ) ;llW v A u A E v A Ap r g s x

• Both σj , κl are independent random variable that are realized at the beginning of the period:

G

G

• Districts, therefore, differ in their expected ideology and in ideological dispersion.

• The shocks are not observed by the candidates;

• The distribution of the shocks is known by the candidates.

10

1 1,

2 2l j j

j jU

h h

1 1,

2 2j U

d dk

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• These differences, moreover, may change over time.

‒ is a r.v. with density φ(h;A).

• We assume districts are symmetric with respect to the distribution of ideological components.

1,..., nh h h

I.3.2 Vote shares

• The votes received by L in district j given pL and pR are:

The votes received by R will be one minus the above.

• We consider two alternative voting systems:

‒ Proportional: a candidate is elected with a probability equal to the share of votes he/she receives;

‒ Majoritarian: A candidate is elected if he wins a majority in a majority of electoral districts.

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( ; , ) ( ; , )1

.2

jR

j

Ljj jW p v A W ph v A

13

I.3.3 Candidates

• Candidates maximize: , where R is a constant and Ii

τ is 1 if the candidate is in office, zero otherwise. Candidates are not myopic.

I.3.4 Equilibrium

• For all the electoral systems we consider, we focus on symmetric Markov equilibria (SME) in WSU strategies.

• A SME can be described by a collection of proposal functions r(b;A,h), b(b;A,h), g(b;A,h), s(b;A,h) and a value function v(b;A,h) (in short p(b;A,h)).

1

t iI R

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II.1 The proportional system

• In a proportional system candidate L maximizes the expected share of votes:

• Given v, L chooses a platform to solve:

• Note that: .

,11

(( , ; ) ; , ) ( ; , )2

jnjj j j

Lh RPA L Rp p v

hW p v A W p v A

n

1

( , , , )

( , ; ) ; ( ; , )max

s.t. ( , , ; ) , 0 & [ , ].

j jn j

j

Rj

r g x sj

h u A E v A A W v A

B r g x b s s j x x x

pr g s x

1

max ( , , ; )n

j jj j

j

h s h B r g x b

15

• Given v, L chooses a platform to solve:

• On the other hand, given r,g,x, the value function is:

Definition. A political equilibrium in a proportional system (PS) is a collection of policies p and a value function v such that p solves (A) given v; and v satisfies (B) given p.

( , ,( ; , ) ( ; , ) ( ; , ); )

( ; , ) ( ; , )

( ; , )( ; )

, ;

( ; )r b A h g bB bnA h x b A h

r b A h g b A h

x b Av b A

v A

u AE A

h

(A)

(B)

,

1( ),

( , ; ) ( , , ; )max

s.t. ( , , ; ) 0 &

max

.

;

[ , ]

jn

j

jr g x

jr g r g x xu A B b E A

B r g x

h

x

v Ah

b x x

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Proposition. In a proportional voting rule system, a well-behaved symmetric political equilibrium exists.

An equilibrium is well-behaved if v is continuous and weakly concave in b for any A.

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When is a political equilibrium Pareto efficient?

• Let us define the MCPF as:

i.e. the marginal increase in income that compensates for a

marginal increase in tax revenues.

• Dynamic efficiency requires:

( ; , ), ( ; , )

.( ; ,

( ; )1 1 ( ; , )

1 ( ; , 1) )M

u rr b A h

n r

b A h g b A h

rR r b A h

r

CPb A h

F b A

Opportunity Cost of Resoutces at tExpected Cost of Resource

s

( ; , ) ( ( ; , ), , )MCPF b A h E MCPF x b A h A h A

• In a PS we have a “similar” condition:

• The equilibrium is dynamically efficient only

if, , therefore, generically it is dynamically

inefficient.

• Proposition. In a PS, policies can not be rationalized by

any set of Pareto weights. The MCPF is a submartingale.18

j i

h h i

( ; , ) ( ( ; , ), , )

max 1

( , )

j

j

i

MCPF b A h E MCPF x b A h

hE A h A

nh

A h A

Expected benefit

Expected cost of reducing future pork tranfers.

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III.1 The majoritarian system: The MMS case

• Given platforms pL, pR, L wins in district j with probability:

• In this case the probability that L wins the election is:

( ; , ) 1

, ; , Pr 1/ 22

Pr ( ; , ) ( ;

( ; , )

, ) .

jL jj j

L R j jj jR

jj j jR

j

L j j

W p v Ap p v A h

W p v

W p v A W p v

A

A

( )

1

2

, ; , 1 , ; ,, ; , .

!j

l kn

L R j L R jl a k C amL R a

nj

p p v A p p v Ap p v A

j

P

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• Given v, L chooses a platform to solve:

• On the other hand, given p, the value function is:

Definition. A political equilibrium in a MMS is a collection of policies p and a value function v such that p solves (A) given v; and vi satisfies (B) given p for any i.

(A)

(B)

, ; ,max .

s.t. ( , , ; ) , 0 & [ , ].

mR

j jp

vp A

B r g x b s s j x x

p

x

( ; , ) ( ; , ) ( ; , )

( ; , )( ;

( )

,.)

;

;m m mi

m

mi m

i

r b A h g b A h s b A h

x b

u AE

AAv b

v AhA

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• Let’s assume that there are only 3 districts, L, M and R and

that, with and .

• h is uniform and i.i.d. over

• As σ↑ the probability that L wins (looses) the R (L) district

converge to 0. So both candidate will focus on district M.

Proposition. There is a σ* such that for σ>σ*, a unique well-

behaved MME exists. Policies are chosen to maximize M

district’s utility:

The MCPF is a martingale.

L R R 0M

( , , )

, ; ( , , ; ) ;max

. . ( , , ; ) 0, [ , ]

m

r g x

u r g A B r g x b E v x A A

s t B r g x b x x x

(1, , ), ( ,1, ), ( , ,1)

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• With respect to a PS, there are two differences.

‒ For any b,A, PS induces a higher g. Compare the objective

functions in a MMS and PS:

This point was first made in Persson and Tabellini [1999]

‒ A PS, however, is more dynamically inefficient.

( , , )

, ; ( , , ; ) ;max

. . ( , , ; ) 0,

max

[ , ]

mi

r g x

jh

hu r g A B r g x b E v x A A

s t B r g x b x x x

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Proposition. For σ>σ*, a political equilibrium in a MMS converges to a steady state in which g, r are:

In a PS, g converges to a stationary distribution, non degenerate with support in:

1

1 1 2( ) 0, 1 2 ,

(1 ) 1

2

A ng A rd

pa rn

n

1

1 1(1 ) 1, ( ) A n

m mp ng A r

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• Lets go back to the general case.

• In general a MMS may not be Pareto efficient

• Let

• We have:

1 1 1 0

( , ) ( ( , ), )

( , ) ( ( , ), )

( , );

; ) ( , );

.

(

m mni i

m mn ni j jj j

m

m m

i

MCPF b A E MCPF x

b A x b A A

b A x b A AE

v x

b A A A

b A A Ax

( , ) ( , )m mi i

b A b AW

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Proposition. A MMS is Pareto efficient whenever the

candidates use the fully symmetric strategies. In general, the

inefficiency → to zero as d→0.

• The candidate’s problem must solve:

• depends on b,A only through:

In a symmetric equilibrium: ΔWj=0 for any b,A,

1

( , , , )max

. . 0 for all , ( , , ; ), & [

( , ,

,

; ;

]

)ni

r g x s ii i

ii

mi

s t s i s B r g x

u r g A s E

b x x

xA vb A A

x

, ; ,mL Rp p v A

( ; , ) ( ; , )j j jL j R jd W d W p v A W p v A

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IV. Comparing electoral Systems

• If we fix a state A,b and a v, we have a static model:

• In this case, PS induces higher g, lower r, higher utilitarian welfare:

( , , )

, ; ( , , ; ) ;max

. . ( , , ; ) 0,

1

1

[ ,

2

]r g x

u r g A B r g x b E v x A A

s t B r g x b x x x

11

1 2

MMSMM

PSP Sr

MMr

rSS

r

SP

uMCPF

B

uMCPF

B

11 11

1 2P SS MMAg

p

Ag

p

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Consider now the dynamic case with endogenous b.

Proposition 9. There is a ω* such that for ω< ω* Eg and Er in the invariant distributions are, respectively, higher and lower in a MMS than in a PS.

1(1 ) 1 n

m nr

1 2

(1 ) 1 2

nr

n

0

r

t

r

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V. Empirical Implications

• Our work may contribute an understanding why:

‒ Empirical evidence on electoral rules is mixed;

‒ positive correlation between budget deficits and MMS.

• Two conceptual limitations on the two most common empirical approaches:

‒ Panel datasets: even if the economies are at the steady state, ignoring the dynamic nature of the data process may generate biased findings;

‒ Country studies: cannot fully explain the differences across sovereign countries where public debt may diverge substantially in the long term.

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VI. Conclusion

• We have characterize dynamic model of elections under proportional and majoritarian rules.

• We have shown:

‒ PS tend to accumulate more debt than MS

‒ Though politicians may desire higher g under PS, they can afford less g, even in the steady state.

‒ Previous static models focused on what politician desire.

• Contrary to static models, policies in dynamic electoral models are not Pareto optimal in any sense.