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1
Dynamic Electoral Competition and
Constitutional Design
Marco Battaglini Princeton University,
CEPR and NBER
2
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.
3
• 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.
4
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):
7
( , , ; ) ( ) (1 ) iiR r xB r g x b pg b s
x x
8
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
9
( ; ; ) ( ; , )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
11
• 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.
12
( ; , ) ( ; , )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
14
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
16
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.
17
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.
19
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
20
• 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
21
• 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)
22
• 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
23
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
24
• 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
25
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
26
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
27
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
28
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.
29
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.