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Legislative Bargaining and the Dynamics of Public Investment

Marco Battaglini

Princeton University

Salvatore Nunnari Caltech

Thomas Palfrey Caltech

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Motivation

•  Most public goods provided by governments are durable.

•  A large literature has studied static public good provision: much less is known about dynamic environments.

–  Do legislatures provide public goods efficiently in a dynamic setting?

–  To what extent this depends on the voting rule adopted?

•  Are the models we use right? What equilibrium concepts should be used?

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•  This work: a first attempt to answer these questions.

•  We consider a legislature of n representatives who allocate resources between private consumption (pork) and a public good that can be accumulated over time.

•  Three alternative voting rules are considered: •  Dictatorship •  Simple Majority •  Unanimity

•  We derive the Symmetric Markov Perfect Equilibrium and test the theoretical predictions in the laboratory.

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•  Highly structured dynamic environment such as the one in this paper pose a challenge for empirical analysis.

•  The control of laboratory experiments allows us to directly test the main comparative static implication of the theory.

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Outline of the Talk

I.  The Model

II.  The Planner’s Solution III.  The Political Equilibrium Under Different Voting Rules IV.  Experimental Design and Implementation V.  Results VI.  Conclusions

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I.  The Model The Economy

•  A continuum of infinitely-lived citizens live in n identical districts.

•  There are two goods - a durable public good g (initial level g0 , depreciation rate d=0), and a private good x.

•  Public good technology: gt=gt-1+It

•  The MRT between x and g is 1. •  Each citizen's per period utility function is

xti +u( gt ). •  The discount factor is δ.

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Public Policies •  In each t, a fixed endowment of W units of private good.

•  An allocation in a given period is n+1-tuple:

( g, x1 , x2 ,… xn )

•  A public policy has to satisfy two feasibility constraints:

•  And a budget constraint:

where y =g+I.

�

xti ≥ 0 ∀i , 1 0 t t tg I g t−= + ≥ ∀

[ ]1

ni

i

x y g W=

+ − ≤∑

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II.  The Planner’s Solution

•  The planner’s problem can be written in the recursive form:

•  Since BC is binding and y >0, we can rewrite:

The planner will invest in g as long as nu′(g)+δv′P(g)>1

�

vP (g) = maxy,xx j∑ + nu(y) + δvP (y)

s.t x j∑ + y − (1− d)g ≤W , x i ≥ 0 ∀i,y ≥ 0⎧ ⎨ ⎩

⎫ ⎬ ⎭

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( ) ( )max

((1 )

1 ) 0.P

jy

nu y v yW d

W d g yx g y

δ+ − + +⎧ ⎫⎨ ⎬= + −

−− ≥⎩ ⎭∑

•  Optimal investment is:

where y*P , is such that:

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y*p

g*p

W 1

y*p

yP (g)

g

Steady state:

45%

�

yP* = u'[ ]−1 1−δ(1− d)

n⎛ ⎝

⎞ ⎠

{ }( ) min ,P PI g W y g∗= −( ) ( ) 1P P Pnu y v yδ

∗ ∗′ ′+ =

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III. The Political Equilibrium •  Public decisions are made by a legislature of representatives

from each of the n districts.

•  One legislator is randomly selected to make a policy proposal (x , y). He cannot propose negative transfers to districts.

•  The affirmative votes of q < n representatives are required to pass legislation.

•  If the proposal is accepted by q legislators, the plan is implemented and the legislature adjourns until next period.

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•  If the proposal is rejected then a status quo policy

y=g, xi=wi = W/n is implemented.

•  We look for a Markov Perfect Equilibrium in stage undominated strategies.

•  Equilibrium strategies: {yL(g), sL(g)}. The proposer receives:

xL(g)=W+g-yL(g)-(q-1)sL(g)

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•  The proposer’s maximization problem is:

•  The first constraint is the incentive compatibility constraint.

•  The set of binding constraints is state dependent and the value function is not typically concave in g.

•  A pure strategy equilibrium exists and can be characterized.

,

( 1) ( ) ( )max ( ) ( ) ( ) ( )

0, 0

LW

L Lny s

W g y q s u y v ys u y v y u g v g

y s

δδ δ

+ − − − + +⎧ ⎫⎪ ⎪+ + ≥ + +⎨ ⎬⎪ ⎪≥ ≥⎩ ⎭

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•  The equilibrium investment function can be represented as a function of two constants y1* and yL*, with yL*>y1*:

•  When g ≤ g1(y1*), the proposer can ignore the other districts.

•  When g > g1(y1*) the proposer cannot ignore others:

-  For g in (g1(y1*), g2(yL*)] he finds optimal to “buy” their approval by increasing g and investing to

-  For g > g2(yL*) he provides pork to a MWC of districts and invests to yL*

)(~ gy

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yL (g) =

y1∗ g ≤ g1(y1

∗)

y(g) g∈ g1(y1

∗),g2 (yL∗ )( ⎤⎦

yL∗ else

⎧

⎨⎪⎪

⎩⎪⎪

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•  The steady state crucially depends on q. –  Dictatorship: q=1 –  Simple Majority: q=(n+1)/2 –  Unanimity: q=n

y*L

g*2

y*1

g*1

y(g)

yL (g)

g

[ ] 1* / (1 )'Ln q dy u

nδ− − −⎛ ⎞= ⎜ ⎟⎝ ⎠

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•  The SS in the Planner’s solution:

•  The SS in the Political Equilibrium:

Observations: •  The equilibrium level of public investment is inefficiently

low for q

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•  Even with unanimity, convergence to the steady state is

inefficiently slow (the proposer can appropriate rents).

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g

[ ] 1* 1 (1 )'Ldy u

nδ− − −⎛ ⎞= ⎜ ⎟⎝ ⎠

y

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•  For any q, efficient investment is a subgame perfect

equilibrium of the legislative bargaining game.

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0

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 10

Unanimity

Simple Maj

Dictatorship

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Round

Theoretical Predictions for Experimental Parameters

SP D SM U

y* 400 1.38 29.83 400

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IV. Experimental design

•  The experiments were all conducted at the Social Science Experimental Laboratory (SSEL) using Caltech students.

•  Six sessions were run, using a total of 90 subjects. No subject participated in more than one session.

•  In all sessions d=0, δ=0.75, , n=5, W=20.

•  Three voting rules: –  Dictatorship: no voting required. –  Simple Majority: q=3. –  Unanimity: q=5.

�

u(g) = 2 g

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•  Discounted payoffs were induced by a random termination rule by rolling a die after each round in front of the room.

•  Each session: 10 matches. Length of a match: 1-13 rounds.

•  Committees stayed the same throughout the rounds of a given match, and subjects were randomly rematched into committees between matches.

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V.  Experimental Results Median Time Paths of g: U vs. SM vs. D

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Median investment proposals: 5 Member Committees – U vs. SM vs. D

Simple Majority

Unanimity

Dictatorship

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In summary:

FINDING 1. Higher q leads to higher public good production.

FINDING 2. All voting rules lead to significantly inefficient public good levels.

FINDING 3. In all voting rules, there is overinvestment relative to the equilibrium in the early rounds, followed by significant disinvestment (D and SM) or no further investment (U) approaching the theoretical predictions. Why do we observe overinvestment? To answer this question we analyze individual voting and proposing behavior.

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V.1 Strategic behavior

V.1.1Proposal behavior: D vs. SM vs. U

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Distribution of pork transfers: D vs. SM vs. U

When pork is proposed, are transfers egalitarian?

Even distribution of pork in U

80% of pork to MWC in SM

75% of pork to proposer in D

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In summary:

FINDING 4. Regardless of the voting rule, most proposals are either (i) invest the entire budget; or (ii) universal private allocations with positive investment. FINDING 5. •  In D, proposer takes 75% of pork for himself and

distributes the rest evenly. •  In SM, proposer favors a MWC of voters (80% of

pork to mwc). •  In U, proposer distributes pork evenly.

V.2 Voting behavior: SM vs. U

Logit Estimates. Dependant variable: Pr{vote=yes}

Net expected utility matters

Cost and benefit matter in equal measure

V.3 Behavioral factors

Logit Estimates. Dependant variable: Pr{vote=yes}

Giving stuff pays off

Little evidence for preference for equality

Mixed evidence on “greed”

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Private allocation offered to previous round’s proposer: D vs. SM vs. U

•  ht-1 is insignificant; it-1 is significant only in U. •  Greedt-1 is significant in D and in SM but not in U.

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In summary: FINDING 6. Voters are forward looking. FINDING 7. The vast majority of proposals pass. FINDING 8. Factors other than expected utility matter. Some evidence history in previous periods matters.

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Conclusions

•  We proposed a simple model of investiment in a durable public good by a legislature and analyzed the political equilibrium under three different voting rules, dictatorship (D), simple majority (SM) and unanimity (U).

•  In laboratory experiments we observe: -  Laboratory trajectories of public good and comparative

static close to theoretical predictions of a MPE.

-  As predicted, U outperforms SM and both U and SM performs much better than D.

-  As predicted by the theory, there are significant inefficiencies under all voting rules.

-  A number of behavioral factors affect players’ choices.