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Lobbying as a Guard against Extremism

Galina Zudenkova

Abstract

This paper analyzes endogenous lobbying over a unidimensional policy issue. Individuals differ in policy preferences and decide either to take part in lobbying activities or not. They are assumed to be group rule-utilitarian such that they follow the rule that, if followed by everyone else in their special interest group, would maximize their group’s aggregate welfare. Once formed, lobbies make contributions to the incumbent government in exchange for a policy favor as in a common-agency model. I show the existence of equilibrium with two organized lobbies. Individuals with more extreme preferences are more likely to join lobbying activities. Therefore, the lobbyists are rather extremists than moderates. However, the competition between those extreme lobbies results in a more moderate policy outcome relative to that initially preferred by the biased government. Lobbies therefore guard against extremism, while acting as moderators of the government’s preferences.

JEL Classification: D72

Acknowledgments

The author is grateful to Daron Acemoglu, Antonio Cabrales, Filipe Campante, Luis Corchón, Jacques Crémer, M. A. de Frutos, Raquel Fernández, Rebecca Morton, Ignacio Ortuño Ortn, Albert Solé-Ollé, Allard Van Der Made, Bengt-Arne Wickström, two anonymous referees and Johann Brunner, the editor, for helpful comments, suggestions and encouragement; and to Cung Truong Hoang for research assistance. The grant from Karin-Islinger-Stiftung foundation is gratefully acknowledged. The usual disclaimer applies.

Appendix

Proof of Proposition 1

The lobbies’ truthful contribution schedules are

(4)CPy=maxUPybP,0,CAy=maxUAybA,0.

The constants bP and bA in the lobbies’ truthful contribution schedules satisfy

(5)γyA2+CAyA=γy2+CPy+CAy,γyP2+CPyP=γy2+CPy+CAy.

Plugging eq. (4) into eq. (5) yields

UPybP=γyA2+γy2+UAyAUAy,
UAybA=γyP2+γy2+UPyPUPy,

where the first line is lobby P’s equilibrium contribution CP, and the second line is lobby A’s equilibrium contribution CA. ■

Proof of Lemma 2

The equilibrium policy level is equal to

y=argmaxy0,1γy2+UAy+UPy=argmaxy0,1γy2αy2+α2y13α31πy2+1π2y131π3.

The first-order condition of this maximization problem amounts to

1+α2π2+2γ22+απy=0.

The second-order condition is satisfied and so

yα,π=1+α2π2+2γ22+απ.

If lobby P did not contribute to the government, the following policy would emerge:

yA=argmaxy0,1γy2+UAy=argmaxy0,1γy2αy2+α2y13α3.

The first-order condition is given by

α2+2γ21+αy=0.

The second-order condition is satisfied and therefore

yAα=α2+2γ21+α.

Likewise, if lobby A did not contribute then the government would choose the following policy:

yP=argmaxy0,1γy2+UPy=argmaxy0,1γy21πy2+1π2y131π3.

The first-order condition amounts to

1π2+2γ22πy=0.

The second-order condition is satisfied and so

yPπ=1π2+2γ22π.

According to Proposition 1, the lobbies’ equilibrium contributions are equal to

CACAy=γyP2+γy2+UPyPUPy,
CPCPy=γyA2+γy2+UAyAUAy.

Plugging yα,π, yAα and yPπ yields

CA=γ1π2+2γ22π2+γ1+α2π2+2γ22+απ2+1π1π2+2γ22π2+1π21π2+2γ22π131π31π1+α2π2+2γ22+απ2+1π21+α2π2+2γ22+απ131π3=α212α+παπ2+2γ242π2+απ2,

and

CP=γα2+2γ21+α2+γ1+α2π2+2γ22+απ2+αα2+2γ21+α2+α2α2+2γ21+α13α3α1+α2π2+2γ22+απ2+α21+α2π2+2γ22+απ13α3=1π21+α+π+παα22γ241+α2+απ2

Proof of Proposition 2

The threshold rule α0,γ maximizes the aggregate utility of anti-policy individuals from the policy outcome y given by

0γuxydx=γy2+γ2y13γ3.

Plugging in the expression for yα,π from Lemma 2 yields

(6)γ1+α2π2+2γ22+απ2+γ21+α2π2+2γ22+απ13γ3.

The first derivative of eq. (6) with respect to α is equal to

(7)γ1α4+α+2γ+2αππ21+απαγ+π22+απ3.

The first derivative (7) is strictly positive for all α0,γ only when 0<γ12 and γπ<12γ+γ. In this case, α=γ maximizes eq. (6). The first derivative (7) is never strictly negative for all α0,γ. The first derivative (7) equals zero in the following cases:

α=0 when γ=0 and 0π1,

α=γ when 0<γ12 and π=12γ+γ,

α=2+5+2γ4π+π when 0<γ12 and 12γ+γ<π1, or when 12<γ<1 and γπ1,

α=1+3 when γ=1 and π=1.The second derivative of eq. (6) is given by2αγγ1+α4+α2γ2απ+π222+απ3γ4+2α2π1+απαγ+π22+απ3+3γ1+α4+α2γ2απ+π21+απαγ+π22+απ4.

Evaluating it at the critical points yields that the second-order condition holds for α=γ, α=2+5+2γ4π+π and α=1+3. Therefore, the argmax of eq. (6) is as follows. For γ=0, α=0 for all πγ,1. For γ0,12,

α={γifπγ,12γ+γ,2+5+2γ4π+πifπ12γ+γ,1.

For γ12,1, α=2+5+2γ4π+π for all πγ,1. For γ=1, α=1+3 for π=1.

The threshold rule πγ,1 maximizes the aggregate utility of pro-policy individuals from the policy outcome y given by

γ1uxydx=1γy2+1γ2y131γ3.

Plugging in the expression for yα,π from Lemma 2 yields

(8)1γ1+α2π2+2γ22+απ2+1γ21+α2π2+2γ22+απ131γ3.

The first derivative of eq. (8) with respect to π is equal to

(9)1γ1+2γ+απ24π1α2+α1+γ1+γππ22+απ3.

The first derivative (9) is never strictly positive for all πγ,1. The first derivative (9) is strictly negative for all πγ,1 only when 12<γ<1 and γ1+2γ<αγ. In this case, π=γ maximizes eq. (8). The first derivative (9) equals zero in the following cases:

π=23 when γ=0 and α=0,

π=2+α3+4α2γ when 0<γ12 and 0αγ, or when 12<γ<1 and 0αγ1+2γ,

π=1 when γ=1 and 0α1.

The second derivative of eq. (8) is equal to

1γ1+2γ+απ24π1+γ2π22+απ3+31γ1+2γ+απ24π1α2+α1+γ1+γππ22+απ41γ4+2απ1α2+α1+γ1+γππ22+απ3.

Evaluating it at the critical points yields that the second-order condition holds for π=23 and π=2+α3+4α2γ. Thus, the argmax of eq. (8) is as follows. For γ=0, π=23 for α=0. For γ0,12, π=2+α3+4α2γ for all α0,γ. For γ12,1,

π={2+α3+4α2γifα0,γ1+2γ,γifαγ1+2γ,γ.

For γ=1, π=1 for all α0,γ.

I find next the equilibrium threshold rules α0,γ and πγ,1 given the government’s preferred policy γ0,1. Given the argmax of eq. (6) and argmax of eq. (8), I find that

for γ=0, α=0 and π=23,

for γ0,12, α=γ and π=2+γ3+2γ,

for γ12,1, α=2+γ+52γ and π=γ,

for γ=1, α=1+3 and π=1.

This can be summarized as follows. For γ0,12, α=γ and π=2+γ3+2γ. For γ12,1, α=2+γ+52γ and π=γ. Plugging α and π into yα,π, CAα,π and CPα,π in Lemma 2 yields the equilibrium policy level y and the equilibrium contributions CA and CP. In particular, for γ0,12, those are equal to

y=2+γ3+2γ,
CA=γ243+2γ6+γ3+2γ4243+2γ3+2γγ,
CP=423+2γ+γ4+γ23+2γ241+γ.

For γ12,1, they amount to

y=2+γ+52γ,
CA=9452γ+γ252γ+γ6242γ,
CP=1γ2452052γ+γ452γ+γ18452γ1+γ.

Rules Characterized by Two Thresholds

I show first that given the rules α_,α and π,1, the anti-policy group’s aggregate utility from the policy outcome is maximized at α_=0. Consider the following rules: α_,α and π,1. The lobbies’ gross objective functions are

UAy=α_αuxydx=αy2+α2y13α3+α_y2α_2y+13α_3,
UPy=π1uxydx=1πy2+1π2y131π3.

The equilibrium policy level is equal to

yα_,α,π=argmaxy0,1(γy2αy2+α2y13α3+α_y2α_2y+13α_3
1πy2+1π2y131π3)=1+α2α_2π2+2γ22+αα_π.

The aggregate utility of anti-policy individuals from the policy outcome yα_,α,π is equal to

0γuxydx=γ1+α2α_2π2+2γ22+αα_π2+γ21+α2α_2π2+2γ22+αα_π13γ3.

Plugging in the expressions for α and π from Proposition 2 yields

0γuxydx=
{γ1+γ2α_22+γ3+2γ2+2γ23+2γα_2+γ21+γ2α_22+γ3+2γ2+2γ23+2γα_13γ3forγ0,12,γ1+2+γ+52γ2α_2γ2+2γ252γα_2+γ21+2+γ+52γ2α_2γ2+2γ252γα_13γ3forγ12,1,

which is a decreasing function of α_0,α. It follows that the anti-policy group’s aggregate utility from the policy outcome is maximized at α_=0.

I show next that given the rules 0,α and π,π, the pro-policy group’s aggregate utility from the policy outcome is maximized at π=1. Given the rules 0,α and π,π, the lobbies’ gross objective functions are

UAy=0αuxydx=αy2+α2y13α3,
UPy=ππuxydx=πy2+π2y13π3+πy2π2y+13π3.

The equilibrium policy level is equal to

yα,π,π=argmaxy0,1(γy2αy2+α2y13α3
πy2+π2y13π3+πy2π2y+13π3)=α2+2γπ2+π221+απ+π.

The aggregate utility of pro-policy individuals from the policy outcome yα,π,π equals

γ1uxydx=1γα2+2γπ2+π221+απ+π2+1γ2α2+2γπ2+π221+απ+π131γ3.

Plugging in the expressions for α and π from Proposition 2 yields

γ1ux(y)dx={(1γ)(γ2+2γ(2+γ3+2γ)2+π¯22(3+2γ+π¯1))2+(1γ2)γ2+2γ(2+γ3+2γ)2+π¯22(3+2γ+π¯1)13(1γ3)forγ[0,12],(1γ)((2+γ+52γ)2+2γγ2+π¯22(52γ+π¯1))2+(1γ2)(2+γ+52γ)2+2γγ2+π¯22(52γ+π¯1)13(1γ3)forγ(12,1],

which is an increasing function of ππ,1. It implies that the pro-policy group’s aggregate utility from the policy outcome is maximized at π=1.

Proof of Corollary 1

The first derivatives of α, π, y, CA and CP with respect to γ are equal to

dαdγ={1forγ0,12,1152γforγ12,1,dπdγ={113+2γforγ0,12,1forγ12,1,
dydγ={113+2γforγ0,12,1152γforγ12,1,
dCAdγ={γ168+γ276+γ164+27γ3+2γ96+γ16+γ8+3γ43+2γγ3+2γ2forγ0,12,43γ2γ2135γ+52γ279+γ424γ226+3γγ16452γ2γ2forγ12,1,
dCPdγ={2+γ221+γ3+2γ21+γ5+4γ+2+γ3+2γ2+3γ41+γ23+2γforγ0,12,1γ52γ283+γγ653γ249635+γ685+γ27γ245452γ52γ1+γ2forγ12,1.

These derivatives are such that dαdγ>0, dπdγ>0, dydγ>0, dCAdγ>0 and dCPdγ<0.

When the government is biased in favor of low policy levels, i. e., γ0,12, then the lobby sizes are equal to

A=γ,P=3+2γ1γ.

It follows that P>A, i. e., the pro-policy lobby P is more numerous than the anti-policy lobby A. The equilibrium policy level is higher than that preferred by the government:

2+γ3+2γ>γforγ0,12.

Lobby P contributes more than lobby A:

CPCA=423+2γ+γ4+γ23+2γ241+γγ243+2γ6+γ3+2γ4243+2γ3+2γγ>0forγ0,12.

When the government is biased in favor of high policy levels, γ12,1, then the lobby sizes are equal to

A=2+γ+52γ,P=1γ.

It follows that A>P, i. e., the anti-policy lobby A is more numerous than the pro-policy lobby P. The equilibrium policy level is lower than that preferred by the government:

2+γ+52γ<γforγ12,1.

Lobby A contributes more than lobby P:

CACP=9452γ+γ252γ+γ6242γ1γ2452052γ+γ452γ+γ18452γ1+γ>0forγ12,1.

When the government in unbiased, γ=12, the lobby sizes are equal to A=P=12, i. e., the lobbies are of the same size. They contribute the same amount to the government: CA=CP=196. The equilibrium policy level is equal to that preferred by the government: y=12.

Government Utility in Equilibrium

The government utility in equilibrium is equal to

Uγy=γy2+CA+CP=
{γ3+2γ28+γ44+γ16+γ48γ92+γ52+7γ41+γ3+2γγforγ0,12,1γ52γ89γ79+γγ19199+γ217+γ7γ7342γ52γ1+γforγ12,1.

It is non-negative, increases in γ for γ0,12 and decreases in γ for γ12,1.

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Published Online: 2016-12-28

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