Lobbying as a Guard against Extremism

Proof of Proposition 1

The lobbies’ truthful contribution schedules are

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

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

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

The first-order condition of this maximization problem amounts to

The second-order condition is satisfied and so

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

The first-order condition is given by

The second-order condition is satisfied and therefore

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

The first-order condition amounts to

The second-order condition is satisfied and so

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

Plugging y∗α,π, yAα and yPπ yields

and

Proof of Proposition 2

The threshold rule α∗∈0,γ maximizes the aggregate utility of anti-policy individuals from the policy outcome y∗ given by

Plugging in the expression for y∗α,π from Lemma 2 yields

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

The first derivative (7) is strictly positive for all α∈0,γ only when 0<γ≤12 and γ≤π<1−2γ+γ. In this case, α=γ maximizes eq. (6). The first derivative (7) is never strictly negative for all α∈0,γ. The first derivative (7) equals zero in the following cases:

α=0 when γ=0 and 0≤π≤1,

α=γ when 0<γ≤12 and π=1−2γ+γ,

α=−2+5+2γ−4π+π when 0<γ≤12 and 1−2γ+γ<π≤1, or when 12<γ<1 and γ≤π≤1,

α=−1+3 when γ=1 and π=1.The second derivative of eq. (6) is given by−2α−γγ−1+α4+α−2γ−2απ+π222+α−π3−γ4+2α−2π1+α−πα−γ+π22+α−π3+3γ−1+α4+α−2γ−2απ+π21+α−πα−γ+π22+α−π4.

Evaluating it at the critical points yields that the second-order condition holds for α=γ, α=−2+5+2γ−4π+π and α=−1+3. Therefore, the argmax of eq. (6) is as follows. For γ=0, α=0 for all π∈γ,1. For γ∈0,12,

For γ∈12,1, α=−2+5+2γ−4π+π for all π∈γ,1. For γ=1, α=−1+3 for π=1.

The threshold rule π∗∈γ,1 maximizes the aggregate utility of pro-policy individuals from the policy outcome y∗ given by

Plugging in the expression for y∗α,π from Lemma 2 yields

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

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

– π=2−3 when γ=0 and α=0,

– π=2+α−3+4α−2γ when 0<γ≤12 and 0≤α≤γ, or when 12<γ<1 and 0≤α≤γ−−1+2γ,

– π=1 when γ=1 and 0≤α≤1.

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

Evaluating it at the critical points yields that the second-order condition holds for π=2−3 and π=2+α−3+4α−2γ. Thus, the argmax of eq. (8) is as follows. For γ=0, π=2−3 for α=0. For γ∈0,12, π=2+α−3+4α−2γ for all α∈0,γ. For γ∈12,1,

For γ=1, π=1 for all α∈0,γ.

I find next the equilibrium threshold rules α∗∈0,γ and π∗∈γ,1 given the government’s preferred policy γ∈0,1. Given the argmax of eq. (6) and argmax of eq. (8), I find that

for γ=0, α∗=0 and π∗=2−3,

for γ∈0,12, α∗=γ and π∗=2+γ−3+2γ,

for γ∈12,1, α∗=−2+γ+5−2γ and π∗=γ,

for γ=1, α∗=−1+3 and π∗=1.

This can be summarized as follows. For γ∈0,12, α∗=γ and π∗=2+γ−3+2γ. For γ∈12,1, α∗=−2+γ+5−2γ and π∗=γ. Plugging α∗ and π∗ into y∗α,π, CA∗α,π and CP∗α,π in Lemma 2 yields the equilibrium policy level y∗ and the equilibrium contributions CA∗ and CP∗. In particular, for γ∈0,12, those are equal to

For γ∈12,1, they amount to

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

The equilibrium policy level is equal to

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

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

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

The equilibrium policy level is equal to

The aggregate utility of pro-policy individuals from the policy outcome y∗α∗,π∗,π‾ equals

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

which is an increasing function of π‾∈π∗,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

These derivatives are such that dα∗dγ>0, dπ∗dγ>0, dy∗dγ>0, dCA∗dγ>0 and dCP∗dγ<0.

When the government is biased in favor of low policy levels, i. e., γ∈0,12, then the lobby sizes are equal to

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:

Lobby P contributes more than lobby A:

When the government is biased in favor of high policy levels, γ∈12,1, then the lobby sizes are equal to

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:

Lobby A contributes more than lobby P:

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

It is non-negative, increases in γ for γ∈0,12 and decreases in γ for γ∈12,1.