Endogenous market regulation in a signaling model of lobby formation

Abstract

This paper aims at explaining industry protection in a context in which the government cannot observe the state of market demand. We develop an asymmetric information model and use the tools of contract theory in order to understand (1) how the level of industry protection is endogenously determined, and (2) why some industries decide to engage in large lobbying costs to become politically active. Our model offers plausible explanations to phenomena such as the “loser’s paradox”, where weak industries receive the most protection although strong industries are the ones that spend more resources on lobbying activities. The model also allows for an analysis of the influence that lobbying costs have on the decision to organize actively as a lobby.

Appendix

Proof of Proposition 3.1

We rewrite the condition $\pi ^{\prime }(X,\overline{\theta })>\pi ^{\prime }(X,\underline{\theta })$ for all $X\$ as

$$\begin{aligned} u^{\prime }\left( \frac{X}{n},\overline{\theta }\right) -u^{\prime }\left( \frac{X}{n}, \underline{\theta }\right) >\left[ u^{\prime \prime }\left( \frac{X}{n},\underline{\theta }\right) -u^{\prime \prime }\left( \frac{X}{n},\overline{\theta }\right) \right] \frac{X}{n}. \end{aligned}$$

The left hand side of the above inequality is positive. We now multiply the right hand term into brackets by $\frac{\gamma }{1+\gamma }$, which is in between zero and one, to obtain:

$$\begin{aligned} u^{\prime }\left( \frac{X}{n},\overline{\theta }\right) -u^{\prime }\left( \frac{X}{n}, \underline{\theta }\right) >\frac{\gamma }{1+\gamma }\left[ u^{\prime \prime }\left( \frac{X}{n},\underline{\theta }\right) -u^{\prime \prime }\left( \frac{X}{n},\overline{ \theta }\right) \right] \frac{X}{n}. \end{aligned}$$

We add and subtract c to the left hand side of the above expression, multiply both sides of the inequality by $1+\gamma >0$ and reorder the expression to obtain:

$$\begin{aligned}&\gamma u^{\prime \prime }\left( \frac{X}{n},\overline{\theta }\right) \frac{X}{n}+\left( 1+\gamma \right) \left[ u^{\prime }\left( \frac{X}{n},\overline{\theta }\right) -c\right] \\&\quad >\gamma u^{\prime \prime }\left( \frac{X}{n},\underline{\theta }\right) \frac{X}{n} +\left( 1+\gamma \right) \left[ u^{\prime }\left( \frac{X}{n},\underline{\theta } \right) -c\right] . \end{aligned}$$

Provided that $\pi ^{\prime }(X,\theta )=u^{\prime }(\frac{X}{n},\theta )+u^{\prime \prime }(\frac{X}{n},\theta )\frac{X}{n}-c$ and $s^{\prime }(X,\theta )=-u^{\prime \prime }(\frac{X}{n},\theta )\frac{X}{ n^{2}}$ for all $\theta$, we write the above inequality as:

$$\begin{aligned} ns^{\prime }(X,\overline{\theta })+\left( 1+\gamma \right) \pi ^{\prime }(X, \overline{\theta })>ns^{\prime }(X,\underline{\theta })+\left( 1+\gamma \right) \pi ^{\prime }(X,\underline{\theta }). \end{aligned}$$

That is, we have $w^{\prime }(X,\overline{\theta })+\gamma \pi ^{\prime }(X, \overline{\theta })>w^{\prime }(X,\underline{\theta })+\gamma \pi ^{\prime }(X,\underline{\theta })\$for all X. The condition that characterizes a maximum is $w^{\prime }(X,\theta )+\gamma \pi ^{\prime }(X,\theta )=0$ for $\theta =\overline{\theta },\underline{\theta }$, and function $w(X,\theta )+\gamma \pi (X,\theta )$ is strictly concave. Therefore $X_{r}^{*}( \overline{\theta })>X_{r}^{*}(\underline{\theta })$, as we wanted to prove.

Proof of Proposition 3.2

Constraints (c) and (b) imply constraint (a) since:

$$\begin{aligned} \pi (\overline{X}_{r},\overline{\theta })-\overline{Z}_{r}\ge \pi ( \underline{X}_{r},\overline{\theta })-\underline{Z}_{r}>\pi (\underline{X} _{r},\underline{\theta })-\underline{Z}_{r}\ge 0. \end{aligned}$$

Therefore, the participation constraint for the high type industry is redundant.

We call $\lambda$, $\mu$ and $\delta$ the Lagrange multipliers associated to inequalities (b), (c) and (d) respectively. The first order conditions of the maximization program are the following:

$$\begin{aligned} \dfrac{\partial L}{\partial \overline{X}_{r}}= & {} qw^{\prime }(\overline{X} _{r},\overline{\theta })+\mu \pi ^{\prime }(\overline{X}_{r},\overline{\theta })-\delta \pi ^{\prime }(\overline{X}_{r},\underline{\theta })=0, \end{aligned}$$

(7.1)

$$\begin{aligned} \dfrac{\partial L}{\partial \underline{X}_{r}}= & {} (1-q)w^{\prime }( \underline{X}_{r},\underline{\theta })+\left( \lambda +\delta \right) \pi ^{\prime }(\underline{X}_{r},\underline{\theta })-\mu \pi ^{\prime }( \underline{X}_{r},\overline{\theta })=0, \end{aligned}$$

(7.2)

$$\begin{aligned} \dfrac{\partial L}{\partial \overline{Z}_{r}}= & {} q\gamma -\mu +\delta =0, \end{aligned}$$

(7.3)

$$\begin{aligned} \dfrac{\partial L}{\partial \underline{Z}_{r}}= & {} (1-q)\gamma -\left( \lambda +\delta \right) +\mu =0. \end{aligned}$$

(7.4)

From (7.3), we obtain $\delta =\mu -q\gamma$. Substituting $\delta$ into (7.4) we find $\lambda =\gamma >0$, implying that the acceptance constraint for the low type holds with equality. The inequalities in (c) and (d) and jointly with the assumption $\pi (X, \overline{\theta })>\pi (X,\underline{\theta })$ for all X, allow us to write the following chain of inequalities:

$$\begin{aligned} \pi (\overline{X}_{r},\overline{\theta })-\overline{Z}_{r}\ge \pi ( \underline{X}_{r},\overline{\theta })-\underline{Z}_{r}>\pi (\underline{X} _{r},\underline{\theta })-\underline{Z}_{r}\ge \pi (\overline{X}_{r}, \underline{\theta })-\overline{Z}_{r}, \end{aligned}$$

(7.5)

from which we derive:

$$\begin{aligned} \pi (\overline{X}_{r},\overline{\theta })-\pi (\overline{X}_{r},\underline{ \theta })\ge \pi (\underline{X}_{r},\overline{\theta })-\pi (\underline{X} _{r},\underline{\theta }). \end{aligned}$$

(7.6)

Since function $\pi (X,\overline{\theta })-\pi (X,\underline{\theta })$ is strictly increasing in X, the inequality in (7.6) implies that $\overline{X }_{r}\ge \underline{X}_{r}$. Now observe that $\mu =0$ would imply $\delta <0$, by Eq. (7.3). Then, it holds that $\mu >0$, meaning that expression (7.5) holds with equality. If the third inequality were also an equality (i.e. if $\delta >0$), from Eq. (7.6) we would obtain $\overline{X}_{r}= \underline{X}_{r}$. Therefore, $\delta >0$ if and only if $\overline{X}_{r}= \underline{X}_{r}$.

In order to prove that $\overline{X}_{r}>\underline{X}_{r}$, we just need to show that $\overline{X}_{r}\ne \underline{X}_{r}$ provided that we already know that $\overline{X}_{r}\ge \underline{X}_{r}$. We proceed by contradiction. Suppose that $\overline{X}_{r}=\underline{X}_{r}$, i.e., that $\delta >0$. Then, Eq. (d) holds with equality. Since $\overline{X}_{r}= \underline{X}_{r}$, we must have $\overline{Z}_{r}=\underline{Z}_{r}$. If $\overline{X}_{r}=\underline{X}_{r}=X$ and $\overline{Z}_{r}=\underline{Z} _{r}=Z$, Eqs. (7.1) and (7.2) remain respectively as:

$$\begin{aligned} qw^{\prime }(X,\overline{\theta })+\mu \pi ^{\prime }(X,\overline{\theta } )-\delta \pi ^{\prime }(X,\underline{\theta })= & {} 0, \\ (1-q)w^{\prime }(X,\underline{\theta })+\left( \lambda +\delta \right) \pi ^{\prime }(X,\underline{\theta })-\mu \pi ^{\prime }(X,\overline{\theta })= & {} 0. \end{aligned}$$

We plug $\mu =\delta +q\gamma$ from Eq. (7.3) into the first equation above, and $\lambda +\delta =\mu +\gamma (1-q)$ from Eq. (7.4) into the second, and solve both equations for $\delta$ and $\mu$ respectively. We obtain

$$\begin{aligned} \delta =\frac{-q\left[ w^{\prime }(X,\overline{\theta })+\gamma \pi ^{\prime }(X,\overline{\theta })\right] }{\pi ^{\prime }(X,\overline{\theta })-\pi ^{\prime }(X,\underline{\theta })}, \end{aligned}$$

(7.7)

and

$$\begin{aligned} \mu =\frac{(1-q)\left[ w^{\prime }(X,\underline{\theta })+\gamma \pi ^{\prime }(X,\underline{\theta })\right] }{\pi ^{\prime }(X,\overline{\theta })-\pi ^{\prime }(X,\underline{\theta })}. \end{aligned}$$

(7.8)

We know that $\delta >0$ by hypothesis and $\mu >0$ by Eq.(7.3). We also know that $\pi ^{\prime }(X,\overline{\theta })-\pi ^{\prime }(X,\underline{ \theta })>0$. Therefore, Eqs. (7.7) and (7.8) imply respectively that $w^{\prime }(X,\overline{\theta })+\gamma \pi ^{\prime }(X,\overline{\theta } )<0$ and $w^{\prime }(X,\underline{\theta })+\gamma \pi ^{\prime }(X, \underline{\theta })>0$, which cannot be true under our assumptions. Then, $\delta =0$.

As $\delta =0$, the incentive constraint for the industry of type $\underline{\theta }$ holds with strict inequality. Next we show that the menu of contracts $\left\{ (\overline{X}_{r},\overline{Z}_{r}),(\underline{X}_{r},\underline{Z}_{r})\right\}$ that solve the maximization program is characterized by the following equations:

$$\begin{aligned} \pi (\underline{X}_{r},\underline{\theta })-\underline{Z}_{r}= & {} 0, \end{aligned}$$

(7.9)

$$\begin{aligned} \pi (\overline{X}_{r},\overline{\theta })-\overline{Z}_{r}= & {} \pi ( \underline{X}_{r},\overline{\theta })-\underline{Z}_{r}, \end{aligned}$$

(7.10)

$$\begin{aligned} w^{\prime }(\overline{X}_{r},\overline{\theta })+\gamma \pi ^{\prime }( \overline{X}_{r},\overline{\theta })= & {} 0, \end{aligned}$$

(7.11)

$$\begin{aligned} q\gamma \left[ \pi ^{\prime }(\underline{X}_{r},\overline{\theta })-\pi ^{\prime }(\underline{X}_{r},\underline{\theta })\right]= & {} (1-q)\left[ w^{\prime }(\underline{X}_{r},\underline{\theta })+\gamma \pi ^{\prime }( \underline{X}_{r},\underline{\theta })\right] \end{aligned}$$

(7.12)

Eqs. (7.9) and (7.10) follow from $\lambda >0$ and $\mu >0$ respectively. Eq. (7.11) follows from Eq. (7.1) and $\delta =0$. Finally, Eq. (7.12) follows from substituting $\mu =\delta +q\gamma$ into Eq. (7.2) and taking into account that $\delta =0$.

Notice that Eq. (7.11) is the equilibrium condition in the symmetric information case. Then $\overline{X}_{r}=X_{r}^{*}(\overline{\theta })$. From Eqs. (7.9) and (7.10), and from the condition $\pi (X,\overline{\theta } )>\pi (X,\underline{\theta })$ for all X, we conclude that the industry of type $\overline{\theta }$ earns informational rents, since

$$\begin{aligned} \pi (\overline{X}_{r},\overline{\theta })-\overline{Z}_{r}=\pi (\underline{X} _{r},\overline{\theta })-\underline{Z}_{r}>\pi (\underline{X}_{r},\underline{ \theta })-\underline{Z}_{r}=0. \end{aligned}$$

Given that the production quota $\overline{X}_{r}$ coincides with the symmetric information quota, $X_{r}^{*}(\overline{\theta })$, the industry of type $\overline{\theta }$ obtains informational rents if and only if $\overline{Z}_{r}<Z_{r}^{*}(\underline{\theta })$. From Eq. (7.12) we have that $(1-q)\left[ w^{\prime }(\underline{X}_{r},\underline{ \theta })+\gamma \pi ^{\prime }(\underline{X}_{r},\underline{\theta })\right] >0$. Provided that function $w(X,\underline{\theta })+\gamma \pi (X, \underline{\theta })$ is strictly concave and reaches its maximum at quota $X_{r}^{*}(\underline{\theta })$, the condition $w^{\prime }(\underline{X} _{r},\underline{\theta })+\gamma \pi ^{\prime }(\underline{X}_{r},\underline{ \theta })>0$ implies $\underline{X}_{r}<X_{r}^{*}(\underline{\theta })$.

Now we prove that $\underline{Z}_{r}>Z_{r}^{*}(\underline{\theta })$. Observe that $\underline{Z}_{r}=\pi (\underline{X}_{r},\underline{\theta })$ and $Z_{r}^{*}(\underline{\theta })=\pi (X_{r}^{*}(\underline{\theta }),\underline{\theta })$. Function $\pi (X,\underline{\theta })$ is concave and reaches a maximum at $X_{M}$ (the monopoly output). Provided that $\underline{X}_{r}<X_{r}^{*}(\underline{\theta })$, we just need to prove that $\underline{X}_{r}>X_{M}$ in order to obtain $\pi (\underline{X}_{r}, \underline{\theta })>\pi (X_{r}^{*}(\underline{\theta }),\underline{ \theta })$. Suppose that $\underline{X}_{r}\le X_{M}$. We show that in this case the regulator is not maximizing its utility. Consider $\underline{X} _{r}=X_{M}-z$, with $z\ge 0$. The value for the transfer is $\pi (X_{M}-z, \underline{\theta })$. Since function $\pi (X,\underline{\theta })$ has an inverted U shape, there exists $h\ge 0$ such that $\pi (X_{M}+h,\underline{\theta })=\pi (X_{M}-z,\underline{\theta })$. Observe that with quota $X_{M}+h$ the regulator receives the same transfer, but the consumer surplus is higher, since $s(X,\underline{\theta })$ is strictly increasing in X and $X_{M}+h\ge X_{M}-z$. Therefore, any quota lower than the monopoly output cannot be part of the regulator’s proposal. We conclude that $\underline{Z}_{r}>Z_{r}^{*}(\underline{\theta })$

Proof of Proposition 4.1

The optimal quota proposed by a lobby is $X_{l}^{*}(\theta )$ such that $w^{\prime }(X_{l}^{*}(\theta ), \overline{\theta })+\gamma \pi ^{\prime }(X_{l}^{*}(\theta ),\overline{ \theta })=0$. Then, $X_{l}^{*}(\theta )=X_{r}^{*}(\theta )$. On the other hand, the policy-maker’s utility is higher under quota $X_{r}^{*}(\theta )$ than when the market is liberalized. Namely, $w(X_{l}^{*}(\theta ),\theta )+\gamma Z_{r}^{*}(\theta )>w^{*}(\theta )$. From here, we obtain

$$\begin{aligned} Z_{l}^{*}(\theta )=\frac{1}{\gamma }\left[ w^{*}(\theta )-w(X_{l}^{*}(\theta ),\theta )\right] <Z_{r}^{*}(\theta ). \end{aligned}$$

Proof of Proposition 4.2

We prove that the set of policy pairs $\left\{ (\overline{X}_{l},\overline{Z}_{l}),(\underline{X}_{l},\underline{Z}_{l})\right\}$ defined below together with beliefs $\mu (X_{l},Z_{l})=1$ for $\left( X_{l},Z_{l}\right) =\left( \underline{X}_{l},\underline{Z}_{l}\right)$ and $\mu (X_{l},Z_{l})=0$ for all $\left( X_{l},Z_{l}\right) \ne \left( \underline{X}_{l},\underline{Z} _{l}\right)$ constitute a sequential separating equilibrium of our signaling game that also satisfies the Intuitive Criterion (Cho and Kreps 1987). The pair $(\overline{X}_{l},\overline{Z}_{l})$ is:

$$\begin{aligned} (\overline{X}_{l},\overline{Z}_{l})=\left( X_{l}^{*}(\overline{\theta } ),Z_{l}^{*}(\overline{\theta })\right) , \end{aligned}$$

and the pair $(\underline{X}_{l},\underline{Z}_{l})$ is the solution to the following equations system:

$$\begin{aligned} \begin{array}{l} \underline{Z}_{l}=\frac{1}{\gamma }\left[ w^{*}(\underline{\theta })-w( \underline{X}_{l},\underline{\theta })\right] , \\ \pi (\underline{X}_{l},\overline{\theta })-\underline{Z}_{l}=\pi (\overline{X }_{l},\overline{\theta })-\overline{Z}_{l}. \end{array} \end{aligned}$$

First we show that any pair $(\overline{X}_{l},\overline{Z}_{l})$ different from $\left( X_{l}^{*}(\overline{\theta }),Z_{l}^{*}(\overline{ \theta })\right)$ cannot be part of a separating equilibrium. We proceed by contradiction. Suppose that $(\overline{X}_{l},\overline{Z}_{l})\ne \left( X_{l}^{*}(\overline{\theta }),Z_{l}^{*}(\overline{\theta })\right)$ and that $(\overline{X}_{l},\overline{Z}_{l})$ belongs to a separating equilibrium. Then, by definition, $\mu (\overline{X}_{l},\overline{Z}_{l})=0$ , i.e., the policy-maker is sure that the lobby is of type $\overline{\theta }$ if the proposal is $(\overline{X}_{l},\overline{Z}_{l})$. This proposal is always accepted whenever $w(\overline{X}_{l},\overline{\theta })+\gamma \overline{Z}_{l}\ge w^{*}(\overline{\theta })$. However, if the policy pair $\left( X_{l}^{*}(\overline{\theta }),Z_{l}^{*}(\overline{ \theta })\right)$ were also accepted, provided that this pair maximizes the lobby’s payoff, then $(\overline{X}_{l},\overline{Z}_{l})$ could not belong to a separating equilibrium. Therefore, we just need to show that the policy $\left( X_{l}^{*}(\overline{\theta }),Z_{l}^{*}(\overline{\theta } )\right)$ is always accepted, for any belief $\mu ^{*}=\mu (X_{l}^{*}(\overline{\theta }),Z_{l}^{*}(\overline{\theta }))$ that this policy may induce on the policy-maker. Namely, it must hold that:

$$\begin{aligned}&\mu ^{*}\left[ w(X_{l}^{*}(\overline{\theta }),\underline{\theta } )+\gamma Z_{l}^{*}(\overline{\theta })\right] +\left( 1-\mu ^{*}\right) \left[ w(X_{l}^{*}(\overline{\theta }),\overline{\theta } )+\gamma Z_{l}^{*}(\overline{\theta })\right] \\&\quad \ge \mu ^{*}w^{*}(\underline{\theta })+\left( 1-\mu ^{*}\right) w^{*}(\overline{\theta }). \end{aligned}$$

Namely, the expected utility for the policy-maker in the pair $\left( X_{l}^{*}(\overline{\theta }),Z_{l}^{*}(\overline{\theta })\right)$ is higher than the expected utility of rejecting the lobby’s proposal. Taking into account that the symmetric information pair $\left( X_{l}^{*}(\overline{\theta }),Z_{l}^{*}(\overline{\theta })\right)$ satisfies $w^{*}(\overline{\theta })=w(X_{l}^{*}(\overline{\theta }),\overline{ \theta })+\gamma Z_{l}^{*}(\overline{\theta })$ we can simplify and write the above inequality as:

$$\begin{aligned} \mu ^{*}\left[ \gamma Z_{l}^{*}(\overline{\theta })-w^{*}( \underline{\theta })+w(X_{l}^{*}(\overline{\theta }),\underline{\theta }) \right] \ge 0 \end{aligned}$$

As $\mu ^{*}\ge 0$, it just need to be proven that $Z_{l}^{*}( \overline{\theta })\ge \frac{1}{\gamma }\left[ w^{*}(\underline{\theta } )-w(X_{l}^{*}(\overline{\theta }),\underline{\theta })\right]$. For this purpose, it is sufficient to show that for any quota X, the minimum transfer that induces the policy-maker to accept a proposal depends positively on type $\theta$. Let this transfer function be $z(X,\theta )= \frac{1}{\gamma }\left[ w^{*}(\theta )-w(X,\theta )\right]$. We can write it as:

$$\begin{aligned} z(X,\theta )=\frac{n}{\gamma }\left\{ \int \limits _{\frac{X}{n}}^{x_{i}(c,\theta )} \left[ u^{\prime }(x,\theta )-c\right] dx\right\} . \end{aligned}$$

The inequality $z(X,\overline{\theta })>z(X,\underline{\theta })$ holds provided that $u^{\prime }(y,\overline{\theta })>u^{\prime }(y,\underline{ \theta })$ for all y. Hence,

$$\begin{aligned} Z_{l}^{*}(\overline{\theta })=z(X_{l}^{*}(\overline{\theta }), \overline{\theta })>z(X_{l}^{*}(\overline{\theta }),\underline{\theta })= \frac{1}{\gamma }\left[ w^{*}(\underline{\theta })-w(X_{l}^{*}( \overline{\theta }),\underline{\theta })\right] , \end{aligned}$$

so the pair $\left( X_{l}^{*}(\overline{\theta }),Z_{l}^{*}( \overline{\theta })\right)$ is always accepted, as we wanted to show. Therefore $(\overline{X}_{l},\overline{Z}_{l})=\left( X_{l}^{*}( \overline{\theta }),Z_{l}^{*}(\overline{\theta })\right)$.

Next we characterize the pair $(\underline{X}_{l},\underline{Z}_{l})$. For this pair to be part of a separating equilibrium, the following conditions are necessary: (a) a lobby of type $\overline{\theta }$ is not interested in making the proposal $(\underline{X}_{l},\underline{Z}_{l})$; (b) if the policy-maker believes that the lobby is type $\underline{\theta }$, it accepts the pair $(\underline{X}_{l},\underline{Z}_{l})$; and (c) a lobby of type $\underline{\theta }$ is better off under the pair $(\underline{X}_{l}, \underline{Z}_{l})$ than under $(\overline{X}_{l},\overline{Z}_{l})=\left( X_{l}^{*}(\overline{\theta }),Z_{l}^{*}(\overline{\theta })\right)$, i.e., it does not face incentives to masquerade as a high type. Each one of these conditions can be represented by a set of policy pairs. Thus, we define the corresponding sets $A,\, B$ and C as follows:

$$\begin{aligned} A= & {} \left\{ (X_{l},Z_{l})\mid \pi (X_{l}^{*}(\overline{\theta }), \overline{\theta })-Z_{l}^{*}(\overline{\theta })\ge \pi (X_{l},\overline{\theta })-Z_{l}\right\} , \\ B= & {} \left\{ (X_{l},Z_{l})\mid w(X_{l},\underline{\theta })+\gamma Z_{l}\ge w^{*}(\underline{\theta })\right\} , \\ C= & {} \left\{ (X_{l},Z_{l})\mid \pi (X_{l},\underline{\theta })-Z_{l}\ge \pi (X_{l}^{*}(\overline{\theta }),\underline{\theta })-Z_{l}^{*}( \overline{\theta })\right\} . \end{aligned}$$

The policy proposal $(\underline{X}_{l},\underline{Z}_{l})$ is the solution to the program:

$$\begin{aligned} \left\{ \begin{array}{ll} Max_{\left\{ X_{l},Z_{l}\right\} } &{}\quad \pi (X_{l},\underline{\theta })-Z_{l} \\ s.t. &{}\quad (X_{l},Z_{l})\in A\cap B\cap C. \end{array} \right. \end{aligned}$$

Let $\lambda$, $\rho$ and $\delta$ be the Lagrange multipliers associated to the constraints represented by sets A, B and C, respectively. We solve this program under the assumption that $\delta =0$, and check later that the inequality in set C is not binding for the pair that solves the program. The pair $(\underline{X}_{l},\underline{Z}_{l})$ that solves the program fulfills the following F.O.C.:

$$\begin{aligned} \dfrac{\partial L}{\partial \underline{X}_{l}}= & {} \pi ^{\prime }(\underline{X}_{l},\underline{\theta })-\lambda \pi ^{\prime }(\underline{X}_{l}, \overline{\theta })+\rho \pi ^{\prime }(\underline{X}_{l},\underline{\theta } )+\rho s^{\prime }(\underline{X}_{l},\underline{\theta })=0, \end{aligned}$$

(7.13)

$$\begin{aligned} \dfrac{\partial L}{\partial \underline{Z}_{l}}= & {} -1+\lambda +\gamma \rho =0. \end{aligned}$$

(7.14)

From equation (7.14) we deduce that it is not possible that both $\lambda$ and $\rho$ are equal to zero. In particular, we prove next that both are positive, implying that the constraints represented by sets A and B hold with equality. Suppose that $\rho =0$. Then, necessarily $\lambda =1$ by Eq (7.14). But in this case, equation (7.13) remains as $\pi ^{\prime }( \underline{X}_{l},\underline{\theta })-\pi ^{\prime }(\underline{X}_{l}, \overline{\theta })=0$, which is a contradiction with our assumptions. Suppose now that $\lambda =0$. Again by equation (7.14) it must be that $\rho =\frac{1}{\gamma }$. Substituting $\lambda =0$ and $\rho =\frac{1}{ \gamma }$ into Eq (7.13) yields $w^{\prime }(\underline{X}_{l},\underline{ \theta })+\gamma \pi ^{\prime }(\underline{X}_{l},\underline{\theta })=0$, thus implying $\underline{X}_{l}=X_{l}^{*}(\underline{\theta })$. However, in that case the lobby of type $\overline{\theta }$ will propose the quota $X_{l}^{*}(\underline{\theta })$ instead of $X_{l}^{*}( \overline{\theta })=\overline{X}_{l}$, so a separating equilibrium would not exist. We conclude that $\lambda$, $\rho >0$, so the pair $(\underline{X} _{l},\underline{Z}_{l})$ is characterized by Eqs. (7.13) and (7.14) together with the constraints in sets A and B holding with equality. Substituting the value of $\lambda$ obtained from Eq. (7.14) into Eq. (7.13) yields:

$$\begin{aligned} \left( 1+\rho \right) \left[ \pi ^{\prime }(\underline{X}_{l},\underline{ \theta })-\pi ^{\prime }(\underline{X}_{l},\overline{\theta })\right] +\rho \left[ ns^{\prime }(\underline{X}_{l},\underline{\theta })+\left( \gamma +1\right) \pi ^{\prime }(\underline{X}_{l},\overline{\theta })\right] =0. \end{aligned}$$

As $\pi ^{\prime }(\underline{X}_{l},\underline{\theta })-\pi ^{\prime }( \underline{X}_{l},\overline{\theta })<0$ by hypothesis, for the above equation to hold it must be true that

$$\begin{aligned} ns^{\prime }(\underline{X}_{l},\underline{\theta })+\left( \gamma +1\right) \pi ^{\prime }(\underline{X}_{l},\overline{\theta })>0. \end{aligned}$$

(7.15)

From the constraints in A and B we derive the following expression, that determines the value for $\underline{X}_{l}:$

$$\begin{aligned} ns(\underline{X}_{l},\underline{\theta })+\pi (\underline{X}_{l},\underline{ \theta })+\gamma \pi (\underline{X}_{l},\overline{\theta })+\gamma \left[ \overline{Z}_{l}-\pi (\overline{X}_{l},\overline{\theta })\right] -w^{*}( \underline{\theta })=0. \end{aligned}$$

(7.16)

Let us define functions $f(x)=ns(x,\underline{\theta })+\left( \gamma +1\right) \pi (x,\overline{\theta })+K$, $h(x)=ns(x,\underline{\theta })+\pi (x,\underline{\theta })+\gamma \pi (x,\overline{\theta })+K$, and $g(x)=w(x, \underline{\theta })+\gamma \pi (x,\underline{\theta })+K$, with $K=\gamma \left[ \overline{Z}_{l}-\pi (\overline{X}_{l},\overline{\theta })\right] -w^{*}(\underline{\theta })$. Observe that functions f(.), h(.) and g(.) are strictly concave in x and our assumptions on function $\pi$ imply that $f(x)>h(x)>g(x)$ and $f^{\prime }(x)>h^{\prime }(x)>g^{\prime }(x)$. Eqs. (7.15) and (7.16) can be expressed respectively as $f^{\prime }( \underline{X}_{l})>0$ and $h(\underline{X}_{l})=0$. Note that $g^{\prime }(X_{l}^{*}(\underline{\theta }))=0$. Let us define $\widetilde{y}$ and $\widetilde{z}$ such that $g(\widetilde{y})=0$, $g^{\prime }(\widetilde{y} )>0,\, f(\widetilde{z})=0$, and $f^{\prime }(\widetilde{z})>0$. Clearly $\widetilde{y}<X_{l}^{*}(\underline{\theta })$ provided that $X_{l}^{*}(\underline{\theta })$ is the maximum of g(.). From the properties of functions $f(.),\,h(.)$ and g(.) there exists $\widetilde{x}\in \left( \widetilde{z},\widetilde{y}\right)$ for which $f^{\prime }(\widetilde{x})>0$ and $h(\widetilde{x})=0$. It holds that $\widetilde{x}=\underline{X}_{l}$. As long as $\widetilde{y}<X_{l}^{*}(\underline{\theta })$ and $\underline{X}_{l}<\widetilde{y}$ we conclude that $\underline{X} _{l}<X_{l}^{*}(\underline{\theta })$.

In order to prove that $\underline{Z}_{l}>Z_{l}^{*}(\underline{\theta })$, recall that the acceptance constraints of the policy-maker are binding both under symmetric and asymmetric information. Then we have that $\underline{Z}_{l}=\frac{1}{\gamma }\left[ w^{*}(\underline{\theta })-w( \underline{X}_{l},\underline{\theta })\right]$. From the properties of w(.) and the fact that $\underline{X}_{l}<X_{l}^{*}(\underline{\theta } )$ we obtain that $\underline{Z}_{l}>Z_{l}^{*}(\underline{\theta })$.

Now we check that the value for the Lagrange multiplier $\delta$ is zero. To see this, just notice that $(\underline{X}_{l},\underline{Z}_{l})$ is a maximum in set $A\cap B$, and that the pair $\left( X_{l}^{*}(\overline{ \theta }),Z_{l}^{*}(\overline{\theta })\right)$ also belongs to $A\cap B$. Then, we have $\pi (\underline{X}_{l},\underline{\theta })-\underline{Z} _{l}>\pi (X_{l}^{*}(\overline{\theta }),\underline{\theta })-Z_{l}^{*}(\overline{\theta })$, i.e., $\delta =0$.

Finally, observe that the beliefs and strategies satisfy the so called Intuitive Criterion. The pair $\left( \underline{X}_{l},\underline{Z} _{l}\right)$ is the only one that maximizes the lobby’s utility in the set $A\cap B$. Hence, it is not “intuitive” that the policy-maker believes that the lobby is of type $\underline{\theta }$ when observing a pair $(\widetilde{X}_{l},\widetilde{Z}_{l})$ different from $\left( \underline{X} _{l},\underline{Z}_{l}\right)$. Suppose that the set of proposals and beliefs $\left\{ (\overline{X}_{l},\overline{Z}_{l}),(\widetilde{X}_{l}, \widetilde{Z}_{l})\right\}$, with $(\widetilde{X}_{l},\widetilde{Z}_{l})\in A\cap B$ and $\mu (\widetilde{X}_{l},\widetilde{Z}_{l})=1$, $\mu (\overline{X }_{l},\overline{Z}_{l})=0$ is a sequential separating equilibrium. If a lobby deviates from such an equilibrium, then it must be a lobby of type $\underline{\theta }$. The reason is that a lobby of type $\overline{\theta }$ have no incentives to deviate from $(\overline{X}_{l},\overline{Z} _{l})=\left( X_{l}^{*}(\overline{\theta }),Z_{l}^{*}(\overline{ \theta })\right)$ for any beliefs that such a deviation induces on the policy-maker. Then, the pair $(\underline{X}_{l},\underline{Z}_{l})$ is the only one for which the described beliefs satisfy the Intuitive Criterion.

Proof of Proposition 5.1

Let organization costs C be such that inequalities (5.1) and (5.2) in the main text hold. If the policy-maker’s beliefs are $\sigma (1)=1$ and $\sigma (0)=0$, a separating equilibrium exist in which only industries facing strong demand get organized. For instance, consider the consumer’s utility function $u(x_{i},\theta )=\theta x_{i}-\dfrac{1}{2}x_{i}^{2}$ and assume $c=0$. The optimal quotas in each one of the regulatory frameworks considered are given by:

$$\begin{aligned} X_{r}^{*}(\overline{\theta })&=X_{l}^{*}(\overline{\theta })= \overline{X}_{r}=\overline{X}_{l}=n\frac{1+\gamma }{1+2\gamma }\overline{\theta }, \end{aligned}$$

(7.17a)

$$\begin{aligned} X_{r}^{*}(\underline{\theta })&=X_{l}^{*}(\underline{\theta })=n \frac{1+\gamma }{1+2\gamma }\underline{\theta }, \end{aligned}$$

(7.17b)

$$\begin{aligned} \underline{X}_{r}&=X_{r}^{*}(\underline{\theta })-\delta _{r}, \end{aligned}$$

(7.17c)

$$\begin{aligned} \underline{X}_{l}&=X_{l}^{*}(\underline{\theta })-\delta _{l}, \end{aligned}$$

(7.17d)

where

$$\begin{aligned} \delta _{r}=n\frac{q}{1-q}\frac{\gamma }{1+2\gamma }\left( \overline{\theta } -\underline{\theta }\right) \end{aligned}$$

and

$$\begin{aligned} \delta _{l}=n\frac{\gamma }{1+2\gamma }\left\{ \left[ \frac{2}{\gamma } \underline{\theta }\left( \overline{\theta }-\underline{\theta }\right) \right] ^{\frac{1}{2}}-\left( \overline{\theta }-\underline{\theta }\right) \right\} . \end{aligned}$$

Functions $\delta _{r}$ and $\delta _{l}$ represent distortions with respect to the symmetric information quotas established on weak demand industries, both for the case of regulation proposed by the policy-maker $\left( \delta _{r}\right)$ as in the case of regulation proposed by the industry lobby $\left( \delta _{l}\right)$. Consider now the following parameters: $\gamma =2$; $\underline{\theta }=1; \overline{\theta }=2$. The incentive conditions hold whenever $-\frac{2}{5}n\le C\le \frac{4}{5}n$.

Next we show that a separating equilibrium in which industries facing weak demand choose $v=1$ and industries facing strong demand choose $v=0$ does not exist. The beliefs supporting such an equilibrium would be $\sigma (1)=0$ and $\sigma (0)=1$, with the following incentive conditions:

$$\begin{aligned} \pi (X_{l}^{*}(\underline{\theta }),\underline{\theta })-Z_{l}^{*}( \underline{\theta })-C\ge \pi (X_{r}^{*}(\overline{\theta }),\underline{ \theta })-Z_{r}^{*}(\overline{\theta }), \end{aligned}$$

(7.18)

and

$$\begin{aligned} 0=\pi (X_{r}^{*}(\overline{\theta }),\overline{\theta })-Z_{r}^{*}( \overline{\theta })\ge \pi (X_{l}^{*}(\underline{\theta }),\overline{ \theta })-Z_{l}^{*}(\underline{\theta })-C. \end{aligned}$$

(7.19)

The right hand side of inequality (7.18) must equal zero, as long as a negative payoff (earned by a low type industry when imposed a regulatory pair intended for high types) can always be avoided by rejecting the regulation. In inequality (7.19), the beliefs induced when the policy-maker observes $v=1$ (the industry faces low demand), require the high type to make the symmetric information proposal of a low type.

We combine inequalities (7.18) and (7.19) to obtain the following chain of inequalities:

$$\begin{aligned} \pi (X_{l}^{*}(\underline{\theta }),\underline{\theta })-Z_{l}^{*}( \underline{\theta })-C\ge & {} \pi (X_{r}^{*}(\overline{\theta }), \underline{\theta })-Z_{r}^{*}(\overline{\theta })=0 \\= & {} \pi (X_{r}^{*}(\overline{\theta }),\overline{\theta })-Z_{r}^{*}( \overline{\theta })\ge \pi (X_{l}^{*}(\underline{\theta }),\overline{ \theta })-Z_{l}^{*}(\underline{\theta })-C. \end{aligned}$$

A necessary condition for this separating equilibrium to exist is:

$$\begin{aligned} \pi (X_{l}^{*}(\underline{\theta }),\underline{\theta })-Z_{l}^{*}( \underline{\theta })-C\ge \pi (X_{l}^{*}(\underline{\theta }),\overline{ \theta })-Z_{l}^{*}(\underline{\theta })-C. \end{aligned}$$

However, this condition can never hold since it contradicts the assumption that, for any quota X, $\pi (X,\underline{\theta })<\pi (X,\overline{ \theta })$.

Proof of Proposition 5.2

A pooling equilibrium in which both types of industry select $v=1$ exist when the conditions (5.3) and (5.4) hold, jointly with beliefs $\sigma (1)=\sigma (0)=q$. Consider the consumer’s utility function in the proof of Proposition 5.1. For the following set of parameters: $c=0$; $q=\frac{1}{2}$; $\gamma =2$; $\underline{\theta }=1$; $\overline{\theta }=2$ a pooling equilibrium in which both types of industry choose $v=1$ exists when $C\le \frac{1}{5}n$. Using a similar reasoning we find that a pooling equilibrium in which both choose $v=0$ exists for $C\ge \frac{3}{5}n$.