Lobbying & Costly Lobbying

Lobbying & Costly Lobbying Special Interest Politics – Chapters 4-5 G. Grossman & E. HelpmanMIT Press 2001 Presented by: Victor Bennett Richard Wang Feb 13 2006

Overview Lobbying • One Lobby • Two States of the World • Three States of the World • Continuous Information • Ex Ante Welfare • Two Lobbies • Like Bias • Opposite Bias • Multidimensional Information • More General Lobbying Game

Overview Costly Lobbying • Fixed Exogenous Costs • Out to SIG’s control • Variable Endogenous Costs • Depend on Actions of the SIG • Policymaker-imposed costs • Costs as a choice variable for the policymaker How do these costs affect equilibria under different model conditions?

Basic Model Setting p Policy Variable q State of the World d Bias (d>0, unless otherwise stated)

Basic Model Setting Assumptions: Lobbyist knows the state of the world (q) but the policymaker does not The policymaker has a prior belief on the state of the world (realization of a random variable, ): ~ U[qmin,qmax]

One Lobby, Two States of the World The Setting • Two states – Low or High qє {qL,qH}, qL<qH • Policymaker: • Initially regards qL,qH are equally likely • Sets p =q when the lobbying reveals the true state • Sets p = E when uncertain about the state • Lobbyist knows the true state • Discuss 2 equilibria: Full Revelation and Babbling

One Lobby, Two States of the World Full Revelation Equilibrium • The lobbyist tells the truth to inform the policymaker. • The policymaker: • Believes the state of the world as told by the lobbyist • Sets the policy p = qHor p = qL when the state is High or Low, respectively.

One Lobby, Two States of the World • When the true state is qH, the lobbyist will tell the truth because: U(p=qH, qH) > U(p=qH, qL) => (qH+d)-qH < (qH+d)-qL • When the true state is qL, the lobbyist will tell the truth iff: U(p=qL, qL) > U(p=qH, qL) => (qL+d)-qL≤ qH -(qL+d) => d ≤ (qH- qL)/2 (4.1)

One Lobby, Two States of the World • Eq (4.1) measures the degree of alignment between the interests of the policymaker and the lobbyist. • When Eq (4.1) is satisfied, the equilibrium is fully revealing. • If Eq (4.1) is not satisfied, the lobbyist’s report lacks credibility.

One Lobby, Two States of the World Babbling Equilibrium • The policymaker: • Distrusts the lobbyist about the reported state • The policymaker remains uninformed. • Sets the policy p = (qH+qL)/2 • The lobbyist has no incentive to report truthfully.

One Lobby, Three States of the World The Setting • Low, Medium, High States: qє {qL,qM ,qH}, qL<qM <qH • Policymaker: • Initially regards qL, qM,qH are equally likely • Sets p =q when the lobbying reveals the true state • Sets p = E when uncertain about the state • Lobbyist knows the true state • We will show 3 equilibriums: • Full Revelation Equilibrium • Partial Transmission Equilibrium • Babbling Equilibrium

One Lobby, Three States of the World Full Revelation Equilibrium • The lobbyist tells the truth to inform the policymaker. • The policymaker: • Believes the state of the world as told by the lobbyist • Sets the policy p = qH, p = qM, or p = qL when the reported state is High, Medium, or Low, respectively.

One Lobby, Three States of the World • When the true state is qL, the lobbyist will tell the truth iff: (i) U(p=qL, qL) > U(p=qM, qL) and (ii) U(p=qL, qL) > U(p=qH, qL) • Since qL<qM <qH, reportingqHwhen the state isqLis unattractive to the lobbyist anytime it is unattractive to report the state asqM. Therefore, we only need to consider (i): => (qL+d) – qL ≤ qM - (qL+d) => d ≤ (qM- qL)/2 (4.2)

One Lobby, Three States of the World • When the true state is qM, the lobbyist has no incentive to report qL because the SIG prefers a policy larger than qM. • The lobbyist will tell the truth at state qM iff: U(p=qM, qM) > U(p=qH, qM) => (qM+d) - qM≤ qH - (qM+d) => d ≤ (qH- qM)/2 (4.3)

One Lobby, Three States of the World • When the true state is qH, the lobbyist has no incentive to report either state qM or qL because these will result in a policy level that is lower than p = qH. • Therefore, there is no restriction needed for truthful reporting in state qH.

One Lobby, Three States of the World Partial Transmission Equilibrium • When either (4.2) or (4.3) is violated. • Lobbyist cannot communicate full information to policymaker. • Lobbyist communicates more-limited information.

One Lobby, Three States of the World • Say, (4.3) is violated. • Lobbyist communicate the state as “Low” or “Not Low”. • Truthful report of “Not Low” requires: (qM+d) - qL≥ (qM+qH)/2 - (qM+d) => d ≥ (qH- qM)/4 - (qM- qL)/2 (4.4) • Truthful report of “Low” requires: (qL+d) - qL≤ (qM+qH)/2 - (qL+d) => d ≤ (qH- qM)/4 + (qM- qL)/2 (4.5)

One Lobby, Three States of the World Babbling Equilibrium • The policymaker: • Distrusts the lobbyist about the reported state • Sets the policy p = (qH+qM+qL)/3 whether the state is High, Medium, or Low. • The lobbyist has no incentive to report truthfully. • The policymaker remains uninformed.

One Lobby, Three States of the World Which Equilibrium? • For both the policymaker and the lobbyist, the ex ante expected utilities for each equilibrium: EU(Full) > EU(Partial) > EU(Babbling) • The lobbyist and the policymaker might coordinate on Full Revelation Equilibrium.

One Lobby, Continuous Information • In the discrete state case, for a lobbyist to distinguish between all possible states, the bias, d, must be smaller than one-half of the distance between any of the states. • In the case where the state variable is continuous, the lobbyist can never communicate to the policymaker the fine details of the state. • Compromise: The lobbyist can credibly report to the policymaker a range that contains the true state – Partition Equilibrium.

One Lobby, Continuous Information The Setting: • The policymaker has a prior belief on the state of the world (random variable, ): ~ U[qmin,qmax] • Lobbyist knows the state of the world (q, the realized value of ) but cannot credibly communicate q to the policymaker. • Lobbyist indicates a range (R) that contains the true value of q. Example: Lobbyist report q in R1qmin≤ q ≤ q1 Lobbyist report q in R2q1 ≤ q ≤ q2 Lobbyist report q in Rnqn-1 ≤ q ≤ qn • Policymaker will set p = (qk+qk-1)/2 when the lobbyist report Rk.

One Lobby, Continuous Information Objective: Find values of q1, q2 … qn such that the policymaker sees the lobbyist’s report as credible. Question: What values of qk-1, qk, and d does the lobbyist prefer to tell the truth when q is in Rk?

One Lobby, Continuous Information Idea: • Suppose q is in R1, the greatest temptation for the lobbyist to lie is when q is a bit less than q1. So we set: (q1+d) - (qmin+q1)/2≤ (q2+q1)/2 - (q1+d) => q2 ≥ 2q1+4d-qmin(4.6) • Now suppose q is in R2. To prevent false report that q is in R1, we set: (q1+d) - (qmin+q1)/2≥ (q2+q1)/2 - (q1+d) => q2 ≤ 2q1+4d-qmin(4.6’) • (4.6) & (4.6’) => q2 = 2q1+4d-qmin (4.7)

One Lobby, Continuous Information • Extending the argument to R3, R4, and so on, we have: qj = 2qj-1+ 4d –qj-2(4.8) • The top most value must coincide with the maximum support of the distribution: qn = qmax(4.9) • (4.8) & (4.9): qj = (j/n)qmax+ ((n-j)/n)qmin - 2j(n-j)d (4.10) • Eqm condition requires that q1 > qmin, which is satisfied iff: 2n(n-1)d < qmax - qmin (4.11)

One Lobby, Continuous Information • Inequality (4.11) is a necessary and sufficient condition for the existence of a lobbying equilibrium with n different reports. • Three observations: • n=1 always exists -> Babbling Equilibrium • The smaller is d, the larger is the maximum number of feasible partitions, n. • If an equilibrium with n reports exists, then an equilibrium with k reports also exists for all k < n. • Question: Given n equilibria, which one will the lobbyist and policymaker agree to coordinate on?

Ex Ante Welfare • If the policymaker and lobbyist agree on their rankings of the equilibria, the players might be able to coordinate on a particular equilibrium that yields each of them the highest ex ante welfare. • The expected welfares in an n-partition equilibrium are: • Policymaker: EGn = (4.12) • Lobbyist: EUn = (4.13) • Both players do agree on their ranking of possible equilibria.

Ex Ante Welfare • Using (4.10) to obtain the form of qj and qj-1, combining with (4.12) and (4.13) and simplifying, we have: (4.14) • RHS of (4.14) is an increasing function of n for all n that satisfy (4.11). • Therefore, both parties would agree, ex ante, the equilibrium using the maximum n allowable by (4.11) is the best among all equilibrium outcomes.

Two Lobbies The Setting • Same information assumptions as one lobby case, except we have two lobbies now. • The two lobbies may have different direction of biases: • Like Bias: Both di,dj > 0 or < 0; |di| < |dj|; i ≠ j • Opposite Bias: sign(di) ≠ sign(dj); |di|≤|dj|; i ≠ j • Three types of messages: • Secret: Each lobbyist is ignorant of the alternative info source • Private: Each lobbyist is aware that another has offered/will offer advice but ignorant on the content • Public: Subquent lobbyist can condition the report on the info that the policymaker already has.

2 Lobbies, Like Bias, Secret Message Outcome • No strategic interaction between the lobbyists. • Each lobbyist will act according to the prescription of one of the equilibria discussed in the one lobby case. • The policymaker will take action based on the combined info from the two lobbyists.

2 Lobbies, Like Bias, Secret Message Example • Lobbyist 1 sends either m1 (indicates q ≤ q1) or m2(indicates q ≥ q1). • Lobbyist 2 sends either 1 (indicates q ≤ 1) or 2(indicates q ≥ 1), where q1 < 1.

2 Lobbies, Like Bias, Secret Message m1 m2 qmin q1 qmax 1 1 2

2 Lobbies, Like Bias, Private Message Full Information Equilibrium • Many different outcome possible, including Full Information Equilibrium. • Policymaker: • Believes each lobbyist report precisely and truthfully. • Sets optimal strategy: p = min{m,m^} • Lobbyists: • Each lobbyist’s report can be used to discipline the other’s • Truthful revelation is an equilibrium

2 Lobbies, Like Bias, Private Message Full Information Equilibrium • Example: • q = 5 • Lobbyist 1 expects m^=5 (truth telling by Lobbyist 2) • No gain for lobbyist if report m>5 (p = 5) • Lobbyist 1 will worsen own situation if report m<5 • So truth telling by Lobbyist 1 • Similarly, truth telling by Lobbyist 2 • But Full Information Equilibrium is fragile.

2 Lobbies, Like Bias, Private Message Full Information Equilibrium • Illustration: • Lobbyist 1 might believe that Lobbyist 2 announces m^>5 (e.g. error by Lobbyist 2) • If Lobbyist 2 really did report m^>5, then Lobbyist 1 would wish he had reported m>5 • If Lobbyist 2 report m^=5, there is no loss for Lobbyist 1 to report m>5 (p is still at 5) • Weakly dominant strategy for each Lobbyist to reveal his ideal point in every state.

2 Lobbies, Like Bias, Public Message • Lobbyists report their information sequentially. • Second Lobbyist learns what the first reported. • Full Revelation Equilibrium cannot occur. • Illustration: • Suppose that there is a full revelation equilibrium • Policymaker believes she learned the true state and set p = q • Lobbyist 1 will have an incentive to report q’> q, and if Lobbyist 2 follows suit, the policymaker would believe and set p = q’ • Both Lobbyists will be better off, while the policymaker will be worse off. • Therefore, full revelation equilibrium cannot occur.

2 Lobbies, Like Bias, Public Message • Every equilibrium with 2 lobbies can be represented as a Partition Equilibrium (Krishna and Morgan 2001). • In Partition Equilibrium, the policymaker learns the true q lies in one of a finite number of non-overlapping ranges. • Policymaker’s information comes from the combined information of the two lobbyists.

2 Lobbies, Like Bias, Public Message Example: • When each lobbyist partition the set of possible values into 2 subsets, the combined information leads to a 3-partition equilibrium. • Setup: • Policymaker’s Belief: q lies between 0 and 24 with equal probability. • Lobbyists’ Bias: d1=1, d2=2

2 Lobbies, Like Bias, Public Message Example • Setup continued: • Lobbyist 1 reports first. • Lobbyist 1 (L1) sends either m1 (indicates q ≤ q1) or m2(indicates q ≥ q1). • Lobbyist 2 (L2) sends either m^1 (indicates q ≤ q^1) or m^2(indicates q ≥ q ^1), where q^1 <q1. • Hypothesize q1 be reasonably large, so Lobbyist 2 wants to distinguish between “very low” (between qminandq^1) or “reasonably low” (between q^1andq1).

2 Lobbies, Like Bias, Public Message • Scenarios: • If L1 announces m2, • L2 reporting m^2 will add no new info • L2 reporting m^1 indicates one of report is false – not equilibrium • If 1 announces m1, • 2 reporting m^1 informs policymaker the state is “very low”; policymaker sets p=p(m1,m^1) • 2 reporting m^2 informs policymaker the state is “reasonably low”; policymaker sets p=p(m1,m^2) p(m1,m^1) p(m2,m^2) p(m1,m^2) m1 m2 0 24 q1 1 1 2

2 Lobbies, Like Bias, Public Message • Equilibrium conditions: L1 and L2 should have no incentive to report falsely. • For L2, this requires at state q = q^1: (q^1+d2) – p(m1,m^1) = p(m1,m^2) -(q^1+d2); d2 =2; => q^1 = q1/2 – 4 (4.15) • For L1, this requires at state q = q1: (q1+d1) – (q1+q^1)/2 = (q1+24)/2 -(q1+d1); d1 =1; => q1 = 10 + q^1/2 (4.16) p(m1,m^1) p(m2,m^2) p(m1,m^2) m1 m2 0 24 q1 1 1 2

2 Lobbies, Like Bias, Public Message • Solution: • Solving (4.15) and (4.16) simultaneously, we have: q^1 = 4/3 q1 = 32/3 2/3 52/3 18/3 0 24 q1 1 4/3 32/3

2 Lobbies, Like Bias, Public Message • Solution: • Another possible equilibrium (reverse role of L1 and L2): q^1 = 28/3 q1 = 8/3 4/3 50/3 18/3 0 24 q1 1 8/3 28/3

2 Lobbies, Like Bias, Public Message Other findings by Krishna and Morgan (2001): • It does not matter which lobbyist report first. The set of possible equilibria unchanged by sequence. • For given parameter values, there exists a maximum number, n, of subset in an equilibrium partition. • As d1 or d2 approaches zero (less bias from policymaker), n increases and more detailed the information convey. • Max n with two lobbies ≤ Max n of the more moderate of the two lobbyists • Ex ante welfares for all parties are higher when the more moderate of the two lobbyists lobbies than that when both lobbyist lobby.

Two Lobbies, Opposite Bias • Opposite Bias: sign(di) ≠ sign(dj); ; |di|≤|dj|; i ≠ j • Policymaker cannot gain complete information (Krishna and Morgan 2001). • However, policymaker can learn more from the two lobbyist together than from either one alone.

Two Lobbies, Opposite Bias Example: • d1 = -3; d2 = 3 • q lies between 0 and 24. • With only either one lobbyist, outcome would be either babbling equilibrium or 2-partition equilibrium with: q1 = 18 and q ^1 = 6 • 3-partition equilibrium is not possible with either one lobbyist alone, but 3-partition equilibrium is possible when both lobbyist lobby together.

Two Lobbies, Opposite Bias Illustration: • d1 = -3; d2 = 3 • q lies between 0 and 24 • L1 reports first, then L2 • Let q1 > q ^1 p(m1,m^1) p(m2,m^2) p(m1,m^2) m1 m2 0 24 q1 1 1 2

Two Lobbies, Opposite Bias Illustration: • To ensure truth telling by L1 and L2: L2: (q ^1+3) - q ^1/2 = (q1+q ^1)/2 – (q ^1+3) => q ^1 = q1/2 – 6 (4.17) L1: (q1-3) – (q1+q ^1)/2 = (q1+24)/2 – (q1-3) => q1 = q ^1/2 + 18 (4.18) p(m1,m^1) p(m2,m^2) p(m1,m^2) m1 m2 0 24 q1 1 1 2

Two Lobbies, Opposite Bias Illustration: • Solving (4.17) and (4.18) simultaneously, we have a 3-partition equilibrium with: q ^1 = 4 and q1 = 20; p = 2, 12, and 22. • All parties have higher ex ante welfare if both lobbyists lobby than only one lobbyist lobby. • Krishna and Morgan (2001) showed partial revelation equilibrium exists whenever both lobbyists are non-extreme. 2 12 22 0 24 q1 1 20 4

Multidimensional Information Battaglini (2000): • Increasing the dimensionality of the policy problem may improve the prospects for information transmission. • Setting: • Policy: 2-dimensional vector p = (p1, p2) • State: 2-dimensional vector q = (q1, q2) • Bias: 2-dimensional vector di = (di1, di2); di≠adi • Policymaker’s utility: G(p,q) = - ∑(pj - qj)2 • Lobbyist’s utility: Ui(p,q) = - ∑(pj - qj- dij)2 • Each lobbyist knows q but the policymaker doesn’t. • Each lobbyist meet with the policymaker in private.

Multidimensional Information • Given di≠adi, there exists an equilibrium with full revelation of q: • L1 reports m = d21q1 + d22q2 • L2 reports m = d11q1 + d12q2 • Based on L1 and L2 reports, policymaker draws two lines: LL: d21q1 + d22q2= 0 L^L^: d11q1 + d12q2 = 0 • Ensures truth telling by both lobbyists (see Figure 4.5)

Lobbying & Costly Lobbying