Lobbying & Costly Lobbying Special Interest Politics Chapters 4-5 G. Grossman & E. Helpman MIT 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 SIGs 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 Players: Policymaker Utilities: G(p,)=-(p-)2 p

Lobby / Special Interest Group (SIG) U(p,)=-(p--)2 Policy Variable State of the World Bias (>0, unless otherwise stated) Basic Model Setting Assumptions: Lobbyist knows the state of the world () but the policymaker does not The policymaker has a prior belief on the state of the ~ world (realization of a random variable, ): ~ ~ U[min,max]

One Lobby, Two States of the World The Setting Two states Low or High {L,H}, L

One Lobby, Two States of the World When the true state is H, the lobbyist will tell the truth because: U(p=H, H) > U(p=H, L) => (H+)-H < (H+)-L When the true state is L, the lobbyist will tell the truth iff: U(p=L, L) > U(p=H, L) => (L+)-L H -(L+) => (H- L)/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 lobbyists 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 = (H+L)/2 The lobbyist has no incentive to report truthfully. One Lobby, Three States of the World

The Setting Low, Medium, High States: {L,M ,H}, L

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 = H, p = M, or p = L when the reported state is High, Medium, or Low, respectively. One Lobby, Three States of the World When the true state is L, the lobbyist will tell the truth iff: (i)

U(p=L, L) > U(p=M, L) (ii) U(p=L, L) > U(p=H, L) and Since L => (L+) L M - (L+) (M- L)/2

(4.2) One Lobby, Three States of the World When the true state is M, the lobbyist has no incentive to report L because the SIG prefers a policy larger than M. The lobbyist will tell the truth at state M iff: U(p=M, M) > U(p=H, M) => => (M+) - M H - (M+) (H- M)/2 (4.3) One Lobby, Three States of the World When the true state is H, the lobbyist has no incentive to report either state M or L because these will result in

a policy level that is lower than p = H. Therefore, there is no restriction needed for truthful reporting in state H. 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: (M+) - L (M+H)/2 - (M+) => (H- M)/4 - (M- L)/2 (4.4)

Truthful report of Low requires: (L+) - L (M+H)/2 - (L+) => (H- M)/4 + (M- L)/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 = (H+M+L)/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, , 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[min,max] ~ Lobbyist knows the state of the world (the realized value of ) but cannot credibly communicate to the policymaker. Lobbyist indicates a range (R) that contains the true value of . Example: Lobbyist report in R1 min 1 Lobbyist report in R2 1 2 Lobbyist report in Rn n-1 n

Policymaker will set p = (k+k-1)/2 when the lobbyist report Rk. One Lobby, Continuous Information Objective: Find values of 1, 2 n such that the policymaker sees the lobbyists report as credible. Question: What values of k-1, k, and does the lobbyist prefer to tell the truth when is in Rk? One Lobby, Continuous Information Idea: Suppose is in R1, the greatest temptation for the lobbyist to lie is when is a bit less than 1. So we set: (1+) - (min+1)/2 (2+1)/2 - (1+) =>

2 21+4-min (4.6) Now suppose is in R2. To prevent false report that is in R1, we set: (1+) - (min+1)/2 (2+1)/2 - (1+) => 2 21+4-min (4.6) (4.6) & (4.6) => 2 = 21+4-min (4.7) One Lobby, Continuous Information Extending the argument to R3, R4, and so on, we have: j = 2j-1+ 4j-2 (4.8) The top most value must coincide with the maximum support of the distribution:

n = max (4.9) (4.8) & (4.9): j = (j/n)max+ ((n-j)/n)min - 2j(n-j) (4.10) Eqm condition requires that 1 > min, which is satisfied iff: 2n(n-1)< max - min (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 , 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: Lobbyist: EGn = EUn =

n 1 ( j j 1 ) 3 12( max min ) j 1 n 1 ( j j 1 ) 3 2 12( max min ) j 1 Both players do agree on their ranking of possible equilibria. (4.12)

(4.13) Ex Ante Welfare Using (4.10) to obtain the form of j and j-1, combining with (4.12) and (4.13) and simplifying, we have: n ( j j 1 )3 ( max min ) 2 2 2

4 ( n 1) ( max min ) n2 j 1 (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: Opposite Bias: Both i,j > 0 or < 0; |i| < |j|; i j sign(i) sign(j); |i||j|; 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 1) or m2 (indicates 1). Lobbyist 2 sends either m1 (indicates 1) or m2 (indicates 1), where 1 < 1. 2 Lobbies, Like Bias, Secret Message m1

min m m2 1 1 2 Message Inference m1 and m1

min 1 m2 and m2 m2 and 1 1 max 1 1 m m max 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 lobbyists report can be used to discipline the others Truthful revelation is an equilibrium 2 Lobbies, Like Bias, Private Message Full Information Equilibrium Example:

=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 = Lobbyist 1 will have an incentive to report > , and if Lobbyist 2 follows suit, the policymaker would believe and set p = 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

lies in one of a finite number of non-overlapping ranges. Policymakers 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 3partition equilibrium. Setup: Policymakers Belief: lies between 0 and 24 with equal probability. Lobbyists Bias: 1=1, 2=2 2 Lobbies, Like Bias, Public Message Example Setup continued: Lobbyist 1 reports first. Lobbyist 1 (L1) sends either m1 (indicates 1) or m2 (indicates 1).

Lobbyist 2 (L2) sends either m^1 (indicates ^1) or m^2 (indicates ^1), where ^1 <1. Hypothesize 1 be reasonably large, so Lobbyist 2 wants to distinguish between very low (between min and ^1) or reasonably low (between ^1 and 1). 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(m1,m^2) m1 m2 1 m p(m2,m^2) 1

m 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 = ^1: (^1+2) p(m1,m^1) = p(m1,m^2) - (^1+2); => 2 =2; ^1 = 1/2 4 (4.15) For L1, this requires at state = 1:

(1+1) (1+^1)/2 = (1+24)/2 - (1+1); => 1 =1; 1 = 10 + ^1/2 p(m1,m^1) p(m1,m^2) m1 p(m2,m^2) m2 1

m (4.16) 1 m 2 2 Lobbies, Like Bias, Public Message Solution: Solving (4.15) and (4.16) simultaneously, we have:

^1 = 4/3 1 2/3 1 4/3 = 32/3 52/3 18/3 32/3

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

18/3 1 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 1 or 2 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(i) sign(j); ; |i||j|; 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: 1 = -3; 2 = 3 lies between 0 and 24. With only either one lobbyist, outcome would be either babbling equilibrium or 2-partition equilibrium with: 1 = 18 and ^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: 1 = -3; 2 = 3 lies between 0 and 24 L1 reports first, then L2 Let 1 > ^1 p(m1,m^1) p(m1,m^2) m1 m2 1 m

p(m2,m^2) 1 m 2 Two Lobbies, Opposite Bias Illustration: To ensure truth telling by L1 and L2: L2: ( ^1+3) - ^1/2 = ( 1+ ^1)/2 ( ^1+3) =>

L1: ^1 = 1/2 6 (4.17) ( 1-3) ( 1+ ^1)/2 = ( 1+24)/2 ( 1-3) => 1 = ^1/2 + 18 p(m1,m^1) p(m1,m^2) m1 p(m2,m^2)

m2 1 m (4.18) 1 m 2

Two Lobbies, Opposite Bias Illustration: Solving (4.17) and (4.18) simultaneously, we have a 3partition equilibrium with: ^1 = 4 and 1 = 20; p = 2, 12, and 22. 2 12 1 4 22

20 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 nonextreme. Multidimensional Information Battaglini (2000): Increasing the dimensionality of the policy problem may improve the prospects for information transmission. Setting: Policy: State:

2-dimensional vector p = (p1, p2) 2-dimensional vector = (1, 2) i = (i1, i2); i i G(p,) = - (pj - j)2 Ui(p,) = - (pj - j- ij)2 Bias: 2-dimensional vector Policymakers utility: Lobbyists utility: Each lobbyist knows but the policymaker doesnt.

Each lobbyist meet with the policymaker in private. Multidimensional Information Given i i, there exists an equilibrium with full revelation of : L1 reports m = 211 + 222 L2 reports m = 111 + 122 Based on L1 and L2 reports, policymaker draws two lines: LL: 211 + 222 = 0 L^L^: 111 + 122 = 0 Ensures truth telling by both lobbyists (see Figure 4.5)

Multidimensional Information M^ 2 Reported by L2 L^ M L 1 M L Reported by L1

M^ L^ Figure 4.5 Multidimensional Information Other results by Battaglini (2000): Small reporting mistakes by lobbyists Probabilities of errors are small and independent for two lobbyist Lobbyists unaware of the errors => There exists an equilibrium with (nearly) full revelation of when the dimension of uncertainty is 2 but not 1 (i.e. both lobbyists observes the state with error.) Can generalize to other utility functions Relies on the alignment of the different policy dimensions with the different dimensions of policy uncertainty. General Lobbying Game

The Setting: One policymaker and one lobbyist Welfare Functions - policymaker: G(p,); lobbyist: U(p,) Properties of G, U: Differentiable Strictly concave in p Gp and Up > 0 Properties of : CDF: F( PDF: f()=F(> 0 for all between min and max. = 0 otherwise General Lobbying Game The Setting: Lobbyists ideal policy > Policymakers ideal policy arg maxp U(p,) > arg maxp G(p,) for all [min,max] Lobbyist:

Learns and communicates m M to Policymaker. Strategy: m = () Policymaker: Suspects Lobbyists strategy: ^() Forms posterior belief distribution: F^(|m,^ General Lobbying Game The Setting: Policymaker chooses p to maximize expected welfare: pG(m|^) = arg maxp G(p,)dF^(|m,^ Lobbyist will construct () such that U[pG(m|^) ,] is maximized. In equilibrium, Policymakers belief must be consistent with the incentive faced by the lobbyist: ^() = () . General Lobbying Game No Full Revelation Equilibrium

Babbling Equilibrium exists Partition Equilibrium (Crawford and Sobel 1982) Lobbyist chooses one of finite n messages: m {m1,m2, ., mn} Policymaker interprets mi indicates i-1 i {1, 2, , n }, 0 = min and n = max Policymaker chooses the policy: i p(mi) = arg maxp G ( p, ) f ( | m )d i i 1

General Lobbying Game Truth telling requires: U[p(mi),i] = U[p(mi+1),i] Maximum number of partition exists, nmax. The more aligned the interests between the policymaker and the lobbyist are, the greater the nmax. If the policymaker and the lobbyist can agree on the ranking of equilibrium, the can agree to coordinate on nmax. Costly Lobbying Fixed Exogenous Costs Out to SIGs 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? Single SIG, Dichotomous Information Players: Utilities: Policymaker SIG Costs G(p,)=-(p-)2 U(p,l)=-(p--)2- lf (H L)/2 l With low costs there is an equilibrium where the act of lobbying transmits

information and both actors are better off. Requirement that SIG is willing to pay lobbying cost in H If (H L)(2+H L) k1 Requirement that SIG wont lobby in L lf (H L)(2+H L) k2 Because k1>k2, there is a range where both are satisfied and the equilibrium exists. Single SIG, Dichotomous Information Players: Utilities: Policymaker SIG Costs G(p,)=-(p-)2 U(p,l)=-(p--)2- lf > (H L)/2

l High costs mean that the SIG cant tie its own hands, so if the policymaker believes the SIG will lobby if H, the SIG lobbies, but is actually worse off in expectation if p = (H L)/2 always and costs are 0. With this condition of high costs, the pure strategy equilibria are either to make the SIG worse off, or offer no additional information to the policy maker A mixed strategy, however, might provide information if the SIG always lobbies if H and mixes when L. Single SIG, Dichotomous Information Players: Policymaker Utilities: G(p,)=-(p-)2

SIG U(p,l)=-(p--)2-l lf > (H L)/2 When will a mixed strategy occur? Beliefs of Policymaker P(Lobbying| H)=1, P(Lobbying| L)= < 1 P(H|Lobbying)= 1/(1+ ), P(H|Lobbying)= /(1+ ) plobby = (H+L)/(1+), pno lobby = L For mixing, SIG must be indifferent when L, so = (H L)(+- 2 lf)(1/ lf) - 1 Costs Single SIG, Continuous Information Players: Policymaker Utilities:

G(p,)=-(p-)2 SIG U(p,l)=-(p--)2-l From 4.1.4 we know that with costless lobbying a moderately biased group can convey information, but no group can communicate exactly. The cost allows the policymaker to split the space into two where the act of lobbying per se indicates which partition contains if there exists a point 1 for which the SIG is indifferent between paying lf and the associated policy. If we assume that the policy makers beliefs are P(1|Lobbying)=1, P(1| Lobbying)=0 then the condition for informative equilibrium is -(12-1/2-)2-lf = -(-1/2-)2 1 = max/2 - 2 + lf /(max/2 ) Two Lobbies, Dichotomous Information, Like Biases Players:

Policymaker SIGs Utilities: G(p,)=-(p-)2 U (p,l)=-(p--)2-l Biases 2>1>0 When firms have like bias, there exist equilibria where one SIG lobbies and the other free rides If lf (H-L)(21+H-L) and If lf (H-L)(21+H-L) and lf (H-L)(21-H+L)

lf (H-L)(21-H+L) SIG1 only lobbies when H and SIG2 never lobbies SIG2 only lobbies when H and SIG1 never lobbies Is there an equilibrium where both lobby? Yes, but not a very satisfactory one. If all four conditions are met, and the Policymaker on believes reports of H from both, both lobby on H and not L. One lobbying never happens, so the beliefs are never tested, which means that this survives as a PBE. Two Lobbies, Dichotomous Information, Opposite Biases Players: Policymaker

SIGs Utilities: G(p,)=-(p-)2 U (p,l)=-(p--)2-l Biases 2>0>1 Despite conflicting interests there exists an equilibrium where one SIG lobbies and the other free rides If lf (H-L)(21+H-L) and lf (H-L)(21-H+L) For example, if these equations are satisfied, SIG1 represents the world, and the policy maker ignores 2. SIG1 only lobbies when H and SIG2 never lobbies

Is there a symmetric equilibrium where both lobby? Yes! Two Lobbies, Dichotomous Information, Opposite Biases Players: Policymaker SIGs Utilities: G(p,)=-(p-)2 U (p,l)=-(p--)2-l Biases 2>0>1 A symmetric equilibrium exists where each SIG lobbies in the case it cares most about, and not in the other case, and the policymaker keeps its priors if both or neither lobby. (H-L)(-(H-L)/4) < lf < (H+L)(-(H-L)/4) lf must be in the range where SIG1 is unwilling to lobby in L to force a higher

policy, and unwilling to not lobby at H to avoid the cost. Similar for SIG2. Two Lobbies, Dichotomous Information, Unknown Biases Players: Policymaker SIGs Utilities: G(p,)=-(p-)2 U (p,l)=-(p--)2-l Biases 2>1 Lohmann (1993) suggests that extremists may always be willing to lobby but moderates will only do it when situations merit it. Thus, if some groups are active, they are likely the extremists, but if all are active then conditions may be more extreme.

An equilibrium exists where both lobby in H but not in L because they are both prefer higher policy levels than the policymaker. Endogenous Costs, Single SIG, Dichotomous Information Players: Policymaker SIG Bias Utilities: G(p,)=-(p-)2 U(p,l)=-(p--)2- > (H L)/2 l With fixed costs full revelation equilibria occur only for a range of possible values of lf. With variable costs, however, full revelation equilibria always exist. The SIG will never lobby in L, and in H will spend enough to tie his hands to show that this is the case. -2 -(H-L-)2-lH is necessary and sufficient so the SIG doesnt pretend the

state is H. If (qH qL)(2d+qH qL) k1 lf (qH qL)(2d+qH qL) k2 Just as in the fixed cost state, a cost between k1 and k2 is necessary for separating equilibrium. In this case, however, the SIG can choose the cost, and thus can always make themselves credible. Endogenous Costs, One SIG, Three States of Nature Players: Policymaker SIG Utilities: G(p,)=-(p-)2 U(p,l)=-(p--)2l Costs lL=0, lH> lM

Suppose that the policy maker expects the SIGs to not lobby at all if L, to pay lM if M, and to pay lH when H. So we want to find the smallest values that meet these constraints. lM=(M L)[2-(M-L)] However, to keep the SIG from stepping up a campaign in M to force policy appropriate for H, we need the following constraint as well: lH = 2(H L) (H M)2 (M L)2 This assures that the additional cost of lobbying for the higher level is to burdensome to be worth it when it is not merited, specifically when M. Endogenous Costs, One SIG, Continuous States of Nature Players: Policymaker Utilities: G(p,)=-(p-)2

SIG U(p,l)=-(p--)2-l Costs l(max)>l(min)=0 l()= 2 The policymaker believes that l() uniquely indicates a policy level p = . Additionally, the marginal cost of signaling a higher level must be equal to the associated marginal gain from a higher policy level. This is enumerated above. Combining these two yields: l()=2(-min) The SIG has no incentive to lie because its utility for a policy it supported, x, is: U(x) = -(x )2 - 2(x- min) Which reaches its maximum at , the true state of the world, so the SIG reveals truthfully when there is no fixed costs. Fixed and Variable Costs, Continuous States of Nature

Players: Policymaker Utilities: G(p,)=-(p-)2 SIG U(p,l)=-(p--)2-l Costs l(max)>l(min)=0 l()= 2 Suppose the policy maker becomes fully informed. In state min, the policy makers utility is very close, i.e. U() U(min). For a small enough difference, the difference in utility for the SIG between lobbying and not is too low to justify spending lf. Contradiction. There is not full

revelation. However an equilibrium with truthful revelation of all states 1 exists under the following conditions: 1 = min + 2((2 + lf) ) USIG(1|Lobbying) = USIG(1|Not Lobbying) l()= 2 marginal benefit of signaling = marginal cost of signaling, i.e. for all 1 l()= Policymaker-Imposed Access Costs When the information provided by SIGs is valuable to policymakers, why would they charge access costs? Time is a scarce resource and policy maker wants to make sure their time is well used Policymaker my have utility over contributions per se My use access costs to differentiate levels of bias Also whether access costs are paid before or after realization of matters. Since post realization fees regress to the cases we just studied, Well focus on imperfectly informed SIGs.

Access Costs, Known Bias Players: Policymaker Utilities: G(p,)=-(p-)2+(1-)c- SIG U(p,c)=-E(p--)2-c With known bias, the policymaker can extract the maximum the SIG is willing to pay. To check for equilibria, however, we must verify that this will be enough to be worth the policymakers time. cmax = [(n2max 1)/12]*[(((max min)2)/ n2max) - 42] Notice that bias is inversely correlated to cmax. This is because information transmitted for a given n is better and the maximum number of partitions grows as bias decreases.

Also notable is that the total gain to the policymaker from charging c is cmax+(1-)cmax = cmax. This indicates that if cmax > then the policymaker charges cmax and gets it, otherwise the policymaker makes herself inaccessible. Access Costs, Known Bias Players: Policymaker Utilities: G(p,)=-(p-)2+(1-)c- SIG U(p,c)=-E(p--)2-c What does this suggest? Low bias makes lobbying worth while to you. Higher bias will make the policymaker pay you so little heed, you are willing to

spend little, to nothing, for the opportunity to lobby. Access Costs, Unknown Bias Players: Policymaker Utilities: G(p,)=-(p-)2+(1-)c- SIG U(p,c)=-E(p--)2-c For tractability, the paper constrains so that a two partition equilibrium exists even for the highest and lowest biases. min > (max min)/12 max < (max min)/4 Here lobbying is compelling for groups with high or low bias, opposite from

before, where propensity to lobby decreased monotonically with . We need a point 1() where the SIG is indifferent between reporting low or high when its bias is . The indifference conditions is: 1() + pl = ph - 1() 1() = (1/2)(pl + ph) Updating with Bayes Rule (where is the average of s that result in lobbying): pL = E[L|L reported] = (3min + max)/4 Access Costs, Unknown Bias Players: Policymaker Utilities: G(p,)=-(p-)2+(1-)c- SIG U(p,c)=-E(p--)2-c

What does the firm gain from lobbying? The difference between the policy it would report, and the policy the policymaker would enact as a default from its priors: B(, ) = (1/2)(-)2 2 + (1/16)(max min)2 This function reaches its min at , which means that the group with the average bias for lobbying SIGs. It reaches its maxima at the extrema. The IR requirements, will always be satisfied at B(i, ) = c. An equilibrium at a given c is thus characterized an equation guaranteeing consistency of beliefs for the policy maker. Some algebra and noting the symmetry of the indifference points about yields: = (1/2)(min + max) But how does the policymaker choose the c youve been talking about? Access Costs, Unknown Bias Players: Policymaker

Utilities: G(p,)=-(p-)2+(1-)c- SIG U(p,c)=-E(p--)2-c Raising c narrows the range where the SIGs will lobby, with the following effects: 1. Reducing the expected impact of . 2. Changes the likelihood of lobby, which lowers the quality of p 3. Increases revenues in some cases The policymaker will maximize utility given and .