1 Introduction 2 The Model
Extended Abstract Delegation of Authority in non contractible cost setting Sneha Gaddam University of Leicester April 16, 2015
1
Introduction
Delegation of decision-making authority based on informational structure affects formal authority in firms and organizations (Aghion and Tirole, 1997). This decision-making power is also important in many real-life situations like tax auditing, political economics, environmental economics etc. In this paper, I focus on the delegation of decision-making power in a principalagent setting where the agents suffer from non-contractible, non-monetary costs when they commit a mistake. As an application, tax avoidance is a good example. Suppose a decision needs to be taken about auditing a tax payer. The tax payer’s income report is observed by two agents. Tax payer may or may not disclose the true income. Each agent separately receives a private signal from nature with some precision about the tax payer’s true income. Auditing is costly and auditing an honest tax payer is defined as a mistake. Besides the monetary costs involved in auditing, the agents would suffer from a non-contractible, non-monetary cost if they commit a mistake. A principal who may care only about the tax revenues or may also have non-contractible cost must now delegate the auditing power to one of these two agents. The principal in this example could be a politician who cares about her vote bank and therefore cares about not annoying the honest tax payers. She should delegate the decision-making power based on the agents’ signal precisions and also on their non-contractible costs.
2
The Model
Two agents, A and B see a signal yL where the true state could be {yL , yH }. An example of this could be the income report submitted by a tax payer. I focus on the pooling equilibrium where the tax payer whether low income earner (yL ) or a high income earner (yH ) always submits yL on the income report. Each of the agents has private information about this signal. The private information of A is precise with probability α and that of B is precise with a probability of β. The monetary cost of checking or auditing the signal is c. The agents suffer
1
from a non-contractible and non-monetary cost of γA and γB respectively if they audit and the tax payer turns out to be honest. One of the agents acts as a sender where he sends his private information to the other agent acting as the receiver. I focus on the truth-telling equilibria. A principal must decide who to delegate the decision making power to given the agents’ signal precisions and the level of non-contractible costs. Notations: t ∈ [0, 1] - Tax Rate F > 0 - Penalty to be paid by the tax payer if discovered to have reported fake income. A - Audit N - Not Audit η > 0 - Share of the total revenue that goes to the principal 1 − η > 0 - Share of the total revenue that goes to A and B δ > 0 - Share of the revenue from (1 − η) that goes to region A 1 − δ > 0 - Share of the revenue from (1 − η) that foes to region B π ∈ [0, 1] - Common prior of A and B over the true state of the exogenous signal, that is, y = yH α ∈ (0.5, 1] - Accuracy of the private signal of region A β ∈ (0.5, 1] - Accuracy of the private signal of region B Let x = t(yH − yL ) + F − c, zA = c +
γA (1−η)δ
and zB = c +
γB (1−η)(1−δ)
I look at all the decision rules (in pure strategies) possible, focus on the truth-telling equilibrium and then derive some preliminary results. Table 1: Decision Rules yˆA
yB
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
D11
D12
D13
D14
D15
D16
yH
yH
A
A
A
A
A
N
N
A
A
N
N
N
N
A
N
N
yL
yH
A
N
A
N
A
N
A
N
A
A
N
N
N
N
A
A
yH
yL
N
A
A
N
A
N
A
A
N
N
A
N
A
N
A
N
yL
yL
N
N
N
N
A
N
A
A
A
N
N
A
A
A
N
A
Proposition 1 It is irrelevant whom the principal chooses to give the decision making power between the agents A and B regardless of their signal precisions provided the non-contractible costs of both the agents are symmetrical (γA = γB ) and the revenues are shared symmetrically between the two agents. Proof case(i) πx > (1 − π)z: Let B be the decision maker. Pick any arbitrary (α, β). There are 2
three possibilities for the signal precisions. (i) β > α (ii)β = α (iii)β < α The agents A and B receive the signals yH or yL with the probabilities as outlined in the table 2 below.
Table 2: Decision Rules B as a decision maker
A as a decision maker
p
yˆA
yB
D1
D2
D3
D5
p
yˆB
yA
D1
D2
D3
D5
p1
yH
yH
A
A
A
A
p1
yH
yH
A
A
A
A
p2
yL
yH
A
N
A
A
p2
yH
yL
N
A
A
A
p3
yH
yL
N
A
A
A
p3
yL
yH
A
N
A
A
p4
yL
yL
N
N
N
A
p4
yL
yL
N
N
N
A
Under case i) β > α, the decision rules that give truth-telling are D1 , D3 and D5 in the region above the 45 degree line. The expected revenues to the third party from these rules chosen by the decision maker B are: from D1 , η{(p1 + p2 ){π(tyH + F − c) + (1 − π)(tyL − c)} + (p3 + p4 )tyL } which is η{(p1 + p2 ){π(t(yH − yL ) + F ) − c} + tyL } from D3 , η{(p1 + p2 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D5 , η{π(t(yH − yL ) + F ) − c + tyL } where η is the share of the total tax revenues the third party gets and
(1−η) 2
are the proportion
of tax revenues that the regions A and B keep. With A as the decision maker, the decision rules D2 , D3 and D5 give truth-telling equilibrium. The expected revenues to the principal from each of those rules are: from D2 , η{(p1 + p2 ){π(t(yH − yL ) + F ) − c} + tyL } from D3 , η{(p1 + p2 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D5 , η{π(t(yH − yL ) + F ) − c + tyL } It is clear that the expected revenues are exactly the same with A or B as decision makers.
Under case ii) β < α, the decision rules that give truth-telling are D2 , D3 and D5 in the region below the 45 degree line. The expected revenues to the principal from these rules chosen by the decision maker B are: 3
from D2 , η{(p1 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D3 , η{(p1 + p2 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D5 , η{π(t(yH − yL ) + F ) − c + tyL } With A as the decision maker, the decision rules D1 , D3 and D5 give truth-telling equilibrium. The expected revenues to the principal from each of those rules are: from D1 , η{(p1 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D3 , η{(p1 + p2 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D5 , η{π(t(yH − yL ) + F ) − c + tyL } In this case too, the expected revenues to the principal are exactly the same with A or B as decision makers.
Under case iii) β = α, the decision rules that give truth-telling are D3 and D5 regardless of who the decision maker is. So, once again, the expected revenues to the principal are exactly the same with A or B as decision makers.
case(ii) (1 − π)z > πx: Similar reasoning for the case when the expected costs of auditing are higher than expected benefits from auditing ((1 − π)z > πx) will lead to the result that it is irrelevant who the decision making authority is delegated to, given that the agents’ noncontractible costs are the same and they share the revenues equally. The above analysis is possible because of the assumption of symmetry of the contours of the decision rules about the 45 degree line and an algebraic proof confirming this symmetry is provided below: Proof of Symmetry: The aim of this sub section is to prove that in the case of πx > (1 − π)z, the curves πxα πxα+(1−π)z(1−α)
(function of the increasing concave curve separating D1 and D3 ,I name it
(1−π)zα πx(1−α)+(1−π)zα (function of the increasing convex curve separating D2 from D3 , πx(1−α) D3 )), πx(1−α)+(1−π)zα (function of the decreasing concave curve forming the upper
f (D1 : D3 )), f (D2 :
boundary of D5 , f (D5 )) are symmetric about the 45 degree lines. f (D1 : D3 ) =
πxα πxα + (1 − π)z(1 − α)
f (D2 : D3 ) =
(1 − π)zα πx(1 − α) + (1 − π)zα
f (D5 ) =
πx(1 − α) πx(1 − α) + (1 − π)zα 4
Verifying if the inverse of f (D1 : D3 ) = f (D2 : D3 ): β=
πxα πxα + (1 − π)z(1 − α)
β{πxα + (1 − π)z(1 − α)} = πxα α(πxβ) + (1 − π)zβ − (1 − π)zβα = πxα α[πx(1 − β) + (1 − π)zβ] = (1 − π)zβ α=
(1 − π)zβ πx(1 − β) + (1 − π)zβ
β=
(1 − π)zα πx(1 − α) + (1 − π)zα
Hence, f −1 (D1 : D3 ) = f (D2 : D3 ) Determining if f (D1 : D3 ) and f (D2 : D3 ) are one-to-one functions: πxα πxβ = πxβ + (1 − π)z(1 − β) πxα + (1 − π)z(1 − α) (πxα + (1 − π)z(1 − α))β = α(πxβ + (1 − π)z(1 − β)) (1 − α)β = (1 − β)α β=α Therefore, f (D1 : D3 ) is a one-to-one function (1 − π)zβ (1 − π)zα = πx(1 − β) + (1 − π)zβ πx(1 − α) + (1 − π)zα αβ(1 − π)z + α(1 − β)πx = πxβ(1 − α) + (1 − π)zαβ β=α Therefore, f (D2 : D3 ) is a one-to-one function Inverse of f (D5 ): β=
πx(1 − α) πx(1 − α) + (1 − π)zα
β(πx(1 − α) + (1 − π)zα) = πx(1 − α) α((1 − π)zβ − πxβ + πx) = πx − πxβ α=
πx(1 − β) πx(1 − β) + (1 − π)zβ
β=
πx(1 − α) πx(1 − α) + (1 − π)zα 5
Therefore, f −1 (D5 ) = f (D5 ) Determining if f (D5 ) is a one-to-one function: πx(1 − α) πx(1 − β) = πx(1 − β) + (1 − π)zβ πx(1 − α) + (1 − π)zα (1 − α)[πx(1 − β) + (1 − π)zβ] = (1 − β)[πx(1 − α) + (1 − π)zα] (1 − α)β = (1 − β)α β=α Hence, f (D5 ) is symmetric about the 45 degree line.
This section presented a preliminary result where the symmetric case has been analysed.
2.1
Asymmetric Case
This section analyses a sub case of asymmetry. Asymmetry arises due to one of the agents having a higher non-contractible costs than the other.
2.1.1
Analysis of truth-telling equilibrium under asymmetric case(γA > γB ) when B is decision maker
The symmetric case analysed in the previous sections has been extended to an asymmetric case. The asymmetric nature of this case arises due to the assumption γA > γB (the non-contractible cost of agent A is greater than that of agent B. γB is equal to the γB used under the symmetric case. That is, γA is higher than the non-contractible cost under the symmetric case). This is a deviation from the assumption in the symmetric case(where γA = γB ). However, the revenues between the two agents are still shared symmetrically, that is δ = 11 . Under this asymmetric case, γA > γB holds such that πx > (1 − π)zA and πx > (1 − π)zB . This means that the incentive is still in favour of auditing since the expected benefits from auditing are greater than the expected costs of auditing, whoever the decision maker is, A or B. When B is the decision maker, the truth-telling equilibrium is depicted in the graph captioned, “Asymmetric case: πx > (1 − π)zA and πx > (1 − π)zB , B as decision maker”. The decision rule D3 , which is to audit always except when both agents receive yL , is smaller compared to the area of D3 under the symmetric case. The intuition for this is, since the non-contractible cost of agent A is higher than that of agent B, the net expected benefits from auditing to A
6
are lower than those of B (πx − (1 − π)zA < πx − (1 − π)zB ). Although, the incentive is still in favour of auditing, it is not as high as it was under the symmetric case for agent A. Therefore, there is no incentive for A to tell the truth always in a region where both the signals are good enough but the net expected benefits from auditing to A are not as high as they were under the symmetric case. Here B might still want to audit, but A might not take the same decision if they were the decision maker. This is the reason for the non-truth-telling region between D1 and D3 , thereby reducing the truth-telling region of D3 .
2.1.2
Analysis of truth-telling equilibrium under asymmetric case(γA > γB ) when A is decision maker
If A is the decision maker and B is the sender, the truth-telling equilibrium is as depicted in the graph captioned “Asymmetric case: πx > (1 − π)zA and πx > (1 − π)zB , A as decision maker”. The expected costs of auditing to A are higher than they were under the symmetric case. The size of the truth-telling region where the decision rule D5 happens is smaller in this case than it was in the symmetric case and the case where B was the decision maker. This is the region where signals are uninformative. In the symmetric case or in the asymmetric case with B as the decision maker, when the signals were uninformative, it was optimal to always audit because the expected benefits from auditing are higher compared to the expected costs of auditing. In this case where A is the decision maker, the expected benefits from auditing are still higher than the expected costs from auditing. But the non-contractible cost(γA ) of agent A is higher than it was in the symmetric case. This means that although the expected benefits to A from auditing are higher than the expected costs of auditing, they are not as high as they were in the symmetric case. Hence when A is given the decision making power, they would still audit in a small region where the signals are informative, but the incentive to always audit starts decreasing when the signals start becoming informative. But if B was the decision maker, then they would like to always audit in this region like in the symmetric case. This explains the reason for the gap in between the truth-telling regions D3 and D5 .
7
Further work in progress is to analyse all possible cases of asymmetry and determine the optimal delegation choice of the principal.
References Aghion, P. and Tirole, J. (1997). Formal and real authority in organizations. The Journal of Political Economy, 105 (37), 1–29.
10
1
Introduction
Delegation of decision-making authority based on informational structure affects formal authority in firms and organizations (Aghion and Tirole, 1997). This decision-making power is also important in many real-life situations like tax auditing, political economics, environmental economics etc. In this paper, I focus on the delegation of decision-making power in a principalagent setting where the agents suffer from non-contractible, non-monetary costs when they commit a mistake. As an application, tax avoidance is a good example. Suppose a decision needs to be taken about auditing a tax payer. The tax payer’s income report is observed by two agents. Tax payer may or may not disclose the true income. Each agent separately receives a private signal from nature with some precision about the tax payer’s true income. Auditing is costly and auditing an honest tax payer is defined as a mistake. Besides the monetary costs involved in auditing, the agents would suffer from a non-contractible, non-monetary cost if they commit a mistake. A principal who may care only about the tax revenues or may also have non-contractible cost must now delegate the auditing power to one of these two agents. The principal in this example could be a politician who cares about her vote bank and therefore cares about not annoying the honest tax payers. She should delegate the decision-making power based on the agents’ signal precisions and also on their non-contractible costs.
2
The Model
Two agents, A and B see a signal yL where the true state could be {yL , yH }. An example of this could be the income report submitted by a tax payer. I focus on the pooling equilibrium where the tax payer whether low income earner (yL ) or a high income earner (yH ) always submits yL on the income report. Each of the agents has private information about this signal. The private information of A is precise with probability α and that of B is precise with a probability of β. The monetary cost of checking or auditing the signal is c. The agents suffer
1
from a non-contractible and non-monetary cost of γA and γB respectively if they audit and the tax payer turns out to be honest. One of the agents acts as a sender where he sends his private information to the other agent acting as the receiver. I focus on the truth-telling equilibria. A principal must decide who to delegate the decision making power to given the agents’ signal precisions and the level of non-contractible costs. Notations: t ∈ [0, 1] - Tax Rate F > 0 - Penalty to be paid by the tax payer if discovered to have reported fake income. A - Audit N - Not Audit η > 0 - Share of the total revenue that goes to the principal 1 − η > 0 - Share of the total revenue that goes to A and B δ > 0 - Share of the revenue from (1 − η) that goes to region A 1 − δ > 0 - Share of the revenue from (1 − η) that foes to region B π ∈ [0, 1] - Common prior of A and B over the true state of the exogenous signal, that is, y = yH α ∈ (0.5, 1] - Accuracy of the private signal of region A β ∈ (0.5, 1] - Accuracy of the private signal of region B Let x = t(yH − yL ) + F − c, zA = c +
γA (1−η)δ
and zB = c +
γB (1−η)(1−δ)
I look at all the decision rules (in pure strategies) possible, focus on the truth-telling equilibrium and then derive some preliminary results. Table 1: Decision Rules yˆA
yB
D1
D2
D3
D4
D5
D6
D7
D8
D9
D10
D11
D12
D13
D14
D15
D16
yH
yH
A
A
A
A
A
N
N
A
A
N
N
N
N
A
N
N
yL
yH
A
N
A
N
A
N
A
N
A
A
N
N
N
N
A
A
yH
yL
N
A
A
N
A
N
A
A
N
N
A
N
A
N
A
N
yL
yL
N
N
N
N
A
N
A
A
A
N
N
A
A
A
N
A
Proposition 1 It is irrelevant whom the principal chooses to give the decision making power between the agents A and B regardless of their signal precisions provided the non-contractible costs of both the agents are symmetrical (γA = γB ) and the revenues are shared symmetrically between the two agents. Proof case(i) πx > (1 − π)z: Let B be the decision maker. Pick any arbitrary (α, β). There are 2
three possibilities for the signal precisions. (i) β > α (ii)β = α (iii)β < α The agents A and B receive the signals yH or yL with the probabilities as outlined in the table 2 below.
Table 2: Decision Rules B as a decision maker
A as a decision maker
p
yˆA
yB
D1
D2
D3
D5
p
yˆB
yA
D1
D2
D3
D5
p1
yH
yH
A
A
A
A
p1
yH
yH
A
A
A
A
p2
yL
yH
A
N
A
A
p2
yH
yL
N
A
A
A
p3
yH
yL
N
A
A
A
p3
yL
yH
A
N
A
A
p4
yL
yL
N
N
N
A
p4
yL
yL
N
N
N
A
Under case i) β > α, the decision rules that give truth-telling are D1 , D3 and D5 in the region above the 45 degree line. The expected revenues to the third party from these rules chosen by the decision maker B are: from D1 , η{(p1 + p2 ){π(tyH + F − c) + (1 − π)(tyL − c)} + (p3 + p4 )tyL } which is η{(p1 + p2 ){π(t(yH − yL ) + F ) − c} + tyL } from D3 , η{(p1 + p2 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D5 , η{π(t(yH − yL ) + F ) − c + tyL } where η is the share of the total tax revenues the third party gets and
(1−η) 2
are the proportion
of tax revenues that the regions A and B keep. With A as the decision maker, the decision rules D2 , D3 and D5 give truth-telling equilibrium. The expected revenues to the principal from each of those rules are: from D2 , η{(p1 + p2 ){π(t(yH − yL ) + F ) − c} + tyL } from D3 , η{(p1 + p2 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D5 , η{π(t(yH − yL ) + F ) − c + tyL } It is clear that the expected revenues are exactly the same with A or B as decision makers.
Under case ii) β < α, the decision rules that give truth-telling are D2 , D3 and D5 in the region below the 45 degree line. The expected revenues to the principal from these rules chosen by the decision maker B are: 3
from D2 , η{(p1 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D3 , η{(p1 + p2 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D5 , η{π(t(yH − yL ) + F ) − c + tyL } With A as the decision maker, the decision rules D1 , D3 and D5 give truth-telling equilibrium. The expected revenues to the principal from each of those rules are: from D1 , η{(p1 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D3 , η{(p1 + p2 + p3 ){π(t(yH − yL ) + F ) − c} + tyL } from D5 , η{π(t(yH − yL ) + F ) − c + tyL } In this case too, the expected revenues to the principal are exactly the same with A or B as decision makers.
Under case iii) β = α, the decision rules that give truth-telling are D3 and D5 regardless of who the decision maker is. So, once again, the expected revenues to the principal are exactly the same with A or B as decision makers.
case(ii) (1 − π)z > πx: Similar reasoning for the case when the expected costs of auditing are higher than expected benefits from auditing ((1 − π)z > πx) will lead to the result that it is irrelevant who the decision making authority is delegated to, given that the agents’ noncontractible costs are the same and they share the revenues equally. The above analysis is possible because of the assumption of symmetry of the contours of the decision rules about the 45 degree line and an algebraic proof confirming this symmetry is provided below: Proof of Symmetry: The aim of this sub section is to prove that in the case of πx > (1 − π)z, the curves πxα πxα+(1−π)z(1−α)
(function of the increasing concave curve separating D1 and D3 ,I name it
(1−π)zα πx(1−α)+(1−π)zα (function of the increasing convex curve separating D2 from D3 , πx(1−α) D3 )), πx(1−α)+(1−π)zα (function of the decreasing concave curve forming the upper
f (D1 : D3 )), f (D2 :
boundary of D5 , f (D5 )) are symmetric about the 45 degree lines. f (D1 : D3 ) =
πxα πxα + (1 − π)z(1 − α)
f (D2 : D3 ) =
(1 − π)zα πx(1 − α) + (1 − π)zα
f (D5 ) =
πx(1 − α) πx(1 − α) + (1 − π)zα 4
Verifying if the inverse of f (D1 : D3 ) = f (D2 : D3 ): β=
πxα πxα + (1 − π)z(1 − α)
β{πxα + (1 − π)z(1 − α)} = πxα α(πxβ) + (1 − π)zβ − (1 − π)zβα = πxα α[πx(1 − β) + (1 − π)zβ] = (1 − π)zβ α=
(1 − π)zβ πx(1 − β) + (1 − π)zβ
β=
(1 − π)zα πx(1 − α) + (1 − π)zα
Hence, f −1 (D1 : D3 ) = f (D2 : D3 ) Determining if f (D1 : D3 ) and f (D2 : D3 ) are one-to-one functions: πxα πxβ = πxβ + (1 − π)z(1 − β) πxα + (1 − π)z(1 − α) (πxα + (1 − π)z(1 − α))β = α(πxβ + (1 − π)z(1 − β)) (1 − α)β = (1 − β)α β=α Therefore, f (D1 : D3 ) is a one-to-one function (1 − π)zβ (1 − π)zα = πx(1 − β) + (1 − π)zβ πx(1 − α) + (1 − π)zα αβ(1 − π)z + α(1 − β)πx = πxβ(1 − α) + (1 − π)zαβ β=α Therefore, f (D2 : D3 ) is a one-to-one function Inverse of f (D5 ): β=
πx(1 − α) πx(1 − α) + (1 − π)zα
β(πx(1 − α) + (1 − π)zα) = πx(1 − α) α((1 − π)zβ − πxβ + πx) = πx − πxβ α=
πx(1 − β) πx(1 − β) + (1 − π)zβ
β=
πx(1 − α) πx(1 − α) + (1 − π)zα 5
Therefore, f −1 (D5 ) = f (D5 ) Determining if f (D5 ) is a one-to-one function: πx(1 − α) πx(1 − β) = πx(1 − β) + (1 − π)zβ πx(1 − α) + (1 − π)zα (1 − α)[πx(1 − β) + (1 − π)zβ] = (1 − β)[πx(1 − α) + (1 − π)zα] (1 − α)β = (1 − β)α β=α Hence, f (D5 ) is symmetric about the 45 degree line.
This section presented a preliminary result where the symmetric case has been analysed.
2.1
Asymmetric Case
This section analyses a sub case of asymmetry. Asymmetry arises due to one of the agents having a higher non-contractible costs than the other.
2.1.1
Analysis of truth-telling equilibrium under asymmetric case(γA > γB ) when B is decision maker
The symmetric case analysed in the previous sections has been extended to an asymmetric case. The asymmetric nature of this case arises due to the assumption γA > γB (the non-contractible cost of agent A is greater than that of agent B. γB is equal to the γB used under the symmetric case. That is, γA is higher than the non-contractible cost under the symmetric case). This is a deviation from the assumption in the symmetric case(where γA = γB ). However, the revenues between the two agents are still shared symmetrically, that is δ = 11 . Under this asymmetric case, γA > γB holds such that πx > (1 − π)zA and πx > (1 − π)zB . This means that the incentive is still in favour of auditing since the expected benefits from auditing are greater than the expected costs of auditing, whoever the decision maker is, A or B. When B is the decision maker, the truth-telling equilibrium is depicted in the graph captioned, “Asymmetric case: πx > (1 − π)zA and πx > (1 − π)zB , B as decision maker”. The decision rule D3 , which is to audit always except when both agents receive yL , is smaller compared to the area of D3 under the symmetric case. The intuition for this is, since the non-contractible cost of agent A is higher than that of agent B, the net expected benefits from auditing to A
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are lower than those of B (πx − (1 − π)zA < πx − (1 − π)zB ). Although, the incentive is still in favour of auditing, it is not as high as it was under the symmetric case for agent A. Therefore, there is no incentive for A to tell the truth always in a region where both the signals are good enough but the net expected benefits from auditing to A are not as high as they were under the symmetric case. Here B might still want to audit, but A might not take the same decision if they were the decision maker. This is the reason for the non-truth-telling region between D1 and D3 , thereby reducing the truth-telling region of D3 .
2.1.2
Analysis of truth-telling equilibrium under asymmetric case(γA > γB ) when A is decision maker
If A is the decision maker and B is the sender, the truth-telling equilibrium is as depicted in the graph captioned “Asymmetric case: πx > (1 − π)zA and πx > (1 − π)zB , A as decision maker”. The expected costs of auditing to A are higher than they were under the symmetric case. The size of the truth-telling region where the decision rule D5 happens is smaller in this case than it was in the symmetric case and the case where B was the decision maker. This is the region where signals are uninformative. In the symmetric case or in the asymmetric case with B as the decision maker, when the signals were uninformative, it was optimal to always audit because the expected benefits from auditing are higher compared to the expected costs of auditing. In this case where A is the decision maker, the expected benefits from auditing are still higher than the expected costs from auditing. But the non-contractible cost(γA ) of agent A is higher than it was in the symmetric case. This means that although the expected benefits to A from auditing are higher than the expected costs of auditing, they are not as high as they were in the symmetric case. Hence when A is given the decision making power, they would still audit in a small region where the signals are informative, but the incentive to always audit starts decreasing when the signals start becoming informative. But if B was the decision maker, then they would like to always audit in this region like in the symmetric case. This explains the reason for the gap in between the truth-telling regions D3 and D5 .
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Further work in progress is to analyse all possible cases of asymmetry and determine the optimal delegation choice of the principal.
References Aghion, P. and Tirole, J. (1997). Formal and real authority in organizations. The Journal of Political Economy, 105 (37), 1–29.
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