Bundling of Authority and Accountability

in Organizations

Susheng Wang1

August 2015

Abstract: The authority to make decisions is often bundled with accountability. Decision

makers are often made accountable for failures. We propose an incomplete-contract approach

to investigate four systems of accountability: no accountability, personal accountability, collec-

tive accountability, and reverse accountability. We consider an organization defined by three

assignments: assignments of income, authority and accountability. These assignments are

endogenously and jointly determined in equilibrium. This setup allows us to investigate

whether and when authority and accountability are bundled in equilibrium. Our main findings

are: (1) authority and accountability are usually, but not always, bundled in equilibrium; (2)

personal and collective accountability systems are equally efficient; (3) it can be optimal to

have no accountability, but only if the authorities are incapable; it is always optimal to impose

accountability on capable authorities; (4) authority tends to be assigned to persons whose

marginal costs of control are relatively small; and (5) reverse accountability is generally inferi-

or to normal accountability; reverse accountability can be optimal only if everyone’s marginal

cost of control is sufficiently large.

Keywords: authority, accountability, organizations, incomplete-contract approach

JEL classification: M12, M14, L23

1 Address: Hong Kong University of Science and Technology. Email: s.wang@ust.hk.

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1. Introduction

In recent years, accountability has been gaining increasing attention. People have come to

realize that accountability can play an important role in business activities, especially after the

financial crisis in 2008. The common view nowadays is that authority should be bundled with

accountability. But is this approach truly economically efficient? Could having no accountabil-

ity be efficient under some circumstances? We show that authority should in general be bun-

dled with accountability, with a few exceptions.

According to BusinessDictionary.com, authority is “institutionalized and legal power in-

herent in a particular job, function, or position that is meant to enable its holder to successful-

ly carry out his or her responsibilities,” and accountability is “the obligation of an individual or

organization to account for its activities, accept responsibility for them, and to disclose the

results in a transparent manner. It also includes the responsibility for money or other entrust-

ed property.” Specifically, in our model, accountability for a project failure takes the form of a

penalty beyond the direct consequences of the failure. A failure may or may not be the result of

making the wrong decision, but the chances of failure can be reduced if the authority is dili-

gent enough. When a project fails, everyone involved suffers, including the decision maker and

non-decision makers. Accountability imposes an extra penalty on someone, who may or may

be the authority. The penalty can be a loss of money, a demotion, a moral cost, a loss of repu-

tation, a public apology, etc.

Accountability can take several forms depending on who is accountable for a failure. The

most popular form of accountability is personal accountability, by which the authority is solely

accountable for the failure. A second popular form of accountability is no accountability, by

which no one is accountable for the failure. A third popular form of accountability is collective

accountability, by which everyone involved, including non-decision makers, is accountable for

the failure. We also consider a fourth form of accountability, called reverse accountability, by

which the authority is not accountable for the failure but others are. Reverse accountability

does not exist in practice, but is theoretically feasible and can help us understand the role of

accountability.

Authority is bundled with accountability in the cases of personal and collective accounta-

bility but not in the cases of reverse and no accountability. In real life, authority is often bun-

dled with accountability, i.e., the authority is made accountable for failures. Why is that? Why

is personal accountability the most popular in practice? Why is no accountability or collective

accountability adopted in some cases? Is reverse accountability always inferior? To address

these questions, we must determine which accountability system is optimal under what condi-

tions.

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We propose an incomplete-contract approach to investigate four popular accountability

systems: personal accountability, no accountability, collective accountability, and reverse

accountability. We consider an organization defined by three assignments: assignments of

income, authority and accountability. These three assignments are endogenously and jointly

determined in equilibrium. This setup allows us to investigate whether and when authority

and accountability are bundled in equilibrium. Our main findings are: (1) authority and ac-

countability are usually, but not always, bundled in equilibrium; (2) personal and collective

accountability systems are equally efficient; (3) it can be optimal to have no accountability, but

only if the authorities are incapable; it is always optimal to impose accountability on capable

authorities; (4) authority tends to be assigned to a manager whose marginal cost of control is

small; and (5) reverse accountability is generally inferior to normal accountability; reverse

accountability can be optimal if everyone’s marginal cost of control is sufficiently large.

The incomplete-contract approach was proposed by Grossman, Hart and Moore (Gross-

man & Hart, 1986; Hart 1988; Hart & Moore, 1990). Based on Coase’s (1960) idea of resolving

conflicts of interest through an organizational approach, the incomplete-contract approach

focuses on the assignment of control rights in organizations. Following Grossman & Hart

(1986), Wang & Zhu (2005) focus on two assignments, the assignments of income and control,

and discuss the conditions under which these two assignments are bundled in equilibrium. We

introduce a third assignment into the model: the assignment of accountability. The three

assignments are endogenously and jointly determined in equilibrium. Our focus is on the

other two assignments: the assignments of authority and accountability. We discuss the condi-

tions under which these two assignments are bundled in equilibrium.

The three forms of accountability addressed in our theory are widely applied in practice,

with personal accountability being the most popular. The best known mechanism of accounta-

bility is political elections. Elected authorities often base their decisions on public opinion and

those representatives who fail to fulfil their election promises are voted out of office. The US

constitution is built on accountability. The frequency of elections, the separation of powers,

and the strength of parliament ensure that the government is accountable to the people. Firms

are allowed the freedom to conduct business activity, but society is also increasingly demand-

ing that they be socially accountable. Corporate governance is built on accountability in that,

the CEO is accountable to the board of directors.

Having no accountability is another popular approach. Individual shareholders give the

CEO the right to invest their money, but the CEO is not accountable to individual shareholders.

He or she is only accountable to large institutional shareholders. When we see a doctor, we

offer the doctor the right to decide on a cure, but the doctor is not usually accountable for

failure, unless it involves medical misconduct or negligence. When a school hires a teacher, it

offers the teacher the right to teach at the school, but he or she is not accountable if students

fail the course. When we buy a product, if there is no warranty, the seller is not accountable for

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any problem with the product. A warranty ties the seller’s right of income with accountability.

In a democracy, voters are given the right to vote but are not accountable for making a bad

choice, although they suffer the consequences. When we invest in mutual funds, we give the

fund manager the right to invest our money but he or she is not accountable for poor fund

performance.

Collective accountability is also popular. In some regimes in the past, the death penalty

may be given not only to a criminal but also his or her whole family and even extended rela-

tives, depending on the seriousness of the crime committed. For many illegal activities, such

as the cocaine and ivory trade and human trafficking, both the sellers and buyers are punisha-

ble.2 Our theory intends to offer a basic understanding to these applications.

Although accountability is well discussed in the literature, few equilibrium theories exist

that tie several major mechanisms in organizations in equilibrium. To our knowledge, we are

the first to offer such a theory based on an incomplete-contract approach (an organizational

approach). The literature was pioneered by Garfinkel (1967), who describes an interesting

example in which a man who undergoes a sex change to become a woman. She has the right to

do so, but she is accountable for any misunderstanding from the change. Accountability in-

duces her to signal in female dress and explain in subtle detail to her friends. Heritage (1984)

offers a detailed discussion of accountability following Garfinkel’s observations and thoughts,

which lays a theoretical foundation of ethnomethodology. Perakyla (1998) observes that doc-

tors in Finland will try to write down their diagnoses in detail in order to address accountabil-

ity. Ferejohn (1999) emphasizes endogenous accountability, by which “public officials are

induced to make their actions relatively controllable by their principals, in order to attract

resources and support.” Acar et al. (2008) study an interesting case of accountability under

joint authority in public-private partnerships. They identify five potential functions of ac-

countability. Yang (2012) offers a recent survey of accountability in public organizations and

emphasizes accountability as an endogenous actor. In our model, the assignment of accounta-

bility is endogenously and jointly determined with other assignments, such as the assignments

of authority and income, based on economic efficiency.

The rest of the paper proceeds as follows. Section 2 lays out the model. Section 3 derives

and analyzes the solution. Section 4 offers concluding remarks. The proofs are in the Appendix.

2 A project in our model may be a socially good project or a socially bad project including crimes. “Social wel-

fare” in our model is the total welfare of all trading parties involved.

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2. The Model

Our model follows Grossman & Hart (1986) and Wang & Zhu (2005). Grossman & Hart

(1986) propose a model with ex ante investments and ex post controls. They allow an endoge-

nous assignment of control conditional on a given assignment of income. Wang & Zhu (2005)

extend Grossman & Hart’s (1986) model by allowing an endogenous assignment of income, in

addition to an endogenous assignment of control. We further extend Wang & Zhu’s (2005)

model by introducing accountability and an endogenous assignment of accountability, in

addition to endogenous assignments of income and control. Our purpose is to study the opti-

mal assignment of accountability.

Project

Two risk-neutral managers and engage in a project. They simultaneously provide

ex ante investments and in the first period, with private costs and .

These investments are uncontractable. They form a joint investment The project

lasts two periods. At the end of the second period, the project yields revenue if it suc-

ceeds or nothing if it fails.

In the second period, there is a further action to be taken. The controlling manager (the

authority) takes this action which determines the success probability of the project, such

that is the ex post expected revenue. For convenience, we will simply assume that the au-

thority chooses in the second period. We can think of as quality; after the initial

investment, in the second stage of production, the authority controls the quality of the product

to improve sales. Assume that is uncontractable ex ante but contractable ex post.

The sequence of events is illustrated in the time line of Figure 1.

0 1 2

Contracting 21, ee qAllocatingrevenue

Uncertainty realized;Renegotiation possible

Ex Ante Ex Post

Figure 1. The Timing of Events

Authority

Although is uncontractable ex ante, we assume that the right to determine its value ex

post is contractable ex ante. That is, the authority or the control right over is contractable ex

ante. The idea of contractable authority over an uncontractable variable is the key contribution

of Grossman & Hart (1986).

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Let be the cost of control where the cost is dependent on the size of the project

The cost of control represents the cost of administration, information gathering, and time

involved in making a decision.

The authority personally benefits from being in charge. Let the benefit of control be

which is the monetary gain from being the authority. But authority may come with accounta-

bility. Suppose that accountability takes the form of a monetary penalty . If the project

fails, besides the loss implied by the income-sharing scheme, the manager who is accountable

for the failure must also pay .

Contracting

At the end of the second period, revenue is divided between the two managers based on

a revenue-sharing assignment in their contractual agreement. A contract contains the assign-

ments of income, authority and accountability. The assignment of income is defined by a

revenue-sharing scheme , where is the income of if revenue is . The

assignment of authority indicates who has the right to decide the value of ex post; it does not

fix the value of , since is uncontractable ex ante. The assignment of accountability imposes

a penalty when the project fails. When the project fails, everyone suffers as the income-sharing

assignment offers the penalty imposes an extra income loss on the manag-

er who is accountable for the failure.

The ex ante contract is renegotiable ex post. Since is contractable ex post, its value can

be fixed in the ex post contract. The authority has an advantage in bargaining power when the

ex ante contract is open for renegotiation; but once the value of is determined in the ex post

contract, she loses her control over . Hence, admissible contracts for the two types of con-

tracts are of the following forms:

The ex ante contract defines bargaining powers for ex post renegotiation on the one hand,

and will take into account the impact of renegotiation on the other. Hence, after taking into

account the impact of renegotiation, the equilibrium ex ante contract must be renegotiation-

proof, i.e., there will be no renegotiation in equilibrium.

Three key differences exist between our model and the models of Grossman & Hart (1986)

and Wang & Zhu (2005). First, we introduce accountability and the assignment of ac-

countability is endogenously determined jointly with the assignments of income and authority.

Second, we introduce the cost of control . Third, the benefit of control in our

model is different from that in Grossman & Hart (1986) and Wang & Zhu (2005). Our is

personal benefit, which does not cost the firm. The benefit of control in Grossman & Hart

(1986) and Wang & Zhu (2005) represents the ability to steal, which will cost the firm. Our

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benefit of control has characteristics similar to those of the value of accountability and the cost

of control, and offers comparable terms.

Accountability

There are four admissible assignments of accountability:

(1) Personal accountability: the authority is accountable.

(2) No accountability: no one is accountable.

(3) Collective accountability: everyone is accountable.

(4) Reverse accountability: the authority is not accountable, but others are.

Accountability may induce the manager to work diligently. To assign authority and ac-

countability properly, one needs to balance among the marginal cost of control, the ability to

control, and the marginal benefit of control.

Although reverse accountability is socially senseless, it is a feasible option theoretically.

We can gain a better understanding by studying this special accountability system.

Assumptions

Output is random ex ante with density function . Let be the ex

ante revenue

and denote ,

Assumption 1. is strictly increasing in and

Assumption 2. and are convex and strictly increasing in and ,

respectively.

Assumption 3. is concave and strictly increasing in and

Parametric Functions

In order to find a closed-form solution, we will use the following set of parametric func-

tions:3

3 Among these functions, the revenue-sharing scheme is endogenous. The linearity of this sharing

scheme is not an assumption. Given Assumptions 1-3 and that admissible revenue-sharing schemes are arbitrary

Lebesgue-integrable functions, the equilibrium revenue-sharing scheme must be linear. See Wang & Zhu (2005).

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where represents the share, represents accountability, represents the benefit, and

represents the cost. The cost function satisfies Assumption 2. Let the joint investment

be of the form

(2)

where and are positive constants, representing control abilities. satisfies As-

sumption 1. Also, let

(3)

where and Thus,

(4)

which satisfies Assumption 3. Assume and . Without the loss of

generality, assume meaning that has better control than . For convenience,

denote

With we have and .

3. The Solution and Analysis

If the ex ante investments and are contractable, the equilibrium solution is Pareto

optimal and is called the first-best (FB) solution. If the ex ante investments and are

uncontractable and the renegotiation-proof condition is not binding (the renegotiation-proof

condition is automatically satisfied without further restrictions on choices), the equilibrium

solution is not Pareto optimal and is called the second-best (SB) solution. If the ex ante in-

vestments and are uncontractable and the renegotiation-proof condition is binding (the

renegotiation-proof condition implies a restriction on choices), the equilibrium solution is

inferior to the SB solution and is called the third-best (TB) solution. The solution in our model

is either SB or TB.

The following lemma from Wang & Zhu (2005) is useful for our derivation of the solution.

Lemma 1. Suppose that is the authority and ∗ is the SB revenue share. Under the condi-

tion that she must be given a revenue share satisfying for some if ∗ her

optimal share is ∗ but if ∗ her optimal share is

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Figure 2. Lemma 1

The intuition of Lemma 1 is clear. Under condition if ∗ is feasible, giving ∗ yields

the SB outcome, which is the most efficient solution in our model. If ∗ is not feasible, the

optimal revenue share should be as close to ∗ as possible, which is .

As will be shown later, the condition is a renegotiation-proof condition when is

the authority.

3.1. Personal Accountability

In this subsection, we restrict ourselves to two admissible accountability systems: person-

al accountability and no accountability. The other two accountability systems will be consid-

ered in the next two subsections. We solve the problem in three steps.

Step 1. The Assignment of Income

In the first step, conditional on the assignments of authority and accountability given, we

find the equilibrium assignment of income. Suppose is the authority with personal ac-

countability. At time , given her revenue share in the ex ante contract, assesses her

income

(5)

Based on this income, will choose if ; but if , may choose

.

However, for the case with there is an alternative for : may demand re-

negotiation. Since is not a socially optimal choice, both parties can benefit from renego-

tiation. With , if bargaining fails, will choose and receive and

will receive nothing; if bargaining leads to an agreement on , will receive

and will receive . The agreement implies social welfare

maximization

∈[ , 1]

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which implies ∗ since . Hence, by the Nash bargaining solution, through rene-

gotiation will receive

and will receive

Notice that the term is what will gain as the authority; she actually receives revenue

share from bargaining. Since , will bargain instead of choos-

ing if However, since , by Lemma 1, it is better to offer

revenue share to in the first place in the ex ante contract. With ,

will voluntarily choose ∗ without renegotiation.

In summary, if will choose ∗ . If will receive revenue

share by choosing or she will receive revenue share by bargaining. is

better off bargaining if But by Lemma 1, since the bargaining

solution is worse than giving the revenue share ex ante. Thus, should never

be given a revenue share less than . Hence, the two parties’ ex post payoffs are

In equilibrium, the authority will always choose voluntarily, the penalty is never

levied, and the contract never needs to be renegotiated.

Without taking into account the costs of ex ante investments, the ex ante payoffs are

In the first period, the two parties choose and in a Nash equilibrium. The Nash equilibri-

um is determined by two first-order conditions, called IC conditions:

At given the assignments of authority and accountability, an income assignment is

determined by bargaining. Bargaining leads to social welfare maximization. Hence, the ex ante

problem is

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, , ,(6)

We now solve problem (6). Under condition problem (6) can be solved in two

steps. We first solve for ∗ ∗ from the following problem:

∗ ,(7)

Then, given ∗ ∗ ∗ and ∗ are determined by the two IC conditions:

∗ ∗∗ ∗ ∗ ∗∗ ∗By the constraint in (7), we have ∗ ∗ . This solution is SB. We can see that, for the SB

solution, social welfare, investments and income shares are independent of accountability.

But if the revenue shares are already determined, which are and

Then, the two IC conditions determine investments

and problem (6) has no optimization, implying social welfare

This solution is TB. From (9), with the parametric functions in (1)-(3), we can see that ac-

countability reduces but raises .

With the parametric functions in (1)-(3), we can find a closed-form solution. If is in

control, when ∗ , we have an SB solution

∗ ∗∗ ∗

∗When ∗ , we have a TB solution

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Symmetrically, if is in control, when ∗ , we have an SB solution

∗ ∗∗ ∗

∗When ∗ , we have a TB solution

Step 2. The Assignment of Authority

In the second step, given the above equilibrium assignment of income conditional on the

assignments of authority and accountability given, we find the most efficient assignment of

authority.

Since our solutions are either SB or TB, an SB solution is the most efficient solution. Giv-

en the parametric functions in (1)-(3), we find that

with personal accountability, and

without accountability, where “iff” is short for “if and only if”.

If only a TB solution can be found, to find the most efficient solution, we need to compare

with , with and without accountability. We use the symbol to indicate that having one

manager take control is more efficient than having the other manager take control. We find

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The preferences in (17) are depicted in Figures 3 and 4. As shown in Figure 3, given personal

accountability as defined by the solution is:

(a) In zone where ’s marginal cost of control is low, having take control is the most

efficient. The solution is SB.

(b) In zone where ’s marginal cost of control is low, having take control is the most

efficient. The solution is SB.

(c) In zone where both and ’s marginal costs of control are low, having either or

take control is the most efficient. The solution is SB.

(d) In zone where ’s marginal cost of control is relatively low, having take control is

the most efficient. The solution is TB.

(e) In zone where ’s marginal cost of control is relatively low, having take control is

the most efficient. The solution is TB.

Symmetrically, the preferences under no accountability are similar and illustrated in Figure 4.

1 2

4 5

Figure 3. Control Zones with Personal Accountability

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2

4 5

Figure 4. Control Zones without Accountability

Several observations can be made from the above two contrasting solutions. First, from

(11) and (13), we can see that, for the SB solution, social welfare, investments and income

shares are independent of Hence, the SB solutions with and without accountability are

equally efficient.

Second, by comparing Figure 3 with Figure 4, the SB zones and with accountability

are larger than the matching zones without accountability. Hence, if the marginal costs of

control are not large such that the solution is SB, accountability can improve efficiency.

Third, given an accountability system, authority tends to be given to the manager whose

marginal cost of control is lower.

Step 3. The Assignment of Accountability

In the final step, given the above equilibrium assignments of income and authority that

are conditional on a given assignment of accountability, we now derive the most efficient

assignment of accountability. Note that we restrict ourselves to two admissible accountability

systems in this subsection: personal accountability and no accountability.

To find the most efficient assignment of accountability, we not only need the preferences

in (17), but also the following preferences:

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For example, to show that having in control is the most efficient, we need to have either of

the following two sets of results, i.e., find conditions under which one of the following sets of

results holds:

• “ ” under accountability, and “accountability no accountability” under control;

• “ ” under no accountability, and “no accountability accountability” under

control.

The result is stated in the following proposition.

Proposition 1 (Personal Accountability). The most efficient solution is indicated in Figure 5.

Specifically,

(a) In zone should take control with or without accountability. The solution is SB.

(b) In zones and should take control with or without accountability. The solution is

SB.

(c) In zone should take control with accountability. The solution is SB.

(d) In zones and , should take control with accountability. The solution is SB.

(e) In zone should take control without accountability. The solution is TB.

(f) In zone should take control with accountability. The solution is TB.

(g) In zones and should take control with accountability. The solution is TB.

(h) In zone or should take control, where takes control with or without account-

ability but takes control with accountability. The solution is SB.

(i) In zone or should take control, where takes control with accountability and

takes control with or without accountability. The solution is SB.

(j) In zone or should take control with accountability. The solution is SB.

(k) In zone or should take control with or without accountability. The solution is

SB.

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2 4

5

6

7 8 9 10

13

Figure 5. Efficient Control Zones

We make several observations from Figure 5. First, Figure 5 indicates that authority is

usually, but not always, bundled with accountability. In zones 2, 4, 5, 6, 8, 9 and 10, authority

is bundled with accountability; in zones 1, 7, 11, 12, 13 and 14, it is equally efficient to bundle or

not bundle authority with accountability; only in zone 3, when a relatively incapable manager

is in control, is authority not bundled with accountability. Our result is consistent with reality

where the bundling of authority with accountability is most popular but not bundling is also

popular.

Second, in zones 1, 7, 11, 12, 13 and 14 where either or ’s marginal cost of control is

small, accountability does not matter – it is equally efficient to bundle or not bundle authority

with accountability.

Third, zones and are SB with accountability but are TB without accountability.

Hence, when either or ’s marginal cost of control is medium, accountability can improve

efficiency.

Fourth, only in zone is having no accountability the most efficient. For since ,

imposing accountability is always optimal. That is, it can be optimal to have no accountability,

but only if the authority is incapable. It is always optimal to impose accountability on a capa-

ble authority.

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Fifth, zones and are SB. Hence, when either or ’s mar-

ginal cost of control is small, the solution is SB.

Sixth, comparing ’s control zones with ’s control zones, we find that the authority

typically has a relatively low marginal cost of control. That is, authority tends to be given to the

manager who has a relatively low marginal cost of control.

Seventh, as increases, ’s control zones expand. Hence, authority tends to be given to

the manager who has a relatively large marginal benefit of control. The explanation lies in the

ex ante investments ∗ and ; a larger implies a larger ex ante investment, meaning that

authority tends to be given to the manager who invests more.

Finally, having no accountability may be efficient because accountability imposes extra

risk and potential cost on decision making. When a high cost of control coupled with poor

control ability sufficiently dampens an authority’s incentive for control, it becomes optimal to

have no accountability.

3.2. Collective Accountability

We now consider collective accountability. It turns out that the solution with collective ac-

countability is the same as that with personal accountability.

Proposition 2 (Collective Accountability). Collective accountability implies the same solu-

tion as that with personal accountability.

Proposition 2 means that collective accountability is equivalent to personal accountability.

But, collective accountability is less popular than personal accountability in practice. The

reason may be that collective accountability is viewed as socially undesirable in some cases.

For example, it is now socially unacceptable to hand down the death penalty to the whole

family of a criminal as a deterrent to serious crimes.

3.3. Reverse Accountability

We now consider reverse accountability. We call the other three systems of accountability

(personal, collective and no accountability) normal accountability. It turns out that reverse

accountability is generally inferior to normal accountability.

Proposition 3 (Reverse Accountability). Reverse accountability is inferior to normal ac-

countability if the marginal costs of control are not very large. Specifically, if or

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reverse accountability is inferior; however, if and reverse ac-

countability can be better than normal accountability.

This result is expected, since reverse accountability “wrongly” imposes an extra cost on

the non-controlling party. This type of accountability induces the non-controlling party to

work hard, which may exaggerate the authority’s incentive to reduce her own cost of control.

However, in cases when the marginal costs of control are very large, reverse accountabil-

ity may be the most efficient. Efficiency increases if the non-controlling party’s investment can

be greatly boosted by accountability. This positive effect on efficiency may outweigh the nega-

tive effect of having no accountability. That is, reverse accountability can be efficient due to its

effect on the incentives of the non-controlling party.

4. Concluding Remarks

In this paper, we propose an equilibrium model to address accountability, taking into ac-

count its connection with other major mechanisms in an organization. We emphasize the

connection among the assignments of income, authority and accountability in equilibrium. We

consider a few popular systems of accountability. We identify the most efficient solution con-

ditional on the marginal cost of control, the marginal benefit of control, and the ability to

control.

Although a project may fail in our model due to bad luck ex post, the authority can influ-

ence this “luck” through her choices. Accountability can further affect her diligence in decision

making. In fact, the project is always successful in equilibrium.

Accountability in practice comes in many forms: electoral, hierarchical, peer, reputational,

market, financial, and legal accountability. We do not discuss how accountability is imple-

mented. In an extended model, the way accountability is implemented may play a role. Joint

authority is also not discussed in this paper due to the technical complexity involved.

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Appendix

Proof of Proposition 1

The Optimal Assignment of Income, Conditional on Given Assign-ments of Authority and Accountability

Given the parametric functions in (1)-(3), if is in control, when , problem

(7) becomes

∗ ,

After introducing a Lagrange multiplier , the first-order conditions (FOCs) are

Substituting the solution of from these FOCs into the constraint in (19) yields

Then,

∗ ∗ (20)

and the SB revenue shares, as defined in (8), are

∗ ∗Then, social welfare is

∗ ∗ ∗ ∗ ∗If is in control, when , as defined by problem (9), the investments are

and the revenue shares are

Then, social welfare is

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The Optimal Assignment of Authority, Conditional on a Given As-signment of Accountability

Conditional on the above solution, we now identify the optimal assignment of authority.

On the one hand, since the SB solution is the most efficient in our model, if ∗ ,

authority should be given to if ∗ , authority should be given to and if ∗ and ∗ , authority can be given to either or . Condition ∗

means

If there is no accountability, the condition becomes

These two conditions are indicated in Figure 6.

On the other hand, if ∗ and ∗ authority should be given to if and

only if

or

If there is no accountability, the condition becomes

These two conditions are also indicated in Figure 6.

In summary,

(1) If (21) holds, putting in control with personal accountability is SB and hence opti-

mal. In particular, if (21) holds for both and putting either or in control with

personal accountability is optimal. Similarly, If (22) holds, putting in control without ac-

countability is SB and hence optimal. In particular, if (22) holds for both and putting

either or in control without accountability is optimal.

(2) In the zone defined by for and , putting in control

with personal accountability is optimal if and only if (23) holds, otherwise putting in con-

trol with personal accountability is optimal. Similarly, in the zone defined by for

and , putting in control without accountability is optimal if and only if (24) holds,

otherwise putting in control without accountability is optimal. In this case, the solution is

TB.

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The Optimal Assignment of Accountability

We now determine the optimal assignment of accountability, conditional on the above

equilibrium assignments of income and authority. Note that the admissible assignments of

accountability are personal accountability and no accountability.

3 4

Figure 6. Efficient Control Zones

In zones 1, 7, 11, 12, 13 and 14 of Figure 6, since accountability has no effect on social wel-

fare, personal accountability and no accountability are equally efficient. In zones 1, 7, 11, 12, 13

and 14, since the solution is SB, both personal accountability and no accountability are opti-

mal. In zones 2, 8, 9 and 10, personal accountability leads to the SB solution and hence is

optimal. Hence, in zones 2, 8, 9 and 10, personal accountability is optimal.

For the TB solution, accountability matters. If ∗ under control, by (12), per-

sonal accountability is better than no accountability if

or

That is, for the TB solution, under control, accountability improves efficiency if and only if

Hence, personal accountability is optimal in zone 4, and no accountability is

optimal in zone 3.

22/25

Similarly, under control, since , we have

implying that no accountability can never be optimal under control in the TB zones. Hence,

in both zones 5 and 6, personal accountability is optimal.

In the special zone of Figure 6, if it is in control, no accountability is optimal; if is

in control, personal accountability is optimal. Which combination is better? The social wel-

fares when is in control without accountability and when is in control with personal

accountability are respectively

We find if and only if

or

This inequality determines in zone whether putting in control without accountability is

better than putting in control with personal accountability. This condition is indicated in

Figure 5. Consider the three relevant lines in (23)-(25) writtenas:

where the first line is defined by the preference with accountability; the second line is

defined by the preference without accountability; and the third line is defined by the

preference when is in control without accountability and is in control with

accountability. On line we find Hence, the line defined by starts

from the intersection point of the lines defined by and . Since

the line defined by is steeper than the line defined by but is less steep than the line de-

fined by . Hence, the line defined by is a dividing line of the special zone . The final

solution is shown in Figure 5.

23/25

Proof of Proposition 2

Suppose is the authority. At time , given revenue share in the ex ante contract,

assesses its income:

(26)

Hence, will choose if ; but if , may choose .

However, there is an alternative: can demand renegotiation. Since is not a so-

cially optimal choice, both parties can benefit from renegotiation. With , if bar-

gaining fails, will choose and receive whereas will receive ; but if

bargaining leads to an agreement on , will receive

and will receive . The agreement implies social welfare maximization

∈[ , ]which implies ∗ since . Hence, by the Nash bargaining solution, will

receive

and will receive

Notice that the term is what will gain as the authority; she actually receives revenue

share from bargaining. Since , will bargain instead of

choosing if However, since , by Lemma 1, it is better to

offer revenue share to in the first place in the ex ante contract. With

, will voluntarily choose ∗ without renegotiation.

Hence, the two parties’ ex post payoffs are

In equilibrium, will always choose the penalties and are never used, and the

contract never needs to be renegotiated. Ignoring the costs of ex ante investments, the ex ante

payoffs are

24/25

In the first period, the two parties choose and in a Nash equilibrium. The Nash equilibri-

um is determined by two IC conditions:

At given the assignments of authority and accountability, an income assignment is

determined by bargaining. Bargaining leads to social welfare maximization. Hence, the ex ante

problem is

, , ,(27)

Under condition problem (27) can be solved in two steps. We first solve for ∗ ∗ from the following problem:

∗ ,(28)

Then, given ∗ ∗ ∗ and ∗ are determined by the two IC conditions:

∗ ∗∗ ∗ ∗ ∗∗ ∗By the constraint in (28), we have ∗ ∗ . The solution is SB.

But if then the revenue shares and are already determined, which are

and Then, the two IC conditions determine the ex ante invest-

ments

and problem (27) cannot be optimized. Hence, social welfare is

The solution is TB. From (30), with the parametric functions in (1)-(3), we can see that, under

control, accountability reduces but raises .

25/25

From (28)-(31), we find that the solution in this case with collective accountability is the

same as that with personal accountability.

Proof of Proposition 3

Since the proof is long and tedious and the result is considered to be unimportant, it is not

included but is available upon request.

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