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Transparency and Decentralization in Hierarchies*
Christian Hofmann LMU Munich
Munich School of Management [email protected]
Raffi J. Indjejikian University of Michigan Ross School of Business
Current version: April 2019
* We are grateful to workshop participants at the University of Amsterdam for many helpful suggestions.
Transparency and Decentralization in Hierarchies
Abstract We examine how accounting practices that render performance accomplishments transparent versus opaque within an organization comport with the decision to delegate decision-making authority. We consider a principal-agent model where the principal either contracts with all parties directly or tasks an agent-manager to contract with agent-workers. The performance measurement system either makes worker performance transparent throughout the organization or discloses worker performance only to the respective worker and the worker’s superiors. While transparency is always beneficial when the principal retains all contracting authority, we find that transparency can be costly when the principal delegates contracting authority. We also identify conditions under which transparency and delegating contracting authority are complementary elements of organizational control, thus contributing to the literature that documents complementarities among organizational design choices. Keywords: Hierarchy; Incentive; Multi-agent contracting; Relative performance evaluation; Complementarity JEL-code: L22, M12, M4
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1. INTRODUCTION
The accounting and economics literatures have long highlighted the importance of
balancing decision rights assignment, performance measurement, and incentive compensation for
managerial incentives (e.g., Jensen and Meckling 1976; Milgrom and Roberts 1995; Brickley,
Smith, and Zimmerman 2015). Using this framework of the three-legged stool, empirical and
theoretical studies examine the number of performance measures, aggregate versus disaggregate
measures, objective versus subjective measures, and the weights assigned to these measures as
key features of performance measurement.1 More recently, the concept of “transparency” has
become popular to characterize performance measurement practice in public firms, health care,
and government services.2 For example, organizations establish transparent performance
measurement by designing open-space environments (“war rooms”) with public performance
scoreboards; rotating employees across teams or jobs; or granting access to the organization’s
information system (“open-book accounting”). As transparent performance measurement and
decentralized decision-making are two common themes of the three-legged stool, we study
whether there is a complementary relation between transparency and decentralization.3
We model a three-tier hierarchy comprising an owner-principal, a manager-agent, and
two worker-agents. The agents’ productive efforts are not directly observable and verifiable by
the principal. Given the agents’ hierarchy, transparency refers to the monitoring of the workers’
accomplishments. With transparent monitoring, each worker sees the performance of the other
1 Lambert (2001), Dutta (2008), and Ederhof et al. (2011) survey this literature. 2 For instance, Amazon is well known for its transparent workplaces (Kantor and Streitfeld 2015). The National Patient Safety Foundation (2015) argues that greater transparency among clinicians within an organization (i.e., “free flow of information”) improves outcomes, reduces medical errors, results in more satisfied patients, and lowers costs of care. See also Gailmard and Patty (2018). 3 For instance, the management and practitioner literatures often describe the modern workplace as being transparent and decentralized. For many years, Bain & Co. has considered benchmarking and employee engagement amongst the most important management tools (Rigby and Bilodeau 2015). The concept of the blockchain technology comprises broad data transparency along with a decentralized autonomous organization (Murray et al. 2018).
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worker, whereas with non-transparent or obfuscate monitoring, each worker only sees his own
performance. The authority to contract with the workers is wielded either by the principal under
centralized contracting or delegated to the manager under decentralized contracting. In our
analysis, we compare transparent and obfuscate monitoring of the workers’ accomplishments
with both centralized and decentralized contracting, and we identify conditions under which
transparent monitoring and decentralized contracting are economic complements, i.e., the
incremental benefit of transparent monitoring over obfuscate monitoring is larger under
decentralization as compared to centralization.
As a motivating example, consider the assignment of decision rights and the design of
performance measurement in a consulting firm comprising managing partner (i.e., principal),
senior consultant (i.e., manager), and junior consultants (i.e., workers). Transparent monitoring
corresponds to public appraisals of junior consultants’ performance (e.g., public performance
rankings), whereas obfuscate monitoring corresponds to individual performance review sessions
between the senior consultant and the junior consultant.4 Hiring and contracting with junior
consultants can be performed by the managing partner or delegated to the senior consultant. With
decentralized contracting, the managing partner specifies a divisional bonus-pool and tasks the
senior consultant to distribute the bonus-pool between the junior consultants and himself.
Our first set of results highlights the benefits and costs of transparency under
centralization and decentralization. With centralized contracting, transparent monitoring means
that each worker sees the performance of the other worker, enabling the principal to filter
common risk from the workers’ compensation by means of relative performance evaluation
(RPE). Thus, transparent monitoring generates a benefit to the principal as the workers demand a 4 While performance ranking is widespread among Amazon, google, and PwC, other firms such as Microsoft, General Electric, and Accenture consulting have dropped “ranking and yanking” its employees. Correspondingly, while some firms apply open-book accounting, others erect “Chinese walls” to limit access to information systems.
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lower risk premium and the principal can efficiently set stronger incentives. As these benefits do
not arise with obfuscate monitoring and because there are no costs of transparency (the principal
can always decide to ignore the performance information, despite disclosing it to the workers),
transparent monitoring dominates obfuscate monitoring under centralized contracting.
In the context of our model, delegating contracting authority to the manager gives rise to
a “control loss” because the manager’s interests do not fully align with the principal’s interests
(Calvo and Wellisz 1978; Melumad et al. 1995). The control loss refers to less productive
workers and varies with the information available to the workers. There is a cost of transparency
under decentralized contracting when the control loss with transparent monitoring exceeds the
control loss with obfuscate monitoring. Intuitively, a manager endowed with contracting
authority may use the larger information flow under transparent monitoring to his benefit, which
is not necessarily congruent with the principal’s benefit. Consequently, the principal prefers
transparent monitoring over obfuscate monitoring only when the benefits of transparency exceed
the costs of transparency.
We find that the principal’s preference for transparent or obfuscate monitoring varies
non-trivially with whether the workers’ performances are positively or negatively correlated.
With positive correlation, the principal prefers transparent monitoring only if the workers’
incentive problem is modest (e.g., with high quality worker performance measures), whereas
with negative correlation the principal prefers transparent monitoring also when the workers’
incentive problem is significant (or, with noisier worker performance measures). Intuitively, the
control loss derives from a risk-averse manager having to compensate the workers; the
associated risk premium increases in the noisiness of the workers’ compensation and is larger
when the workers’ compensation is positively rather than negatively correlated.
4
Our second set of results highlights the conditions under which transparency and
decentralization are complementary mechanisms for the principal. With positive correlation, we
find that complementarity only prevails when the workers’ incentive problem is modest and the
workers’ performance measures are weakly correlated. In contrast, with negative correlation,
complementarity can also arise when the workers’ incentive problem is significant and the
workers’ performance measures are highly correlated. More generally, complementarity more
likely prevails with negative as compared to positive correlation of the workers’ performances.
Our final set of results identifies conditions under which the principal prefers
decentralized contracting relative to centralized contracting. With decentralized contracting, the
manager and the workers form a partnership regarding the manager’s productive effort, which
alleviates the manager’s incentive problem and allows the principal to efficiently set stronger
incentives, thus resulting in a more productive manager. As the benefits of decentralized
contracting increase in the manager’s marginal productivity, the principal prefers decentralized
contracting to centralized contracting when the manager’s marginal productivity is substantial.
Our study contributes to the literature on transparency in organizations and the
complementarity among organizational control choices. The accounting and economics
literatures substantiate the benefits and costs associated with transparency. For instance,
Christensen (1981) demonstrates that the principal can be better off by preventing the agent from
observing pre-decision information. In dynamic agency relations subject to renegotiation,
Indjejikian and Nanda (1999) show that the principal can actually prefer aggregate information.
Assuming a sequential production setting, Winter (2006; 2010) shows that the disclosure of
workers’ performance to other workers allows the principal to reduce agency costs. More
specifically, pay transparency can trigger feelings of envy, disappointment, or anger (Cullen and
5
Perez-Truglia 2018), suggesting a reasoning to obfuscate payments to employees. Our study
differs, because these studies consider centralized contracting whereas we focus on decentralized
contracting.5
The accounting and economics literatures also address the presence or absence of
complementarity among organizational control choices (e.g., Roberts 2004; Brynjolfsson and
Milgrom 2013). Generally, decentralization can beneficially “empower” divisional managers
who have better knowledge about their division than headquarters (Zimmerman 2014). However,
transparency is a double-edged sword in that managers can also use the transparency of
divisional accomplishments to the detriment of the firm, and particularly so when headquarters
appoints them with vast decision rights (e.g., Ross 2014).6
Hofmann and Indjejikian (2019) identify conditions under which decentralization and the
quality of the manager’s performance measure are substitute choices for the principal. We
contribute to this work by identifying conditions under which decentralization and the disclosure
of the manager’s accomplishments (i.e., the performance transparency for those workers
reporting to the manager) are complementary choices. While Hofmann and Indjejikian (2019)
consider the quality of the manager’s performance measure and, thus, the benefits of
decentralization, this study addresses the disclosure of workers’ performance measures and, thus,
5 There is a broad behavioral literature that addresses the consequences of transparent workplaces. For instance, Bol et al. (2016) argue and find that transparency about evaluation outcomes reduce managers’ tendency to compress subjective performance ratings. See also Evans et al. 2016. 6 Following Schnackenberg and Tomlinson (2016), the quality of information in a sender-receiver relationship relates to disclosure, accuracy, and clarity. Disclosure refers to the availability, accessibility, and visibility of information to agent-receivers; accuracy refers to the precision, reliability, and validity of the information disclosed by agent-senders; clarity refers to the understandability and comprehensibility of the information to receivers. In this context, disclosure (i.e., transparency) can be conceptualized as the receiver-specific signal accuracy with either zero or positive precision.
6
the costs of decentralization. In turn, our study identifies conditions under which obfuscate
monitoring dominates transparent monitoring.7
Finally, we contribute to the literature on relative performance evaluation (RPE) in
hierarchies. Various theoretical and empirical studies consider the use of RPE at the CEO level
(e.g., Antle and Smith 1986; Albuquerque 2009; Gong et al. 2011). Empirical evidence
supporting use of RPE at lower hierarchical ranks is mixed. For a Korean postal service
organization, Matsumura and Shin (2006) find evidence consistent with RPE at lower levels of
the organization. In contrast, examining workforce incentives, Bandiera et al. (2005) study a
leading farm in the United Kingdom that switched from using RPE to piece rates. Holzhacker et
al. (2019) find evidence consistent with the idea that RPE can discourage cooperation among
peers. Given the cost of transparency associated with decentralized contracting, our findings
suggest that RPE at lower hierarchical ranks is less prevalent than RPE at higher ranks, and
particularly so when the employees are engaged in highly similar work environments.
The rest of the paper is organized as follows. In Section 2, we present the model. In
Section 3, we solve for the compensation arrangements assuming centralized and decentralized
contracting as well as transparent and obfuscate monitoring. In Section 4, we address the
incremental benefit of transparent relative to obfuscate monitoring with both centralized and
decentralized contracting, study the complementarity between transparent monitoring and
decentralized contracting, and present conditions for when the principal prefers decentralized
contracting. In Section 5 we conclude.
7 Feltham and Hofmann (2012) demonstrate the benefit of obfuscate monitoring when the agent-workers can collude, assuming that the long-term relation between the workers essentially enforces any side-contracts between the workers.
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2. MODEL
We consider a single-period model with three main actors, a principal, a manager, and
two workers. The principal is risk neutral while the manager and workers are risk averse with
preferences characterized by negative exponential utility functions with risk aversion coefficients
mr and wr . The project’s output is generated sufficiently late such that output is not available for
contracting. Specifically, the project’s expected output is given by
1 2E( ) ( )m m w w wx b a b a a= + + , (1)
where ma and the wa s represent the activities of manager and workers with marginal
productivities mb and wb . We assume output x is completed in sequence so that the activities of
the manager (at least partially) precede the activities of the workers. For instance, the firm uses a
two-stage technology where the manager provides intermediate inputs that the workers use to
produce final outputs. Sequential production introduces the possibility that agents involved in
later stages of the production process receive signals about the effort provided by agents
involved in earlier stages of the production process (e.g., Baliga and Sjöström 1998; Strausz
1999; Jelovac and Macho-Stadler 2002; Winter 2006). To focus on the information
consequences of sequential production, we ignore complementarities between the manager’s and
the workers’ activities and assume that the activities are additively separable. The manager
provides costly effort of 2 / 2ma and each worker provides costly effort of 2 / 2wa .
The principal makes three decisions. First, the principal decides whether to employ a
“manager” with responsibility to contract with both workers (decentralized contracting) or to
retain contracting authority for both workers (centralized contracting). For emphasis, we ignore
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the possibility that the principal delegates contracting authority for one worker to the manager
and retains contracting authority for the other worker.
Second, the principal decides about the transparency of the firm’s monitoring system.
Specifically, the principal grants the agents access to the following signals of the manager’s and
the worker’s performance:
m mmmy am ε= + , (2a)
1 1 1w w w wy am ε θ= + + and 2 2 2w w w wy am ε aθ= + + , (2b)
where the m s reflect performance measure sensitivity to the agents’ actions, the mε in (2a) is id
2(0, )mN σ , the swε in (2b) are iid 2(0, )wN σ , θ in (2b) is id 2(0, )N θσ , and { 1, 1}a ∈ + − reflects
the covariance between the workers’ performance measures. For instance, workers operating in a
similar environment have positively correlated performance measures ( 1a = + implies
1 2Cov[ , ] 0w wy y > ), whereas allocation of joint revenues or costs yields negatively correlated
performance measures ( 1a = − implies 1 2Cov[ , ] 0w wy y < ). In our analysis, we consider the
following monitoring systems: (a) the principal lets each agent see each other agents’
performance (transparent monitoring); (b) the principal does not allow the workers to see the
performance of the other worker (obfuscate monitoring). One important insight of our analysis is
that the optimal transparency of the firm’s monitoring system depends in a non-trivial way on the
locus of contracting authority within the hierarchy.8
8 With three agents, where each agent can be granted access to see the performance of none, one, or both other agents, there are 64 monitoring systems characterized by different agent – observed performance measure combinations. Clearly, motivating productive effort requires that an agent sees his own performance. As the subsequent analysis will show, the benefits of delegating contracting authority to the manager necessitate that the workers see the manager’s performance. In turn, motivating worker effort with delegated contracting requires that the manager sees the workers’ performance. Thus, the only transparency choice left open is whether a worker sees the performance of the other worker.
9
Third, the principal selects the manager’s compensation contract, mc , commensurate with
the authority delegated and set in a manner to ensure the contract provides the manager his
reservation wage (we scale the manager’s reservation wage so that his reservation certainty
equivalent is zero). Finally, we assume the manager’s contract is a linear function of all
verifiable performance measures described in (2), including potentially all performance measures
pertaining to all workers available to the manager.
Because the principal’s project is completed in sequence, the manager begins production
by providing effort ma . In line with Hortala-Vallve and Villalba (2010), the workers observe the
manager’s action choice.9 After the manager provides ma , the workers receive their contracts.
With centralized contracting, the principal chooses wic whereas, with decentralized contracting,
the manager chooses wic (with 1,2i = ), all set to ensure the contracts provide the workers their
reservation wages (we scale the workers’ reservation wages so that their reservation certainty
equivalents are zero). Finally, and in line with our assumption for the manager, we assume the
workers’ contracts are also linear functions of potentially all performance measures described in
(2) available to the workers.10
9 What is important is that the workers have some information about ma prior to accepting the contract by the manager. Our results are largely unaffected when assuming that the principal, the manager, and both workers observe “soft” unverifiable information about ma (e.g., Feltham et al. 2016 and Hofmann and Indjejikian 2018). 10 A mechanism where the workers truthfully reveal their information to the principal can render this information verifiable and contractible (Ma 1988; Ma et al. 1988). Because the equilibria for such mechanisms are often neither technically robust nor descriptively realistic, we preclude such mechanisms (Hermalin and Katz 1991; Aghion, Dewatripont, and Rey 1994).
10
3. INCENTIVES WITH CENTRALIZED AND DECENTRALIZED CONTRACTING
In this section, we describe the agents’ effort choices and their contracts set by the
manager and the principal. Assuming transparent monitoring, we write the agents’ compensation
as
1 1 2 2m m m m m w m wc f v y y yδ δ= + + + and (3a)
wi wi wi wi wij wj wi mc f v y y yδ δ= + + + , for , 1, 2i j = and i j≠ (3b)
for the manager and the workers, respectively, where mf and the wf s are the fixed components
of the compensation, mv and the wv s are the incentive rates for an agent’s own performance, and
the mδ s and the wδ s are incentive rates for other agents’ performances. With obfuscate
monitoring the workers do not see each other’s performance measure, implying 0wijδ = .
In what follows, we characterize the optimal contracts under centralized and
decentralized contracting and for both transparent monitoring and obfuscate monitoring. In
Section 4, we use the principal’s expected profits to study the presence or absence of
complementarities associated with transparency and decentralization.
3.1 Centralized contracting
With centralized contracting, the derivation of the agents’ efforts and incentive rates and
the principal’s expected profit is straightforward. With either transparent or obfuscate
monitoring, the agents see their own performance, implying that the agents’ efforts are
m m ma vm= and wi w wia vm= , for 1, 2i = .
Transparent monitoring
With transparent monitoring, we have:
11
Lemma 1: In an organization characterized by centralized contracting and transparent
monitoring, the agents’ incentive rates and the principal’s expected profit are:
/1
CT m mm
m
bvAm
=+
and 0CTmiδ = for 1,2i = ; (4a)
2/
1 (1 )CT w wwi
w
bvA
mb
=+ −
, CTwijCTwiv
δab= − , and 0CT
wiδ = for , 1, 2i j = and i j≠ ; (4b)
2 2
22(1 ) 1 (1 )CT CT CT m w
m wm w
b bA A
π π πb
= + = ++ + −
, (4c)
where 2
2m m
mm
rA σm
= and 2
2 2( )w
w
wrwA θσ σ
m+= reflect the severity of the manager’s and the workers’
incentive problems, 2
2 2w
θ
θ
σσ σ
b+
= is the fraction of common risk to total risk in the worker’s
performance measure, and the superscript “CT” refers to the combination of centralized
contracting and transparent monitoring.
Lemma 1 presents standard results. The manager’s effort incentives, CTmv , reflect the
usual incentive-risk trade-off and 0CTmiδ = because the workers’ performance measures are not
informative about the manager’s effort. As the workers’ performance measures are each
conditionally informative about the other worker’s effort (Antle and Demski, 1988), we can
describe a worker’s compensation as ) ( )( )(CT CT CT CTwi wi wi wij wi wij wi wjc f v y y yδ δ+ −= + + − , where
wi wjy y− reflects filtering of common risk via relative performance evaluation (RPE) and CTwijδ
reflects the relevance of RPE. Risk filtering is efficient as the optimal incentive rate CTwijδ
minimizes the workers’ compensation risk for given effort incentives, yielding
2 2 2 2Var[ ] ( )(1 )wi w wic vθσ σ b= + − .
12
The worker’s effort incentives, CTwiv , and the expected profit from both workers, CT
wπ ,
illustrate the benefits associated with transparent monitoring. By efficiently filtering risk via
RPE, transparent monitoring reduces the severity of the worker’s incentive problem to
2(1 )wA b− , thus allowing the principal to economically set stronger incentives. Finally, we note
that the sign of the covariance between the workers’ performance measures, a , is
inconsequential for the workers’ effort incentives and the principal’s expected profit.
Obfuscate monitoring
With obfuscate monitoring the workers do not see each other’s performance measure.
Due to separability, the manager’s incentive rates and the principal’s expected profit from the
manager’s effort are as in expressions (4a) and (4c). For the workers, obfuscate monitoring
implies 0wijδ = in expression (3b). The results with centralized contracting and obfuscate
monitoring follow easily from Lemma 1 as /
1CO w wwi
w
bvAm
=+
and 2
1CO ww
w
bA
π =+
, where the
superscript “CO” refers to the combination of centralized contracting and obfuscate monitoring.
Intuitively, with obfuscate monitoring risk filtering via RPE is infeasible as the workers do not
see each other’s performance measure.
3.2 Decentralized contracting
With decentralized contracting, the principal sets the contract for the manager and
delegates to the manager the authority to contract with the workers along with the requirement to
pay the workers’ compensation. Thus, for the manager to accept this authority the principal has
13
to compensate the manager (in expectation) for the workers’ pay and any risk associated with
compensating the workers.11
With either transparent or obfuscate monitoring, the workers see their own performance
measure, implying that the workers’ efforts take their standard form:
wi w wia vm= , for 1, 2i = . (5)
Transparent monitoring
The manager chooses the workers’ compensation contracts to maximize his expected
compensation net of his risk premium, subject to the workers’ effort choice, expression (5), and
the workers’ accepting the manager’s contract offer. The key difference between the manager
choosing the workers’ incentive rates and the principal choosing the workers’ incentive rates is
that the principal is risk neutral, whereas the manager is risk averse. Thus, the manager’s choice
of the workers’ incentive rates is partly driven by his motivation to reduce his compensation risk
by shifting this risk down to the workers.
Expressions (6a) and (6b) illustrate the effect of the manager’s risk tolerance on the
workers’ incentive rates:
2
2 2( ) ( )
( ) ( )1
1 1 1 1mi m w
wi mi wi wjiww w
r Av vrA A
δ bδ δ
b b−
= + × − −+ − + −
and (6a)
wij mj wj wij mi wi wjim
wi w wi wi
v vrv r v vδ δ δ δ δ
ab ab − − − −
= − + + (6b)
for , 1, 2i j = and i j≠ . Expressions (6a) and (6b) show that the manager’s choice of the
worker’s incentive rates reflects the manager’s incentive rates for the wy s as chosen by the
11 Our characterization of the manager’s and the workers’ payments corresponds to divisional bonus-pool arrangements where the principal determines the size of the bonus pool and delegates to the manager the authority to decide about the distribution of the pool between the workers and himself.
14
principal, 1mδ and 2mδ . If the manager were risk neutral, 0mr = , the worker’s effort incentives,
wiv in (6a), and the relevance of RPE, wijδ in (6b), display their standard form as in Lemma 1.
That is, wiv reflects the usual incentive-risk trade-off and wijδ yields efficient risk filtering.12
The second term in (6a) highlights that the usual incentive-risk trade-off for the
worker’s effort incentives is expanded by the manager’s motivation to shift his compensation
risk to the workers, the magnitude of which varies with the manager’s residual risk after
contracting with the worker, mi wi wjivδ δ− − . Intuitively, setting 0wiv > has the additional
benefit of reducing the manager’s risk associated with wiy . The risk shifting via wiv varies with
the use of RPE in compensating the other worker and is augmented (alleviated) when 0wjiδ < (
0wjiδ > ). Thus, risk sharing is more (less) pronounced for a larger (smaller) residual incentive
risk (i.e., ( )0wjiδ < > ). Expression (6a) suggests the possibility for stronger effort incentives
relative to centralized contracting; however, as the principal ultimately bears the cost of the
manager’s risk premium, below we show that the principal reduces miδ and, thus, the worker’s
effort incentives.
The second and third terms in (6b) show that the risk-averse manager also adjusts the
use of RPE in the worker’s contract to account for the effect of RPE on the manager’s
compensation risk associated with the wy s. From the second term, the manager chooses a larger
wijδ as this has the additional benefit of reducing the manager’s risk associated with the other
12 The principal has no inherent interest in the magnitude of a worker’s performance measure and uses wy only to
motivate wa . In contrast, the manager is inherently interested in wy as his compensation varies with wy .
Consequently, with decentralized contracting, the principal’s benefit per unit of scaled performance, /w wb m , is
replaced by the manager’s benefit per unit performance, miδ .
15
worker’s performance, wjy . From the third term, the manager chooses a larger wijδ when the
workers’ performance measures are positively correlated ( 1a = + ) as increasing wijδ reduces the
covariance risk associated with compensating both agents. The latter effect reverses when
worker performances are negatively correlated ( 1a = − ).
Solving (6a) and (6b) along with the first-order conditions for wiδ for the workers’
incentive rates, using 1 2m m mδ δ δ= = (due to symmetry), we have:
Lemma 2: In an organization characterized by decentralized contracting and transparent
monitoring, the manager’s choice of the workers’ incentive rates is:
2
2 2
1 [1 ( )]
1 [1(
1
1 ) 2 (1 )]
m
w
m
w
wDTwi
wm
rr
rwr
Av
A A
ab b
b aδ
b b
+ +=
+
+
− + −+
−
+, (7a)
2
2(1 (1 ))1 [1 (1 )]
)(m
w
wr
w
D
r
Twij mDT
wwi
ArArv
δ ab ab bab
ab b
++ −
++ ×
+ −= −
+, and (7b)
2m
m w
rm
DTi r rw vδ
+= for , 1, 2i j = and i j≠ , (7c)
where 2
2 2( )w
w
wrwA θσ σ
m+= reflects the severity of the workers’ incentive problems,
2
2 2w
θ
θ
σσ σ
b+
= is the
fraction of common risk to total risk in the worker’s performance measure, and the superscript
“DT” refers to the combination of decentralized contracting and transparent monitoring.
Proof: See Appendix.
Lemma 2 further illustrates that the manager chooses the worker’s incentive rates to
reduce his compensation risk by shifting this risk down to the workers. This is particularly
apparent with his own performance, my , as including my in the worker’s contract via DTwiδ
16
reduces the manager’s compensation risk associated with my to 2Var[ ] Var[ ]w
m w
rm m m mr r
v y v y+
< .
As the manager’s risk premium increases in his risk aversion, the motivation to shift
compensation risk increases in mr . For instance, ceteris paribus, a more risk-averse manager
chooses stronger incentives associated with worker performance (i.e., ( / ) / 0DTwi m mv rδ∂ ∂ > ).
Lemma 2 also illustrates that the worker’s incentive rates for the wy s vary with the sign
of the covariance between the workers’ performance measures, a . In line with above, the
manager sets stronger incentives if the workers’ performance measures are positively rather than
negatively correlated (i.e., ( 1) ( 1)DT DTwi wiv va a= > = − for given mδ ). Intuitively, given 0mδ > ,
with positive correlation the manager bears a larger risk when compensating both workers as
compared with negative correlation, yielding a stronger motivation to shift this risk down to the
workers. Thus, different to centralized contracting, the sign of the covariance affects the
worker’s effort incentives.
In parallel to contracting with the workers, the manager chooses his effort, which is
standard and given by
m m ma vm= . (8)
When setting the manager’s compensation contract, the principal anticipates the
manager’s choice of the workers’ incentive rates, expressions (7), and the manager’s choice of
effort, expression (8). We characterize the manager’s contract and the principal’s expected profit
in Lemma 3:
Lemma 3: In an organization characterized by decentralized contracting and transparent
monitoring, the manager’s incentive rates and the principal’s expected profit are:
17
2
/1 ( )w
m w
m m
r
DTr
m rm
bA
v m
++
= and (9a)
2
2 2
/ 1 [1 (1 )]
1 (1 )[1 (1 )]
( )m
w
m
w
rw
mw wr
rw w
D
r
Tb A
A A
abδ
m b
ab b
+ + + −
+ + + + −= ; (9b)
2
2
2
22
22
12 ( )
11 (1 ) (1 )(1 (1 ))1 [ (
1
1 1 )]
( )
w
m w
m m
mw w
w
mr
m r r
w
r rrr rr
DT DT DTm w
ww w
w
A
b
A
A
A
b
A
π π π
abb ab b
ab
ba
b
++
+ − + − + + + + + + + −
= + =
+
, (9c)
where 2
2m m
mm
rA σm
= and 2
2 2( )w
w
wrwA θσ σ
m+= reflect the severity of the manager’s and the workers’
incentive problems, 2
2 2w
θ
θ
σσ σ
b+
= is the fraction of common risk to total risk in the worker’s
performance measure, and the superscript “DT” refers to the combination of decentralized
contracting and transparent monitoring.
Proof: See Appendix.
The manager’s incentive rate for his own performance, DTmv , and the principal’s expected
profit from the manager, DTmπ , reflect that with decentralized contracting the manager shares the
compensation risk associated with my with his workers. As the workers see the manager’s
action, the manager and the two workers essentially form a “partnership” with an aggregate risk
tolerance of 1 22 m w
m wm w
r rr rr r
++ = . Thus, the benefit of decentralized contracting is that
decentralization essentially alleviates the severity of the manager’s incentive problem to
18
2
2 2 2m m w w
m w m wmm m
r r rr r r r
A Aσm + +
= < and the principal offers stronger incentives and realizes a larger profit
from the manager.
The expression for DTmδ , together with DT
wiv and DTwjiδ in (7a) and (7b), reflect that with
decentralized contracting incentives cascade from the manager to the workers. In other words,
the principal sets 0DTmδ > to motivate the workers’ efforts indirectly by motivating the manager
to contract with his workers. Accounting for the manager’s choice of the workers’ incentive rates
in Lemma 2, the manager bears a residual risk associated with worker performance of
2
2Var[( ) ] Var[ ](1 )
1 (1 ) (1 2 ) (1 )m m
w w
DT DTr
DT DTm wi wj
wi wi m w
wr
r ri
AA
v y yab b
aδ δ δ
b b
+ −
+− −
+ + + −=
, (10)
which is non-zero except for knife-edge cases (with negative correlation, 1a = − , and
21 2 / (1 2 1 4 )w wA Ab = − + + + ). As the principal ultimately bears the cost of the manager’s risk
premium, the manager’s residual risk associated with worker performance entails a cost of
delegated contracting.
While expression (10) shows that it is generally costly to the principal to motivate a risk-
averse manager to contract with his workers, the specific effect of mr on DTmδ is ambiguous. For
instance, we find that DTmδ increases in mr if wA is sufficiently small. Intuitively, with a small
wA , the residual compensation risk imposed on the manager is inconsequential. As a more risk-
averse manager shifts more incentive risk to the workers ( ( / ) / 0DTwi m mv rδ∂ ∂ > , see above), the
cost to the principal of indirectly motivating the workers decreases and the principal optimally
19
chooses a larger incentive rate DTmδ . When wA is large, the residual compensation risk imposed
on the manager is substantial and the benefit of motivating the workers is reduced.
Finally, even though the manager shifts more compensation risk to the workers if the
workers’ performance measures are positively rather than negatively correlated (
( 1) ( 1)DT DTwi wiv va a= > = − for given mδ , see above), with positive correlation the principal
chooses a weaker incentive rate DTmδ as compared to negative correlation (i.e.,
( 1) ( 1)DT DTm mδ a δ a= < = − ). Intuitively, as the manager bears a larger risk when compensating
both workers with positive correlation as compared to negative correlation, in the former case the
principal more strongly reduces the incentive rate DTmδ to economize on the risk premium
demanded by the manager.
The expression for the principal’s expected profit from the workers, DTwπ , reflects the
benefit of transparent monitoring and the cost of delegated contracting. In line with above,
transparent monitoring enables filtering of common risk from the workers’ compensation via
RPE; the benefit of transparent monitoring is captured by the second term 2(1 )wA b− in the
denominator of DTwπ . The cost of delegating contracting authority to the manager is captured by
the third term in the denominator of DTwπ and comprises (i) the manager’s residual compensation
risk associated with worker performance (expression (10)) and (ii) that the workers’ effort
incentives are weaker under decentralized contracting compared with centralized contracting
(i.e., DT CTw wa a< ). Thus, the principal economizing on DT
mδ to reduce the manager’s risk
premium dominates the manager’s incentive to shift incentive risk down to the workers.
20
The expression DTwπ also highlights the interrelation between the benefit of transparent
monitoring and the cost of delegated contracting as both vary with the fraction of common risk to
total risk in the worker’s performance measure, b . We will investigate this interrelation in
Section 4 where we study the complementarity between transparency and delegated contracting.
Obfuscate monitoring
With obfuscate monitoring, the workers do not see each other’s performance measure,
implying 0wijδ = in expression (3b). As the manager’s choice of the workers’ incentive rates
continues to be affected by his motivation to reduce his compensation risk, obfuscate monitoring
limits the means available to the manager to shift this risk down to the workers. Expression (11)
illustrates the ensuing effect of the manager’s risk tolerance on the workers’ incentive rates:
( )( )1 1
mi m wwi mi wi mj wj
w w w
r Av v vA r A
δ δ δ ab= + × − + −+ +
for , 1, 2i j = and i j≠ . (11)
When setting the workers’ incentives with obfuscate monitoring, expression (11) shows
that the usual incentive-risk trade-off continues to be expanded by the manager’s motivation to
shift his compensation risk to the workers. As in the case of transparent monitoring, setting
0wiv > has the additional benefit of reducing the manager’s risk associated with the worker’s
performance. Even though the manager cannot use RPE to filter risk from a worker’s
compensation, expression (11) illustrates that the risk-averse manager considers the co-variance
between the workers’ performance measures as the manager has to bear the risk of compensating
both workers. Similar to transparent monitoring, the manager has stronger (weaker) incentives to
shift incentive risk to the worker with positively (negatively) correlated performance measures.
Solving expressions (11) for wiv yields the manager’s choice of the worker’s incentive
rate. The principal anticipates this incentive rate when setting the manager’s compensation
21
contract. We characterize the manager’s and the principal’s choices for the incentive rates related
to the wy s and the principal’s expected profit from the workers’ effort in Lemma 4. Due to
separability, the manager’s and the workers’ incentive rates for my and the principal’s expected
profit from the manager’s effort are as in expressions (7c), (9a), and (9c).
Lemma 4: In an organization characterized by decentralized contracting and obfuscate
monitoring, the agents’ incentive rates for the workers’ performance measures and the
principal’s expected profit from the workers’ efforts are:
1 (1 )
1 )(1
m
w
m
w
rr
mr
w
rDOwi
w w
Av
A A
aδ
b
ab
+
+ ++
+= , 0
DOwijDOwiv
δ= for , 1, 2i j = and i j≠ (12a)
/ 1 (1 )
1 (1 )(1 )
( )m
w
m
w
rw w wr
rw wr
DOmi
b A
A A
m aδ
b
ab=
+ +
+ + + for 1,2i = ; (12b)
2
2
1 (1 )1 (1 )
m
mw
w
DOw
w
w
w wr w
rrr
b
AA AA
π
abab
+ + + + +
= , (12c)
where 2
2 2( )w
w
wrwA θσ σ
m+= reflects the severity of the workers’ incentive problems,
2
2 2w
θ
θ
σσ σ
b+
= is the
fraction of common risk to total risk in the worker’s performance measure, and the superscript
“DO” refers to the combination of decentralized contracting and obfuscate monitoring.
Proof: See Appendix.
Lemma 4 shows that, with obfuscate monitoring, incentives related to the wy s cascade
from the manager to the workers. By setting 0DOmiδ > the principal motivates the worker’s effort
22
indirectly by motivating the manager to contract with his worker. Similar to transparent
monitoring, the manager bears a residual risk associated with worker performance of
Var[( ) ] Var[ ]1 (1 (1 ))m
w
wr
DO DO DOmi w
wi wi mi w
riv y yA
Aδ
abδ
+ +− =
+. (13)
The expression for the principal’s expected profit from the workers, DOwπ , reflects that
there are no benefits of transparency as obfuscate monitoring precludes filtering of common risk
from the workers’ compensation via RPE. The cost of delegated contracting is captured by the
third term in the denominator of DOwπ and comprises (i) the manager’s residual compensation
risk associated with worker performance (expression (13)) and (ii) that the workers’ effort
incentives are weaker under decentralized contracting compared with centralized contracting
(i.e., DO COw wa a< ). Finally, even though obfuscate monitoring precludes filtering of common risk
from the workers’ compensation, the expression for DOwπ highlights that the presence of
common risk as well as the sign of the covariance affect the cost of delegated contracting.
4. TRANSPARENCY IN HIERARCHIES
In this section, we first study the principal’s choice for transparent monitoring with either
centralized contracting and decentralized contracting. Next, we explore conditions for when
transparent monitoring and decentralized contracting are complementary mechanisms for the
principal, i.e., the incremental benefit of transparent monitoring over obfuscate monitoring is
larger under decentralization than centralization. We close this section by highlighting conditions
under which the principal is better off by delegating contracting authority to the manager.
23
4.1 Transparency with centralized contracting
With centralized contracting, the principal’s expected profit from the manager’s effort is
unaffected by the transparency about the workers’ contributions. The expected profit from the
workers’ efforts under transparent monitoring is given in Lemma 1 as 2
21 (1 )CO ww
w
bA
πb
=+ −
and
the expected profit from the workers’ efforts under obfuscate monitoring is given by
2
1CO ww
w
bA
π =+
. Thus, the following proposition is straightforward:
Proposition 1: In an organization characterized by centralized contracting, the principal prefers
transparent monitoring to obfuscate monitoring, i.e., CT COπ π≥ and the relation is strict when
0θσ ≠ .
With centralized contracting, transparency about the workers’ contributions allows the
principal to filter risk from the workers’ performance evaluation such that the principal
economically sets stronger incentives. In other words, transparent monitoring is associated with
benefits of transparency whereas obfuscate monitoring is not.
4.2 Transparency with decentralized contracting
With decentralized contracting, the principal’s expected profit from the manager’s effort
continues to be unaffected by the transparency about the workers’ contributions. The expected
profit from the workers’ efforts under transparent monitoring, DTwπ , is defined in expression (9c)
and the expected profit from the workers’ efforts under obfuscate monitoring, DOwπ , is defined in
expression (12c). The principal prefers transparent monitoring to obfuscate monitoring when
DT DOw wπ π≥ .
24
While only transparent monitoring is associated with benefits of transparency, the cost of
delegated contracting (i.e., the third terms in the denominator of (9c) and (12c)) differs between
transparent and obfuscate monitoring. Hence, a sufficient condition for the principal to prefer
transparent monitoring over obfuscate monitoring is when the cost of delegation with obfuscate
monitoring exceeds the cost of delegation with transparent monitoring. In turn, when the cost of
delegation with transparent monitoring exceeds the cost of delegation with obfuscate monitoring,
there is scope for the principal to prefer obfuscate monitoring. For instance, with positive
correlation, 1a = + , obfuscate monitoring implies a lower cost of delegation than transparent
monitoring when the severity of the worker’s incentive problem as captured by 2
2w
w
σm
is either large
or small, i.e., if 2 2 2
2 2 2 22( )max{ ; }w w
w w m wr rθ
θ θ
σ σ σm σ σ σ+
> or 2 2 2
2 2 2 22( )min{ ; }w w
w w m wr rθ
θ θ
σ σ σm σ σ σ+
< . When 2
2w
w
σm
is either
large or small, RPE is used primarily for risk-sharing purposes by the manager rather than to
filter common risk from the workers’ performance evaluation. Intuitively, when 2
2w
w
σm
is either
large or small, little incentive risk is imposed on the workers (specifically, Var[ ]wi wv y is hump-
shaped in 2wσ ).
Proposition 2 extends the comparison of transparent monitoring with obfuscate
monitoring by also considering the benefits of transparency associated with transparent
monitoring.
Proposition 2: In an organization characterized by decentralized contracting, the principal
prefers transparent monitoring to obfuscate monitoring, i.e., DT DOπ π≥ ,
(i) with positive correlation, 1a = + , and
(a) 2
21
2w
mw rσm
< if 2θσ is sufficiently small or sufficiently large;
25
(b) 2
21 /1
2[ , )m ww
m mw
r rr r
σm
+∈ iff 2θσ is sufficiently small;
(ii) with negative correlation, 1a = − , and
(a) 2
21 /m ww
mw
r rr
σm
+≤ ;
(b) 2
21 /m ww
mw
r rr
σm
+> iff 2θσ is sufficiently large.
Proof: See Appendix.
Following Proposition 2, with positive correlation and an easy worker incentive problem,
2
21
2w
mw rσm
< , the principal prefers transparent monitoring when either the benefit of transparency is
substantial (i.e., 2θσ is large) or when the cost of delegated contracting is modest (i.e., 2
θσ is
small). With a severe worker incentive problem, 2
21 /m ww
mw
r rr
σm
+> , the benefit of transparency is less
than the incremental cost of delegation, implying that the principal always prefers obfuscate
monitoring. Recall that with negative correlation, there is a lower cost of delegation from
transparent monitoring. Hence, the principal always prefers transparent monitoring when the
workers’ incentive problem is easy. When the workers’ incentive problem is severe, the principal
prefers transparent monitoring when the benefit of transparency is substantial (i.e., 2θσ is large).
We conclude by noting that the conditions in Proposition 2 for when the principal prefers
transparent monitoring are less strict with negative correlation as compared to positive
correlation. This implies, for instance, in a firm comprising multiple units run by workers, that
RPE is more likely to be used in firms where the firm’s accounting system allocates joint costs to
the units (implying negatively correlated performance measures) as compared to firms where the
26
units sell products in markets that are subject to similar economic shocks (implying positively
correlated performance measures).
4.3 Complementarity between transparency and decentralized contracting
Complementarity between transparency and decentralized contracting requires that the
incremental benefit of transparent monitoring over obfuscate monitoring given delegated
contracting, DT DOπ π− , exceeds the incremental benefit of transparent monitoring over
obfuscate monitoring given centralized contracting, CT COπ π− . As 0CT COπ π− ≥ (Proposition
1), complementarity between transparency and decentralized contracting implies stronger
conditions relative to the conditions for when the principal prefers transparent monitoring to
obfuscate monitoring under decentralized contracting (Proposition 2). Further, as the principal’s
expected profit from the manager’s effort is unaffected by the transparency about the workers’
contributions with either centralized or decentralized contracting (i.e., DT DOm mπ π= and
CT COm mπ π= ), complementarity follows from comparing the principal’s expected profits from the
workers’ efforts, DT DO CT COw w w wπ π π π− > − . Proposition 3 summarizes our results.
Proposition 3: Decentralized contracting and transparent monitoring are complements,
DT DO CT COπ π π π− > − ,
(i) with positive correlation iff 2
21 /m ww
mw
r rr
σm
+< and 2θσ sufficiently small;
(ii) with negative correlation if 2
21 /m ww
mw
r rr
σm
+≤ and 2θσ sufficiently small.
Complementarity is more likely to prevail with negative correlation than with positive
correlation.
Proof: See Appendix.
27
With either positive or negative correlation, complementarity prevails only with a low
cost of delegation (i.e., 2
2w
w
σm
small) and a modest benefit of transparency (i.e., 2θσ small).
Intuitively, centralized contracting allows for more effective filtering of common risk, which is
of relevance when 2θσ is large. In line with Proposition 2, complementarity is more likely to
prevail with negative than positive correlation as the sign of the correlation affects the cost of
delegation but does not affect the benefit of transparency. That is, with positive correlation,
complementarity prevails only when the workers’ incentive problem is easy (2
21 /m ww
mw
r rr
σm
+< ),
whereas with negative correlation, complementarity can also prevail when the workers’ incentive
problem is more demanding (2
21 /m ww
mw
r rr
σm
+> ).
4.4 Conditions for decentralized contracting
While Proposition 3 states conditions under which decentralized contracting and
transparent monitoring are complements, these conditions are not sufficient for the principal to
actually prefer delegation of contracting authority to the manager. Specifically, Proposition 3
implies ( )DT CT DO COπ π π π> + − which is less strict than DT CTπ π> when 0DO COπ π− < .
In other words, while the incremental benefit of transparent rather than obfuscate monitoring
with decentralized contracting exceeds the incremental benefit of transparent monitoring with
centralized contracting (i.e., DT DO CT COπ π π π− > − ), it is possible that the principal prefers
centralized contracting with transparent monitoring over decentralized contracting (i.e.,
CT DTπ π> ).
28
Following Lemma 1 and 3, the benefit of decentralized contracting is that the severity of
the manager’s incentive problem is alleviated, implying an increment benefit from decentralized
contracting to the tune of
2222
2 2
20
(
( )
2 1 ) )2( )1 ( ) 2(1 ( )( ) 1
m
m w
w w
m w m w
m
m
rm m r rDT CT m
m m r rm mr r r r m
bA
b AbA A A
π π +
+ +
=−+ +
− = >+ +
. (14)
In turn, decentralized contracting involves a loss of control as the workers’ contracts follow from
the manager, whose motivation is not perfectly aligned with the principal’s motivation,
0CT DTw wπ π− > . The principal prefers decentralized contracting relative to centralized
contracting when the incremental benefit from a more productive manager exceeds the loss of
control from less productive workers, DT CT CT DTm m w wπ π π π− > − . Following (14), the incremental
benefit increases in the manager’s marginal productivity, mb , implying that the principal prefers
decentralized contracting when mb is sufficiently large. Proposition 4 summarizes.
Proposition 4: With transparent monitoring, the principal prefers decentralized contracting
over centralized contracting, DT CT CT DTm m w wπ π π π− > − , when the manager’s marginal
productivity is sufficiently large.
Proof: As outlined in the text.
A straightforward corollary from Proposition 4 is that the possibility to render the
workers’ performance opaque to other workers enlarges the conditions under which
decentralized contracting is beneficial to the principal. Intuitively, obfuscate monitoring is
potentially beneficial under decentralized contracting but costly under centralized contracting. A
direct empirical implication is that controlling for the variation of within-firm transparency
reduces the probability to identify cases where the firm delegates decision-making authority.
29
5. CONCLUSION
In many organizations, information about employees’ performance is not uniformly
distributed across the organization’s employees. While some organizations make employee
performance fully transparent and let this information freely flow through the organization, other
organizations are much more restrictive in terms of information flow. Our analysis suggests that
the free flow of information interacts with the decentralization of decision making, introducing
the possibility for a cost of transparency. Intuitively, a self-interested agent empowered with
contracting-authority may opportunistically make use of the free flow of information through the
organization. Despite this cost of transparency, we identify conditions under which transparency
and delegation of contracting authority are complementary elements of organizational control.
The intuition is, however, more general and applies to other agents within the organization. For
example, it would be interesting for future research to address the complementarity between
transparency and delegation of contracting authority in settings where agent-workers collude by
using information that is jointly available to them.
30
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APPENDIX
Proof of Lemma 2:
Given m mmmy am ε= + , 1 1 1w w w wy am ε θ= + + , and 2 2 2w w w wy am ε aθ= + + described
in (2), we write the agents’ compensation as:
1 1 2 2m m m m m w m wc f v y y yδ δ= + + + and (A.1a)
wi wi wi wi wij wj wi mc f v y y yδ δ= + + + , for , 1, 2i j = and i j≠ (A.1b)
where mf and the wf s are the fixed components of the agents’ compensation, mv and the wv s
are the incentive rates corresponding to the agents’ own performance metric, and the δ s are the
incentive rates corresponding to other agents’ performance metrics.
The certainty equivalent of worker 1, 2i = (with 1, 2j = and j i≠ ) is given by
21 1 2 2
2 2 21 2
/ 2
V )
E[ ] E[ ] E[ ]
( Var[ ] 2 Cov] [ , ]ar ar[ ] V [2
wi wi wi m wi w wi w wi
wwi wi wij wj wi wij w w wi m
CE f y v y v y ar v y y v y y y
δ
δ δ δ
= + + + −
− + + +. (A.2a)
Substituting (A.1a) and (A.1b) into (A.2) and differentiating with respect to wia yields the
worker’s incentive compatibility constraint,
wi w wia vm= . (A.2b)
The worker’s individual rationality constraint is given by 0wiCE ≥ .
The certainty equivalent of the manager is given by
1 2 1 2 1 1 21 1
2 22 2 12 2 1 2
2 21 1 21 1 2 2 12 2
1 1 21 2 2 12 1 2
( )E[ ] ( )E[ ]
( )E[ ] (( ) Var[ ]2
( ) Var[ ] ( ) Var[ ]2( )( )Cov[ , ]) ,
/ 2
m m w w m w w m m w w w
mm w w w m m w w m
m w w w m w w w
m w w m w w w w
CE f f f v y v yrv y a v y
v y v yv v y y
δ δ δ δ
δ δ δ δ
δ δ δ δδ δ δ δ
= − − + − − + − −
+ − − − − − −
+ − − + − −
+ − − − −
(A.3a)
35
reflecting that the workers’ compensation is paid for by the manager. Using the workers’
incentive compatibility constraints, (A.2a), and the binding individual rationality constraints,
0wiCE = , in (A.3a) yields
22 2 2 2 2 2
1 1 2 2 1 2
2 21 2 1 1 21 1
22 2 12 2
1 1 21 2 2 12 1 2
2 21 21 1
1 1E[ ]2 2 2
(( ) Var[ ] ( ) V
]
ar[ ]2
r
( ) Var[ ]2( )( )Cov[ , ])
((2
)Va [
mm m m m m w w m w w w w w w
mm w w m m w w w
m w w w
m w w m w w w w
ww w w
aCE f v y v v v v
r v y v y
v yv v y y
r v y
δ m δ m m m
δ δ δ δ
δ δδ δ δ δ
δ
= + + + − − −
− − − + − −
+ − −
+ − − − −
− + 2 22 12 2
2 21 12 2 21 1 2 1 2
( Var[ ]
2( Cov[ , ] ( Var[ ]) .
)
+ ) )
w w w
w w w w w w w w m
v y
v v y y y
δ
δ δ δ δ+
++
+ +
(A.3b)
We note that (A.3b) uses the fact that the workers see the manager’s effort, which implies that
E[ ]my is unaffected by whether the manager or the workers form these expectations.
The first-order conditions for the workers’ incentive rates regarding the manager’s
performance measure my imply
2m
wi mm w
r vr r
δ =+
for 1, 2i = .
The first-order conditions for the workers’ incentive rates regarding the workers’ performance
measures are given by
and
( )
( )
2 21 2
1 1 2
2( )Var[ ] 2( )Cov[ , ]2
2 Var[ ] 2 Cov[ , ] 02
m mmi w w wi mi wi wji wi mj wj wij w w
wi
wwi w wij w w
CE rv v y v y yv
r v y y y
δ m m δ δ δ δ
δ
∂= − − − − − − − −
∂
− + =
36
( )
( )
1 2
1 2
2( )Var[ ] 2( )Cov[ , ]2
2 Var[ ] 2 Cov[ , ] 0 ,2
m mmj wj wij wj mi wi wji w w
wij
wwij wj wi w w
CE r v y v y y
r y v y y
δ δ δ δδ
δ
∂= − − − − − − −
∂
− + =
for , 1, 2i j = and i j≠ . Using 1 2Var[ ] Var[ ] Var[ ]w w wy y y= = , 2Var[ ]w w
ww
r yAm
= , and
1 2Cov[ , ] Var[ ]w w wy y yab= , where 2 2
2 2Var[ ] wwyθ θ
θb
σ σσ σ
=+
= , the first-order conditions
0m
wi
CEv
∂=
∂ and 0m
wij
CEδ
∂=
∂ simplify to
( )( ) ( ) ( ) 0mmi wi mi wi wji mj wj wij w wi wij w
w
rv v v A v Ar
δ δ δ δ δ ab δ ab− + − − + − − − + = (A.4a)
( )( ) ( ) ( ) 0mmj wj wij mi wi wji wij wi
w
r v v vr
δ δ δ δ ab δ ab− − + − − − + = . (A.4b)
Solving (A.4b) for /wij wivδ yields (6b). Substituting the solution for /wji wjvδ into (A.4a) and
solving for wiv yields (6a). By symmetry, the manager’s incentive rates for the workers’
performance measures are identical (i.e., 1 2m m mδ δ δ= = ). Solving (6a) and (6b) for wiv and
/wij wivδ yields (7a) and (7b).
Proof of Lemma 3:
From (A.3b), the manager’s incentive compatibility constraint is
m m ma vm= . (A.5)
The principal chooses the contract parameters of the manager to maximize her expected
net profit from the manager and the workers, 1 2( ) E[ ]m m w w w mb a b a a c+ + − , subject to the
manager’s and the workers’ incentive compatibility constraints, (A.2b) and (A.5), the manager’s
37
choice of the workers’ incentive rates, (7a) and (7b), and the manager’s individual rationality
constraint that the certainty equivalent in (A.3b) is greater or equal to zero. The principal’s
unconstrained maximization problem is given by
2 2 2 2 2 21 2
2 21 2 1 1 21 1
22 2 12 2
1 1 21 2 2 12 1 2
2 2 21 21 1
2
2
11 1 12 2 2
(( ) V
r
( )
)Var[
ar[ ] ( ) Var[ ]2
( ) Va [ ]2( )( )Cov[ , ])
(( (2
]
m m w w w w
mm w w m m w w w
m w w w
m w w m w w w
w
w
ww w w
m m m
w
w w w v v v
r v y v y
v yv v
r
b v b v
y y
v y v
vπ
δ
m m m
δ δ δ δ
δ δδ δ δ
m m
δ
δ
− −
− − − +
+
= + + −
− −
− −
+ − − − −
− + ++ 212 2
2 21 12 2 21 1 2 1 2
Var[ ]
2( Cov[ , ] ( Var[ ])
)
+ ) )
w w
w w w w w w w w m
y
v v y y yδ δ δ δ+ + +
(A.6a)
where wiv and wijδ according to (7a) and (7b). Substituting (7a) to (7c) into (A.6a), using
1 2Var[ ] Var[ ] Var[ ]w w wy y y= = , 2Var[ ]w w
ww
r yAm
= , and 1 2Cov[ , ] Var[ ]w w wy y yab= , and
simplifying, yields
2 2 22
22
2 2
2 22
2 2
1 ( )2
1 (1 1 )2
1 1 (1 2 1 )
1 (1
1
(
)(1 1 )12
)
1 (1) 2 1 )
( ) ( )
( )
( ( )
w
m w
m
w
m
w
m
w
m
w
rm m m m m m mr r
rwrw
w mrw w wr
rw wr
mrw wr
b v r v
AbA A
A A
A A
π m m σ
ab bm δ
m b ab b
ab bδ
b ab b
+− +
+ + + −
+
=
+
⋅+ − + + + −
+ + + −−
+ − + + −
+
(A.6b)
Differentiating (A.6b) with respect to mv and mδ and solving the associated first-order
conditions yields (9a) and (9b). Substituting (9a) and (9b) into (A.6b) yields (9c), particularly
( )( )( )
22 2
2 2 2 2
( )
( ) ( )
1 (1 1 )
1 1 (1 2 1 ) 1 (1 )(1 1 )( )
m
w
m m
w w
rw wr
r rw w w wr r
DTw
b A
A A A A
aπ
b b
b ab b ab b
+ + + −
+ − + + + − +=
+ + + −,
38
which simplifies to the expression in (9c).
Proof of Lemma 4:
With obfuscate monitoring, 0wijδ = in (A.2a) and (A.3). The workers’ effort choice is
given by (A.2b). Thus, the manager’s certainty equivalent in (A.3b) simplifies to
22 2 2 2 2 2
1 1 2 2 1 2
2 21 2 1 1 1
22 2 2 1 1 2 2 1 2
2 2 2 21 1 2 2 1 2
1 1E[ ]2 2 2
(( ) Var[ ] ( ) Var[ ]2
( ) Var[ ] 2( )( )Cov[ , ])
( Var[ ] ( VarVar[ )2
]
mm m m m m w w m w w w w w w
mm w w m m w w
m w w m w m w w w
ww w w w w w
aCE f v y v v v v
r v y v y
v y v v y yr v y v y
δ m δ m m m
δ δ δ
δ δ δ
δ δ
= + + + − − −
− − − + −
+ − + − −
− + + + [ ]) .my
(A.7)
The first-order condition for the workers’ incentive rates regarding their performance measure
are given by
2 21 2( 2( )Var[ ] 2( )Cov[ , ])
2
(2 ) 0Va [2
r ]
m mmi w w wi mi wi wi mj wj w w
wi
wwi wi
CE rv v y v y yv
r v y
δ m m δ δ∂
= − − − − − −∂
− =
for , 1, 2i j = and i j≠ . Using 1 2Var[ ] Var[ ] Var[ ]w w wy y y= = , 2Var[ ]w w
ww
r yAm
= , and
1 2Cov[ , ] Var[ ]w w wy y yab= , where 2 2
2 2Var[ ] wwyθ θ
θb
σ σσ σ
=+
= , the first-order condition simplifies
to
(( ) ( ) ) 0mmi wi mi wi mj wj w wi w
w
rv v v A v Ar
δ δ δ ab− + − + − − = . (A.8)
Solving (A.8) for wiv yields (11). By symmetry, the manager’s incentive rates for the workers’
performance measures are identical (i.e., 1 2m m mδ δ δ= = ). Thus, solving (11) for wiv yields
(12a).
39
Consistent with the proof of Lemma 3, the manager’s choice of effort is given by (A.5)
and the principal’s unconstrained maximization problem is given by
2 2 2 2 2 21 2
2 21 2 1 1
22 2 1 2 1 2
2 2 2 21 1 2 1 2
1 2
2
1 1 12 2 2
(( ) Var[ ] ( ) Var[ ]2
( ) Var[ ] 2( )( )Cov[ , ])
( V )
( )
ar[ ] ( Var[ ]2
Var[ ] )
m m m w w w w
m
m m w w w w
mm w w m w w
m w w m w m w w w
ww w w w w w m
b v b v v v v
r v y v y
v y v v y y
v
r v y v y y
m m m
δ δ δ
δ
π m m
δ δ
δ δ
− −
− − − + −
+ − + − −
− +
+ + −
+
=
+
(A.9a)
where wiv according to (12a). Substituting (12a) into (A.9a), using
1 2Var[ ] Var[ ] Var[ ]w w wy y y= = , 2Var[ ]w w
ww
r yAm
= , and 1 2Cov[ , ] Var[ ]w w wy y yab= , and
simplifying, yields
2 2 22
2 21 (1 )
1 (1 1
1 ( )2
1 (1 ) (1 )122 () )1
w
m w
m m
w w
m m
w w
rm m m m m m mr r
r rw wr rw
w m mr rw r r
w
w w w w
A
A A A A
b v r v
A Ab
π
ab
m m σ
abm δ δ
ab am b
+− +
+ +
=
+ + + + + +
+ +⋅ −
+
+
(A.9b)
Differentiating (A.9b) with respect to mv and mδ and solving the associated first-order
conditions yields (9a) and (12b). Substituting (12b) into (A.9b) yields
( )( )( )
22 1 (1 )
1 (1 ) 1 (1 )(1 )
m
w
m m
w w
rw wr
r rw w w wr
Ow
r
Db A
A A A A
ab
bπ
a ab
+=
+
+ + + + + +,
which simplifies to the expression in (12c).
Proof of Proposition 2:
The principal’s expected profit from the manager’s effort is the same with transparent and
obfuscate monitoring. Hence, DT DOπ π≥ simplifies to DT DOw wπ π≥ .
40
To prove the statement in the text, note that the cost of delegated contracting with
transparent monitoring is given by
22
2(1Cost (1 )(1 (1 ))
1 [1 ( ))
1 ]m m
mw w
w
r rTM rr
ww
wr
r
AAA
abab ab
bab
b
+ − = + + + + + + −
and the cost of delegated contracting with obfuscate monitoring is given by
2
Cost (1 )1 (1 )
m
mw
w
rOM r
ww
wr
r
AAA
abab
= + + +
.
A sufficient condition for Cost CostOM TM≤ is
2 22
21
1 (1 ) 1 [1 (1 )]( )
m m
w w
r rr r
ww
w w
A AA Aa
b
ab b
ab
b
+ − ≤ + + + + + −
. (A.10)
When 2 01( )wAa bb + − > , (A.10) simplifies to ( )2 2 2(1 (1 ) ) 0m
w
rw wr
A Aab b b ab b− + + − − − < ,
which is only satisfied if 1a = + . Using 1a = + and 2
2 2Var[ ] 1
1w w w w
w w
r y rwA σ
bm m −= = and simplifying
yields
( )( )2 2
2 2 01 (1 ) (1 )(1 )
w w
w wm wr rσ σ
m mb b b b
b− + − − <
−,
which is satisfied if either { }2
2 )1
1(max ;w
w mw r rσ b b
b bm−
+> or { }2
2 )1
1(min ;w
w mw r rσ b b
b bm−
+< .
When 2 01( )wAa bb + − < , requiring 1a = − , (A.10) simplifies to
( )2 2 22 (1 2 (2 )(1 ) ) 0m
w
rw wr
A Aab b ab b ab b+ − + + + + + − < . (A.11)
41
As the second term in (A.11) is positive, the condition (A.11) is satisfied iff w wA A< , where wA
solves (A.11) for equality.
Using (9c) and (12c), DT DOw wπ π≥ iff
2
22
22
2
2
11 (1 ) (1 )(1 (1 ))1 [1 (1 )]
1 (1 )1 (1 )
( )m m
mw w
w
m
mw
w
ww w
w
ww w
w
r rrr rr
w
rrrr w
AA AA
AA A
b
A
b
bb
abab
abb ab ab
ab
+ − + − + + + + + + + −
+ + + + +
≥
(A.12a)
which simplifies to
( )
2
22
22
(1 )1 1
1(1 ) 1 (1 )1 1 1
( )
( )( )
m
mw
w
m m
w w m
w
r wrr
wr
r r wr r r
wr
AA
AA
abab
ab bb ab abab b
+
+ +
+ − > − + + + + + + + −
(A.12b)
Substituting 2
2 2 1w
θ
θ
σ wbσ σ w
= =+ +
and 2 2
2 2 (1 )w y w w
w w
r rwA σ σ
m mw= = + , where
2
2w
θσwσ
= , (A.12b) is satisfied
iff 2 3 4 50 1 2 3 4 5 0R v v v v v vw w w w w= + + + + + > , where
2 2 2
2 2 24 2
0
1 1( ) ( )
m m
w wm w w w w w w
m mw w w w
w w
r rr rr r r r
r rrr r
v σ σ σm m m
+ + = − + −
,
( )( )
2 2 2
2 2 2
2
2
2 21
2 3
( )[2 (1 ) 5(1 ) (2 ) 2 (1 ) ( )
( ) 7 (2 4 ) ( ) ]
m w w m m m m w w m m w w
w w w w w w ww w w
m m m w w
w w w w
r r r r r r r r r rr r r r r r r
r r r rr r r
v σ σ σm m m
σm
a a a
a
= − + + − + + + + +
+ + +,
42
( ) ( )( )( ) ( )
2 2
2 2
2 2
2 2
2
2 2 3 4
2
3
1 2 6 (1 ) 2 11 6 4 (2 ) ( )
2( ) (2 3 ) 5 (1 ) ( ) 3( ) 1 8 4 (1 2 ) ( )
m m m w w m m m m w w
w w w w w w ww w
m m m m w w m m m w w
w w w w w w ww w
r r r r r r r r rr r r r r r r
r r r r r r r r rr r r r r r r
v σ σm m
σ σm m
a a
a a
+ + − + + + + − +
− + + + − + + +
=,
( )( )( ) ( )
2 2
2 2
2 2
2 2
2 2
2 3 3 4
3 1 (1 ) 2(2 ) ( ) 6 (13 2 ) 2 (1 ( ) ) ( )
2( ) 4 (2 3 ) 7 4 (4 3 ) ( ) ( ) 15 50 27 (1 2 ) ( )
m m w w m m m m m w w
w w w w w w ww w
m m m m m w w m m m w w
w w w w w w w ww w
r r r r r r r r rr r r r r r r
r r r r r r r r r rr r r r r r r r
v σ σm m
σ σm m
a a a
a a
+ + + − + + − + − −
− + + + + − + + +
=
( )( )
( )
2 2
2 2
2
2
2
2
42 2
2 2 2
2 3
[2(1 )(1 2 ) 5 12 4( ) 4 (1 3 ( )
2 1 8 16( ) (1 8 16( ) ( )
4 2(3 7 ) 7 (1 2 ) ( ) ( )
)
)
]
m w w m m m m m w w
w w w w w ww w
m m m m m w w
w w w w w w
m m m w w
w w w w
r r r r r r r rr r r r r r
r r r r r rr r r r r
r r r rr r r
v σ σm m
σm
σm
a a
a
a
+ + + + + + + +
− − + + + + +
− + + +
=
, and
( )( )2 2 2
2 2 22
5 1 (( ) ) (1 2 ) 1 (4 6 ) 1 2m w w m m w w m w w
w w w ww w w
r r r r r r rr r r rv σ σ σ
m m ma+ + + + − += − .
(i) With positive correlation, 1a = + , 1 0v < . (a) When 2
21
2w
mw rσm
< , then 0 0v > and 5 0v > ,
implying that 0R > if w is sufficiently small or sufficiently large. (b) When 2
21 /1
2[ , )m ww
m mw
r rr r
σm
+∈
, then 0 0v > , 2 0v < , 3 0v < , 4 0v < , and 5 0v < , implying that 0R > if w is sufficiently small.
(c) When 2
21 /m ww
mw
r rr
σm
+≥ , then 1 0v ≤ , 2 0v < , 3 0v < , 4 0v < , and 5 0v < , implying that 0R ≤ .
(ii) With negative correlation, 1a = − , 1 0v > , 2 0v > , 3 0v > , 4 0v > , and 5 0v = . (a)
When 2
21 /m ww
mw
r rr
σm
+≤ , then 0 0v ≥ and 0R > for all w . (b) When 2
21 /m ww
mw
r rr
σm
+> then 0 0v < and
0R > if w is sufficiently large.
Proof of Proposition 3:
Using (4c), (9c), and (12c), DT DO CT COπ π π π− > − iff
43
2
22
22
2 2 2
22
11 (1 ) (1 )(1 (1 ))1 [1 (1 )]
1 (1 )1 (
( )
11 (1 )
1 )
m m
mw w
w
m
mw
w
ww w
w
w
r rrr rr
w
rrr
w w
www
w wwr
AA
b
AA
b bAA
AA AA
b
bb
abb ab a
aba
ab
b
b
b
+ − + − + + + + + + + −
− ≥ −++ −
+ + + + +
(A.13)
Substituting 2
2 2 1w
θ
θ
σ wbσ σ w
= =+ +
and 2 2
2 2 (1 )w y w w
w w
r rwA σ σ
m mw= = + , where
2
2w
θσwσ
= , (A.13) is
satisfied iff 2 3 4 5 6 70 1 2 3 4 5 6 7 0cR u u u u u u u uw w w w w w w= + + + + + + + ≥ , where
2 2 2 2
2 2 2 22 3
02
1 1( ) (1 ) ( )
m m
w ww w w w m w w w w
m mww w w w
w w
r rr rr r r r r
r rrr r
u σ σ σ σm m m m
+ + − + + −
=
,
( )2 2 2 2 2 2
2 2 2 2 2 2
2 2 2
2 2 2
2 2 2 3
2 2 2
1 (1 )[ 5(1 ) 7(1 ) 7( ) ( ) 9( ) ( )
(1 ) 2(1 ) (3 2 ) 2 (1 2 ( ) ) ) ( 3(( ) 2
w w w w m w w m m w w m w w m w w
w w w w ww w w w w w
w w m m m w w m m m w w m m
w w w w w w w ww w w
r r r r r r r r r r rr r r r r
r r r r r r r r r r rr r r r r r r r
u σ σ σ σ σ σm m m m m m
σ σ σm m m
a
− + − + − + + +
+ + + + + + + + + +
=
( )2
23)( ) ]w w
w
r σm
,
( )( )( )
2
2
2
2
2
2
2
2 3 2 2
2 3
2
2 3 3
2
(1 ) 2 1 3 ( ) 3 (1 )
1 2( ) ( ) 4 (4 7 2( ) ) ( )
2 1 10 3( ) 4( ) 5 (1 5 4( ) ( ) ) ( )
15 2 35( ) 6 (7
m m m m w w
w w w w w
m m m m m w w
w w w w w w
m m m m m m w w
w w w w w w w
m m m
w w w
r r r r rr r r r
r r r r r rr r r r r
r r r r r r rr r r r r r
r r rr r r
u σm
σm
σm
a
a
a
a
− + − + + + +
+ − − − − + +
+ + + − − + + +
− − −
=
+ ( )( ) ( )
2
2
2 2
2 2
2 4
2 2 5 2 6
12 7( ) ) ( )
2 1 6 32( ) 7 (1 2 ) ( ) 1 4 37( ) 8 (2 3 ) ( )
m m w w
w w w
m m m m w w m m m m m w w
w w w w w w w w ww w
r r rr r
r r r r r r r r r r rr r r r r r r r r
σm
σ σm m
a a
+ +
− + + + + − + + + +
,
44
( )( )( )
2
2
2
2
2
2
2
2 3 2 3 2
2 3 2 3
3
3
1 (1 ) 4 (1 )(1 )
3 23 26( ) 3( ) (20 37 22( ) 3( ) ) ( )
2 4 28( ) 16( ) (9 44 43( ) 15( ) ) ( )
5 80 105(
m m w w
w w w
m m m m m m w w
w w w w w w w
m m m m m w w
w w w w w w
m m m
w w w
r r rr r
r r r r r r rr r r r r r
r r r r r rr r r r r
r r rr r r
u σm
σm
σm
a τ a
a
a
− − + − + + +
+ − − − − + + +
− − − + +
+ −
=
+
−
+
( )( )( )
2
2
2
2
2
2
2 2 4
2 2 5
2 2 6
) (95 174 111( ) ) ( )
2 6 36 80( ) (20 80 81( ) ) ( )
7 28 91( ) (1 54 79( ) ) ( )
m m w w
w w w
m m m m w w
w w w w w
m m m m m w w
w w w w w w
r r rr r
r r r r rr r r r
r r r r r rr r r r r
σm
σm
σm
a
τ a
a
− + +
+ + + +− +
− + + + + +
,
( )( )( )
2
2
2
2
2
2
2
2
2 3 2
2
4
2
[2 4(1 ) (1 )
10 3 2(1 )(4 )(1 2 3( ) )
2 4(3 5 ) (1 )(7 32 49( ) 25( ) ) ( )
21 172 165( ) 4 (25 54 41( ) ) (
w w m m
w ww
m m m m w w
w w w w w
m m m m w w
w w w w w
m m m m m w w
w w w w w
r r rr r
r r r r rr r r r
r r r r rr r r r
r r r r r rr r r r r
u σm
σm
σm
σ
a
a
a
a
− + + +
+ − + + + + + +
+ − + + + + + +
+ + + + + +
=
( )( )
2
2
2
2
2
3
2 4
2 2 5
)
2 14 41 (2 3 ) 5 (6 24 25( ) ) ( )
19 76 142( ) 2 (3 50 69( ) ) ( ) ]
w
m m m m m w w
w w w w w w
m m m m m w w
w w w w w w
r r r r r rr r r r r
r r r r r rr r r r r
m
σm
σm
a
a
+ + + + +
+ + +
+
+ + +
,
( )( )( )
2
2
2
2
2
2
2
2
2
52
3
2
( ) [(1 ) 3 4( ) 4(1 )
2 (4 ) (1 )(2 9 26( ) 18( ) )
2(11 ) (1 )(49 142 131( ) ) ( )
2 3(3 4 ) (1 )(25 100 108( )
w w m m m
w w ww
m m m m w w
w w w w w
m m m m w w
w w w w w
m m m m
w w w w
r r r rr r r
r r r r rr r r r
r r r r rr r r r
r r r rr r r r
u σm
σm
σm
a a
a
a
a
− + + + +
+ − + + + + + +
+ − + + + + +
+ − + + + + +
=
( )( )
2
2
2
2
2 3
2 4
) ( )
4(3 2 ) (1 )(13 108 136( ) ) ( ) ]
w w
w
m m m m w w
w w w w w
r
r r r r rr r r r
σm
σm
a+ − + + + +
,
( ) ( )( ) ( )
2 2
2 2
2 2
2 2
3 2 26
2 2 2 3
( ) [2(1 ) 1 4( ) 2(1 ) 1 2(1 )(5 22 23( ) )
2 2 (1 )(11 44 48( ) ) ( ) 4 1 (1 )(3 16 18( ) ) ( ) ]
w w m m m m m w w
w w w w ww w
m m w w m m w w
w w w ww w
r r r r r r rr r r r r
r r r r r rr r r r
u σ σm m
σ σm m
a a a
a a
= − + + + + + + + + +
+ − + + + + + + + + +,
( )2 2 2
2 2 24 2 2
7 ( ) 1 1 2(1 ) 4( ) 4(1 2 ( )( ) 1)w w m m m m w w w w
w w w ww w w
r r r r r r rr r r ru σ σ σ
m m ma a− + + + + + += + .
(i) With positive correlation, 1a = + , 1,...,7 0u+ < , where the index “+” indicates 1a = + ;
thus, 0cR > iff 0 0u > , which is the case if 2
21 /m ww
mw
r rr
σm
+< and 2θσ is sufficiently small. (ii) With
45
negative correlation, 1a = − , 1,...,5u− are of ambiguous sign, 6 0u− < , and 7 0u− = , where the index “-
” indicates 1a = − ; thus, 2
21 /m ww
mw
r rr
σm
+< and 2θσ sufficiently small is a sufficient condition for
0cR > .
Complementarity is more likely to prevail with negative correlation than with positive
correlation if c cR R− +> where ( 1)c cR R a− = = − and ( 1)c cR R a+ = = + . Comparing the
coefficients derived above yields 1,...,7 1,...,7 0u u− +− > .
