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Disclosure of status in an agency setting
Accepted Manuscript Title: Disclosure of Status in an Agency Setting Author: Anthony M. Marino Oguzhan Ozbas PII: DOI: Reference:
S0167-2681(14)00143-7 http://dx.doi.org/doi:10.1016/j.jebo.2014.05.002 JEBO 3369
To appear in:
Journal
Received date: Revised date: Accepted date:
5-4-2013 8-5-2014 9-5-2014
of
Economic
Behavior
&
Organization
Please cite this article as: Anthony M. Marino, Oguzhan Ozbas, Disclosure of Status in an Agency Setting, Journal of Economic Behavior and Organization (2014), http://dx.doi.org/10.1016/j.jebo.2014.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights JEBO.pdf
Highlights
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We develop a model in which the principal and the agent share private information about the value of the agent for a multi-agent organization. The principal can disclose private information and make public the relative standing or status of all agents in the organization. We study whether it is better in terms of profit and utility to disclose or to not disclose status to the group of agents. Conditions for the optimality of disclosure versus non-disclosure are characterized for the cases of exogenous and endogenous human capital.
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Disclosure of Status in an Agency Setting∗
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Anthony M. Marino†
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May 2014
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Oguzhan Ozbas‡
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Abstract
We develop a model in which the principal and the agent share private information about the
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value of the agent for a multi-agent organization. The principal can disclose private information
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and make public the relative standing or status of all agents in the organization. We study whether it is better in terms of profit and utility to disclose or to not disclose status to the group of agents. Conditions for the optimality of disclosure versus non-disclosure are characterized for
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the cases of exogenous and endogenous human capital. Keywords: Status; Disclosure; Human Capital. JEL Codes: D1, D2, J3, L, L2, M5.
∗
We thank William Neilson (the editor) and two anonymous referees for helpful comments. We also benefited from the comments of Ricardo Alonso, Odilon Câmara, Heikki Rantakari, and workshop participants at the University of Southern California and Queens University. Financial support from the Marshall General Research Fund is gratefully acknowledged. † Marshall School of Business, University of Southern California; [email protected] ‡ Koç University and Marshall School of Business, University of Southern California; [email protected]
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1. Introduction In most multi-agent organizations, the principal and an individual agent share private information regarding the value of that agent for the organization. Such information might arise in the course
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of day-to-day interaction between the principal and the agent or come from periodic performance
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reviews. The organization then has a choice in most situations to make the private information public, and thus disclose the relative standing or status of all members of the organization. In
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this paper, we study this novel organizational design question in the context of an endogenously optimal agency contract while building on recent theoretical and empirical works on status.
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Our analysis is motivated by the observation that firms devote significant management attention
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to thinking about the nature of evaluative feedback given to workers, and specifically about whether to disclose rankings as part of the feedback process to improve organizational performance. The
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right disclosure policy is not clear a priori, as a higher than expected rank can incentivise one
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worker while at the same time a lower than expected rank can disincentivise another worker going forward. Indeed, McGregor (2006) discusses the struggles that many prominent companies have recently had in implementing evaluation systems that use rankings. She points out that as many
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as one third of U.S. corporations reveal rankings to their employees. However, companies such as Goldman Sachs Group Inc., General Electric Co., and Chemtura have had doubts concerning the use of rankings to motivate employees and have adjusted the implementation of such systems. The benefits and costs to an agent of status revelation can in principle emanate from an organizational reward structure related to status and/or through status entering the agent’s utility function directly. In a field experiment involving employees at Amazon’s crowd-sourcing website, Mechanical Turk, Barankay (2010) carefully shuts down the former channel to test the direct utility channel and shows that on average relative performance feedback reduces future effort. In another field experiment involving salespeople at a furniture company, Barankay (2011) again finds that 1 Page 3 of 46
such feedback reduces performance. Similarly, Ashraf et al. (2014) find that disclosures that are carefully designed to induce social comparisons reduce effort, especially among low ability workers in a nationwide health worker training program in Zambia. In a laboratory setting where students
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can signal ability to others, McManus and Rao (2013) also find that social comparisons reduce effort. In contrast, Azmat and Iriberri (2010) study the effect of relative performance feedback
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on student effort and find that it improves performance. These field studies suggest that status
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directly affects utility, and the conflicting results on performance suggest that the effect of status revelation on organizational effort incentives and performance can depend on organization specific
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circumstances. Our theoretical analysis identifies such conditions.
As hypothesized in much of the literature in economics and psychology, we assume that status
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and wages are complements in the agent’s utility. Given this assumption, we consider multiple identical agents in an organization where there is a hidden action agency problem for the principal
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with regard to each agent. The principal designs an optimal incentive contract for each agent and
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commits to a disclosure or a non-disclosure policy at the time of hiring the agents. Initially, we assume that each agent is endowed with a certain probability of having high human capital and
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study the effect of disclosing status to the group of agents after the agents’ human capital levels are realized. We find that disclosure is optimal for the principal if the firm faces a sufficiently favorable production situation — one in which the typical agent exhibits a high sensitivity of cash flow production to effort and/or a low sensitivity of effort cost to effort. In such situations, status concerns play a positive motivational role and increase the principal’s profit. Conversely, if the firm faces a sufficiently unfavorable production situation, then status concerns play a negative motivational role and non-disclosure is optimal for the principal. Interestingly, we find that agents always prefer disclosure to non-disclosure because more information about status allows them to better condition their effort choices. Thus, our model points to status concerns as a possible source
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of tension between agents and principals regarding the appropriate amount of organization-wide transparency in principal-agent relationships. In the basic model, we also ask how changes in human capital affect the principal’s profit. We
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find that under either disclosure rule, an increase in the typical agent’s probability of having high human capital can have a strong negative motivational effect through toughened competition for
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status and decrease the principal’s expected profit when there is rapidly rising or falling returns
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to status in utility. In all other cases, an increase in human capital increases expected profit as it normally does in standard models.
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In a final section of the paper, we extend the model by allowing the agent to exert costly effort to increase the probability of having high human capital. If the firm faces a favorable production
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situation and there are increasing returns to status, then disclosure always results in greater human capital effort and expected profit as long as returns to status are not rising too rapidly. On the
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other hand, if the production situation is unfavorable and there are decreasing returns to status,
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then non-disclosure generates greater human capital effort and expected profit as long as returns to status are not falling too rapidly.
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Our work is broadly related to a growing literature on social status. Departing from traditional preferences over absolute consumption, this literature considers social status as an additional argument in the utility function, which is similar to how psychologists and sociologists view preferences. Various experimental studies confirm the existence of preferences for status by inducing status in the lab (See Heffetz and Frank (2011) for an excellent survey.). Motivated by Maslow’s (1943) hierarchy of needs, we use a preference specification featuring a positive interaction between status and wealth. Consistent with Maslow’s theory that status and self-actualization motives are high-level psychological needs that come only after basic physiological necessities are met, the positive interaction between status and wealth implies that more wealthy
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agents care more about status. While advances from related fields continue to refine thinking in psychology in important ways, the broad outlines of Maslow’s hierarchical approach to understanding human motivation remain central (Kenrick et al. (2010)).
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Our other modeling assumptions are intended to capture key features of status that consistently appear in empirical and experimental research. First and foremost, status is positional with respect
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to a desirable trait. We take human capital to be that desirable trait in an organization. Status
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is also non-tradable and only acquired through actions. In our model, we assume that agents can engage in costly effort to accumulate human capital and thereby acquire status. Lastly, status is
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based on social perceptions and visibility. This is our main motivation for focusing on disclosure policies.
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Our paper is related to recent works on behavioral agency theory that consider non-material incentives such as status and respect in optimal contracts (Auriol and Renault (2008), and Ellingsen
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and Johannesson (2008)).1 These works also explore implications for organizational design as we
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do. While our setting incorporates many of the standard features of status, our modeling approach is different in several important respects. In contrast to the usual modeling approach of endowing
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the principal with precise control over the allocation of a status budget, our approach instead endows the agent with a chance to achieve status through costly effort. Our emphasis on disclosure policies as an organizational design problem is also unique in endogenizing the source of status. A related behavioral literature on other regarding preferences focuses on concerns for relative consumption and studies the effects of wage externalities across workers. These works represent a different but complementary approach to the study of interrelated preferences in organizations (see Itoh (2004) and his references). Finally, another related stream of literature studies the optimality of interim relative perfor1
Moldovanu et al. (2007) and Dubey and Geanakoplos (2010) consider agents concerned with status and formulate optimal status partitions in their respective problems.
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mance feedback in the context of dynamic tournaments (Lizzeri et al. (2002), Crutzen et al. (2010), Ederer (2010), Aoyagi (2010), and Goltsman and Mukherjee (2011)). In a tournament model with status, Ederer and Patacconi (2010) show how a firm would prefer to reward few winners without
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explicitly identifying many losers if workers’ utility depends on own wage and effort as well as a modal reference wage. In contrast to our paper, in these papers the benefits and costs of status
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revelation feed through the monetary reward structure of the tournament as opposed to being gen-
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erated by non-monetary status in the agent’s utility. We view our work as complementary given the pervasiveness of non-pecuniary status-like awards in the real world (Ellingsen and Johannesson,
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(2007)).
The paper is organized as follows. Section 2 presents the model. Section 3 compares disclo-
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sure and non-disclosure policies, and conducts comparative statics with respect to human capital. Section 4 endogenizes human capital accumulation, Section 5 discusses robustness, and Section 6
2. The Basic Model
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concludes.
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Consider a firm with a principal and two agents. In stage 1, the two identical agents join the firm and each is endowed with a firm specific human capital level which can be high, or low, With probability an agent’s capital is high, which is realized after the agent joins the firm. Only the agent and the principal observe the agent’s realized human capital. One can think of as any measure of worth of an individual in the organization which is intrinsic to the individual. In our model, is a critical component of the agent’s status, which we will initially assume is exogenous and later make endogenous. Importantly, is not a symbol or an award assigned by the principal as in Auriol and Renault (2008) nor is it bestowed by society, but instead it is a representation of the skill of the agent in doing her/his job within the organization. This measure could arise from
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third parties or from the observation of the principal. For a teacher, might represent a score from teaching evaluations generated by students or it could be the average score of the teacher’s students in taking standardized tests. For a sales person, could be a measure of that person’s interpersonal
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skills and this information might be communicated to the agent and the principal by customers. Most organizations have periodic performance reviews in which the agent submits to the principal
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information pertinent to job performance, and the principal generates an overall score, for the
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agent. In many principal-agent settings, the principal learns such information about agents. The disclosure policy in this paper revolves around The disclosure issue is whether to divulge to each
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agent the human capital levels of all agents (the other agent).
We will measure the relative standing or status of an agent by the share of total human capital
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and assume that it enters an agent’s utility function in a positive way. Given the two outcomes for there are three status outcomes ∈ { } for each agent. Status is medium if an agent
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attains the same capital as the other agent
≡
1 = = + + 2
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The stage 1 probability of this state conditional on an agent attaining a high capital is 2 while it is (1 − )2 conditional on an agent attaining a low capital. Status is low if an agent attains a low capital and the other agent attains a high capital
≡
− 1 = − where ≡ + 2 2( + )
The stage 1 probability of this state is (1 − ). Status is high if an agent obtains a high capital
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and the other agent attains a low capital
1 = + + 2
The probability of this outcome is (1 − ) from a stage 1 perspective.
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≡
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ˆ ∈ { 0} with Prob(ˆ = ) =
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Given a human capital parameter , in stage 2 an agent produces a cash flow, which is given by
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where is the agent effort in cash flow generation and ∈ (0 1) is the effort elasticity parameter. We assume that ∈ (0 1). All agents have the same elasticity parameter , but those with
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greater human capital or ability generate more expected cash flow for a given amount of effort.
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Also, as discussed before, status is directly related to productivity and more than a token.
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We further assume that compensation can only be based the agent’s cash flow, ruling out three other potential conditioning variables: the agent’s human capital, and the other agent’s cash flow and human capital. In our model, it turns out that the principal would not condition compensation
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on the agent’s human capital even if there were no third party verifiability problems associated with measuring human capital.
Given the two states of cash flow, the principal’s compensation contract for the agent is given by2
2
= + ˆ
Given the two state cash flow process, this linear contract is fully general.
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The principal and the agent are risk neutral and the agent’s utility function is given by
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= () −
The function 1 represents the agent’s personal cost of effort, and () represents a status
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function which is assumed to be complementary with the wage in the agent’s utility. We assume that is increasing in Thus, the agent is risk neutral in the wage, given status, and utility is
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strongly monotonic in status and the wage. Preferences are strictly convex in the status function
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and the wage so that there is a diminishing marginal rate of substitution between and However, because the curvature of is general, there can be an increasing or decreasing marginal
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rate of substitution between the wage and status . If is concave, non-increasing marginal returns to status, then the marginal rate of substitution between and is decreasing or constant.
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If is sufficiently strictly convex, increasing marginal returns to status, then the marginal rate of
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substitution between and can be increasing. Finally, under this formulation, the status function represents the marginal utility of another dollar. Because it is multiplicative with the wage in
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utility, there are increasing returns to scale in and with respect to utility, gross of effort cost. The complementarity assumption has been utilized by others in examining status and utility (Auriol and Renault (2008), and Hopkins and Kornienko (2004)). The agent is assumed to be risk neutral in income, given status. The complementarity assumption asserts that the marginal utility of income is increasing in status such that richer agents enjoy extra status more than poorer agents and would be willing to exert more effort to increase status. As pointed out in Auriol and Renault (2008), this behavior is consistent with Maslow’s pyramid of needs (1943, 1970) and his modern reinterpretation, Kenrick et al. (2010), from which it can be argued that individuals with lower income are concerned with material needs and do not pay as much attention to status, while those
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with greater income have met their material needs and are much more preoccupied with raising their status.3 At this point it is useful to summarize the sequence of decisions from the perspective of an
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agent. (i) The principal meets with the agent along with another agent while precommitting to a disclosure policy and informing the agent that a compensation contract will be offered after the
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principal observes the agent’s ; (ii) The agent is endowed with a level of human capital which
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both the principal and the agent observe; (iii) The principal sets and subject to the agent’s participation and limited liability constraints, fully discloses in the case of disclosure, or discloses
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nothing in the case of non-disclosure; (iv) The agent exerts effort ; (v) The returns to the principal and the agent accrue. The disclosure policy takes one of two forms. Full disclosure is defined as
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the case where the principal makes public the status levels of all agents, and non-disclosure is the
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2.1. Disclosure and Non-disclosure
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situation where the principal does not reveal any information about status.4
In the case of full disclosure, an agent knows his status level in addition to his human capital
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level before exerting effort. The agent solves
max ()( + ) − {}
The solution is given by
1 ( ) = [ ()] −
(IC )
3
For some indirect empirical evidence that there is complementarity between and see Auriol and Renault (2008), pp. 309-310. 4 We do not consider partial disclosure where the principal sends a signal about the other agent’s Note that many real world disclosure policies operate as full or no disclosure.
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and the second order condition for this problem is met under our assumptions.5 Condition (IC ) is a constraint in the principal’s problem. In addition, the principal faces participation and limited liability constraints as follows: ()( + ) − ≥ 0
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(P )
(LL)
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where in (P ) we assume that the agent’s reservation utility is zero.
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≥ 0
Under the assumption of a zero reservation utility, the agent’s participation constraint is non-
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agent’s human capital and status is described by
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binding so that the principal’s optimal = 06 The principal’s choice of compensation, given the
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The first order condition is
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max [ ()] − (1 − ) {}
=
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(1)
At a point where the first order condition is met, the second order condition is satisfied. To see this, note that if we define
≡
−
then the principal’s problem is equivalent to maximizing − +1 The first order condition yields = ( + 1) and the second order condition is that ( − 1) −2 − ( + 1) 0 which can 5
The second order condition for this problem is met if the second derivative of − is negative. Under our assumption that ∈ (0 1) and 1 this derivative is negative. We assume that parameters are such that ∈ (0 1) 6 Note that − − 0 suffices to guarantee that P is nonbinding. Substituting for − from IC , this condition is (1 − ) 0 which is true. If P is nonbinding then it is clear that = 0
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be rewritten as ( − 1) 2 ( + 1) Substituting = ( + 1) into the latter condition yields −1 0 which is true. Note that the agent’s effort depends on and while the incentive pay does not. Thus,
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our modelling choices help isolate a pure effort channel in principal-agent relationships with status concerns.
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The principal’s reduced form profit under full disclosure from an agent who knows his human
+1 2 0 (1 + )2 +1
(2)
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( ) = · +1 (()) where ≡
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capital and status is written as
+1 (()) +1 (1 + )
(3)
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( ) =
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Further, the agent’s reduced form utility can be written as
This completes optimization in stage 2, under full disclosure. With non-disclosure, the principal does not reveal an agent’s human capital level to any other
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agent. The analysis of this case is identical to that of disclosure with [()|] replacing () where
[()| ] = ( ) + (1 − )( ) and
[()| ] = ( ) + (1 − )( )
It is again true that the optimal beta is = 7 Note that does not convey any information about status , and, thus, the agent does not form a new status expectation after observing the 7 Using the same reasoning as in the case of disclosure, it can be shown that the second order conditon to the principal’s problem is met at a point where the first order condition is satisfied.
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principal’s choice of compensation, consistent with non-disclosure. The principal’s reduced form profit from employing an agent who knows only his own human capital is
+1 [ [()|]] +1 (1 + )
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() =
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The reduced form utility of the agent is
(4)
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() = · +1 [ [()|]]
(5)
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It is worth noting that disclosure has nuanced motivational effects in our model. We see that if the state of the world is such that there is a high and a low human capital agent, then the
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high (low) capital agent has high (low) status and has stronger (weaker) effort incentives under disclosure than under non-disclosure. In this case, disclosure raises the effort incentives of the high
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human capital agent and dampens those of the low human capital agent. This result is similar to
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the results in the dynamic tournament literature discussed in the introduction. However, what is further possible in our model is that when both agents are of high capability, disclosure dampens
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effort incentives because agents know that their status level is while with non-disclosure they attach a probability of (1 − ) to status being In the opposite case where both agents are of low capability, disclosure has a positive effect on effort incentives because agents know that the low status outcome is not possible, whereas that remains a possibility under non-disclosure. Therefore, in our model, disclosure does not just motivate high capability agents and demotivate low capability agents.
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3. Comparisons and Comparative Statics 3.1. Comparisons
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In this sub-section, we are interested in examining two questions. If the principal were to precommit to a disclosure policy and contract with the agent, then which disclosure policy would result in
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greater ex ante expected profit for the principal and which policy would result in greater ex ante
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expected utility for the agent? Let ( ) ≡ = Consider
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[ ( )] = {+1 [ + (1 − ) ]
[ ( )] =
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+(1 − )+1 [ + (1 − ) ]} {+1 [ +1 + (1 − ) +1 ] (1 + ) (7)
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d
+(1 − )+1 [ +1 + (1 − ) +1 ]}
and
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[ ()] = {+1 [ + (1 − ) ] +(1 − )+1 [ + (1 − ) ] }
[ ()] =
{+1 [ + (1 − ) ] +1 (1 + )
+(1 − )+1 [ + (1 − ) ] +1 }
(8)
Proposition 1. The following comparisons can be made. ¤ £ (i) Given 0 we have () T [ ( )] as T 1
£ ¤ (ii) ( ) [ ()]
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(All proofs are provided in the Appendix.) We adopt the following definition to describe the economic environment. Definition. The situation where 0 1 is defined as an unfavorable situation for the firm,
The parameter =
−
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and 1 is defined as a favorable situation for the firm. 0 is monotonically increasing in ∈ (0 1) and monotonically decreasing
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in 1 If the situation is unfavorable, 1 then the firm’s expected cash flow technology is not
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very responsive to the agent’s effort and/or the disutility of effort is very responsive to increases in effort exertion. If the situation is favorable, 1, then increases in the agent’s effort produce
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large increases in expected cash flow and/or very small increases in effort cost.
Proposition 1 basically relies on Jensen’s inequality to show that in unfavorable production
unfavorable situation, the function =
−
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situations non-disclosure is better for the principal and in favorable ones disclosure is better. In the 1 is concave in and the sign of ( )−( ) is
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that of the difference between a convex combination of the images of this concave function and the
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concave image of the same convex combination of values. It then follows that by concavity of and Jensen’s inequality this difference is negative and non-disclosure dominates for the principal.
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A type of risk aversion is created for the principal, with respect to the status variable which induces him to shy away from disclosing status to the agent. In the favorable production situation, the opposite is true in that the function is convex in as opposed to concave. In this case, the principal prefers variability with respect to the status variable , and disclosure is optimal. One can think about Proposition 1 (i) through the motivational effects of disclosure for agents ¡ ¢ with status concerns. If 1, the principal’s optimal contract = induces greater expected
probability of high cash flow when status levels of all agents are made public. If 1, the opposite
result holds when the principal does not disclose any information about status. The sign of the
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difference in expected probability of high cash flow is given by
= { [ + (1 − ) − ( + (1 − ) ) ]
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{ [ ] − [ ]}
cr
+(1 − ) [ + (1 − ) − ( + (1 − ) ) ]}
which is positive when 1 and negative when 1. And, because the optimal contract pays
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¢ ¡ the same = fraction of high cash flow under both status disclosure policies, the comparisons
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for expected probability of high cash flow carry over to expected profits that the principal retains after paying the agent.
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Proposition 1 also shows that while non-disclosure may be better for the principal with an unfavorable situation, it is never better than disclosure for the agent. For an agent, ( ) and
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( ) are based on a convex function of , +1 ( + 1) 1 In the case of non-disclosure, the
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agent’s expected utility ( ) is effectively the convex image of a convex combination of values whereas in the case of disclosure, the agent’s utility ( ) is the same convex combination of image
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values of +1 with all other terms equal. It then follows that by convexity of +1 the latter expectation dominates and disclosure is better than non-disclosure. The intuition here is that the agent can make better effort decisions under disclosure versus non-disclosure because effort is based on actual status as opposed to the expectation of status as would be the case with non-disclosure. Given the same across solutions, ex-ante expected utility is greater under disclosure regardless of the favorability versus unfavorability of production. However, if there is an unfavorable production situation, the principal would prefer non-disclosure and prefer the agent to make the “wrong” effort choice by basing effort on expected status. This yields greater ex-ante expected cash flow, given that the agent obtains rents through the non-
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binding participation constraint and that the principal is maximizing his share of expected cash flow rather than total surplus. In sum, for firms with favorable production situations, disclosure raises both profit and utility so
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that total surplus is greater than it is under non-disclosure. In the case of firms facing unfavorable production situations, non-disclosure raises profit, but lowers utility, so that the effect on total
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surplus is ambiguous. It is easy to show that, depending on the parameters, total surplus can be
us
greater or less under non-disclosure with an unfavorable production situation. If the production situation is sufficiently unfavorable, then total surplus is always greater under non-disclosure. Let
an
the total surplus under disclosure be given by ( ) and let the same under non-disclosure be denoted () Using the results of Proposition 1 we can summarize the results on total surplus
M
in the following.
[ ()]
d
Corollary 1. If 1 [ ( )] [ ()] If 1 is sufficiently small, then [ ( )]
te
The core issue regarding disclosure is whether an actor’s equilibrium payoff function is concave or convex in the hidden information variable, For the agent, the equilibrium payoff function
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
is convex in the hidden information variable regardless of whether the situation is unfavorable ( 1) or favorable ( 1). For the principal, concavity is implied in our constant elasticity setting by unfavorability ( 1) and convexity is implied by favorability ( 1) The unfavorability condition is true in situations where it is hard for the agent to generate a large response in the probability of success by exerting more effort, and, when he does exert more effort, the cost of effort responds greatly. In sales this might correspond to a situation where the client is currently not purchasing the sales person’s product and is very much wed to the status quo product. It is very costly and difficult for the sales person to change the client’s mind. In academics, an example might be a difficult required teaching assignment where the students must take a course, which
16 Page 18 of 46
they abhor on average because they are ignorant of the future useful applications, and they are quite vocal about their feelings. The teacher then faces a probability of success which is not very responsive to effort and at the same time personal effort cost rises rapidly with increases in effort.
ip t
In the sales example, a favorable situation might be created when the client has experienced the sales person’s product for some time and intensely likes the product. In the academic example, a
cr
favorable situation is created when the teacher is given the assignment to teach a known useful and
us
very popular elective course.
an
3.2. Comparative Statics of
In standard principal-agent models, an increase in the agent’s talent ( in our model) would lead to
M
an increase in the principal’s expected profit in equilibrium as long as the marginal cost of talent is sufficiently low for the principal. As we show below, this standard result may not hold when agents
d
have status concerns. Specifically, the principal may be worse off with more talented agents, even
te
when the principal faces no marginal cost of hiring talent in the traditional sense — recall that the participation constraint is non-binding at a solution. The reason is that status concerns affect the
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
principal’s ability to induce effort in our model. In one set of results, we show that if agents have a sufficiently risk-seeking preference for status (convex ), the positive talent effect can be swamped by the negative motivational effect of lower status due to an increase in in the interval [12 1], leading the principal to locally prefer less talent over more talent and globally prefer an interior under either disclosure rule. In a parallel set of results, a similar local effect arises if agents have a sufficiently risk-averse preference for status (concave ). Again in this case, the positive talent effect can be swamped by the negative motivational effect of lower status due to an increase in in the interval [0 12]. Proposition 2 contains these results for the case of disclosure, and Proposition 3 presents analogous results for
17 Page 19 of 46
the case of non-disclosure.
3.2.1. Comparative Statics of : Disclosure
ip t
Expected profit under disclosure from (7) can be rewritten as
cr
[ ( )] = [(+1 )+(+1 ++1 −2+1 )+(+1 ++1 −+1 −+1 )2 ]
us
so that [ ( )] is a quadratic function of Differentiation with respect to yields the following
an
linear function of
M
[ ( )] = (+1 ++1 −2+1 )+2[ (+1 ++1 )−+1 −+1 ] (9)
If the slope coefficient of this function is negative (non-negative), then ( ( )) is a strictly
te
d
concave (convex) function of Define the following augmented share term
( ) ≡
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
+1 +1 + +1
where it can be verified that ( ) ∈ (05 1) (0) = and 0 ( ) 0 by ∈ (0 1) and Then, from (9),
2 [ ( )]2 0 if ( ) + (1 − ( )) 1
(10)
2 [ ( )]2 ≥ 0 if ( ) + (1 − ( )) ≤ 1
(11)
Using (9)-(11), we present Proposition 2. For a given we have that the following hold.
18 Page 20 of 46
(i) Let (10) hold Then [ ( )] is strictly concave in If
[ + (1 − ) ] ∈ (1 2]
ip t
(12)
then [ ( )] 0 ∈ (0 1) and [ ( )] has a boundary maximum at = 1 If
cr
+ (1 − ) 2 then [ ( )] has an interior maximum at a ˆ( ) ∈ (05 1)
us
(ii) Let (11) hold. Then [ ( )] is convex in If
(13)
an
+ (1 − ) ∈ [2(1 − ) 1]
then [ ( )] 0 ∈ (0 1) and [ ( )] has a boundary maximum at = 1 If
M
+ (1 − ) 2(1 − ) then [ ( )] is strictly convex and has an interior minimum at a ˜( ) ∈ (0 (1 − ( ))) where (1 − ( )) 05
te
d
Given values of the parameters and the curvature of [ ( )] in depends on the agent’s preferences for status through the three image values of the function Condition (10) for concavity occurs if the ratios of = are large as would be the case if the three image values
Ac ce p
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= formed a convex first order spline function. Such a function would take on the form ˆ : [ ] → [( ) ( )], where
ˆ () =
⎧ ⎪ ⎪ ⎨ ( ) + (1 − )( ) for = + (1 − ) ∈ [0 1]
⎪ ⎪ ⎩ ( ) + (1 − )( ) for = + (1 − ) ∈ [0 1]
A strictly convex spline function ˆ would satisfy the condition
− − ⇔ − 1 1 − − −
(14)
19 Page 21 of 46
so that each ratio is large.8 The converse condition (11) for convexity is true if the ratios = are small as would be the case with a concave first order spline function. We call the former case where (10) is met the case of increasing marginal returns to status (IRS) and the
ip t
latter case where (11) holds the case of decreasing marginal returns to status (DRS). Proposition 2 tells us that when (10) holds, the profit function is strictly concave in and this
cr
curvature is induced by IRS. If + (1 − ) 2 and IRS are great, expected profit
us
reaches a unique maximum at an interior ˆ which is greater than one half. In this case, expected profit actually decreases in for ˆ If conversely, 1 + (1 − ) ≤ 2 and IRS
an
are not too great, expected profit strictly increases in and reaches a maximum at = 1 When (11) holds, as would be the case with DRS, then profit is convex in . If + (1 −
M
) 2(1 − ) and DRS are great, then expected profit is strictly convex in and there is an interior ˜ less than one half, where expected profit reaches a minimum. For ˜ profit decreases
d
in If conversely, 2(1 − ) ≤ + (1 − ) ≤ 1 and DRS are not too great, expected
te
profit strictly increases in and reaches a maximum at = 1 The counterintuitive cases, where expected profit decreases in , appear whenever returns to
Ac ce p
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status are strongly rising or strongly falling. In the former case it can happen at high , and in the latter case it can happen at low . The intuition for these results can be explained by noting that, given , the probability of having heterogeneous agents is (1 − ) which is maximized at = 05. With IRS, heterogeneity has a positive effect on expected profit because the greater effort supplied by the relatively high status agent exceeds the lower effort supplied by the relatively low status agent. If IRS are sufficiently strong and gets above 12 , the positive effect of heterogeneity on expected profit declines rapidly and expected profit peaks at an interior high . With DRS, heterogeneity has a negative effect on expected profit, and if DRS are sufficiently strong, expected 8
In (14), we use the fact that ( − ) = ( − ) = given our assumptions.
20 Page 22 of 46
profit dips at an interior low . Proposition 2 can also be interpreted as saying that if the principal could choose the ex ante probability that an agent would have high human capital, he would choose = 1 except in one
ip t
case. This case occurs when there are strong increasing returns to status in the sense that the converse of (12) holds, + (1 − ) 2 which in turn implies (10). This condition
cr
insures that the principal’s profit function is strictly concave in with an interior maximum ˆ ∈
us
(05 1) Here, the optimal ˆ might be achieved through an ex ante screening process set up by the firm. The design of such a process is outside the scope of this paper, but the result that = 1
an
is not always preferred is interesting and counterintuitive. A numerical example of this case can be generated by positing the convex status function () = 2 and assuming = 2 (disclosure is
M
optimal), = 1 and = 9 Then = 2718 = 1221 and = 60509 We can compute expected profit as
d
[ ( )] = 0007 + 26667 − 212882
te
£ ¤ where is constant. In this example the strictly concave ( ) has a unique maximum at
approximately ˆ = 0626 If we make less convex by taking () = 5 then = 1284 =
Ac ce p
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1051 and = 1568 [ ( )] is increasing and strictly concave in with a maximum at = 1 where [ ( )] = 0002 + 1791 − 05912 Numerical examples of part (ii) of Proposition 2 can be constructed by adopting a first order spline form for The case where [ ( )] is strictly convex with an interior minimum is obtained by setting = 055 = 045 = 15 (disclosure is optimal), = 3 = 01 and = 31 9
Numerical examples are displayed using rounding. Note that our examples meet all of the assumptions of the model. For the Cobb-Douglas technology, expected profit as a function of ∈ (0 1) depends on the function and parameters and The parameter can be any real number greater than zero. Further, only needs to be increasing, , = , need to be fractional with , and only needs to be greater than zero.
21 Page 23 of 46
We have [ ( )] = 0706 − 0183 + 06432 In this case, expected profit is strictly convex with a unique minimum at approximately ˜ = 0142
ip t
Finally, [ ( )] is strictly convex with a boundary maximum at = 1, if = 06 = 04 =
an
3.2.2. Comparative Statics of : Non-disclosure
us
[ ( )] = 0523 + 0474 + 04502
cr
15 (disclosure is optimal), = 3 = 01 and = 31 We have
M
The case of non-disclosure has similar comparative static effects of as the case of disclosure. The analysis is complicated by the fact that the agent does not know status perfectly and must take
d
conditional expectations. The derivative of expected profit in is no longer a simple linear function
te
of In this case,
[ ()] = [+1 ( + (1 − ) ) − +1 ( + (1 − ) ) ]
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
− [+1 ( + (1 − ) ) −1 ( − )
+(1 − )+1 ( + (1 − ) ) −1 ( − )]
(15)
We present
Proposition 3. We have that the following hold. (i) If
1 −1
and
= are sufficiently large, then [ ()] has an interior
maximum ˚ ∈ (0 1) [ ()] is positive for small and negative for large (ii) If ≤
1 −1
and
= are sufficiently small, then [ ()] has a boundary
maximum at = 1 [ ()] is negative for small and positive for large
22 Page 24 of 46
(iii) For sufficiently small, [ ()] 0 for all ∈ (0 1) Our results for non-disclosure mirror those for disclosure. In Proposition 3, the conditions (i) are those of IRS and the conditions (ii) are those of DRS. Condition (iii) states that in unfavorable
ip t
situations where non-disclosure is optimal, expected profit increases in The intuition for this result is gleaned from (15), where we see that for small the second term is zero while the first
cr
term is positive. Here the positive direct effect of an increase in which dictates that there is
us
higher probability of the high type as opposed to a low type (first term), dominates the negative indirect effect of variations in expected status, conditional on the state where the agent is either of
an
the high or low type (second term). These indirect effects are present because a rise in raises the probability that a high type will have medium status and that low type will have low status.
M
Proposition 3 points out that it is possible for expected profit to be decreasing in if marginal returns to status are strongly rising or strongly falling. Further, with rising marginal returns to
d
status, the diminishing profit case can occur at high while with falling marginal returns it can
te
occur at low Finally, the counterintuitive case, where the principal would not want to set = 1 if he could control it, occurs if there are sufficient rising marginal returns to status. In this case,
Ac ce p
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[ ()] has an interior maximum in and the intuition is the same as in the case of disclosure. A numerical example, where () = 6 = 05 (non-disclosure is optimal), = 1 and = 9 illustrates this case Then = 20086 = 1822 and = 221406. We have
[ ()] = 0854(221406 − 201321)05 + 0032 (1 − ) (20086 − 18263)05
= 0729 Here, where is a constant and [ ()] has a unique maximum at approximately ˚ 1 −1
= 0100 = 05 To illustrate Part (iii) for this example, take = 005 and [ ()]
is strictly increasing in
23 Page 25 of 46
Part (ii) of Proposition 3 can be illustrated using a first order spline form for If we take = 055 = 045 = 05 (non-disclosure is optimal), = 1 = 002 and = 105 then we have
1 −1
= 20 = 05 This function is strictly convex with interior minimum at approxi-
cr
with
ip t
[ ()] = 0408 (1050 − 0050)05 + 0302 (1 − ) (1 − 0980)05
4. Endogenous Human Capital Acquisition
an
we can take = 01 and [ ()] is strictly increasing in
us
mately = 0150 and boundary maximum at = 1 To illustrate Part (iii) in this same example,
M
In this section, we extend the model to consider the case where the agent can exert effort in stage 1 of the game to build human capital. In stage 2, the cash flow production process takes place as
d
before.
te
Formally, when the two identical agents join the firm in stage 1, each exerts effort to increase the probability of having high human capital within the firm. With probability () the agent achieves high capital and with probability 1 − () the agent achieves low capital, . The
Ac ce p
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function is strictly concave with 0 0 00 0 Only the agent and the principal observe the agent’s realized productivity, which is assumed to be non-contractible due to verifiability problems. While an agent can deduce the other agent’s effort for productivity, because agents are identical, an agent does not know the other agent’s realized human capital. To compute endogenous probabilities for status shares in the extended model, let 1 and 2 denote the respective effort levels of the first and second agents. The probability of a high status share for the first agent is (1 )(1 − (2 )) the probability of a medium status share is (1 )(2 ) conditional on the agent attaining a high productivity, while it is (1 − (1 ))(1 − (2 )) conditional on the agent attaining a low productivity, and, finally, the probability of a low status share 24 Page 26 of 46
is (1 − (1 )) (2 ). Endogenous probabilities for status shares for the second agent are defined analogously. In the extended model, the sequence of decisions from the perspective of an agent is as follows:
ip t
(i) The principal meets with the agent along with another agent while precommitting to a disclosure policy and informing the agent that a compensation contract with a fixed wage and some positive
cr
fraction of cash flow will be offered after the principal observes the agent’s ; (ii) The agent
us
rationally expecting such payments exerts effort ; (iii) The agent’s is realized; (iv) The principal sets and subject to the agent’s participation and limited liability constraints, fully discloses
an
in the case of disclosure, or discloses nothing in the case of non-disclosure; (v) The agent exerts effort ; (vi) The returns to the principal and the agent accrue.
M
First, consider the case of disclosure. Utilizing reduced form (3), the stage 1 problem for the
d
agent is to solve
{ }
te
max ( )[( )(1 − ( )) + ( )( )] +(1 − ( ))[( )(1 − ( )) + ( )( )] −
Ac ce p
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where the cost of human capital effort takes the same form as cash flow effort. As in the basic problem, if the agent’s reservation utility is zero, then the solution to this problem satisfies the agent’s ex ante participation constraint.10 After simplification this problem can be rewritten as
max { }
{( )+1 [ +1 (1 − ( )) + +1 ( )] 1+
+(1 − ( ))+1 [ +1 (1 − ( )) + +1 ( )]} − The agent’s objective function is of the generic form () + − with 0 and 0 0 00 0 (0) = 0 The agent’s FOC is 0 = −1 where 0 because is strictly concave and (0) = 0 Participation is met if 0 − 0 Substituting −1 for 0 from the FOC, participation is met if −1 − 0 which is true by 1 10
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The first order condition is
0 =
{+1 [ +1 (1 − ( )) + +1 ( )] 1+ (16)
ip t
−+1 [( +1 (1 − ( )) + +1 ( )]}0 ( ) − −1
cr
Because the function (·) is strictly concave, the effort cost function is strictly convex, and the
us
coefficient multiplying 0 (·) is positive, the second order condition is met.
Next, consider non-disclosure. Utilizing reduced form (5), the first order condition to agent’s
{ +1 [ (1 − ( )) + ( )] +1 1+
M
0 =
an
stage 1 problem can be written
(17)
d
−+1 [ (1 − ( )) + ( )] +1 }0 ( ) − −1
te
As in (16), the second order condition is met because the function (·) is strictly concave, the effort cost function is strictly convex, and the coefficient multiplying 0 (·) is positive.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
4.1. Characterization of a Symmetric Nash Equilibrium for Disclosure and Non-disclosure In each of the regimes of disclosure and non-disclosure, the question of whether a symmetric Nash equilibrium exists in the choice variables arises and the properties of such an equilibrium are of interest. Using (16) and (17) define
() ≡ +1 [ +1 (1 − ()) + +1 ()] − +1 [ +1 (1 − ()) + +1 ()]
(18)
() ≡ +1 [ (1 − ()) + ()] +1 − +1 [ (1 − ()) + ()] +1
(19)
and
26 Page 28 of 46
In symmetric Nash equilibria for disclosure and non-disclosure, respectively, we have that an agent’s selection of effort is determined by
()0 () − −1 = 0 for = 1+
(20)
ip t
Ω () ≡
cr
The following assumption places boundary restrictions on the function () (A.1) (0) = 0 (∞) = 0, (∞) = 1 and 0 (0) 0
us
In addition, we will utilize
an
(A.2) The functions () and − are sufficiently concave so as to imply that Ω0 () 0 for =
M
Assumption (A.1) is fairly standard and assumption (A.2) guarantees a decreasing marginal net benefit function in each problem.11
d
We have
te
Lemma 1. A unique symmetric Nash Equilibrium in exists under each of the disclosure rules, if (A.1)-(A.2) hold. 11
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
For disclosure (A.2) implies that 0
{ +1 (+1 − +1 ) + +1 (+1 − +1 )}(0 )2 (1 + ) { +1 [ +1 (1 − ) + +1 ] − +1 [+1 (1 − ) + +1 ]}00 + (1 + ) −( − 1)−2 for all
and for non-disclosure it implies that 0
{( + 1)+1 [ (1 − ) + ] (− + ) (1 + )
−( + 1)+1 [ (1 − ) + ] (− + )}(0 )2 { +1 [ (1 − ) + ] +1 − +1 [ (1 − ) + ] +1 }00 + (1 + ) −( − 1)−2 for all
27 Page 29 of 46
4.2. Comparisons Let denote the solution to (16) and let denote the solution to (17). Further let = ( ) and = ( ) In the following proposition we consider the case where the firm faces a favorable
ip t
production situation and is convex.
Proposition 4. Assume that (A.1)-(A.2) hold. Let 1 and let be convex. Then
cr
(i) , and
us
(ii) [ ( )] [ ()] if (10) and (12) hold, 2 + (1 − ) 1
is sufficiently concave such that the condition
( − )
M
( − )
an
Next, we consider the case where the firm has an unfavorable production situation and where
(21)
d
holds. By − = − = concavity of implies that ( − ) ( − ) but (21)
te
insists that the more restrictive condition ( − )
(
− ) is met. We have
Proposition 5. Assume that (A.1)-(A.2) hold. Let 1 and let be sufficiently concave such
Ac ce p
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that (21) holds. Then (i) .
(ii) [ ()] [ ( )] if (11) and (13) hold, 2(1 − ) ≤ + (1 − ) ≤ 1 Propositions 4 and 5 extend our earlier results to the case where the agent can exert costly effort to increase the probability of having high human capital in the firm. Proposition 4 says that if the principal faces a favorable production situation and is convex, then commitment to a policy of disclosure elicits greater human capital effort than does a policy of non-disclosure. Conditions (10) and (12) state, from Proposition 2, that there are rising marginal returns to status, but that they are not too strongly rising. When these conditions are met, a greater level of effort under
28 Page 30 of 46
disclosure translates into a greater probability of a high human capital level which in turn implies a greater expected profit. That is, (10) and (12) from Proposition 2 insure that expected profit under disclosure is strictly increasing in We can guarantee that disclosure is best only in this
ip t
special case for the generalized model. The same logic applies to Proposition 5 where the principal faces an unfavorable production
cr
situation. Here, non-disclosure leads to greater human capital effort which leads to greater if
us
is sufficiently concave so that condition (21) is met, but it must be true that falling returns to status are not so extreme that conditions (11) and (13) are violated.12 Under these assumptions,
an
greater effort leads to greater which implies greater expected profit under non-disclosure. When rising or falling returns to status are extreme and violate (10) and (12) or (11) and (13),
M
then while the results on effort still hold, there is no guarantee that an increase in will in fact lead
d
to a rise in expected profit. The results on profit in Propositions 4 and 5 can then be overturned.
te
5. Model Assumptions and Robustness
Our basic results regarding the optimality of disclosure and non-disclosure are based on the assump-
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
tions that the probability of a high outcome and the cost of effort functions are of the constant elasticity form, that agents are risk neutral, that there are two agents and that the cash flow process has two states. The two agent assumption is not critical, but for disclosure policy to matter through status concerns and affect the agent’s expectation of status in the organization, the number of agents cannot be too large. For an -agent organization, our model predicts that the effect of disclosure policy would decline with as the sample distribution of human capital levels converges to that of the population. It is easy to show that assumptions (21), (11), and (13) are not inconsistent. Let = (1−−10 ) = 6 = 4 and = 5 In this case, both assumptions are met with the right side of (21) given by 63888(10)−3 and the left side of (21) is 11578(10)−2 The term + (1 − ) = 099855 and 2(1 − ) = 089898 Each of these is less than unity. 12
29 Page 31 of 46
The two state assumption allows us to keep the optimal contract simple, as our goal is to concentrate on the disclosure decision. The constant elasticity assumption for cash flow production and effort cost (with or without risk neutrality) has the effect of making the optimal contract,
ip t
independent of the disclosure rule. This allows us to isolate the effect of disclosure policy on profit and total surplus through a pure effort channel that is not confounded with variation in the optimal
cr
contract.
us
If one deviates from the assumption that the probability of a high outcome and the cost of effort functions are of the constant elasticity form, the principal’s optimal contract can then vary with
an
the disclosure policy. In order to rule out signaling by the principal through the contract under non-disclosure in this more general model, a different timeline and information structure than those
M
used in the present paper would be necessary. In that more general model, the principal would set a contract before knowing each agent’s human capital, which is initially private information to the
d
agent. In the case of disclosure, the contract would be a menu where each part of the menu depends
te
on the agent’s status and human capital level. In the case of non-disclosure, there would be no such conditioning on status but only on human capital level. The issue of the optimality of disclosure
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
versus non-disclosure for either the principal or the agent would depend again on the curvature of the actor’s payoff function in the hidden information variable . If the actor’s payoff function is concave, then non-disclosure would be optimal for that actor and conversely if it is convex. The new aspect is that variations in across disclosure rules can affect that curvature. As a result, non-disclosure can become optimal for the agent, in contrast to the present model in which the agent always prefers disclosure.13 13
Marino (2014) obtains the result that the agent can be harmed by disclosure in a case where there is general hidden information about the agent’s utility having to do with the agent’s fit in the organization and the contract varies with the disclosure rule. The situation where the agent can affect the hidden information about fit is not considered.
30 Page 32 of 46
6. Conclusion We study the choice of disclosure policies as an organizational design problem in the presence of agents with status concerns. Our work has novel testable implications.
ip t
If the organization’s production situation is favorable, specifically 1 then disclosure has a
cr
positive motivational effect on agents and maximizes the principal’s profits net of wages. Otherwise, non-disclosure is the principal’s preferred policy. Unfavorability, specifically 1 is descriptive of
us
agency situations in which the agent’s effort does not impact the probability of success greatly but it does highly impact effort cost. Favorability is descriptive of the opposite set of circumstances
an
where the sensitivity of cash flow production to effort is high and/or the sensitivity of personal effort
M
cost to effort is low. In our model, unfavorability leads to concavity of the principal’s equilibrium payoff in the status function, making non-disclosure optimal, while favorability leads to convexity
d
of the principal’s payoff in the status function, making disclosure optimal.
te
In addition, agents always prefer disclosure to non-disclosure because more information allows them to better condition their effort choices. Our model thus points to status concerns as a potential source of disagreement between principals and agents about the proper level of organization-wide
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
transparency in principal-agent relationships. Tensions are likely to arise in unfavorable production situations.
With status concerns, hiring more talented employees may not be always better for the principal. As more talented employees reduce each other’s chances of achieving high status and dull each other’s motivation to exert effort, the principal may want to avoid hiring more talented employees. This local effect can also be global when agents’ returns to status are sufficiently increasing. Disclosure policies can also affect firm-specific investments made by agents with status concerns. If the organization is in a sufficiently favorable production situation and agents’ returns to status are increasing, then disclosure can induce more firm-specific investment by agents and lead to more 31 Page 33 of 46
profits for the principal. Similar results obtain with non-disclosure when the production situation
te
d
M
an
us
cr
ip t
is sufficiently unfavorable and agents’ returns to status are decreasing.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
32 Page 34 of 46
Appendix
i o n h ( ) − [ ()]
cr
= {+1 [ + (1 − ) − ( + (1 − ) ) ]
ip t
Proof of Proposition 1: (i) We have
(1.a)
us
+(1 − )+1 [ + (1 − ) − ( + (1 − ) ) ]}
an
Rewriting (1.a), non-disclosure dominates if
M
+ (1 − ) ( + (1 − ) ) and
(2.a)
d
+ (1 − ) ( + (1 − ) )
te
These conditions hold if 1 That is, is a concave function of . (ii) Disclosure dominates if the reverse inequality in (2.a) holds, and this holds if 1 £ ¤ (iii) We have that ( ) − [ ()] is given by
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
{+1 [(1 − ) +1 + +1 ] + (1 − )+1 [1 − ) +1 + +1 ] (1 + )
−+1 [(1 − ) + ] +1 + (1 − )+1 [(1 − ) + ] +1 }
(3.a)
Because ( + 1) 1 +1 is a convex function of Thus, the expression (3.a) is positive and the result holds. ¥ Proof of Corollary 1: For 1 the result is obvious from Proposition 1. Consider the
33 Page 35 of 46
second statement We have that
[()] +1 (1 + ) ()] = [()] [ + (1 + )
cr
ip t
( ) = [()] +
[ [()|]] +1 (1 + ) [()|]] = [ [()|]] [ + (1 + )
an M
The sign of [ − ] = sign of
()]} − {[ [()|]] [ + [()|]]} +1 +1
te
The sign of this expression is that of
d
{[()] [ +
us
() = [ [()|]] +
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
{[ + (1 − ) )] − [ + (1 − ) ] }
+
+1 +1 {[ + (1 − ) ] − [ + (1 − ) ] +1 } +1
+(1 − ) {[ + (1 − ) ] − [ + (1 − ) ] } +
+1 +1 (1 − ){[ + (1 − ) ] − [ + (1 − ) )] +1 } +1
where ≡ ( ) for ∈ { } and ∈ { } The first two lines of this expression (high branch) are symmetric to the last two lines, so let us consider the first two lines of the high branch.
34 Page 36 of 46
The limit of the high branch as → 0 is 0 and the limit as → 1 is
ip t
1 2 {[ 2 + (1 − ) 2 )] − [ + (1 − ) )]2 } 0 2
cr
Take
{ {[ + (1 − ) )] − [ + (1 − ) )] } →0
{ ln + (1 − ) ln − ln[ + (1 − ) ]}
an
=
+1 +1 {[ + (1 − ) ] − [ + (1 − ) ] +1 }} +1
us
+
M
By concavity, ln + (1 − ) ln − ln[ + (1 − ) ] 0Using these facts, it follows that the high branch is negative for small 1 and positive for larger 1 The same results
d
hold for the last two lines of the low branch, so that the proof is complete. ¥
write
te
Proof of Proposition 2: (i) Strict concavity follows directly from (10). From (9), we can
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
h i ( ) R 0 iff (1 − 2)(( ) + (1 − ( )) ) R 2(1 − ( ) − )
(4.a)
Let = 05 then the left side () of (4.a) is zero and (4.a) becomes R · 2(1 − ( )) By (1 − ( )) 05 " ” holds. Next, let ∈ (1 − ( ) 05) In this case, The right side () of (4.a) is negative while the 0 so that holds. This result does not depend on the condition (10). Next, let ∈ (0 1 − ( )) Given (10), the result holds if (1 − 2) 2(1 − ( ) − ) which is true if 1 2(1 − ( )) The latter is true, so that the result holds. We have shown that
35 Page 37 of 46
£ ¤ ( ) 0 for ∈ (0 05) Next, let ∈ (05 1) Rewrite (4.a) as 2 − 2(1 − ( )) R ( ) + (1 − ( )) = ( ) 2 − 1
ip t
( ) =
The 1 for all ∈ (0 1) and for all ≥ 0 Further, lim = +∞, lim = →05
→1
cr
2( ) 0 and 0 If (12) holds, then ( ) ( ) for all ∈
£ ¤ £ ¤ (05 1) so that ( ) 0 for ∈ (05 1) We have established that ( )
us
£ ¤ 0 for ∈ (0 1) and that there is a boundary maximum of ( ) at = 1 Finally, let
an
condition (13) hold. From the above analysis, ∃ ˆ( ) ∈ (05 1) such that (ˆ ) = ( )
( ) ( ) for ∈ (05 ˆ) and ( ) ( ) for ∈ (ˆ 1) We have that ( ) +(1−( )) −2(1−( )) . 2[( ) +(1−( )) − ]
This completes the proof of (i).
M
ˆ( ) =
(ii) Convexity follows directly from (11). Begin by letting ∈ (0 1 − ( )) Here, 05 and
te
d
(1 − ( ) − ) 0 so that (4.a) can be expressed
( ) = ( ) + (1 − ( )) R
2(1 − ( ) − ) = ( ) 0 1 − 2
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
By (11), ( ) ≤ 1 and ( ) 1 for all ∈ (0 1) and ≥ 0 We have that ( ) 0 lim = 2(1 − ( )), →0
lim
→(1−)
= 0 and ( ) 0 If (13) holds, then, for
all ∈ (0 1 − ( )) 2(1 − ( )) ( ) and the result holds. If the converse of (13) holds, then from the above limit and monotonicity conditions, ∃ ˜ ∈ (0 1 − ) such that ( ) ( ) for ∈ (0 ˜) ( ) = (˜ ), and ( ) ( ) for ∈ (˜ (1 − )) Finally if ∈ (1 − 05) then we have from (4.a) the result because this result did not depend on (10). Finally let ∈ (05 1) We rewrite (4.a) as
( ) =
2 − 2(1 − ) R ( ) + (1 − ( )) = ( ) 2 − 1 36 Page 38 of 46
We have that ( ) 1 for all ∈ (0 1) ≥ 0 By (11), ( ) ≤ 1 so that ( ) ( ) for all ∈ (05 1) and the result holds. ¥ Proof of Proposition 3: It will be useful to rewrite condition (15) as follows:
1 +(1−)
1 + [1 −
]
R
+1 [ + (1 − )]
ip t
+1 [(1 − ) + ]
(5.a)
cr
[ ()] R 0 iff
1 1 − [ (1−)+ − 1]
The right side of (5.a) is a positive fraction and is decreasing in On the left side we have
1 +(1−)
] ∈ [0 1) Before proving the main part of Proposition 3,
us
1 [ (1−)+ − 1] ≥ 0 and [1 −
and, for all ∈ (˚ 1]
−
1 1 − [ (1−)+
1 then there exists a ˚ such that 1− [ (1−˜)+˜ −1] = 0
1 Proof of Lemma for Proposition 3: 1 − [ (1−)+ − 1] 0 iff
Γ( ) Clearly, lim Γ( ) = ∞ lim Γ( )= →1
d
→0
− 1] 0 and ( ()) 0
M
Lemma for Proposition 3. If
an
let us establish the following result.
−
1 [
1 (1−)+
−1]
≡
and Γ( ) 0 given that Γ
te
is continuous in the results hold. Using (5.a), [ ()] 0 for all ∈ (˚ 1] ¥ Proof of Proposition 3 (i): In light of the Lemma, take the limit of (5.a) as → 0 We obtain
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
after rearranging
[ ()] R 0 iff
With
2 2
fixed and
( )
1 + (1 −
)
R(
+1 )
(6.a)
→ ∞ → 1 this inequality becomes and [ ()] 0 for small
We have ruled out boundary maxima and [ ()] is continuous on a compact set. Thus, an interior maximum exists. ¥ (ii): Assume ≤
−
1 In this case, 1 − [ (1−˜)+˜ − 1] 0, for all ∈ [0 1] The limit of
(5.a) as tends to zero is depicted in the right side of (6.a). Fix and let both and be small. We have from (6.a) that [ ()] 0 as → 0Next, consider the limit of
37 Page 39 of 46
(5.a) as tends to unity. Here, (5.a) holds if (Note that ≤
¶ +1 µ
¶
R
implies 1 − ( − 1) 0)
1 1 − (
− 1)
¶ +1 µ
¶
µ ¶ R1+ 1−
us
µ
cr
If both and are small then [ ()] 0 as → 1
ip t
µ
−
We have shown that [ ()] is negative at = 0 and positive at = 1 Also [ ()] |=0 =
an
[ ( )] [ ( )] = [ ()] |=1 We need only show that [ ()] |=1 dominates [ ()] for any ∈ (0 1) under the stated conditions. With the stated assumptions and these
M
results, we need only show
)] + +1 (1 − )[ + (1 − )]
With
becomes
te
d
+1 +1 [ + (1 − )(
small, [ + (1 − )( )] → 1 and [ + (1 − )] → (1 − ) so that the condition
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
+1 +1 + (1 − )+1 (1 − )
This is true.
(iii): From (15), we see that the second term tends to zero as → 0 The first term is positive by and
+ (1 − ) − ( + (1 − ) ) = (1 − )( − ) + ( − ) 0 This completes the proof. ¥
38 Page 40 of 46
Proof of Lemma 1: If a symmetric Nash equilibrium exists it is unique by (A.2) because Ω () are strictly decreasing in Next, for existence note that lim Ω () = [( )+1 −( ) +1 ]0 (0)− →0
0 0 for = where ≡ ( ) for ∈ { } and ∈ { } Moreover, lim
→∞
ip t
Ω () = [( )+1 − ( ) +1 ]0 (∞) − ∞ 0 for = By continuity of Ω () an equilibrium exists. ¥
us
tion of (20), R if () R () for all Now () () if
cr
Proof of Proposition 4: (i) Because under (A.2) Ω () are strictly decreasing and by inspec-
an
{[( )) +1 (1 − ()) + ( ) +1 ()] −[ (1 − ()) + ()] +1 }
M
{[( ) +1 (1 − ()) + ( ) +1 ()]
(7.a)
d
−[ (1 − ()) + ()] +1 }, for all
te
where ≡ ( ) for ∈ { } and ∈ { } Let ≡ and note that (6.a) is met if +1 satisfies the condition that the expression
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
( + ∆) +1 (1 − ()) + +1 () − [( + ∆)(1 − ()) + ()] +1
(8.a)
is increasing in and ∆ This is true because the left side of (7.a) is of the form (8.a) with = and = + ∆2 where ∆2 = ( − ) and the right side of (7.a) is of the form (8.a) with = and = + ∆1 where ∆1 = ( − ) and ∆2 ∆1 To see that ∆2 ∆1 note by − = − = and convex in , ( − ) ( − ) With ∆2 ∆1 Expression (8.a) is increasing in if its derivative in is positive. Computing this derivative it is given by ( + 1)[( + ∆) (1 − ) + ] − ( + 1)[( + ∆)(1 − ) + ] 0 39 Page 41 of 46
by ( + 1) convex in 1 Analogously, taking the derivative of (8.a) with respect to ∆ we have
by ( + ∆) [( + ∆)(1 − ) + ( + 1)] It follows that
cr
(ii) Consider
ip t
( + 1)( + ∆) (1 − ) − ( + 1)[( + ∆)(1 − ) + ] (1 − ) 0
us
n h i o ( ) − [ ()]
an
= {[ ( ( ) + (1 − )( ) ) − ( + (1 − ) ) ] +[(1 − ) ( ( ) + (1 − )( ) )
M
−(1 − ) ( + (1 − ) ) ]}
d
where in equilibrium = ( ) for = and = 1 2. We have that From Propo-
te
¤ £ ¤ £ sition 1, ( ) |= − [ ()] |= 0 It then suffices to show that ( ) |= −
£ ¤ £ ¤ ( ) |= 0 A sufficient condition for the latter is that ( ) 0 for all ∈ [0 1] which is true under (10) and (12), by Proposition 2(i). ¥
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Proof of Proposition 5: (i) With an unproductive agent, we have that ( + 1) =
−
∈ (1 2)
In this case the function () +1 is convex and its first derivative, ( + 1)() is a concave function. Condition (7.a) from the proof of Proposition 4 tells us that non-disclosure generates greater effort if
{[( ) +1 (1 − ()) + ( ) +1 ()] − [ (1 − ()) + ()] +1 } {[( ) +1 (1 − ()) + ( ) +1 ()] − [ (1 − ()) + ()] +1 } for all (9.a)
40 Page 42 of 46
Condition (9.a) is met if the function +1 satisfies the condition that the expression
(10.a)
ip t
( + ∆) +1 (1 − ( )) + +1 ( ) − [( + ∆)(1 − ( )) + ( )] +1
is decreasing in and increasing in ∆ This is true because the left side of (9.a) is of the form (10.a)
cr
with = and = + ∆2 where ∆2 = ( − ) and the right side of (9.a) is of the form (10.a) with = and = + ∆1 where ∆1 = ( − ) and ∆2 ∆1 by condition
us
(21). Taking the derivative of (10.a) with respect to we obtain that (10.a) is decreasing in if
an
( + 1)( + ∆) (1 − ()) + ( + 1) () − ( + 1)[( + ∆)(1 − ()) + ()] 0
(11.a)
M
Condition (11.a) is the condition for concavity of ( + 1) in The proof of Proposition 4 shows
d
that (10.a) is increasing in ∆ Thus, (9.a) holds, and .
te
(ii) We have that
h i [ ( ) − ( ())]
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
= {[ ( ( ) + (1 − )( ) ) − ( + (1 − ) ) ] +[(1 − ) ( ( ) + (1 − )( ) ) −(1 − ) ( + (1 − ) ) ]}
(12.a)
£ ¤ Further from part (i), From Proposition 1, ( ) |= − [ ()] |= 0 It then
£ ¤ £ ¤ suffices to show that ( ) |= − ( ) |= 0 From the proof of Proposition 4,
¤ £ the latter condition is true if ( ) 0 for all ∈ [0 1] Sufficient conditions are then
(11) and (13), from Proposition 2. ¥
41 Page 43 of 46
References [1] Aoyagi, Masaki, 2010. Information feedback in a dynamic tournament. Games and Economic
ip t
Behavior 70, 242-260. [2] Ashraf, Nava, Oriana Bandiera, and Scott Lee, 2014. Awards unbundled: evidence from a
cr
natural field experiment. Journal of Economic Behavior and Organization, Forthcoming.
us
[3] Auriol, Emmanuelle and Régis Renault, 2008. “Status and incentives.” RAND Journal of Economics 39, 305-326.
an
[4] Barankay, Iwan, 2010. Rankings and social tournaments: evidence from a field experiment.
M
University of Pennsylvania Working Paper, July 2010.
[5] Barankay, Iwan, 2011. Gender differences in productivity responses to performance rankings:
d
evidence from a randomized workplace experiment. University of Pennsylvania Working Paper,
te
September 2011.
[6] Crutzen, Benoit S. Y., Otto H. Swank and Bauke Visser, 2010. Confidence management: on
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
interpersonal comparisons in teams. Forthcoming, Journal of Economics and Management Strategy, March 2010.
[7] Dubey, Pradeep, John Geanakoplos, 2010. Grading exams: 100, 99, 98, . . . or A, B, C?. Games and Economic Behavior 69, 72-94.
[8] Ederer, Florian, 2010. “Feedback and motivation in dynamic tournaments. Journal of Economics and Management Strategy 19, 733-769. [9] Ederer, Florian and Andrea Patacconi, 2010. Interpersonal comparison, status and ambition in organizations. Journal of Economic Behavior and Organization 75, 348-363.
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[10] Ellingsen, Tore and Magnus Johannesson, 2007. Paying respect. Journal of Economic Perspectives 21, 135-150. [11] Ellingsen, Tore and Magnus Johannesson, 2008. Pride and prejudice: the human side of incen-
ip t
tive theory. American Economic Review 98, 990-1008.
cr
[12] Goltsman, Maria and Arijit Mukherjee, 2011. Interim performance feedback in multistage
us
tournaments: the optimality of partial disclosure.” Journal of Labor Economics 29, 229-265. [13] Heffetz, Ori and Frank, Robert H, 2011. Preferences for status: evidence and economic im-
an
plications. In: Jess Benhabib, Alberto Bisin, Matthew Jackson (Eds). Handbook of Social
M
Economics, Volume 1A, Amsterdam: Elsevier, 69-91.
[14] Hopkins, Ed and Tatiana Kornienko, 2004. Running to keep the same place: consumer choice
d
as a game of status. American Economic Review 94, 1085-1107.
55, 18-45.
te
[15] Itoh, Hideshi, 2004. Moral hazard and other-regarding preferences. Japanese Economic Review
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[16] Kenrick, Douglas T., Vladas Griskevicius, Steven L. Neuberg, and Mark Schaller, 2010. Renovating the pyramid of needs: contemporary extensions built upon ancient foundations. Perspectives on Psychological Science 5, 292-314. [17] Lizzeri, Alessandro, Margaret Meyer and Nicola Persico, 2002. The incentive effects of interim performance evaluations. Caress Working Paper, August 2002. [18] Marino, Anthony M., 2014. Transparency in agency: the constant elasticity case and extensions. International Journal of Industrial Organization 33, 9-21. [19] McGregor, J., 2006. The Struggle To Measure Performance. Business Week, January 9, 2006.
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[20] McManus, T. Clay and Justin M. Rao, 2013. Signaling smarts: revealed preferences for self and social perceptions of intelligence. University of Tennessee Working Paper, 2013.
ip t
[21] Maslow, A. H., 1943. A theory of human motivation. Psychological Review 50, 370-396. [22] Maslow, A.H., 1970. Motivation and Personality (2nd ed.). New York, NY: Harper and Row.
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[23] Moldovanu, Benny, Aner Sela, and Xianwen Shi, 2007. Contests for status. Journal of Political
te
d
M
an
us
Economy 115, 338-363.
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44 Page 46 of 46
S0167-2681(14)00143-7 http://dx.doi.org/doi:10.1016/j.jebo.2014.05.002 JEBO 3369
To appear in:
Journal
Received date: Revised date: Accepted date:
5-4-2013 8-5-2014 9-5-2014
of
Economic
Behavior
&
Organization
Please cite this article as: Anthony M. Marino, Oguzhan Ozbas, Disclosure of Status in an Agency Setting, Journal of Economic Behavior and Organization (2014), http://dx.doi.org/10.1016/j.jebo.2014.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights JEBO.pdf
Highlights
ip t
cr
us an M
d
te
We develop a model in which the principal and the agent share private information about the value of the agent for a multi-agent organization. The principal can disclose private information and make public the relative standing or status of all agents in the organization. We study whether it is better in terms of profit and utility to disclose or to not disclose status to the group of agents. Conditions for the optimality of disclosure versus non-disclosure are characterized for the cases of exogenous and endogenous human capital.
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Page 1 of 46
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Disclosure of Status in an Agency Setting∗
cr
Anthony M. Marino†
an
May 2014
us
Oguzhan Ozbas‡
M
Abstract
We develop a model in which the principal and the agent share private information about the
d
value of the agent for a multi-agent organization. The principal can disclose private information
te
and make public the relative standing or status of all agents in the organization. We study whether it is better in terms of profit and utility to disclose or to not disclose status to the group of agents. Conditions for the optimality of disclosure versus non-disclosure are characterized for
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the cases of exogenous and endogenous human capital. Keywords: Status; Disclosure; Human Capital. JEL Codes: D1, D2, J3, L, L2, M5.
∗
We thank William Neilson (the editor) and two anonymous referees for helpful comments. We also benefited from the comments of Ricardo Alonso, Odilon Câmara, Heikki Rantakari, and workshop participants at the University of Southern California and Queens University. Financial support from the Marshall General Research Fund is gratefully acknowledged. † Marshall School of Business, University of Southern California; [email protected] ‡ Koç University and Marshall School of Business, University of Southern California; [email protected]
Page 2 of 46
1. Introduction In most multi-agent organizations, the principal and an individual agent share private information regarding the value of that agent for the organization. Such information might arise in the course
ip t
of day-to-day interaction between the principal and the agent or come from periodic performance
cr
reviews. The organization then has a choice in most situations to make the private information public, and thus disclose the relative standing or status of all members of the organization. In
us
this paper, we study this novel organizational design question in the context of an endogenously optimal agency contract while building on recent theoretical and empirical works on status.
an
Our analysis is motivated by the observation that firms devote significant management attention
M
to thinking about the nature of evaluative feedback given to workers, and specifically about whether to disclose rankings as part of the feedback process to improve organizational performance. The
d
right disclosure policy is not clear a priori, as a higher than expected rank can incentivise one
te
worker while at the same time a lower than expected rank can disincentivise another worker going forward. Indeed, McGregor (2006) discusses the struggles that many prominent companies have recently had in implementing evaluation systems that use rankings. She points out that as many
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as one third of U.S. corporations reveal rankings to their employees. However, companies such as Goldman Sachs Group Inc., General Electric Co., and Chemtura have had doubts concerning the use of rankings to motivate employees and have adjusted the implementation of such systems. The benefits and costs to an agent of status revelation can in principle emanate from an organizational reward structure related to status and/or through status entering the agent’s utility function directly. In a field experiment involving employees at Amazon’s crowd-sourcing website, Mechanical Turk, Barankay (2010) carefully shuts down the former channel to test the direct utility channel and shows that on average relative performance feedback reduces future effort. In another field experiment involving salespeople at a furniture company, Barankay (2011) again finds that 1 Page 3 of 46
such feedback reduces performance. Similarly, Ashraf et al. (2014) find that disclosures that are carefully designed to induce social comparisons reduce effort, especially among low ability workers in a nationwide health worker training program in Zambia. In a laboratory setting where students
ip t
can signal ability to others, McManus and Rao (2013) also find that social comparisons reduce effort. In contrast, Azmat and Iriberri (2010) study the effect of relative performance feedback
cr
on student effort and find that it improves performance. These field studies suggest that status
us
directly affects utility, and the conflicting results on performance suggest that the effect of status revelation on organizational effort incentives and performance can depend on organization specific
an
circumstances. Our theoretical analysis identifies such conditions.
As hypothesized in much of the literature in economics and psychology, we assume that status
M
and wages are complements in the agent’s utility. Given this assumption, we consider multiple identical agents in an organization where there is a hidden action agency problem for the principal
d
with regard to each agent. The principal designs an optimal incentive contract for each agent and
te
commits to a disclosure or a non-disclosure policy at the time of hiring the agents. Initially, we assume that each agent is endowed with a certain probability of having high human capital and
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study the effect of disclosing status to the group of agents after the agents’ human capital levels are realized. We find that disclosure is optimal for the principal if the firm faces a sufficiently favorable production situation — one in which the typical agent exhibits a high sensitivity of cash flow production to effort and/or a low sensitivity of effort cost to effort. In such situations, status concerns play a positive motivational role and increase the principal’s profit. Conversely, if the firm faces a sufficiently unfavorable production situation, then status concerns play a negative motivational role and non-disclosure is optimal for the principal. Interestingly, we find that agents always prefer disclosure to non-disclosure because more information about status allows them to better condition their effort choices. Thus, our model points to status concerns as a possible source
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of tension between agents and principals regarding the appropriate amount of organization-wide transparency in principal-agent relationships. In the basic model, we also ask how changes in human capital affect the principal’s profit. We
ip t
find that under either disclosure rule, an increase in the typical agent’s probability of having high human capital can have a strong negative motivational effect through toughened competition for
cr
status and decrease the principal’s expected profit when there is rapidly rising or falling returns
us
to status in utility. In all other cases, an increase in human capital increases expected profit as it normally does in standard models.
an
In a final section of the paper, we extend the model by allowing the agent to exert costly effort to increase the probability of having high human capital. If the firm faces a favorable production
M
situation and there are increasing returns to status, then disclosure always results in greater human capital effort and expected profit as long as returns to status are not rising too rapidly. On the
d
other hand, if the production situation is unfavorable and there are decreasing returns to status,
te
then non-disclosure generates greater human capital effort and expected profit as long as returns to status are not falling too rapidly.
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Our work is broadly related to a growing literature on social status. Departing from traditional preferences over absolute consumption, this literature considers social status as an additional argument in the utility function, which is similar to how psychologists and sociologists view preferences. Various experimental studies confirm the existence of preferences for status by inducing status in the lab (See Heffetz and Frank (2011) for an excellent survey.). Motivated by Maslow’s (1943) hierarchy of needs, we use a preference specification featuring a positive interaction between status and wealth. Consistent with Maslow’s theory that status and self-actualization motives are high-level psychological needs that come only after basic physiological necessities are met, the positive interaction between status and wealth implies that more wealthy
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agents care more about status. While advances from related fields continue to refine thinking in psychology in important ways, the broad outlines of Maslow’s hierarchical approach to understanding human motivation remain central (Kenrick et al. (2010)).
ip t
Our other modeling assumptions are intended to capture key features of status that consistently appear in empirical and experimental research. First and foremost, status is positional with respect
cr
to a desirable trait. We take human capital to be that desirable trait in an organization. Status
us
is also non-tradable and only acquired through actions. In our model, we assume that agents can engage in costly effort to accumulate human capital and thereby acquire status. Lastly, status is
an
based on social perceptions and visibility. This is our main motivation for focusing on disclosure policies.
M
Our paper is related to recent works on behavioral agency theory that consider non-material incentives such as status and respect in optimal contracts (Auriol and Renault (2008), and Ellingsen
d
and Johannesson (2008)).1 These works also explore implications for organizational design as we
te
do. While our setting incorporates many of the standard features of status, our modeling approach is different in several important respects. In contrast to the usual modeling approach of endowing
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the principal with precise control over the allocation of a status budget, our approach instead endows the agent with a chance to achieve status through costly effort. Our emphasis on disclosure policies as an organizational design problem is also unique in endogenizing the source of status. A related behavioral literature on other regarding preferences focuses on concerns for relative consumption and studies the effects of wage externalities across workers. These works represent a different but complementary approach to the study of interrelated preferences in organizations (see Itoh (2004) and his references). Finally, another related stream of literature studies the optimality of interim relative perfor1
Moldovanu et al. (2007) and Dubey and Geanakoplos (2010) consider agents concerned with status and formulate optimal status partitions in their respective problems.
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mance feedback in the context of dynamic tournaments (Lizzeri et al. (2002), Crutzen et al. (2010), Ederer (2010), Aoyagi (2010), and Goltsman and Mukherjee (2011)). In a tournament model with status, Ederer and Patacconi (2010) show how a firm would prefer to reward few winners without
ip t
explicitly identifying many losers if workers’ utility depends on own wage and effort as well as a modal reference wage. In contrast to our paper, in these papers the benefits and costs of status
cr
revelation feed through the monetary reward structure of the tournament as opposed to being gen-
us
erated by non-monetary status in the agent’s utility. We view our work as complementary given the pervasiveness of non-pecuniary status-like awards in the real world (Ellingsen and Johannesson,
an
(2007)).
The paper is organized as follows. Section 2 presents the model. Section 3 compares disclo-
M
sure and non-disclosure policies, and conducts comparative statics with respect to human capital. Section 4 endogenizes human capital accumulation, Section 5 discusses robustness, and Section 6
2. The Basic Model
te
d
concludes.
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Consider a firm with a principal and two agents. In stage 1, the two identical agents join the firm and each is endowed with a firm specific human capital level which can be high, or low, With probability an agent’s capital is high, which is realized after the agent joins the firm. Only the agent and the principal observe the agent’s realized human capital. One can think of as any measure of worth of an individual in the organization which is intrinsic to the individual. In our model, is a critical component of the agent’s status, which we will initially assume is exogenous and later make endogenous. Importantly, is not a symbol or an award assigned by the principal as in Auriol and Renault (2008) nor is it bestowed by society, but instead it is a representation of the skill of the agent in doing her/his job within the organization. This measure could arise from
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third parties or from the observation of the principal. For a teacher, might represent a score from teaching evaluations generated by students or it could be the average score of the teacher’s students in taking standardized tests. For a sales person, could be a measure of that person’s interpersonal
ip t
skills and this information might be communicated to the agent and the principal by customers. Most organizations have periodic performance reviews in which the agent submits to the principal
cr
information pertinent to job performance, and the principal generates an overall score, for the
us
agent. In many principal-agent settings, the principal learns such information about agents. The disclosure policy in this paper revolves around The disclosure issue is whether to divulge to each
an
agent the human capital levels of all agents (the other agent).
We will measure the relative standing or status of an agent by the share of total human capital
M
and assume that it enters an agent’s utility function in a positive way. Given the two outcomes for there are three status outcomes ∈ { } for each agent. Status is medium if an agent
te
d
attains the same capital as the other agent
≡
1 = = + + 2
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The stage 1 probability of this state conditional on an agent attaining a high capital is 2 while it is (1 − )2 conditional on an agent attaining a low capital. Status is low if an agent attains a low capital and the other agent attains a high capital
≡
− 1 = − where ≡ + 2 2( + )
The stage 1 probability of this state is (1 − ). Status is high if an agent obtains a high capital
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and the other agent attains a low capital
1 = + + 2
The probability of this outcome is (1 − ) from a stage 1 perspective.
ip t
≡
us
ˆ ∈ { 0} with Prob(ˆ = ) =
cr
Given a human capital parameter , in stage 2 an agent produces a cash flow, which is given by
an
where is the agent effort in cash flow generation and ∈ (0 1) is the effort elasticity parameter. We assume that ∈ (0 1). All agents have the same elasticity parameter , but those with
M
greater human capital or ability generate more expected cash flow for a given amount of effort.
d
Also, as discussed before, status is directly related to productivity and more than a token.
te
We further assume that compensation can only be based the agent’s cash flow, ruling out three other potential conditioning variables: the agent’s human capital, and the other agent’s cash flow and human capital. In our model, it turns out that the principal would not condition compensation
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on the agent’s human capital even if there were no third party verifiability problems associated with measuring human capital.
Given the two states of cash flow, the principal’s compensation contract for the agent is given by2
2
= + ˆ
Given the two state cash flow process, this linear contract is fully general.
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The principal and the agent are risk neutral and the agent’s utility function is given by
ip t
= () −
The function 1 represents the agent’s personal cost of effort, and () represents a status
cr
function which is assumed to be complementary with the wage in the agent’s utility. We assume that is increasing in Thus, the agent is risk neutral in the wage, given status, and utility is
us
strongly monotonic in status and the wage. Preferences are strictly convex in the status function
an
and the wage so that there is a diminishing marginal rate of substitution between and However, because the curvature of is general, there can be an increasing or decreasing marginal
M
rate of substitution between the wage and status . If is concave, non-increasing marginal returns to status, then the marginal rate of substitution between and is decreasing or constant.
d
If is sufficiently strictly convex, increasing marginal returns to status, then the marginal rate of
te
substitution between and can be increasing. Finally, under this formulation, the status function represents the marginal utility of another dollar. Because it is multiplicative with the wage in
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utility, there are increasing returns to scale in and with respect to utility, gross of effort cost. The complementarity assumption has been utilized by others in examining status and utility (Auriol and Renault (2008), and Hopkins and Kornienko (2004)). The agent is assumed to be risk neutral in income, given status. The complementarity assumption asserts that the marginal utility of income is increasing in status such that richer agents enjoy extra status more than poorer agents and would be willing to exert more effort to increase status. As pointed out in Auriol and Renault (2008), this behavior is consistent with Maslow’s pyramid of needs (1943, 1970) and his modern reinterpretation, Kenrick et al. (2010), from which it can be argued that individuals with lower income are concerned with material needs and do not pay as much attention to status, while those
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with greater income have met their material needs and are much more preoccupied with raising their status.3 At this point it is useful to summarize the sequence of decisions from the perspective of an
ip t
agent. (i) The principal meets with the agent along with another agent while precommitting to a disclosure policy and informing the agent that a compensation contract will be offered after the
cr
principal observes the agent’s ; (ii) The agent is endowed with a level of human capital which
us
both the principal and the agent observe; (iii) The principal sets and subject to the agent’s participation and limited liability constraints, fully discloses in the case of disclosure, or discloses
an
nothing in the case of non-disclosure; (iv) The agent exerts effort ; (v) The returns to the principal and the agent accrue. The disclosure policy takes one of two forms. Full disclosure is defined as
M
the case where the principal makes public the status levels of all agents, and non-disclosure is the
te
2.1. Disclosure and Non-disclosure
d
situation where the principal does not reveal any information about status.4
In the case of full disclosure, an agent knows his status level in addition to his human capital
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level before exerting effort. The agent solves
max ()( + ) − {}
The solution is given by
1 ( ) = [ ()] −
(IC )
3
For some indirect empirical evidence that there is complementarity between and see Auriol and Renault (2008), pp. 309-310. 4 We do not consider partial disclosure where the principal sends a signal about the other agent’s Note that many real world disclosure policies operate as full or no disclosure.
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and the second order condition for this problem is met under our assumptions.5 Condition (IC ) is a constraint in the principal’s problem. In addition, the principal faces participation and limited liability constraints as follows: ()( + ) − ≥ 0
ip t
(P )
(LL)
us
where in (P ) we assume that the agent’s reservation utility is zero.
cr
≥ 0
Under the assumption of a zero reservation utility, the agent’s participation constraint is non-
M
agent’s human capital and status is described by
an
binding so that the principal’s optimal = 06 The principal’s choice of compensation, given the
te
The first order condition is
d
max [ ()] − (1 − ) {}
=
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(1)
At a point where the first order condition is met, the second order condition is satisfied. To see this, note that if we define
≡
−
then the principal’s problem is equivalent to maximizing − +1 The first order condition yields = ( + 1) and the second order condition is that ( − 1) −2 − ( + 1) 0 which can 5
The second order condition for this problem is met if the second derivative of − is negative. Under our assumption that ∈ (0 1) and 1 this derivative is negative. We assume that parameters are such that ∈ (0 1) 6 Note that − − 0 suffices to guarantee that P is nonbinding. Substituting for − from IC , this condition is (1 − ) 0 which is true. If P is nonbinding then it is clear that = 0
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be rewritten as ( − 1) 2 ( + 1) Substituting = ( + 1) into the latter condition yields −1 0 which is true. Note that the agent’s effort depends on and while the incentive pay does not. Thus,
ip t
our modelling choices help isolate a pure effort channel in principal-agent relationships with status concerns.
cr
The principal’s reduced form profit under full disclosure from an agent who knows his human
+1 2 0 (1 + )2 +1
(2)
an
( ) = · +1 (()) where ≡
us
capital and status is written as
+1 (()) +1 (1 + )
(3)
te
d
( ) =
M
Further, the agent’s reduced form utility can be written as
This completes optimization in stage 2, under full disclosure. With non-disclosure, the principal does not reveal an agent’s human capital level to any other
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agent. The analysis of this case is identical to that of disclosure with [()|] replacing () where
[()| ] = ( ) + (1 − )( ) and
[()| ] = ( ) + (1 − )( )
It is again true that the optimal beta is = 7 Note that does not convey any information about status , and, thus, the agent does not form a new status expectation after observing the 7 Using the same reasoning as in the case of disclosure, it can be shown that the second order conditon to the principal’s problem is met at a point where the first order condition is satisfied.
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principal’s choice of compensation, consistent with non-disclosure. The principal’s reduced form profit from employing an agent who knows only his own human capital is
+1 [ [()|]] +1 (1 + )
us
() =
ip t
The reduced form utility of the agent is
(4)
cr
() = · +1 [ [()|]]
(5)
an
It is worth noting that disclosure has nuanced motivational effects in our model. We see that if the state of the world is such that there is a high and a low human capital agent, then the
M
high (low) capital agent has high (low) status and has stronger (weaker) effort incentives under disclosure than under non-disclosure. In this case, disclosure raises the effort incentives of the high
d
human capital agent and dampens those of the low human capital agent. This result is similar to
te
the results in the dynamic tournament literature discussed in the introduction. However, what is further possible in our model is that when both agents are of high capability, disclosure dampens
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effort incentives because agents know that their status level is while with non-disclosure they attach a probability of (1 − ) to status being In the opposite case where both agents are of low capability, disclosure has a positive effect on effort incentives because agents know that the low status outcome is not possible, whereas that remains a possibility under non-disclosure. Therefore, in our model, disclosure does not just motivate high capability agents and demotivate low capability agents.
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3. Comparisons and Comparative Statics 3.1. Comparisons
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In this sub-section, we are interested in examining two questions. If the principal were to precommit to a disclosure policy and contract with the agent, then which disclosure policy would result in
cr
greater ex ante expected profit for the principal and which policy would result in greater ex ante
us
expected utility for the agent? Let ( ) ≡ = Consider
an
[ ( )] = {+1 [ + (1 − ) ]
[ ( )] =
M
+(1 − )+1 [ + (1 − ) ]} {+1 [ +1 + (1 − ) +1 ] (1 + ) (7)
te
d
+(1 − )+1 [ +1 + (1 − ) +1 ]}
and
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[ ()] = {+1 [ + (1 − ) ] +(1 − )+1 [ + (1 − ) ] }
[ ()] =
{+1 [ + (1 − ) ] +1 (1 + )
+(1 − )+1 [ + (1 − ) ] +1 }
(8)
Proposition 1. The following comparisons can be made. ¤ £ (i) Given 0 we have () T [ ( )] as T 1
£ ¤ (ii) ( ) [ ()]
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(All proofs are provided in the Appendix.) We adopt the following definition to describe the economic environment. Definition. The situation where 0 1 is defined as an unfavorable situation for the firm,
The parameter =
−
ip t
and 1 is defined as a favorable situation for the firm. 0 is monotonically increasing in ∈ (0 1) and monotonically decreasing
cr
in 1 If the situation is unfavorable, 1 then the firm’s expected cash flow technology is not
us
very responsive to the agent’s effort and/or the disutility of effort is very responsive to increases in effort exertion. If the situation is favorable, 1, then increases in the agent’s effort produce
an
large increases in expected cash flow and/or very small increases in effort cost.
Proposition 1 basically relies on Jensen’s inequality to show that in unfavorable production
unfavorable situation, the function =
−
M
situations non-disclosure is better for the principal and in favorable ones disclosure is better. In the 1 is concave in and the sign of ( )−( ) is
d
that of the difference between a convex combination of the images of this concave function and the
te
concave image of the same convex combination of values. It then follows that by concavity of and Jensen’s inequality this difference is negative and non-disclosure dominates for the principal.
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A type of risk aversion is created for the principal, with respect to the status variable which induces him to shy away from disclosing status to the agent. In the favorable production situation, the opposite is true in that the function is convex in as opposed to concave. In this case, the principal prefers variability with respect to the status variable , and disclosure is optimal. One can think about Proposition 1 (i) through the motivational effects of disclosure for agents ¡ ¢ with status concerns. If 1, the principal’s optimal contract = induces greater expected
probability of high cash flow when status levels of all agents are made public. If 1, the opposite
result holds when the principal does not disclose any information about status. The sign of the
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difference in expected probability of high cash flow is given by
= { [ + (1 − ) − ( + (1 − ) ) ]
ip t
{ [ ] − [ ]}
cr
+(1 − ) [ + (1 − ) − ( + (1 − ) ) ]}
which is positive when 1 and negative when 1. And, because the optimal contract pays
us
¢ ¡ the same = fraction of high cash flow under both status disclosure policies, the comparisons
an
for expected probability of high cash flow carry over to expected profits that the principal retains after paying the agent.
M
Proposition 1 also shows that while non-disclosure may be better for the principal with an unfavorable situation, it is never better than disclosure for the agent. For an agent, ( ) and
d
( ) are based on a convex function of , +1 ( + 1) 1 In the case of non-disclosure, the
te
agent’s expected utility ( ) is effectively the convex image of a convex combination of values whereas in the case of disclosure, the agent’s utility ( ) is the same convex combination of image
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values of +1 with all other terms equal. It then follows that by convexity of +1 the latter expectation dominates and disclosure is better than non-disclosure. The intuition here is that the agent can make better effort decisions under disclosure versus non-disclosure because effort is based on actual status as opposed to the expectation of status as would be the case with non-disclosure. Given the same across solutions, ex-ante expected utility is greater under disclosure regardless of the favorability versus unfavorability of production. However, if there is an unfavorable production situation, the principal would prefer non-disclosure and prefer the agent to make the “wrong” effort choice by basing effort on expected status. This yields greater ex-ante expected cash flow, given that the agent obtains rents through the non-
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binding participation constraint and that the principal is maximizing his share of expected cash flow rather than total surplus. In sum, for firms with favorable production situations, disclosure raises both profit and utility so
ip t
that total surplus is greater than it is under non-disclosure. In the case of firms facing unfavorable production situations, non-disclosure raises profit, but lowers utility, so that the effect on total
cr
surplus is ambiguous. It is easy to show that, depending on the parameters, total surplus can be
us
greater or less under non-disclosure with an unfavorable production situation. If the production situation is sufficiently unfavorable, then total surplus is always greater under non-disclosure. Let
an
the total surplus under disclosure be given by ( ) and let the same under non-disclosure be denoted () Using the results of Proposition 1 we can summarize the results on total surplus
M
in the following.
[ ()]
d
Corollary 1. If 1 [ ( )] [ ()] If 1 is sufficiently small, then [ ( )]
te
The core issue regarding disclosure is whether an actor’s equilibrium payoff function is concave or convex in the hidden information variable, For the agent, the equilibrium payoff function
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is convex in the hidden information variable regardless of whether the situation is unfavorable ( 1) or favorable ( 1). For the principal, concavity is implied in our constant elasticity setting by unfavorability ( 1) and convexity is implied by favorability ( 1) The unfavorability condition is true in situations where it is hard for the agent to generate a large response in the probability of success by exerting more effort, and, when he does exert more effort, the cost of effort responds greatly. In sales this might correspond to a situation where the client is currently not purchasing the sales person’s product and is very much wed to the status quo product. It is very costly and difficult for the sales person to change the client’s mind. In academics, an example might be a difficult required teaching assignment where the students must take a course, which
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they abhor on average because they are ignorant of the future useful applications, and they are quite vocal about their feelings. The teacher then faces a probability of success which is not very responsive to effort and at the same time personal effort cost rises rapidly with increases in effort.
ip t
In the sales example, a favorable situation might be created when the client has experienced the sales person’s product for some time and intensely likes the product. In the academic example, a
cr
favorable situation is created when the teacher is given the assignment to teach a known useful and
us
very popular elective course.
an
3.2. Comparative Statics of
In standard principal-agent models, an increase in the agent’s talent ( in our model) would lead to
M
an increase in the principal’s expected profit in equilibrium as long as the marginal cost of talent is sufficiently low for the principal. As we show below, this standard result may not hold when agents
d
have status concerns. Specifically, the principal may be worse off with more talented agents, even
te
when the principal faces no marginal cost of hiring talent in the traditional sense — recall that the participation constraint is non-binding at a solution. The reason is that status concerns affect the
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principal’s ability to induce effort in our model. In one set of results, we show that if agents have a sufficiently risk-seeking preference for status (convex ), the positive talent effect can be swamped by the negative motivational effect of lower status due to an increase in in the interval [12 1], leading the principal to locally prefer less talent over more talent and globally prefer an interior under either disclosure rule. In a parallel set of results, a similar local effect arises if agents have a sufficiently risk-averse preference for status (concave ). Again in this case, the positive talent effect can be swamped by the negative motivational effect of lower status due to an increase in in the interval [0 12]. Proposition 2 contains these results for the case of disclosure, and Proposition 3 presents analogous results for
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the case of non-disclosure.
3.2.1. Comparative Statics of : Disclosure
ip t
Expected profit under disclosure from (7) can be rewritten as
cr
[ ( )] = [(+1 )+(+1 ++1 −2+1 )+(+1 ++1 −+1 −+1 )2 ]
us
so that [ ( )] is a quadratic function of Differentiation with respect to yields the following
an
linear function of
M
[ ( )] = (+1 ++1 −2+1 )+2[ (+1 ++1 )−+1 −+1 ] (9)
If the slope coefficient of this function is negative (non-negative), then ( ( )) is a strictly
te
d
concave (convex) function of Define the following augmented share term
( ) ≡
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+1 +1 + +1
where it can be verified that ( ) ∈ (05 1) (0) = and 0 ( ) 0 by ∈ (0 1) and Then, from (9),
2 [ ( )]2 0 if ( ) + (1 − ( )) 1
(10)
2 [ ( )]2 ≥ 0 if ( ) + (1 − ( )) ≤ 1
(11)
Using (9)-(11), we present Proposition 2. For a given we have that the following hold.
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(i) Let (10) hold Then [ ( )] is strictly concave in If
[ + (1 − ) ] ∈ (1 2]
ip t
(12)
then [ ( )] 0 ∈ (0 1) and [ ( )] has a boundary maximum at = 1 If
cr
+ (1 − ) 2 then [ ( )] has an interior maximum at a ˆ( ) ∈ (05 1)
us
(ii) Let (11) hold. Then [ ( )] is convex in If
(13)
an
+ (1 − ) ∈ [2(1 − ) 1]
then [ ( )] 0 ∈ (0 1) and [ ( )] has a boundary maximum at = 1 If
M
+ (1 − ) 2(1 − ) then [ ( )] is strictly convex and has an interior minimum at a ˜( ) ∈ (0 (1 − ( ))) where (1 − ( )) 05
te
d
Given values of the parameters and the curvature of [ ( )] in depends on the agent’s preferences for status through the three image values of the function Condition (10) for concavity occurs if the ratios of = are large as would be the case if the three image values
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
= formed a convex first order spline function. Such a function would take on the form ˆ : [ ] → [( ) ( )], where
ˆ () =
⎧ ⎪ ⎪ ⎨ ( ) + (1 − )( ) for = + (1 − ) ∈ [0 1]
⎪ ⎪ ⎩ ( ) + (1 − )( ) for = + (1 − ) ∈ [0 1]
A strictly convex spline function ˆ would satisfy the condition
− − ⇔ − 1 1 − − −
(14)
19 Page 21 of 46
so that each ratio is large.8 The converse condition (11) for convexity is true if the ratios = are small as would be the case with a concave first order spline function. We call the former case where (10) is met the case of increasing marginal returns to status (IRS) and the
ip t
latter case where (11) holds the case of decreasing marginal returns to status (DRS). Proposition 2 tells us that when (10) holds, the profit function is strictly concave in and this
cr
curvature is induced by IRS. If + (1 − ) 2 and IRS are great, expected profit
us
reaches a unique maximum at an interior ˆ which is greater than one half. In this case, expected profit actually decreases in for ˆ If conversely, 1 + (1 − ) ≤ 2 and IRS
an
are not too great, expected profit strictly increases in and reaches a maximum at = 1 When (11) holds, as would be the case with DRS, then profit is convex in . If + (1 −
M
) 2(1 − ) and DRS are great, then expected profit is strictly convex in and there is an interior ˜ less than one half, where expected profit reaches a minimum. For ˜ profit decreases
d
in If conversely, 2(1 − ) ≤ + (1 − ) ≤ 1 and DRS are not too great, expected
te
profit strictly increases in and reaches a maximum at = 1 The counterintuitive cases, where expected profit decreases in , appear whenever returns to
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
status are strongly rising or strongly falling. In the former case it can happen at high , and in the latter case it can happen at low . The intuition for these results can be explained by noting that, given , the probability of having heterogeneous agents is (1 − ) which is maximized at = 05. With IRS, heterogeneity has a positive effect on expected profit because the greater effort supplied by the relatively high status agent exceeds the lower effort supplied by the relatively low status agent. If IRS are sufficiently strong and gets above 12 , the positive effect of heterogeneity on expected profit declines rapidly and expected profit peaks at an interior high . With DRS, heterogeneity has a negative effect on expected profit, and if DRS are sufficiently strong, expected 8
In (14), we use the fact that ( − ) = ( − ) = given our assumptions.
20 Page 22 of 46
profit dips at an interior low . Proposition 2 can also be interpreted as saying that if the principal could choose the ex ante probability that an agent would have high human capital, he would choose = 1 except in one
ip t
case. This case occurs when there are strong increasing returns to status in the sense that the converse of (12) holds, + (1 − ) 2 which in turn implies (10). This condition
cr
insures that the principal’s profit function is strictly concave in with an interior maximum ˆ ∈
us
(05 1) Here, the optimal ˆ might be achieved through an ex ante screening process set up by the firm. The design of such a process is outside the scope of this paper, but the result that = 1
an
is not always preferred is interesting and counterintuitive. A numerical example of this case can be generated by positing the convex status function () = 2 and assuming = 2 (disclosure is
M
optimal), = 1 and = 9 Then = 2718 = 1221 and = 60509 We can compute expected profit as
d
[ ( )] = 0007 + 26667 − 212882
te
£ ¤ where is constant. In this example the strictly concave ( ) has a unique maximum at
approximately ˆ = 0626 If we make less convex by taking () = 5 then = 1284 =
Ac ce p
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1051 and = 1568 [ ( )] is increasing and strictly concave in with a maximum at = 1 where [ ( )] = 0002 + 1791 − 05912 Numerical examples of part (ii) of Proposition 2 can be constructed by adopting a first order spline form for The case where [ ( )] is strictly convex with an interior minimum is obtained by setting = 055 = 045 = 15 (disclosure is optimal), = 3 = 01 and = 31 9
Numerical examples are displayed using rounding. Note that our examples meet all of the assumptions of the model. For the Cobb-Douglas technology, expected profit as a function of ∈ (0 1) depends on the function and parameters and The parameter can be any real number greater than zero. Further, only needs to be increasing, , = , need to be fractional with , and only needs to be greater than zero.
21 Page 23 of 46
We have [ ( )] = 0706 − 0183 + 06432 In this case, expected profit is strictly convex with a unique minimum at approximately ˜ = 0142
ip t
Finally, [ ( )] is strictly convex with a boundary maximum at = 1, if = 06 = 04 =
an
3.2.2. Comparative Statics of : Non-disclosure
us
[ ( )] = 0523 + 0474 + 04502
cr
15 (disclosure is optimal), = 3 = 01 and = 31 We have
M
The case of non-disclosure has similar comparative static effects of as the case of disclosure. The analysis is complicated by the fact that the agent does not know status perfectly and must take
d
conditional expectations. The derivative of expected profit in is no longer a simple linear function
te
of In this case,
[ ()] = [+1 ( + (1 − ) ) − +1 ( + (1 − ) ) ]
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
− [+1 ( + (1 − ) ) −1 ( − )
+(1 − )+1 ( + (1 − ) ) −1 ( − )]
(15)
We present
Proposition 3. We have that the following hold. (i) If
1 −1
and
= are sufficiently large, then [ ()] has an interior
maximum ˚ ∈ (0 1) [ ()] is positive for small and negative for large (ii) If ≤
1 −1
and
= are sufficiently small, then [ ()] has a boundary
maximum at = 1 [ ()] is negative for small and positive for large
22 Page 24 of 46
(iii) For sufficiently small, [ ()] 0 for all ∈ (0 1) Our results for non-disclosure mirror those for disclosure. In Proposition 3, the conditions (i) are those of IRS and the conditions (ii) are those of DRS. Condition (iii) states that in unfavorable
ip t
situations where non-disclosure is optimal, expected profit increases in The intuition for this result is gleaned from (15), where we see that for small the second term is zero while the first
cr
term is positive. Here the positive direct effect of an increase in which dictates that there is
us
higher probability of the high type as opposed to a low type (first term), dominates the negative indirect effect of variations in expected status, conditional on the state where the agent is either of
an
the high or low type (second term). These indirect effects are present because a rise in raises the probability that a high type will have medium status and that low type will have low status.
M
Proposition 3 points out that it is possible for expected profit to be decreasing in if marginal returns to status are strongly rising or strongly falling. Further, with rising marginal returns to
d
status, the diminishing profit case can occur at high while with falling marginal returns it can
te
occur at low Finally, the counterintuitive case, where the principal would not want to set = 1 if he could control it, occurs if there are sufficient rising marginal returns to status. In this case,
Ac ce p
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[ ()] has an interior maximum in and the intuition is the same as in the case of disclosure. A numerical example, where () = 6 = 05 (non-disclosure is optimal), = 1 and = 9 illustrates this case Then = 20086 = 1822 and = 221406. We have
[ ()] = 0854(221406 − 201321)05 + 0032 (1 − ) (20086 − 18263)05
= 0729 Here, where is a constant and [ ()] has a unique maximum at approximately ˚ 1 −1
= 0100 = 05 To illustrate Part (iii) for this example, take = 005 and [ ()]
is strictly increasing in
23 Page 25 of 46
Part (ii) of Proposition 3 can be illustrated using a first order spline form for If we take = 055 = 045 = 05 (non-disclosure is optimal), = 1 = 002 and = 105 then we have
1 −1
= 20 = 05 This function is strictly convex with interior minimum at approxi-
cr
with
ip t
[ ()] = 0408 (1050 − 0050)05 + 0302 (1 − ) (1 − 0980)05
4. Endogenous Human Capital Acquisition
an
we can take = 01 and [ ()] is strictly increasing in
us
mately = 0150 and boundary maximum at = 1 To illustrate Part (iii) in this same example,
M
In this section, we extend the model to consider the case where the agent can exert effort in stage 1 of the game to build human capital. In stage 2, the cash flow production process takes place as
d
before.
te
Formally, when the two identical agents join the firm in stage 1, each exerts effort to increase the probability of having high human capital within the firm. With probability () the agent achieves high capital and with probability 1 − () the agent achieves low capital, . The
Ac ce p
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function is strictly concave with 0 0 00 0 Only the agent and the principal observe the agent’s realized productivity, which is assumed to be non-contractible due to verifiability problems. While an agent can deduce the other agent’s effort for productivity, because agents are identical, an agent does not know the other agent’s realized human capital. To compute endogenous probabilities for status shares in the extended model, let 1 and 2 denote the respective effort levels of the first and second agents. The probability of a high status share for the first agent is (1 )(1 − (2 )) the probability of a medium status share is (1 )(2 ) conditional on the agent attaining a high productivity, while it is (1 − (1 ))(1 − (2 )) conditional on the agent attaining a low productivity, and, finally, the probability of a low status share 24 Page 26 of 46
is (1 − (1 )) (2 ). Endogenous probabilities for status shares for the second agent are defined analogously. In the extended model, the sequence of decisions from the perspective of an agent is as follows:
ip t
(i) The principal meets with the agent along with another agent while precommitting to a disclosure policy and informing the agent that a compensation contract with a fixed wage and some positive
cr
fraction of cash flow will be offered after the principal observes the agent’s ; (ii) The agent
us
rationally expecting such payments exerts effort ; (iii) The agent’s is realized; (iv) The principal sets and subject to the agent’s participation and limited liability constraints, fully discloses
an
in the case of disclosure, or discloses nothing in the case of non-disclosure; (v) The agent exerts effort ; (vi) The returns to the principal and the agent accrue.
M
First, consider the case of disclosure. Utilizing reduced form (3), the stage 1 problem for the
d
agent is to solve
{ }
te
max ( )[( )(1 − ( )) + ( )( )] +(1 − ( ))[( )(1 − ( )) + ( )( )] −
Ac ce p
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where the cost of human capital effort takes the same form as cash flow effort. As in the basic problem, if the agent’s reservation utility is zero, then the solution to this problem satisfies the agent’s ex ante participation constraint.10 After simplification this problem can be rewritten as
max { }
{( )+1 [ +1 (1 − ( )) + +1 ( )] 1+
+(1 − ( ))+1 [ +1 (1 − ( )) + +1 ( )]} − The agent’s objective function is of the generic form () + − with 0 and 0 0 00 0 (0) = 0 The agent’s FOC is 0 = −1 where 0 because is strictly concave and (0) = 0 Participation is met if 0 − 0 Substituting −1 for 0 from the FOC, participation is met if −1 − 0 which is true by 1 10
25 Page 27 of 46
The first order condition is
0 =
{+1 [ +1 (1 − ( )) + +1 ( )] 1+ (16)
ip t
−+1 [( +1 (1 − ( )) + +1 ( )]}0 ( ) − −1
cr
Because the function (·) is strictly concave, the effort cost function is strictly convex, and the
us
coefficient multiplying 0 (·) is positive, the second order condition is met.
Next, consider non-disclosure. Utilizing reduced form (5), the first order condition to agent’s
{ +1 [ (1 − ( )) + ( )] +1 1+
M
0 =
an
stage 1 problem can be written
(17)
d
−+1 [ (1 − ( )) + ( )] +1 }0 ( ) − −1
te
As in (16), the second order condition is met because the function (·) is strictly concave, the effort cost function is strictly convex, and the coefficient multiplying 0 (·) is positive.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
4.1. Characterization of a Symmetric Nash Equilibrium for Disclosure and Non-disclosure In each of the regimes of disclosure and non-disclosure, the question of whether a symmetric Nash equilibrium exists in the choice variables arises and the properties of such an equilibrium are of interest. Using (16) and (17) define
() ≡ +1 [ +1 (1 − ()) + +1 ()] − +1 [ +1 (1 − ()) + +1 ()]
(18)
() ≡ +1 [ (1 − ()) + ()] +1 − +1 [ (1 − ()) + ()] +1
(19)
and
26 Page 28 of 46
In symmetric Nash equilibria for disclosure and non-disclosure, respectively, we have that an agent’s selection of effort is determined by
()0 () − −1 = 0 for = 1+
(20)
ip t
Ω () ≡
cr
The following assumption places boundary restrictions on the function () (A.1) (0) = 0 (∞) = 0, (∞) = 1 and 0 (0) 0
us
In addition, we will utilize
an
(A.2) The functions () and − are sufficiently concave so as to imply that Ω0 () 0 for =
M
Assumption (A.1) is fairly standard and assumption (A.2) guarantees a decreasing marginal net benefit function in each problem.11
d
We have
te
Lemma 1. A unique symmetric Nash Equilibrium in exists under each of the disclosure rules, if (A.1)-(A.2) hold. 11
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
For disclosure (A.2) implies that 0
{ +1 (+1 − +1 ) + +1 (+1 − +1 )}(0 )2 (1 + ) { +1 [ +1 (1 − ) + +1 ] − +1 [+1 (1 − ) + +1 ]}00 + (1 + ) −( − 1)−2 for all
and for non-disclosure it implies that 0
{( + 1)+1 [ (1 − ) + ] (− + ) (1 + )
−( + 1)+1 [ (1 − ) + ] (− + )}(0 )2 { +1 [ (1 − ) + ] +1 − +1 [ (1 − ) + ] +1 }00 + (1 + ) −( − 1)−2 for all
27 Page 29 of 46
4.2. Comparisons Let denote the solution to (16) and let denote the solution to (17). Further let = ( ) and = ( ) In the following proposition we consider the case where the firm faces a favorable
ip t
production situation and is convex.
Proposition 4. Assume that (A.1)-(A.2) hold. Let 1 and let be convex. Then
cr
(i) , and
us
(ii) [ ( )] [ ()] if (10) and (12) hold, 2 + (1 − ) 1
is sufficiently concave such that the condition
( − )
M
( − )
an
Next, we consider the case where the firm has an unfavorable production situation and where
(21)
d
holds. By − = − = concavity of implies that ( − ) ( − ) but (21)
te
insists that the more restrictive condition ( − )
(
− ) is met. We have
Proposition 5. Assume that (A.1)-(A.2) hold. Let 1 and let be sufficiently concave such
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
that (21) holds. Then (i) .
(ii) [ ()] [ ( )] if (11) and (13) hold, 2(1 − ) ≤ + (1 − ) ≤ 1 Propositions 4 and 5 extend our earlier results to the case where the agent can exert costly effort to increase the probability of having high human capital in the firm. Proposition 4 says that if the principal faces a favorable production situation and is convex, then commitment to a policy of disclosure elicits greater human capital effort than does a policy of non-disclosure. Conditions (10) and (12) state, from Proposition 2, that there are rising marginal returns to status, but that they are not too strongly rising. When these conditions are met, a greater level of effort under
28 Page 30 of 46
disclosure translates into a greater probability of a high human capital level which in turn implies a greater expected profit. That is, (10) and (12) from Proposition 2 insure that expected profit under disclosure is strictly increasing in We can guarantee that disclosure is best only in this
ip t
special case for the generalized model. The same logic applies to Proposition 5 where the principal faces an unfavorable production
cr
situation. Here, non-disclosure leads to greater human capital effort which leads to greater if
us
is sufficiently concave so that condition (21) is met, but it must be true that falling returns to status are not so extreme that conditions (11) and (13) are violated.12 Under these assumptions,
an
greater effort leads to greater which implies greater expected profit under non-disclosure. When rising or falling returns to status are extreme and violate (10) and (12) or (11) and (13),
M
then while the results on effort still hold, there is no guarantee that an increase in will in fact lead
d
to a rise in expected profit. The results on profit in Propositions 4 and 5 can then be overturned.
te
5. Model Assumptions and Robustness
Our basic results regarding the optimality of disclosure and non-disclosure are based on the assump-
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
tions that the probability of a high outcome and the cost of effort functions are of the constant elasticity form, that agents are risk neutral, that there are two agents and that the cash flow process has two states. The two agent assumption is not critical, but for disclosure policy to matter through status concerns and affect the agent’s expectation of status in the organization, the number of agents cannot be too large. For an -agent organization, our model predicts that the effect of disclosure policy would decline with as the sample distribution of human capital levels converges to that of the population. It is easy to show that assumptions (21), (11), and (13) are not inconsistent. Let = (1−−10 ) = 6 = 4 and = 5 In this case, both assumptions are met with the right side of (21) given by 63888(10)−3 and the left side of (21) is 11578(10)−2 The term + (1 − ) = 099855 and 2(1 − ) = 089898 Each of these is less than unity. 12
29 Page 31 of 46
The two state assumption allows us to keep the optimal contract simple, as our goal is to concentrate on the disclosure decision. The constant elasticity assumption for cash flow production and effort cost (with or without risk neutrality) has the effect of making the optimal contract,
ip t
independent of the disclosure rule. This allows us to isolate the effect of disclosure policy on profit and total surplus through a pure effort channel that is not confounded with variation in the optimal
cr
contract.
us
If one deviates from the assumption that the probability of a high outcome and the cost of effort functions are of the constant elasticity form, the principal’s optimal contract can then vary with
an
the disclosure policy. In order to rule out signaling by the principal through the contract under non-disclosure in this more general model, a different timeline and information structure than those
M
used in the present paper would be necessary. In that more general model, the principal would set a contract before knowing each agent’s human capital, which is initially private information to the
d
agent. In the case of disclosure, the contract would be a menu where each part of the menu depends
te
on the agent’s status and human capital level. In the case of non-disclosure, there would be no such conditioning on status but only on human capital level. The issue of the optimality of disclosure
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
versus non-disclosure for either the principal or the agent would depend again on the curvature of the actor’s payoff function in the hidden information variable . If the actor’s payoff function is concave, then non-disclosure would be optimal for that actor and conversely if it is convex. The new aspect is that variations in across disclosure rules can affect that curvature. As a result, non-disclosure can become optimal for the agent, in contrast to the present model in which the agent always prefers disclosure.13 13
Marino (2014) obtains the result that the agent can be harmed by disclosure in a case where there is general hidden information about the agent’s utility having to do with the agent’s fit in the organization and the contract varies with the disclosure rule. The situation where the agent can affect the hidden information about fit is not considered.
30 Page 32 of 46
6. Conclusion We study the choice of disclosure policies as an organizational design problem in the presence of agents with status concerns. Our work has novel testable implications.
ip t
If the organization’s production situation is favorable, specifically 1 then disclosure has a
cr
positive motivational effect on agents and maximizes the principal’s profits net of wages. Otherwise, non-disclosure is the principal’s preferred policy. Unfavorability, specifically 1 is descriptive of
us
agency situations in which the agent’s effort does not impact the probability of success greatly but it does highly impact effort cost. Favorability is descriptive of the opposite set of circumstances
an
where the sensitivity of cash flow production to effort is high and/or the sensitivity of personal effort
M
cost to effort is low. In our model, unfavorability leads to concavity of the principal’s equilibrium payoff in the status function, making non-disclosure optimal, while favorability leads to convexity
d
of the principal’s payoff in the status function, making disclosure optimal.
te
In addition, agents always prefer disclosure to non-disclosure because more information allows them to better condition their effort choices. Our model thus points to status concerns as a potential source of disagreement between principals and agents about the proper level of organization-wide
Ac ce p
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transparency in principal-agent relationships. Tensions are likely to arise in unfavorable production situations.
With status concerns, hiring more talented employees may not be always better for the principal. As more talented employees reduce each other’s chances of achieving high status and dull each other’s motivation to exert effort, the principal may want to avoid hiring more talented employees. This local effect can also be global when agents’ returns to status are sufficiently increasing. Disclosure policies can also affect firm-specific investments made by agents with status concerns. If the organization is in a sufficiently favorable production situation and agents’ returns to status are increasing, then disclosure can induce more firm-specific investment by agents and lead to more 31 Page 33 of 46
profits for the principal. Similar results obtain with non-disclosure when the production situation
te
d
M
an
us
cr
ip t
is sufficiently unfavorable and agents’ returns to status are decreasing.
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
32 Page 34 of 46
Appendix
i o n h ( ) − [ ()]
cr
= {+1 [ + (1 − ) − ( + (1 − ) ) ]
ip t
Proof of Proposition 1: (i) We have
(1.a)
us
+(1 − )+1 [ + (1 − ) − ( + (1 − ) ) ]}
an
Rewriting (1.a), non-disclosure dominates if
M
+ (1 − ) ( + (1 − ) ) and
(2.a)
d
+ (1 − ) ( + (1 − ) )
te
These conditions hold if 1 That is, is a concave function of . (ii) Disclosure dominates if the reverse inequality in (2.a) holds, and this holds if 1 £ ¤ (iii) We have that ( ) − [ ()] is given by
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
{+1 [(1 − ) +1 + +1 ] + (1 − )+1 [1 − ) +1 + +1 ] (1 + )
−+1 [(1 − ) + ] +1 + (1 − )+1 [(1 − ) + ] +1 }
(3.a)
Because ( + 1) 1 +1 is a convex function of Thus, the expression (3.a) is positive and the result holds. ¥ Proof of Corollary 1: For 1 the result is obvious from Proposition 1. Consider the
33 Page 35 of 46
second statement We have that
[()] +1 (1 + ) ()] = [()] [ + (1 + )
cr
ip t
( ) = [()] +
[ [()|]] +1 (1 + ) [()|]] = [ [()|]] [ + (1 + )
an M
The sign of [ − ] = sign of
()]} − {[ [()|]] [ + [()|]]} +1 +1
te
The sign of this expression is that of
d
{[()] [ +
us
() = [ [()|]] +
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
{[ + (1 − ) )] − [ + (1 − ) ] }
+
+1 +1 {[ + (1 − ) ] − [ + (1 − ) ] +1 } +1
+(1 − ) {[ + (1 − ) ] − [ + (1 − ) ] } +
+1 +1 (1 − ){[ + (1 − ) ] − [ + (1 − ) )] +1 } +1
where ≡ ( ) for ∈ { } and ∈ { } The first two lines of this expression (high branch) are symmetric to the last two lines, so let us consider the first two lines of the high branch.
34 Page 36 of 46
The limit of the high branch as → 0 is 0 and the limit as → 1 is
ip t
1 2 {[ 2 + (1 − ) 2 )] − [ + (1 − ) )]2 } 0 2
cr
Take
{ {[ + (1 − ) )] − [ + (1 − ) )] } →0
{ ln + (1 − ) ln − ln[ + (1 − ) ]}
an
=
+1 +1 {[ + (1 − ) ] − [ + (1 − ) ] +1 }} +1
us
+
M
By concavity, ln + (1 − ) ln − ln[ + (1 − ) ] 0Using these facts, it follows that the high branch is negative for small 1 and positive for larger 1 The same results
d
hold for the last two lines of the low branch, so that the proof is complete. ¥
write
te
Proof of Proposition 2: (i) Strict concavity follows directly from (10). From (9), we can
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
h i ( ) R 0 iff (1 − 2)(( ) + (1 − ( )) ) R 2(1 − ( ) − )
(4.a)
Let = 05 then the left side () of (4.a) is zero and (4.a) becomes R · 2(1 − ( )) By (1 − ( )) 05 " ” holds. Next, let ∈ (1 − ( ) 05) In this case, The right side () of (4.a) is negative while the 0 so that holds. This result does not depend on the condition (10). Next, let ∈ (0 1 − ( )) Given (10), the result holds if (1 − 2) 2(1 − ( ) − ) which is true if 1 2(1 − ( )) The latter is true, so that the result holds. We have shown that
35 Page 37 of 46
£ ¤ ( ) 0 for ∈ (0 05) Next, let ∈ (05 1) Rewrite (4.a) as 2 − 2(1 − ( )) R ( ) + (1 − ( )) = ( ) 2 − 1
ip t
( ) =
The 1 for all ∈ (0 1) and for all ≥ 0 Further, lim = +∞, lim = →05
→1
cr
2( ) 0 and 0 If (12) holds, then ( ) ( ) for all ∈
£ ¤ £ ¤ (05 1) so that ( ) 0 for ∈ (05 1) We have established that ( )
us
£ ¤ 0 for ∈ (0 1) and that there is a boundary maximum of ( ) at = 1 Finally, let
an
condition (13) hold. From the above analysis, ∃ ˆ( ) ∈ (05 1) such that (ˆ ) = ( )
( ) ( ) for ∈ (05 ˆ) and ( ) ( ) for ∈ (ˆ 1) We have that ( ) +(1−( )) −2(1−( )) . 2[( ) +(1−( )) − ]
This completes the proof of (i).
M
ˆ( ) =
(ii) Convexity follows directly from (11). Begin by letting ∈ (0 1 − ( )) Here, 05 and
te
d
(1 − ( ) − ) 0 so that (4.a) can be expressed
( ) = ( ) + (1 − ( )) R
2(1 − ( ) − ) = ( ) 0 1 − 2
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
By (11), ( ) ≤ 1 and ( ) 1 for all ∈ (0 1) and ≥ 0 We have that ( ) 0 lim = 2(1 − ( )), →0
lim
→(1−)
= 0 and ( ) 0 If (13) holds, then, for
all ∈ (0 1 − ( )) 2(1 − ( )) ( ) and the result holds. If the converse of (13) holds, then from the above limit and monotonicity conditions, ∃ ˜ ∈ (0 1 − ) such that ( ) ( ) for ∈ (0 ˜) ( ) = (˜ ), and ( ) ( ) for ∈ (˜ (1 − )) Finally if ∈ (1 − 05) then we have from (4.a) the result because this result did not depend on (10). Finally let ∈ (05 1) We rewrite (4.a) as
( ) =
2 − 2(1 − ) R ( ) + (1 − ( )) = ( ) 2 − 1 36 Page 38 of 46
We have that ( ) 1 for all ∈ (0 1) ≥ 0 By (11), ( ) ≤ 1 so that ( ) ( ) for all ∈ (05 1) and the result holds. ¥ Proof of Proposition 3: It will be useful to rewrite condition (15) as follows:
1 +(1−)
1 + [1 −
]
R
+1 [ + (1 − )]
ip t
+1 [(1 − ) + ]
(5.a)
cr
[ ()] R 0 iff
1 1 − [ (1−)+ − 1]
The right side of (5.a) is a positive fraction and is decreasing in On the left side we have
1 +(1−)
] ∈ [0 1) Before proving the main part of Proposition 3,
us
1 [ (1−)+ − 1] ≥ 0 and [1 −
and, for all ∈ (˚ 1]
−
1 1 − [ (1−)+
1 then there exists a ˚ such that 1− [ (1−˜)+˜ −1] = 0
1 Proof of Lemma for Proposition 3: 1 − [ (1−)+ − 1] 0 iff
Γ( ) Clearly, lim Γ( ) = ∞ lim Γ( )= →1
d
→0
− 1] 0 and ( ()) 0
M
Lemma for Proposition 3. If
an
let us establish the following result.
−
1 [
1 (1−)+
−1]
≡
and Γ( ) 0 given that Γ
te
is continuous in the results hold. Using (5.a), [ ()] 0 for all ∈ (˚ 1] ¥ Proof of Proposition 3 (i): In light of the Lemma, take the limit of (5.a) as → 0 We obtain
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
after rearranging
[ ()] R 0 iff
With
2 2
fixed and
( )
1 + (1 −
)
R(
+1 )
(6.a)
→ ∞ → 1 this inequality becomes and [ ()] 0 for small
We have ruled out boundary maxima and [ ()] is continuous on a compact set. Thus, an interior maximum exists. ¥ (ii): Assume ≤
−
1 In this case, 1 − [ (1−˜)+˜ − 1] 0, for all ∈ [0 1] The limit of
(5.a) as tends to zero is depicted in the right side of (6.a). Fix and let both and be small. We have from (6.a) that [ ()] 0 as → 0Next, consider the limit of
37 Page 39 of 46
(5.a) as tends to unity. Here, (5.a) holds if (Note that ≤
¶ +1 µ
¶
R
implies 1 − ( − 1) 0)
1 1 − (
− 1)
¶ +1 µ
¶
µ ¶ R1+ 1−
us
µ
cr
If both and are small then [ ()] 0 as → 1
ip t
µ
−
We have shown that [ ()] is negative at = 0 and positive at = 1 Also [ ()] |=0 =
an
[ ( )] [ ( )] = [ ()] |=1 We need only show that [ ()] |=1 dominates [ ()] for any ∈ (0 1) under the stated conditions. With the stated assumptions and these
M
results, we need only show
)] + +1 (1 − )[ + (1 − )]
With
becomes
te
d
+1 +1 [ + (1 − )(
small, [ + (1 − )( )] → 1 and [ + (1 − )] → (1 − ) so that the condition
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
+1 +1 + (1 − )+1 (1 − )
This is true.
(iii): From (15), we see that the second term tends to zero as → 0 The first term is positive by and
+ (1 − ) − ( + (1 − ) ) = (1 − )( − ) + ( − ) 0 This completes the proof. ¥
38 Page 40 of 46
Proof of Lemma 1: If a symmetric Nash equilibrium exists it is unique by (A.2) because Ω () are strictly decreasing in Next, for existence note that lim Ω () = [( )+1 −( ) +1 ]0 (0)− →0
0 0 for = where ≡ ( ) for ∈ { } and ∈ { } Moreover, lim
→∞
ip t
Ω () = [( )+1 − ( ) +1 ]0 (∞) − ∞ 0 for = By continuity of Ω () an equilibrium exists. ¥
us
tion of (20), R if () R () for all Now () () if
cr
Proof of Proposition 4: (i) Because under (A.2) Ω () are strictly decreasing and by inspec-
an
{[( )) +1 (1 − ()) + ( ) +1 ()] −[ (1 − ()) + ()] +1 }
M
{[( ) +1 (1 − ()) + ( ) +1 ()]
(7.a)
d
−[ (1 − ()) + ()] +1 }, for all
te
where ≡ ( ) for ∈ { } and ∈ { } Let ≡ and note that (6.a) is met if +1 satisfies the condition that the expression
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
( + ∆) +1 (1 − ()) + +1 () − [( + ∆)(1 − ()) + ()] +1
(8.a)
is increasing in and ∆ This is true because the left side of (7.a) is of the form (8.a) with = and = + ∆2 where ∆2 = ( − ) and the right side of (7.a) is of the form (8.a) with = and = + ∆1 where ∆1 = ( − ) and ∆2 ∆1 To see that ∆2 ∆1 note by − = − = and convex in , ( − ) ( − ) With ∆2 ∆1 Expression (8.a) is increasing in if its derivative in is positive. Computing this derivative it is given by ( + 1)[( + ∆) (1 − ) + ] − ( + 1)[( + ∆)(1 − ) + ] 0 39 Page 41 of 46
by ( + 1) convex in 1 Analogously, taking the derivative of (8.a) with respect to ∆ we have
by ( + ∆) [( + ∆)(1 − ) + ( + 1)] It follows that
cr
(ii) Consider
ip t
( + 1)( + ∆) (1 − ) − ( + 1)[( + ∆)(1 − ) + ] (1 − ) 0
us
n h i o ( ) − [ ()]
an
= {[ ( ( ) + (1 − )( ) ) − ( + (1 − ) ) ] +[(1 − ) ( ( ) + (1 − )( ) )
M
−(1 − ) ( + (1 − ) ) ]}
d
where in equilibrium = ( ) for = and = 1 2. We have that From Propo-
te
¤ £ ¤ £ sition 1, ( ) |= − [ ()] |= 0 It then suffices to show that ( ) |= −
£ ¤ £ ¤ ( ) |= 0 A sufficient condition for the latter is that ( ) 0 for all ∈ [0 1] which is true under (10) and (12), by Proposition 2(i). ¥
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Proof of Proposition 5: (i) With an unproductive agent, we have that ( + 1) =
−
∈ (1 2)
In this case the function () +1 is convex and its first derivative, ( + 1)() is a concave function. Condition (7.a) from the proof of Proposition 4 tells us that non-disclosure generates greater effort if
{[( ) +1 (1 − ()) + ( ) +1 ()] − [ (1 − ()) + ()] +1 } {[( ) +1 (1 − ()) + ( ) +1 ()] − [ (1 − ()) + ()] +1 } for all (9.a)
40 Page 42 of 46
Condition (9.a) is met if the function +1 satisfies the condition that the expression
(10.a)
ip t
( + ∆) +1 (1 − ( )) + +1 ( ) − [( + ∆)(1 − ( )) + ( )] +1
is decreasing in and increasing in ∆ This is true because the left side of (9.a) is of the form (10.a)
cr
with = and = + ∆2 where ∆2 = ( − ) and the right side of (9.a) is of the form (10.a) with = and = + ∆1 where ∆1 = ( − ) and ∆2 ∆1 by condition
us
(21). Taking the derivative of (10.a) with respect to we obtain that (10.a) is decreasing in if
an
( + 1)( + ∆) (1 − ()) + ( + 1) () − ( + 1)[( + ∆)(1 − ()) + ()] 0
(11.a)
M
Condition (11.a) is the condition for concavity of ( + 1) in The proof of Proposition 4 shows
d
that (10.a) is increasing in ∆ Thus, (9.a) holds, and .
te
(ii) We have that
h i [ ( ) − ( ())]
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
= {[ ( ( ) + (1 − )( ) ) − ( + (1 − ) ) ] +[(1 − ) ( ( ) + (1 − )( ) ) −(1 − ) ( + (1 − ) ) ]}
(12.a)
£ ¤ Further from part (i), From Proposition 1, ( ) |= − [ ()] |= 0 It then
£ ¤ £ ¤ suffices to show that ( ) |= − ( ) |= 0 From the proof of Proposition 4,
¤ £ the latter condition is true if ( ) 0 for all ∈ [0 1] Sufficient conditions are then
(11) and (13), from Proposition 2. ¥
41 Page 43 of 46
References [1] Aoyagi, Masaki, 2010. Information feedback in a dynamic tournament. Games and Economic
ip t
Behavior 70, 242-260. [2] Ashraf, Nava, Oriana Bandiera, and Scott Lee, 2014. Awards unbundled: evidence from a
cr
natural field experiment. Journal of Economic Behavior and Organization, Forthcoming.
us
[3] Auriol, Emmanuelle and Régis Renault, 2008. “Status and incentives.” RAND Journal of Economics 39, 305-326.
an
[4] Barankay, Iwan, 2010. Rankings and social tournaments: evidence from a field experiment.
M
University of Pennsylvania Working Paper, July 2010.
[5] Barankay, Iwan, 2011. Gender differences in productivity responses to performance rankings:
d
evidence from a randomized workplace experiment. University of Pennsylvania Working Paper,
te
September 2011.
[6] Crutzen, Benoit S. Y., Otto H. Swank and Bauke Visser, 2010. Confidence management: on
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
interpersonal comparisons in teams. Forthcoming, Journal of Economics and Management Strategy, March 2010.
[7] Dubey, Pradeep, John Geanakoplos, 2010. Grading exams: 100, 99, 98, . . . or A, B, C?. Games and Economic Behavior 69, 72-94.
[8] Ederer, Florian, 2010. “Feedback and motivation in dynamic tournaments. Journal of Economics and Management Strategy 19, 733-769. [9] Ederer, Florian and Andrea Patacconi, 2010. Interpersonal comparison, status and ambition in organizations. Journal of Economic Behavior and Organization 75, 348-363.
42 Page 44 of 46
[10] Ellingsen, Tore and Magnus Johannesson, 2007. Paying respect. Journal of Economic Perspectives 21, 135-150. [11] Ellingsen, Tore and Magnus Johannesson, 2008. Pride and prejudice: the human side of incen-
ip t
tive theory. American Economic Review 98, 990-1008.
cr
[12] Goltsman, Maria and Arijit Mukherjee, 2011. Interim performance feedback in multistage
us
tournaments: the optimality of partial disclosure.” Journal of Labor Economics 29, 229-265. [13] Heffetz, Ori and Frank, Robert H, 2011. Preferences for status: evidence and economic im-
an
plications. In: Jess Benhabib, Alberto Bisin, Matthew Jackson (Eds). Handbook of Social
M
Economics, Volume 1A, Amsterdam: Elsevier, 69-91.
[14] Hopkins, Ed and Tatiana Kornienko, 2004. Running to keep the same place: consumer choice
d
as a game of status. American Economic Review 94, 1085-1107.
55, 18-45.
te
[15] Itoh, Hideshi, 2004. Moral hazard and other-regarding preferences. Japanese Economic Review
Ac ce p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
[16] Kenrick, Douglas T., Vladas Griskevicius, Steven L. Neuberg, and Mark Schaller, 2010. Renovating the pyramid of needs: contemporary extensions built upon ancient foundations. Perspectives on Psychological Science 5, 292-314. [17] Lizzeri, Alessandro, Margaret Meyer and Nicola Persico, 2002. The incentive effects of interim performance evaluations. Caress Working Paper, August 2002. [18] Marino, Anthony M., 2014. Transparency in agency: the constant elasticity case and extensions. International Journal of Industrial Organization 33, 9-21. [19] McGregor, J., 2006. The Struggle To Measure Performance. Business Week, January 9, 2006.
43 Page 45 of 46
[20] McManus, T. Clay and Justin M. Rao, 2013. Signaling smarts: revealed preferences for self and social perceptions of intelligence. University of Tennessee Working Paper, 2013.
ip t
[21] Maslow, A. H., 1943. A theory of human motivation. Psychological Review 50, 370-396. [22] Maslow, A.H., 1970. Motivation and Personality (2nd ed.). New York, NY: Harper and Row.
cr
[23] Moldovanu, Benny, Aner Sela, and Xianwen Shi, 2007. Contests for status. Journal of Political
te
d
M
an
us
Economy 115, 338-363.
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