American Economic Journal: Microeconomics 2012, 4(2): 146–183
http://dx.doi.org/10.1257/mic.4.2.146
Contractual and Organizational Structure
with Reciprocal Agents†
By Florian Englmaier and Stephen Leider*
We solve for the optimal contract when agents are reciprocal, demonstrating that generous compensation can substitute for performancebased pay. Our results suggest several factors that make firms more
likely to use reciprocal incentives. Reciprocity is most powerful when
output is a poor signal of effort and when the agent is highly reciprocal and/or productive. Similarly, reciprocal incentives are attractive
when firm managers have strong incentive pay and discretion over
employee compensation. While reciprocal incentives can be optimal
even when identical firms compete, a reciprocity contract is most
likely when one firm has a match-specific productivity advantage
with the agent. (JEL D23, D86, J33, M12, M52)
O
nly a fraction of real world labor contracts make substantial use of explicit monetary incentives, and those that do often have only weak monetary incentives.
Lemieux, MacLeod, and Parent (2009) find that in the United States only 37  percent
of individuals have some kind of pay-for-performance component to their compensation, and that among those who do the median magnitude of these incentives is only
3.5 percent of their base wage.1 This prevalence of weak (monetary) incentives in real
world contracts, while apparently still obtaining nontrivial effort, is not well explained
by the standard approach in contract theory which emphasizes the use of output realizations as a (noisy) measure of employee effort to overcome the moral hazard problem. In these models, by tying the level of the agent’s compensation to the amount
of output produced, the agent is given monetary incentives to increase his output.2
* Englmaier: University of Konstanz, PO Box 144, 78457 Konstanz, Germany (e-mail: florian.englmaier@
uni-konstanz.de); Leider: University of Michigan Ross School of Business, 701 Tappan Street R4486, Ann Arbor,
MI (e-mail: [email protected]). We thank Greg Barron, Tobias Böhm, Nicolas Klein, Georg Gebhardt, Fuhito
Kojima, Meg Meyer, Andy Newman, Al Roth, Klaus Schmidt, Carmit Segal, Ferdinand von Siemens, Christopher
Weber, Christian Zehnder, and the participants at the European Summer Symposium for Economic Theory 2008
and the European Economic Associations Meetings in Barcelona 2009 for comments and suggestions. Florian
Englmaier thanks the Harvard Business School and the Study Center Gerzensee of the Swiss National Bank for
their hospitality and the German Science Foundation (DFG) for financial support via grant EN 784/2-1 and through
SFB/TR 15. Stephen Leider thanks the National Science Foundation and the Sperry Fund for financial support. †
To comment on this article in the online discussion forum, or to view additional materials, visit the article page
at http://dx.doi.org/10.1257/mic.4.2.146.
1 Belfield and Marsden (2003) and Piekkola (2005) report similar results for the UK (20 percent of employees,
and 3.5 percent of base wages) and Finland (38 percent of employees and 3.6 percent of base wages). See also
Prendergast (1999) for a general survey of empirical tests of the standard contracting model.
2 Holmström (1979), Grossman and Hart (1983), and Holmström and Milgrom (1991). Models that emphasize
multitasking in work tasks have been one explanation why weak, rather than strong, incentives prevail. However,
such models typically assume directly that the agent is willing to do some positive amount of work absent monetary
incentives. The use of career incentives is another potential explanation for the empirical observation.
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147
Alternative sources of incentives in the workplace have been considered before.
Akerlof (1982) is an early example. He models the labor relation as a gift exchange
where agents respond to generous treatment by the firm (i.e., generous wage packages) by exerting more than minimal effort. While Akerlof’s model is based on the
effect of work “norms,’’ our paper will build on explicit models of social preferences. In particular we will employ “reciprocity,’’ a concept formally described by
Rabin (1993) for normal form games and by Dufwenberg and Kirchsteiger (2004)
and Falk and Fischbacher (2006) for sequential games.3 We use a standard principal-agent framework: a risk-neutral firm hires a risk-averse worker to exert nonverifiable effort. As in standard agency models, the worker’s risk aversion makes
imposing monetary incentives costly. We then add the assumption that the agent is
reciprocal—i.e., the agent’s utility increases in the principal’s revenue whenever
the firm provides the agent with a rent in excess of his outside option. Thus, when
the firm is generous to the agent by giving him something valuable (additional
compensation), the agent desires to provide, in turn, something of value to the firm
(higher effort). The agent’s reciprocal attitude can be used by the firm to align the
agent’s preferences with those of the firm, thus generating intrinsic motivation.
Making use of those the firm can substitute away from using explicit incentives,
hence offering a less performance-sensitive contract. While the standard incentive contract, i.e., the contract solving the problem with purely selfish agents as
laid out by Holmström (1979) or Grossman and Hart (1983), is always feasible
if the agent is sufficiently reciprocal it is optimal to use at least some degree of
reciprocal incentives. We also derive two main comparative statics on the nature
of the optimal contract to document, under which conditions reciprocal incentives
are relatively more effective and hence are more likely to be employed. When the
agent is more intrinsically reciprocal, or when the effect of the agent’s effort on
the principal’s revenue is greater, then more reciprocal incentives will be used,
i.e., the contract will use weaker explicit incentives as measured by the sensitivity of the agent’s remuneration.4 In contrast, the more informative the signal is
with respect to the agent’s effort choice, the more explicit incentives will be used.
Hence, we can show that explicit monetary and implicit reciprocal incentives are
substitutes in the principal’s optimal incentive mix.
These results suggest that reciprocal incentives (i.e., contracts with limited performance-based pay and generous average compensation) should be empirically
most prevalent in job settings where explicit incentives work poorly due to a noisy
production function, where firms have the greatest ability to identify and screen
for reciprocal individuals, and where output is highly valuable to the principal. We
then examine further two additional aspects that can influence the attractiveness of
reciprocal incentives. We first analyze how a firm can maximize the efficiency of
reciprocal incentives by fine tuning aspects of its organizational structure. We then
consider the role of labor market competition where reciprocal workers are scarce.
See also the paper by Cox, Friedman, and Sadiraj (2008), grounding reciprocity in neoclassical demand theory.
Englmaier and Leider (2010, 2012) test the predictions of our model with respect to the relation of the initial
gift and the effect of the agent’s effort on the principal’s revenue, both in a lab experiment and in a field experiment,
and find supportive evidence in both cases.
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To study the role of organizational form we develop an extension of our basic
contracting model, capturing the basic structure of a firm’s hierarchy. In the model
extension we consider three simple organizational relationships between two managers within the firm to illustrate the role of managerial compensation and decisionmaking for the reciprocity of subordinates. We assume throughout that the agent
feels more reciprocal towards the person deciding about his contract, i.e., the one
making the gift. First, we consider a situation with a firm owner and a middle manager. The manager determines the agent’s contract, while the owner takes no action.
We then vary the strength of the manager’s incentives, i.e., his pay-performance sensitivity. This captures how important the agent’s effort is for the manager’s pay. We
show, that stronger managerial incentives make reciprocal incentives more effective,
implying weaker explicit incentives for the worker. Second, we consider a situation where the two managers are supervising different agents who work independently. Each manager determines the contract for his agent. We then vary whether
the managers’ incentives are narrow (depending only on the revenue generated by
their agent) or broad (depending on the total revenue). It turns out that more narrow
incentives improve the effectiveness of reciprocal incentives as they better target
the worker’s return gift to “his’’ manager. Thirdly, we consider a situation where
one manager is the agent’s direct supervisor (whose compensation is highly dependent on the agent’s output) and the other is a senior manager (whose compensation
is weakly dependent on the agent’s output). We consider the effect of having the
agent’s pay set centrally (by the senior manager) versus giving the supervisor discretion over pay, i.e., having him set the contract and show that the manager whose
pay is more dependent on the agent’s performance should be responsible for setting
pay. As opposed to the results of our basic contracting model, the predictions we
derive in the extended version of the model are based on observable features of firm
hierarchies and thus are in principle testable with field data.
We then present present a model with a simple form of labor market competition.
Two firms make sequential contract offers to either a reciprocal agent or a selfish
agent. We show that when one firm has a substantial productivity advantage, it is too
costly for the second firm to bid away the reciprocal agent (as his effort generates a
smaller reciprocal utility for any given contractual “gift’’). However, even when the
firms are ex ante identical if a reciprocal contract is optimal in the one-firm model it
will always be optimal for the first firm to provide a reciprocal contract in the twofirm case. If the agent is highly reciprocal this may require the first firm to provide
extra rents to prevent the agent from being bid away, however even in the limit hiring
the reciprocal agent leads to a strict decrease in the firm’s wage bill.
Our approach, connecting gift exchange with organizational structure and labor
market competition, suggests a new direction for research in the growing empirical literature on gift exchange in the field. Existing research finds mixed evidence
for gift exchange, depending on the setting. Falk (2007) documents an increase in
donations if solicitation letters include a present for potential donors and Leuven
et al. (2005) provide survey evidence that firms with a more reciprocal work force
are more likely to provide their workers with general training (as they deem it more
likely that this gift will be repaid within an ongoing relation). However, Gneezy
and List (2006) ran a field experiment hiring students for a day job and find that the
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149
effect of gift exchange in the field is only minor, fast disappearing and overall not a
viable employment strategy. In contrast, Kube, Maréchal, and Puppe (2006), in a setting similar to Gneezy and List’s, find weak evidence for the presence of (positive)
gift exchange and strong evidence for negative reciprocity as a response to wage cuts.
Also Bellemare and Shearer (2009) analyze gift exchange within a real tree planting
firm in British Columbia and find persistent effects of gift exchange. Finally, Dohmen
et al. (2009) provide detailed survey evidence from the German Socio-Economic
Panel (GSOEP) for the prevalence of reciprocity on a representative level. They find
substantial heterogeneity in the degree of reciprocity, measured by the participants’
response to six different survey questions and distinguish between positive, which is
the focus of our analysis, and negative reciprocity. They proxy for effort with measures
of working hours and overtime hours and find that positively reciprocal workers exert
more of both. Moreover they find that positively reciprocal people are less likely to be
unemployed. Dur, Non, and Roelfsema (2010) analyze a model where the principal
has the option, in addition to providing explicit incentives, to induce effort by directing (costly) attention to the agent. This attention is not contractible and the steeper the
explicit incentives for the agent are the less incentives remain for the principal to direct
attention to the agent. Dur, Non, and Roelfsema (2010) argue that promotion incentives help to overcome this double moral hazard problem. They also use information
from the GSOEP and show that the stylized facts are (partially) consistent with their
model. Our theoretical results suggest that future empirical research should focus on
finding evidence for gift exchange by studying job settings where reciprocal motivations are strongest: highly (firm-specific) productive workers, jobs where output is a
poor signal of effort, and firms that give supervisors both strong performance pay and
discretion over employee compensation.
The rest of this paper is structured as follows. The next section sets out and analyzes our contracting model, the following Section III extends a simplified version of
our contracting model into a model of organizational structure. Section IV presents
results on the robustness of our analysis to competition and Section V concludes.
The Appendices contain the multiple action case and additional proofs.
I. The Baseline Model
While the basic idea of reciprocity is both simple and appealing—a reciprocal
person wants to repay kindness with kindness and mischief with mischief—­applying
models of reciprocity is somewhat intricate. Rabin (1993), for simultaneous move
games, and Dufwenberg and Kirchsteiger (2004) and Falk and Fischbacher (2006),
for sequential games, have developed powerful and very general models to analyze
any game. Our objective is more modest in scope: To develop a somewhat simplified model that is still sufficiently rich to capture the key features of reciprocity in a
principal-agent setting. Hence, we do not need the extra machinery that allows the
aforementioned papers to analyze general games.5 Principal-agent problems have a
5 Outcome based theories, most notably “inequity aversion’’ by Fehr and Schmidt (1999) or the closely related
paper by Bolton and Ockenfels (2000) have been developed to capture important aspects of the gift exchange
intuition. See e.g., Dur and Glazer (2008) and Englmaier and Wambach (2010) for applications of outcome based
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American Economic Journal: Microeconomicsmay 2012
number of features that make it easier to apply a reciprocity-type model (compared
to a generic game). To that end, we will use a somewhat simplified model that is still
able to capture the essence of reciprocity for the purpose of our problem at hand. In
doing so we will employ several assumptions, natural for the labor market contracting situation we are interested in, which will help to simplify our analysis. 1) The
contract communicates expectations, hence there is less scope for beliefs to influence the equilibrium. 2) The agent’s outside option in the labor market, is the natural reference point for what constitutes a “fair’’ action of the firm. 3) The agent’s
individual rationality constraint ensures that no destructive reciprocity equilibrium
exists as it is not optimal for the firm to make an unkind offer that would be accepted
and result in destructive negative reciprocal behavior by the worker.
A. Model Setup
We first consider a moral hazard model with one risk-neutral principal who wants
to maximize expected profits and one risk-averse agent who cares about reciprocity.
We assume that there are n states of the world that are characterized by outputs q​ ​i​
with i = 1, … , n respectively. The agent can take one of two actions6 (effort levels)​
a​1​and ​a​2​with ​a​1​ < ​a2​​and corresponding costs from effort c(⋅) with c(​a​2​) > c(​a1​​).
The two actions imply respective probabilities of the states π
​ ​i​(​a1​​) and ​πi​​(​a2​​) where
for the respective expected return of the principal ∑ ​π​i​(​a2​​)​qi​​ > ∑ ​π​i​(​a1​​)​qi​​ holds.
As is standard, we assume that the monotone likelihood ratio property (MLRP)
​π​j​(​a2​​)
​πi​ ​(​a​2​)
​  > ​ _
 
 
​ 
for all j > i. Hence, higher states are better signals of
holds, i.e., that _
​ ​π​ ​(​a​​)  
​πi​ ​(​a​1​)
j 1
high effort.
 ) is a set of wage payments w(​qi​​) in each state, as well as a (nonA contract (​wi​​, ​ a​
binding) request for an action a​
​  . In a real world context we expect that a firm communicates to a worker the level of performance expected of him (e.g., in a job description
or in a code of conduct), which puts his compensation into context. While a​
​   is not
enforceable, it serves to fix the agent’s beliefs about the principal’s intended generosity (since the expected utility of a contract depends on the agent’s action).
The agent’s inherent concern for reciprocity is measured by η ∈ [0, +∞). The
 ), is given by
agent’s utility given that he takes action a′, under the contract (w, ​ a​
(
)
_
 ) u(w(​qi​​)) − c(​ a​
 ) − ​u ​ 
 ) = ∑ ​πi​ ​(a′ ) u(w(​qi​​)) + η ∑ ​πi​ ​(​ a​
U(a′, ​ a​
(
)
× ∑ ​π​i​(a′ ) ​qi​​ − c(a′ ).
models to contracting and incentives, Englmaier (2005) for a survey of this literature, as well as Fehr and Schmidt
(2003) for a broader survey of social preferences and decision making. These theories simplify the problem by
only conditioning on the final allocations of resources. Hence, inequity aversion trades off capturing “reciprocity’’
directly in preferences for greater tractability. Inequity aversion does capture directly, however, the separate and
important phenomenon of “concern for fairness’’ and does remarkably well in explaining observed behavior in
laboratory studies. However, with weak or no monetary incentives, inequity aversion will only induce effort if either
1) the agent begins with a larger payoff than the manager, and the manager receives > 50 percent of the profit from
the agent’s effort (thus effort reduces the agent’s advantageous inequality) or 2) the agent begins with a smaller
payoff than the manager, and the agent receives > 50 percent of the profit from his own effort (thus effort reduces
the agent’s disadvantageous inequality). While often true in laboratory gift exchange games, these conditions are
generally not met in real world labor market settings.
6 We show how our results generalize to the general multi-action case in Appendix A.
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We assume that u(⋅) is a real-valued, continuous, strictly increasing, conw​)  , ∞) where w
​_ ​ is finite, and
cave function defined on some open interval (u(​_
(w) = − ∞
.
li​m​   w →​_w​u 
​ 
The utility function consists of three parts:
(i) Expected utility from the monetary wage payment ∑ ​πi​ ​(a′ ) u(w(​qi​​)).
(
)(
)
 )u(w(​qi​​)) − c(​ a​
 ) − ​u0​​ ∑ ​π​i​(a′ ) ​qi​​ .
(ii) Reciprocal utility η ∑ ​π​i​(​ a​
(iii) Effort costs c(a′ ).
For simplicity, we assume that in the reciprocal portion of the agent’s payoffs he
values the return gift of his effort as the increase in the expected revenue of the principal, i.e., that he is risk neutral towards the effect his effort has on the principal’s
revenue. We believe that this simplifying assumption is justified, as recent experimental papers have shown that subjects are on average risk neutral in their choices
over lotteries that others will face (see Güth, Levati, and Ploner 2008 and Brennan
et al. 2008).7
With this utility function a “generous’’ contract is one that provides a rent to the
agent, i.e., an expected monetary utility in excess of the agent’s outside option.8
This reservation utility, provided for by the labor market, is a natural reference
point for what constitutes a “fair’’ action of the firm. Whenever the firm allows
the worker to earn an expected rent in excess of his next best alternative in the
market, the worker will interpret this as a kind gesture. Furthermore, if the principal gives the agent a larger rent, he will be seen as more kind. Hence, a more
generous contract will induce the agent to feel more reciprocal, i.e., he will derive
greater marginal and absolute utility from the principal’s revenue.9 In the basic
model we assume that the outside option is fixed exogenously. In Section III, we
explicitly discuss the effects of endogenizing the outside option in a competitive
labor market.
7 However, if the agent was risk averse over the principal’s revenue (i.e., if we replace ​∑   ​​ ​πi​ ​​(a′ ) ​qi​​ with​
∑   ​​ ​π​i​​(a′ ) f (​qi​​) for a concave function f ) all the results of our base model will carry through (with appropriate adjust-
ment to the affected assumptions—e.g., assuming that E[ f (q) | a] is concave in a instead of E[q | a]). The effect of the
agent’s effort on the principal is fully captured in our model by the difference in average revenue between actions
 ] − E[ f (q) | ​a1​​]. This
(ΔER). If the agent is risk averse this term would be replaced by the analogous term E[ f (q) | ​ a​
 ] > E[ f (q) | ​a1​​]), though
difference will always be positive (since MLRP implies FOSD, and therefore E[ f (q) | ​ a​
potentially smaller in magnitude than ΔER. Since ΔER always appears as ηΔER, any result for an agent who is
risk neutral over principal revenue can be replicated for a risk-averse agent by increasing η. The intuition of the
extended model also still applies, although the specific simple functional forms we used for the manager’s payoffs
would need to be changed.
8 We believe the assumption that the outside option is the reference point for a “fair’’ offer is generally representative, especially for jobs where there is a standard wage. The general intuition of our model would carry through
if instead a contract is “fair’’ only if it provides a specific rent over the outside option, although some results may
require a sufficiently large η. In this case the standard selfish contract will not be acceptable to a reciprocal agent,
therefore if the required rent is large enough the principal may prefer to hire a selfish agent.
9 Note that, for tractability and to focus on our main idea, we use the principal’s gross revenue, instead of
revenue net of the wage payment. We want to as simply as possible capture the intuition that leaving a rent to the
agent aligns his interests to the principal’s interests. Our main results should still carry through, as it is clear that
Observations 3 and 4 below still hold if the agent has reciprocal utility over the principal’s net profits.
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American Economic Journal: Microeconomicsmay 2012
We follow the approach of Grossman and Hart (1983) and focus on the implementation problem for any particular action. In the two action case we focus on
implementing a​ ​2​, as ​a​1​can be implemented trivially. The principal’s problem is to
solve the following program:
​   
min ​  ∑ ​πi​ ​(​a2​​)w(​qi​​)
 
w(​qi​​), ​ a​
_
 ) ≥ ​u ​   [IC ] U(​a​2​, ​ a​
 ) ≥ U(​a1​​, ​ a​
 ) s.t. [IR] U(​a​2​, ​ a​
  = ​a2​​.
[EB] ​ a​
The first and second constraints are the standard individual rationality [IR] and
incentive compatibility [IC] constraints. The third constraint requires that beliefs are
in equilibrium [EB], i.e., that it is proper for the agent to interpret the intended kind . Effectively,
ness of the principal’s contract offer as his expected rent under action ​ a​
[EB] is an equilibrium selection assumption where we are making a substantive
restriction (each contract maps to one belief—ruling out babbling equilibria and
“destructive’’ equilibria where incorrect beliefs make a potentially generous contract seem unkind) and a restriction without loss of generality (equating the belief
induced to the actual communication). The [IC] and [EB] constraints together ensure
that the principal is asking for the action he is attempting to implement.
However, note that it is sufficient to simply assume [EB] and solve to imple  = ​a2​​ :
ment ​ a​
_
 )w(​qi​​) s.t. U(​ a​
 , ​ a​
 ) ≥ ​u ​  , ​ a​
 ) ≥ U(​a1​​, ​ a​
 ).
  
 ​  ∑ ​π​i​(​ a​
  and U(​ a​
​
min 
w(​qi​​)
Now we make the standard transformation
​w​i​ = h(u(​wi​​)), h(⋅) = ​u−1
​ ​(⋅), h′(⋅) > 0, h″(⋅) > 0.
The following notational definitions will be helpful:
 
 ) = ​∑ ​    ​​πi​ ​(​ a​
 )​qi​​
ER(​ a​
​u​i​ = u(w(​qi​​))
 
 
 
 
 
 ) − c(​a1​​) ΔER = ​
 )q​ i​​ − ​
Δc = c(​ a​
∑  ​    ​​πi​ ​(​ a​
∑ ​    ​​πi​ ​(​a1​​)​qi​​.
Additionally, we can transform the [IR] into a more useful form:
 
(
)
 
_
_
 , ​ a​
 ) = ​
 )​ui​​ + η ​∑ ​    ​​π​i​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​  ER(​ a​
 ) − c(​ a​
 ) ≥ ​u ​ 
U(​ a​
∑ ​    ​​πi​ ​(​ a​
 
 
(
 
)
_
 )) ​∑ ​    ​​πi​ ​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​  ≥ 0
(1 + ηER(​ a​
 
 
_
 )​ui​​ − c(​ a​
 ) − ​u ​   ≥ 0
​
∑  ​    ​​π​i​(​ a​
 
englmaier and leider: Reciprocity and Contracts
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153
and similarly for the [IC]
 , ​ a​
 )  ≥ U(​a1​​, ​ a​
 )
U(​ a​
(
)
_
 ) − ​πi​​(​a​1​)) ​ui​​ + η ∑ ​π​i​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​  (ΔER)  ≥ Δc.
∑ (​πi​​(​ a​
Thus the principal’s full problem is given by
 )h(​ui​​) ​
  
m​uin 
  ​  ∑ ​π​i​(​ a​
​​
i
_
 )​ui​​ − c(​ a​
 ) − ​u ​   ≥ 0
s.t. ∑ ​π​i​(​ a​
(
)
_
 ) − ​πi​​(​a​1​)) ​ui​​ + η ∑ ​π​i​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​  (ΔER) ≥ Δc.
and ∑ (​π​i​(​ a​
As in the standard problem, we now have a convex minimization problem with a
finite number of linear inequality constraints. Therefore the Karush-Kuhn-Tucker
conditions are necessary and sufficient. However, first we wish to build intuition
about the principal’s problem. Examining the [IC], we can see that when the agent
is reciprocal η > 0, the principal has two means of providing incentives. First, the
principal can provide monetary incentives as in the standard problem. Second, the
principal can provide the agent with a positive rent, thus engendering reciprocity.
The former is costly as it exposes the risk-averse agent to a risky income stream
for which the agent has to be compensated with a risk premium, while the latter
involves an excess payment in order to provide the agent with a rent. The principal’s
task is to mix these two incentives optimally.
B. Extreme Contracts
Before solving for the generally optimal contract, it is useful to make the following observations about the standard contract (solving the problem with purely
selfish agents which holds the agent down to his outside option) and a flat-wage
contract—two extremes in terms of monetary and reciprocal incentives.
Observation 1: For η = 0 the optimal contract is given by the solution to the
standard problem.
This is straightforward to see since for η = 0 the principal’s problem collapses
to the standard problem
 )h(​ui​​)
min  ∑ ​πi​ ​(​ a​
_
 )​ui​​ − c(​ a​
 )] ≥ ​u ​   and ∑  [​πi​​(​ a​
 ) − ​πi​​(a)] ​ui​​ ≥ Δc
s.t. ∑  [​πi​​(​ a​
[
]
​π​i​(a)
​ ​ + ​λIC
​ ​ 1 − ​ _
 
 
​ .
with the classic solution: h′(​ui​​) = ​λIR
 )
​π​ ​(​ a​
i
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American Economic Journal: Microeconomicsmay 2012
Hence, our expansion of the problem to consider reciprocal agents nests the standard problem as a special case.
Observation 2: The optimal standard contract implements a​
​   for any η.
It is well known that the optimal contract in the standard problem holds the agent down
_
to his outside option, EU = ​u ​.  Hence, the agent has zero rent, and the r­ecip­rocal
portion of his utility drops out. Therefore, since the standard contract meets the [IC]
of the standard problem, it also meets the [IC] of the full problem. This means that a
reciprocal agent who has been given zero rent acts like a selfish agent. Thus if a firm
were to use a standard incentive contract for its workers, even any reciprocal individuals within the workforce would still exert effort. The firm would not observe too-low
productivity, it would merely have a (potentially) higher-than-necessary labor cost.
More generally, we can prove the following lemma for any contract:
 ) implements ​ a​
  for ​η1​​, then it implements ​ a ​ for all​
Lemma 1: If a contract (​w​i​, ​ a​
η​2​ ≥ ​η1​​.
Proof:
It is clear that the [IR] is invariant to η. Similarly, the RHS of the [IC] does not vary
_
 ) − ​πi​​(​a1​​)) ​ui​​+ η(∑​πi​ ​(​ a​
 )​ui​​− c(​ a​
 ) − ​u ​)  (ΔER)])
​ ​= [ ∑(​πi​​(​ a​
with η. The LHS (LH​SIC
_
 )​ui​​ − c(​ a​
 ) − ​u ​)  (ΔER) ≥ 0.
is increasing in η: ∂LH​SIC
​ ​/∂η = (∑​πi​ ​(​ a​
Therefore, when the agent is more reciprocal the set of feasible contracts increases,
and thus the principal’s agency costs are nonincreasing in η.
 )h(​u​  *i​  ​) denote the value function to the princi ) = mi​nu​​ ​∑ i ​ ​ ​πi​​​(​ a​
Let V(η, ΔER; ​ a​
pal’s problem (i.e., the expected wage bill), where ​u*​​is the optimal contract. Then
the lemma immediately implies the following proposition.
 ) is nonincreasing in η for all ​ a​
 .
Proposition 1: V(η, ΔER; ​ a​
Hence, it will be cheaper for the principal to induce a more reciprocal agent to
exert effort than a less reciprocal agent. Moreover, since in the [IC] η is multiplied
by ΔER, we can directly conclude that increasing the value of output to the principal
(i.e., increasing ΔER ) has the same consequences.
 ) implements ​ a​  for ΔE​R1​​, then it
Corollary 1: 1. For any η, if a contract (​wi​​, ​ a​
 ) is nonincreasing in
 for all ΔE​R2​​ ≥ ΔE​R1​​. 2. For any η, V(η, ΔER; ​ a​
implements ​ a​
ΔER for all a​
​  .
Focusing now on the opposite extreme of a flat wage, where there are no explicit
monetary incentives, we can also easily see that reciprocal agents can still be induced
to work hard. Thus, reciprocal motivations are a separate lever to provide incentives
and can be sufficient on their own.
Observation 3: For η > 0 a flat wage with a large enough rent u​
​ ˜  will imple .
ment ​ a​
englmaier and leider: Reciprocity and Contracts
Vol. 4 No. 2
155
_
˜  for all i, where ​ u​
˜  > 0. Since we assume a positive
 ) + ​ u​
Consider u​ ​i​ = ​u ​  + c(​ a​
rent, the [IR] will be met. Consider the [IC]:
 ) − ​πi​​(a)) ​ui​​ + η[∑​πi​ ​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​]  (ΔER)  ≥ Δc ∑(​πi​​(​ a​
_
_
(
)
˜ ) ∑(​πi​​(​ a​
 ) + ​ u​
 ) − ​πi​​(a))
(​u ​   + c(​ a​
[
_
]
_
˜ ) − c(​ a​
 )(​u ​   + c(​ a​
 ) + ​ u​
 ) − ​u ​  (ΔER)  ≥ Δc
+ η ∑​πi​ ​(​ a​
˜ ) (ΔER)  ≥ Δc.
η(​ u​
˜  large enough: ​ u​
˜  ≥ Δc/ηΔER. Thus, if agents are
Clearly the [IC] can be met for ​ u​
sufficiently reciprocal, a firm can still induce effort even if contracts are completely
invariant w.r.t. signals of good performance. In fact, for very reciprocal agents the
principal can induce effort quite cheaply, approaching the First Best implementation
cost.
Observation 4: For η → ∞ the First Best is arbitrarily closely approximated
with a flat wage with an infinitesimal rent.
From equation for u​
​ ˜ above, it is clear that as η increases the rent needed to induce
the agent to exert effort decreases. In particular, in the limit the rent needed is zero.
Hence, the principal’s cost to implement a​
​  is merely the outside option plus the cost
of effort, equal to the First Best cost.
C. The Optimal Contract
Returning to the Principal’s full problem, we can now form the Lagrangian and
solve for the optimal ​ui​​.
The First Order Condition for u​ ​i​in the principal’s problem is given by
 )h′(​ui​​) − ​λI​R​​πi​​(​ a​
 ) − ​λIC
 ) − ​πi​​(a) + η ​
 )(ER(​ a​
 ) − ER(a))]
​ ​[​πi​​(​ a​
π​
0 = ​
π​i​(​ a​
i​(​ a​
[
]
​π​i​(a)
 ) − ER(a)) .
h′(​u​i​) =  ​λIR
​ ​ + ​λIC
​ ​ 1 − ​ _
 
 ​ 
+ η(ER(​ a​
 )
​πi​ ​(​ a​
​ ​ × [IC] = 0 with [IR] and
with complementary slackness λ
​ ​IR​ × [IR] = 0 and ​λIC
[IC] denoting the [IR] and [IC] constraints respectively. For further analysis the following lemma will be helpful.
Lemma 2: IC binds at the optimum for all η.
Proof:
First suppose the [IC] and the [IR] are both slack. Then an alternative contract​
u​  ′i​ ​ = ​ui​​ − ϵ will still meet the [IR] and [IC] for small enough ϵ, yet cost the ­principal
156
American Economic Journal: Microeconomicsmay 2012
strictly less. Next, suppose the [IC] is slack, but the [IR] binds. In this case, the
contract provides no rent to the agent, hence the reciprocal portion of his utility drops out. Therefore the agent acts like a selfish agent, and the logic from the
standard problem carries through: if the [IC] were slack the FOC’s would imply a
flat wage, however a flat wage does not provide any monetary incentives to exert
effort—violating the [IC]. Therefore the [IC] must bind for all η.
Hence, there are two classes of optima that we must consider:
1) Contracts where both the [IC] and the [IR] bind.
2) Contracts where the [IC] binds and the [IR] is slack.
To show that the latter case is relevant, we can also directly prove that for η sufficiently large, the optimal contract will provide the agent with a rent.
Proposition 2: There exists a finite η​ *​​ such that for all η ≤ ​η​*​ the optimal
contract is a standard contract (i.e., provides zero rent) and for all η > ​η*​​ the
optimal contract is a reciprocity contract (i.e., provides the agent with a strictly
positive rent).
The full proof is in Appendix BA, however we can easily demonstrate that for
some parameter range a reciprocity contract is optimal, and thus provide an upper
limit for ​η*​​. Since the standard contract (and its cost to the principal) is invariant to η, to show that a reciprocity contract is sometimes optimal it is sufficient
to show that some (nonoptimal) reciprocity contract implements a​
​   at lower cost
˜  = ∑ ​π​i​(​ a​
 )h(​us​t, i​) be
to the principal. Let ​us​t​ be the standard contract, and let ​ w​
the principal’s expected cost for the standard contract. Consider the flat wage​
˜  − ϵ) = ​ u​
˜  for some small ϵ. Clearly, this contract has strictly lower cost
u​i​ = u(​ w​
for the principal. If ϵ is small enough, this flat wage will still satisfy the [IR] and
provide the agent with a positive rent (since the agent is risk averse). Moreover, it
will satisfy the [IC] if
_
Δc
__
˜  − c(​ a​
 ) − ​u ​)  (ΔER) ≥ Δc ⇔ η ≥ ​ 
  
   
 ​ .
η(​ u​
_
˜  − c(​ a​
 ) − ​u ​)  (ΔER)
(​ u​
The right-hand side is finite, hence a large (but finite) η is sufficient to satisfy the
[IC]. In that case, the flat wage u​
​ ˜  will implement action a​
​   at a strictly lower cost
to the principal than the standard contract. Therefore the optimal positive-rent
contract will have strictly lower cost than the optimal zero-rent contract (i.e., the
standard contract), hence the optimal contract will be a reciprocity contract.
Having demonstrated that there is an open set of parameters such that a reciprocity contract is optimal, we can proceed to describe its characteristics. Indeed,
the reciprocity contract is sufficiently tractable that we can describe a number of
its features.10
10 In Englmaier and Leider (2008), we derive the exact form of the optimal contract for the classes of CRRA and
CARA utility, and provide an implicit definition of ​η​*​.
englmaier and leider: Reciprocity and Contracts
Vol. 4 No. 2
157
Proposition 3: For η > ​η*​​, the optimal contract has the following property, for
i > j :
​π​i​(​a1​​)
 
 ​ 
+ ηΔER
1 − ​ _
u′(​
w
​
)
​
 )
​πi​ ​(​ a​
j
_
__
​ 
   ​.
 ​ 
 
= ​   
u′(​wi​​)
​π​j​(​a​1​)
_
1 − ​ 
 
 ​ 
+ ηΔER
 )
​πj​ ​(​ a​
Proof:
We solve for the optimal contract by setting λ
​ I​R​ = 0 (since we know that for
1
   ​
  = ​λIC
​​
η > ​η*​​ the [IR] will be satisfied and slack). Therefore the FOC is ​ _
u′(​wi​​)
​_
πi​ ​(​a​1​)
​  + ηΔER). If we take the ratio of any two FOC’s, λ
​ ​IC​cancels and we
× (1 − ​ ​π​ ​(​ a​ )  
i
​π​i​(​a​1​)
1 − ​ _
 
 ​ 
 + ηΔER
u′(​wj​​)
 )
​π​i​(​ a​
_
get _
​  u′(​w​​)  
​  = ​ 
  
  
 ​. Hence, the ratio of the marginal utilities between two
​π​ ​(​a​​)
i
 + ηΔER
1 − ​ _ ​ 
 
 )
​πj​ ​(​ a​
j
1
states, and thus the incentive intensity, depends on the ratio of their likelihood
ratios as well as a constant denoting the marginal impact on reciprocity of increasing the rent.
Since we can identify features of the optimal contract without solving explicitly for ​λIC
​ ​, we can derive general results for the optimal contract. Our guiding
intuition is that the principal is mixing optimally between two incentive mechanisms—explicit monetary incentives (which depend on the wage spread across
states, but not the wage level) and reciprocal incentives (which depend on the
overall wage level, but not directly on the spread). Therefore, we would expect
that as one incentive mechanism becomes more cost-efficient, the principal should
use it relatively more.
D. Comparative Statics: Reciprocity
Our model suggests that either standard or reciprocal incentives can be optimal
depending on the environment. To identify the situations where reciprocity incentives are most likely to be used, we identify comparative statics for the strength
of reciprocity. Specifically, we examine how the use of reciprocal incentives vary
based on the intrinsic reciprocity of the agent, the productivity of the agent, and the
informativeness of output as a signal of effort.
When the agent is more reciprocal it takes a smaller rent to induce any given
effort, making reciprocal incentives cheaper. Therefore, the principal should use
reciprocal incentives more relative to explicit incentives. Indeed, we find that
as the agent becomes more reciprocal, the optimal contract gets flatter between
states, in marginal utilities (for the general optimal contract), and therefore also
in wage levels.
u′(​wj​​)
 
 
​ 
is decreasing in η for all
Proposition 4: For η > ​η​*​, in the optimal contract ​ _
u′(​wi​​)
u′(​wj​​)
_
​ 
= 1.
(i, j) where i > j, with li​mη​ →∞​  ​  u′(​w​​)  
i
158
American Economic Journal: Microeconomicsmay 2012
Proof:
We can simply take the derivative of the marginal utility ratios, since they do not
depend on ​λIC
​ ​ :
[​  ]
[​ 
]
​π​i​(​a​1​)
_
1 − ​ 
  
​  + ηΔER
u′(​
w
​
)
​
 )
​π​i​(​ a​
j
∂
∂
__
_
_
_
 ​ 
 ​  
 
 = ​    ​     
  
​   ​  
​πj​ ​(​a1​​)
∂η u′(​wi​​)
∂η 1 − ​ _
  
​  + ηΔER
 )
​πj​ ​(​ a​
1 − ​ _​)
  − (1 − ​ _​)
 ) ΔER
(
(
___
 ​
   
= 
​     
​πj​ ​(​a1​​)
  
 )
​πj​ ​(​ a​
​π​i​(​a​1​)
  
 )
​π​i​(​ a​
​π​ ​(​a​​)
  
​  + ηΔER​)​​
(1 − ​ _
 )
​π​ ​(​ a​
j
2
1
j
​π​ ​(​a​​)
​π​​(​a​​)  
_
​  − ​ _
  
​  ΔER
(​
 
 )
​π​ ​(​ a​
 ) )
​π​ ​(​ a​
__
    ​ < 0,
= 
​   
​π​ ​(​a​​)
  
​  + ηΔER​)​​
(1 − ​ _
 )
​π​ ​(​ a​
i
1
j
i
1
j
j
1
2
j
( ​π​ ​(​ a​)
)
​πj​ ​(​a​1​)
​π​​(​a​​)
​  i  1   
​  
− ​ _
  
​ which is
where the sign of the expression is solely determined by _
 
i
​πj​ ​(​ a​)
negative by MLRP. The limit claim follows directly from l’Hôpital’s rule.
In order to help intuition, we will repeatedly refer to the cases of CRRA and
CARA utility as the results there are very easily interpretable in terms of wage ratios
and wage differences instead of the somewhat abstract marginal utilities ratios in
​w​​
the general case. In particular, for CRRA (CARA) utility, _
​ ​wij​ ​​ (​wi​​ − ​wj​​) is decreasing
​_
wi​​
11
in η, with li​m​η→∞​ ​  ​wj​​ ​  = 1 (li​m​η→∞​ ​wi​​  − ​w​j​  = 0). Since the rent needed to induce
effort gets smaller as the agent gets more reciprocal, the cost of reciprocal incentives
decreases. Therefore in the optimal contract the principal substitutes away from
explicit incentives (since they are now relatively more expensive compared to reciprocal incentives).
We can also directly prove that increasing the value of the agent’s effort similarly
leads to weaker incentives. Thus, making the return “gift’’ of effort more valuable to
the principal (for the same cost of effort) increases the agent’s willingness to work
hard in exactly the same way as making him intrinsically more reciprocal.
u′(​wj​​)
u′(​wj​​)
 
 
​ 
is decreasing in ΔER, with li​m​ΔER→∞​  _
​  u′(​w​​)  
​ 
= 1.12
Corollary 2: For η > ​η*​​, ​ _
u′(​w​​)
i
i
Proof:
Note that in the above equations, η always appears as ηΔER, hence an increase in
ΔER is equivalent to an increase in η.
All the proofs for the CRRA and CARA results can be found in Englmaier and Leider (2008).
For CRRA (CARA) utility, (​wi​​/​w​j​) (​wi​​ − ​wj​​) is decreasing in ΔER, with li​
m​
wj​​ 
ΔER→∞​  ​wi​​/​
= 1 (li​m​ΔER→∞​ ​wi​ ​  − ​w​j​  = 0).
11 12 englmaier and leider: Reciprocity and Contracts
Vol. 4 No. 2
159
E. Comparative Statics: Information
We can also demonstrate that when output is a more informative signal of effort,
contracts use more explicit incentives. We are unaware of any standard instrument
to examine directly the strength of incentives between states for general nonlinear
contracts and more than two outputs. Hence, we begin by developing some notation
to allow us to compare the incentives of two contracts.
Let w and x be two contracts specifying wage payments in states i = 1, … , n.
Definition 1: Define the partial order over contracts w ​
≻​mu​  x, “w implies
sharper marginal utilities in the extremes than x” if ∃ i′ ≥ ​ i ​˜   > ​ j ​˜   ≥ j′
u′(​wj​​)
u′(​xj​​)
​  ≥ ​ _
 
 
​  ∀i ≥ ​ i ​˜,  j ≤ ​ j ​˜ with strictness for i = i′ or j = j′.
s.t. _
​  u′(​w​​)  
u′(​x​​)
i
i
This relation requires that for some set of “high’’ states and some set of “low’’
states, the ratio of marginal utilities for any high state and any low state is higher in
w than in x, and strict for at least one high state and one low state. Thus the “sharper’’
contract provides greater rewards for achieving high output states relative to low
output states. For example increasing the bonus size for a step-function bonus, or
increasing the commission rate for a commission with a sales-hurdle makes the
monetary incentives “sharper.’’13
In some cases, we will be able to make stronger comparisons between contracts,
such that every state is either a high state or a low state.14
Definition 2: Define the partial order w ​≻m​ u, ​ k​˘ ​  x, “w has sharper marginal utili˘  − 1.
˘ than x” if w ​≻​ ​ x for ​ i ​˜   = ​ k​˘ and ​  j ​˜   = ​ k​
ties about state ​ k​
mu
As with the previous comparative static, for CRRA and CARA utilities the
general relation for marginal utilities will imply a relation for ratio and difference
(respectively) of wage levels: “sharper’’ contracts give a larger increase in wages
between low output states and high output states.
Having developed a concept to compare incentive intensity across contracts,
we also need to specify in what sense the principal’s information “improves.’’ We
focus on the standard comparison in the literature: that the likelihood ratios of
one information structure are a mean preserving spread of the other information
structure.15
Note that ≻
​ ​mu​ is transitive, if w ​≻​mu​  x for (​​  i ​˜​ 1​, ​​  j ​˜1​  ​) and x ​≻​mu​  y for (​​  i ​˜​ 2​, ​​  j ​˜2​  ​), then w ​≻​mu​  y for (​  i ​˜,  ​  j ​˜) 
˜
˜
= (max [​​  i ​1​  ​, ​​  i ​2​  ​], min [​​  j ​˜1​  ​, ​​  j ​˜2​  ​]), where there will be strictness for i′ = max [​i​  ′1 ​ ​, ​i​  ′2 ​]​  and j′ = min [ ​j​  ′1 ​,​  ​j​  ′2 ​ ​  ]. Additionally,
note that if w ​≻mu
​ ​  x for some (​i′​1​, ​​  i ​˜1​  ​, ​​  j ​˜1​  ​, ​j′​1​) then you cannot have w ​≻mu
​ ​  x for some other (​i′​2​, ​​  i ​˜2​  ​, ​​  j ​˜2​  ​, ​j′​2​). To see
u′(​w1​​)
u′(​x1​​)
_
˜
˜
this, assume without loss of generality that ​​ i ​​1  ​  ≥ ​​  i ​ ​2​. Then w ​≻​mu​  x implies ​ 
 
 ​  
> ​ _
 
 ​, while x ​≻​mu​  w
u′(​wi1
​ ​)
u′(​xi​1​)
u′(​
w
​
)
​
u′(​
x
​
)
​
1
1
​ 
implies _
 
 ​  
≤ ​ _
 
 ​, 
a contradiction.
13 u′(​wi1
​ ​)
u′(​xi​1​)
˘ fixed.
Again, note that ​≻mu, ​ 
​ k​˘ ​is transitive holding ​ k​
See, for example, Kim (1995).
14 15 160
American Economic Journal: Microeconomicsmay 2012
Let ​Π​F​ and ​ΠG​ ​ be two discrete probability distributions over the states, where
  are distributed according to F and G
the likelihood ratios for implementing ​ a​
­respectively. Recall that G is a mean preserving spread of F if F and G have the
same means and ​∫ y​  ​    ​G(t)dt ≥ ​∫ y​  ​    ​F(t)dt for all y ∈ ℜ and strict for some range
with positive measure.
From Rothschild and Stiglitz (1970), we can focus on the step function S(x)
= G(x) − F(x), and the corresponding step function s(​zk​​) = ​gk​​ − ​fk​​. Where for
​
z​k​ ∈ {z | z is a likelihood ratio of either ​Π​F​or ​ΠG​ ​ }
we write that
[
]
[
]
​π​  Fj​  ​
​π​  Gj​  ​
_
​
f​k​ = Pr 1 − ​ _
 ​   = ​z​k​ .
 
 
​
and
g
​
​
 
​
= Pr
1 − ​ 
 ​
 
= ​
z
​
k
k
​​ π​​  Fj​  ​
​​ π​​  Gj​  ​
Additionally, if we restrict attention to a specific class of mean-preserving spreads,
we will be able to make stronger comparisons between contracts.
˘ s.t. S(x) ≥ 0 for x < ​ x​˘ 
Definition 3: G differs from F via a balanced MPS if ∃ ​ x​
˘
˘ . Let ​ k​ be the lowest state with a likelihood ratio greater than
and S(x) ≤ 0 for x > ​ x​
or equal to x​
​ ˘ .
Hence, in a balanced MPS, the likelihood ratios for all states with a likelihood
˘ do not increase, while the likelihood ratios for all the states with
ratio smaller than ​ x​
˘ do not decrease. Increasing the variance of a normal
a likelihood ratio larger than ​ x​
distribution is an example of a balanced MPS.
In order to isolate the effect of information we will also assume that the productivity of effort is the same under both information structures, i.e., that ΔE​RF​ ​ 
= ΔE​R​G​.
Proposition 5: If G is an MPS of F, ΔE​RF​ ​ = ΔE​RG​ ​, ​w​*​(G), x​​*​(F)
are the optimal contracts given the distribution of likelihood ratios, and
​ ​*​(G) ​≻m​ u​ ​x*​​(F ). Additionally, if G is a balanced MPS
η ≥ max [​η​*​(F), ​η*​ ​(G)] then w
*
*
​ k​˘ ​ ​x​​(F).
of F, then ​w​​(G) ​≻mu, ​ 
Proof:
See Appendix BB.
As output becomes a more informative signal about effort this means that the
agent taking the high effort action makes it increasingly more likely that a high
output state is obtained rather than a low output state. The optimal contract exploits
this by increasing the (relative) reward in high states and decreasing the (relative)
reward in low states, i.e., the optimal contract provides “sharper’’ monetary incentives. For the cases of CRRA and CARA utility, the corresponding results for wage
ratios and wage differences follow immediately.
Vol. 4 No. 2
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161
Thus as the output realization becomes a better signal of effort, the optimal contract uses more explicit incentives, in the sense of larger changes in either marginal
utility or wages between low output and high output states.
F. Multiple Action Case
In Appendix A, we show how our results extend to the case of multiple action
levels for the agent. We show that without any further distributional assumptions
Observations 1 through 4 directly extend to the multiple action case. We can also
show that the principal’s value function is nonincreasing in η, that a downward [IC]
always binds and hence the principal does not provide excessive incentives, and that
for sufficiently high values of η a reciprocity contract is indeed the optimal contract.
Furthermore we show that if we assume CDFC, introduced by Rogerson (1985), all
our results, including the comparative statics, extend to the multi action case.
Generally it is hard in our framework, without imposing strong assumptions,
to make statements on which effort level will be implemented. However, we can
show that increases in the agent’s concern for reciprocity will push towards the
first best outcome.
˜  s.t. if η ≥ ​ η​
˜  implementing ​am​ ​ domiProposition 6: If ​aFB
​ ​ ≥ ​am​ ​ > ​an​​, then ∃ ​ η​
nates implementing ​an​​. Therefore, if η is sufficiently large, the first best action will
be implemented.
Proof:
_
 + c(​a​k​)) and denote the wage for implementing action ​ak​​ under
Let ​w​  FB
k​  ​ = h(​u ​ 
 ) is continuous and nonincreasing in η for
the first best contract. Recall that V(η, ​ a​
_
 , and that li​m​   η→∞
  ​
 (​ak​​) = h(​u ​  + c(​a​k​)). Additionally, since ER(a) is concave
V
all ​ a​
w​  FB
and c(⋅) is convex, we have that ER(​am​ ​) − ​w​  FB
m​  ​ > ER(​an​​) − ​
n​  ​.
​ ​) − ER(​an​​), then the claim is true for all η, since the
If V(0, ​a​m​) < ​w​  FB
n​  ​ + ER(​am
expected wage bill to implement ​a​n​for a particular η can be no smaller than the first
best wage bill, and therefore we have ER(​a​m​) − V(η, ​am​ ​) > ER(​an​​) − V(η, ​an​​)—
i.e., the profit from implementing ​a​m​is larger than the profit for implementing ​an​​.
˜  be defined such that V(​ η​
˜ , ​am​ ​) = ​w​  FB
​ ​) − ER(​an​​). Then
Otherwise, let ​ η​
n​  ​ + ER(​am
˜  we will have V(η, ​a​m​) < ​w​  FB
​ n ​ + ER(​am​ ​) − ER(​an​​), and therefore
for any η ≥ ​ η​
ER(​a​m​) − V(η, ​am​ ​) > ER(​an​​) − V(η, ​an​​).
This means that as η increases implementing lower actions will be ruled out as a
potential optimum in favor of implementing higher actions, with the set of possible
optimum actions eventually shrinking to include just the first best action. More
­generally, a higher η will lead to a weakly higher effort choice when V(η, a) does
not decrease faster in η for lower actions than for higher actions—however this condition is difficult to ensure.
II. A Model of Organizational Structure
In this section we analyze how a firm can maximize the efficiency of reciprocal
incentives by fine tuning aspects of its organizational structure. Our baseline model
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American Economic Journal: Microeconomicsmay 2012
assumes that either the principal is a single individual, or if the principal is a firm
that the agent feels reciprocal to the firm as a whole. However, rather than treating
the firm as a monolith, the agent could have different reciprocal feelings towards
particular individuals on the firm side. In particular it seems reasonable to suppose
that an individual may give more “credit’’ for kindness to the particular manager
that sets his pay. Moreover, Coffman (2011) finds that individuals place the majority of the blame for unkind actions on the specific individual who ultimately made
the decision, even when the previous actions of another person restricted his choice
set.16 If the agent feels more reciprocal towards particular individuals within the
firm, then the compensation setting and decision-making features of the organization will affect the impact of reciprocal sentiments. If the firm decides to make use
of reciprocal incentives, specific properties of the organizational structure like the
allocation of decision rights over pay (“who gives the gift’’) or the use of performance pay across the hierarchy can be adjusted to give reciprocal incentives the
most leverage.17
To see this, consider the following stylized model. There are two risk-neutral
managers, I and II, whose payoffs are (different) monotonic functions of firm
revenue ( ​fI​​(ER) and ​f​II​(ER) respectively). For simplicity we assume that only one
manager takes an action that is payoff relevant to the agent, and that the agent
gives all of the “credit’’ for rent he receives to the one manager who took the
action, i.e., he feels reciprocal only towards the decider.18 Since we are focusing
on the agency problem of the reciprocal agent, we assume that whichever manager takes an action will set the agent’s contract to maximize overall firm profit.
In particular, we will consider three simple organizational relationships between
I and II to illustrate the role of managerial compensation and decision-making for
the reciprocity of subordinates:
• I is the firm owner and II a manager. The manager determines the agent’s contract, while the owner takes no action. We then vary the strength of the manager’s incentives.
• I and II are managers of different agents (who work independently). Each manager determines the contract for his agent. We then vary whether the managers’
incentives are narrow (depending only on the revenue generated by their agent)
or broad (depending on the total revenue).
16 Pazy and Ganzach (2009) examine the interaction of pay structure with the degree to which Perceived
Organizational Support (POS) and Perceived Supervisor Support (PSS) influence performance and organizational commitment. They document that there are substantial interactions between pay structure and POS and
PSS respectively. In line with our assumption that middle managers rather than firm owners are the key references for an employee’s reciprocity, they find that in a variable pay context PSS has a significant effect on performance while POS has not.
17 Note that in setting up our model we disregard considerations that feature prominently in the literature on
optimal delegation, see e.g., Holmström (1984) or Alonso and Matouschek (2008). Obviously, delegation motives
surely play an important role in both the decision of how strong and broad managerial incentives are and how much
discretion is left to managers.
18 It would be sufficient to generate the following results to instead assume that the decision of one manager
has a larger impact on the agent’s payoff than the other manager, and that the agent assigns a larger fraction of the
perceived kindness to the manager whose action was relatively more important. The simpler model we consider in
the text is the limit version of this assumption.
englmaier and leider: Reciprocity and Contracts
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163
• I is the agent’s direct supervisor (whose compensation is highly dependent on
the agent’s output) and II is a senior manager (whose compensation is weakly
dependent on the agent’s output). We consider the effect of having the agent’s
pay set centrally (by II ) versus giving the supervisor discretion over pay (i.e.,
I sets the contract).
Furthermore we assume that the agent’s preference intensity for reciprocity is
person-independent, i.e., the same η applies towards managers I and II respectively.
Then we can write the agent’s preferences as follows:
[​
(​
)]
_
 ) − ​u ​  ( ​f​I​(ER(a)))
 ) = ​
∑  ​  ​​πi​ ​​(a)​u​i​ − c(a) + η DI​​ × ∑  ​​π
​   i​ ​​(a​
​   )​u​i​ − c(​ a​
U(a, ​ a​
 
[​
 
(​
)]
_
 ) − ​u ​  ( ​f​II​(ER(a))).
+ η DII​ ​ × ∑  ​​π
​   i​ ​​(a​
​   )​u​i​ − c(​ a​
 
with ​D​k​ = 1 if manager k determines the agent’s contract, for k ∈ {I, II }
= 
0 otherwise.
For simplicity we focus on the two action case where η > ​η*​​, i.e., a reciprocity
contract is optimal. We also abstract away from agency problems at the managerial
level; we assume that all managers set the agent’s pay optimally with respect to the
firm’s overall objective function.
A. Strength of Managerial Incentives
We first consider the case where principal I is the owner of the firm and principal
II is the manager. We assume that the manager sets the agent’s pay while the owner
​ ​II​ = 1. To
takes no action (but choosing the manager’s contract), i.e., D
​ ​I​ = 0 and D
have a simple metric for the strength of the manager’s incentives, we assume that the
owner and the manager split the revenue via a linear compensation scheme:
fII​ ​ (ER) = α + β × ER(a).
​f​I​(ER) = (1 − β) × ER(a) − α ​
Hence, the manager will have stronger incentives when β is larger. It is straightforward t o show that reciprocity will have a stronger incentive effect for the agent when
β is larger, and therefore the costs of agency will be smaller. Let V(η, β) denote the
value function at the optimum of the owner’s problem given the manager’s incentives.
Proposition 7: For a given η > ​η*​​(β), V(η, β) is decreasing in β.
Proof:
Since ​D​I​ = 0, the reciprocity terms relating to the owner drop out of the agent’s
utility. Therefore, this model is isomorphic to the base model, replacing ER(a) with
α + β ⋅ ER(a), and therefore replacing ΔER with β ⋅ ΔER. Note that α cancels—
the agent only cares about the change in the manager’s payoff due to effort, not the
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American Economic Journal: Microeconomicsmay 2012
absolute level. Therefore increasing β is equivalent to increasing ΔER, and thus the
set of implementable contracts increase (since reciprocal incentives are stronger)
and therefore the costs of agency decrease. This holds even if α depends on β, since
V is invariant to α.
Additionally, the comparative static for reciprocity also extends.
u′(​wj​​)
​  u′(​w​​)  
​ 
is decreasing in β for all i > j.19
Corollary 3: For η > ​η*​​(β), _
i
Since only the manager but not the owner factors into the agent’s reciprocity,
increasing the strength of his incentives makes the agent’s effort more valuable to
him, and therefore for any amount of rent the agent is more willing to exert effort.
Thus at the optimum the agent’s explicit monetary incentives will be weaker when
the manager’s incentives are stronger.
B. Breadth of Managerial Incentives
We now consider the case where managers I and II are division managers—each
oversees an agent who works independently and generates separate output E​R​I​and E​
R​II​. We assume the two agents are identical. Each manager determines the contract for
his agent. We consider two kinds of incentives for the managers: narrow incentives
where each manager is compensated for the output of his worker, and broad incentives
where both managers are compensated for the total output of both workers.
E​RI​​ + E​RII​ ​
_
Case 1 (Broad incentives): ​f​k​(E​RI​​, E​RII​ ​) = ​ 
 
 
 
​
2
Case 2 (Narrow incentives): ​f​k​(E​RI​​, E​RII​ ​) = E​Rk​​.
Since the agents’ outputs are independent we focus without loss of generality
on the agency problem for agent II. For this agent we then have that manager II is
the only manager who makes a payoff-relevant decision for him (i.e., D
​ I​​ = 0 and​
D​II​ = 1). Therefore the reciprocity term for manager I drops out of the agent’s utility function. Thus, broadening managerial incentives decreases the impact that high
effort will have on the payoff of the manager that the agent feels reciprocal towards.
Therefore, the reciprocal sentiments of the agent will have the greatest impact when
managerial incentives are narrow. Let ​Vi​​(η) denote the value function of the principal’s problem in Case 1.
Proposition 8: For a given η > max(​η​  *1​,​  ​η​  *2​)​  and for any contract for the other
agent, ​V1​​(η) > ​V2​​(η).
Proof:
 I​​) denote the expected revenue given the action of agent I. Then Case
Let E​RI​​(​​ a​
2 is equivalent to setting α = 0 and β = 1 in the previous model, while Case 1 is
​w​​
For CRRA (CARA) utility, _
​ ​wi​  ​​ (​w​i​ − ​wj​​) is decreasing in β for all i > j.
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165
 I​​)
E​RI​​(​​ a​
equivalent to setting α = ​ _
   
​ and β = ​ _12 ​.  Since V decreases in β, the result fol2
lows immediately.
Since broader incentives are less responsive to the agent’s effort, the agent is less
able to return the principal’s kindness when he has broad incentives. Therefore,
reciprocal incentives are less effective. Similarly, the optimal monetary incentives
for the agent will be weaker when the manager’s incentives are narrower.
u′(​wj​​)
​  u′(​w​​)  
​ 
is smaller in Case 2 than in Case 1 for
Corollary 4: For η > max(​η​  *1​,​  ​η​  *2​)​  , _
i
all i > j.20
C. Managerial Discretion
Lastly, we consider the role of who has decision rights over the agent’s pay. To
that end we compare the case where the agent’s compensation is determined centrally by a senior manager I (e.g., by the HR department, by a firm-wide compensation policy, etc.) or where the agent’s supervisor II has discretion over the agent’s
pay. To capture this difference we distinguish I and II by the responsiveness of their
payoffs to this agent’s output. We assume that the supervisor’s payoffs are relatively
more sensitive to an increase in output than the senior manager who is further away
in the hierarchy of the firm. That is, we assume
 )) − ​f​II​(ER(​a1​​)) > ​f​I​(ER(​ a​
 )) − ​f​I​(ER(​a1​​)).
​f​II​(ER(​ a​
We now consider two cases:
Case 1 (Central Policy): Manager I determines the agent’s contract, while manager II takes no action.
Case 2 (Managerial Discretion): Manager II determines the agent’s contract,
while manager I takes no action.
Since the payoffs of manager I are affected less when the agent works hard than the
payoffs of manager II, it is straightforward to show that reciprocity will be more
effective as an incentive for the agent when manager II sets his pay (and thus gets
credit for any rents).
Proposition 9: For a given η > max(​η​  *1​,​  ​η​  *2​)​  , ​V​1​(η) > ​V2​​(η).
Proof:
In each case, the reciprocity terms for the manager who doesn’t make a decision
drop out of the agent’s utility function, therefore each Case k is isomorphic to the
 )) − ​fk​(​ ER(​a1​​)). Thus the results from the
base model, replacing ΔER with f​​k(​ ER(​ a​
base case immediately imply the result, since ΔE​RII​ ​ > ΔE​RI​​.
​w​​
For CRRA (CARA) utility, _
​ ​wi​ ​​  ( w
​ ​i​ − ​wj​​) is smaller in Case 2 than in Case 1 for all i > j.
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American Economic Journal: Microeconomicsmay 2012
Again, this implies that the optimal monetary incentives will be weaker in Case 2
than in Case 1.
u′(​wj​​)
​  u′(​w​​)  
​ 
is smaller in Case 2 than in Case 1 for
Corollary 5: For η > max(​η​  *1​,​  ​η​  *2​)​  , _
i
all i > j.21
These results indicate that the organizational features of the firm can significantly
influence the effectiveness of reciprocity as an incentive device. Taken together they
suggest that reciprocity will be most effective when the manager who has decision
rights over the compensation for the agent also has strong, narrow incentives related
to the agent’s productivity.
D. Empirical Evidence
One problem with most behavioral contracting models (and with optimal contracting models more generally) is the difficulty of generating empirically falsifiable predictions. Results depend on hard to observe variables (preference
parameters, stochastic properties of the problem, costs to monitor, etc.) and not
even data on all predictions (incentive intensity, details of the contract, etc.)
are readily accessible to the researcher. Our basic model is not different in this
respect. Results also depend on hard to observe parameters like η, the effect of
agent’s effort on the principal’s payoff, or the stochastic properties of the problem.
However, the above described extended version of the model allows us to make
predictions about properties of organizational structure. Englmaier and Leider
(2008) analyze the 1998 WERS UK survey and identify firms that are likely to
have a more reciprocal workforce. They find evidence that suggests that, in these
firms, it is more likely that middle managers have decision rights over worker pay
(i.e., the managers are directly “sending a gift’’), there is more and more narrowly
targeted incentive pay for those managers (i.e., a high value of the return gift,
effort, to the initial sender), and there is less usage of incentive pay for workers.
Huang and Cappelli (2010) use a national sample of US establishments which
includes information on employee selection. They show that firms that screen job
candidates to find workers with a stronger “work ethic’’ employ policies that rely
less on monitoring and explicit incentives. Their measure and their results are also
consistent with a story along our lines.
III. A Model with Competition
Our analysis has so far focused on analyzing a model with one firm and one
worker, with the worker’s outside option (and benchmark for generosity) set exogenously. Our model then generates an optimal contract that provides positive rents
to the worker. In the presence of competition, however, a worker’s outside option
will depend endogenously on the offers he receives from other firms. A natural
​w​​
For CRRA (CARA) utility, _
​ ​wi​ ​​  ( w
​ ​i​ − ​wj​​) is smaller in Case 2 than in Case 1 for all i > j.
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167
question, therefore, is whether the rent-providing reciprocity contracts we have
identified can survive in the presence of competition, or whether the competing
firms will bid up the wages so that the final contract is perceived as merely “neutral’’ compared to the second best offer and hence workers’ reciprocal inclination
is no longer profitably employable.
While a general analysis of the effect of reciprocity on incentive contracts in
labor markets is beyond the scope of this paper, we present here a simple analysis
of a particular form of labor market competition.22 In particular, we assume there
are two risk-neutral firms, and two workers.23 One worker is selfish (​ηS​​ = 0) while
the other worker is reciprocal (​ηR​ ​ > 0) and we assume their types to be observable
to the firm. The two workers are otherwise identical. To allow scope for reciprocity, we assume that the reciprocal worker is sufficiently reciprocal that he would be
offered a reciprocity contract in the one-firm setting (i.e., η​ R​ ​ > ​η*​​ ). In order to focus
on the competition between the firms, we restrict attention to the two action case
from before and we will assume that the firms take turns making contract offers to
the worker of their choice, and the worker will accept the offer if it is superior to
his current offer. The firms stop when each has had an offer accepted by a different
worker. The sequential wage offers allow us to make a straightforward assumption about how the reciprocal worker interprets the intended generosity of a wage
offer: the generosity is equal to the difference in expected monetary wages (given
the requested action) of the new offer and the agent’s current accepted offer (or the
exogenous outside option in the case of the first wage offer).
Similar to models of efficiency wages, when there is sufficient friction in the
labor market reciprocity contracts remain quite powerful. We examine here the case
of match specific productivity, i.e., substantial differences in the workers’ abilities
to contribute to firms’ revenues (measured as differing ΔER). If the productivity
difference is large enough then the one-firm reciprocity contract remains optimal.
We expect that other forms of labor market frictions, such as search costs, firm
information advantages about worker ability, or equilibrium unemployment would
have similar effects.
A. Differences in Firm Productivity
One straightforward result is that the reciprocity contract can be sustained with
firm competition if one of the firms has a sufficient advantage in productivity. We
will model that here by assuming that the two firms are identical except that firm
one receives a greater increase in revenues from the agent taking the high action
(ΔE​R​1​ > ΔE​R2​​). Here Firm 1 has an advantage in providing reciprocal incentives
to the reciprocal agent, as for the same cost of effort the agent can give a larger gift
back to the firm (and therefore will receive a larger increase in his utility). For intermediate levels of reciprocity, this means that Firm 2 would not offer the r­ eciprocal
See also Kosfeld and von Siemens (2011) for an analysis of labor market competition with employees who
are conditional cooperators.
23 This is similar to the assumption of excess labor supply common to many models of efficiency wages. Note
that we make the slightly weaker assumption of no excess demand for labor.
22 168
American Economic Journal: Microeconomicsmay 2012
worker a reciprocity contract even in the one-firm setting, and is therefore indifferent between employing the two workers. Therefore, if Firm 1 has a sufficient
advantage, we can have an equilibrium with a reciprocity contract. For what follows,
let η​ ​  *1​​  denote the threshold from which on a reciprocity contract is optimal in the
one-firm case.
​  E​R1​​/ΔE​R2​​then there exists an equilibrium
Proposition 10: If η​ ​  *1​ ​  < ​ηR​ ​  < ​η​  *1​Δ
where Firm 1 offers the reciprocal worker the one-firm reciprocity contract, and
Firm 2 offers the selfish worker the one-firm standard contract.
Proof:
As noted previously, because ηΔER always appear together, they are jointly
determined. Therefore, the reciprocity thresholds for the one-firm problem for each
​  E​R1​​  = ​η​  *2​Δ
​  E​R2​​. Thus, if ​η​  *1​ ​  < ​ηR​ ​  < ​η​  *1​ ​  ΔE​R1​​/ΔE​R2​​ only
firm are such that ​η​  *1​Δ
Firm 1 can benefit from offering the reciprocal worker the reciprocity contract.
Therefore, it is an equilibrium for the two firms to divide the workers and offer each
the one-firm optimal contract.
B. Competition with Identical Firms
However, the reciprocity contract can also be sustained even under competition
between ex ante identical firms (i.e., ΔE​R​1​ = ΔE​R2​​). When one firm offers the
reciprocal agent a contract with a positive rent, this makes it more expensive for
the other firm to offer a competing contract that would generate reciprocal incentives. Not only must the second firm offer a contract that is sufficiently attractive
in total utility for the agent to accept it, the reference for generosity in monetary
utility is also higher. Because the agents have concave utility functions, providing
the same monetary rent in utility terms requires a larger increment in wages. If
the monetary rents of the first contract are sufficiently high, then the second firm
may prefer to employ the selfish agent rather than try to win the services of the
reciprocal agent.
To be more formal, we can represent the implementation problem for the second
firm offering a contract to the reciprocal agent who already has a contract offer that
_
provides expected monetary utility, inclusive
of effort costs, of ​​ u ​​ (which is being
_
compared to a generosity benchmark of ​​ ​u ​​​0 ​). The second firm then needs to solve the
following problem:
 
 )h(​ui​​)
  
  ​ ​∑ ​  ​​​  πi​ ​(​ a​
​
m​uin 
​​ 
i
 
 
(
)
 
_
 )​ui​​ + η ​∑ ​  ​​​  πi​ ​(​ a​
 )​ui​​ − c(​ a​
 ) − ​ 
 ) − c(​ a​
 )
​u ​​  ER(​ a​
s.t. ​∑ ​  ​​​  πi​ ​(​ a​
 
_
_
 
_
 )
≥ ​ ​u ​​   + η(​ ​u ​​   − ​​ ​u ​​​0 ​) ER(​ a​
 
(
 
)
_
 ) − ​πi​​(​a​1​)) ​ui​​ + η ​∑ ​  ​​​  πi​ ​(​ a​
 )​ui​​ − c(​ a​
 ) − ​ 
∑ ​  ​​  (​πi​​(​ a​
​u ​​  (ΔER) ≥ Δc,
and ​
 
 
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englmaier and leider: Reciprocity and Contracts
169
_ _
where the value function at the optimal contract is V
​ ​2​(η, ​ ​u ​​,  ​​ ​u ​​​0 ​). Note that if
_
_
_
​ ​u ​​   = ​​ ​u ​​​0 ​ = ​
u ​
this
problem
corresponds
to
the
one-firm
problem. It is clear that
 
_
increasing ​ ​u ​​ tightens both constraints, making fewer contracts feasible. Therefore,
we have the following lemma:
 ) implements ​ a​
 given
contract with expected
Lemma 3: If a contract
(​w​i​, ​ a​
_
_ a current
_
  for all ​​ u ​​′  ≤ ​ ​u ​​. 
monetary utility ​​ u ​​,  then it implements ​ a​
This immediately implies:
_ _
_
_
Proposition 11: ​V2​​(η, ​ ​u ​​,  ​​ ​u ​​​0 ​) is nondecreasing in ​​ u ​​ for all η, ​​ ​u ​​​0 ​.
_
We can then define a parameter ​​​ u ​​​* ​(η) which is the minimal monetary utility of the
existing contract sufficient to make the second firm indifferent
_ _ between _employing
_
the reciprocal agent and the selfish agent, i.e., so that ​V2​​(η, ​​ ​u ​​​* ​, ​u ​)  = ​V​2​(0, ​u ​,  ​u ​) . If the
first firm offers the reciprocal agent a contract that provides him with at least as much
_
monetary utility as ​​ ​u ​​​* ​then the second firm will prefer to hire the selfish worker.
For intermediate values of ​η​R​, it will be the_ case that the one-firm reciprocal
contract will provide at least as much utility as ​​​ u ​​​* ​. The following example captures
such a situation.
Example 1: Let u(​w​i​) = ln(​wi​​), and ​ηR​ ​ = 1. Suppose there are two effort levels with c(​a​1​) = 0 and c(​a​2​) = 3, and two output levels: ​qL​ ​ = 5 and ​q​H​ = 10,
with Pr(q = 10 | ​a​1​) = ​ _13 ​ and Pr(q = 10 | ​a2​​) = ​ _23 ​.  Lastly, suppose that the exog_
enous initial outside option is ​u ​  = 1. The selfish worker can be induced to implement the high action with a contract providing utility of (​u​1​ = − 2,  ​u2​​ = 7) with
an expected wage of 731.14. The one-firm reciprocal contract provides utility
(4.78, 5.96) with an expected wage bill for the principal of 297.31. This contract
gives the agent an expected monetary utility of 2.56, and an expected total utility
of 15.60. If the second firm wants to offer the reciprocal agent a reciprocity contract, it would need to offer a contract that provides sufficient utility for the agent
to accept, and provide enough additional rent to induce reciprocal incentives (as
the selfish agent is now a less expensive worker to use for a standard contract).
The least-cost reciprocity contract is (6.34, 7.52), with an expected wage bill of
1,420.94. Hence, in equilibrium, the first firm will offer the reciprocal agent the
one-firm reciprocity contract, and the second firm will offer the selfish agent the
one-firm standard contract.
However, for larger values of ​ηR​ ​ the one-firm reciprocity contract does not
increase the costs for the second firm enough: both because it provides a smaller
rent to the agent, and because the value to the second firm of hiring the reciprocal
agent increases. In the above example, if η​ ​R​ = 10 then at the one-firm reciprocal
contract the second firm would still prefer to hire the reciprocal worker than the
selfish. When this is the case, the first firm will want to offer a contract that provides
a sufficiently large rent to the agent that it prevents a counter-offer from the second
firm. The first firm’s problem is therefore:
170
American Economic Journal: Microeconomicsmay 2012
 
 
 
 )h(​ui​​)
​  
 
  ​ ​
 ∑ 
 
 ​  ​​​  πi​ ​(​ a​
min ​u​i​
 
 
_
 )​ui​​ − c(​ a​
 ) − ​u ​   ≥ 0 s.t. ​
∑  ​  ​​​  πi​ ​(​ a​
 ) − ​πi​​(​a1​​)) ​ui​​ + η ​∑ ​  ​​​  πi​ ​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​  (ΔER) ≥ Δc
and ​∑ ​  ​​  (π
​ i​​(​ a​
 )​ui​​ − c(​ a​
 )  ≥ ​​ ​u ​​​* ​
and ​∑ ​  ​  ​​πi​ ​(​ a​
 
 
 
 
 
_
(
 
 
)
_
_
with value function ​V1​​(η, ​​ ​u ​​​* ​). If the third constraint is slack then the one-firm contract is sufficient. If it binds then the contract will have to offer a higher expected
utility to the agent than the one-firm contract. This will lead to a greater use of reciprocal incentives and less use of monetary incentives—i.e., a flatter wage. Of course,
these additional rents are costly to the first firm. We therefore must demonstrate that,
for any ​η​R​, the additional rents required are small enough that the first firm would
still want to hire the reciprocal agent.
Proposition 12: For any η​ R​ ​ > ​η*​​there exists an equilibrium where the first firm
offers the reciprocal agent a reciprocity contract and the second firm offers the selfish agent the one-firm standard contract.
Proof:
The solution to the first firm’s problem is the least cost strategy to hire the reciprocal agent: it is the cheapest contract that prevents a counter-offer. Analogously,
any strategy that required the first firm to counter an offer of the second firm’s would
be even more expensive. We therefore need to show that the first firm prefers to hire
the reciprocal worker rather than the selfish worker. If the third constraint is slack
(i.e., the one-firm contract suffices) then this is clearly true. This will also be true
if the third constraint binds because the first firm always has a lower cost with the
_
reciprocal worker than the second firm, and ​​ ​u ​​​* ​is defined such that the second firm
is indifferent between hiring the reciprocal worker and the selfish worker. To see
this, note that the contract the second firm would use to hire the reciprocal worker
is a feasible solution to the first firm’s problem, however the incentive constraint
_
_
to the_first firm’s problem
would be slack because ​​​ u ​​​* ​ > ​u ​.  Therefore the solution
_
_
_ _
has a lower expected wage bill, and we have V
​ 1​​(η, ​​ ​u ​​​* ​) < ​V2​​(η, ​​ ​u ​​​* ​, ​u ​)   = ​V​2​(0, ​u ​,  ​u ​)  .
Therefore the first firm prefers to offer a contract to the reciprocal agent.
This result means that reciprocity contracts can be optimal even when two ex
ante identical firms are competing for a single reciprocal worker. Furthermore,
­competition can lead to larger rents going to the reciprocal worker compared to
the one-firm case. This means that unlike in the one-firm case, increased reciprocity diminishes the profit of the first firm, as the size of the rents necessary to preempt the second firm increase. However, even in the limit as η​ ​R​  → ∞ the first firm
receives a benefit from hiring the reciprocal worker.
Proposition 13:
_
_
​V​1​(η, ​​ ​u ​​​* ​) → h(​​ ​u ​​​* ​).
As
η​
​ ​  → ∞,
R
_*
_ _
_
 )]
​​​  u ​​​ ​  → ​ _12 ​[ u(​V2​​(0, ​u ​,  ​u ​)  ) + ​u ​  − c(​ a​
and
Vol. 4 No. 2
englmaier and leider: Reciprocity and Contracts
171
Proof:
See Appendix B, part C.
Obviously, this is not to claim that it will be possible to profitably employ reciprocal motivation under all circumstances. The above analysis also highlights that
there are limits to the robustness of our initial results. In a totally frictionless market
with ex ante identical firms and excess labor demand it seems reasonable to expect
that firms will bid up wages to the workers’ expected marginal products, leaving no
scope for reciprocal motivation.
IV. Conclusion
The importance of fairness and social preferences, especially for the work relation, has long been documented. We focus on reciprocity, a widely studied form of
social preferences, particularly in the experimental literature. Despite its prominence
in experimental economics, there is so far little theoretical work on how reciprocity
affects the employment relation in the presence of moral hazard.24 We solve for the
optimal contract in the basic principal-agent problem and show that reciprocal motivations and explicit performance-based pay are substitutes, and that using reciprocal
incentives decreases agency costs. Furthermore, we show that the optimal contract
entails an optimal mix of both incentive forms: using explicit incentives less when
the agent is more reciprocal, and using them more when output is a more informative
signal of effort. Our results highlight the importance of the magnitude of the benefit to
the principal from high effort in enhancing the role for reciprocal incentives.
This last finding allows us to extend our model to capture features of organizational structure. We predict a collocation of incentive pay and decision rights in a
firm’s hierarchy in firms that try to use reciprocal incentives. Furthermore, in those
firms incentive pay for non-managers should be less prevalent, i.e., they rely more
on reciprocal incentives, and the scope of performance pay for managers should be
more narrowly defined. We also consider reciprocity contracts in the presence of
labor market competition between employers. We find that reciprocal incentives are
viable when ex ante identical firms compete, however a reciprocity contract is most
attractive when the agent has a large productivity advantage at one firm.
Together our results suggest that reciprocal incentives can vary in usefulness
depending on the characteristics of the job and the firm. We therefore suggest
that future empirical attempts to measure the effect of gift exchange in the field
focus on settings where reciprocity contracts are most powerful: when production
is noisy, when workers are highly productive, when firms can screen effectively
for reciprocal workers, when managers have strong explicit incentives, and when
labor market competition is weak. Studying these settings may help clarify the
existing empirical literature.
24 Dufwenberg and Kirchsteiger (2000) is a notable exception. They study wage setting games with non-contractible effort choices and show that employers refuse to hire workers who offer their services at less than the prevailing wage. Note that this finding corresponds to our assumption that rules out destructive reciprocity equilibria.
Moreover, they also assume that principals are not motivated by reciprocity. However, as opposed to our study, they
do not consider state contingent incentive contracts.
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American Economic Journal: Microeconomicsmay 2012
Appendix A: The Multiple Action Case
A. General Results without Additional Distributional Assumptions
Consider the case where the agent is choosing between a set of actions
a​ ​1​ < ​a2​​ < ⋯ < ​am​ ​. We assume that c(a) is increasing and convex in a, and that
ER(a) is increasing and concave in a. We begin by considering which results extend
from the two action case without any further assumptions on the distribution of output. As before, we focus on the implementation problem. The principal’s problem
to implement ​ak​​is
 
  
∑  ​    ​​πi​ ​(​a​k​)w(​qi​​)
​ 
min ​ ​
 
w(​qi​​), ​ a​
 
_
 ) ≥ ​u ​;  [IC] U(​a​k​, ​ a​
 ) ≥ U(a′, ​ a​
 ) ∀a′; [EB] ​ a​
  = ​ak​​.
s.t. [IR] U(​ak​​, ​ a​
Substituting in a​
​   = ​a​k​to satisfy [EB] and transforming the [IR] and [IC] we get
 )h(​ui​​)
  
  ​  ​∑  ​​π
​  ​i​​(​ a​
​
m
in 
 
​u​​
i
_
 )​ui​​ − c(​ a​
 ) − ​u ​   ≥ 0
​  ​i​​(​ a​
s.t. ​∑    ​​π
(
)
_
 ) − ​πi​​(a′ )) ​ui​​ + η ∑ ​π​i​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​  (ER(​ a​
 ) − ER(a′ ))
s.t. ∑ (​πi​​(​ a​
 ) − c(a′ )∀a′.
≥ c(​ a​
With the FOC’s analogous to the two-action case,
[
]
​π​i​(​a​k​)
 ) − ER(​ak​​)) .
​ −k​​ 1 − ​ _
 ​ 
 
+ η(ER(​ a​
h′(​ui​​) = ​λI​R​ + ​∑   ​ ​​  λIC
 )
​πi​ ​(​ a​
k
First, notice that Observations 1—optimality of the standard contract for
η = 0—and 2—the standard contract is implementable for all η—extend directly
from the two-action case. For η = 0 the agent is selfish, and in general the standard
contract provides zero rent to the agent, so the reciprocal portion of his utility drops
 for any η. However, if ​ a​
 is not
out. Therefore the standard contract will implement ​ a​
 for a given ​η1​​
the highest action, it is not the case that any contract that implements ​ a​
will still implement a​
​   for any η​ 2​​ > ​η1​​. This is because the reciprocal portion of the
agent’s utility for an upwards [IC] is negative: working harder would give a larger
gift back to the principal. Hence, if an upwards [IC] is binding, then increasing η
would make the agent strictly prefer the higher action. However, we can show that
for any increase in η, a translation of the initial contract where all the utility payments are shifted down by a constant δ will still implement a​
​  .
  given η​ 1​​ then for any η​ ​2​ > ​η1​​, ∃ δ  ≥  0 such that
Lemma 4: If ​ui​​ implements ​ a​
​  .
​vi​​ = ​ui​​ − δ implements a​
englmaier and leider: Reciprocity and Contracts
Vol. 4 No. 2
(
)[
]
173
Proof:
​η​1​
_
 )​ui​​ − ​u ​  − c(​ a​
 ) . For the [IR] we have
Let δ =  1 − ​ _
​η2​  ​ ​  ∑ ​π​i​(​ a​
_
[
]
_
 )​vi​​ − c(​ a​
 ) − ​u ​   = ∑ ​π​i​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​  − δ
∑ ​πi​ ​(​ a​
[
]
_
 )​ui​​ − c(​ a​
 ) − ​u ​  = ∑ ​π​i​(​ a​
(
)[
 
​η​1​
_
 )​ui​​ − ​u ​   − c(​ a​
 )
− 1 − ​ _
​   ​    ​​πi​ ​(​ a​
​η2​  ​ ​  ∑
( )[
 
 
]
​η​​
_
 )​ui​​ − ​u ​   − c(​ a​
 )  ≥ 0.
= 
_
​  ​η1​  ​​   ​∑ ​    ​​πi​ ​(​ a​
2
 
]
Hence, we can hold constant the value the agent places on the rent by scaling
down the rent by the ratio of ​η1​​to ​η2​​.
And for any [IC] we have
 ) − ​
 )​vi​​ − c(​ a​
 ) − ​u ​)
 ) − ER(a′ ))
∑(​πi​​(​ a​
πi​​(a′ )) ​vi​​ + ​η2​​(∑ ​πi​ ​(​ a​
  (ER(​ a​
_
 ) − c(a′ ))
− (c(​ a​
( )[
]
​η​1​
_
 ) − ​πi​​(a′ )) ​ui​​ + ​η2​​ ​ _
 )​ui​​ − c(​ a​
 ) − ​u ​ 
=  ∑(​πi​​(​ a​
​η2​  ​ ​  ∑ ​π​i​(​ a​
 ) − ER(a′ )) − (c(​ a​
 ) − c(a′ ))
× (ER(​ a​
[
]
_
 ) − ​πi​​(a′ )) ​ui​​ + ​η1​​ ∑​π​i​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​ 
=  ∑(​πi​​(​ a​
 ) − ER(a′ )) − (c(​ a​
 ) − c(a′ )) ≥ 0.
× (ER(​ a​
Since η times the rent is held constant, as is the spread of wage payments across
states, if the first contract meets the [IC] then the second contract will as well.
Thus a cheaper contract with the same explicit incentives will implement a​
​  . This
immediately implies that the cost to the principal of the second-best optimal contract is nonincreasing in η, as it was in the two-action case.
 ) is nonincreasing in η for all ​ a​
 .
Proposition 14: V(η, ​ a​
Proof:
For any ​η2​​ > ​η1​​, apply Lemma 1 to the optimal contract given ​η1​​. Therefore, the
optimal contract under η​ ​2​must have at least as small a cost to the principal as under η​ 1​​,
and will have strictly lower cost if the optimal contract under ​η1​​yields a positive rent.
Observation 3—there exists a high enough flat wage that implements an effort
level for η > 0—and 4—for η → ∞ the First Best is arbitrarily closely approximated with a flat wage—also extend, though we have to be careful that the flat
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American Economic Journal: Microeconomicsmay 2012
wage does not provide too large a rent (which would induce the agent to change to a
higher action). Since effort costs are convex and expected revenue are concave, for
_
˜  the two relevant ICs when a​
 ) + ​ u​
​   = ​a​k​are:
a flat wage u​ i​​ = ​u ​  + c(​ a​
˜ (ER(​ a​
 ) − ER(​ak−1
 ) − c(​ak−1
​ ​)) ≥ c(​ a​
​ ​)
η​  u​
˜ (ER(​ a​
 ) − ER(​ak+1
 ) − c(​ak+1
​ ​)) ≥ c(​ a​
​ ​).
and η​  u​
 )
 ) − c(​ak​−1​)
c(​ak​+1​) − c(​ a​
c(​ a​
˜  ≥ ​ __
 if __
  
  
 ​≥ ​ u​
  
  
 ​
.
Therefore, the flat wage will implement ​ a​
​ η(ER(​
 ))
 ) − ER(​a​ ​))
a​ ​) − ER(​ a​
η(ER(​ a​
k+1
k−1
The interval is nonempty due to the the convexity of c and the concavity of ER, demonstrating Observation 3, and the lower limit shrinks to zero in the limit as η → ∞,
demonstrating Observation 4.
As in the two-action case, the lowest effort level can be implemented with a flat
wage, and has the same cost as in the first best.
_
 ) is the optimal contract for all η.
  = ​a1​​, then ​ui​​ = ​u ​  + c(​ a​
Lemma 5: If ​ a​
Proof:
The [IR] holds with equality by construction. For each [IC] we have
 ) − ​πi​​(a′ )) ​ui​​ + η(∑ ​πi​ ​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​)
 ) − ER(a′ ))
∑(​πi​​(​ a​
  (ER(​ a​
_
_
 ) − c(a′)
≥ c(​ a​
(_
)
_
 ))∑(​πi​​(​ a​
 ) − ​πi​​(a′ )) + η (​u ​   + c(​ a​
 ))∑ ​πi​ ​(​ a​
 ) − c(​ a​
 ) − ​u ​ 
(​u ​   + c(​ a​
 ) − ER(a′ )) ≥ c(​ a​
 ) − c(a′ )
× (ER(​ a​
 ) − c(a′ ),
0 ≥ c(​ a​
 ) < c(a′ ) for all a′. Since this contract has a cost to the prinwhich is true since c(​ a​
cipal equal to the first-best cost, it is optimal.
Next, we can show the standard result that for any optimal contract at least one
downward [IC] will bind. Since both explicit and reciprocal incentives are costly,
this ensures that at the optimum the principal is not providing excessive incentives.
  we have
Lemma 6: At the optimum, for at least one a′ < ​ a​
 ) − ​πi​​(a′ )) ​ui​​ + η(∑ ​π​i​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​)
 ) − ER(a′ ))
∑(​πi​​(​ a​
  (ER(​ a​
_
 ) − c(a′ ).
= 
c(​ a​
Proof:
Suppose instead that all downward [IC]s are slack. Then, since the principal’s
problem is a convex problem, we can drop all of these constraints without affecting
englmaier and leider: Reciprocity and Contracts
Vol. 4 No. 2
175
  is the lowest-cost action, and
the optimal solution. However, in this new problem ​ a​
therefore a flat wage such that the [IR] binds is optimal. However, in the full problem
 —therefore the assumption leads to a contradiction.
this contract will not implement ​ a​
Lastly, we can show that for high enough η, the optimal contract is a reciprocity
contract.
Proposition 15: There exists a finite η​ ​*​ such that for all η ≤ ​η​*​ the optimal
contract is a standard contract (i.e., provides zero rent) and for all η > ​η​*​ the
optimal contract is a reciprocity contract (i.e., provides the agent with a strictly
positive rent).
Proof:
As before, we provide an upper bound on η​ ​*​by showing that a flat wage can
 strictly cheaper than the standard contract. Let ​us​t​be the standard contract,
induce ​ a​
˙  − ϵ) = ​ u​
˙  for
 )h(​us​t, i​). Again we consider the flat wage ​ui​​ = u(​ w​
and let w​
​ ˙  = ∑​πi​ ​(​ a​
some small ϵ. As before the [IR] will be strictly satisfied for ϵ small enough, and to
  = ​ak​​).
satisfy the [IC]s we simply must satisfy the [IC]s for ​ak​−1​and ​ak​+1​ (where ​ a​
_
˙  − c(​ a​
 ) − ​u ​)  (ER(​ a​
 ) − ER(​ak​−1​)) ≥ c(​ a​
 ) − c(​ak​−1​)
η(​ u​
_
˙  − c(​ a​
 ) − ​u ​)  (ER(​ a​
 ) − ER(​ak​+1​)) ≥ c(​ a​
 ) − c(​ak​+1​)
η(​ u​
which implies
 )
 ) − c(​ak​−1​)
c(​ak+1
​ ​ − c(​ a​
c(​ a​
___
___
 ​     
   
 ​ ≥ η ≥ ​ 
    
   
 ​ .
_
_
˙  − c(​ a​
˙  − c(​ a​
 ) − ​u ​)  (ER(​ak​+1​) − ER(​ a​
 ))
 ) − ​u ​)  (ER(​ a​
 ) − ER(​ak​−1​))
(​ u​
(​ u​
The above condition will be satisfied for some finite η. Therefore, the optimal
­positive-rent contract will have strictly lower cost than the optimal zero-rent contract (i.e., the standard contract), hence the optimal contract will be a reciprocity
contract. For larger η, we know from Lemma 4 above, even smaller flat wages will
continue to implement a​
​  .
B. Assuming CDFC
We can extend all of the results from the two-action case if we can find assumptions
that ensure that at the optimum only one [IC] binds (and therefore the FOC’s have
the same structure). To do so we need to demonstrate that for the optimal contract
the agent’s effort decision is a concave problem. Fortunately, the combination of the
Monotone Likelihood Ratio Property (MLRP) and the Convexity of the Distribution
Function Condition (CDFC), introduced by Rogerson (1985), is sufficient.
  ​  ​​​
 πi​ ​(a) denote the distribution function at state k given the action a.
Let F
​ ​k​(a) = ​∑ i=1
k
Definition 4: Convexity of the Distribution Function Condition (CDFC):
CDFC holds if for a < a′ <  a″ and for some α where 0 ≤ α ≤ 1 we have
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American Economic Journal: Microeconomicsmay 2012
c(a′ ) = αc(a″ ) + (1 − α)c(a) then that implies ​F​k​(a′ ) ≤ α ​F​k​(a″ ) + (1 − α)​Fk​​(a)
∀k = 1, … , n.
Note that if CDFC holds, then ER(a) is concave in a.
​  i​ ​​(a′)​qi​​ = ​∑    ​​F
​  i​​​(a′ )(​qi​​ − ​qi​+1​) + ​qn​​
ER(a′ ) = ​∑    ​​π
(​
)
(​
)
  ​ ​​Fi​​​(a″ )(​q​
q​i+1​) + ​q​n​
≥ α ∑   ​ ​​Fi​​​(a)(​q​i​ − ​q​i+1​) + ​q​n​ + (1 − α) ∑
i​ − ​
 
= αER(a) + (1 − α) ER(a″ ).
 
We can then prove that with MLRP and CDFC, the optimal contract will be
nondecreasing.
Proposition 16: If MLRP and CDFC hold, then the optimal incentive scheme u​ ​  *i​  ​
satisfies ​u​1​ ≤ ​u2​​ ≤ ⋯ ≤ ​un​​.
Proof:
Consider the optimal contract for the relaxed problem where the agent can only
 }. We know at the optimum of this problem that
choose an action from {a | a ≤ ​ a​
 
(
)
​π​i​(​ak​​)
 ) − ER(​ak​​)) .
 ​ 
 
+ η(ER(​ a​
h′(​ui​​) = ​λI​R​ + ​∑ ​  ​​​  λk​ ​ 1 − ​ _
 )
​πi​ ​(​ a​
 
Since MLRP holds, at the optimum u​ i​​ is nondecreasing. To show that this contract
coincides with the optimal contract, it suffices to show that all of the [IC]s for higher
actions are satisfied.
Suppose to the contrary that a higher action is strictly preferred, i.e., for some
  we have
a″ > ​ a​
 
_
 )​ui​​ + η(​πi​​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​)  ER(​ a​
 ) − c(​ a​
 )
∑  ​ ​  ​​πi​ ​(​ a​
​
 
 
_
 )​ui​​ − c(​ a​
 ) − ​u ​)  ER(a″ ) − c(a″ ).
< ​∑ ​  ​​​  πi​ ​(a″ )​u​i​ + η(​πi​​(​ a​
 
  we have
We know that a downward [IC] binds, i.e., for some a′ < ​ a​
_
 )​ui​​ + η(​πi​​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​)  ER(​ a​
 ) − c(​ a​
 )
​
∑   ​ ​​πi​ ​​(​ a​
 
 
_
 )​ui​​ − c(​ a​
 ) − ​u ​)  ER(a′ ) − c(a′ ).
= ​∑ ​  ​​​  πi​ ​(a′ )​ui​​ + η(​πi​​(​ a​
 
englmaier and leider: Reciprocity and Contracts
Vol. 4 No. 2
177
 ) = αc(a″ ) + (1 − α) c(a′ ), and from CDFC this
Also, for some α we have c(​ a​
 ) ≤ α​Fk​​(a″ ) + (1 − α)​Fk​​(a′ ). Therefore
implies F
​ k​​(​ a​
 
_
 )​ui​​ + η(​πi​​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​)  ER(​ a​
 ) − c(​ a​
 )
​
∑  ​ ​​​  πi​ ​(​ a​
 
 
_
 )(​ui​​ − ​u​i+1​) + ​u​n​ + η(​πi​​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​)  ER(​ a​
 ) − c(​ a​
 )
= ​∑ ​ ​​​  Fi​​(​ a​
 
[​
 
_
 )​ui​​ − c(​ a​
 ) − ​u ​)  ER(a″ ) − c(a″ )]
≥ α ∑ ​ ​​​  Fi​​(a″ )(​ui​​ − ​u​i+1​) + ​un​​ + η(​πi​​(​ a​
 
[
_
]
 ) − ​u ​)  ER(a′ ) − c(a′ )
+ (1 − α) ∑ ​Fi​​(a′ )(​ui​​ − ​ui​+1​) +  ​un​​ + η(​π​i​(a​
​   )​u​i​  −   c(​ a​
[​
_
]
 )​ui​​ − c(​ a​
 ) − ​u ​)  ER(a″ ) − c(a″ )
= α ∑  ​ ​​  πi​ ​​(a″ )​ui​​ + η(​πi​​(​ a​
 
[​
 
_
]
 )​ui​​ − c(​ a​
 ) − ​u ​)  ER(a′ ) − c(a′ ) .
+ (1 − α) ∑ ​ ​​​  πi​ ​(a′ )​ui​​ + η(​πi​​(​ a​
 
However, this contradicts the assumption that a″ is strictly preferred.
Additionally, since the optimal contract will be monotone, this ensures that the
agent’s problem is concave. Hence, only the local downward [IC] will bind.
Proposition 17: If MLRP and CDFC hold, then at the optimum for implement  = ​ak​​, only the [IC] for ​ak​−1​binds.
ing ​ a​
Proof:
_
 )​ui​​ − c(​ a​
 ) − ​u ​)  ER(a) −
The agent’s problem is ma​
x​   a​  ∑ ​πi​ ​(a)​ui​​ + η(∑ ​πi​ ​(​ a​
 
 )​ui​​ −
c(a) which is equivalent to ma​
x​  a​  ∑ ​Fi​​(a)(​u​i​ − ​ui​+1​) + ​un​​ + η(∑​πi​ ​(​ a​
_
 ) − ​u ​)  ER(a) − c(a). Since we know u​ ​i​is nondecreasing, therefore the first term
c(​ a​
is concave by CDFC. Similarly, we showed that ER(a) is concave in a, and c is
convex. Therefore the agent’s problem is concave, and so the local downward [IC] is
sufficient for a global maximum.
Since only one [IC] is binding, then the multi-action case is equivalent to the two  = ​ak​​and replacing ​a1​​with ​a​k−1​. Therefore all our results from
action case where ​ a​
the two-action model extend.
Corollary 6: If MLRP and CDFC hold, then Propositions 2 to 5 contracts and
Corollary 2 holds.
Appendix B: Additional Proofs
A. Proof of Proposition 2 and Proposition 15: Existence of ​η​*​
By construction the Principal’s problem is a convex minimization with linear
constraints. Therefore, for a given set of parameters, the optimal solution is unique.
178
American Economic Journal: Microeconomicsmay 2012
Moreover, by Proposition 1 (for the two-action case) or Proposition 14 (for the multiple-action case) we know that the value function for the principal’s problem is nonincreasing in η. We will show below in Proposition 18 that the value function V(η)
is absolutely continuous25 in η, for the moment suppose this result holds. Since we
know from the sketch of the proof in the main text that there is an η large enough that
the value function is strictly smaller than for the standard η = 0 case, given absolute
continuity, there must be some ​η*​​such that for η ≤ ​η*​​ V(η) = V(0) (and therefore the
standard contract is optimal), and for η > ​η*​​it holds that V(η) < V(0) (and therefore a
reciprocity contract is optimal). Moreover, since the solution for any η is unique, ​η​*​ is
the point such that the optimal contract derived by ignoring the [IR] coincides with the
standard contract. Additionally, when η > ​η*​​we know that the optimal contract provides a strictly positive rent. Hence, in this region, we know that if we solve for the optimal contract by ignoring the [IR], we know that the solution will in fact meet the [IR].
Therefore to complete the proof, we prove the absolute continuity of V for the
multiple action case.
Proof of Absolute Continuity:
Recall that in the multi-action case we have: N is the number of states, M is the
number of actions, a ∈ A is the set of actions and u ∈ (u(​w​
_ ), ∞) is the set of utilities.

The Principal’s Implementation Problem for a​
​    is the program P:
 
 
_
 )h(​ui​​) s.t. [IR] ​∑   ​ ​​  πi​ ​​(​ a​
 )​ui​​ − c(​ a​
 ) − ​u ​   ≥ 0
​ max    ​  − ​∑   ​ ​​  πi​ ​​(​ a​
 
 
u
i
(​
i
) (​
_
) (​
i
)
 )​ui​​ − c(​ a​
 ) + η ∑   ​  ​ ​πi​ ​​(​ a​
 )u​ i​​ − c(​ a​
 ) − ​u ​  ∑   ​ ​​  πi​ ​​(​ a​
 )​qi​​
  and [IC ] ​∑   ​ ​​  πi​ ​​(​ a​
i
(​
i
_
)
 )​ui​​ − c(​ a​
 ) − ​u ​  ∑   ​ ​​  πi​ ​​(a)​q​i​ ∀a ∈ A.
≥ ∑
​    ​ ​​  πi​ ​​(a)​u​i​ − c(a) + η ∑   ​ ​​  π​i​​(​ a​
i
i
i
 )h(​u​  *i​  ​)) denote the value function, where ​u*​​is a
 ) = − ​  
  ​(− ​∑
  ​  
​ ​πi​​​(​ a​
Let V(η; ​ a​
max 
u
i
solution to P.
Proposition 18: V(η) is absolutely continuous in η.
Proof:
_N
_
  ​ where ​ U​  is a
Consider the constrained program P′ where u ∈ U ′ = (u(w), ​U​ ​]​
finite number such that
_
 ) > − ​
​h(​ U​ ) − (1 − ​
π​) _
​w​   ∀ a ∈ A,
V(0; ​ a​
π 
_
_
_
​ ​ ​πi​​(a) i.e., the smallest state probability. Hence, U​
​    is a payment
where _
π
​ ​  =  mi​na, i
_
sufficiently large so that the cheapest possible contract involving a payment of U​
​   is
25 of ​η​*​.
Although Proposition 18 provides us with absolute continuity, only continuity is necessary for the existence
englmaier and leider: Reciprocity and Contracts
Vol. 4 No. 2
179
strictly dominated_ by the standard contract. Hence, any contract involving any payment larger than U​
​   will also be strictly dominated.
Therefore the program P′ is equivalent to P, since their solution sets will be identical (any contract feasible in P but not in P′ is strictly dominated by a contract that
is feasible in both). Hence, V′(η) = V(η).
_ N
u​ , U​ ​]​
​   ​ and where
Now _
consider the constrained program P″ where u ∈ U ′′ =  [ ​_
 . Any contract that involves a utility payment smaller
​    + ​_
π​​u_   ​ = ​u0​​ + ​ c​
(1 − ​_
π​)  U​
than ​u_ ​ violates the [IR], and thus is infeasible. Therefore the set of feasible contracts is identical between P′ and P ″, so the programs are equivalent. Hence,
V ″(η) = V′ (η) = V(η).
Since U ″ is a compact, convex set; the continuity, concavity and differential conditions on the objective function and the constraints are met, and a feasible point
exists where all the constraints are slack, the conditions of Milgrom and Segal
(2002, Corollary 5) are met, proving absolute continuity. Hence, V(η) is absolutely
continuous in η, and thus the existence of η​ ​*​follows.
B. Proof of Proposition 5
Proposition 5: If G is an MPS of F, ΔE​RF​ ​ = ΔE​RG​ ​, ​w*​​(G), ​x*​​(F ) are the
optimal contracts given the distribution of likelihood ratios, and η ≥ max [​η*​​(F ),
​ ​*​(G) ​≻m​ u​​  x​*​(F ). Additionally, if G is a balanced MPS of F, then
​η​*​(G)] then w
*
​w​​(G) ​≻m​ u, ​  i ​˘​ ​  x*​​(F ).
Proof:
From Rothschild and Stiglitz (1970), we say G differs from F by a single MPS if​
g​k​ = ​fk​​except for four states (denoted 1 to 4 for simplicity with ​z​1​ < ​z2​​ < ​z3​​ < ​z4​​ )
4
​  k=1
  ​ ​
   ​​zk​​(​g​k​ − ​fk​​)
and ​g​1​ − ​f1​​ = − (​g2​​ − ​f2​_​) ≥ 0, ​g4​​ − ​f4​​ = − (​g3​​ − ​f3​​) ≥ 0 and ∑
​   z4​​ − ​z3​​ (i.e., the change in the likelihood ratios).
= 0 Let δ​_ ​ = ​z2​​ − ​z1​​and δ​ = ​
From Lemma 1 in Rothschild and Stiglitz (1970) (with corrected proof in Leshno,
Levy and Spector 1997), there exists a sequence of distributions F
​ ​0​, … , ​FL​​such that​
F​0​ = F, ​FL​​ = G, and ​Fl​​differs from ​Fl​−1​by only a single MPS.
For each single MPS there is going to be a state j whose likelihood ratio
decreases, and another state i > j whose likelihood ratio increases (the likelihood
ratios of all the other states will stay the same). Therefore, because the lowest state
that changes has its likelihood ratio decrease and the highest state that changes
has its likelihood ratio increase, we can let ​ j ​˜  be the highest state such that for all
​π​  Fj​  ​
​π​  Gj​  ​
states j ≤ ​ j ​˜  we have 1 − ​ _F ​ ≥ 1 − ​ _G ​ and let j′ be the highest state j ≤ ​ j ​˜  such
that
​​ π​​  j​  ​
​π​  Fj​  ​
​π​  Gj​  ​
1 − ​ _
 ​ 
  > 1 − ​ _
 ​.  Similarly
​​ π​​  Fj​  ​
​​ π​​  Gj​  ​
​​ π​​  j​  ​
let ​ i ​˜  be the lowest state such that for all states
​π​  i​  ​
​π​  i​  ​
_
˜  such that
i ≥ ​ i ​˜  we have 1 − ​ _
 ​ ≤ 1 − ​ 
 ​ and let i′ be the lowest state i ≤ ​ i ​
​​ π​​  F​  ​
​​ π​​  G​  ​
F
G
i
i
​π​  i​  ​
​π​  i​  ​
˜  the likelihood ratio does
1 − ​ _
 ​ 
  < 1 − ​ _
 ​.  Therefore for any state lower than ​ j ​
​​ π​​  Fi​  ​
​​ π​​  Gi​  ​
not increase, and for any state higher than ​ i ​˜ the likelihood ratio does not increase.
Then for any pair of states (i, j) with i ≥ ​ i ​˜ and j ≤ ​ j ​˜ either (1) neither states’ likelihood ratio changed from F to G, (2) state i’s likelihood ratio increased and state j’s
F
G
180
American Economic Journal: Microeconomicsmay 2012
likelihood ratio stayed the same, (3) state j’s likelihood ratio decreased and state i’s
likelihood ratio stayed the same, or (4) state i’s likelihood ratio
increased* and state j’s
u′(​w​  *j​  ​(G))
u′(​w​  j​  ​(F))
_
​  
= ​ _
 
 
​ 
since the
likelihood ratio decreased. In the first case we will have ​ u′(​w​  *​  ​(G))  
u′(​w​  *i​  ​(F))
i
contracts will be the same for those states.
For the second case, we will have
​π​  i​  ​
_
_
​π​  Fi​  ​
 )
 )
1 − ​  G  ​  + ηΔER(​ a​
1 − ​ _F  ​  + ​
δ​ + ηΔER(​ 
 
a​
u′ w​  *j​  ​(G)
(​ π​​ )​  i​  ​
(​ π​​ )​  i​  ​
_
__
___
​  *
 ​ = ​    
  
=  ​    
   
   
 ​
​π​  Fj​  ​
​π​  Gj​  ​
u′ w​  i​  ​(G)
 )
 )
1 − ​ _  ​  + ηΔER(​ a​
1 − ​ _  ​  + ηΔER(​ a​
(​
(​
)
 
 ​
)
G
(​ π​​ )​  Gj​  ​
(​ π​​ )​  Fj​  ​
​π​  i​  ​
 )
1 − ​ _
  ​  + ηΔER(​ a​
u′(​x​  *j​  ​(F ))
 )​  Fi​  ​
(​ 
π​​
_
   
   
 
 ​  
.
> 
​ __
 ​
 
= ​ 
​π​  Fj​  ​
u′(​x​  *​i*​ ​(​ ​ F ))
 )
1 − ​ _F  ​  + ηΔER(​ a​
F
(​ π​​ )​  j​  ​
For the third case, we will have
​π​  i​  ​
​π​  i​  ​
 )
 )
  ​  + ηΔER(​ a​
1 − ​ _
1 − ​ _
  ​  + ηΔER(​ a​
*
G
u′
(​
w
​ 
​ 
(
​
G)
)
 
(​ 
π​​
)
​ 
​ 
​
(​ 
π​​ )​  Fi​  ​
j
i
_ ​
__
___
​ 
 ​
= ​ 
 
  
=  ​    
   
   
   
 ​
​π​  Fj​  ​
​π​  Gj​  ​
u′(​w​  *i​  ​(G))
_
 
 
1 − ​ 
 
 
​
 
− ​
δ
​ + ηΔER(​ 
 
a​
)
1 − ​ _
 
 
​
 
+ ηΔER(​ 
a​
)
_
F
G
G
(​ π​​ )​  j​  ​
F
(​ π​​ )​  j​  ​
​π​  i​  ​
 )
1 − ​ _
  ​  + ηΔER(​ a​
u′(​x​  *j​  ​(F ))
(​ 
π​​ )​  Fi​  ​
__
   
 ​  
 
.
> 
​    
 ​ = ​ _
F
​π​  j​  ​
u′(​x​  *i​  ​(F ))
 )
1 − ​ _F  ​  + ηΔER(​ a​
F
(​ π​​ )​  j​  ​
Lastly, for the fourth case, we will have
_
​π​  Gi​  ​
​π​  Fi​  ​
 )
 )
  ​  + ηΔER(​ a​
1 − ​ _
1 − ​ _
  ​  + ​
δ​ + ηΔER(​ 
 
a​
*
G
F
u′(​
w
​ 
​ 
(
​
G))
(​ π​​ )​  i​  ​
(​ π​​ )​  i​  ​
j
_
__
___
​ 
 ​
 
= ​ 
 
   
   
   
 ​
 ​  
= ​    
​π​  Fj​  ​
​π​  Gj​  ​
u′(​w​  *i​  ​(G))
_
 
 
1 − ​ 
 
 
​
 
− ​
δ
​ + ηΔER(​ 
 
a​
)
1 − ​ _
 
 
 
​
+ ηΔER(​ 
a​
)
_
F
G
(​ π​​ )​  j​  ​
(​ π​​ )​  j​  ​
​π​  i​  ​
 )
1 − ​ _
  ​  + ηΔER(​ a​
u′(​x​  *j​  ​(F ))
(​ 
π​​ )​  Fi​  ​
__
   
 
 ​  
,
> ​    
 ​ = ​ _
F
​π​  j​  ​
u′(​x​  *​i*​ ​(​​  F ))
_
 
1 − ​ 
 
 
 
​
+ ηΔER(​ 
a​
)
F
F
(​ π​​ )​  j​  ​
_
where ​δ_ ​  >  0 and  δ​ ​ >  0. One of the cases with strictness will apply if either i = i′
or j = j′.
˘ the lowest state above
Additionally, note that if G is a balanced MPS of F with ​ k​
˘ the likelihood ratio does not decrease and
the balance point, then for all states i ≥ ​ k​
˘
˘ the likelihood ratio does not increase
˜
therefore ​  i ​   = ​ k​ . Similarly, for all states j < ​ k​
˘
˜
and hence ​ j ​   = ​ k​  − 1.
englmaier and leider: Reciprocity and Contracts
Vol. 4 No. 2
181
C. Proof of Proposition 13
_
_ _
_
 )] and
Proposition
ηR​ ​  → ∞, ​​ ​u ​​​  *​  → ​ _12 ​[ u(​V2​​(0, ​u ​,  ​u ​)  ) + ​u ​  − c(​ a​
_*
_* 13: As ​
​V1​​(η, ​​ ​u ​​​ ​) → h(​​ ​u ​​​ ​).
Proof:
We first want to show that if ​η​R​ is sufficiently large then the second firm would
use a flat wage as its offer to the reciprocal agent. Specifically, if η​ R​ ​ is large
˜ ) that would satisfy the IR constraint with equality also
enough the flat wage (​ u​
_
_
˜  − c(​ a​
˜  + ​ηR​ ​(​ u​
 ) − ​ ​u ​​  )ER(​ a​
 ) − c(​ a​
 ) = ​ ​u ​​  +
satisfies
the
IC.
The
IR
gives us that ​ u​
_
_
 ), or
​ηR​ ​(​​  u ​​  − ​u ​)  ER(​ a​
 )
 )
1 + 2​ηR​ ​ER(​ a​
​ηR​ ​ER(​ a​
_ __
_
˜
 ) + ​ 
​  u​ = c(​ 
a​
​u ​​ ​     
   ​ − ​u ​ ​   _
  
 
 ​ .
 )
 )
1 + ​ηR​​ER(​ a​
1 + ​ηR​​ER(​ a​
Plugging this into the IC we get
(
)
 )
​η​ ​ER(​ a​
(​η​ ​ΔER) (​ ​u ​​ − ​u ​) (​  _
  
  ​) ≥ Δc
 )
1 + ​η​​ER(​ a​
_ 1 + 2​ηR​ ​ER(​ a​
_
 )
 )
​ηR​ ​ER(​ a​
_
(​η​R​ΔER) ​​  u ​​ ​   __
  
   ​ − ​u ​ ​   _
  
 
 ​ − ​ ​u ​​  ≥ Δc
 )
 )
1 + ​ηR​​ER(​ a​
1 + ​ηR​​ER(​ a​
_
 
R
_
_
 
R
R
_
 )ΔER(​ ​u ​​   − ​u ​)  − ​η​R​ER(​ a​
 )Δc − Δc ≥ 0.
​η​  2R ​ ​ER(​ a​
_
Let ​​ ​u ​​​O  F​ denote the average net expected utility that the one-firm optimal reciprocity contract provides, and note that in the model with competition we will always
_
_
have ​​  u ​​  ≥ ​​ ​u ​​​O  F​. Then we have
_
_
 )ΔER(​ ​u ​​   − ​u ​)  − ​η​R​ER(​ a​
 )Δc − Δc ​  R(​ a​
​η​  2R ​E
_
_
 )ΔER(​​ ​u ​​​O  F​ − ​u ​)  − ​η​R​ER(​ a​
 )Δc − Δc ≥  0.
​  R(​ a​
≥ ​η​  2R ​E
Therefore the IC will be satisfied if we have
____
_
_
_
 )Δc + ​√​(    
 )Δc)​2​ + 4ER(​ a​
 )ΔER(​​ ​u ​​​O  ​F − ​u ​)  Δc ​
 ER(​ a​
ER(​ a​
_
​η​R​ ≥ ​ _____
     
    
_  ​
 )ΔER(​​ ​u ​​​O  ​F − ​u ​) 
2ER(​ a​
_
˜  
˜ , the first firm will set ​ ​u ​​  such that ​ u​
Given that the second firm will offer ​ u​
_ _
= u(​V2​​(0, ​u ​,  ​u ​)  ), which implies that
_
 )
 )
 )
​ηR​ ​ER(​ a​
η​R​ER(​ a​
1 + ​η​R​ER(​ a​
_ _ 1 + ​
_
 )​ __
​​ ​u ​​​* ​ = u(​V2​​(0, ​u ​,  ​u ​) ) ​ __
  
  
  
  
  
  
 ​ − c(​ a​
 ​ + ​u ​​  __
 ​ .
 )
 )
 )
1 + 2​ηR​ ​ER(​ a​
1 + 2​ηR​ ​ER(​ a​
1 + 2​ηR​ ​ER(​ a​
182
American Economic Journal: Microeconomicsmay 2012
_
_ _
_
 )]. By a similar arguThis means we have li​m​   ​ηR​ ​→∞​​​​  
u ​​​* ​  = ​ _12 ​[ u(​V​2​(0, ​u ​,  ​u ​)  ) + ​u ​  − c(​ a​
ment as above, we can show that in the_first firm’s problem, if ​ηR​ ​ is large enough,
a flat wage providing
equal to ​​ ​u ​​​* ​ will satisfy the IC. Therefore we have
_*
_utility
*
 
​ 1​​(η, ​​ ​u ​​​ ​)  =  h(​​ ​u ​​​ ​).
li​m​  ​ηR​ ​→∞​​V
References
Akerlof, George A. 1982. “Labor Contracts as Partial Gift Exchange.” Quarterly Journal of Econom-
ics 97 (4): 543–69.
Alonso, Ricardo, and Niko Matouschek. 2008. “Optimal Delegation.” Review of Economic Studies 75
(1): 259–93.
Belfield, Richard, and David Marsden. 2003. “Performance Pay, Monitoring Environments, and Estab-
lishment Performance.” International Journal of Manpower 24 (4): 452–71.
Bellemare, Charles, and Bruce Shearer. 2009. “Gift Giving and Worker Productivity: Evidence from a
Firm-Level Experiment.” Games and Economic Behavior 67 (1): 233–44.
Ben-Ner, Avner, Louis Putterman, Fanmin Kong, and Dan Magan. 2004. “Reciprocity in a Two-Part
Dictator Game.” Journal of Economic Behavior & Organization 53 (3): 333–52.
Bolton, Gary E., and Axel Ockenfels. 2000. “ERC: A Theory of Equity, Reciprocity, and Competition.”
American Economic Review 90 (1): 166–93.
Brennan, Geoffrey, Luis G. Gonzalez, Werner Güth, and M. Vittoria Levati. 2008. “Attitudes toward
Private and Collective Risk in Individual and Strategic Choice Situations.” Journal of Economic
Behavior and Organization 67 (1): 253–62.
Coffman, Lucas C. 2011. “Intermediation Reduces Punishment (And Reward).” American Economic
Journal: Microeconomics 3 (4): 77–106.
Cox, James C., Daniel Friedman, and Vjollca Sadiraj. 2008. “Revealed Altruism.” Econometrica 76
(1): 31–69.
Dohmen, Thomas, Armin Falk, David Huffman, and Uwe Sunde. 2009. “Homo Reciprocans: Survey
Evidence on Behavioural Outcomes.” Economic Journal 119 (536): 592–612.
Dufwenberg, Martin, and Georg Kirchsteiger. 2000. “Reciprocity and Wage Undercutting.” European
Economic Review 44 (4–6): 1069–78.
Dufwenberg, Martin, and Georg Kirchsteiger. 2004. “A Theory of Sequential Reciprocity.” Games and
Economic Behavior 47 (2): 268–98.
Dur, Robert, and Amihai Glazer. 2008. “Optimal Contracts When a Worker Envies His Boss.” Journal
of Law, Economics, and Organization 24 (1): 120–37.
Dur, Robert, Arjan Non, and Hein Roelfsema. 2010. “Reciprocity and Incentive Pay in the Workplace.”
Journal of Economic Psychology 31 (4): 676–86.
Englmaier, Florian. 2005. “Moral Hazard, Contracts and Social Preferences: A Survey.” In Psychology,
Rationality and Economic Behaviour: Challenging Standard Assumptions, edited by Bina Agarwal
and Alessandro Vercelli, 125–39. Houndmills, UK: Palgrave Macmillan.
Englmaier, Florian, and Stephen Leider. 2008. “Contractual and Organizational Structure with Reciprocal Agents.” CESifo Working Paper 2415.
Englmaier, Florian, and Stephen Leider. 2010. “Gift Exchange in the Lab––It is Not (Only) How
Much You Give ...” CESifo Working Paper 2944.
Englmaier, Florian, and Stephen Leider. 2012. “Managerial Payoff and Gift Exchange in the Field.”
CESifo Working Paper 3707.
Englmaier, Florian, and Achim Wambach. 2010. “Optimal Incentive Contracts under Inequity Aversion.” Games and Economic Behavior 69 (2): 312–28.
Falk, Armin. 2007. “Gift Exchange in the Field.” Econometrica 75 (5): 1501–11.
Falk, Armin, and Urs Fischbacher. 2006. “A Theory of Reciprocity.” Games and Economic Behavior
54 (2): 293–315.
Fehr, Ernst, Georg Kirchsteiger, and Arno Riedl. 1998. “Gift Exchange and Reciprocity in Competitive Experimental Markets.” European Economic Review 42 (1): 1–34.
Fehr, Ernst, and Klaus M. Schmidt. 1999. “A Theory of Fairness, Competition, and Cooperation.”
Quarterly Journal of Economics 114 (3): 817–68.
Fehr, Erst, and Klaus M. Schmidt. 2003. “Theories of Fairness and Reciprocity: Evidence and Economic Applications.” In Advances in Economics and Econometrics, Theory and Applications,
Vol. 4 No. 2
englmaier and leider: Reciprocity and Contracts
183
Eighth World Congress,Vol. 1, edited by Mathias Dewatripont, Lars Peter Hansen, and Stephen J.
Turnovsky, 208–57. Cambridge, UK: Cambridge University Press.
Gneezy, Uri, and John A. List. 2006. “Putting Behavioral Economics to Work: Testing for Gift
Exchange in Labor Markets Using Field Experiments.” Econometrica 74 (5): 1365–84.
Grossman, Sanford J., and Oliver D. Hart. 1983. “An Analysis of the Principal-Agent Problem.”
Econometrica 51 (1): 7–45.
Güth, Werner, M. Vittoria Levati, and Matteo Ploner. 2008. “On the Social Dimension of Time and
Risk Preferences: An Experimental Study.” Economic Inquiry 46 (2): 261–72.
Holmström, Bengt. 1979. “Moral Hazard and Observability.” Bell Journal of Economics 10 (1): 74–91.
Holmström, Bengt. 1984. “On the Theory of Delegation.” In Bayesian Models in Economic Theory,
edited by Marcel Boyer and Richard E. Kihlstrom, 115–41. Amsterdam: North-Holland.
Holmström, Bengt, and Paul Milgrom. 1991. “Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design.” Journal of Law, Economics, and Organization 7 (Special Issue): 24–52.
Huang, Fali, and Peter Cappelli. 2010. “Applicant Screening and Performance-Related Outcomes.”
American Economic Review 100 (2): 214–18.
Kim, Son Ku. 1995. “Efficiency of an Information System in an Agency Model.” Econometrica 63
(1): 89–102.
Kosfeld, Michael, and Ferdinand A. von Siemens. 2011. “Competition, Cooperation, and Corporate
Culture.” RAND Journal of Economics 42 (1): 23–43.
Kube, Sebastian, Michel André Maréchal, and Clemens Puppe. 2006. “Putting Reciprocity to Work––
Positive versus Negative Responses in the Field.” Unpublished.
Lemieux, Thomas, W. Bentley MacLeod, and Daniel Parent. 2009. “Performance Pay and Wage
Inequality.” Quarterly Journal of Economics 124 (1): 1–49
Leshno, Moshe, Haim Levy, and Yishay Spector. 1997. “A Comment on Rothschild and Stiglitz’s
‘Increasing Risk: I: A Definition.’” Journal of Economic Theory 77 (1): 223–28.
Leuven, Edwin, Hessel Oosterbeek, Randolph Sloof, and Chris van Klaveren. 2005. “Worker Reciprocity and Employer Investment in Training.” Economica 72 (285): 137–49.
Milgrom, Paul, and Ilya Segal. 2002. “Envelope Theorems for Arbitrary Choice Sets.” Econometrica
70 (2): 583–601.
Pazy, Asya, and Yoav Ganzach. 2009. “Pay Contingency and the Effects of Perceived Organizational and
Supervisor Support on Performance and Commitment.” Journal of Management 35 (4): 1007–25.
Piekkola, Hannu. 2005. “Performance-Related Pay and Firm Performance in Finland.” International
Journal of Manpower 26 (7–8): 619–35.
Prendergast, Canice. 1999. “The Provision of Incentives in Firms.” Journal of Economic Literature
37 (1): 7–63.
Rabin, Matthew. 1993. “Incorporating Fairness into Game Theory and Economics.” American Economic Review 83 (5): 1281–1302.
Rogerson, William P. 1985. “The First-Order Approach to Principal-Agent Problems.” Econometrica
53 (6): 1357–67.
Rothschild, Michael, and Joseph E. Stiglitz. 1970. “Increasing Risk: I. A Definition.” Journal of Economic Theory 2 (3): 225–43.