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Working Paper Series
Mechanism Design and Assignment
Models
WP 03-09
Edward Simpson Prescott
Federal Reserve Bank of Richmond
Robert M. Townsend
Federal Reserve Bank of Chicago
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Mechanism Design and Assignment Models
Edward Simpson Prescott
Federal Reserve Bank of Richmond
Robert M. Townsend¤y
University of Chicago
Federal Reserve Bank of Chicago
Federal Reserve Bank of Richmond Working Paper No. 03-09
July 29, 2003
Abstract
This mechanism design paper studies the assignment of people to projects over
time. Inability to communicate interim shocks is a force for long-term assignments,
though exceptions exist for high risk aversion. In contrast, costless reporting of
interim shocks makes switching powerful for virtually all environments. Switching
elicits honest reports and mitigates incentive constraints allowing, in particular, bene…cial concealment of project quality. Properties of the production technology are
also shown to matter. Substitutability of intertemporal e¤ort is a force for long-term
assignments while complementarity with Nash equilibrium strategies is a force for job
rotation.
JEL Classi…cation: D82, L23
Keywords: Assignment, private information, communication
¤
We would like to thank seminar participants at CEMFI, the Federal Reserve Bank of Richmond, and
the University of Chicago for helpful comments. The views expressed in this paper do not necessarily
express the views of the Federal Reserve Banks of Chicago or Richmond or the Federal Reserve System.
y
Prescott: Federal Reserve Bank of Richmond, P.O. Box 27622, Richmond, VA 23261. Email: Edward.Prescott@rich.frb.org. Townsend: University of Chicago, Dept. of Economics, 1126 E. 59th St.,
Chicago, IL 60637. Email: rtownsen@midway.uchicago.edu.
1
1
Introduction
This paper examines a variant on the classical assignment problem of Koopmans and Beckmann (1957). There is a mechanism designer who must use people to evaluate and operate a
set of projects. Each project operates over more than one stage, but in any single stage the
designer may assign only one person to it. The designer’s information about each project is
limited; he observes a project’s output, but not its interim shock nor its labor inputs. Only
the agent assigned to a project at a particular stage observes that stage’s relevant variable,
be it the interim shock or his own labor input. In addition to setting standard contractual terms such as output-dependent consumption, the designer has the ability to reassign
agents among the projects. Providing conditions under which these reassignments occur
is the goal of this paper. The conditions we examine include communication possibilities,
agent heterogeneity, conditions on preferences, and technological complementarities.
Organizations regularly face assignment problems. Conglomerates must decide how to
allocate executives across divisions. Firms must decide how to allocate managers across
departments. Managers must decide how to allocate employees across jobs. Frequently,
these decisions have time and contingent components. How long should a manager be
assigned to a project? Under what conditions, should he be reassigned? Regular periodic
job rotation is one strategy undertaken by many organizations. Executives are rotated
across divisions, and managers are often rotated across functional areas. Even within a
function employees may be rotated. For example, many large banks rotate their loan
o¢cers among lending o¢ces.1 This solution to the assignment problem is costly. Jobspeci…c knowledge is lost and time is spent learning details speci…c to the new assignment.
Yet, despite these costs organizations still regularly reassign people.
There are several theories of job reassignment. In Meyer (1994), reassignments help an
organization learn about the ability of workers. Reassignments can also provide training
for managers who are later promoted. In Ickes and Samuelson (1987), rotation can solve
“a ratchet e¤ect.” For incentive reasons long-term contracts are bene…cial, but rotation is
the only way the organization can commit to the long-term contracts.
1
Rotation may occur at the economy level as well. Germany requires that its …rm periodically hire new
auditors, presumably to prevent collusion between management and auditors.
2
Our goal in this paper is to identify additional forces – complementary to those identi…ed by the literature – that lead to reassignments. The forces we identify include limited
information revelation, information scrambling, as well as properties of production technologies. Unlike Ickes and Samuelson (1987), we do not rely on limited commitment by the
mechanism designer.
Section 2 lays out the general environment and compares it with the Koopmans and
Beckmann (1957) assignment problem. The remaining sections study various simpli…cations
to it. Sections 3 and 4 contain two-stage models with an interim shock in the …rst stage
followed by a labor e¤ort in the second. The interim shock is observed only by the agent
initially assigned to the project, while the labor e¤ort is observed only by the agent assigned
to the project in the second stage. Between the two stages the agents can be reassigned to
di¤erent projects.
In Section 3, the agent cannot report interim shocks to the mechanism designer. This
lack of information greatly impedes reassignments. Yet, despite this impediment, we still
…nd conditions such as high risk aversion under which reassignment is optimal. The problem
studied in this section is related to literatures that examine the value of allowing the agent
to know information that is private to him and not observed by the designer.
In Section 4, agents send reports to the designer, who uses these reports to make
second-stage assignments. We …nd that reassignment is always optimal with this communication and often strictly dominates allocations without reassignment. Also, we …nd
that the designer may not want to inform the agent about the interim state of his newly
assigned project. Instead, the designer may want to make agent reassignment random, in
e¤ect scrambling the information communicated to the agent. Under some conditions this
scrambling is valuable because it relaxes second-stage moral hazard constraints.
Section 5 studies a model in which the interim shock stage is replaced by one in which
the agent takes a hidden e¤ort. In this extension, we …nd that optimality of reassignment
depends on the substitution and complementarity properties of production over the two
project stages. Substitution is a force for staying put, as the agent takes full responsibility for all stages of production e¤ort. Complementarity is a force for reassignment; the
Nash equilibrium in e¤orts alleviates incentive constraints. Finally, Section 6 discusses
extensions.
3
2
The Environment
There is a continuum of agents and a continuum of projects, both of measure one. The
continuum assumption should be viewed as an approximation to the large number of people
and projects that make up a …rm or an economy. This abstraction avoids the need to worry
about aggregate uncertainty that may arise when there is a …nite number of agents and
shocks are identically and independently distributed.
Production on a project takes two stages. In the …rst stage there is an interim shock µ
to each project. Each project’s shock is drawn from the same probability distribution h(µ)
and there is a …nite number of possible realizations of µ: Shocks are independent across
projects, but with the continuum assumption h(µ) can also be viewed as the fraction of
projects experiencing shock µ. In the second stage of production, each project requires a
labor input b 2 B: In some sections of the text the set B is …nite, while it is a continuum
in others. This set typically contains the zero e¤ort point. On each project, the shock and
the labor input determine the conditional probability distribution of the project’s output
q 2 Q; the set Q is …nite. We write the conditional distribution as p(qjb; µ). Shocks µ and
labor inputs b applied to one project do not a¤ect production on other projects. Later, in
Section 5 below, we modify this technology by replacing the interim shock with an initial
e¤ort.
In each stage only one agent may be assigned to a particular project. The agent assigned
to a project in the …rst stage observes the initial shock µ. The agent assigned to a project
in the second stage supplies the labor input b: Both the initial shock and the labor input
are private information, while the output is publicly observed. The designer may reassign
agents between stages using whatever information he has available. We assume, however,
that an agent assigned to a new project in the second stage does not observe its interim
shock.
There is a …nite number of agent types i. Each type is a positive fraction of the
population ®i > 0: An agent’s preferences are
Ui(c) ¡ Vi (b);
where c is his consumption, with c 2 <+. We assume that the functions Ui and Vi are
4
increasing, that Ui is weakly concave, and that Vi is weakly convex. Since we will be
solving for Pareto optima, we assign Pareto weights ¸i > 0 to type-i agents.
Because all projects are ex ante identical, the initial assignment of agents does not
matter. Any agent can be assigned to any project. The interesting assignment problem
occurs later, after the interim shock µ has been realized. At that point, the designer may
reassign agents using whatever information he has available.
2.1
Full Information and the Classical Assignment Problem
If shocks µ and e¤ort levels b were publicly observed, then this problem would be a version
of the classical assignment problem. Let ± i(µ) be the probability a type-i agent is assigned
to a project of quality µ. Equivalently, ± i(µ) is the fraction of type-i agents assigned to a
project of quality µ. Let ci (µ) and bi(µ) be the consumption and e¤ort, respectively, of a
type-i agent who is assigned to a project of quality µ. The assignment problem is
Program 1 :
max
±i (µ);ci (µ);bi (µ)
s.t.
X
i
®i
X
i
X
µ
®i ¸i
X
Ã
µ
±i (µ)(Ui (ci(µ)) ¡ V i(bi(µ)))
± i(µ) ci (µ) ¡
8µ;
X
i
8i;
X
p(qjbi (µ); µ)q
q
®i± i(µ) = h(µ);
X
± i(µ) = 1:
!
· 0;
(1)
(2)
(3)
µ
Equation (1) is a standard resource constraint; total consumption is less than total
output. Equations (2) are the assignment constraints. They guarantee that the number of
agents assigned to project type µ equals the number of such projects in the population. The
last constraint, equation (3), ensures that the ± i are probability measures, or equivalently,
that all agents of type i are assigned to some project type.
Solutions to Program 1 are easily characterized. Because of the public information,
agents work as hard as they are told. Agents assigned to more productive projects work
5
harder than agents assigned to less productive projects. Agents are fully insured over
output and with separable preferences consumption does not depend on the assignment µ.
Any di¤erences in consumption levels among agents in the cross section will be the result
of di¤erences in preferences and the Pareto weights.
Program 1 is a version of the classical assignment problem of Koopmans and Beckmann
(1957).2 In the simplest application of that problem, there are n heterogeneous jobs and
n heterogeneous agents, with di¤ering quality of matches. The problem is to match each
worker with a job in a way that maximizes the objective function. In our problem, the
quality or productivity of the job corresponds to the interim shock µ, while the heterogeneity
in agents correspond to the agent type i. Heterogeneity in preferences or Pareto weights,
provides reasons to reassign agents. For example, low Pareto weight agents or agents with
less aversion to work are assigned to high productivity projects. If agents are homogeneous,
however, then the assignment problem is of little interest. With separability in preferences,
consumption is not tied to project assignment µ. Furthermore, even though e¤ort may be
higher for productive projects, projects are randomly assigned so as to maximize ex ante
expected utility and maintain equal utility treatment.
Private information alters the analysis. The designer’s assignment rule a¤ects his ability
to elicit information about the projects, information that he knows a priori in the classical
assignment problem. These factors a¤ect the quality of matches. They even make the
assignment problem important when the agents are homogenous, that is, even when the
driving forces for matches in the classical model are not operating.
3
No Communication
In this section, we assume that shocks µ in the …rst stage are private to the agent initially
assigned to the project, and that e¤ort b is private to the agent assigned to the project in
the second stage. Furthermore, we assume that an agent cannot report the shock on his
initial project. Despite his ignorance of the interim state of a project, the designer may
reassign agents across projects after the …rst stage. If he does this, the reassignment must,
by necessity, be random across project shocks µ:
2
Also, see the survey in Sattinger (1993).
6
This environment is the most di¢cult one for generating reassignments of agents. In
the classical problem, the designer knows the quality of a project; his problem is simply to
match heterogenous agents to heterogenous projects. But in this problem, not only is the
designer ignorant of this information, he does not receive a report on it.
As we will see in the next section, reporting is so valuable that preventing it by …at
here requires some justi…cation. The applications we have in mind for this section have
production processes in which it is di¢cult to completely describe the interim state of
production. For example, agricultural shocks not only include whether it rained or not, but
also information on how much it rained and where the runo¤ was. This kind of information,
particularly if not formally measured, is di¢cult to communicate. Reporting a state is not
simply a matter of reporting a value from the real line; it requires that a sender and receiver
make a substantial investment in time and expertise to establish a communication channel
between them (Arrow (1971)). As an approximation, we make the extreme assumption
that an agent cannot communicate any information on the interim stage of the project to
the designer. It is too extreme a description for many applications, but it illustrates the
point nicely and makes the analysis manageable.3
An inability to communicate interim shocks should be particularly important when
some sort of coordination is needed across stages in production. Consider a production
technology on each project, p(qjb; µ); that requires a di¤erent e¤ort b for each realization of
µ; if the wrong e¤ort is chosen then output is low. An example of such a technology is one
in which the type of fertilizer to be applied to a crop depends on the moisture content of the
soil, in‡uenced by but not solely dependent on previous rainfall. If agents cannot report
on the interim state then switching is ine¤ective. The agent could not tell the designer the
moisture content of the soil, so whoever was assigned to that plot in the second stage would
not know how much fertilizer to apply. In this section, we analyze a production technology
3
The results in this section are related to literatures that examine the value of allowing an agent to
gather information that the principal does not observe. Results are mixed in the accounting literature.
Christensen (1981) and Penno (1989) contain examples where the agent knowing certain information is
bad. Baiman and Sivaramakrishnan (1991) study a similar model with the addition of second-stage moral
hazard. They …nd conditions on the production function in which it is always more desirable for the agent
to know more information. Finally, in a monopolist problem with varying demand, Lewis and Sappington
(1994) …nd conditions under which it is, and is not, desirable for the monopolist to help potential buyers
observe a parameter indicating how much they like the monopolist’s project.
7
where some coordination is desirable, in the sense that it is valuable to work harder on more
productive projects, but the coordination is not as valuable as in the fertilizer example.
Speci…cally, we work with a simple deterministic second-stage production function q =
bµ; and unless otherwise speci…ed all possible realizations of µ are positive: We also assume
that agents are homogenous so we drop subscripts i and remove explicit references to
fractions ®i and ¸i from the problem.
We start by analyzing no-switching contracts. If agents are not switched then ±(µ) =
h(µ), that is, the probability that an agent is assigned to a project of type µ is simply
the fraction h(µ) of those projects in the population. The problem is a standard hidden
information problem. The assigned agent knows µ and decides on e¤ort b; but the designer
observes neither of them.
The designer chooses state-contingent e¤ort, b(µ), and output-dependent consumption,
c(q): Because of the deterministic production function, however, consumption can be made
explicitly a function of µ. Notice that c(q) = c[b(µ)µ]; so we write c as if it depended directly
on µ, namely, c(µ) ´ c[b(µ)µ].
Because of the private information, incentive constraints are needed to guarantee that
an agent who receives shock µ takes e¤ort b(µ). If the project type is µ and the agent
contemplates deviating to another e¤ort consistent with the output produced by a µ0 -type
agent, then to rationalize his output he must produce the output q that would be consistent
0
0
with e¤ort b(µ ) at shock µ , even though the true shock is µ.
0
The requisite deviating
0
e¤ort would be b(µ )µ =µ. The incentive constraints (5), below, prevent this deviation.
One related concern is that an agent will decide to work a level of b that is inconsistent
with the output of any type. In this case, the designer knows the agent has deviated. For
simplicity, we abstract from the details of the punishments and assume that the designer can
automatically prevent deviations that have a positive probability of revealing a deviation.4
4
Alternatively, we could restrict the designer’s punishments by setting consumption equal to zero. For
preferences in which U(0) = ¡1, like log utility, the e¤ect is the same. For other preferences, a constraint
that limits agents’ utility to be at least equal to receiving zero consumption and taking zero e¤ort would
be necessary. This constraint is easy to analyze. The analogous constraint for the switching case, however,
is not so simple to analyze.
8
The programming problem is
Program 2:
max
c(µ);b(µ)
s.t.
X
h(µ)(U(c(µ)) ¡ V (b(µ)))
X
h(µ)(c(µ) ¡ µb(µ)) · 0;
µ
µ
U(c(µ)) ¡ V (b(µ)) ¸ U(c(µ 0 )) ¡ V (
b(µ 0 )µ0
); 8µ; µ 0 :
µ
(4)
(5)
Equation (4) is the resource constraint. It guarantees that consumption does not exceed
output. Equations (5) are the incentive constraints discussed earlier.
If agents are switched, neither the designer nor the agent newly assigned to a project
know its interim shock. Consequently, all agents must be assigned the same e¤ort level b.
Furthermore, there is no incentive constraint. For any deviation from the recommended
e¤ort b there is a positive probability that the deviation will be revealed by an output impossible in equilibrium, causing the unmodeled non-pecuniary punishment to be imposed.5
As in the no-switching model, the designer can infer µ from q. Here, the inference follows
from µ = q=b: Therefore, as in the no-switching model, we write c(µ).
The problem if agents are switched is
Program 3:
max
c(µ);b
s.t.
X
h(µ)U(c(µ)) ¡ V (b)
µ
X
µ
h(µ)(c(µ) ¡ µb) · 0:
(6)
Equation (6) is the resource constraint.
5
The random variable µ can only take on a …nite number of values, none of which are zero. If an e¤ort
less than b is taken then the lowest value of µ will reveal the deviation. Similarly, for an e¤ort greater than
b the highest value of µ plays that role. Finally, depending on the possible values of µ; intermediate values
of µ may be revealing as well.
9
All agents work the same e¤ort. The common e¤ort level is chosen so that the marginal
disutility of e¤ort equals expected marginal product, that is, V 0 (b) = E(µ). Finally, with
risk aversion consumption is constant.
Each regime has an advantage and a disadvantage relative to the other. If agents are
not switched, their e¤orts can be tailored to the relative productivities of each project
but there are incentive constraints. If agents are switched, the same e¤ort is applied to
all projects regardless of relative productivities but there are no incentive constraints. As
the next two propositions demonstrate, risk aversion is an important factor in determining
which regime is better.
Proposition 1 If agents are risk neutral with respect to consumption then no-switching
dominates switching.
Proof: With risk neutrality incentive constraints do not bind in the no-switching model.
Let b(µ) satisfy V 0(b(µ)) = µ and give each agent his own product, c(µ) = µb(µ). This
contract is incentive compatible because it satis…es the …rst-order conditions to the agents’
problems. It also is a solution to the full-information problem. Consequently, it is better
than the best switching contract. Q.E.D.
With risk neutrality there is no distortion from the incentive constraints. Without the
incentive distortion, the ability to tailor e¤ort levels to marginal productivities is unambiguously good. This is not true when agents are risk averse. The designer wants to reduce
the variability of consumption but that limits the incentives of productive agents to work
hard. If agents are risk averse enough, it is more important to remove the disincentive to
work than it is to tailor e¤orts to marginal productivities.
We prove the theorem for preferences where any variation in consumption is undesirable.
Preferences are Leontief if U (fc(µ)g) = minµ U(c(µ)). Under Leontief preferences there will
not be any state-dependent variation in consumption, that is, c(µ) will be a constant. If
it is not, any consumption paid out in excess of the minimal received consumption level is
wasted. These preferences punish variation in consumption and in this sense are analogous
to an extreme form of risk aversion. After the proof, we provide an example using a
standard utility function in which switching still dominates.
10
Under Leontief preferences, low-productivity types work relatively hard and high-productivity
types work relatively little to ensure that there is no variation in output-dependent consumption. With switching, output can be made higher by having both agents apply identical e¤ort. The loss of output from low-productivity types who were working hard is more
than compensated by the increased output from high-productivity types.
Proposition 2 If preferences are Leontief and if µ j > 0 for all j, then switching dominates.
Proof: See Appendix.
Figure 1 illustrates Proposition 2 for the case where there are two shocks that occur with
equal probabilities. Under the no-switching contract each agent knows the quality of his
project. With Leontief preferences agents are so risk averse that they dislike any variation in
their consumption. This assumption forces solutions to the no-switching model to satisfy
c(µ) = c. Because consumption is not state-contingent, the solution to the optimal noswitching problem is characterized by b(µ) satisfying µb(µ) = µ0 b(µ0 ) for all µ, µ 0, that is, all
agents must produce the same output regardless of their shock µ. Consequently, the less
productive agents work the hardest. In Figure 1 both agents produce qns .
A switching contract that does better is for both agents to work bs = 0:5b(µl )+ 0:5b(µ h):
Disutility of e¤ort is less and total output is higher because the increase in the high-type’s
output is greater than the decrease in the low-type’s output.
The result is not just applicable for the extreme preferences of the proposition. For
example, if preferences are U(c) = ¡(c + 0:01)¡0:18 ¡ 0:5b1:22, with two types, and µ l = 1;
µ h = 2; then switching dominates.
There is a simple case where we can prove that not switching dominates switching,
regardless of the preferences. Assume that there are two possible shocks, and unlike in
Proposition 2 above, assume that µ l = 0 and µ h > 0. Production is still q = bµ. Because
the lowest value of µ is zero, an agent can always work zero and pretend he received the µ l
shock. This creates an incentive constraint under switching that is
h(µl )U (c(µl )) + h(µ h)U(c(µ h)) ¡ V (b) ¸ U(c(µ l )) ¡ V (0);
(7)
where the summation over µ indicates the uncertainty the agent has over the interim state
of his newly assigned project. Here, on the right-hand side, the agent contemplates taking
11
q
q = bθ h
qs(θh )
.
}
qns
Decrease
in output
{.
Increase in
output
qs(θl)
bs
b(θh )
q = bθl
b(θl)
b
Figure 1: Illustration of proof of Proposition 2 for case with two types and h(µl ) = h(µh ) =
0:5: Under no-switching, agents work b(µl ) and b(µ h) and total output is qns . Under
switching both agents work bs and total output is 0:5qs (µ l ) + 0:5qs (µ h). Total output under
switching is greater because qs (µ h) ¡ qns > qns ¡ qs (µ l ).
the deviating e¤ort, b = 0, ensuring that output will be zero. In equilibrium, zero output
is produced by agents with the low shock, so c = c(µ l ). On the left-hand side, e¤ort b is
taken so that output is q = bµ and consumption depends on the type of project they are
assigned to.
The resource constraint is
h(µ l )c(µ l ) + h(µ h)c(µ h) · h(µ h)µ hb:
(8)
For b > 0 the optimal, incentive-compatible switching contract is characterized by higher
consumption for higher output, that is, c(µ h) > c(µ l ):
If the agents are not switched, they know their type µ, and then we have the earlier
private information case. The optimal no-switching contract is characterized by b(µ l ) = 0
12
and b(µh ) > 0: For µ l , the problem is trivial in the sense that there is no value to e¤ort.
The incentive constraints, (5), only apply to the high type µh so
(9)
U(c(µh )) ¡ V (b(µh)) ¸ U(c(µ l )) ¡ V (0):
The resource constraint is
(10)
h(µl )c(µ l ) + h(µh)c(µ h) · h(µh)µhb(µ h):
In this special case, no-switching dominates switching.
Proposition 3 If there are two shocks, µ l = 0, and if the optimal switching contract is
characterized by c(µ h) > c(µl ), then no-switching strictly dominates switching.
Proof: Take an optimal switching contract (b; c(µ l ); c(µh )). Now consider the noswitching contract b(µ l ) = 0, b(µ h) = b, with c(µ) unchanged. Under this contract the
resource constraint is unchanged; equations (8) and (10) are identical.
Furthermore,
U(c(µ h)) > h(µl )U(c(µl )) + h(µ h)U (c(µh)). Substituting this inequality and b = b(µ h)
into the left-hand side of equation (7) delivers the no-switching incentive constraint (9).
We thus have a feasible no-switching contract that is better than the optimal switching
contract. Q.E.D.
4
The Roles of Information and Communication
Without communication, the optimality of reassignment depends heavily on risk aversion.
In this section, we examine the importance of the previous no-communication assumption
by allowing an agent to report to the designer on his initial project’s shock. The combination of reporting and switching is useful for three reasons: it reveals information, it helps
to make better matches, and it allows for the scrambling of information communicated to
the agent.
To study the heterogeneity issue, we drop the representative consumer simpli…cation
and return to heterogenous agents. As in the environment of Section 2, let there be a …nite
number of observably di¤erent types, with each type i constituting a positive fraction,
®i > 0, of the population. An agent’s type is public information. Pareto weights are
P
described by ¸i , with i ¸i = 1. All agents of a given type are treated identically ex ante.
13
4.1
Information revelation
In this section, we study the prototypical moral hazard problem by adding a stochastic
element to second-stage production. The production function is p(qjb; µ) > 0 for all q; b; µ, so
p is a non-degenerate probability distribution. Second-stage e¤ort b is private information.
The set of e¤orts, B; is a …nite set.
For reasons illustrated shortly, we allow for some randomization in contractual terms. As
in the no-switching model, the contract contains a recommended e¤ort level that depends
on the interim state of the project. Now, however, this recommendation may be random.
It is described by the conditional probability distribution ¼i(bjµ) for type i. Because of
the randomized e¤ort, consumption needs to be a function not only of the interim state
µ and the output realization q; but also the realized recommended e¤ort b. We write
the compensation schedule of type i as ci(q; b; µ). We could have allowed randomization
in the compensation schedule as well but with separable preferences consumption will be
degenerate so we dropped explicit consideration of this source of randomization.
We …rst consider no-switching contracts. When a type-i agent stays on his project with
probability one, he knows the state µ. By the Revelation Principle the contract needs to
satisfy incentive constraints that induce truthful reporting of µ and then, given a truthful
report, other constraints that ensure that the agent takes the recommended e¤ort. The
truth-telling constraints are
8i; µ;
¸
X
q;b
X
q;b
p(qjb; µ)¼i (bjµ)[Ui (ci(q; b; µ)) ¡ Vi(b)]
(11)
p(qjÁ(b); µ)¼ i(bjµ 0 )[Ui(ci (q; b; µ 0)) ¡ Vi(Á(b))]; 8µ0 6= µ; 8Á : B ! B:
Constraints (11) ensure that telling the truth, µ; and then taking the resulting recommended
e¤ort b, is preferable to lying, i.e., sending a report µ 0 6= µ, and then taking any deviation
strategy, Á; which maps recommended e¤ort b to alternative e¤ort b0 . For more details on
these constraints, see Myerson (1982) for the original treatment or Prescott (2003) for an
exposition in a similar model.
In addition to constraints (11), the Revelation Principle requires constraints that ensure
that an agent who truthfully reports µ takes recommended e¤ort b: For all i; µ; and b such
14
that ¼i (bjµ) > 0,
X
X
p(qjb; µ)[Ui(ci(q; b; µ)) ¡ Vi (b)] ¸
p(qj^b; µ)[Ui(ci (q; b; µ)) ¡ Vi (^b)]; 8^b:
q
(12)
q
With communication a strikingly simple mechanism improves upon no-switching contracts.
Proposition 4 Regardless of risk aversion, heterogeneity in preferences and welfare weights,
and the speci…cation of technology p(qjb; µ) > 0, switching and telling the agent the state
of his newly assigned project µ weakly dominates not switching him. Dominance is strict if
incentive constraints (11) bind.
Proof: Consider the following contract: After agents report on their interim shocks,
the designer switches them and then makes their new assignment and compensation independent of the report they sent. Under this contract, an agent’s utility does not depend
on his report so he reports the true shock. Next, assume that the quality of each agent’s
assigned project is randomly drawn from the distribution h(µ) and the designer tells each
agent the quality of his new project µ. The compensation schedule is still described by
ci(q; b; µ); but now µ is the quality of an agent’s newly assigned project.
Because the designer and the agent know the state of the project, there are no truthtelling constraints as in equations (11). The only incentive constraints left are those on the
agent’s e¤ort, which are identical to constraints (12) in the no-switching scheme. Thus, the
set of no-switching contracts is a subset of the switching contracts. Consequently, switching
weakly dominates no-switching. The dominance is strict if truth-telling constraints (11)
bind in the no-switching optimum, as these are eliminated in the switching regime. Q.E.D.
In the …rst-stage of the switching scheme, agents are just information monitors. They
report the true shock because they are indi¤erent to what they observe and what they
report.6 The arrangement is essentially a moral-hazard economy, with the added feature
that there is a random, publicly observed shock to the production technology. It should be
noted that this result does not depend on the second-stage moral hazard. The information
revelation result still applies to the deterministic production technology analyzed in the
previous section.
6
A similar idea is used in Hirao (1993).
15
4.2
Information scrambling
In the contract described above the designer tells the agent the interim state of his newly
assigned project. That property of the contract was imposed by …at. While su¢cient
to illustrate the information revelation role of switching, it need not be optimal. Indeed,
sometimes the designer would choose not to tell the agent the state of his newly assigned
project. In this case, not only does switching remove truth-telling constraints but it also
scrambles information. The agent now has to infer the quality of his newly assigned project.
As we will see below, scrambling weakens second-stage incentive constraints. Ignorance is
bliss here.
In the switching stage, the designer assigns agents to projects according to the assignment probabilities ± i(µ). As before, the designer recommends an e¤ort level b; according
to the possibly stochastic rule ¼i (bjµ).
If the designer does not tell the agent the quality of his new project then the agent has
to infer it. He has three pieces of information from which to form his inference: his type i,
the assignment rule ± i(µ), and the recommended e¤ort rule ¼ i(bjµ): A type-i agent who is
recommended e¤ort b forms a posterior over project quality of pri(µjb). The posterior is
related to the other objects by the relationship pri (µjb) = ±i (µ)¼i(bjµ)=¼ i(b), where ¼i (b) is
the unconditional probability that a type-i agent is recommended e¤ort b.
The incentive constraint can be written directly in terms of the posterior probabilities,
pri(µjb); but it is more convenient to substitute out for this term. Again, there are no
truth-telling constraints. The moral hazard incentive constraints are for all b such that
¼i (bjµ)±i (µ) > 0;
X
q;µ
p(qjb; µ)¼i (bjµ)±i (µ)[Ui (ci(q; b; µ))¡Vi(b)] ¸
X
q;µ
p(qjbb; µ)¼ i(bjµ)± i(µ)[Ui (ci (q; b; µ))¡Vi (^b)]; 8^b;
(13)
where the ¼i (b) cancel out of both sides.
Compare these moral hazard constraints, (13), with the moral hazard constraints, (12),
used by the other two schemes. For a given b; (13) is a convex combination of all the
incentive constraints (12) corresponding to µ for which b was recommended. We can now
prove the following theorem.
16
Proposition 5 A switching contract where the designer does not tell the agent the shock
µ of his newly assigned project weakly dominates a switching contract where the designer
tells the agent the value of µ of his newly assigned project.
Proof: Any contract satisfying (12) for each µ will satisfy (13), but not necessarily vice
versa. Q.E.D.
If each agent assigned to a di¤erent quality project was recommended a di¤erent e¤ort
level then there would not be any value to scrambling. Each agent would be able to infer
from his recommended e¤ort level the quality of his new project. But, if agents cannot
make that inference, because more than one project quality is assigned a particular e¤ort
level, then the dominance in Proposition 5 may be strict. The numerical example below
demonstrates.
4.3
Private information and the assignment problem
We now formulate the economy-wide mechanism design problem and then present an example that illustrates not one but three reasons for switching: the classical role of matching
heterogenous agents to heterogenous jobs, information revelation, and information scrambling.
We take as a starting point the observation that at no cost the designer can induce
truthful revelation of project qualities, µ, by the schemes described above. The program is
Program 4:
max
±i (µ)¸0;¼i (bjµ)¸0;ci (q;b;µ)
X
i
®i¸i
X
q;b;µ
p(qjb; µ)¼ i(bjµ)± i(µ)[Ui (ci (q; b; µ)) ¡ V i(b)]
X X
s.t. ®i
p(qjb; µ)¼i (bjµ)±i (µ)(ci (q; b; µ) ¡ q) · 0;
i
(14)
q;b;µ
(13) (the incentive constraints),
X
8µ;
®i ±i (µ) = h(µ);
(15)
8i; µ 3 ±i (µ) > 0;
(16)
i
X
17
b
¼i (bjµ) = 1:
Constraint (14) is the resource constraint. Constraints (15) are the assignment constraints
and constraints (16) guarantee that the ¼i (bjµ) are probability measures.
4.4
A numerical example
Let there be two types of agents, i = 1 (type 1s) and i = 2 (type 2s). The Pareto weights
on the types are ¸1 = 0:2 and ¸2 = 0:8. The low status agents are a much larger fraction
of the population than the high status agents. Speci…cally, ®1 = 0:8 and ®2 = 0:2.
Preferences are separable in consumption and e¤ort. Utility from consumption is U(c) =
c0:5 =0:5. Agents may choose from only three possible e¤orts, b1; b2; or b3. The e¤ort portion
of the utility function is described by V (b1) = V (b2) = 0, and V (b3) = 0:5.
There are three di¤erent types of projects, indexed by µ1 ; µ2 , and µ 3. Project types
are random and drawn from the distribution h(µ 1) = h(µ 2) = 0:4, and h(µ 3) = 0:2. Each
type of project may produce either a low output, ql = 0; or a high output, qh = 1. The
probability distribution of the outputs is a function of the project type µ and the e¤ort
b and is described by p(qjb; µ). Output on each project is independent of other projects.
Table 1 describes the p(qjb; µ) production function used in the example.
b1
b2
b3
µ1
µ2
µ3
ql
qh
0:99 0:01
0:70 0:30
0:40 0:60
ql
qh
0:70 0:30
0:99 0:01
0:40 0:60
ql
qh
0:99 0:01
0:99 0:01
0:99 0:01
b1
b2
b3
b1
b2
b3
Table 1: A production technology, p(qjb; µ), that generates switching.
The …rst two projects, indexed by µ1 and µ2 , are the most productive as long as e¤ort
b3 is taken. The two types of projects are identical except that bi, i = 1; 2 has a di¤erent
e¤ect on each project. If bi is worked on a µi , i = 1; 2, project then the project is extremely
unproductive. The third project is so unproductive that the highest e¤ort has no marginal
e¤ect at all.
Table 2 lists the solution to the example. The top half of the table lists the optimal
allocation for type-1 agents and the bottom half lists it for type-2 agents. The second
column in each half lists the probability of being assigned to each project type, and the
18
third column contains the e¤ort taken conditioned on the project assignment. Finally, the
fourth and …fth columns list consumption given outputs ql and qh respectively.
Type 1’s allocation
Action c(q = 0) c(q = 1)
b3
0
0.32
b3
0
0.32
-
µ1
µ2
µ3
± 1(µ)
0.5
0.5
0
µ1
µ2
µ3
Type 2’s allocation
± 2(µ) Action c(q = 0) c(q = 1)
0
0
1
b1 or b2
1.65
1.66
Table 2: Optimal Allocation as a Function of the Agent’s Type
Type-1 agents are assigned randomly to the productive projects, µ 1 and µ2 , while the
high-status type-2 agents are always assigned to the low quality projects, µ3 . On the
productive projects the type-1 agents are induced to work hard, b3, by a contract that
gives them 0 consumption if output is low and 0.32 units of consumption if output is high.
Type-2 agents work either b1 or b2, since both are just as unproductive and gives the same
disutility. Consistent with their high status, they receive large consumption transfers. Also,
because they is no incentive problem on their e¤ort, they are virtually fully insured over
outputs.7 Type-2 agent’s are truly the idle rich.
Randomizing over µ i, i = 1; 2, in a type-1 agent’s project assignment mitigates incentive
constraints. Half of the time a type-1 agent is assigned to a µ 1 project and half of the time
he is assigned to a µ 2 project. Since he does not know his assigned project type, he forms
a posterior probability based on his information set. His knowledge at that point is based
on his type and his recommended e¤ort. He infers from the equilibrium distribution that
since he is a type-1 agent and was recommended e¤ort b3 there is a 50% chance he has been
assigned to a µ 1 project and a 50% chance he has been assigned to a µ 2 project. He knows
for sure that he is not working a µ 3 project because only type-2 agents are assigned to µ 3
7
The two consumption levels di¤er only because of the grid-based numerical methods we used to solve
the example.
19
projects. Now consider the choice facing this agent after being recommended e¤ort b3. If he
takes either e¤ort b1 or b2 he gets less disutility but there is a 12 (:99)+ 12 (:70) = 0:845 chance
of producing the low output, as is evident from Table 1. The dependence of consumption
on output is su¢cient to preclude this strategy.
The 0.845 chance of the low output is directly due to the scrambling of project assignments. Consider the case where the designer tells a type-1 agent the state of his project.
If he is assigned to a µ1 project and is recommended e¤ort b3 then an incentive compatible
contract must stop him from taking e¤ort b1 or b2. The very unproductive e¤ort, b1 , is
easy to prevent, but the other e¤ort, b2, is much harder to detect since it produces the
low output 70% rather than 99% of the time. Compare the lower number with the 0.845
chance of the low output if the agent deviates under the scrambling regime. The incentives
necessary now to make b3 incentive compatible are costlier than when there is scrambling.
This result can be con…rmed if the above program is solved when there are no µ2 projects,
80% of the projects are type µ1 , and the rest are type µ3 . Now, implementing the b3 e¤ort
is so costly that the program gives up on implementing it. Instead, e¤ort b2 is taken on all
type µ 1 projects and output declines.
Finally, switching is also bene…cial because it allows for the standard better match between agents and projects. Without switching some type-2 agents would work high quality
projects, while some type-1 agents would work low quality projects. Such an assignment
is clearly not optimal because given the distribution of Pareto weights it is desirable for
type-1 agents to work hard. Switching ensures that workers who work hard are assigned to
projects where their hard work is productive and it ensures that idle workers are assigned
to projects where hard work is not productive.
5
The Role of E¤orts
In the previous sections, the quality of a project was determined by an initial random shock.
With communication that random shock could be elicited at no cost by the designer. An
agent’s role in the …rst stage of a production process was simply to gather information.
There are many situations, however, where the quality of a project would be determined
by the e¤orts taken by an agent. In this section, we study this question by replacing the
20
interim shock µ in the …rst stage with an initial e¤ort level a. This e¤ort level is taken
by the agent initially assigned to a project and is private information to him. As with the
interim shock, an agent assigned to a new project in the second stage does not observe the
e¤ort a taken on it by the initial agent, and must take the recommended second stage e¤ort
b; regardless of e¤ort a. We focus our analysis on complementarity in production between
the two e¤orts.
To keep this problem tractable, we restrict our analysis to symmetric equilibria in which
there are no heterogeneous types, all the agents are treated identically ex ante, and all face
the same e¤ort pro…le over the two stages. Output on a particular project is a function
solely of e¤orts a and b taken on that project and a project-speci…c random shock. These
latter shocks across projects are uncorrelated. Speci…cally, if agents are not switched, the
production function on an agent’s project is written simply as p(qja; b), where p denotes the
probability of output q given e¤orts a and b in the two stages. If agents are switched, then
it is necessary to keep track of output on both projects to which an agent was assigned.
The production function from his perspective is p(q1 ja; b¤)p(q2 ja¤; b); where a¤ and b¤ are
the …rst and second stage e¤orts recommended to others, either after the agent leaves or
before he arrives, respectively. We adopt the convention that from the agent’s perspective,
q1 refers to his initial project whether or not he is switched and q2 refers to his second-stage
project if he is switched from his original project. Similarly, the consumption of an agent
who is not switched is c(q1 ), while the consumption of a switched agent is c (q1; q2).
Agents receive utility from consumption and disutility from e¤orts. Utility is separable
and written U(c) ¡ V (a; b), where U is strictly concave and V is strictly convex.
For analytical reasons we make several simpli…cations. First, we model switching as a
discrete decision made by the designer at the time of contracting. Thus, all participants in
the economy know beforehand whether or not they will be switched. Our second simpli…cation is to allow only the designer to send messages (recommending e¤orts) immediately
after contracting and not at an interim stage. This assumption precludes the designer from
recommending an interim-stage e¤ort a and then, after the interim stage e¤ort is taken,
sending a random message which recommends a …nal-stage e¤ort b.8 Finally, as noted, we
8
Examples can be generated where such a strategy is bene…cial. Unfortunately, it greatly complicates
the analysis of the switching problem.
21
also restrict our focus to symmetric contracts. Symmetry here means that agents face the
same contract and are recommended the same sequence of e¤orts. Essentially, our framework reduces the incentive constraints to a symmetric, pure strategy, Nash equilibrium in
the game played between the two agents. However, realizations of output may still di¤er
across the two projects.
We start by considering the no-switching contract. The optimal no-switching contract
solves
Program 5:
max
c(q 1);a;b
s.t.
X
q1
X
q1
P
q1
p(q1ja; b)U (c(q1 )) ¡ V (a; b)
p(q1ja; b)(c(q1) ¡ q1) · 0;
p(q1ja; b)U(c(q1)) ¡ V (a; b) ¸
X
q1
p(q1jba; bb)U(c(q1)) ¡ V (b
a; bb); 8b
a; bb:
(17)
(18)
Equation (17) is the resource constraint and equations (18) are the incentive constraints.
If agents are switched the problem changes. Now consumption can depend on output of
the two projects where an agent has worked. Since the treatment of the agents is symmetric,
we can keep the problem relatively simple.
Program 6:
max
c(q1 ;q2);a;b
s.t.
X
q1;q2
X
q1;q2
X
q1;q2
¸
X
q1;q2
p(q1 ja; b)p(q2ja; b)U (c(q1 ; q2)) ¡ V (a; b)
p(q1 ja; b)p(q2ja; b)(c(q1 ; q2 ) ¡ q1 ) · 0;
p(q1ja; b)p(q2ja; b)U(c(q1; q2)) ¡ V (a; b)
p(q1j^
a ; b)p(q2ja; ^b)U(c1(q1; q2 )) ¡ V (^a; ^b); 8^a; ^b:
22
(19)
(20)
Equation (19) is the resource constraint. Notice that the only output entering it is q1.
Some kind of formulation like this is needed because under switching, two di¤erent agents
work on a given project but there is really only one project per agent and we need to avoid
double counting. Put di¤erently, since the projects are identical and all agents are assigned
the same e¤ort levels we could have just as easily replaced q1 with q2.
The incentive constraints (20) re‡ect the ability of the agent to a¤ect output on his
initial project through e¤ort a and output on his second project through e¤ort b. The
other agent’s e¤orts on both of these projects are taken as given, that is, b when the agent
in question contemplates a or ba, and a when the agent in question contemplates b or bb.
5.1
Substitutes
We now consider an environment in which the initial and interim e¤orts a and b are perfect
substitutes in the production function, that is, on each project the probability distribution
of output is described by p(qja + b). It is sometimes useful to write total e¤ort as e ´ a + b.
A function satis…es the monotone likelihood ratio condition (MLRC) if p0 (qje)=p(qje) is
nondecreasing in q. Let P (qje) be the cumulative distribution function. Then, a function
satis…es the convexity of the distribution function condition (CDFC) if P 00 (qje) ¸ 0 for all
q and e. Finally we assume that V (a; b) = V (a) + V (b). These assumptions are su¢cient
for the use of the …rst-order approach in single agent problems, which will be used in the
proof.
Proposition 6 If initial and interim stage e¤orts are perfect substitutes in production, the
production function satis…es MLRC and CDFC, and V (a; b) = V (a)+V (b); then the optimal
no-switching symmetric equilibria strictly dominates all switching symmetric equilibria.
Proof: If an agent is switched, then he faces the option of deviating on project one,
project two, or both. The …rst-order conditions to the agent’s problem are:
X
q1;q2
X
q1 ;q2
p0(q1 ja + b)p(q2ja + b)U(c(q1; q2)) ¡ V 0(a) = 0;
(21)
p(q1ja + b)p0 (q2ja + b)U (c(q1 ; q2)) ¡ V 0 (b) = 0:
(22)
23
The …rst-order approach to incentive problems is not necessarily su¢cient in the switching
case. Nevertheless, these conditions are still necessary for a solution and that is all we need
for our proof. Our strategy is to show that given a contract satisfying (21) and (22), a
better, incentive compatible, no-switching contract can be designed. To do this, take any
contract satisfying (21) and (22). Construct a no-switching contract e
c(q1) by
U (ec(q1 )) =
X
q2
or, equivalently,
p(q2 ja + b)U(c(q1 ; q2 )); 8q1;
X
c(q1 ) = U ¡1(
e
p(q2ja + b)U(c(q1; q2))); 8q1;
(23)
(24)
q2
where U
¡1
is the inverse of the utility function U . Substitution of (23) into (21) delivers
X
q1
p0 (q1ja + b)U(e
c(q1)) ¡ V 0 (a) = 0:
(25)
Let Ve (e) = mina;b V (a) + V (b) subject to a + b = e. This object is the indirect disutility
e
of e¤ort received by the agent. By the Theorem of Maximum (actually minimum here) V
is a convex function like V . Because of the symmetry, any solution to the agent’s problem
will be characterized by a = b = e=2; which implies that Ve 0(e) = V 0 (e=2) = V 0(a) = V 0 (b).
Substituting into (25) delivers
X 0
p (q1 je)U (e
c(q1 )) ¡ Ve 0 (e) = 0:
(26)
q1
This is the …rst-order condition to the agent’s problem expressed in terms of total e¤ort
e. Furthermore, the …rst-order approach is su¢cient for the no-switching problem because
of the assumptions of MLRC and CDFC (see Rogerson (1985) or Hart and Holmstrom
(1987)). Therefore, the contract (e
c(q1); a; b) is incentive compatible in the no-switching
P
regime. Furthermore, by concavity of U, e
c(q1) < q2 p(q2ja + b)c(q1; q2)) which means
that there are more resources available in the economy. By a continuity argument, there
is a feasible no-switching contract that improves upon (e
c(q1); a; b). Therefore, no-switching
strictly dominates switching. Q.E.D.
No-switching contracts are powerful in this model because they allow incentives to be
focused on one project rather than two and thereby reduce consumption variation.
24
5.2
Complements
In contrast, switching can be valuable for production functions with complementarity in
the two inputs. For this result, we consider the production function p(qjf (a; b)), where
complementarities are expressed through the functional form of f . A probability function p1 exhibits …rst-order stochastic dominance over another probability function p2 if
P
P
x p1(x)g(x) ¸
x p2(x)g(x) for all weakly increasing functions g. We assume that p
exhibits …rst-order stochastic dominance in the value of f , that is, for f(a; b) > f(e
a; eb),
p(qjf(a; b)) …rst-order stochastically dominates p(qjf(e
a; eb)). The function f is symmetric
in the two inputs. More importantly, it is characterized by expected output being highest
along the diagonal of (a; b) space, increasing as both inputs are increased, but dropping
rapidly as one moves o¤ the diagonal. Speci…cally,
Assumption 1 The function f (a; b) is symmetric. Higher e¤orts a and b with a = b, that
is, upward movement along the diagonal, increase f(a; b). However, increasing one e¤ort
while maintaining the other - movement o¤ the diagonal - reduces f(a; b) the larger the
di¤erence between a and b. Formally, the value of f decreases as ja ¡ bj increases.
One implication of Assumption 1 is that for a 6= b, f(a; b) < f(a; a) and f(a; b) < f(b; b).
An example of a production function that satis…es Assumption 1 is f (a; b) = min(a; b) ¡
ja ¡ bj:
Proposition 7 If p(qjf (a; b)) satis…es …rst-order stochastic dominance, f satis…es Assumption 1, V (a; b) = V (a) + V (b), and the consumption sharing rule in the solution to
the no-switching program, c(q1), is weakly monotonically increasing then the best switching
contract strictly dominates the best no-switching contract.
Proof: See Appendix.
When an agent is switched, he plays a Nash-like game in e¤orts with the other agents
assigned to his projects. As a consequence, all of his feasible deviations push him to
o¤-diagonal e¤ort pairs, which are extremely unproductive and relatively inexpensive to
prevent. In contrast, an agent who is not switched can deviate in both stages so as to take
on-diagonal e¤ort pairs. These are more productive and thus more expensive to prevent.
25
6
Discussion
The previous sections isolated forces for reassignment in the assignment model. In many
applications several of these forces, even opposing ones, may operate simultaneously. One
interesting class of models builds upon the previous sections by having the initial e¤ort
a determine the interim state µ stochastically by the production function h(µja): As in
Sections 2 and 3, µ is private information. Next, like the earlier models, the agents report
on their shock, reassignments are made, and then the agents take their private secondstage e¤orts b that determine the publicly observed outputs q according to the production
function p(qjb; µ); the shock µ is the interim state of the project an agent is assigned to in
the second stage.
Analyzing this model is di¢cult. Even writing down the switching version of the model
is not straightforward. One version that can be analyzed, however, is the case where q = bµ.
If agents are not switched, there are lots of incentive constraints. Working backwards there
are no incentive constraints on the second-stage e¤ort if there is a truthful report and the
initial action was taken. The deterministic production function allows the designer to infer
e¤ort b from the output and µ. However, there are still a myriad of incentive constraints
on the report on µ as well as on the initial action.
The following switching contract eliminates the truth-telling constraints on µ: Upon
receipt of the agents’ reports, the designer randomly assigns agents to projects of quality
µ by the distribution h(µja) and then recommends each agent an e¤ort b that may depend
on the quality of their new project. In equilibrium the designer knows the state of each
project so – just like in the the no-switching contract considered above – deviations in b can
be detected and thus prevented. But now the same process can also be used to eliminate
the truth-telling constraints. Unlike in the no-switching contract, if an agent lies about the
state of his original project he cannot continue the cover-up by adjusting his e¤ort from the
recommended level. Instead, some other agent will work the recommended e¤ort and then
produce an output that reveals the initial deception. Consequently, µ is public information
for all intents and purposes and the problem collapses to an almost standard moral hazard
problem where the only incentive problem is on the initial e¤ort a.
In this example, perfect inference is only possible because of the deterministic pro26
duction function. For more general production functions p(qjb; µ), where inference is less
than perfect, the analysis is not so clear. Forces like those studied in Section 5.1 would
push towards no-switching assignments. Limits on communication could work in a similar
direction. Still, which force would dominate seems hard to ascertain at this point.
A
Proofs
Proposition 2 If preferences are Leontief and if µ j > 0 for all j, then switching dominates.
Proof: Leontief preferences forces solutions to the no-switching model to satisfy c(µ) =
c. Because consumption is not state-contingent, a solution to the no-switching problem is
characterized by b(µ) satisfying µb(µ) = µ0 b(µ 0), for all µ, µ 0, that is, all agents must produce
the same output regardless of their shock µ. The less productive a worker is the harder he
works. Formally, b(eµ) > b(µ) for e
µ < µ and total output for the no-switching contract, qns ;
is
qns ´
X
h(µ i)µ ib(µ i):
i
27
Now consider a switching contract with b =
P
i h(µ i )b(µ i );
the previous average e¤ort of
the no-switching contract. Then total output for switching, qs is
X
X
qs =
h(µ i )µi (
h(µ j)b(µ j ))
all j
i
=
X
h(µ i )µi h(µ i)b(µ i) +
i
X
X
h(µ i)µi (
h(µj )b(µ j))
j6=i
j6=i
i
j 6=i
(adding and subtracting a common term)
X
X
X
=
[h(µ i)µ i h(µi )b(µi ) +
h(µ j )µi h(µ i)b(µ i) ¡
h(µ j )µ ih(µ i)b(µ i)]
i
+
=
X
X
i
i
X
h(µ i)µi (
h(µj )b(µ j))
j 6=i
µ i h(µi )b(µi ) ¡
= qns ¡
= qns ¡
= qns +
XX
i
i
n X
i
X
h(µ j)µ i h(µi )b(µi ) +
j6=i
h(µ j )µ ih(µ i)b(µ i) +
X
i
j6=i
n X
i
X
X
i
X
h(µ i)µ i (
h(µ j )b(µj ))
j6=i
X
h(µ i )µi (
h(µj )b(µ j ))
h(µ i )h(µj )(µ i b(µ i) + µj b(µ j )) +
j6=i
n X
i
X
h(µ i )h(µj )(µ i b(µ j ) + µ j b(µ i ))
i=2 j =1
i=2 j=1
i=2 j=1
> qns:
XX
h(µ i)h(µ j )(µ i ¡ µj )(b(µ j ) ¡ b(µ i))
The last inequality holds because, again, µ i > µj and b(µ j ) > b(µ i). As a consequence,
the switching contract produces more output, which means (common) consumption c is
higher than under the no-switching contract. Furthermore, since e¤ort b is a convex combination of the no-switching e¤ort levels and since V is convex, disutility from e¤ort is
less with switching. Therefore, the best switching contract is strictly better than the best
no-switching contract. Q.E.D.
Proposition 7 If p(qjf (a; b)) satis…es …rst-order stochastic dominance, f satis…es Assumption 1, V (a; b) = V (a) + V (b), and the consumption sharing rule in the solution to
the no-switching program, c(q1), is weakly monotonically increasing then the best switching
contract strictly dominates the best no-switching contract.
Proof: We start with an optimal solution to the no-switching problem (a; b; c(q1)). It
is characterized by a = b. Now consider the following switching contract. The agent is
28
switched with probability one, e¤ort on project one is a; e¤ort on project two is b, and
consumption is
c(q1; q2 ) = U ¡1(0:5U(c(q1)) + 0:5U(c(q2)));
(27)
where c(q1) and c(q2) are the terms of the no-switch contract applied to both of the projects
the agent works. When the two outputs di¤er, consumption is set to a level that gives the
same utility as if the contract randomized over the two outputs in the no-switching contract.
If q1 = q2, then c(q1; q2) = c(q1), that is, consumption is unchanged. This contract gives
the same utility as the no-switching contract because
X
q1
=
X
q1
= 0:5
=
p(q1jf(a; b))
X
X
q1
p(q1jf(a; b))
q1
X
q2
X
q2
(28)
p(q2 jf (a; b))U(c(q1; q2))
p(q2 jf (a; b)) (0:5U (c(q1 )) + 0:5U(c(q2)))
p(q1jf (a; b))U(c(q1)) + 0:5
p(q1jf(a; b))U (c(q1 )):
X
q1
p(q2jf (a; b))U(c(q2))
The last line holds because of the symmetry in the problem. Furthermore, this contract
also uses less resources than the no-switching contract. Our strategy is to show that it is
incentive compatible under the switching regime. A continuity argument will then show
that better switching contracts exist.
Consider the deviating strategy ba, bb. The consumption part of utility is
X
p(q1jf(b
a; b))
q1
p(q1jf(b
a; b
a))
q1
·
·
X
X
q2
p(q1jf(b
a; b
a))
X
p(q2 jf(a; bb))U (c(q1 ; q2))
q
Ã2
X
q2
!
p(q2 jf (a; bb))U(c(q1; q2))
q2
X
(29)
p(q2jf(bb; bb))U (c(q1 ; q2)):
The …rst inequality holds because f(b
a; b
a) ¸ f (ba; b), p satis…es …rst-order stochastic domP
inance, and the term q2 p(q2jf(a; bb))U(c(q1; q2 )) is an increasing function in q1. (To see
this, substitute (27) for c(q1; q2) and recall that c(q1) is weakly increasing by assumption.)
The second inequality holds based on a similar argument with respect to q2.
29
Now, consider the entire utility from taking deviating strategy ba, bb. Utility from this
strategy is
X
q1
·
X
q1
= 1=2
p(q1jf(b
a ; b))
p(q1jf(b
a ; ba))
Ã
X
q
+1=2
q2
· 1=2
q1
+1=2
=
Ã
XX
q1
q2
q2
X
q2
p(q2jf (a; bb))U(c(q1; q2)) ¡ V (ba) ¡ V (bb)
p(q2jf(bb; bb))U(c(q1; q2)) ¡ V (ba) ¡ V (bb)
(30)
!
p(q1jf(b
a; b
a))U(c(q1)) ¡ V (b
a) ¡ V (ba)
Ã1
X
Ã
X
X
!
(31)
!
(32)
p(q2jf(bb; bb))U(c(q2)) ¡ V (bb) ¡ V (bb)
!
p(q1jf(a; b))U(c(q1)) ¡ V (a) ¡ V (b)
X
q2
p(q2jf(a; b))U(c(q2 )) ¡ V (a) ¡ V (b)
p(q1jf(a; b))p(q2jf(a; b))U(c(q1; q2 )) ¡ V (a) ¡ V (b);
(33)
which is the incentive constraint. Inequality (30) follows from (29). Equality (31) follows
from substituting (27) for c(q1; q2 ) and using (28). Inequality (32) follows from the noswitching incentive constraints. Finally, (33) is obtained by using (28).
The constructed switching contract is feasible, gives the agent the same utility, and uses
less resources. By continuity, a feasible switching contract exists that increases the agent’s
utility. Q.E.D.
30
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31
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