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Working Paper Series
Working Paper No. 01-5 August 2001
DECISION PROCESSES, AGENCY PROBLEMS, AND INFORMATION: AN ECONOMIC ANALYSIS
OF BUDGET PROCEDURES
Anthony M. Marino and John G. Matsusaka
This paper can be downloaded without charge at:
The FBE Working Paper Series Index: http://www.marshall.usc.edu/web/FBE.cfm?doc_id=1491
Social Science Research Network Electronic Paper Collection:
http://papers.ssrn.com/abstract_id=239252
Decision Processes, Agency Problems, and Information:
An Economic Analysis of Budget Procedures
Anthony M. Marino and John G. Matsusaka
University of Southern California
Many organizations attempt to manage agency problems not with
incentive contracts but by keeping the principal involved in the decision
process, that is, by limiting delegation. This paper develops a model to
investigate the economics of several decision processes that are commonly
used to set budgets in both the public and private sector. The key tradeoff
is that partial delegation allows the principal to reject those projects he
dislikes, but causes the agent to distort the information he transmits to the
principal.
August 2001
Comments are appreciated. We thank Tom Gilligan, Randall Kroszner, Fulvio Ortu, and
Jan Zabojnik for helpful discussions and workshop participants at Cornell University,
Hong Kong UST, MIT, University of Chicago, University of Hong Kong, University of
Kansas, and USC for constructive feedback. Research support was provided by the
Earhart Foundation, USC, and the Stigler Center at the University of Chicago. Please
contact the authors at [email protected] and [email protected] or
Department of Finance and Business Economics, Marshall School of Business,
University of Southern California, Los Angeles, CA 90089-1427. Updates will be posted
on http://www-rcf.usc.edu/~amarino.
1
I. Introduction
Managing principal-agent problems is a perennial challenge for organizations in
both the public and private sector. A common strategy is to structure the decisionmaking
process so that the principal retains some authority or to otherwise limit the amount of
delegation. Consider, for example, the different ways school districts set their budgets.
While complete delegation is most familiar—the principals (voters) leave the spending
decisions entirely to their agents (school board)—some school districts (in Oregon, New
York, and other states) employ referendums: the school board’s budget proposal must be
approved by the voters to go into effect. Other school districts allows voters to approve
the board’s spending plan in a modified form, or set the budget independent of the board
using voter initiatives. Also, in some school districts, the board’s budget proposal is
subject to ex ante caps on total expenditure or total taxes (California).1
Similar variation appears in corporations, or what might be more familiar to some
readers, universities. Here the principal is the CEO (dean) and the agent is a division
manager (department chair). Hiring decisions of staff (and junior faculty) are fully
delegated typically, while decisions concerning executives (senior faculty) are subject to
veto. Expenditures of a routine nature are delegated, while expenditures for new
1 Swiss cantons, which are analogous to American states, display an equally wide variation in decision
processes including complete delegation, mandatory referendums on new projects above a spending
threshold, initiatives, and town meetings (Feld and Matsusaka, 2000). For an extensive discussion of
various institutional processes employed by the U.S. Congress to contain delegation problems, see Kiewiet
and McCubbins (1991).
2
products, plants, and facilities are subject to veto. The trend in some universities toward
faculty-controlled expense accounts is a move toward delegation, and away from a
situation where travel expenditures, computers, and the like required approval.
The rich variety of decision processes observed in practice poses important
research questions. What explains why different organizations adopt different decision
processes and why different procedures are used within a given organization? And how
do the actual decisions that emerge depend on the procedure used to make them? Existing
theory provides only fragmentary guidance on these questions.
Our goal in this paper is to develop a model of the budgeting process that will fill
in part of the picture. We are especially interested in shedding some light on the tradeoffs
between decision processes that are actually observed in practice. Our analysis turns on
the superior information of the agent, which presumably is one reason the agency
relationship is established in the first place. If the principal is worried that the agent might
do the wrong thing, at first glance it would seem to be always in the principal’s interest to
retain the right to intervene in the decision. In the worst case where the principal is
completely uninformed, he can always rubber-stamp the agent’s proposal. The question
then boils down to, what is the cost of retaining a right to intervene? One potential cost
was emphasized recently by Aghion and Tirole (1997): if the agent can be overruled, he
might inefficiently reduce his information-collection effort. We focus on a problem that
has received less attention: the agent may distort the information he transmits to the
principal if he fears being overruled.2
2 Another distortion can arise at the implementation stage: Zabojnik (forthcoming) explores inefficiencies
that may occur when the agent is forced to implement a project that he particularly dislikes.
3
The essential tradeoff can be seen by comparing full delegation with a decision
process in which the principal retains a veto right. The budget process begins with the
agent proposing a spending plan. Under full delegation, the proposal goes into effect
immediately, while under the veto process, the principal has the right to reject it. We
assume the agent derives private benefits from spending and is therefore more willing to
go forward with a project than the principal is. Since the agent has superior information,
the principal will attempt to infer something about the project’s quality from the agent’s
spending proposal. If it is optimal to invest more in high quality projects, the principal
will view a large spending proposal as indicative of a high quality project. An agent with
a low quality project, then, may propose an excessively large budget in order gain the
principal’s approval. When the principal cannot separate good from bad proposals, the
principal can be worse off with a veto because projects become inefficiently large. One
implication is that spending actually can be higher under a veto process than under
delegation.
With this intuition in mind, we explore several issues. First, in what situations do
the benefits of delegation exceed the costs? A key result is that the principal prefers to
delegate in situations where an agent with a low quality project would mimic an agent
with a high quality project (that is, a pooling equilibrium). The principal prefers to retain
veto rights when mimicking does not occur (a separating equilibrium). The desirability of
delegation then depends on whether or not pooling occurs. Several implications follow:
(1) Delegation is optimal for projects with low variance (“routine”) while veto is better
for those with high variance (“innovative”). The reason is that it is less costly for the
agent to mimic a project that is just “a little” better than his project, than one that is
4
“much” better. (2) Delegation does not necessarily become better as the agent’s
preferences move into alignment with the principal’s preferences. It is possible for a
worsening of the agency problem to make pooling less likely because it requires a larger
proposal to successfully pool. (3) The information distortion problems at the center of our
analysis are less important when the agent’s information can be verified by an outsider.
Therefore, the model suggests that delegation is better when the agent’s information is
“soft” than when it is “hard.”
Second, we explore variants of the two basic budgeting processes. One variant is
to set an upper bound on the amount of spending ex ante. In our model, such a restriction
always reduces spending when the decision is delegated, but can increase spending when
the principal retains veto rights. This can happen if the spending threshold makes it easier
for an agent with a low quality project to pool. An implication is that spending limits are
more effective when coupled with delegated decisionmaking, and can be
counterproductive (increase spending) when coupled with a veto process.
Another variant allows the principal to approve the proposal in a modified form
rather than accept or reject it as is. Here again, an agent with a low quality project may
have an incentive to propose a large project in order to pool and avoid being overridden
by the principal. This process can deliver the principal’s unconstrained optimal outcome
in some circumstances, but is dominated by veto and delegation in others.
A third variant provides complete delegation for spending below a given amount
and veto otherwise. In our model, such an arrangement is generally superior to pure
delegation or pure veto, not because it helps the principal avoid time costs of trivial
5
matters, but because it reduces the likelihood of pooling by allowing an agent with a low
quality project to separate.
We also explore an alternative information structure—what happens if the
principal is also informed about the project’s quality? Casual intuition suggests that
delegation becomes less desirable if the principal is informed. This is not necessarily true
in our model. When the principal is informed, an uninformed agent may make too
aggressive a proposal, essentially relying on the principal to identify project flaws that he
did not see. This can create a pooling situation, and if so, the principal may find
delegation more desirable when he is informed.
Although the main purpose of our paper is to identify tradeoffs between existing
budget processes, it is also interesting to identify what an optimal budget process would
be in our model environment. We provide one answer to this by characterizing the
optimal decision process/mechanism using the revelation principle. It turns out that the
actual decision processes we study can be viewed as optimal mechanisms for some
parameter configurations, or as very close approximations. Delegation is the exception—
it can never be an optimal mechanism.
Our paper is related to several others in the literature. The pioneering work in the
area is the series of papers by Gilligan and Krehbiel (1987, 1989, 1990) that studied the
tradeoff between open and closed rules in parliaments.3 We follow that path by focusing
on actual decision processes, but expand the scope to consider delegation and other
processes that are employed outside of legislatures. Aghion and Tirole (1997) develop a
model to study the tradeoffs between complete delegation and complete non-delegation.
3 See Gilligan (1993) for a survey and more complete list of articles.
6
While they focus on how limiting delegation can affect the agent’s information collection
efforts, we study the effect on information transmission. We also incorporate institutional
details of actual decision processes. Harris and Raviv (1996, 1998) and Baron (2000)
study the properties of optimal decision processes using a mechanism design approach, in
corporate and legislative environments, respectively.4 Their approaches are
complementary to ours (especially where we consider optimal mechanisms) but the
emphasis is different. We are primarily interested in understanding the consequences of
observed budgeting practices (that may or may not be optimal) and not as much in
identifying the globally optimal practice.
The paper is organized as follows. Section II describes the model. Section III
develops the tradeoff between two simple decision processes, delegation and veto.
Section IV investigates three variants of the basic decision processes: spending limits,
override, and threshold delegation. Section V extends the model to consider an informed
principal. Section VI compares our approach to a mechanism design approach. Section
VII concludes.
4 See also the recent paper by Bernardo, Cai, and Luo (2000) that considers the interaction between
incentive contracts and budgeting procedures.
7
II. The Model
The model features a principal who employs an agent to evaluate projects and
make proposals. The principal provides the funding for the investment. For concreteness,
think of a school board deciding whether to construct a new school building. The
principal is the taxpayer who will ultimately provide the funds for the building, and the
agent is the school board. An alternative “corporate” interpretation is that the principal is
a capital budgeting committee and the agent is a division manager.
A. Sequence of Actions
There are three periods. In period 0, the principal adopts a decision process. In
period 1, the agent possibly receives information about a project’s value, and proposes a
level of funding. In period 2, the principal can reject the proposal (unless the decision is
fully delegated), and if approved, the investment is made and the project pays off.
B. Information
The underlying “quality” of the project (say, a new school building) is θ ∈ {H, L}
with probabilities π and 1 – π respectively, where H > L, and .][ ME ≡θ We can think of
θ as parameterizing the expected enrollments over the life of a new building.
The agent has private information: with probability p he knows the project’s
quality. Let I ∈ {L, M, H} indicate the agent’s information where M indicates no
information. At this point we assume that the principal is uninformed: he knows only the
distribution of θ. We consider the case of an informed principal below.
8
C. Project Return
A project’s gross return is )(nfθ , where n is the scale of the project and f is
increasing and concave with 0)0( =f . In our school board example, n could represent
the square feet of the building. The principal provides the funds for the project at a
(normalized) cost of 1 per unit.
D. Principal and Agent Utility Functions
The principal and agent are risk neutral. Since the principal pays the cost of the
project, his utility function is
(1) .)( nnfv −= θ
The utility function of the agent is assumed to be
(2) nvu α+= ,
where .10 <≤ α This formulation has two important features, both of which are fairly
standard in the literature. First, the agent cares about the principal’s utility, but second, he
also derives a payoff from project size per se. This payoff function could be a primitive,
or it could be induced by external sanctions. For example, if the school board has to stand
for re-election, its members are not oblivious to the will of the voters, but there are
enough frictions in the voting market that incumbent officeholders can capture some
rents. In a corporate context, (2) could be treated as the reduced form of a model in which
9
equilibrium contracts do not perfectly align principal and agent incentives. We shall
sometimes refer to α as the severity of the agency problem.
The agent’s utility function can be restated as
(3) .)1()( nnfu αθ −−=
A comparison of (2) and (3) indicates that the principal and agent in our formulation
differ only in their private opportunity cost of funds. The agent’s opportunity cost of a
unit of n is α−1 and the principal’s cost is 1. The consequence of this specification is
that the agent prefers a larger n than the principal does, other things equal.
It is necessary below to calculate principal and agent expected utilities conditional
on beliefs about the value of θ. We will express these expected utilities as uI and vI where
I is an actor’s expected value of θ conditional on his information. For example, uH is the
agent’s utility conditional on knowing the quality of the project is H. When the agent has
no information, his utility is uM.
To create an interesting conflict between the principal and agent, we assume they
may disagree about whether a project is worth funding. Both would like to go forward (at
least for some n) if the project is known to be high quality and neither wants the project if
it is low quality. They disagree when there is no information (I = M): The agent would
like to proceed but principal would like to stop. The formal statement of these
assumptions is this:
Assumptions. Principal’s utility function: 0vL < for all n, 0vM < for all n, 0max >Hn v .
Agent’s utility function: 0uL < for all n, ,0max >Mn u 0max >Hn u .
10
III. Two Simple Decision Processes: Delegation and Veto
We highlight the basic tradeoffs by comparing two simple decision processes. The
first is (complete) delegation: the agent is given the power to approve the project at
whatever scale he chooses and cannot be overruled by the principal. The second is veto:
the agent proposes a scale and the principal can either approve it without modification or
reject it completely.5 Both processes are common in practice. The public sector version of
veto is a mandatory referendum. For example, in many school districts the school board’s
budget proposal does not go into effect if a majority of the voters reject it (see Romer,
Rosenthal, and Munley, 1992). Most states require referendums for debt issues (Bohn and
Inman, 1996) and most Swiss cantons require referendums for new public expenditures
(Feld and Matsusaka, 2000).6 Something like a veto process appears in corporate
budgeting as well.7 Later, we will consider other and more complicated decision
processes.
5 Our depiction of the veto process follows Romer and Rosenthal (1979) as extended by Banks (1990).
6 The parliamentary closed rule is another example (Gilligan and Krehbiel, 1987).
7 Bower’s (1970) well known study of capital budgeting practices characterizes one (allegedly
representative) company as follows (page 65): “The (executive committee) review varied in thoroughness
depending in large measure on the extent of the project’s controversialism, but always the result of the
review was ‘go’ or ‘no go.’ The definition of a project did not change.” The interesting feature here is that
executive committee did not modify proposals.
11
A. Complete Delegation
Under complete delegation, the project goes forward at the agent’s optimal
spending level. Let *In be the optimal scale for the agent (the maximand of (3))
conditional on his information, I. When nonzero, the optimal scale solves α−=′ 1)( *InfI .
The optimal scale is increasing in I. By assumption, 0* =Ln . The principal’s (period 0)
expected utility under complete delegation (D) is then
(4) ).()()1(]|[ **0 HHMM nvpnvpDvE π+−=
The first term is less than zero. We assume that 0)( * >HH nv so that the principal’s
expected utility from delegation is not trivially negative.
B. Veto
Under the veto process, the principal can reject the proposal. He will do so if he
infers from the proposal that the agent has no information. The uninformed agent (“M-
agent”) takes this into account when making his proposal. In particular, he may propose
the scale that an H-agent would have chosen, that is, the M-agent may pool with the H-
agent.
A number of different outcomes are possible depending on the parameter
configurations, but the interesting economics can be seen by comparing equilibria in
which agent proposals pool with those in which they separate. The most transparent cases
attain when the principal is willing to accept the H-agent’s optimal project size
conditional on knowing that the agent has },{ HMI ∈ . Therefore, we assume that
12
0)( * >HR nv , where )1/())1)(1(( πππ ppLpHR +−−−+= is the expected project value
conditional on LI ≠ . Given this, there are two (Perfect Bayesian) equilibria
distinguished by one simple condition.
Pooling equilibrium: When 0)( * >HM nu , the H-agent and M-agent both propose *Hn , the
principal accepts a proposal of *Hn , and rejects all other proposals.
This is an equilibrium because no agent type gains from making a different
proposal, and the principal cannot do better with an alternative adoption strategy. The
proof is straightforward. Obviously, the H-agent, who is receiving his globally optimal
outcome, will not deviate. The M-agent’s payoff is positive in equilibrium, but zero if he
deviates because his proposal will be rejected. Finally, the principal’s behavior is optimal
along the equilibrium path because 0)( * >HR nv , and his rejection of proposals off the
equilibrium path is optimal if he believes those deviations come from an M-agent, which
is the only reasonable conjecture.8
8 This is the only equilibrium that survives the usual refinements. For example, there is a Perfect Bayesian
equilibrium where the agents pool at δ−*Hn , the principal accepts this proposal, and rejects all others. For
the principal to find it optimal to reject a deviation of *Hn , he must believe the agent’s conditional expected
type is less than R (since 0)*( >HR nv ). However, beliefs that give such a large weight to M are eliminated
by the standard refinements, such as the intuitive criterion and D1, since only the H-agent would like to
deviate to *Hn . Equilibria that pool at δ+*
Hn do not survive refinements for similar reasons.
13
The important feature of this equilibrium is that the agent may ask for a larger
budget than he would like in an effort to mislead the principal about the project’s
prospects. This is somewhat counterintuitive: the principal knows that the agent is
excessively fond of spending, but the agent fears that his proposal will be rejected if it is
too small. The agent’s incentive to pad his proposal when the principal is involved in the
decision drives the key tradeoffs in the model. We are unaware of systematic evidence
that can tell us how often this happens, but the history of online grocer Webvan provides
an interesting case study.
Webvan opened for business in July 1999. Customers went to the company’s web
site to order groceries, which were then home-delivered in the company’s signature blue-
green vans. The company raised $1.2 billion on the equity markets, more than any online
retailer except Amazon.com. However, Webvan lost approximately $100 million each
month, ran out of cash, and closed down and liquidated in July 2001. The problem,
according to analysts, was that Webvan set out to enter 26 markets before it had figured
out how to turn a profit in one: “They needed to operate in one market, get their model
perfect and show they could have a positive cash flow. No one has ever gone public with
a national rollout with zero markets performing to plan.” Why didn’t Webvan start in a
single market and incur only minor losses until it figured out how to operate in the black?
According to a venture capitalist who backed Webvan, “It’s easy to say, `Man, you could
have done a few less markets,’ but there was a huge Catch-22. There was a unique
opportunity to raise a huge amount of capital in the public market so we could build a
business far faster than Sam Walton rolled out Wal-Mart. But to raise money, you had to
get above the noise level, build a brand name, and make big promises to investors.” This
14
is exactly the sort of behavior our model tries to capture: Webvan’s backers felt they
needed to propose a bigger project than they really wanted in order to attract funding. 9
In equilibrium, the project goes forward at a scale of *Hn if the agent’s
information is M or H. The principal’s expected payoff under veto (V) in this equilibrium
is then
(5) ).()()1(]|[ **0 HHHM nvpnvpVvE π+−=
Separating equilibrium: When 0)( * <HM nu , the H-agent proposes *Hn , the M-agent
proposes *Hnn ≠ , the principal accepts a proposal of *
Hn , and rejects all other proposals.
The proof is identical to the one above, except that here the M-agent would rather
not have the project at all than operate it at the H-agent’s preferred scale. In equilibrium
the project goes forward only if the agent knows that I = H. Then the principal’s expected
return is
(6) ).(]|[ *
0 HH nvpVvE π=
To summarize, there are two possible equilibria, and which one attains depends
on whether the uninformed agent earns a positive return from mimicking the H-agent’s
9 The first quote is from Lauren Levitan, an analyst with Robertson Stephens, and the second is from David
Beirne with Benchmark Capital. Both were taken from “ Some Hard Lessons for Online Grocer,” New York
Times, February 19, 2001.
15
proposal.10 The possibility of both pooling and separating behavior here is a general
feature of signaling games of this type, and does not depend on discreteness in our model
or the particular parameter configurations we have assumed. For example, see the
continuous models of Crawford and Sobel (1982) and Banks (1990).
C. Comparison of Delegation and Veto
Now we compare the two decision processes from the principal’s point of view.
The principal chooses a decision process in period 0, and we assume that he can commit
to it. In practice, it is difficult to find an environment where the principal can unalterably
commit to a decision process since some sort of escape route is usually available. But
reasonably effective commitment devices are often present. In our school board example,
the referendum is provided by the state legislature or constitution, making it costly to add
or remove. For all practical purposes, whatever decision process is in place at the start of
a budget cycle cannot be altered by the electorate until the next budget cycle at the
earliest. Commitment also can be supported by reputation or repeated play, as discussed
by Baker, Gibbons, and Murphy (1999).
Casual intuition suggests that the principal would always prefer veto to
delegation, since it can be exercised at no opportunity cost. This turns out not to be true
in some situations.
Proposition 1. The principal prefers delegation when the veto equilibrium pools
( 0)( * >HM nu ), and prefers veto when the veto equilibrium separates ( 0)( * <HM nu ).
10 If 0)( * =HM nu , then there are both pooling and separating equilibria.
16
The proof follows from comparison of (4), (5), and (6). The intuition is this:
Under both delegation and veto, the project goes forward in the H state at scale *Hn and
does not go forward in the L state. The difference in decision processes appears in the M
state. In this state, the principal’s payoff is negative for any 0>n , and increasingly so as
n rises. Under delegation, the project is implemented at a scale of *Mn . Under veto with
pooling, the project also goes ahead, but at an even larger scale, *Hn , which is worse for
the principal. In contrast, under veto with separation, the project does not go forward,
which is ideal for the principal.11
The basic tradeoff can be summarized as follows: the benefit of veto is that it
allows the principal to reject some projects he dislikes; the cost is that agents will pad
their proposals to make them appear more valuable. Whether delegation or veto is
optimal depends on how willing the agent is to pad his proposal.
A conventional explanation for delegation is to economize on the principal’s time.
Proposition 1 identifies another possible rationale for delegation, and suggests that the
principal may wish to delegate even if the opportunity cost of his time is small.
11 Note that the agent’s utility also depends on the choice of decision process. In general, this could be
relevant for the principal if it affects the wage paid to the agent. We abstract from this issue by assuming
that the agent’s participation constraint does not bind. See Aghion and Tirole (1997) for a model where the
participation constraint plays an important role in the choice of a decision process.
17
D. When is Delegation Better than Veto?
The next question is what determines whether delegation or veto is optimal for the
principal? Proposition 1 indicates that the answer depends on whether the M-agent pools
with the H-agent or separates under the veto process. Formally, delegation is better when
0)( * >HM nu . Several observations follow.
(1) Since )( *HM nu does not depend on p, an immediate result is that the optimal decision
process does not depend on the probability that the agent is informed. Casual intuition
might suggest that full delegation is better when the agent is more likely to be
informed. This is not the case in our model because the optimality of delegation does
not turn on the agent’s information—he knows the same amount regardless of the
decision process—but on the likelihood of pooling.12
(2) Veto becomes better when H rises holding constant M. An increase in H causes *Hn to
rise, and thereby )( *HM nu to fall. A large enough fall in )( *
HM nu makes veto optimal.
Intuitively, an increase in H holding constant M makes it more costly for the M-agent
to mimic the H-agent’s proposal. What this says, less formally, is that as the project’s
upside (or variance) increases, veto becomes more appealing for the principal. The
reason is that it becomes less likely that an uninformed agent would choose to pool.
12 An increase in p will decrease the magnitude of the difference between the principal’s payoff under the
two decision processes, although it will not change the sign of the difference. Changes in p are irrelevant
only if they do not violate our parametric assumptions, particularly that 0)*( >HR nv .
18
This logic suggests that delegation might be better for routine tasks with little upside
potential, while veto is optimal for new and innovative projects.
(3) Casual intuition again suggests that as the agency problem becomes more severe, veto
is a better choice. This is not necessarily true in our model: an increase in α can make
delegation or veto optimal. To see why, note that the condition 0)( * >HM nu can be
restated using the definitions of Mu and *Hn as )(/ *
HnHM ε> , where
)(/)()( nfnfnn ′=ε is the elasticity of f. An increase in α causes an increase in *Hn ,
which can raise or lower ε depending on the precise form of f.13 More intuitively, a
rise in α increases *Hn , which makes pooling less attractive for the M-agent, but the
rise in α also reduces the M-agent’s cost of n. The net effect depends on which of
these two forces dominates. The bottom line is that there is not a simple connection
between severity of the agency problem and the desirability of veto.
(4) The general sense of the model is that veto is problematic when it causes the agent to
distort his proposal. We expect this problem to be most severe in situations where the
agent’s information is “soft” in the sense that it cannot be verified by anyone else.
When information is “hard” the principal could set up an auditing processes that
13 One specification in which delegation becomes a better choice when α rises is 2
)( nnnf −= , with the
restriction 5.0<n . Here, ε is decreasing in n. If nenf −−= 1)( with the restriction 1<n , then ε is
increasing in n, and an increase in α has the opposite effect.
19
might allow him to tap the agent’s information with less risk of distortion in the
project. Our model suggests that delegation is better for the principal when
information is soft, and veto is better when information is hard.14
E. Spending Level
Another interesting question is how does the decision process affect the size of
the project? Empirical evidence suggests that spending outcomes are different under full
delegation and veto.15 We can express project size/spending in terms of our model as
E0[n]. The next proposition follows from simple calculation.
Proposition 2. The expected value of spending is higher under delegation than under veto
when the veto equilibrium separates, and is lower when the veto equilibrium pools.
(1) Casual reasoning suggests that when the agent is more inclined toward spending than
the principal is, a veto process results in less spending. This would be true in a
complete information model (for example, an extension of Romer and Rosenthal
(1979)), where veto has a pruning effect. It is not necessarily true in our asymmetric
information model. In the separating case, veto does result in lower spending because
the principal shuts down the project when the agent is uninformed. On the other hand,
14 This contrasts with the model of Aghion and Tirole (1997) in which the optimal decision process does
not depend on whether information is hard or soft.
15 See Matsusaka (1995, 2000) and Feld and Matsusaka (2000) for evidence and references to the literature.
20
in the pooling case, spending is higher under veto because the uninformed agent
increases his proposal in order to pool with the H-agent.
(2) Under delegation, a decline in α causes spending to fall. This is not necessarily true
under veto. A decline in α does result in less spending holding constant the type of
equilibrium (pooling or separating). However, a large enough fall in α can transform
a separating equilibrium into a pooling equilibrium and therefore cause spending to
rise. Intuitively, a fall in α reduces the H-agent’s proposal, which could make pooling
more attractive to the M-agent.
(3) An important feature of our depiction of the veto decision process is that proposals do
not necessarily indicate the agent’s underlying demand. A recent study of
congressional water project proposals by DelRossi and Inman (1999) contains some
pertinent evidence. Water project proposals originate with individual congressmen
who presumably see a benefit to their district from the project, and those projects
acceptable to House as a whole are passed in an omnibus water project bill. Prior to
1986, the funds for water projects were taken from general revenues. In 1986, the
rules were changed so that local taxpayers had to contribute part of the funding.16
When the rule change went into effect, congressmen with pending proposals were
allowed to modify their proposals in light of the new cost-sharing requirement.
16 To oversimplify, prior to 1986, the local government only had to contribute the land necessary for the
project, while after 1986 an additional cash contribution on the order of 25 percent of cost was required.
21
In terms of our model, think of Congress as the principal (source of funds) and the
individual representative as the agent. Prior to 1986, the individual representative’s
private cost of funds was approximately zero (that is, 01 ≈−α ) while after 1986 it was
positive ( 25.01 ≈−α ). Therefore, the reform reduced α. The prevailing decision process
was essentially veto—the Congress did not modify a member’s proposal, but could reject
it.17 DelRossi and Inman document that on average, representatives responded to the
reform by cutting back the size of their proposals. This is what we expect if proposals
reflect the agents’ demands. However, 30 of 82 sample projects were increased in size in
the wake of the reform. Such behavior is inconsistent with a complete information (non-
strategic) view of the veto process, but can arise naturally in our model.18
17 In the years studied by DelRossi and Inman, 85 of 335 proposals were rejected from the omnibus bill.
18 It is also inconsistent with the view that a complete delegation process was used. Since proposals appear
to have been strategic, this calls into question DelRossi and Inman’s assumption that the proposal revealed
the individual representative’s demand.
22
IV. Variants of the Simple Decision Processes
A. Ex Ante Spending Limit
A popular variant of the delegation and veto decision processes is to limit the total
amount of spending ex ante. In our school board example, this might take the form of a
restriction on spending growth based on a formula that incorporates enrollments and
inflation, or less directly, a statutory maximum property tax rate. We can model a
spending limit as an upper bound, N, on the project size that is set in period 0 and cannot
be altered thereafter.
Consider a spending limit with delegation first. It is clear that N > *Hn would have
no effect. As N falls below *Hn , the spending limit cuts the size of the H-agent’s project.
This makes the principal better off, at least until N reaches the principal’s optimal
spending level in the H state. Reductions in N below this point will continue to cut
spending, although this benefits the principal only if the gains from reducing the M-
agent’s proposal (if any) exceed the losses from reducing the H-agent’s proposal.
Now consider a spending limit in the context of the veto process. As above, a
spending limit in excess of *Hn does not bind. A spending limit set below *
Hn reduces the
project size in the pooling equilibrium. However, in the separating equilibrium, a
spending limit below *Hn may increase the expected project size. This can happen if the
spending limit reduces the H-agent’s proposal to the point where the M-agent becomes
willing to mimic it, that is, if it transforms a separating equilibrium into a pooling
equilibrium. In this case, delegation becomes more desirable than veto for the principal.
23
Intuitively, by constraining the H-agent, a spending limit makes it harder for him to
separate from the M-agent. This leads to the next proposition.
Proposition 3. (a) A binding spending limit reduces spending under delegation but can
increase spending under veto. (b) For a sufficiently low spending limit, delegation is
always (weakly) optimal.
One thing Proposition 3 suggests, somewhat counterintuitively, is that a spending
limit and the veto process might not be complementary ways to address an agency
problem. Rather, they may be substitutes. In practice, then, we would expect to see
spending limits more often in situations where delegation is used than those where veto is
used. Another empirical implication is that spending limits are more effective (cut
spending by a larger amount) when used in conjunction with a delegation process than
when used with a veto process.
B. Override
In the veto decision process, the principal commits to approve or reject the agent’s
proposal, and does not have the option to adopt the proposal in a modified form. While
this is a reasonable description of the referendum and parliamentary closed rule, in other
situations the principal retains the right to substitute his own proposal for the agent’s
proposal. Here we examine a decision process called override in which the principal is
24
free to reject the agent’s proposal, accept it “as is”, or approve the project at a scale of his
own choosing.19
Under the override decision process, the agent’s actual proposal is formally
irrelevant. All that matters is the agent’s report to the principal of the state, L, M, or H.
The L-agent reports truthfully and the principal does not proceed with the project. The H-
agent also reports truthfully because he has nothing to gain by pretending to be an L-
agent or M-agent. The M-agent faces a choice: if he reports truthfully, then the project
will be rejected, while if he lies and reports H then the project will be implemented at
**Rn , the principal’s optimal scale conditional on },{ HMI ∈ , that is, the solution to
1)( ** =′ RnfR (similarly, define **Hn as the solution to 1)( ** =′ HnfH ).20 As with veto, the
interesting equilibria depend on whether the M-agent pools or separates. Our earlier
assumption that 0)( * >HR nv implies that 0)( ** >RR nv , which narrows the field to two
equilibria.
Pooling equilibrium: When 0)( ** >RM nu , the H-agent and M-agent both report H, the
principal rejects the project if the agent reports L or M, and approves the project at a
scale of **Rn if the agent reports H.
19 This is the non-delegation decision process studied in Aghion and Tirole (1997).
20 We use one asterisk to indicate the agent’s optimal spending levels, and two asterisks to indicate the
principal’s optimal spending levels.
25
Separating equilibrium: When 0)( ** <RM nu , all agents report truthfully, and the principal
rejects the project if the agent reports L or M, and approves the project at a scale of **Hn
if the agent reports H.
In the separating case, the outcome is the principal’s global optimum. Therefore,
override dominates delegation in the separating equilibrium. In the pooling case, the
principal can be better or worse off under override than delegation. To see this, note that
the difference between the principal’s expected utility under delegation (D) and override
(O) is
(7) ( ) ( ).)()()1()()(]|[]|[ ******00 RMMMRHHH nvnvpnvnvpOvEDvE −−+−=− π
Equation (7) is opaque—each of the two terms can be positive or negative. However, it is
not difficult to show that 0]|[]|[ 00 >− OvEDvE for some parameter values, that is,
delegation can be optimal. The easiest way to show this is with a numerical example; see
paragraph (2) below. The intuition is that delegation is optimal when the agency problem
is not too severe (α is not too large) but the costs of pooling are high (roughly, override is
costly when **Rn is far from the principal’s optimal n in both the M and H states).
Proposition 4. The principal prefers override to delegation when the override
equilibrium separates )0)(( ** <RM nu , but prefers delegation for some parameter values
when the override equilibrium pools.
26
Proposition 4 clarifies one issue concerning our key result in Proposition 1. One
might suspect that delegation outperforms veto there because it restricts the principal’s
ability to react to the proposal. Proposition 4 shows that even if we allow the principal to
react in any way to the proposal, delegation can still be optimal, and the reason is the
same: when the principal retains decision rights, the agent distorts his information
transmission, possibly to the principal’s detriment.
(1) What determines whether the principal prefers override or delegation? One way to
answer this question is to identify sufficient conditions for
0)1()()( ****** <−−= RRRM nnMfnu α , because override is better when the override
equilibrium displays separation. A first observation is that a sufficiently low α makes
override optimal. This follows from the fact that **Rn does not depend on α, and is
somewhat contrary to conventional wisdom—the right to override is more valuable
when agency problems are modest. Second, since **Rn is increasing in R, a sufficiently
large R holding constant M makes override optimal. This reinforces the message from
the veto case that delegation is good for projects with a low upside potential, what
might be thought of as routine decisions.
(2) In some situations, the principal may be able to choose between delegation, veto, and
override. A complete comparison of the three processes is algebraically intensive and
yields few additional insights. But it can be shown that each of the processes is
optimal for some parameter configurations. Here is a particular example. Let
2)( nnnf −= for 5.0<n , and let 4=H , 0=L , 5.0=π , α = 0.5, and p = 0.8. This
27
specification meets all of the model assumptions and 0)( ** >HM nu and 0)( * >HR nv .
The override pooling equilibrium generates an expected payoff for the principal of
0.090, delegation generates 0.097, and veto (which separates) generates 0.109.
Override is dominated by delegation—as noted in Proposition 4—and veto is superior
to both override and delegation.
C. Threshold Delegation
Another common decision process is a hybrid of delegation and veto/override that
we call threshold delegation: a spending level is set below which decisions are delegated
and above which they are subject to veto or override. Public sector examples of threshold
delegation appear in Swiss cantons, where new projects that cost more than a certain
amount require approval by referendum, and American states where debt issues above a
threshold amount trigger referendums. Threshold delegation also is used in corporations
and other organizations, where managers can approve small expenditures independently,
but a budgeting committee must approve large expenditures. We shall model this as a
process with a spending level, T, in which the decision is delegated when Tn ≤ , and
subject to override when Tn > .21
We want to identify when threshold delegation can be better for the principal than
delegation and override alone, and bring out its economic logic. Threshold delegation
offers no advantage when the override equilibrium exhibits separation because override
delivers the principal’s optimal outcome. Therefore, we must focus on the case where the
21 The case of threshold delegation with veto (instead of override) is similar.
28
override equilibrium features pooling. The problem with override here is that the M-
agent’s proposal is far too large—larger, even, than the agent would choose if he could
act without constraint. The problem with delegation is that the M-agent and the H-agent
pad their proposals.
Threshold delegation can address both problems. To see this, we need to
characterize equilibrium behavior under threshold delegation. Let 0Mn be the (minimum)
project size that gives the M-agent the same payoff as **Hn , that is, the solution to
)()( **0HMMM nunu = . For the purposes of this discussion we assume that ***
HM nn < so that
**0HM nn < exists. The equilibrium under threshold delegation with ),( **0
HM nnT ∈ is the
following: the M-agent proposes },min{ * Tnn M= , the H-agent proposes Tn > , and the
principal overrides any proposal greater than T with **Hn .22 The M-agent stays below the
threshold because ),( **0HM nnn ∈ gives him a higher payoff than **
Hn . The threshold is
below the H-agent’s optimal project size, so he accepts the override outcome (or T, if it is
greater) instead of proposing the smaller project size that would free him from the
principal’s oversight.
How does the principal fare in this situation? The principal is obviously no worse
off in the H state because the proposal ends up at his optimum. In the M-state, if the
threshold is set below **Rn then the project goes forward at a scale no larger than it would
22 A belief structure that supports this equilibrium is the principal assigning a deviation to the H-agent with
probability 1. These are the only “reasonable” beliefs when *MnT ≥ . When *
MnT < , a pooling equilibrium
in which the threshold is ignored may exist. Since the tradeoffs in that case are already discussed above,
throughout this section we assume that the agents play to the separating equilibrium described in the text.
29
have under either delegation ( *Mn ) or override ( **
Rn ); this is also better for the principal.
Intuitively, threshold delegation in this equilibrium addresses the override “pooling”
problem by allowing the M-agent to separate and addresses the delegation “padding”
problem by overriding the H-agent (and possibly by restraining the M-agent). Thus, we
have the following result.
Proposition 5. The principal prefers threshold delegation with ),( **0RM nnT ∈ to
delegation and override when the override equilibrium pools.
The economic function of threshold delegation in our signaling model is to allow
the M-agent to separate but still restrain the H-agent. The same benefit arises if we
consider veto instead of override. Our argument is similar to that of Harris and Raviv
(1998), in which something akin to threshold delegation is shown to be the optimal
decision process in a mechanism design framework.23
Casual observation suggests that this decision process is common in practice.
Decisions with limited financial consequences are left to the agent while the principal
keeps a hand in expensive decisions. One benefit of this arrangement is to economize on
23 Our approach and that of Harris and Raviv may overstate the benefits of threshold delegation by ignoring
a problem that is important in practice: the agent may (inefficiently) subdivide a large project into several
smaller projects in order to evade the spending threshold. This is a serious problem in Swiss cantons (see
Feld and Matsusaka, 2000) and probably in corporations, too. For example, see Bower’s (1970, pages 15-
16) discussion of a division that built and equipped an entire plant on expense orders in order to avoid the
$50,000 threshold that required approval of top management.
30
the principal’s time—it is not efficient for him to weed out the smallest inefficiencies.
Our analysis suggests that threshold delegation may have another benefit. By allowing
the agent to overspend “a little” on small projects, it prevents even larger distortions that
might occur if the agent had to justify his project to the principal.
A final question of interest is what determines the optimal threshold? Note that
the principal wants to set the threshold as low as possible without inducing the agent to
pool, which means the optimal threshold is 0*MnT = . Several implications can be derived
from the fact that 0Mn is the solution to ****00 )1()()1()( HHMM nnMfnnMf αα −−=−− . First,
*T is decreasing in H, holding constant M. This mirrors our results above: as the project
becomes more “routine,” the agent is given more discretion. Second, *T increases as α
increases. Somewhat counterintuitively, as the agency problem becomes more severe, it
is optimal to give the agent more discretion. The reason is that an increase in α increases
the M-agent’s payoff from **Hn more than his payoff from *T . To prevent pooling, *T
must be increased to make the two payoffs equal again. Third, *T does not depend on p.
31
V. Extension: An Informed Principal
So far we have assumed that the principal is completely uninformed about θ. Here
we extend the model by allowing the principal to learn the state of the world with
probability q after the agent makes his proposal. This extension helps to understand the
role of the agent’s information advantage in determining the decision process since the
agent’s information advantage declines as q rises.
We focus on the comparison of delegation and veto (override is essentially the
same.) Under delegation, equilibrium behavior and the principal’s payoff are the same as
when the principal is uninformed. Under the veto process, the principal now rejects the
proposal if his own information reveals that L=θ . This affects the equilibrium outcome,
the principal’s utility, and more subtly, the agent’s behavior. In particular, an M-agent is
more likely to pool now because he can rely on the principal to some degree to reject the
project with some probability in the L-state.24 The M-agent anticipates that the principal
will learn that the project’s value is L and reject it with probability )1( π−q . Therefore,
the M-agent’s expectation of the state conditional on the principal’s approval is given by
).1/())1)(1(( qqLqHR πππ +−−−+=′
24 The principal also updates his beliefs and becomes more willing to accept a proposal if his information is
M. We assume a parameter configuration such that he still finds it optimal to reject a proposal from a
separated M-agent.
32
Therefore, the M-agent is willing to pool with the H-agent if 0)( * >′ HR nu . The two
equilibria take the following form.
Pooling equilibrium: When 0)( * >′ HR nu , the H-agent and M-agent both propose *Hn , the
principal accepts a proposal of *Hn if his information is H or M, and rejects the proposal
if his information is L or if .*Hnn ≠
Separating equilibrium: When 0)( * <′ HR nu , the H-agent proposes *Hn , the M-agent
proposes *Hnn ≠ , the principal accepts a proposal of *
Hn , and rejects all other proposals.
One difference between this case and one with an uninformed principal is in the
condition for pooling, here 0)( *' >HR nu . Since )()( **
HMHR nunu >′ , an immediate
implication is that pooling is more likely when the principal is informed. The reason is
that the M-agent is more enthusiastic about going forward when he can rely on the
principal to eliminate the L project with some probability. This suggests (as we show
shortly) that an increase in q can change the optimal decision process from veto to
delegation. This is somewhat counterintuitive: we might expect that as the principal
becomes more informed, delegation will become less desirable.
The other difference when the principal is informed is that veto can be optimal
even with a pooling equilibrium. To see this, compare the principal’s payoff under
delegation and veto. In the separating case, veto is obviously better and does not depend
33
on q. This is because the principal’s information is not used. In the pooling case, the
principal’s expected payoff is
(8) ).)()(1)(1())())(1((]|[ ****0 HHHH nnMfqpnnHfpqpVvE −−−+−−+=π
The first term in (8) is positive and the second is negative. Note that ]|[0 VvE is
increasing in q, and is positive for sufficiently large q. Thus, the principal’s payoff can be
higher under veto than delegation even in the pooling case if he is sufficiently informed.
This is because an informed principal can eliminate the objectionable L-project. The
bottom line is stated in the next proposition:
Proposition 6. An increase in the principal’s probability of being informed can make
delegation better than veto, but a sufficiently large increase makes veto optimal.
Public sector decision processes that limit delegation, particularly the initiative
and referendum, have enjoyed increasing popularity in the last 30 years. One explanation
for this trend is the declining cost of information—citizens now have access to as much
information as their representatives. For this reason, some have predicted that so-called
“direct democracy” is the wave of the future.25 Proposition 6 suggests that the situation
25 An extreme example is a feature in the Economist (December 21, 1996) titled, “Full Democracy.” It
argues, “… the defenders of the old-fashioned form of democracy have to face the fact that the world has
changed dramatically since the time when it might have seemed plausible to think the voters’ wishes
needed to be filtered through the finer intelligence of those ‘representatives.’ … As a result, what worked
34
could be more complicated, and that one cost of direct democracy may be distorted
information transmission by government officials.
VI. Optimal Mechanism from a Revelation Game
To this point, the paper has focused on analyzing the benefits and costs of budget
procedures that are observed in practice. In this section, we investigate how these
procedures compare to a theoretically “optimal” decision process. We search for an
optimal process using the revelation principle, which allows us to identify optimal
mechanisms from among the set of mechanisms in which the principle is capable of
committing (costlessly) to a specific project size for each state reported by the agent. This
class of mechanisms assumes a greater ability to commit by the principal than may be
realistic, but seems like a natural starting point.
The revelation principle states that any decision process can be expressed as an
equivalent revelation game in which the agent reports a value of θ and is given an
incentive to report truthfully. The agent’s report, call it },,{ HMLJ ∈ , results in a
spending level. The optimal mechanism is a mapping of reported states into spending that
maximizes the principal’s expected utility, subject to truth-telling constraints. More
formally, it is the Jn defined for },,{ HMLJ ∈ that solves:
reasonably well in the 19th century will not work in the 21st century. Our children may find direct
democracy more efficient, as well as more democratic, than the representative sort” (page 4).
35
(9) )}()1()()1()({max}{
LLMMHHn
nvpnvpnvpJ
ππ −+−+
subject to
(10) )()( KJJJ nunu ≥ for all ,, JKJ ≠
(11) 0≥Jn for all J, 0)( ≥JJ nu for all J.
Condition (10) imposes truth-telling, and condition (11) contains the non-negativity
conditions.
The next proposition (proved in the appendix) characterizes the solution.
Proposition 7. An optimal mechanism Jn takes one of three forms depending on the
parameters: if 0)( ** ≤HM nu , then (a) 0== ML nn and **HH nn = ; if 0)( ** >HM nu then
either (b) 0=Ln and **RHM nnn == ; or (c) 0=Ln , HMM nnn <<≤ *0 , HH nn <** , and
)()( HMMM nunu = .26
The optimal mechanism described in Proposition 7, for the most part, can be
implemented by the actual decision processes studied earlier in the paper. The
mechanism in case (a) can be implemented with an override process. This was noted in
Section IV where we saw that override results in the principal’s unconstrained optimum
26 A necessary condition for Case (b) to hold is ***MR nn ≤ , but there is no simple expression to delineate (b)
and (c).
36
(of (9)) when the agent separates, which is the condition for case (a) to apply
( 0)( ** ≤HM nu ). Case (b) can be implemented with the override process as well, although
in this situation pooling occurs.
Case (c) is more complicated. The truth-telling condition is difficult to satisfy
here, making separation difficult, and the M state is onerous for the principal, making
pooling undesirable. The solution is to grant a relatively small project size to the M-agent,
and allow the H-agent a relatively large project size. Override cannot implement such an
outcome because the principal is unable to commit to approve such a large project in the
H state. Delegation does not work either, because the M-agent spends too much.
Threshold delegation/veto can resolve both of these implementation problems.
First, a threshold of MnT = appropriately caps the M-agent’s project size. Second, by
granting the agent agenda control power, the principal commits to allow spending in the
H state to exceed his personal optimum, **Hn . If *
HH nn < , a spending limit equal to Hn
completes the implementation. If *HH nn > , a spending minimum is necessary.
A simple veto process (without a threshold) is an optimal mechanism only in the
special case where the solution takes the form of (c) in Proposition 7 with 0=Mn and
0)( * =HM nu . Otherwise, as noted above, a threshold is useful.
The only decision process that is never optimal is full delegation. This can be seen
immediately from Proposition 7—the outcomes *Mn and *
Hn can never occur in an
optimal mechanism. However, full delegation is very common in practice. How can this
inconsistency be explained? One possibility that comes to mind is that the analysis omits
the opportunity cost of the principal’s time. If the principal’s time is sufficiently valuable
37
relative to the potential waste from choosing the wrong project size, delegation could be
efficient. Still, this argument for delegation seems more applicable to small projects,
while many large budgeting decisions (such as the federal budget) are fully delegated as
well. Another explanation could involve unmodeled complexity costs. It may be difficult
in practice to determine the optimal threshold and spending limits, especially if they vary
from project to project and over time, as seems likely.
VII. Conclusion
The paper studies the economics of various processes that are used to make
budget decisions in public and private organizations. We develop a model in which a
principal employs an agent to make budget proposals. The agent prefers to spend more
than the principal does, and has superior information about project returns. The principal
chooses how much of the decision to delegate to the agent. The central tradeoff is this:
delegation allows the agent to overspend, but limiting delegation induces the agent to
distort the information he transmits to the principal in order gain approval of a large
project. We show how the tradeoff between these two distortions can help understand the
choice of decision processes and the behavior of the agent under each process.
One important direction for future research is to investigate the relation of
incentive contracts and decision processes.27 Casual observation and empirical evidence
suggest that actual contracts in the private sector often provide agents with very weak
incentives to pursue the principal’s interest (Jensen and Murphy, 1990), and incentive
27 Bernardo, Cai, and Luo (forthcoming) makes some progress on this issue.
38
contracts are virtually unheard of in the public sector. It is unclear why this is so. We
show that a well-chosen decision process can result in the principal’s unconstrained
optimal outcome in some cases, so one explanation could be that adroit management of
the decision process can address agency problems satisfactorily without having to bear
costs of incentive contracts (such as exposing the agent to significant amounts of risk.)
This does not seem like an adequate explanation for the complete absence of incentive
contracts in most political decisions, however. A more complete analysis would provide
some insight on why the corner solution is so popular.
It would also be useful to have a deeper theory of commitment. Our analysis
implicitly assumes that the principal can commit to a decision process. Indeed, we argue
that some decision processes are effective precisely because they commit the principal to
actions that are not in his interest ex post. However, we do not ask why the principal is
able to commit to the particular institutions we study and not others. It may well be that
some decision processes that are theoretically optimal in a world where commitment is
costless (as with a mechanism design framework) are inefficient in reality because of
commitment problems.
39
Appendix: Proof of Proposition 7
For reference, we restate the mechanism design problem. The optimal mechanism
is the Jn defined for },,{ HMLJ ∈ that maximizes
(a.1) ),()1)(1()()1()(][ LLMMHH nvpnvpnvpvE ππ −−+−+=
subject to
(a.2)JK )()( KJJJ nunu ≥ for all ,, JKJ ≠
(a.3) 0≥Jn for all J,
(a.4) 0)( ≥JJ nu for all J.
Proof of Proposition 7:
The optimal mechanism takes one of three forms. We consider them in order.
Solution (a). If 0)( ** ≤HM nu , then the solution is the unconstrained maximum of (a.1)
and, as we show in the text, it can be implemented (with override).
Solutions (b) and (c). If 0)( ** >HM nu , then the solution can take two forms, pooling and
separating. We begin by establishing two properties of an optimal mechanism.
Lemma 1. An optimal mechanism satisfies 0=Ln .
Proof: Assume to the contrary that 0>Ln . We will show that 0=′Ln yields a higher
payoff to the principal and still satisfies the constraints. On the first point, note that
40
]0|[]|[ =′< LL nvEnvE because )(nvL is decreasing for 0≥n . As for the constraints,
(a.4) is satisfied by 0)0( =Lu . Constraints (a.2)LK hold by )()0( KLL nuu > for LK ≠ .
Finally, (a.2)KL hold because 0)0()( =≥ KKK unu for LK ≠ . ||
Lemma 2. An optimal mechanism satisfies HM nn ≤ .
Proof: This result follows from the agent’s truth-telling constraints. Define
)()()( MIHI nunuI −=φ . The truth-telling constraints can be restated for the H-agent
(a.2)HM as 0)( ≥Hφ , and for the M-agent (a.2)MH as 0)( ≤Mφ . Now suppose that
HM nn > . Observe that 0)()(/ <−= MH nfnfdIdφ . If 0)( ≥Hφ , then 0)( >Mφ : the
two constraints cannot be satisfied simultaneously. ||
Lemmas 1 and 2 imply that there are at most two types of solutions to the
revelation game when 0)( ** >HM nu , those that pool, and those that separate with
HM nn < . We next characterize them and show that a locally optimal solution of each
type exists for all parameter values. The proof is completed with a numerical example
showing that both pooling and separating solutions can be globally optimal for some
parameter configurations.
Solution (b)—Pooling.
Consider a mechanism with 0=Ln and nnn HM ′== . For nnn HM ′== , the
payoff function in (a.1) can be simplified as )())1(1(][ nRvpvE R ′−−= π . The
unconstrained maximum is **Rnn =′ . Since *
M**
R nn ≤ by assumption, we have a strictly
41
concave function defined on a compact convex set, }|),{( *MHMHM nnnnn ≤= . The
maximizer *Rn exists and is unique.
Solution (c)—Separating with HMM nnn <<≤ *0 , HH nn <* , and )()( HHMM nunu = .
We shall first characterize the separating solution(s), supposing they exist. Note
that *HM nn < . This must be true for (a.2)HM to hold. With this in mind, we can show that
(a.2)MH holds with equality. Define 00Mn as the solution to Mf( 00
Mn ) – (1- α) 00Mn = Mf( *
Hn ) –
(1- α) *Hn . Suppose there is a solution for which this constraint does not bind. Then a
smaller value of Mn still satisfies (a.2)MH, and also satisfies (a.2)HM because *HM nn < .
However, the smaller value of Mn increases ][vE , which cannot be true for a solution. If
0=Mn , a decrease in Hn has the same effect, because **00HMH nnn >> . Next, we observe
that HMM nnn << * . This follows from the fact that (a.2)MH binds, HM nn < , and u is
single-peaked. The last property is **HH nn > . Suppose that there is a solution for which
this is not the case. Then an increase in Hn increases E[v]. Constraint (a.2)MH holds
because *MH nn > , and (a.2)HM holds because ***
HH nn < .
Next, we show that a solution of this form exists. Note that the problem can be
restated as ][max vE subject to )()( HMMM nunu = for ),0[ *MM nn ∈ and MH nn > . The
set of ),( HM nn defined by the constraint is compact if we add the point ),( **MM nn .
Because the objective function is continuous, a maximum exists. The solution is
obviously not at the “compactification” point because *MM nn < . Although the objective
function is strictly concave, the solution is not in general unique. A sufficient condition
42
for uniqueness is that the constraint function )( MH nn ψ= defined by )()( HMMM nunu =
for ),0[ *MM nn ∈ , MH nn > , is regular convex.28
Finally, we show that both pooling and separating solutions can be globally
optimal for some parameter values. To see precisely that either type of equilibrium is
possible, we use an example. Let f(n) = n – n2, where n is restricted as n < .5. Let H = 4,
L = 0, and π = .25, so that M = 1. Also, let α = .5 and p = .8. In this case, the pooling
solution to (a.5) is nH = nM = .3, with E[v] = .09, whereas the separating solution is nH =
.4, nM = .1,with E[v] = 11. Here, the separating solution is the globally optimal
mechanism. If α is changed to .85, the pooling solution is the same, but the separating
solution is nH = .47, nM = .38, with E[v] = .07. Now the pooling solution is the globally
optimal mechanism. For each example, it is easy to check that all of the assumptions of
our model are met (In particular, uM( **Hn ) > 0). In addition, for this f, the constraint
function ψ is convex, so the separating solution we study is the unique maximum. ||
28 The function ψ is regular convex if the condition 2))(/)(()(/)( HMMMHMMM nunununu ′′≥′′′′ holds, or
equivalently, 0≥′′ψ .
43
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