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clevelandfed.org/research/workpaper/1995/wp9506.pdf
Working Paper 9506
IMPERFECT STATE VERIFICATION
AND FINANCIAL CONTRACTING
by Joseph G. Haubrich
Joseph G. Haubrich is a consultant and economist at the
Federal Reserve Bank of Cleveland. The author thanks
Ian Gale and Stanley Longhofer for helpful comments.
Working papers of the Federal Reserve Bank of Cleveland
are preliminary materials circulated to stimulate discussion
and critical comment. The views stated herein are those of
the author and not necessarily those of the Federal Reserve
Bank of Cleveland or of the Board of Governors of the
Federal Reserve System.
July 1995
clevelandfed.org/research/workpaper/1995/wp9506.pdf
ABSTRACT
Standard work on costly state verification, monitoring, and auditing generally assumes
perfect signals about the underlying state, especially in questions about financial
contracting. Relaxing that assumption has several intriguing consequences. Most
imperfect audits turn out to be useless, and those that are useful cannot be ranked by
conventional criteria such as Blackwell's information measure. Thus, the notion of "more"
or "less" information becomes problematic.
clevelandfed.org/research/workpaper/1995/wp9506.pdf
clevelandfed.org/research/workpaper/1995/wp9506.pdf
1. Introduction
Costly state verification, monitoring, or auditing often shows up in models of
informational economics and finance. Some papers use it to explain individual contracts,
such as Townsend (1979), which introduces the concept, and Mookherjee and Png
(1989), which explores random auditing. Others extend this work to look at financial
institutions, such as Diamond (1984) and Williamson (1986). In all of these cases, the
audit/monitoring technology, or state verification, is perfect in the sense that the true state
is revealed with certainty. Much less work has been done using imperfect monitoring,
where the signal gives only probabilistic information about the state. The reluctance
stems in part from the belief that this generalization would be messy and complicated
without yielding substantially new insights. For example, Dye (1986), in considering a
principal-agent model, finds he can generalize his results to the case of imperfect
monitoring with more restrictive and less plausible assumptions. Mookherjee and Png (p.
414) assert that "it may be verified, by continuity arguments, that our results extend to
the case when there are 'small' errors in auditing."
The situation is in fact quite different for the standard costly state verification
model. When signals about states are uncertain, there is generally no advantage to
monitoring. Information in this class of models is useless unless it is perfect; Even in the
exceptional cases, this result means that the natural metrics on information break down: It
becomes difficult to rank monitoring schemes, even using such powerful tools as
Blackwell's (195 1, 1953) Measure of Informativeness.
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This .suggests that future investigations of imperfect monitoring may have to
proceed on a case-by-case basis, to mix imperfect signals with perfect monitoring, or to
rely on other types of imperfect monitoring, such as those used by Gorton and Haubrich
(1987), where only a minimum effort level can be verified, or Lacker and Weinberg
(1989), where some fraction of output can be hidden.
After presenting a simple two-state example, section 2 proves the general case and
then discusses the exceptions.
Section 3 shows the deficiencies of Blackwell's
information measure in this model, and section 4 concludes.
2. Imperfect State Verification
In the model economy discussed here, a risk-averse (possibly risk-neutral) agent
has private information about his income, and a risk-neutral uninformed principal has the
right to audit or monitor the agent. For concreteness, I consider the agent as a borrower
endowed with an investment project that needs one unit of funds to begin operation. The
borrower has no funds and must raise them externally from a lender, who may invest in
the project or in a riskless alternative asset with gross return R. A project that gets funded
produces a random income level Yi in state i. There are N possible income levels:
Y, < Y, <...< Y,. A, is the probability of state i occurring. The Ai 's thus describe the
lender's prior distribution about the agent's income. The project's outcome is costlessly
observed by the borrower, but this is private information. Without any sort of audit, the
borrower can always claim that the bad state occurred and can transfer the minimum
amount to the principal. To make the problem interesting, we assume that this amount
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isn't enough to keep the principal/investor happy, and that it falls below the reservation
utility he gets from an alternative investment; in other words,
Y; c R .
The lender may pay cost y to obtain a signal s about the income level actually
realized.
We assume 0 < y < Y, , so that income always covers audit costs.
The
conditional probability n, relates the signal and the state; that is, it denotes the
probability of getting signal s given state i. When income truly equals Y; , n, = Pr[sli].
Recall that by the definition of conditional probability,
li, = 1.
One very natural
signal is the sort announcing "The true state is 5," which signals that Y5 is actual income.
The signal is of course uncertain, and li,, tells us the probability that if the state really is
5, we will receive the signal telling us that it is 5. The formulation, however, is more
general. The number of possible signals, S, need not equal the number of income states,
N. We might have four income states, but get a signal saying only that income was good
or bad. Conversely, income may depend on whether it rained or not, signaled by a
barometer reading of low, average, or high. The general signaling literature even allows
continuous signals with finite states (Kihlstrom [1984]).
The sequence of events has two stages. In the first, the project owner proposes
contracts specifying (ex post) state-contingent payoffs to the potential lender. The lender
then decides between lending to the borrower or investing in the riskless asset. When the
borrower obtains funding, the project produces its output and the borrower learns the
state. In the next stage, the borrower announces a state, whereupon the lender flips a coin
and with probability pi conducts an audit. The borrower then makes a transfer to the
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lender. If an audit occurs, the transfer
4,
depends on both the announced state and the
signal received. If no audit occurs, the transfer 2;: depends only on the announced state.
In the second stage, lenders have no income of their own, so that all payments must come
from the project, including not only transfer payments but also audit costs.
An eye toward financial contracting also results in some restrictions on transfers
between the borrower and the investor (or the peasants and the lord). First, the borrower
must have non-negative consumption. The lord cannot take more wheat than the farmer
has grown. Second, the transfer must be non-negative: The investor never makes an
interest payment to the entrepreneur, as the lord never gives wheat to the farmers -- he
hasn't any to give.' This describes a world where the resources to be divided come from
the agent's production, as in Border and Sobel (1987). Plausible alternative worlds exist.
For example, in Mookherjee and Png (1989), negative transfers provide insurance against
bad states.
Throughout this paper, I restrict attention to incentive-compatible direct
mechanisms. Hence, transfers between the agent and principal depend only on the
income level announced by the agent and on the signal received, or
<,,
2;: if no signal is
sent. This may represent a real restriction. Standard statements of the Revelation
Principle (Harris and Townsend [1981]) do not allow messages that depend on the
-
1
-
-
-
-
The agent can never claim to have more than the amount actually produced. Think of
the claim as delivering bushels of wheat to the investor. The farmer may hold back (just
as a businessman may hide profits), but he cannot deliver wheat that doesn't exist. Gale
and Hellwig (1985) emphasize this point.
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player's type. Green and Laffont (1986) show that in some cases using imperfectly
verified information, the Revelation Principle does not hold.
The lender is risk neutral and the agent is weakly risk averse with von-NeumannMorgenstern utility function u, so that u is strictly increasing, differentiable, and concave.
The agent's consumption must be non-negative, with the convention that u(O)=O. This
allows us to express the basic programming problem of the model as follows:
(1)
rnax
(Fh,T,pi)
x,' kixsxis[piu(x c)+ (1
-
- pi)u(y - T ) ]
(Expected utility of
borrower)
Subject to
(2)
V i ~ ~ s ~ i s [ p ; ~ ( ~ - ~ s ) + ( l - p ;2) ~ ( ~ - T (Incentive
)l
compatibility
xs
n . [phu(Y;-F,)+(l-ph)u(T
1s
-Th)l vi,h
or reporting constraints)
(3)
~ , A ; ~ ~ Z , [ P ~ F ~ - P ~ Y + ( ~ - P ~ ) T (Expected
I ~ R
profit for principal)
(4)
q - es2 0
(5)
F-T
(6)
cs2 0
(7)
T. 2 y .
20
(Non-negativity constraints)
The problem, then, is to choose audit probabilities pi, along with transfers
T
and
cs
dependent on the state, signal, and audit, to maximize the borrower's expected utility
subject to i) the incentive compatibility constraints, ii) a participation constraint for the
lender, and iii) the appropriate non-negativity constraints.
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The non-negativity constraints (4-7) bound the set of possible
qs and
and, in
conjunction with the form of the reporting constraints (2), guarantee a compact set.
Maximizing a continuous function, the agent's expected utility (1) over a compact set has
a solution by the Maximum Value Theorem (see Bartle [1976], section 22). Mookherjee
and Png, who allow negative transfers, cannot invoke this theorem and provide a different
existence proof. In either case, actually calculating the optimum is tricky. The nonconvexity of the reporting constraints (2) generally precludes the use of Lagrange
multipliers in these sorts of problems.
A simple 2x2 example exhibits the techniques and intuition. The key is the
interaction between the incentive compatibility and non-negativity constraints.
Let
N=S=2, set the audit cost to zero, and ignore the possibility of random audits. In addition,
let the conditional probability of each signal be strictly greater than zero, so that nis > 0 .
Then,
the
reporting
V, = n,,u(Y, - F,,)
constraint
+ n2,u(Y2 - F,,)
(2)
implies
that
for
state
2,
2 n 2 1 ~ ( Y-2q l )+ n2,u(Y2 - 4 , ) . This says that the
agent's expected utility from telling the truth -- correctly declaring 1=2 -- exceeds his
utility from falsely declaring i=l. The uncertainty arises because the signal may confirm
(s=l) or contradict (s=2) the declared state. The non-negativity constraints force
and
4, I I: . Hence, n,,u(Y,
- 4 , )+ n,,u(Y,
- F,,)
4, 5 I:
2 u(Y2 - Y , ) . The transfer to the
principal can never exceed Y, : The principal cannot receive more in the good state than
in the bad state. If this form of the problem is to have any interest, the principal's
expected profit (3) won't be met by this contract. Partial information does not help, as the
incentive compatibility constraints conflict with the expected profit constraint.
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This result has a very simple, straightforward, and intuitive explanation. The
signal has shifted probabilities around, but it has not changed the nature of the problem.
The principal cannot prove that the bad state did not occur and must therefore settle for
the minimum possible payment.
The technique, and intuition, generalize to more incomes and more signals.
Proposition 1: Let there be N income states and S signals. If zis> 0 (strictly) for all s,
then for
,
,
and pi
solving (1) subject to (2)-(7, we have that
Cia,Csnis[piFs - p i y + (1 - p i q ) ] II:.
That is, expected profits never exceed the
output of the lowest state.
Proof: The incentive compatibility constraints (2) yield
Vi = ~ s~ [ ~ ; ~ ( ~ - ~ ~ ) + ( 1 - ~ ; )2 ~ (C ~s -n 1; s; [) ~1h u ( x - ~ h ) + ( l - ~ h ) u ( ~ - ~ ) l
lS
V i, h for all i in (1, 2, ...N}. In particular,
v; 2 C s a [ p 1 u ( l :-&,I+
(1- p,)u():
-Vl.
Thus, by the non-negativity constraints (4) and (9,
v ; ~ C s x i s [ ~ , u ( ~ - Y , ) + ( l - ~ , )=~ ~( s~~-i K
s [ )~I( ~ . - Y , ) I = u ( Y , - Y , ) .
Risk aversion, via Jensen's inequality, implies that the expected utility of an
uncertain transfer with expected value
certain transfer of
Y, will be lower than the expected utility of a
Y, . Hence, to satisfy the borrower's incentive compatibility constraint
2 u(y - Y, ) , the largest expected transfer to the principal (lender) cannot exceed
Y, .
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Hence, the principal can never extract more than in the lowest output state Y, . I
Of course, with sufficiently low opportunity cost R, the principal would be
satisfied with such a contract. But if we assume as above that R > 6 , that is, if the bank
demands more than the lowest possible output (or the landlord demands more than the
worst possible harvest), no investment will take place. The principal cannot get an
acceptable return from the project. We state this as:
Corollary: If Y, < R , then (1) subject to (2)-(7) has no solution.
An alternative statement, if we had included the initial investment decisions,
would note that the only solution sets the initial investment to zero; the project does not
get funded.
Imperfect information can help if some of the conditional probabilities n, are
zero. Receiving a particular signal may now definitely rule out some states and allow
larger transfers. Returning to the simple two-state example will help to clarify this.
Let the matrix of n, be
lot5 51.
The first thing to notice here is that the
certain state is 2: If you get a signal saying state 2, you are in fact in that state. If the
signal says state 1, you can't be sure. The reporting constraints are then
= u(q
-4,) 2
u(0)
V, =0.5u(Y, -F,,)+O.5u(Y, - F , , ) ~ O . ~ U ( Y , - F ; , ) + O . ~ U ( Y- 4, , ) .
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With F,, not pinned down by the reporting constraint for state 1, it can exceed
allowing F,, to exceed
Y, -- a definite improvement.
Y,,
Of course, it's possible that the
expected profit constraint is not met, but the zero in the first row is a move in the right
direction.
The above example also shows that once we have added a zero in the first row,
then more zeros can help -- in this case creating perfect information. This intuition
generalizes. Putting a zero in the first row means that some signals will rule out the
lowest state. Adding more zeros can rule out more states.
The sense in which information is generically useless in this class of models
should now be clear. Unless there are zeros in the first row (that is, unless the probability
of some signal given state 1 is zero), partial information adds nothing. Consider { xis) as
,
from an absolutely continuous distribution. Then it is
a random vector in S N Sdrawn
only on a set of (Lebesgue) measure zero'that partial information can improve upon the
no-information case.
3. Blackwell's Information Criteria
Using imperfect information leads to a natural desire for quantification. Can we
rank signals by how much information they provide? David Blackwell (195 1, 1953)
provides an affirmative answer by proving the equivalence of several natural measures of
informativeness, such as every information user preferring one signal, or one signal being
noisier than another (see McGuire [I9861 or Kihlstrom [I9841 for expositions). For our
purposes, the most useful formulation is the one using Markov matrices. Let P and Q be
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the conditional probability matrices associated with two signals (p, and 9,). Then P is
more informative than Q (in the sense of Blackwell) if there exists a Markov matrix M
(x
m, = 1 and m, 2 0 ) such that
Unfortunately, the Blackwell measure does not work for the problem at hand.
First, notice that one signal may be more informative than another and yet be unable to
improve upon the outcome if there are no zeros in the first row. An example would be
Second, and conversely, adding zeros does not necessarily make the signal more
informative in the sense of Blackwell. Working out these examples is not difficult. Note,
however, that adding a zero, leaving all other rows unchanged, and redistributing only the
probability mass from the matrix element reduced to zero does not guarantee a more
informative signal in the sense of Blackwell. That is, we cannot find a Markov matrix M
satisfying (6) when
1
In this case, the element m,, = --, so M cannot be Markov.
8
Blackwell's theorem on the comparison of experiments has two parts.
The
sufficiency part shows that any decisionmaker will prefer a more informative signal to a
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less informative one. The necessity part shows that if any decisionmaker prefers one
signal to another, the preferred signal will be more informative.
In the monitoring model used in this paper, a weak form of sufficiency holds.
Adding noise, in the sense of multiplying by a Markov matrix M, may remove a zero
from the first row and hence make things worse. Even if adding noise does not remove
the zero, a variant of the proof in Grossman and Hart's (1983) Proposition 13 shows that
the added randomness does not help. Where signals are useless, adding noise will have
no effect.
Finally, adding noise will never put a zero in the first row unless the
corresponding columns of the M matrix are all zero, which implies that all the
corresponding columns of the transformed matrix must also contain only zeros. This
merely drops one signal from consideration, which does nothing to help monitoring.
Thus, a less informative signal cannot be better in the monitoring model.
The counterexamples (7) and (8) show that the necessity side of the Blackwell
theorem fails in the monitoring model. Risk-averse agents prefer one signal to another,
even though that signal is not more informative in the sense of Blackwell. This may not
be surprising, as the monitoring model deals with incentives -- and thus with control -- in
addition to estimation. Even so, it was only recently that Kim (1995) produced a
counterexample for the principal-agent modeL2
2
See also the interesting work of Singh (1991).
11
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4. Conclusion
Extending models of auditing, monitoring, and costly state verification to cases of
imperfect signals seems a worthwhile goal. However, except in special cases, imperfect
signals cannot improve upon the no-audit case.
Likewise, except under special
circumstances, Blackwell's information measure does not describe the quality of the
information provided by the signal. The special cases may still be worth studying,
however, since these extremes may be the scenarios most likely to occur in the real world
(Friedman [1965]). Realistically, monitoring technology seems sufficiently advanced that
a hugely profitable entrepreneur or farmer probably cannot appear truly destitute, even
though he may hide or divert some funds. In future research, I hope to see whether the
special cases have interesting applications. An alternative approach would be to consider
some sort of two-stage audit. Then, imperfect information could provide a tighter
distribution over the states, which in certain cases could then be verified perfectly.
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