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Working Paper 9301

SHARING WITH A RISK-NEUTRAL AGENT
by Joseph G. Haubrich

Joseph G. Haubrich is an economic advisor at
the Federal Reserve Bank of Cleveland. The
author thanks Patty Beeson for raising the
central questions of this paper, and Charlie
Kahn for helping to answer them.
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.
March 1993

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Abstract

In the standard solution to the principal-agent problem, a risk-neutral agent bears all the
risk. This paper shows that, in fact, multiple solutions exist, and often the risk-neutral
agent is not the sole bearer of risk. Furthermore, as risk aversion approaches zero, the
unique risk-averse solution converges to the risk-neutral solution wherein the agent bears
the least amount of risk. Even a small degree of risk aversion can lead to agents' bearing
significantly less risk than the simple solution suggests.

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I. Introduction
In the classical principal-agent problem, a risk-neutral agent bears all the risk. This
solution, while correct, is misleading. Other solutions exist wherein the agent does not
bear all the risk, and these have some claim to being more "natural," since they are the
limits of solutions to the risk-averse case.
Specifically, in the Grossman-Hart (1983; hereafter referred to as GH) principalagent problem with a finite number of actions and states, many optimal sharing rules exist;
in only one does the agent bear all the risk.1 With a large enough stake in the project, the
agent will not shirk -- and with a finite number of states and actions, this stake need not be
100 percent.
Once agents have some risk aversion, the principal-agent problem has a unique
solution. For the two-state case, the limits can be computed as risk aversion approaches
zero. The risk-averse solutions do not converge to the classic risk-neutral solution,
however, but to the solution with the lowest risk for the agent. Less risk makes a riskaverse agent happier, so he demands a lower risk premium, in turn making the principal
happier. But exceptions occur. There are knife-edge cases in which the optimal action
discretely shifts with an infinitesimal increase in risk aversion. In this case, the sharing
rule, and thus the risk borne by the agent, differ substantially when the principal wants to
induce distinctly different actions.
By increasing the number of actions, these results reduce to the standard
continuous-action principal-agent models (see Holmstrom [1979]). Under reasonable
conditions, the set of risk-neutral solutions shrinks to one.
This should introduce a note of caution to applications of the principal-agent
model. The simple risk-neutral solution is not a good approximation of the optimal
contract, even for arbitrarily low risk aversion. It can be seriously misleading to compare
actual contracts in which risk aversion is important, say in executive compensation, with
the predictions of the risk-neutral principal-agent model . Fortunately, the GH model used

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here can also deliver quantitative predictions, allowing more direct comparisons (Haubrich
[1991]).
II. Other Sharing Rules

A. The Model
First, let us quickly review the assumptions, notation, and approach of the GH
model. For concreteness, assume the principal owns a firm, but that she delegates its
management to the agent. There are a finite number of outcomes (gross profit states) ql
< q2 < ... < q,. The principal, who is risk neutral, only cares about the firm's expected net
profit, defined as gross profit minus any payment to the manager.
In managing the firm, the agent takes an action, often thought of as effort, that the
principal cannot observe. The principal does observe the outcome, however, and, like the
agent, knows how different actions determine the probability of the outcome states. Both
know .ni(a), the probability of outcome ql given action a. This probabilistic setting means
the agent might work hard but still have little output to show for it. In choosing an action,
the agent does not know the ultimate result. Conversely, in seeing the outcome, the
principal cannot deduce the agent's action.
making the expected benefit
Actions belong to the finite set A={al, a2, a3 ... h),
n

xi (a)qi. To avoid the Mirrlees

to the principal from an action equal to B(a)=
i=l

(1976) problem of increasingly bigger penalties imposed with progressively smaller
probabilities, we assume that xi(a) is strictly greater than zero for all states and actions.
The agent likes income but dislikes effort. His utility fbnction U(a,I) depends
positively on his income from the principal, I, and negatively on his action, a. GH find it
usehl to place the following restrictions on U(a,I):

Assumption A1 : U(a,I) has the form G(a) + K(a)V(I), where V(1) is a real-valued,
continuous, strictly increasing, and concave hnction with domain

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[LOO] lim V(1) = -00. G and K are real-valued, continuous hnctions defined on A,
and
I+i

with K strictly positive. In addition, for all al, a2 in A and I, J in (I,oo),G(al) + K(al)V(I)
2 G(a2) + K(a2)V(I) implies G(al) + K(al)V (J) 2 G(a2) + K(a2)V(J).

The agent has a reservation utility U, the expected utility he can achieve working
elsewhere. Sometimes, we consider this as derived from an outside income F, so that g
= v(F). If the principal does not offer him a contract worth at least U, the agent takes

another job. To make the model at all interesting, some income level should induce the
agent to work. GH formalize this as

Assumption A2: [g-G(a)]/K(a) I V(m) for all a in A.

As an example of when this assumption does not hold, consider negative
exponential utility -e-k(l-a) and a g of +5. In this case, even infinite income could not
make the agent work.
If the principal could observe actions, it would be straightforward to determine
what she pays the agent for a particular one. Call this the first-best cost, or Cm(a):

where h = V-1.
As GH put it (p. 1l), "CFB(a)is simply the agent's reservation price for picking
action a." Given this cost, the first-best optimal action maximizes the principal's net
benefit, B(a)-CFB(a).
Of course, the principal cannot observe the agent's actions, nor can she directly

.
base pay on effort. Instead, she chooses an incentive scheme I={11,12,..I,) wherein
payment Ii depends on the observed final state qi. Given this, the agent will choose the

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action that maximizes his expected utility. Knowing how the agent will react, the principal
now can break her problem into two parts. For each action, she calculates the least costly
incentive scheme that induces the agent to choose that course. This gives her the

=x
n

expected cost of motivating the agent to perform a particular action a, C(a)

ni(a)Ii.

i=l

She then chooses the action with the highest net benefit; that is, the one that maximizes
B(a)-C(a).

B. Multiple Solutions
The possibility of multiple solutions arises from looking at the mathematics of the
agent's problem. With risk neutrality, the concave programming problem with a unique
solution becomes a linear programming problem with multiple solutions. When neither the
principal nor the agent cares about risk, risk enters only as it reflects the share held for
incentive purposes. When the principal is not indifferent between the two most desirable
actions, multiple equilibria can result. The larger the gap between the actions, the more
risk the agent can bear. With a risk-averse agent, the principal would minimize the agent's
risk subject to meeting the incentive constraints. For a risk-neutral agent, only the
incentives matter, and any risk configuration consistent with them works.
The traditional solution assigns all the risk to the agent:

The agent bears all the risk for shortfalls in q, and the principal gets

Now, suppose the agent bears less risk and takes only a fraction z of the shortfall in q.

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Income in state i becomes

where t is a constant (discussed in detail below) and T measures the fraction of risk borne
by the agent. Proposition 1 gives sufficient conditions for T being less than one:

PROPOSITION 1: Assume A1-A2 and a risk-neutral agent. If

where

then there exists an optimal contract paying the agent Ii = 7qi - t [B(a*) - CFB(a*)] for
some value oft.

The proof is straightforward and revealing. To emphasize the underlying logic, I
make two simplifling assumptions about utility, both of which are easily generalized.
First, I specialize the risk-neutral income utility to V(1)

= I,

rather than to V(1) = a+PI.

Second, I use the additively separable form of utility, setting U(a,I) = G(a) + V(I), or here,
G(a) + I.

PROOF:
For the optimal action, the principal calculates the least costly method of getting
the agent to choose action a*. The incentive scheme must minimize the principal's
expected payment to the agent while still inducing him to act. This is a programming

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problem, including individual rationality, incentive compatibility, and feasibility
constraints:

subject to

n

(Ic)

n

C ni (a*) [G(a*) + Ii] 2 Cni(a*) [G(a) +Ii] for a t a *,

(FEAS)

Ii I oo for all i.

We now must determine the .r: in equation (1) that will satisfy these conditions.
This means choosing .r: to satisfy

resulting in

(2)

t=

a ( a * ) - CFB(a*)

By construction, a .r: value between zero and one satisfies the individual rationality
constraint (IR). Some values of .r: also satisfy the incentive compatibility constraint (IC),
as I now show. Substituting equation (2) into (I), the incentive scheme becomes

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This makes the incentive compatibility constraint

which simplifies to

Whether or not a risk-neutral agent bears all the risk depends on whether there is a
gap between G(a*)-G(a) and B(a)-B(a*).2 This gap is not solely a matter of chance,
however. The principal chooses a* to maximize B(a)-C(a) or, in the risk-neutral case,
B(a)-CFB(a). Since a* is the optimal action, it satisfies B(a*)-CFB(a*) 2 B(a)-CFB(a).
Rearranging and using the definition of CFB we have

If the inequality in (6) is strict, z can be less than one, meaning that the agent does
not assume all the risk. There are three cases to consider, depending on the sign of each
side of (6).

(i) Both G(a*)-G(a) and B(a)-B(a*) are positive. In this case, a has the larger gross payoff
but is more costly to implement than a*. Clearly, if (6) holds, any z in the relevant range
of [0,I.] satisfies (5).

(ii) If G(a*)-G(a) is positive and B(a)-B(a*) is negative, any z works. In this case, the less
costly action, a*, also has the better payoff.

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(iii) Both G(a*)-G(a) and B(a)-B(a*) are negative. In this case, a* is more costly but has a
better payoff. We usually think of this as the "normal" case. With negative numbers,
division reverses signs, so (5) implies that z, the fraction of risk the agent does bear, can
fall anywhere in the interval

With a more general utility fbnction, this becomes the condition stated in the proposition:

Even equation (7) understates the fbll range of incentive schemes wherein the
principal bears risk. With more than two states, the sharing rule need not be linear, and a
single-parameter z will not capture all possible deviations from the classical case. In
general, the solution set will be the convex hull of extreme points, a multidimensional
"flat" or "face" of the constraint set for the linear programming problem (PI).

111. Convergence
Solutions in which the principal assumes some risk are more than curiosities. As
risk aversion approaches zero, the risk borne by the agent converges to a number less than
one. The traditional solution offers a poor approximation of this, even near zero.
Unfortunately, I have results only for the two-state case -- the sole case with
closed-form solutions for the risk-averse problem. Such strong assumptions seem to be
necessary for convergence results. For instance, GH often assume only two states, or
negative exponential utility. Without strong restrictions, odd things can happen in the

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model: The (IR)
constraint may not bind, higher profits may mean less money for the
agent, or the agent may get more money for less effort.

A. Limiting Cases
With only two states, the single-parameter z fully describes how much risk the
agent bears. Usually, the risk-averse solutions converge to the solution with the smallest z
value (rather than to the classical solution of z=l). Some exceptions exist because the
optimal action can switch at zero, which in turn causes a discrete jump in the risk burden.
To explore convergence, we must first make sure that the utility functions do
indeed converge. If we index the income utility function by risk aversion y, V(y,I), we
embody this convergence as a new assumption.

Assumption A3: As 8 approaches 0, V(y,I) converges uniformly to a+PI, a,P# 0, on the
interval [-q,, q,].

Though natural, this assumption does restrict utility functions. For example, the
converges to zero, a constant function that is
negative exponential function -e-~@-a)
inadmissible by A1 .
The statement of proposition 2 requires a little groundwork. First, the proof uses
the closed-form solution for the two-state case found by GH:

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The derivation of these formulas depends crucially on GH's proposition 6, which proves
that the agent is indifferent between the optimal action a* and some less costly action.
Two possibilities can make convergence problematic. As risk aversion falls, either the
optimal action or the less costly action may change. A change in the optimal action
matters for the convergence result, but whether or not a change in the less costly action
does is unclear. I have produced neither a proof nor a counterexample for this case.
Thus, the statement of

2 reflects these two possibilities.

Now define the unique profit share for a given utility function and risk aversion as
.r(V,y). Further define the minimum T in equation (7)as .r-.

This gives

PROPOSITION 2: Given assumptions A1-A3, if the optimal action and the indifferent
alternative action do not change for risk aversion in the neighborhood of zero, then
lim z(V, Y)= T-.
Y+O

As before, to ease the notational burden and emphasize the logic, I present the
proof for the additively separable case. The generalization to other utility functions is
straightforward.

PROOF:
In the risk-neutral case, we know from equation (6) that T

=

G(a, - G(a, )
'(ai ) - '(a,
'

which clearly depends on the optimal action ak and a particular alternative ai. This implies
an income difference between states of

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In the limit of the risk averse case, the optimal incomes are given by the limits of
equations (8) and (9).

.

Since nl(aj) + n2(aj) = 1, we can express the probabilities in terms of xl(-)Is.
Making this substitution and collecting terms yields

Taking the difference and simplifling, we find

which matches (10).
The equality between equations (10) and (13) proved so far depends on the
constancy of both the optimal action and the alternative action. I conjecture that even if
the alternative action switches in the neighborhood of zero, the equality (and thus the
proposition) still holds.

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B. Action Shifts
Proposition 2 does not hold when the optimal action shifts at zero. Suppose one
action is best at a risk aversion of zero and another at a risk aversion greater than zero.
As the action changes, so too does the sharing rule. The best way to see this is with a
simple two-act example. Here, the principal induces the better action at zero risk
aversion, but pays a flat fee and accepts the lower action for risk aversion greater than
zero.
We start out with B(a*)-C(a*)=B(a)-C(a), or indifference between the two
actions, so that the switch happens at zero. Notice that this sets z equal to one, meaning
that the agent bears all the risk. We next want B(a2)-C(a2)< B(al)-C(al), making the
lower action preferred for y > 0. To do this, set V(1) = I -y12. Then, h(v) =
1 +}
I 2 y. With h(v) in hand, we can assess the second-best costs once we
have calculated vl and v2. The goal is then to show that, in some cases, ac(a2)

a
Y

> 0. ~f

this is true, an increase in y leads to the principal preferring action al, since the cost of
action a2 increases while the rest of the variables, B(a2), B(al), and C(al), remain
unchanged. (C[al] is a fixed payment independent of state.)
Simpli@ingvl and v2 from equations (8) and (9), we have

The last terms in each of these expressions are constant with respect to y, so we may
rewrite them as

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Using the above equations, we can solve for I1 and I2 and thus for C(a2):

Notice that i3111i3y and i312/i3y have the same sign, matching aC(a2)/i3y. Explicitly
calculating the first of these derivatives, we have

The first two terms are positive, while the last can be rewritten as [f - (1 + y)T + P]. As y

-+ 0, the last term approaches I - f2 + P. For

not too large, that term is positive, and

we have the counterexample.
In this counterexample, the agent bears all the risk if he is risk neutral, but assumes
none at all if he is even slightly risk averse. In other words, convergence fails in a
spectacular way. But it may fail in more prosaic fashions as well. The limit of the riskaverse case may be higher or lower than z-.

Figure 1 schematically illustrates these

possibilities.
Mathematically, convergence fails because of a difference between zrisk-neutral case and z for the limit of the risk-averse case. This difference is

for the

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Indifference at zero risk aversion implies G(al) + B(al) = G(ak) + B(ak), leaving
open two distinct possibilities: Either ak or al can be the high-cost, high-benefit action. If
B(ak) > B(al), then G(ak) < G(al), and vice versa. The sign of equation (14) then can go
either way.

C. Increasing the Number of Actions
The lowest share of risk the agent can take, z-,

is decreasing in the gap in the

-

principal's payoff between the chosen and the indifferent act, [G(ak) G(a$]/[B(a$

-

B(ak)]. It seems intuitive that as the number of actions increases, the gap decreases and
hence z-

moves toward one, its value in the continuous action case. But it is possible to

work the convergence so that exceptions occur. If B and G are continuous functions,
some condition on the difference, such as lim 1 ak-ak+l1

= 0, would

ensure the result.

IV. Conclusion
The traditional solution to the risk-neutral principal-agent problem is misleading.
With finite states and finite actions, many solutions exist, and in all but one of these the
principal bears the risk. The traditional solution cannot even claim to be the limiting case
as risk aversion decreases: In fact, it is the solution farthest away from the limit.
These results have two main consequences. First, they caution us against using the
traditional solution as an approximation for the less tractable risk-averse case. This
explains the divergence between Haubrich's (1991) findings and those of Jensen and
Murphy (1990). Second, they also illustrate the range, power, and tractability of GH's
version of the principal-agent model.
Nevertheless, the results presented here should be taken as preliminary -- brief
observations of a rare nocturnal animal. Theorem 1 provides sufficient, but not necessary,
conditions for multiple solutions and does not characterize all possible solutions. The
convergence results require even stronger restrictions and depend on the two-act case.

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Still, I believe the scattered sightings reported here show a surprising -- and noteworthy -aspect of the principal-agent model.

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FOOTNOTES

1. Although GH do not consider multiple solutions in the risk-neutral case, they are quite

carefiil in stating their theorems. Hence, this result does not imply any error in their
work.
2. Haubrich (1991) provides several numerical examples of problems of this type,
showing that solutions do exist and that the theorem is not vacuous.

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REFERENCES

Grossman, Sanford J., and Oliver D. Hart, "An Analysis of the Principal-Agent Problem,"
Econometrics, vol. 5 1, no. 1, January 1983, pp. 7-45.
Haubrich, Joseph G., "Risk Aversion, Performance Pay, and the Principal-Agent
Problem," Federal Reserve Bank of Cleveland, Working Paper No. 9 118, December 1991.
f
Holmstrom, Bengt, "Moral Hazard and Observability," Bell Journal o Economics, vol.
10, 1979, pp. 74-91.
Jensen, Michael C., and Kevin J. Murphy, "Performance Pay and Top-Management
Incentives," Journal o Political Economy, vol. 98, no. 2, April 1990, pp. 225-264.
f
Mirrlees, James A., "The Optimal Structure of Incentives and Authority within an
Organization," Bell Journal o Economics, vol. 7, 1976, pp. 105-131.
f

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Figure 1
Convergence of Sharirlg Rule

T

1

-

Profit share

Tmin

lim T < Tmin

Risk aversion

Source: Author.