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Note: This page uses characters that may not display properly.
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version. 1 Introduction This text presents models and proofs of
the
classic trade-off between risk and incentives regarding the remuneration of
managers by corporate owners. The presentation is rigorous. If you
prefer a more intuitive discussion of the problem click here. Otherwise stay
put. The models of this text are believed to be good representatives of the
formal principal-agency literature of the hidden action type. The analysis
follows Tirole [1988, page 51-55] and part 1 in Hart and Holmström [1987].
The text proceeds as follows: Section 2 outlines the typical agency model
framework its assumptions and notational apparatus. Then in section 3, the
four agency models are presented. Note that footnotes e.g. '[2]'
can be viewed by clicking them. 2 The General Framework Model Problem: The problem is to study incentive and insurance
(risk) issues between a manager [the agent] and some shareholders [the
principals]. The shareholders are owners of a firm and they have hired the
manager to run it. Results are generated the usual way by changing the
underlying assumptions. General Assumptions: The following
assumptions are part of all the following models unless otherwise specified. 1) Shareholders
are risk neutral.[1]
For a graphic explanation of the different risk concepts click here. 2) Shareholders
objective is to maximize the firms expected profit. 3) There is only
one manager. 4) The manager’s
work effort (e) is unobservable by the shareholders (asymmetric information). 5) Firm profit
∏(e,q), manager wage w(∏), and manager utility
U(w,e) are common knowledge. Neither Tirole nor Hart & Holmström explicitly state the following
assumptions. However, it is believed to be important to state them
explicitly. If not, one may easily get the wrong impression that the
predictions of the models are more robust than they actually are. Experienced
economists know this but outsiders do not. 6) All agents (the
manager and the principals) maximize utility. A trivial, tautological
assumption. Says that agents want to get more of what they like. 7) All agents are
perfectly rational. An unrealistic but strongly simplifying assumption. Says
that agents grasp all the information in the model and they have the ability
to calculate the optimal decisions instantaneously and costless. 8) Time is static.
Everything is as if the world only existed in one timeless period. The models
include a many other assumptions, some made explicit below. However, for
expositional reasons they will remain implicit. Notation and Further Assumptions: ·
e is work effort by the manager, and e take on
intervals e Î [emin, emax] = E. ·
q is uncertainty or state of nature affecting firm profits independent
of manager’s work effort e. Also qÎQ, where Q is the
distribution space. Q is assumed to be common knowledge. ·
∏ = ∏(e,q) is random firm profit. Assume ∏’e>0. This assumption is crucial in all agency
theory. Because ∏ depend on q, ∏ is itself stochastic. Assume ∏ Î [∏min, ∏max] = P. ·
w(∏) is the manager’s wage function or salary
scheme. Presumably w’∏>0 (This
will be demonstrated to be the conclusive characteristic of the optimal wage
function in the general agency model, case 4 below). In the principal-agent
literature w(∏) is also interpreted as the wage contract that the
shareholders design. It is assumed that there exists a wage scheme space W, wÎW. ·
U0 is the reservation utility. This is, the expected utility the manager
believes he can get in a job elsewhere, less the cost of searching. In other
words, the reservation utility is the utility he could get by redeploying his
resources at the best alternative use. ·
F(e) is the manager’s disutility from effort. Assume F’e ≥0, F’’ee >0, F’(0)=0, and F’(∞)=∞. ·
U(w,e) = u[w(∏(e,q))] - F(e) is the
utility function of the manager. Assume U’w >0, U’’ww<0. This
assumption implies that the manager is risk averse. Note that the utility
function is separable in income and effort. This simplifies the analysis. ·
f(∏;e) is the density function or mass
function for the stochastic firm profit ∏ given the effort level {e}. The Model: Shareholders expected payoff: πSha = E[∏(e,q) - w(∏(e,q))] =∫ [∏(e,q) - w(∏(e,q))]f(∏;e)d∏ q {∏ÎP} Managers expected payoff: πMan = E[U(w(∏(e,q)),e)] = ∫ u(w(∏(e,q))f(∏;e)d∏ - F(e) q {∏ÎP} The
shareholders must design a wage contract w*(.) and pick a effort level e*
that maximizes their expected payoff: {w*(.), e*} º argmax
πSha = argmax ∫ [∏(e,q) - w(∏(e,q))]f(∏;e)d∏ (1) {w(.)Î W, eÎE} {w(.)Î W, eÎE} subject to, argmax πMan = argmax ∫ u(w(∏(e,q)))f(∏;e)d∏
- F(e) ≥ U0, and (IR,2) {eÎE} {eÎE} ∫u(w(∏(e*,q)))f(∏;e*)d∏
- F(e*) ≥ ∫u(w(∏(e,q)))f(∏;e)d∏ - F(e), " eÎ[emin, emax] (IC,3) {∏ÎsP}
{∏ÎP} The first constraint
is called the individual rationality
constraint (IR) or individual participation constraint. It states that
given the wage contract {w(.)} a rational agent will require the utility from
the effort level {e} that maximizes his utility to be at least equal to his
reservation utility {U0}. This
constraint is comparable to the volunteer trade assumption in the perfect
market economy. Standing alone it cannot prevent the problem from reaching
first best. The second restriction is the incentive compatibility (IC) constraint. It states that given the
observable wage contract {w(.)}, the unobservable level of effort chosen by
the principal {e*} must maximize the utility of the agent over all possible
levels of effort {"eÎ[emin, emax]} for the agent to have an incentive to actual
conduct that level of effort. Without going into detail this restriction
embed the moral hazard issues of the model. The above model provides a
general framework for the corporate principal agent problem. The following
studies this framework in four different cases. 3 The Basic Models Case 1: The Full Information Case Modified Assumptions 1) The
shareholders observe effort {e}. This is, full information or perfect
monitoring. Replaces assumption 4 above. 2) The manager is
risk averse. That is, U’’ww < 0. When the
shareholders can observe the manager’s effort the problem becomes much more
simplified since they can impose any effort level {e} by threatening with
severe punishment should the effort order not be obeyed. The incentive compatibility
constraint (IC) is therefore not relevant. For simplicity assume e is solved.
Then the wage contract becomes the only relevant decision variable. The
problem is therefore that the shareholders must design a wage contract w*(.)
that maximizes their expected payoff: {w*(.)} º argmax ∫ [∏(e,q) - w(∏(e,q))]f(∏;e)d∏ (4) {w(.)ÎW} subject to, argmax ∫ u(w(∏(e,q)))f(∏;e)d∏
- F(e) ≥U0 (IR,5) {eÎE} The individual
rationality constraint must be binding. If not the shareholders would have
incentive to lower the salary {w(∏)} until it becomes binding to save
on their own cost of management services. The Lagrangian may now be formed. L = ∫ {[∏(e,q) - w(∏(e,q))]f(∏(e,q);e) + l[u(w(∏(e,q))) - F(e) - U0]f(∏(e,q);e)}d∏ {∏ÎP} F.O.C. with
respect to w: f(∏(e,q);e) + lu’w(w(∏(e,q)))f(∏(e,q);e) = 0 <=>
u’w(w(∏(e,q))) = 1/l (6) Q.E.D. So the optimal
wage structure is independent of profit. In other words, the agent is given a
constant wage in order to be fully insured. The optimal choice of effort
level may be found by taking the derivative with respect to e. Analysis is
omitted. The intuition is that if the shareholders are risk neutral and the
manager is risk averse, and then all the risk should be borne by the risk
neutral part because he can bear the risk costless. Furthermore, it does not
create any incentive problem because in this model version the effort level
is observable. Threatening with severe punishment can impose the efficient
level of effort. This full information solution is a first best solution. It
does not deviate from what would be obtained in the perfect
market economy model. The “mechanism” that ensures first best is not the
market mechanism but the existence of an omniscient planner [the principal or
the shareholders]. He has all the information necessary to calculate the
first best solution, and the powers [perfect monitoring and punishment
abilities] to ensure that it is actually carried out. The assumption of
symmetric information on effort level does not fit many real world
situations. It would probably not be wrong to assume that most
principal-agent relations of the shareholder-manager type imply a significant
degree of asymmetric information. This is the issue in the following three
models. Case 2: Asymmetric information, and risk neutral
manager. Modified Assumptions 1) The
shareholders cannot observe effort {e}. Back to assumption 4 above. 2) The manager is risk neutral.
This is, U’w > 0, and
U’’ww = 0. 3) The utility
function below is now assumed to be measuring money equivalents. This implies
that utility can be measured cardinally. The reason for this assumption is
that we need to make the manager indifferent between receiving profit or
utility. Modified Notation and Further Assumptions ·
U(w,e) = w(∏(e,q)) - F(e) is the utility function of the manager. This
utility function is clearly risk neutral since U’’ww = 0. This is new! Now the manager’s utility is: U(w,e) = w(∏(e,q)) - F(e) <=> w(∏(e,q)) = U(w,e) + F(e) (7) With this
manager-utility function, the IR constraint becomes: ∫ w(∏(e,q))f(∏;e)d∏ - F(e) ≥ U0 {eÎE} and it must be
binding. Otherwise the manager could just decrease the wage until it becomes
binding. The IR restriction may therefore be written: ∫
U(w,e)f(∏;e)d∏ = U0 (IR,8) {eÎE} Then the
shareholder’s payoff function becomes: πSha = E[∏(e,q) - w(∏(e,q))] = ∫ [∏(e,q) -
w(∏(e,q))]f(∏;e)d∏ q {∏ÎP} (by substitution of (7) above) =
∫ [∏(e,q) - U(w,e) - F(e)]f(∏;e)d∏ {∏ÎP} (by
substitution of (8) above) =
∫ [∏(e,q) - U0 - F(e)]f(∏;e)d∏ {∏ÎP} =
∫ [∏(e,q)]f(∏;e)d∏
- U0 - F(e) {∏ÎP} Solving this
with respect to the optimal effort decision {e*} yields the same first best
outcome as under the full information case 1 above, since only the IR
restriction has been used. This is, for given w: {e*} = argmax[ ∫ [∏(e,q)]f(∏;e)d∏ - U0 - F(e)] (9) {eÎE} The following demonstrates that a wage contract
making the agent the residual claimant of the firm’s profit gives the manager
the full incentive to maximize the principals' profit. This will result in a
solution identical to the first best, full information model. The agent may
be made a residual claimant on the firms profit if he is offered to lease the
firm at an annual price of:[2] p
= argmax[ ∫ ∏(e,q)f(∏;e)d∏
- U0 - F(e)] (10) {eÎE} Note that this
is the value that maximizes the shareholder’s value under full information.
The question is now; does the manager accept this lease? The answer is YES!
His payoff function when he has to pay the lease p and has the right to the
firms profit is: πMan = ∫ [∏(e,q)f(∏;e) -
F(e) -
p]d∏ (Note! Use assumption 3 above) {∏ÎP} <--firm
profit--><-work disu.-><- lease-> =
∫ [∏(e,q)f(∏;e) -
F(e) -
{∏(e,q)f(∏;e) -
U0 - F(e)}]d∏ =
U0 (11) {∏ÎP} Q.E.D. Since the
manager receives his reservation utility he accepts. The shareholders and the
manager get exactly the same payoffs as under full information. However, the
situation is very different. In this model the agent must bear the entire
risk in order to have the full incentive to maximize firm profit. This
strategy was not optimal in case 1, because the manager was risk averse.
Neither was it necessary for inducement of incentives, because of the special
powers of the principal [perfect monitoring and punishment]. In this case,
our manager is risk neutral and is therefore capable of bearing the risk
costlessly. This is fortunate, because the principals have no other ways
[imperfect monitoring and punishment powers] to induce the optimal solution
than to make the agent the residual claimant of the firm’s profit. The nice feature with the model
above is that it induces optimal resource allocation without having an
omniscient principal dictating the actions. In this respect, it is like the
market mechanism. Unfortunately, it assumes risk neutral agents. This might
be reasonable if the agent is rich relative to the size of the firm profit.
However, this is not a fact of real world life. Among the private enterprises
most GDP value is made in big corporations with profits that are many times
higher than their manager’s salaries. The next two models analyze more
realistic situations with risk averse managers. Case 3: Asymmetric information, and an infinitely
risk averse manager. Modified Assumptions 1) The manager is
infinitely risk averse. That is, U’w > 0, and U’’ww <<<
0. 2) The
shareholders cannot observe effort {e}. This is asymmetric information. 3) For simplicity
assume that the distribution of the profit ∏ = ∏(e,q) is such that
∏Î[∏min,∏max]=P, and that
the distribution borders are independent of the effort level. We still have
that ∏’e >0, but
this is so only when we look at the mean value of the profit. This case turns
out to be the simplest in terms of mathematics. An implication of infinitely
risk aversion is that the manager will avoid risk at any cost in mean return.
However, the agent is capable to make a choice between two risky payoffs. To
be specific, the agent always prefers a random wage w1 and a level of effort e1 to another pair (w2,e2) if min(w1) > min(w2) or min(w1) = min(w2) and e1 < e2. Note that in the infinite risk averse case the mean value is not a
relevant decision variable. Now, look at the manager’s payoff function
replicated below for convenience and consider assumption 3 above: Manager’s expected payoff: πMan = E[U(w(∏(e,q)),e)] = ∫ u(w(∏(e,q))f(∏;e)d∏ - F(e) q {∏ÎP} The wage {w}
depends on profit {∏} that again depends on effort {e}. However, according
to assumption 3, the wage only depends on profit with respect to the mean
value, not with respect to the border values {[∏min, ∏max]}. So the minimum wage is independent of work effort. Furthermore,
because our utility function {U} only attaches value to the minimum wage, the
effort level {e} only affects the managers payoff through the disutility term
F(e). Since F’e > 0 the manager will optimize by choosing emin no matter how much his expected salary would
increase if he cared to work more than emin. Given this fact the shareholders cannot do better than to give the
manager his reservation utility evaluated at emin. The resulting payoffs becomes: Shareholder’s
expected payoff: πSha = [∏(emin,q) - w(∏(emin,q))] Manager’s
expected payoff: πMan = u(w(∏(emin,q))) - F(emin) = U0min Q.E.D. |
