Οικονομικό Πανεπιστήμιο Αθηνών
Τμήμα Οικονομικής Επιστήμης
Άνοιξη 2015
Χρηματοοικονομικές αγορές και εταιρική διακυβέρνηση
Lecture 3: The principal-agent problem.
The principal-agent problem is the result of information asymmetries (that develop after
the contract is signed). I.e. after an owner of a firm hires a manager, the owner may be
unable to observe how much effort the manager puts into the job. Similarly, the manager
will often end up having better information than the owner about the opportunities
available to the firm.
The principal hires the agent to perform some task which is costly to the agent. The
principal may be unable to directly observe the action of the agent, but observes some
output which is determined at least in part by the actions of the agent. The agent
undertakes the task as long as his utility exceeds a reservation level of utility (opportunity
cost). The agent decides whether to work hard or not.
The principal’s problem is to design an incentive payment that induces the agent to take
the best action from the viewpoint of the principal.
Let x be the outcome received by the principal. The agent can either work hard and put
effort b, or not, and put effort a. If the outcome is determined by the level of effort then x
= x(a). The cost of action a is c(a), and s(x) the payment.
The utility for the principal is: x - s(x).
The utility for the agent is s(x) - c(a).
The principal wants to choose a function s(.) to maximize his utility subject to the
following constraints.
1. The participation (individual rationality) constraint: The minimum amount of payment
required by the agent that gives him utility u.
2. The incentive compatibility constraint: The incentive scheme is such that the agent
chooses to work hard.
Full information about the agent’s effort: monopoly solution
The monopolist sets the payment schedule.
The monopoly wants to induce the agent to put effort b.
The effort level is observable.
The monopolist’s problem: choose s(.) and the optimal action b such that
Max x(b) - s(x(b))
St.
s(x(b)) - c(b) ≥ u
s(x(b)) - c(b) ≥ s(x(a)) - c(a)
(IR)
(IC)
Solution:
From IR we have s(x(b)) = c(b) + u
The principal chooses the level of b that maximizes:
x(b) - c(b) - u
(1)
From the FOC wrt b we have x’(b) = c’(b).
Let this level be b* . So output is x* = x(b* ).
How can b* be the optimal choice for the agent?
Solution:
The landowner rents the land to the worker for some price R and the worker receives all
output. s(x(b)) = x(b) – R. If the worker maximizes
s(x(b)) - c(b) = x(b) – R- c(b),
he will choose the effort level b* such that MP(b*)=MC(b*) which is the amount of effort
the land owner wants (see (1)). This is the effort outcome which maximizes the profits of
the landowner. The rent is determined by the participation constraint:
s(x(b*)) - c(b*) – R = u
Hidden actions (Moral Hazard): monopoly solution
The agent’s actions are not observable. Let two possible output levels (x1, x2). The agent
can either put work effort of level a or b which influence the probability of occurrence of
the various outputs.
For example,
p1a is the probability that output level x1 is observed if the agent chooses action a.
Also, si = s(xi) the payment if xi is observed.
The expected profit of the principal if the agent chooses action b is:
(x1-s1) p1b + (x2-s2) p2b
p2b = 1 - p1b.
The agent has utility function u(si) and the cost of action (a) for him is cα. The agent will
choose the action b if:
u(s1) p1b + u(s2) p2b – cb ≥ u(s1) p1a + u(s2) p2a - cα (IC)
and
u(s1) p1b + u(s2) p2b – cb ≥ ubar (PC)
The principal wants to max his profits subject to (IC) and (PC).
Both restrictions hold as equalities:
If (PC) holds as inequality, we can decrease both s1 and s2 by the same amount ε. The
(IC) will hold, and the Principal’s profits will increase.
Example. Risk neutral agent.
We have:
Agent’s reservation level of utility: 9
Salary: w
Work level: a = 5 (high a) or a = 0 (low a)
Agent’s utility: U(s1, a) = s1 – a
Here: The agent is a salesman who represents the principal to a client.
The client can place:
One order that is worth (gross) $100, one order that is worth (gross) $400
If a = 5, $400 results with probability 0.6, $100 w.p. 0.4.
If a = 0, $400 results with probability 0.4, $100 w.p. 0.6.
The number of sales is observable.
The principal is risk neutral.
The contract:
The agent is paid $s1 if sale is small.
The agent is paid $s2 if sale is large.
The agent has three options:
(a) Refuse the contract, reservation utility is 11.
(b) Take the contract and put low effort. Exp. Utility is:
0.6 * s1 + 0.4 * s2
(c) Take the contract and put high effort. Exp. Utility is:
0.4 * s1 + 0.6 * s2 – 5
The principal will choose payments $s1, $s2 to maximize his expected profits:
Max 0.4 * (100 - s1) + 0.6 * (400 - s2).
Alternatively, the best contract (for principal) such that the agent puts high effort is:
Min
0.4 * s1 + 0.6 * s2
s.t.
0.4 * s1 + 0.6 * s2 – 5 > 11
0.4 * s1 + 0.6 * s2 – 5 > 0.6 * s1 + 0.4 * s2
The first constraint: The agent participates (IR).
The first constraint: The agent chooses to put high effort (IC).
Solution
s1 = 1
s2 = 26
Both constraints bind (are equalities) at the optimum.
Thus, the optimal contract is:
If $100 sale, wages are 1
If $400 sales, wages are 26
The expected wage is 0.4 * 1 + 0.6 * 26 = 16
Exp. Profit is:
0.4* (100 – s1) + 0.6 * (400 - s2) = 0.4 * (100 - 1) + 0.6 * (400 – 26)
References
Hal R. Varian, “Intermediate Microeconomics. A modern approach”, W.W. Noton &
Company, Fifth edition, pages 653 - 657.
Hal R. Varian, “Microeconomic Analysis”, W.W. Noton & Company, Third edition,
pages 440-454.
David M. Kreps, “A course in microeconomic theory”, Harvester Wheatsheaf, 1990,
pages 579-585.