Principal-agent Modeling
責任代理模式
Dr. Chak-Tong Chau
仇澤棠博士
. Fulbright Professor
中美交流富布萊特教授
我請您們考慮一些問題
A small medical insurance scenario 一個醫療保健的問題
When you have a small illness, do you normally see your doctor?
當你有小病的時候，你會不會自費看醫生？
What about, if your firm pay for your expense?
但是，如果是單位付錢呢，那又怎樣？
我請您們考慮一些問題
A car maintenance scenario 一個汽車維修的問題
Your car is being rented for 2 months. Supposedly, it needs oiling every month. How likely you will remember to do so?
你的汽車是租來用兩個月的，它需要每月潤滑上油一次。你會不會依時地去上油？
How about if this is your own car?
如果這是你自己的汽車，你又會不會去做？
我請您們考慮一些問題
A medical insurance problem 自費醫療保險的問題
When we purchase medical insurance, the insurance company usually requires that you disclose your medical history. Pre-conditions are usually excluded from the coverage.
購買保險的時候，它們通常要求你列出你的病歷。但是如果你有大病的話，很可能保險公司不愿意受保。
我請您們考慮一些問題
If you do in fact have some major medical problems that require expensive treatments, would you disclose these problems?
如果你真的有大病, 你會不會真實地上報？
What do all these tell us about certain human behavior?
這些問題表明了一些什么的人性行為？
Agency Problems and Behavior
代理人的行為与問題
A moral hazard problem (道德危机問題)
when an individual has an incentive to deviate from the contract and take self-interested actions because the other party has insufficient information to know if the contract was honored.
醫療保健 雖然我知道我与雇主的契約明确列出我不要浪費公司的資源。但是用公司的好過用我的嘛！而且公司又不會知道我未能遵守契約。
Agency Problems and Behavior
代理人的行為与問題
A horizon problem 水平界線問題
If one party’s risk or compensation is not the same as the other party’s, the one with a shorter horizon will tend to secretly maximize the short-term benefits, at the expense of the other longer-term party.
汽車維修 我明白汽車不維修壽命不會長。但是，兩個月以后這車子變成怎么樣与我無關了吧。
Agency Problems and Behavior
代理人的行為与問題
An adverse selection problem 逆向選擇問題
The tendency of individuals with private information about something that affects a potential trading partner’s benefits to make offers that are detrimental to the trading partner.
自費醫療保險:雖然我知道保險公司需要知道我的病歷從而決定保險費。但是誠實的代价是較高的費用。此外，我不說，誰知道。
誰是代理人？什么是代理成本？
An agent is someone who has certain special expertise that is desired by the principal to use for his/her benefits. The agent is usually risk adverse, has decision rights to manage, but does not own, the organization’s assets.
代理人(agent) 是任何人在公司有決策權力，但是并非產權的最終所有者。代理人通常有較佳的專長，更好的資訊，和對風險抱保守的態度(risk adverse)。
誰是代理人？什么是代理成本？
There are three (3) types of agency costs. 代理成本有三類：
設計限制性契約的成本 (bonding costs)
建立監督制度的成本 (monitoring costs)
剩餘的損耗 (residual loss)
Note that some costs are bornt by the principal but some are bornt by the agent.
注意的是，有時這些成本是由委托人(principal)負擔。不過有時這些成本是由代理人自己負擔的。
Agency Costs
Bonding costs – costs incurred, before entering the contract, to convince the principal that such agency relationship will not result in the above-mentioned agency problems. Examples are: reputation building, 3rd party guarantor, etc.
Agency Costs
Monitoring costs – costs incurred, after entering the contract, to ensure that such agency problems will not arise. Examples include auditing and inspection costs.
Agency Costs
Residual loss – loss unavoidably arise, despite the bonding and monitoring costs, the contract still cannot yield the utmost benefits, because:
the agency problems do arise, or
due to the suspicion of the agency problems, the principal refuses to pay the agent compensations that fully reflect his/her efforts.
Examples of the Principal-agent Model
$40,000
$40,000
$40,000
$55,000
E3=4
$40,000
$40,000
$55,000
$55,000
E2=5
$40,000
$55,000
$55,000
$55,000
E1=6
S4=
S3=
S2=
S1=
Probabilities and payoffs for 4 different events
Effort level
Examples of the Principal-agent Model
Agent’s Utility Function: Xa½ - e2 100
where:
Xa = agent’s compensations
e = the effort level used by the agent
Question 1: If you were the principal in entering the contract,
which level of effort (e1, e2, or e3) would you demand?
Question 2: If you, the principal, can closely monitor and
observe the agent at all time, what are the amount and
condition of payment? And, what is the expected payoff
for the principal?
Now, let’s assume that you cannot monitor and observe
the agent directly. What would you, as the agent, do?
Now, can you see the agency problems here?
112
111
100
18,496½ - 42 =
E3=4
18,496½ - 52 =
E2=5
18,496½ - 62 =
E1=6
Expected utility of the agent
Effort level
Is it likely to have the “adverse selection” problem?
How about the “moral hazard” problem?
And, the horizon problem? Residual loss?
What can we say, up to this point?
Under condition of unobservability (incomplete information), fixed payments to agents (. workers, employees) most likely do not work.
What are then the alternatives?
We can give the principal a fixed payment instead.
Or, we can come up with an “incentive compatible” conditional contract.
Fixed Payment to the Principal
Consider this new contract under which the principal gets
$32,750 no matter what happens and the agent keeps the
rest. Will this work?
[(55,000½+40,000½)-32,750]-16=
E3=4
[(55,000½+40,000½)-32,750]-25=
E2=5
[(55,000½+40,000½)-32,750]-36=
E1=6
Expected payoff to the agent
Effort level
Fixed Payment to the Principal
Thus, numerically this will work to ensure that the agent gives the highest effort.
However, there is nonetheless a loss to the principal (33,504-32,750)=754 which is in a sense a monitoring cost (maximum cost to pay for an information system to reveal the agent’s effort level).
But the most fundamental problem is that this type of contracts violates the “risk adverse” nature of the agent. Now the agent becomes the principal!
Incentive Compatible Contract – Problem Setup
Maximize (55,000 – R55)Φ55(e1) + (40,000-R40)Φ40 (e1)
Subject to:
R55½Φ55(e1) + R40½Φ40(e1) - e12 = 100
R55½Φ55(e1) + R40½Φ40(e1) - e12 R55½Φ55(e2) + R40½Φ40(e2) – e22
R55½Φ55(e1) + R40½Φ40(e1) - e12 R55½Φ55(e3) + R40½Φ40(e3) – e32
Incentive Compatible Contract – Specific Solutions
Maximize (55,000 – R55) + (40,000-R40)
Subject to:
R55½() + R40½() - 36 = 100
R55½() + R40½() - 36 R55½() + R40½Φ40() – 25
R55½() + R40½() - 36 R55½() + R40½() – 16
Solutions: R55 = 21,609 R40 = 8,464
Expected payoffs: Principal = 33,020
Agent = 18,980
Summary of Different Contracts
18,980
19,250
18,496
33,020
32,750
33,504
Expected Payoffs
8,464
7,250
18,496
31,536
32,750
21,504
40,000 (p=)
21,609
22,250
18,496
33,391
32,750
36,504
55,000 (p=)
Incentive Compat.
Fixed Rent to Prin.
Observable
Incentive Compat.
Fixed Rent to Prin.
Observable
Agent’s Payoff
Principal’s Payoffs
Event under e1
What do we know from these?
The best case scenario for the principal is when he can observe the agent’s effort level directly.
The worst case scenario to the principal appears to be simply charging a fixed rent.
The difference between the two ($754) represents the maximum amount to pay for an information system to reveal the agent’s effort.
The middle, 2nd best solution (incentive compatible contract) may not always be the next best thing though!
Let’s say that we set the two variables, R55 and R40, to be
18,769 and 11,449 respectively.
100
100
95
(18,769½)+(11,449½)-4½ =
E3=4
(18,769½)+(11,449½)-5½ =
E2=5
(18,769½)+(11,449½)-6½ =
E1=6
Expected utility of the agent
Effort level
Now, the principal is telling the agent NOT to work hard!
The $33,159 is actually better than the $33,020 under “incentive compatible” contract!
30,855
(55,000-18,769)+(40,000-11,449) =
E3=4
33,159
(55,000-18,769)+(40,000-11,449) =
E2=5
n/a
Not a feasible solution, agent’s utility < 100
E1=6
Expected utility of the principal
Effort level
A Few Cautionary Remarks
This model presented here is a single-period model. Multiple-period (repeated games) can give very different answers.
There can be multiple principals as well as multiple agents in the model. Such models, however, become extremely complex.
Information systems are not considered here.
Concluding Remarks
The Principal-agent model is theoretical elegant but mathematically tedious to use.
Empirical (real-life) evidence seems to support the model well.
The challenges, in my opinion, include:
to come up with useful, testable hypotheses;
to extend the model to more complex, but real business situations;
to encourage researchers to teach newcomers the basic skill in understanding the model rather than simply to publish in “ivory-tower” type of journals.
Explain the Table.
The higher the effort, the more LIKELY a good payoff.
Emphasize the a good payoff ($55,000) CAN happen despite a low effort E3. Conversely, a high effort does not GUARRANTEE high payoff.
Point out that the X is a positive utility and e is a negative utility (disutility).
Question 1: As the principal, of course I would demand e1 which will give me the best chance to get the higher payoff of $.
Question 2: If I can observe effort at all time, I would require that I will pay the agent, if and only if, I can observe that he/she is indeed giving me e1. Moreover, the amount I pay is solved by:
X = (100 – 62)2 = 18,496
The expected payoff to the principal under this perfect information scenario is: 33,504
(55,)+(40,) – 18,496 = 33,504
The agent will maximize his/her utility and naturally will choose the E3 level. Hence the principal’s utility becomes:
[55,+40,]-18496 = 26,004 which is much, much lower than the 33,504 that we arrived earlier under the complete information case.
33,504-26004 = 7500 which is the residual loss to the principal.
This 32,750 can be solved by setting:
[(55,000 – X)½ x + (40,000 – X) ½ ] – 36 = 100
The objective function is to maximize the principal’s objective function subject to the 3 constraints:
The solutions must satisfy the agent’s minimum compensation requirements,
The utility to the agent under effort level 1 must exceed effort level 2,
The utility to the agent under effort level 1 must exceed effort level 3.
This is an improvement to the fixed payment to principal contract (33,020 – 32750) = 270 but still less than the 33,504.
The objective function is to maximize the principal’s objective function subject to the 3 constraints:
The solutions must satisfy the agent’s minimum compensation requirements,
The utility to the agent under effort level 1 must exceed effort level 2,
The utility to the agent under effort level 1 must exceed effort level 3.
This is an improvement to the fixed payment to principal contract (33,020 – 32750) = 270 but still less than the 33,504.
Note that:
Only e1 is relevant here since this is what we tried to induce the agent to give.
Payoff to Principal under “observable” = (55,000-18,496)*+(40,000-18,496)*=33,504
Agent’s payoff is fixed under “observable” since Principal will pay if e1 is observed, not contingent on the outcome.
The “observable” is called the first-best solution to the principal. This is however the worst to the agent.
This is a “pareto-optimal” solution (zero-sum game) since in all 3 cases, total payoffs to principal and agent add to 52,000. In other words, one cannot gain without hurting the other.
The “incentive compatible” solution is called the second-best solution. But this is not always the case.
