Principal-agent Modeling
責任代理模式
1
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？
2
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？
3
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。
4
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？
5
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。
6
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 shortterm 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。
7
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。
8
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)。
9
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。
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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.
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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.
12
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.

13
Examples of the Principal-agent Model
Probabilities and payoffs for 4 different events
Effort
level
S1=0.3
S2=0.3
S3=0.2
S4=0.2
E1=6
$55,000
$55,000
$55,000
$40,000
E2=5
$55,000
$55,000
$40,000
$40,000
E3=4
$55,000
$40,000
$40,000
$40,000
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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?
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Now, let’s assume that you cannot monitor and observe
the agent directly. What would you, as the agent, do?
Effort level
E1=6
E2=5
E3=4
Expected utility of the agent
18,496½ - 62 =
100
18,496½ - 52 =
111
18,496½ - 42 =
112
Now, can you see the agency problems here?
Is it likely to have the “adverse selection” problem?
How about the “moral hazard” problem?
And, the horizon problem? Residual loss?
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What can we say, up to this point?




Under condition of unobservability (incomplete
information), fixed payments to agents (i.e.
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.
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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?
Effort
level
Expected payoff to the agent
E1=6
[(55,000½x0.8+40,000½x0.2)-32,750]-36=
100.36
E2=5
[(55,000½x0.6+40,000½x0.4)-32,750]-25=
98.56
E3=4
[(55,000½x0.3+40,000½x0.7)-32,750]-16=
88.35
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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!
19
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
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Incentive Compatible Contract – Specific Solutions
Maximize (55,000 – R55)0.8 + (40,000-R40)0.2
Subject to:
R55½(0.8) + R40½(0.2) - 36 = 100
R55½(0.8) + R40½(0.2) - 36  R55½(0.6) + R40½Φ40(0.4) – 25
R55½(0.8) + R40½(0.2) - 36  R55½(0.3) + R40½(0.7) – 16
Solutions:
R55 = 21,609 R40 = 8,464
Expected payoffs:
Principal
=
Agent
=
33,020
18,980
21
Summary of Different Contracts
Event
under
e1
Principal’s Payoffs
Observable
Fixed Rent
to Prin.
55,000
(p=0.8)
36,504
32,750
40,000
(p=0.2)
21,504
Expected
Payoffs
33,504
Incentive
Compat.
Agent’s Payoff
Observable
Fixed Rent
to Prin.
Incentive
Compat.
33,391
18,496
22,250
21,609
32,750
31,536
18,496
7,250
8,464
32,750
33,020
18,496
19,250
18,980
22
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!
23
Let’s say that we set the two variables, R55 and R40, to be
18,769 and 11,449 respectively.
Effort level
Expected utility of the agent
E1=6
(18,769½)0.8+(11,449½)0.2-6½ =
95
E2=5
(18,769½)0.6+(11,449½)0.4-5½ =
100
E3=4
(18,769½)0.3+(11,449½)0.7-4½ =
100
Now, the principal is telling the agent NOT to work hard!
Effort level
Expected utility of the principal
E1=6
Not a feasible solution, agent’s utility < 100
n/a
E2=5
(55,000-18,769)0.6+(40,000-11,449)0.4 =
33,159
E3=4
(55,000-18,769)0.3+(40,000-11,449)0.7 =
30,855
The $33,159 is actually better than the $33,020 under
“incentive compatible” contract!
24
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.
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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.

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