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Information, Control and Games

Shi-Chung Chang

EE-II 245, Tel: 2363-5251 ext. 245

scchang@cc.ee.ntu.edu.tw, http://recipe.ee.ntu.edu.tw/scc.htm

Office Hours: Mon/Wed 1:00-2:00 pm or by appointment

Yi-Nung Yang

(03 ) 2655205, yinyang@ms17.hinet.net

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Moral Hazard and Incentives Theory

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Examples and a Definition

• When you purchase an expensive piece of equipment, e.g., stereo or TV, do you purchase a service contract also?

• Some facts:– With a service contract, one will be less careful

in using the equipment

• What kind of car insurance do you have? Why?

• Owner-Manager• Firm-Saleman

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Moral hazard

• 道德風險– A person who has insurance coverage will have

less incentive to take proper care of an insured object than a person who does not

• Two players are involved:– Principal: The insurance company, like the

leader– Agent: The insured person, like the follower

• Principal- Agent problem ( 代理人問題、僱傭關係問題 )– Owner-Manager– Firm-Saleman

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Difficulties in Principal-Agent problem

– What is good for the agent might not be good for the principal

• Asymmetric inform– The principal might not be able to observe the agent’s

action– Owner-Manager 股東 vs 總經理

• Owners (principal) care about the profits, and • managers (agent) care additionally about the number of hours they work• The appropriateness and quality of managers’ actions could be

“unobservable” to the owner

– Firm-Saleman老闆 vs 售貨員

• What can be done to induce the agent to behave properly?– Incentive schemes

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The essential question in incentive scheme design

• The essential question:– What kind of incentive schemes should the

principal adopt so that his agent would be induced to choose a pre-specified action?

• How to formulate the problem mathematically?

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A Principal-Agent Model

– Consider a simple owner-manager model with one owner (principal) and one manager (agent)

– The principal decides and announces the incentive scheme– The agent decides effort High eH (with disutility dH) or effort

Low eL (with disutility dL), assuming dH > dL?– The states:

• The profit could be good g, medium m, or bad b (g > m > b):

• With eH, Prob(g) = 0.6, Prob(m) = 0.3, and Prob(b) = 0.1• With eL, Prob(g) = 0.1, Prob(m) = 0.3, and Prob(b) = 0.6• How much the agent gets depends on the scheme and the

profit• eH doesn’t guarantee g, and eL doesn’t necessarily end up at b• Conversely, g doesn’t imply eH, and b doesn’t imply eL

• What are possible incentive schemes?

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Examples of Incentive Schemes

• A pure wage scheme– 任何情況下 , agent ( 經理 ) 皆領固定報酬 , i.e.,

w = wg = wm = wb

• A pure franchise scheme– agent 付固定權利金 (franchise fee) f 給 principal

w = profit - f• An intermediate─wage plus bonus scheme

– agent gets a base wage ( 底薪 ) plus bonus ( 分紅 )w = wb + (wm-wb) if the profit = mw = wb + (wg-wb) if the profit = g

• An infeasible effort-based wage scheme (No MH)– The wage depends on the effort of the agent

w = wH if the agent decides effort High eH w = wL if the agent decides effort High eL

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Special Assumptions

• Utility functionu(w) = 2√w

• DisutilitydH = eH = 10dL = eL = 0

• profit, = g, m, b g=200m=100b=50

• Agent’s payoffEu(w)-d (d=eH, eL)

• Principal’s payoffE - w

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A pure wage scheme

• Agent’s strategy– Fixed payment: w = wg = wm = wb

– Agent’s payoff:u(w)-dH if the agent decides effort = eH (= dH)u(w)-dL if the agent decides effort = eL (= dL)

– Q: if you were the agent, will you choose?– With an equal wage, the agent will choose eL

• Principal’s strategy– Known (assuming) the agent’s effort = eL =0– so he sets w0 for the agent, – Principal’s expected payoff

=0.1g+0.3m+0.6b-w = 20+30+20-0 = 80 – Note: if the agent’s effort can be enforced to be eL >0

w is set to be u(w)=eL

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Joseph Schumpeter 熊彼得 • 熊彼得為奧地利出生的經濟史及經濟

思想家– 畢業於維也納大學。二十五歲就以

《理論經濟學的本質與主要內容》一書奠定了學術地位。

– 1931 年赴美，終身在哈佛任教，名列哈佛七賢之一。

– 熊彼得最具代表性的思想，就是「資本主義的創造性破壞」 (The creative destruction of capitalism)

• 熊彼得上課的給分原則 : 對三種人 , 成績給 A

• (1) 女士 , (2) 基督徒 ...• (3) 以及其它所有的人• Q: 為何不是所有的學生選擇 effort

eL

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The Pure Franchise Scheme• Agent’s payoff: E[u(w)] -d

– where w=Profit –f, and

– eH: J1H =E[u(w)] -d= 0.6u(g-f) +0.3u(m-f) +0.1u(b-f) -eH

– eL: J1L = 0.1u(g-f) +0.3u(m-f) +0.6u(b-f) -eL

• Q. Which one would the agent choose, eH or eL?

– eH is preferable if J1H ≥J1L, or 0.5[u(g-f) -u(b-f)] ≥eH -eL

– Under SA: u(w) = 2√w, eH =10, eL =0, g=200, m=100, b=50, we have √(200-f) -√(50-f) ≥10 and f≤50

– Combined the two conditions: 43.75 ≤f≤50

• How about the principal?– The principal’s income =Franchise fee =50 <80 (pure wage)– The agent’s expected payoff = 1.2√150+0.6√50-10=8.94

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Base Wage + bonus• Agent

base wage wb, bonus wm -wb for m,wg -wb for g– eH: J1H = 0.6u(wg) +0.3u(wm) +0.1u(wb) -eH

– eL: J1L = 0.1u(wg) +0.3u(wm) +0.6u(wb) -eL

– eH is preferable if J1H ≥J1L, or 0.5[u(wg) -u(wb)] ≥eH -eL

– SA: u(w) =2√w, eH =10, dL =0, g=200, m=100, b=50, and wb 0, the above becomes √wg ≥10, or wg ≥100

• Principal– eH: J0H =0.6(g-wg) +0.3(m-wm) +0.1(b-wb)– eL: J0L =0.1(g-wg) +0.3(m-wm) +0.6(b-wb)– The principal would be willing to see eH

if J0H ≥J0Lor 0.5[(g-wg) -(b-wb)] ≥0, or wg -wb ≤g-b– SA and wb 0: wg ≤150 ~OK, since previously wg ≥100– With wg =100 and wm =0, J0H =60 + 30 +5=95 >80 >50

• The agent’s expected payoff =1.2√100 -10 =2

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The Infeasible Effort-based Wage scheme• If no MH, i.e., the principal could observe the agent’s effort, the

principal could pay the agent– wH if his effort = eH wL if his effort = eL

• The agent prefers eH if u(wH)-eH ≥ u(wL)-eL

– SA: u(w) =2√w, eH =10, dL =0, g=200, m=100, b=50, and wL 0, the above becomes √wH ≥5, or wH ≥25 (J1H=?)

• The principal– eH: J0H =0.6g +0.3m +0.1b-wH

– eH: J0L =0.1g +0.3m +0.6b-wL

– The principal prefers eH if J0H ≥J0Lor 0.5(g-b) ≥(wH -wL)

– SA and wL 0: 75≥wH ~OK, since previously wH ≥25– J0H =120 + 30 +5-25 = 130 (>95 >80 >50)

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Optimal Incentive Scheme

• It is difficult to find the optimal incentive scheme. However, for the four incentive scheme analyzed before, we can find out which one is the best for the principal

• Q. Which one? How to address this issue?– The pure wage scheme: J0L = 80 and J1L = 0

– The pure franchise scheme: J0H = 50 and J1H = 8.94

– The base wage plus bonus scheme: J0H = 95 and J1H = 2

– The effort-based wage scheme: J0H = 130 and J1H = 0

– The best feasible schemes: Base wage plus bonus, but this is still worse than the infeasible effort-based scheme. Why?

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Necessary condition for elicit eH (hard work)

• Base wage + bonus– 0.5[(g-wg) -(b-wb)] ≥0 or

0.5 (g-b) ≥ 0.5 (wg -wb) (150 ≥ wg ≥ 100)

• Effort-based Wage scheme (if no MH)– 0.5(g-b) ≥(wH -wL) (75 ≥ wH ≥

25)

• 由於 MH, principal 必需提高 incentive, 才能確保 agent 願意 work hard.

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Base wage + bonus

• The principal’s view– (g-b) ≥ 0.5 (wg -wb)

• The agent’s view– 0.5[u(wg) -u(wb)] ≥eH -eL

• bonus = wg -wb

– if u(w) is monotonically increasing in w, 則(g-b)↑ ==> bonus ↑(eH-eL)↓ ==> bonus ↑

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Some General Conclusions

• Result 1: To elicit eH, bonuses are needed for good results– Follow directly from the discussion of the pure wage

scheme

• Corollary 1': The principal is always strictly worse off if there is moral hazard versus when there is not– When there is no moral hazard involved, the principal can

condition the agent’s payment directly on the effort level– No matter what is the outcome, the agent gets the same

level of reward for the same level of effort ~ A pure wage scheme conditioned on the effort

– Now since the principal cannot observe eH, eL, moral hazard comes into the picture, and the principal will be worse off

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Result 2: The higher the profit, the higher the bonus.

• True?– Not exactly true! How should it be modified?

• Consider a more general case where– With eH, Prob(g) = 0.6, Prob(m) = 0.3, and

Prob(b) = 0.1With eL, Prob(g) = 0.1, Prob(m) = p, and Prob(b) = 0.9 - p

– Previously p = 0.3. If p < 0.3, then m and g are more likely a result of hard work. He should reward the agent.

• Q. What is the general condition? Need to define likelihood ratio

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Insurance market

• 假設 & 定義– 原始財富水準 w

– 發生意外機率 , 損失 L

– 為防止意外繳交保費 ,

– 投保額 z ( 即發生意外之後 , 投保人獲償之金額 )

– q 為每單位投保額所需繳交之保費 ( 由保險公司決定 )( 故選擇投保額 z 者 , 需繳交保費 = qz)

– 消費者效用函數 = u(w)

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投保者 ( 消費者 ) 的期望效用極大化

• 投保者 ( 消費者 ) 選擇 z, 以尋求期望效用最大 :即求解 :max (1- )u(w-qz) + u(w-qz-L+z)– 上式對 z 偏微分求解最適投保額 z , 其一階條件為

(1- ) u(w-qz)(-1)q+ u(w-qz-L+z)(-q+1)=0, 或

(1- ) u(w-qz) q= u(w-qz-L+z)(1-q)

• 再假設保險公司在完全競爭市場中– 即利潤 =0, i.e., 收到的保費皆用來支付理賠 qz= z (q=

), 代入上式 , ( 如果 u(.) is a monotonic function) 可得 : u(w-qz)= u(w-qz-L+z) ==> w-qz= w-qz-L+z

– z = L ( 消費者選擇之保額 = L ( 發生意外時之損失 ))

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Moral Hazard ( 道德風險 )

• 若個人發生意外的機率 與其小心程度 x 有關 = (x), for x ≥0, 且– 愈小心的人 , 發生意外的機率愈低 , i.e.,

(x)/x = (x) <0

• 若保險公司無法觀察每人投保人之「小心程度」– 而將每單位保費設為相同的 q,

– 則投保人 ( 消費者 ) 尋求期望效用最大時 , 同時選 z 和 x, 即求解 :max (1- (x))u(w-x-qz) + (x)u(w-x-qz-L+z)

– 其一階條件為(1) (x) (1-q) u(w-x-qz-L+z)- (1- (x)) q u(w-x-qz)=0 (2) - (x) u(w-x-qz) + (1- (x)) u(w-x-qz)(-1)+ (x)u(w-x-qz-L+z)+ (x) u(w-x-qz-L+z)(-1)

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保險為完全競爭市場下之道德風險 (1/2)

• 第 2 個 FOC- (x) u(w-x-qz) + (1- (x)) u(w-x-qz)(-1)+ (x)u(w-x-qz-L+z)+ (x) u(w-x-qz-L+z)(-1)==>(x) [u(w-x-qz-L+z)- u(w-x-qz)] - (1- (x)) u(w-x-qz) -(x) u(w-x-qz-L+z)

• 假設保險公司為完全競爭市場 , i.e., q= (x)– 代入 (1) 式得 :

(1- ) u(w-x- z-L+z)=(1- ) u(w-x- z)故 w-x- z-L+z = w-x- z,

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保險為完全競爭市場下之道德風險 (2/2)

• 所以– 當 x=0, w- z-L+z = w-z, 令 Q = w- z-L+z, R= w-z, Q=R

• 再討論 (2) 式 if x=0 (0) [u(Q)- u(R)] - (1- ) u(R) -u(Q) = - u(R) < 0

• 若 u(Q) = 0 ( 且 q= (x) , 即 Q=R, u(Q)=u(R)) - (x) u(R) - (1- (x)) u(R)+ (x)u(Q)==> - (1- (x)) u(R) = 0, 即 (1- (x)) = 任何數 for x>0

• 結論– if x = 0 ( 完全不需小心 ) ==> - u(Q) < 0, i.e., u(Q)>0

if x >0 ( 稍微小心 ) ==> u(Q)= 0

– 由於 u(Q)>0 (u(Q)= 0 不會成立 ), 故 x = 0即投保者「完全不需小心」發生意外 ==>Moral Hazard