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Information, Control and Games

Explore the concept of moral hazard and incentives theory through examples and a clear definition. Understand the difficulties in the principal-agent problem and learn about different incentive schemes. This article also introduces the influential thinker Joseph Schumpeter and his concept of creative destruction.

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Information, Control and Games

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  1. 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

  2. Moral Hazard and Incentives Theory

  3. 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

  4. 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

  5. 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

  6. 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?

  7. 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?

  8. 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 principalw = 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 agentw = wH if the agent decides effort High eHw = wL if the agent decides effort High eL

  9. 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

  10. 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 w0 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 >0w is set to be u(w)=eL

  11. 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

  12. 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

  13. Base Wage + bonus • Agentbase wage wb, bonus wm -wbfor m,wg -wbfor 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 eHif 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

  14. 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 eHif 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)

  15. 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?

  16. 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.

  17. 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 ↑

  18. 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

  19. 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

  20. 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)

  21. 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))

  22. 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)

  23. 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,

  24. 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

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