3. Managerial Compensation and Optimal Contracting

Jun He May 31, 2026
公司金融Corporate Finance 代理问题Agency Problem 管理层薪酬Managerial Compensation 最优契约Optimal Contracting 自由现金流Free Cash Flow 委托代理Principal-Agent 学习笔记Study Note
Note

权衡理论让股东最大化 \(\text{APV}=V_0+\text{DTS}-\text{CFD}\)，但「股东」内部分为经理人与其他股东，激励未必相容——经理人只最大化自身利益。本章讨论如何对齐二者激励、资本结构如何发挥作用（全程假设经理人行动可观察可验证）。三类代理问题：企业津贴 (perks)、帝国建造 (empire building)、偷懒 (shirking)；四类治理机制：监督、控制权市场、激励契约、资本结构 (§3.1)。§3.2 Jensen-Meckling (1976) 津贴：经理持股比例 \(\alpha<1\) 时取津贴过多（\(g'(p^{\text{M}})=1/\alpha\) vs 一阶最优 \(g'(p^{\text{FB}})=1\)，故 \(p^{\text{M}}\ge p^{\text{FB}}\)）；无风险债通过提高经理持股比例可解决津贴过取，但有风险债反而诱发资产替代、使问题更糟。§3.3 帝国建造：自由现金流不可契约（Jensen 1986），高杠杆（Stulz 1990）用付息压缩自由现金流；Zwiebel (1996) 动态模型解释资本结构为何持续（中等质量经理发债抵御接管）。§3.4 偷懒下的最优契约：Innes (1990) 无成本验证 + 有限责任 → 债务最优（赢者通吃，IC 绑定 \(R_H^\star=X_H-C/\Delta\)）；Townsend (1979) 有成本状态验证 → 截断 \(j^\star\)、低状态审计付清/高状态付固定 \(p\)，仍是债务。§3.5 与资本结构无关的薪酬：Holmstrom-Milgrom (1987) 连续时间 CARA → 线性契约 \(W_1=\alpha+\beta Y_1\)，\(\beta^\star=\frac{1}{1+k\gamma\sigma^2}\)；Gabaix-Landier (2008) 正向匹配——大公司雇佣高才能经理、给更高薪 (3.7)，公司平均规模上升则薪酬上升。

Note

Trade-off theory has equity holders maximize \(\text{APV}=V_0+\text{DTS}-\text{CFD}\), but "equity holders" split into the manager and other shareholders, whose incentives need not be compatible — the manager only maximizes his own benefit. This chapter discusses how to align the two and the role of capital structure (assuming throughout that the manager's actions are observable and verifiable). Three agency problems: perks, empire building, shirking; four governance mechanisms: monitoring, market for corporate control, incentive contracts, capital structure (§3.1). §3.2 Jensen-Meckling (1976) perks: when the manager holds fraction \(\alpha<1\) of equity he takes too much perk (\(g'(p^{\text{M}})=1/\alpha\) vs first-best \(g'(p^{\text{FB}})=1\), so \(p^{\text{M}}\ge p^{\text{FB}}\)); risk-less debt solves perk overtaking by raising the manager's equity share, but risky debt can instead trigger asset substitution and make it worse. §3.3 empire building: free cash flow is non-contractible (Jensen 1986), and higher leverage (Stulz 1990) uses interest payments to squeeze free cash flow; Zwiebel (1996)'s dynamic model explains why capital structure persists (middle-quality managers issue debt to fend off takeovers). §3.4 optimal contracting under shirking: Innes (1990) costless verification + limited liability → debt is optimal (win-or-nothing, binding IC gives \(R_H^\star=X_H-C/\Delta\)); Townsend (1979) costly state verification → cutoff \(j^\star\), audit-and-pay-all in low states / flat \(p\) in high states, still debt. §3.5 compensation unrelated to capital structure: Holmstrom-Milgrom (1987) continuous-time CARA → linear contract \(W_1=\alpha+\beta Y_1\), \(\beta^\star=\frac{1}{1+k\gamma\sigma^2}\); Gabaix-Landier (2008) assortative matching — larger firms hire more talented managers and pay more (3.7), and compensation rises with average firm size.

3.1 Agency Problems

基于权衡理论，股东应最大化调整现值

$$\text{APV}=V_0+\text{DTS}-\text{CFD}$$

其中 \(V_0\) 为全股权基准价值。但「股东」内含两个子群体——经理人与其他股东——其激励未必相容：经理人总是最大化自身利益，未必最大化全体股东利益。本章讨论如何对齐二者激励、资本结构如何发挥作用。全程假设经理人的所有决策与行动对其他股东可观察、可验证（否则一切可契约化，问题就trivial）。

本节聚焦三类代理问题 (agency problems)：

  • 企业津贴 (corporate perks)：经理享用津贴（豪华公寓、私人飞机等）让公司买单；也可给自己开高薪。
  • 帝国建造 (empire building)：经理享受统治庞大公司的感觉，把钱花在低效项目上扩大公司规模，从管理职位中攫取效用。
  • 偷懒 (shirking off)：经理投入低于有效水平的努力。

可用的公司治理机制：

  • 监督 (monitoring)：受债权人、大股东（董事会）、相关监管机构（政府）检查。
  • 控制权市场 (market for corporate control)：来自外部接管的潜在威胁。
  • 激励契约 (incentive contract)：如以期权作为薪酬一部分以对齐管理层激励。
  • 资本结构 (capital structure)：管理层行为受资本结构影响，更高债务下可能更有利于股东。

Based on trade-off theory, equity holders should maximize the adjusted present value

$$\text{APV}=V_0+\text{DTS}-\text{CFD}$$

where \(V_0\) is the all-equity benchmark value. But "equity holders" contains two sub-categories — the manager and all other equity holders — whose incentives need not be compatible: the manager always maximizes his own benefit, not necessarily the benefit of all equity holders. This chapter discusses how to align their incentives and the role of capital structure. We assume throughout that all of the manager's decisions and actions are observable and verifiable by other equity holders (otherwise everything is contractible and the problem is trivial).

This section focuses on three agency problems:

  • Corporate perks: managers take advantage of perks (luxurious apartment, private jet, etc.) and let the company pay the bill; they can also give themselves a huge salary.
  • Empire building: managers enjoy the feeling of ruling an empire-like company, so they spend money on inefficient projects to enlarge the company's scale and extract utility from being on the managerial position.
  • Shirking off: managers may put in a less-than-efficient amount of effort.

The available corporate governance mechanisms:

  • Monitoring: checked by creditors, large equity holders (board of directors), and related regulatory agencies (government).
  • Market for corporate control: a potential threat from an outsider's takeover.
  • Incentive contract: contracts such as options as part of compensation to align managerial incentives.
  • Capital structure: managerial behaviors are affected by capital structure, and may be more favorable to equity holders under a higher level of debt.

3.2 Corporate Perks: Jensen and Meckling (1976)

3.2.1 Set-up

  • 记津贴价值为 \(p\)，津贴 \(p\) 对公司的成本为 \(g(p)\)，假设 \(g'(\cdot)>0\)、\(g''(\cdot)>0\)（津贴越高，对公司的成本越高且加速上升）。
  • 记 \(p=0\) 时的基准公司价值为 \(K\)。故公司价值 \(V(p)=K-g(p)\)。
Tip

Remark 3.1 为抽离其他摩擦与成本，此处假设资本结构仅通过经理的津贴量影响公司价值，不考虑税盾与财务困境。

3.2.2 First-best Perk

社会最优最大化「公司利益 + 经理利益」之和 \(\max_p V(p)+p\)，一阶条件 (3.1)：

3.2.1 Set-up

  • Denote the value of the perk by \(p\), and the cost of perk \(p\) to the firm by \(g(p)\), assuming \(g'(\cdot)>0\), \(g''(\cdot)>0\) (higher perks induce increasingly higher cost to the firm).
  • Denote the benchmark firm value with \(p=0\) by \(K\). So the firm's value is \(V(p)=K-g(p)\).
Tip

Remark 3.1 To abstract away from all other frictions and costs, we assume the capital structure here only takes effect on the firm's value through the manager's perk amount. So tax shield and financial distress are not considered.

3.2.2 First-best Perk

The socially optimal result maximizes the sum of the firm's interest and the manager's interest, \(\max_p V(p)+p\), with first-order condition (3.1):

$$V'(p)+1=0\ \Longrightarrow\ g'(p^{\text{FB}})=1\tag{3.1}$$

其中 \(p^{\text{FB}}\) 解社会优化问题，称「first best」。

3.2.3 Perk Overtaking

设经理持有总股权的 \(\alpha\in[0,1]\) 比例。经理问题 \(\max_p \alpha V(p)+p\)，一阶条件 (3.2)：

where \(p^{\text{FB}}\) solves the social optimization problem and is thus called the "first best" result.

3.2.3 Perk Overtaking

Suppose the manager holds fraction \(\alpha\in[0,1]\) of total equity. The manager's problem is \(\max_p \alpha V(p)+p\), with first-order condition (3.2):

$$\alpha V'(p)+1=0\ \Longrightarrow\ \alpha g'(p^{\text{M}})=1\tag{3.2}$$

由 \(g''>0\)，(3.1) 与 (3.2) 一并给出 \(p^{\text{M}}\ge p^{\text{FB}}\)，当 \(\alpha<1\) 时严格成立。即经理若不完全拥有公司，便取过多津贴。

Tip

Remark 3.2 由构造及 \(g'>0,g''>0\)，津贴取用对其他股东有负外部性。但经理总是最大化自身利益、拒绝将外部性内部化。

3.2.4 Risk-less Debt Solves Perk Overtaking

津贴过取可通过提高经理持股比例解决；一种办法是用无风险债提高债务比、同时保持总资产规模不变（无风险 = 无违约风险，即便最坏情形公司也保持偿付能力）。

  • 公司有两个项目：
  • 安全项目：产出恒为 \(K-g(p)\)。
  • 风险项目：两状态——高状态 \(H\) 以概率 \(\pi\in(0,1)\) 出现，产出 \(H-g(p)\)；低状态 \(L\) 以概率 \(1-\pi\) 出现，产出 \(L-g(p)\)。
  • 安全项目对公司更有效率 (3.3)：

By \(g''>0\), (3.1) and (3.2) together imply \(p^{\text{M}}\ge p^{\text{FB}}\), with strict inequality if \(\alpha<1\). So managers take too much perk if they don't own the firm completely.

Tip

Remark 3.2 By construction and \(g'>0,g''>0\), perk-taking has a negative externality on other shareholders. However, the manager always maximizes his own interest and refuses to internalize the externality.

3.2.4 Risk-less Debt Solves Perk Overtaking

Perk overtaking may be resolved by increasing the manager's percentage of share holding; one way is to increase the debt ratio using risk-less debt while keeping the total asset size fixed (risk-less = no potential risk of defaulting, so the firm remains solvent even in the worst scenario).

  • The firm has two projects:
  • Safe project: outcome is always \(K-g(p)\).
  • Risky project: two states — High state \(H\) with probability \(\pi\in(0,1)\), outcome \(H-g(p)\); Low state \(L\) with probability \(1-\pi\), outcome \(L-g(p)\).
  • The safe project is more efficient for the firm (3.3):

$$K-g(p)>\pi(H-g(p))+(1-\pi)(L-g(p))\ \Longrightarrow\ K>\pi H+(1-\pi)L\tag{3.3}$$

  • 总股权价值为 0.5，完全由经理拥有。两个项目都成本 1，故经理须为任一项目筹集 0.5。
  • 若有无风险债：面值 0.5，必有 \(L-g(p^{\text{FB}})>0.5\)。由于债务无风险，经理的收益恒为「总项目收益 $-0.5$」，故他选安全项目（由 (3.3) 更有效率），其收益为

$$\underbrace{K-g(p)+p}_{=V(p)+p}-0.5,$$

在 \(p=p^{\text{FB}}\) 处取最大。first-best 实现，津贴问题解决。

3.2.5 Risky Debt May Not Solve Perk Overtaking

延续 §3.2.4：若无风险债不可得，则 \(L-g(p^{\text{FB}})<0.5\)，意味公司若选风险项目且 \(L\) 实现则必须违约。设债权人无法就「选哪个项目」缔约，经理只能借到面值 \(F>0.5\) 的风险债。则经理选风险项目当且仅当

  • The total equity value is 0.5, completely owned by the manager. Both projects cost 1, so the manager must raise 0.5 for either project.
  • If risk-less debt is available: the face value is 0.5, and it must be that \(L-g(p^{\text{FB}})>0.5\). Since the debt is risk-less, the manager's payoff is always the total project payoff minus 0.5, so he chooses the safe project (more efficient by (3.3)), with payoff

$$\underbrace{K-g(p)+p}_{=V(p)+p}-0.5,$$

attaining its maximum at \(p=p^{\text{FB}}\). The first-best is realized and the perk problem is solved.

3.2.5 Risky Debt May Not Solve Perk Overtaking

Continuing §3.2.4: if risk-less debt is not available, then \(L-g(p^{\text{FB}})<0.5\), which means the firm has to default if it chooses the risky project and the \(L\) state realizes. Suppose the creditor cannot contract on which project to choose, so the manager only has access to risky debt with face value \(F>0.5\). Then the manager chooses the risky project iff

$$\pi(H-g(p)+p-F)+(1-\pi)p>K-g(\hat p)+\hat p-F\ \Longrightarrow\ F>\frac{K-g(\hat p)+\hat p-\pi H-p+\pi g(p)}{1-\pi}$$

其中 \(p\) 为选风险项目时的最优津贴、\(\hat p\) 为选安全项目时的最优津贴。可见只要 \(F\) 足够高，经理就选风险项目：

  • 风险项目无效率，故解已偏离 first-best。
  • 选风险项目时经理在 \(H,L\) 两状态都享用津贴，却仅在 \(H\) 状态承担津贴成本。故风险项目下的津贴选择由 \(\max_p \pi(H-g(p)+p-F)+(1-\pi)p\) 给出，一阶条件 \(\pi g'(p)=1\)。
  • 由 first-best \(g'(p^{\text{FB}})=1\)，因 \(\pi\in(0,1)\)、\(g''>0\)，得 \(p>p^{\text{FB}}\)，即津贴问题未解决。风险债通过让经理选错项目（资产替代 / 风险转移）反而使情况更糟。

3.2.6 Summary

债务除 DTS 外还有一项好处：提高经理的持股比例，使其激励更对齐于公司、不取过多津贴。然而若债务过高则可能出现资产替代：经理选错项目，且仍取过多津贴——因为违约成为可能，公司违约时他不承担津贴成本。两股力量的权衡可钉定一个最优债务水平。

3.3 Empire Building

3.3.1 Non-contractible Free Cash Flow Problem

有些公司每天产生巨额自由现金流 (free cash flow)（会计概念：调整利息支付、资本投资等后的收入，可由经理自由支配）。此类现金流可被观察但不可契约化（因为外部人事前无法判断哪个项目正 NPV，只有经理知道他是否想选）。Jensen (1986) 指出经理总把自由现金流花在无利可图的项目上，或为帝国建造、或因过度自信。此问题在大型成熟公司最严重，因其产生更多此类现金流。

3.3.2 Higher Leverage as A Solution

如 Stulz (1990) 所指，改变融资政策以控制经理可支配的资源，能降低不当投资的成本。简言之，提高债务比、迫使公司付息，可减少自由现金流，从而降低经理挥霍与帝国建造的成本。

一个自然的问题：想建帝国的经理不愿背负太多债务（限制其投资决策权）。故必有某些约束者 (discipliners)（董事会或其他大股东）为公司规定一定的债务比。

3.3.3 Dynamic Consistency in Capital Structure: Zwiebel (1996)

然而若约束者不能持续地约束资本结构，经理可待债务到期、还清后又重获投资自由。现实中经理有时确实在无约束者指示下控制资本结构，但资本结构却看似长期稳定。经理喜欢自由现金流，为何不在能还债时突然摆脱债务？

资本结构的这种一致性是文献中一个重要理论问题，需动态框架解释。Zwiebel (1996) 讨论一个内嵌前瞻的动态模型，基本逻辑：

  • 若公司不发足够的债，会向市场发出「公司未来投资很可能低效」的信号。
  • 公司随即成为接管目标，迫使其加杠杆以抵御接管。
  • Zwiebel (1996) 设经理有三种质量：
  • 高质量经理有足够好项目维持良好业绩，无需发债即可击退接管。
  • 低质量经理会被接管踢出。
  • 最有趣的是，中等质量经理会发债以捍卫对公司的控制权。

3.4 Shirking off

至此只关注债与股两类传统证券。但股东与经理间的契约可远比这两种复杂。允许更多契约特征有助于在信息不对称时解决经理的激励问题、防止偷懒。

3.4.1 Optimal Contracting with Costless State Verification: Innes (1990)

经理努力不可观察，但产出可无成本直接验证、从而可契约化。

where \(p\) is the optimal perk if the manager chooses the risky project and \(\hat p\) is the optimal perk if he chooses the safe project. So as long as \(F\) is high enough, the manager chooses the risky project:

  • The risky project is not efficient, so the solution is already away from first-best.
  • With the risky project, the manager enjoys the perk under both \(H\) and \(L\) states, but only bears the perk cost under the \(H\) state. So the perk choice with the risky project is given by \(\max_p \pi(H-g(p)+p-F)+(1-\pi)p\), with f.o.c. \(\pi g'(p)=1\).
  • Recall the first-best \(g'(p^{\text{FB}})=1\); since \(\pi\in(0,1)\) and \(g''>0\), we get \(p>p^{\text{FB}}\), i.e. the perk problem is not solved. Risky debt even makes it worse by letting the manager choose the wrong project (asset substitution / risk shifting).

3.2.6 Summary

Debt has another benefit in addition to DTS: increasing the manager's percentage of equity holding so his incentive is more aligned with the corporate and he won't take too much perk. However, if debt is too high, there might be an asset substitution problem: the manager chooses the wrong project and still takes too much perk, because default becomes possible and he won't bear the perk cost when the company defaults on debt. The trade-off between these two forces could pin down an optimal debt level.

3.3 Empire Building

3.3.1 Non-contractible Free Cash Flow Problem

Some firms generate a huge amount of free cash flow (an accounting concept: the income after adjusting for interest payment, capital investment etc., which could be spent at the manager's discretion) every day. Such cash flows can be observed but are not contractible (because outsiders cannot tell which project has positive NPV ex-ante, and only the manager knows if he wants to choose it). Jensen (1986) mentions that managers always spend free cash flows on unprofitable projects, either for empire building or because of over-confidence. Such a free-cash-flow problem is most severe in large and mature firms as they generate more such cash flows.

3.3.2 Higher Leverage as A Solution

As pointed out by Stulz (1990), changing financing policy to control for available resources under the manager's discretion can reduce the costs of inappropriate investment. In a nutshell, raising the debt ratio to force the firm to pay interest could reduce the amount of free cash flow, thus reducing the cost of managerial squandering and empire building.

A natural issue: managers who want to build an empire won't be happy to have much debt, which limits their power in making investment decisions. So there must be some discipliners (board of directors or other large equity holders) who regulate a certain debt ratio for the firm.

3.3.3 Dynamic Consistency in Capital Structure: Zwiebel (1996)

However, if the discipliners are not consistently ruling the capital structure, the manager could wait until the debt matures, pay it off, and then be free again in making investment decisions. In reality, managers sometimes do control the capital structure without directions from discipliners, yet the capital structure seems pretty consistent over time. Managers like free cash flows. Why don't they suddenly get rid of debt whenever they can?

Such consistency in capital structure is an important theoretical issue that requires a dynamic framework to explain. Zwiebel (1996) discusses a dynamic model with forward-looking embedded, whose basic logic is:

  • If the firm does not raise enough debt, it will be a signal to the market that the firm's future investment is very likely to be inefficient.
  • The firm will then become a target for takeover, which forces it to lever up to fight against the takeover.
  • Zwiebel (1996) proposes a model in which managers have three levels of quality:
  • High-quality managers have enough good projects to maintain good performance, so they can fend off takeovers without raising debt.
  • Low-quality managers will be kicked out by takeovers.
  • Most interestingly, middle-quality managers will raise debt to defend their control of the firm.

3.4 Shirking off

Up to now we focused on the two traditional types of securities: debt and equity. However, contracts between shareholders and managers could be way more complex than these two forms. Allowing for more features of the contract could help solve the incentive problem of managers and keep them from shirking off when information is asymmetric.

3.4.1 Optimal Contracting with Costless State Verification: Innes (1990)

The manager's effort is not observable, but the outcome is directly verifiable without extra cost, and thus contractible.

设定（简单委托-代理模型）：

  • 委托人 (principal) = 风险中性投资者，提供全部投资，可观察产出但不可观察代理人努力。
  • 代理人 (agent) = 风险中性经理，不出钱只出努力，且无力从经理处买下项目（否则因双方风险中性，经理可买下公司、轻易实现 first-best，问题trivial）。
  • 三期：
  • \(t=0\)：投资者投入 \(I\) 并与经理签约。
  • \(t=1\)：经理选努力 \(e\)。\(e=1\) 高努力、对经理成本 \(C\)；\(e=0\) 低努力、对经理免费。
  • \(t=2\)：两可能产出状态实现 \(X_H>X_L\)，且 \(\mathbb P(X_H\mid e=1)=p\)、\(\mathbb P(X_H\mid e=0)=p-\Delta\)（\(\Delta>0\)）。
  • 假设努力有效率：\(\Delta\cdot(X_H-X_L)>C\)。
  • 为使投资有吸引力，假设高努力下项目正 NPV：\(pX_H+(1-p)X_L-I-C>0\)。
  • 契约规定：\(X_H\) 实现则投资者获 \(R_H\)；\(X_L\) 实现则获 \(R_L\)，且 \(R_H>R_L\)。

最优契约：设 \(e=1\) 对投资者最优（比较 \(e=1\) 与 \(e=0\) 两个最大化问题后）。则最优契约要实现 \(e=1\)：\(\max_{R_H,R_L} p R_H+(1-p)R_L\)，约束为

Setup (a simple principal-agent model):

  • The principal = the risk-neutral investor, who provides the whole investment and can observe the outcome but not the agent's effort.
  • The agent = the risk-neutral manager, who devotes no money but only effort, and cannot afford to buy the project from the manager (otherwise, since both are risk-neutral, the manager could buy the firm and the first-best is easily realized, making the problem trivial).
  • Three periods:
  • \(t=0\): the investor puts money \(I\) into the project and signs the contract with the manager.
  • \(t=1\): the manager chooses effort \(e\). \(e=1\) is high effort, inducing cost \(C\) to the manager; \(e=0\) is low effort, free to the manager.
  • \(t=2\): two possible outcome states realize, \(X_H>X_L\), with \(\mathbb P(X_H\mid e=1)=p\) and \(\mathbb P(X_H\mid e=0)=p-\Delta\) (\(\Delta>0\)).
  • Assume effort is efficient: \(\Delta\cdot(X_H-X_L)>C\).
  • To make investment attractive, assume the project has positive NPV with high effort: \(pX_H+(1-p)X_L-I-C>0\).
  • The contract regulates: if \(X_H\) realizes, the investor gets \(R_H\); if \(X_L\) realizes, the investor gets \(R_L\), with \(R_H>R_L\).

Optimal contract: suppose \(e=1\) turns out optimal for the investor (after comparing the \(e=1\) and \(e=0\) maximization problems). Then the optimal contract wants \(e=1\) implemented: \(\max_{R_H,R_L} p R_H+(1-p)R_L\), subject to

$$\text{(IR)}\quad p(X_H-R_H)+(1-p)(X_L-R_L)\ge C$$

$$\text{(IC)}\quad \Delta\cdot\left[(X_H-R_H)-(X_L-R_L)\right]\ge C$$

$$\text{(LL)}\quad R_H

(IR) 个体理性保证经理愿参与；(IC) 激励相容保证经理实施 \(e=1\)；(LL) 有限责任保证经理付给投资者不超过项目产出。

Note

求解 / Solution（Innes 最优契约 = 债务） 最优契约尽量少留给经理： - 低状态经理得零：\(R_L^\star=X_L\)。 - IC 绑定，高状态的奖金恰好够诱使 \(e=1\)： $$\Delta\cdot\left[(X_H-R_H)-\underbrace{(X_L-R_L)}_{=0}\right]=C\ \Rightarrow\ \Delta(X_H-R_H)=C\ \Rightarrow\ R_H^\star=X_H-\frac{C}{\Delta}.$$ - 经理收益 \(M\)： $$\begin{cases}M_H=X_H-R_H^\star-C=\dfrac{1-\Delta}{\Delta}C\\[2pt] M_L=X_L-R_L^\star=0\end{cases}$$ 即经理在高状态拿全部剩余、低状态一无所获。此「赢者通吃」契约恰与债务契约相似。\(\blacksquare\)

故由 Innes (1990)，债务最优，与 Jensen-Meckling (1976) 结论相同但机理不同：Innes 着眼于设计契约避免经理偷懒；Jensen-Meckling 着眼于内部化经理津贴过取的负外部性。

上述讨论局限于二元努力、二元产出，过于简化。在连续产出设定下（Innes 1990）：

  • 无单调契约约束时（单调 = 投资者收益随产出不减），「赢者通吃」（产出低于阈值经理得零、超过则得全部）为最优契约。
  • 有单调契约约束时，传统债务为最优契约。（详见 He 2019c 第 18 节。）
Tip

Remark 3.3 Innes (1990) 中经理的行动空间被狭义定义为「仅通过改变概率密度影响均值的努力水平」。可给经理更灵活的行动空间，如 Hébert (2017) 让代理人以某种「熵」成本控制产出分布，得到即便无单调契约假设、债务仍最优的结果。

3.4.2 Optimal Contracting with Costly State Verification: Townsend (1979)

现实中项目/公司产出未必对外部人直接可观察。Townsend (1979) 设投资产出仅在采取有成本审计时才向投资者揭示，并证明债务最优。

设定：

  • 项目需初始投资 \(I\)，全来自风险中性投资者。经理风险中性、零财富、有限责任。
  • 产出现金流 \(\tilde y\) 仅取决于随机状态实现，且私下只对经理揭示。
  • 状态 \(s\) 实现时产出记 \(y_s\)。经理向投资者报告 \(\hat y\le y_s\)。
  • 据 \(\hat y\)，投资者决定是否审计。\(a=1\) 审计，\(a=0\) 不审计。
  • \(a=1\)：投资者付固定成本 \(c>0\)，真实 \(y_s\) 揭示。若 \(\hat y=y_s\)，无事发生、投资者获 \(\hat p\le\hat y\)；若 \(\hat y
  • \(a=0\)：无事发生、投资者获 \(\hat p\le\hat y\)。

审计与否的权衡：更多审计使经理更如实上报、付给投资者更高比例的 \(\hat y\)；但审计有成本、缩小蛋糕。需设计契约：在尽量少用审计的前提下，为投资者攫取尽量多的 \(\hat y\)。

机制设计与显示原理：假设双方完全承诺。经理发信号 \(\hat y\)（真或假地）揭示其私有类型，投资者据此取行动 \(a\) 钉定配置。显示原理指出，对任意「类型上报 + 相应行动」，都存在一个讲真话的直接机制给出完全相同结果（见 He 2019c 第 20 节），故下文只讨论讲真话的直接机制。

投资者的最优契约（问题）：

(IR) individual rationality ensures the manager participates; (IC) incentive compatibility ensures the manager implements \(e=1\); (LL) limited liability ensures the manager never pays the investor more than the project outcome.

Note

Solution (Innes optimal contract = debt) The optimal contract leaves as little as possible to the manager: - In the low state the manager receives zero: \(R_L^\star=X_L\). - IC binds, so the bonus for the high state is just enough to induce \(e=1\): $$\Delta\cdot\left[(X_H-R_H)-\underbrace{(X_L-R_L)}_{=0}\right]=C\ \Rightarrow\ \Delta(X_H-R_H)=C\ \Rightarrow\ R_H^\star=X_H-\frac{C}{\Delta}.$$ - The manager's payoff \(M\): $$\begin{cases}M_H=X_H-R_H^\star-C=\dfrac{1-\Delta}{\Delta}C\\[2pt] M_L=X_L-R_L^\star=0\end{cases}$$ i.e. the manager gets all the remaining value in the high state and nothing in the low state. This "win-or-nothing" contract is exactly similar to a debt contract. \(\blacksquare\)

So by Innes (1990), debt is optimal — the same result as Jensen-Meckling (1976) but with quite different rationales: Innes focuses on optimal contracting to avoid the manager's shirking; Jensen-Meckling focus on internalizing the negative externality of the manager's perk overtaking.

The discussion above is limited to binary effort and binary outcome, which is oversimplified. In a continuous-outcome setting (Innes 1990):

  • Without the monotone contract constraint (monotone = the payoff to the investor is non-decreasing in the outcome), "win-or-nothing" (the manager gets nothing below a threshold and everything above it) is the optimal contract.
  • Under the monotone contract constraint, traditional debt is the optimal contract. (See Section 18 of He 2019c.)
Tip

Remark 3.3 In Innes (1990), the manager's action space is narrowly defined as some effort levels that only affect the mean by affecting the probability density. We can give the manager a more flexible action space; for example, Hébert (2017) proposes a model where agents can control the outcome distribution with some "entropy" cost, which yields the result that debt is optimal even without the monotone contract assumption.

3.4.2 Optimal Contracting with Costly State Verification: Townsend (1979)

In reality, project/firm outcomes may not be directly observable to outsiders. Townsend (1979) proposes a model where the investment outcome is only revealed to investors if costly auditing is taken, and shows that debt is optimal.

Setup:

  • The project requires \(I\) as the initial investment, all from risk-neutral investors. The manager is risk-neutral, has zero wealth and limited liability.
  • The outcome cash flow \(\tilde y\) only depends on the random state realization, and is privately revealed to the manager.
  • Given state \(s\) realized, the realized outcome is denoted \(y_s\). The manager reports \(\hat y\le y_s\) to investors.
  • Depending on \(\hat y\), investors decide whether to audit. \(a=1\) means audit, \(a=0\) no audit.
  • \(a=1\): the investor pays a fixed cost \(c>0\), and true \(y_s\) is revealed. If \(\hat y=y_s\), nothing happens and investors get \(\hat p\le\hat y\); if \(\hat y
  • \(a=0\): nothing happens, investors get \(\hat p\le\hat y\).

Trade-off between auditing and not: more auditing ensures the manager reports more truthfully and pays a higher proportion of \(\hat y\) to investors; but auditing is costly and reduces the size of the pie. We need a contract that grabs as much \(\hat y\) as possible for investors while not using too much auditing.

Mechanism design and the revelation principle: assume both parties have full commitment. The manager sends a signal \(\hat y\) to (truthfully or untruthfully) reveal his private type, and investors take action \(a\) to pin down the allocation. The revelation principle tells us that for any type-reporting and corresponding action, there is a truth-telling direct mechanism that yields exactly the same result (see Section 20 of He 2019c), so we discuss the truth-telling direct mechanism from now on.

The investor's optimal contract (problem):

$$\max_{p_i}\ \sum_i(y_i-p_i)\pi_i$$

$$\text{(IR of investors)}\quad \sum_i p_i\pi_i-\sum_i a_i c\,\pi_i\ge I\tag{3.4}$$

$$\text{(IC of manager, }i>j\Rightarrow y_i\ge y_j)\quad y_i-p_i\ge y_i-p_j\ \text{ for }i>j\text{ if }a_j=0\tag{3.5}$$

$$\text{(LL of manager)}\quad p_i\le y_i$$

(3.4) 保证投资者至少回本；(3.5) 保证经理在不审计的状态下如实上报；(LL) 为有限责任。

结论：存在一个截断状态 \(j^\star\)：

  • 当 \(y_i\ge y_{j^\star}\)，经理只付固定值 \(p_{j^\star}=p\) 给投资者。
  • 当 \(y_i
  • 投资者 IR (3.4) 取等 (3.6)：

$$\sum_{i

Note

推理 / Reasoning（三步说明截断契约最优） Step 1：若状态 \(j^\star\) 不审计，则任何更高状态 \(i>j^\star\) 都不应审计、且 \(p_i=p_{j^\star}\)。（(3.5) 立即给出 \(a_{j^\star}=0\) 时 \(p_i\le p_{j^\star}\)。直觉：经理总能报 \(\hat y=y_{j^\star}\)，因为那里不审计；更高产出时经理不会付超过 \(p_{j^\star}\)，故总是假装持有 \(y_{j^\star}\)、付 \(p_{j^\star}\) 而不被审计。） Step 2：由 Step 1，审计发生在低状态。投资者应在这些低状态最大化 \(p_i\)，即 \(p_i=y_i\)（\(iStep 3：进一步改进以确保 \(y_{j^\star-1}\le p\)。有 \(y_{j^\star-1}p\)，可把截断改为 \(j^\star-1\)、保持同一 \(p\)，则投资者获更多（\(p>y_{j^\star-1}-c\)）、经理也获更多（\(y_{j^\star-1}>p\)）。在经理偏好的契约中，投资者 IR (3.4) 应绑定，即 (3.6)。最优契约具有债务特征。\(\blacksquare\)

可见 Townsend (1979) 的有成本状态验证模型给出债务最优——不要求契约对投资者单调，却作为结果实现单调。Diamond (1984) 等许多研究也得到类似的债务最优结论。

但此类模型引出新问题：

  • 审计是公司的无谓损失 (deadweight loss)，导致投资不足：满足 \(\mathbb E[\tilde y]-c\cdot a(\hat y)
  • 承诺至关重要。即便投资者知道经理如实上报，他们仍须承诺在低状态付成本审计；若不履行，经理便失去如实上报的激励。显示原理依赖承诺，承诺无保障则失效。
Tip

Remark 3.4 在无成本状态验证模型中，债务最优通过经理的努力激励实现；而在有成本状态验证模型中，债务最优不通过努力激励、而通过向投资者揭示真实产出实现。两种建模假设下债务最优的驱动力不同。

至此讨论的都在资本结构空间内，多数研究认为债务式融资能更好地对齐经理与股东的激励。然而对大公司而言改变资本结构成本高昂，且存在不触及资本结构的契约替代方案来对齐激励。例如 Holmstrom and Milgrom (1987) 讨论产出直接可契约的模型。

3.5.1 Linear Optimal Contracting: Holmstrom and Milgrom (1987)

  • 连续时间，时间归一化到 \(t\in[0,1]\)。
  • 产出 \(Y_t\) 可观察，\(dY_t=\mu_t\,dt+\sigma\,dZ_t\)，\(Z_t\) 标准布朗运动，\(\mu_t\) 为经理控制的漂移。
  • 漂移 \(\mu_t\) 对经理产生成本 \(\kappa(\mu_t)\)，\(\kappa'>0\)、\(\kappa''>0\)。
  • 经理具 CARA（指数）效用，只关心 \(t=1\) 的效用，欲最大化

(3.4) ensures investors at least break even; (3.5) ensures the manager reports truthfully in no-auditing states; (LL) is limited liability.

Result: there is a cutoff state \(j^\star\):

  • When \(y_i\ge y_{j^\star}\), the manager only pays a flat value \(p_{j^\star}=p\) to investors.
  • When \(y_i
  • The investor's IR (3.4) holds with equality (3.6):

$$\sum_{i

Note

Reasoning (three steps showing the cutoff contract is optimal) Step 1: if there is no auditing in state \(j^\star\), then for any higher state \(i>j^\star\) there should be no auditing, and \(p_i=p_{j^\star}\). ((3.5) immediately gives \(p_i\le p_{j^\star}\) for \(i>j^\star\) if \(a_{j^\star}=0\). Intuition: the manager can always report \(\hat y=y_{j^\star}\) since no auditing takes place there; for higher outcomes he won't pay more than \(p_{j^\star}\), so he always pretends to have \(y_{j^\star}\) and pays \(p_{j^\star}\) without being audited.) Step 2: by Step 1 auditing takes place in low states, so investors should maximize \(p_i\) in those low states, i.e. \(p_i=y_i\) for \(iStep 3: a further improvement to ensure \(y_{j^\star-1}\le p\). We have \(y_{j^\star-1}p\), the contract could be further improved by changing the cutoff to \(j^\star-1\) with the same \(p\): investors get more (\(p>y_{j^\star-1}-c\)) and the manager also gets more (\(y_{j^\star-1}>p\)). In a contract favored by the manager, the investor's IR (3.4) should bind, i.e. (3.6). The optimal contract has debt features. \(\blacksquare\)

So Townsend (1979)'s costly state verification model yields debt optimality — it does not require the payoff to investors to be monotone in assumption, yet achieves monotonicity as a result. Many other studies such as Diamond (1984) also achieve similar debt-optimality results.

However, such models create some new issues:

  • Auditing is a deadweight loss to the firm, which causes under-investment: projects with \(\mathbb E[\tilde y]-c\cdot a(\hat y)
  • Commitment is crucial. Even if investors know the manager is reporting truthfully, they still commit to pay the cost to audit in low states; if they fail to, the manager has no incentive to report truthfully anymore. The revelation principle relies on commitment, so it fails if commitment is not guaranteed.
Tip

Remark 3.4 In costless state verification models, debt optimality is attained through the effort incentives of the manager; in costly state verification models, debt optimality is attained not through effort incentives but through revealing the true outcome to investors. So the driving forces of debt optimality differ under these two modeling assumptions.

Up to now we discussed models in the space of capital structure, of which most studies argue that debt-like financing aligns the manager's incentive with shareholders' better. However, changing the capital structure is costly for a huge firm, and there are other contracting alternatives to align the manager's incentives without touching the capital structure. For example, Holmstrom and Milgrom (1987) discuss a model where the outcome is directly contractible.

3.5.1 Linear Optimal Contracting: Holmstrom and Milgrom (1987)

  • Continuous time, normalized to \(t\in[0,1]\).
  • Output \(Y_t\) is observable, \(dY_t=\mu_t\,dt+\sigma\,dZ_t\), with \(Z_t\) a standard Brownian motion and \(\mu_t\) the drift controlled by the manager.
  • The drift \(\mu_t\) causes a cost \(\kappa(\mu_t)\) to the manager, with \(\kappa'>0\), \(\kappa''>0\).
  • The manager has CARA (exponential) utility and only cares about the utility at \(t=1\), maximizing

$$U=\max_{\mu_t:\,0\le t\le1}\mathbb E\!\left[-\frac1\gamma\exp\!\left[-\gamma\left(W_1-\int_0^1\kappa(\mu_t)\,dt\right)\right]\right]$$

其中 \(W_1\) 为 \(t=1\) 的工资。委托人求解 \(\max_{W_1}\mathbb E[Y_1-W_1]\) s.t. \(U\ge\underline U\)（经理外部选项）。Holmstrom and Milgrom (1987) 证明：

  • 最优工资为线性形式 \(W_1=\alpha+\beta Y_1\)，其中 \(\alpha\) 由匹配经理外部选项 \(U=\underline U\) 钉定。
  • 经理等价求解 \(\max_\mu \alpha+\beta\mu-\dfrac\gamma2\sigma^2\beta^2-\kappa(\mu)\)。例如 \(\kappa(\mu)=\tfrac12 k\mu^2\) 时 \(\mu^\star=\dfrac\beta k\)。
  • 委托人等价求解（代入 \(\mu^\star=\beta/k\)）

where \(W_1\) is the wage at \(t=1\). The principal solves \(\max_{W_1}\mathbb E[Y_1-W_1]\) s.t. \(U\ge\underline U\) (the manager's outside option). Holmstrom and Milgrom (1987) show:

  • The optimal wage is linear: \(W_1=\alpha+\beta Y_1\), where \(\alpha\) is pinned down by matching the manager's outside option \(U=\underline U\).
  • The manager equivalently solves \(\max_\mu \alpha+\beta\mu-\dfrac\gamma2\sigma^2\beta^2-\kappa(\mu)\). For example, with \(\kappa(\mu)=\tfrac12 k\mu^2\), \(\mu^\star=\dfrac\beta k\).
  • The principal thus equivalently solves (plugging in \(\mu^\star=\beta/k\))

$$\max_\beta\ \frac\beta k-\frac k2\left(\frac\beta k\right)^2-\frac\gamma2\sigma^2\beta^2\ \Longrightarrow\ \beta^\star=\frac{1}{1+k\gamma\sigma^2}$$

Tip

Remark 3.5 虽然上面把问题当作一次性努力决策 \(\mu\) 来解，经理其实可随时改变 \(\mu_t\)，故这是个动态问题。HM (1987) 的卓越结果在于：尽管问题动态，最优契约仍是线性的——这与代理人只决策一次的静态设定颇为不同。（离散时间设定见 He 2019c 第 19 节。）

3.5.2 Managerial Compensation for Their Talents: Assortive Matching Model, Gabaix and Landier (2008)

经理薪酬也可基于其才能，与激励无关。Gabaix and Landier (2008) 用正向匹配 (assortive matching) 模型解释经理基于才能的高薪。

  • 公司按 \(n\in[0,N]\) 索引，规模 \(S(n)\)、\(S'(n)<0\)。
  • 经理按 \(m\in[0,M]\) 索引，才能 \(T(m)\)、\(T'(m)<0\)，且 \(M>N\)。
  • 记均衡工资函数 \(w(m)\)（竞争结果，公司视为给定）、匹配函数 \(m=M(n)\)。竞争市场的均衡工资由下式有效决定：

$$\max_{M(\cdot)}\int_0^N C\cdot S(n)^\gamma\cdot T(M(n))\,$$

其中 \(C\) 为才能 \(T\) 对产出的量化效应，公司 \(n\) 的产出为 \(C\cdot S(n)^\gamma\cdot T(M(n))\)。

  • 有效结果是正向匹配：才能越高的经理为越大的公司工作（用排序不等式易证）。该生产函数隐含「规模越大产出越高」。
  • 公司 \(n\) 求解雇谁：\(\max_m C\cdot S(n)^\gamma\cdot T(m)-w(m)\)，一阶条件 \(C\cdot S(n)^\gamma\cdot T'(m)=w'(m)\)，由此得 (3.7)：
Tip

Remark 3.5 Although we solved the problem above as if it is a one-time effort decision \(\mu\), the manager can actually change \(\mu_t\) anytime they like, so this is a dynamic problem. The remarkable result of HM (1987) is that despite the dynamics, the optimal contract is still linear — quite different from a static setting where the agent only makes a decision once. (For the discrete-time setup, see Section 19 of He 2019c.)

3.5.2 Managerial Compensation for Their Talents: Assortive Matching Model, Gabaix and Landier (2008)

A manager's compensation could also be based on their talents, which has nothing to do with incentives. Gabaix and Landier (2008) discuss an assortative matching model to explain a manager's high compensation based on talents.

  • Firms indexed by \(n\in[0,N]\) have size \(S(n)\), with \(S'(n)<0\).
  • Managers indexed by \(m\in[0,M]\) have talent \(T(m)\), with \(T'(m)<0\), and assume \(M>N\).
  • Denote the equilibrium wage function by \(w(m)\) (a competition result taken as given by firms) and the matching function by \(m=M(n)\). A competitive market has an equilibrium wage function efficiently determined by:

$$\max_{M(\cdot)}\int_0^N C\cdot S(n)^\gamma\cdot T(M(n))\,$$

where \(C\) is the quantitative effect of talent \(T\) on output, and the output of firm \(n\) is \(C\cdot S(n)^\gamma\cdot T(M(n))\).

  • The efficient outcome is assortative matching: a more talented manager works for a larger firm (easily proved by ranking inequality). This production function implicitly assumes that larger size leads to higher output.
  • Firm \(n\) solves which manager to hire: \(\max_m C\cdot S(n)^\gamma\cdot T(m)-w(m)\), with f.o.c. \(C\cdot S(n)^\gamma\cdot T'(m)=w'(m)\), which implies (3.7):

$$w(n)=w(n_0)+\int_{n_0}^n w'(u)\,du=w(n_0)+\int_{n_0}^n C\cdot S(u)^\gamma\cdot T'(u)\,du=w(n_0)+\int_n^{n_0}C\cdot S(u)^\gamma\cdot\underbrace{(-T'(u))}_{>0}\,du\tag{3.7}$$

其中 \(n_0\) 为任意参照公司（最后一项 $>0$ 当 \(nS(n_0)\)）。该结果表明：均衡中更大的公司雇佣才能更高的经理、给更高薪水，是一种有效匹配。(3.7) 还表明：当公司平均规模上升时，对经理的薪酬也上升。

where \(n_0\) is any reference firm (the last term is $>0$ when \(nS(n_0)\)). This result suggests that in equilibrium, a larger company hires a more talented manager and gives him a higher salary, which is efficient matching. (3.7) also suggests that when the average size of companies increases, the compensation to managers also increases.

References

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