19. Application of the Moral Hazard Model: Holmstrom-Milgrom
19. Application of the Moral Hazard Model: Holmstrom-Milgrom
本章导读 §17.5 一般模型与 §18 Innes (1990) 都需大量额外假设才能得到"工资随产出递增"的直觉结果，更糟的是解可能不存在。本章转向一种新设定：代理人具有 CARA 效用，并得到结论——每个可实施（满足 IR、IC）的努力（即所选分布）\(\phi\) 都由唯一的工资表诱导。§19.1 记号：离散期 \(t=1..T\)、离散产出态 \(x_t\in X=\{X_0,..,X_n\}\)、工人所选每期分布 \(\phi_t\in\Phi\subseteq\Delta^n\)（\(\Phi\) 在单纯形内部＝分布非退化）；委托人风险中性 \(w\)，工人 CARA \(u(y)=-e^{-ry}\)（注 19.1：绝对风险厌恶恒定，使模型平稳）。§19.2 单期模型：委托人 \(\max\sum\phi_i(X_i-w_i)\) s.t. IC、IR；命题 19.1（任一 \(\phi\) 要么不可实施，要么存在唯一令 IC、IR 取等的 \(w^\star\)）；命题 19.2（CARA 下改变外部选项 CE，\(\phi^\star\) 不变，只需对每态工资加常数差）。§19.3 \(T\) 期模型：逐期 IC (19.2)；命题 19.3（每期延续值 CE 唯一钉定）、(19.4) 态依存工资、命题 19.4（最优合约每期相同，线性聚合工资 \(w(X^T)=w^\star\cdot A^T=\sum w_i^\star A_i^T\)，\(A_i^T\)＝产出 \(i\) 至第 \(T\) 期的出现次数）。无图。
19. Application of the Moral Hazard Model: Holmstrom-Milgrom
Overview Both the general model of §17.5 and Innes (1990) of §18 require many additional assumptions to get the intuitive "wages increase in output" result, and worse, the solution may not exist. This chapter turns to a different setting: the agent has CARA utility, and the result is that each implementable (IR&IC-satisfying) effort (i.e. chosen distribution) \(\phi\) is induced by a unique wage schedule. §19.1 notation: discrete periods \(t=1..T\), discrete output states \(x_t\in X=\{X_0,..,X_n\}\), the per-period distribution \(\phi_t\in\Phi\subseteq\Delta^n\) chosen by the worker (\(\Phi\) in the interior of the simplex = non-degenerate distribution); principal risk-neutral with utility \(w\), worker CARA \(u(y)=-e^{-ry}\) (Remark 19.1: constant absolute risk aversion makes the model stationary). §19.2 one-period model: principal \(\max\sum\phi_i(X_i-w_i)\) s.t. IC, IR; Proposition 19.1 (any \(\phi\) is either not implementable, or there exists a unique \(w^\star\) satisfying IC and IR with equality); Proposition 19.2 (under CARA, changing the outside option CE leaves \(\phi^\star\) unchanged, just add the constant difference to each state's wage). §19.3 \(T\)-period model: period-by-period IC (19.2); Proposition 19.3 (each period's continuation-value CE uniquely pinned down), (19.4) state-contingent wage, Proposition 19.4 (the optimal contract is the same each period, with a linearly aggregated wage \(w(X^T)=w^\star\cdot A^T=\sum w_i^\star A_i^T\), \(A_i^T\) = the number of occurrences of output \(i\) up to period \(T\)). No figures.
在 §17.5 一般模型与 §18 Innes (1990) 模型中，我们施加了许多额外假设才得到"工资随产出递增"的直觉结果；更麻烦的是解可能不存在。现在考虑一种不同的设定，它以代理人的 CARA 效用为特征，并得出：每个可实施（即满足 IR 与 IC）的努力（所选分布）\(\phi\) 都由唯一的工资表所诱导。
19.1 记号 / Notation
In the general model of §17.5 and the Innes (1990) model of §18, we imposed many additional assumptions to get the intuitive result that wages increase in output; what makes it worse is the possible non-existence of a solution. Now consider a different setting, which features the CARA utility of the agent, and the result that each implementable (i.e. IR&IC-satisfying) effort (chosen distribution) \(\phi\) is induced by a unique wage schedule.
19.1 Notation
记号 / Notation 离散期 \(t=1,\dots,T\)。离散产出态 \(x_t\in X=\{X_0,\dots,X_n\}\)；大写 \(X\) 表示各期相同的潜在产出，小写 \(x\) 表示每期实现的产出。分布 \(\phi_t\in\Phi\subseteq\Delta^n\)（\(t=1,\dots,T\)），由工人选择。\(\Delta^n\) 是 \(n\) 维单纯形，有 \(n+1\) 个顶点对应各 \(X_i\)；在顶点 \(i\) 处 \(X_i\) 以概率 1 实现，两顶点连线上任一点是两结果的严格概率混合。我们把 \(\Phi\) 限制为单纯形 \(\Delta^n\) 内部的子集，即分布非退化；\(c(\phi_t)\) 是选 \(\phi_t\) 的货币成本，把 \(\Phi\) 限于内部等价于令 \(c(\phi_t)\) 在 \(\phi_t\) 趋于单纯形边界时变得任意大；非退化分布使模型非平凡。委托人风险中性，财富 \(w\) 的效用即 \(w\)。工人收入效用属于 CARA 族 \(u(y)=-e^{-ry}\)，\(y\) 为最终收入，故代理人关心最终收入减努力成本 \(w-c(\phi_t)\)。Discrete periods \(t=1,\dots,T\). Discrete output states \(x_t\in X=\{X_0,\dots,X_n\}\); capital \(X\) denotes potential outputs the same for all periods, lower case \(x\) the realized output each period. Distribution \(\phi_t\in\Phi\subseteq\Delta^n\) (\(t=1,\dots,T\)), chosen by the worker. \(\Delta^n\) is the \(n\)-dimensional simplex with \(n+1\) vertices corresponding to each \(X_i\); at vertex \(i\) the outcome \(X_i\) is realized with probability 1, and any point on the line segment between two vertices is a strict probability mixture of those two outcomes. We restrict \(\Phi\) to be a subset in the interior of the simplex \(\Delta^n\), meaning the distribution is non-degenerate; \(c(\phi_t)\) is the monetary cost of choosing \(\phi_t\), and restricting \(\Phi\) to the interior is equivalent to imposing that \(c(\phi_t)\) becomes arbitrarily high as \(\phi_t\) approaches the edge of the simplex; a non-degenerate distribution makes the model non-trivial. The principal is risk-neutral with utility of wealth \(w\) equal to \(w\). The worker's utility of income belongs to the CARA family \(u(y)=-e^{-ry}\), \(y\) being final income, so the agent cares about final income minus the cost of effort, \(w-c(\phi_t)\).
记结果依存工资表为 \(w(x_1,\dots,x_t)\)，记至第 \(t\) 期的全部历史结果为 \(\mathbf{X}^t=(x_1,\dots,x_t)\)，\(\mathbf{X}^t\in\mathcal{H}^t\)，\(\mathcal{H}^t\) 为至第 \(t\) 期所有可能历史之集。则有：
Denote the outcome-contingent wage scheme by \(w(x_1,\dots,x_t)\), and the collection of all history outcomes up to period \(t\) by \(\mathbf{X}^t=(x_1,\dots,x_t)\), \(\mathbf{X}^t\in\mathcal{H}^t\) where \(\mathcal{H}^t\) is the set of all possible histories up to period \(t\). Then we have:
$$ u\big(w(\mathbf{X}^T),\{\phi_t\}\big)=-e^{-r\left(w(\mathbf{X}^T)-\sum_{t=1}^T c(\phi_t)\right)} $$
注 19.1：为何用 CARA / Remark 19.1: why CARA CARA 效用具有这样的性质：绝对风险厌恶 \(-\dfrac{u''(y)}{u'(y)}\) 不随 \(y\) 改变。由 Arrow–Pratt 定理（见 Macroeconomics Notes），相同的绝对风险厌恶意味着对同一赌局有相同的风险溢价，即无论财富多少都对该赌局持相同态度。而 CRRA 意味着在更高财富下对同一赌局的风险溢价更低——人越富对同样的输赢越趋于风险中性。日常生活里 CRRA 似乎更合理，但这里我们用 CARA 把模型变成一个平稳问题：工人无论从此前各期工资累积了多少财富，对风险的态度都相同；这样委托人每期面对相同的问题——这是一个非常重要的简化，对求解至关重要。CARA utility has the property that the absolute risk aversion \(-\dfrac{u''(y)}{u'(y)}\) is constant regardless of \(y\). By the Arrow–Pratt Theorem (see the Macroeconomics Notes), the same absolute risk aversion implies the same risk premium w.r.t. the same gamble, i.e. the same attitude towards a gamble regardless of wealth. CRRA, by contrast, implies a lower risk premium for the same gamble at higher wealth — one becomes more risk-neutral about the same amount of win or loss when richer. CRRA seems more reasonable in daily life, but here we use CARA to make the model a stationary problem: the worker always has the same attitude towards risk regardless of how much wealth he has accumulated from previous periods' wages; this way the principal faces the same problem every period — a very important simplification crucial for the solution.
19.2 单期模型 / One period model
在只有一期的模型里，委托人的问题为下式，受 IC 与 IR 约束，其中 \(\underline w\) 为确定性等价 CE（即外部选项）：
19.2 One period model
In the model with only one period, the principal's problem is below, subject to IC and IR, where \(\underline w\) is the certainty equivalence CE (i.e. the outside option):
$$ \max_{(w_0,\dots,w_n),\,\phi}\ \sum_{i=0}^n\phi_i(X_i-w_i) $$
$$ \phi=(\phi_0,\dots,\phi_n)\in\arg\max_{\phi'}\sum_{i=0}^n\phi'_i\,u(w_i-c(\phi')) \tag{IC} $$
$$ \sum_{i=0}^n\phi_i\,u(w_i-c(\phi))\ge u(\underline w) \tag{IR} $$
命题 19.1 / Proposition 19.1 对任意 \(\phi\in\Phi\)，要么 \(\phi\) 不可实施，要么存在唯一的 \(w^\star=(w_0^\star,w_1^\star,\dots,w_n^\star)\) 使 IC 与 IR 都取等号。For any \(\phi\in\Phi\), either \(\phi\) is not implementable, or there exists a unique \(w^\star=(w_0^\star,w_1^\star,\dots,w_n^\star)\) that satisfies IC and IR with equality.
证明 / Proof 该结果说 IC 与 IR 足以钉死唯一解：对任一可实施的 \(\phi\)，存在唯一的工资表导向 \(\phi\)。考虑由 IC 约束导出的工人问题（IR 取等是委托人所希望的）：This result says IC and IR are enough to pin down a unique solution: for any implementable \(\phi\), there is a unique wage schedule leading to \(\phi\). Consider the worker's problem that comes out of the IC constraint (with IR equality desired by the principal):
$$\max_{\phi}\sum_{i=0}^n\phi_i\,u(w_i-c(\phi))\quad\text{s.t.}\quad \sum_{i=0}^n\phi_i\,u(w_i-c(\phi))=u(\underline w)$$
拉格朗日函数为 \(\mathcal{L}=\sum_{i=0}^n\phi_i u(w_i-c(\phi))+\lambda\big[\sum_{i=0}^n\phi_i u(w_i-c(\phi))-u(\underline w)\big]\)。关于每个 \(\phi_i\) 的一阶条件为：The Lagrangian is \(\mathcal{L}=\sum_{i=0}^n\phi_i u(w_i-c(\phi))+\lambda\big[\sum_{i=0}^n\phi_i u(w_i-c(\phi))-u(\underline w)\big]\). The f.o.c. w.r.t. each \(\phi_i\) is:
$$(1+\lambda)\left[u(w_i-c(\phi))-\sum_{i=0}^n\phi_i u'(w_i-c(\phi))\frac{\partial c(\phi)}{\partial\phi_i}\right]=0 \;\Rightarrow\; u(w_i-c(\phi))=\sum_{i=0}^n\phi_i u'(w_i-c(\phi))\frac{\partial c(\phi)}{\partial\phi_i} \tag{19.1}$$
因 \(u(\cdot)\) 与 \(c(\cdot)\) 是已知函数，可把所需的 \(\phi=(\phi_0,\dots,\phi_n)\) 代入 (19.1)，得到要么 \(w_i\) 无解、要么 \(w_i\) 唯一解。解唯一是因为我们假设 \(u\) 递增且凹，\(u'>0\)、\(u''<0\)，故 (19.1) 左端随 \(w_i\) 递增、右端随 \(w_i\) 递减，至多一个交点。对所有 \(i=0,1,\dots,n\) 重复，便钉死唯一的 \(w^\star=(w_0^\star,\dots,w_n^\star)\)（若每个 \(w_i^\star\) 都存在）。\(\blacksquare\)Since \(u(\cdot)\) and \(c(\cdot)\) are known functions, we can plug the desired \(\phi=(\phi_0,\dots,\phi_n)\) into (19.1) and obtain either no solution for \(w_i\) or a unique solution for \(w_i\). The solution is unique because we assume \(u\) increasing and concave, \(u'>0\), \(u''<0\), so the LHS of (19.1) is increasing in \(w_i\) and the RHS decreasing in \(w_i\), so there is at most one intersection point. Repeat for all \(i=0,1,\dots,n\) to pin down a unique \(w^\star=(w_0^\star,\dots,w_n^\star)\) if every \(w_i^\star\) exists. \(\blacksquare\)
命题 19.2 / Proposition 19.2 设 \((w^\star,\phi^\star)\) 解 CE\(=\underline w\) 的单期问题，则对 CE\(=\hat w\)，\((w^\star+\hat w-\underline w,\,\phi^\star)\) 最优（脚注 19.1：\(w^\star\) 是向量，故 \(w^\star+\hat w-\underline w=(w_0^\star+\hat w-\underline w,\dots,w_n^\star+\hat w-\underline w)\)）。Suppose \((w^\star,\phi^\star)\) solves the one-period program with CE\(=\underline w\). Then for CE\(=\hat w\), \((w^\star+\hat w-\underline w,\,\phi^\star)\) is optimal (footnote 19.1: \(w^\star\) is a vector, so \(w^\star+\hat w-\underline w=(w_0^\star+\hat w-\underline w,\dots,w_n^\star+\hat w-\underline w)\)).
证明 / Proof 该结果说：若改变 CE（外部机会），则 \(\phi^\star\) 不变，唯一改变的是工资表——只需对每个状态的工资加上新旧确定性等价之差。这源于 CARA 的指数形式。考虑：This result says: if you change the CE (outside opportunity), then \(\phi^\star\) won't change; all that changes is the wage schedule — just increase the wage for each state by the difference between the new and old certainty equivalence. This follows from CARA and the exponential form. Consider:
$$\sum_{i=0}^n\phi_i u(w_i-c(\phi))\ge u(\underline w)$$
两边同乘 \(e^{-r(\hat w-\underline w)}>0\)（保号），并用 \(u(y)=-e^{-ry}\) 整理：Multiply both sides by \(e^{-r(\hat w-\underline w)}>0\) (preserving the sign) and rearrange using \(u(y)=-e^{-ry}\):
$$\sum_{i=0}^n\phi_i\left(-e^{-r(w_i+\hat w-\underline w-c(\phi))}\right)\ge -e^{-r\hat w} \;\Rightarrow\; \sum_{i=0}^n\phi_i\,u(w_i+\hat w-\underline w-c(\phi))\ge u(\hat w)$$
故对任一在 CE\(=\underline w\) 下令 IR 取等的 \((w,\phi)\)，\((w+\hat w-\underline w,\phi)\) 在 CE\(=\hat w\) 下令 IR 取等。又 CE 不出现在 IC 中，故确定性等价的变化对 IC 无影响，\((w+\hat w-\underline w,\phi)\) 满足 IC。由命题 19.1，\(w+\hat w-\underline w\) 是 \(\phi\) 唯一可实施的工资表，它只是给原工资 \(w\) 加常数 \(\hat w-\underline w\)，故只给委托人目标函数加常数 \(\underline w-\hat w\)。因此若 \((w^\star,\phi^\star)\) 解 CE\(=\underline w\) 的问题，则 \((w^\star+\hat w-\underline w,\phi^\star)\) 必解 CE\(=\hat w\) 的问题。\(\blacksquare\)So for any \((w,\phi)\) satisfying IR with equality at CE\(=\underline w\), \((w+\hat w-\underline w,\phi)\) satisfies IR with equality at CE\(=\hat w\). Also, CE does not appear in IC, so the change in certainty equivalence has no effect on IC, and \((w+\hat w-\underline w,\phi)\) satisfies IC. By Proposition 19.1, \(w+\hat w-\underline w\) is the only implementable wage schedule for \(\phi\), which simply adds a constant \(\hat w-\underline w\) to the original wage \(w\), so it only adds a constant \(\underline w-\hat w\) to the principal's objective. So if \((w^\star,\phi^\star)\) solves the program with CE\(=\underline w\), then \((w^\star+\hat w-\underline w,\phi^\star)\) must solve the program with CE\(=\hat w\). \(\blacksquare\)
19.3 \(T\) 期模型 / \(T\) periods model
在 \(T\) 期模型里，工人的动态努力策略为 \(\{\phi_t(\mathbf{X}^{t-1})\}_{t=1,\dots,T}^{\mathbf{X}^{t-1}\in\mathcal{H}^{t-1}}\)，即每期产出依赖历史；工资基于 \(\mathbf{X}^T\)，各期产出只在最后支付。又假设 \(T\) 期足够短，无需贴现。
设委托人想实施某一特定的逐期分布序列 \(\{\phi_t(\mathbf{X}^{t-1})\}_{t=1,\dots,T}^{\mathbf{X}^{t-1}\in\mathcal{H}^{t-1}}\)，则需满足 IR 与 IC 以保证该序列可实施。
首先，把工人在第 \(t\) 期视角下的期望效用记为下式（可丢弃常数项后重定义）：
19.3 \(T\) periods model
In the model with \(T\) periods, the worker's dynamic effort strategy is \(\{\phi_t(\mathbf{X}^{t-1})\}_{t=1,\dots,T}^{\mathbf{X}^{t-1}\in\mathcal{H}^{t-1}}\), i.e. each period's output depends on history; the wage is based on \(\mathbf{X}^T\), with each period's output only paid in the end. We also assume the \(T\) periods are short enough that no discounting is necessary.
Suppose the principal wants to implement a particular sequence of period distributions \(\{\phi_t(\mathbf{X}^{t-1})\}_{t=1,\dots,T}^{\mathbf{X}^{t-1}\in\mathcal{H}^{t-1}}\); then he needs to satisfy IR and IC to make sure the particular sequence is implementable.
First, denote the worker's expected utility in the perspective of period \(t\) by the expression below (after dropping constant terms it can be redefined):
$$ U_t(\mathbf{X}^t)=\mathbb{E}\!\left[u\Big(\underbrace{w(\mathbf{X}^T)-\sum_{\tau=t+1}^T c(\phi_\tau)}_{\text{to be determined, changeable}}-\underbrace{\sum_{\tau=0}^t c(\phi_\tau)}_{\text{constant}}\Big)\,\Big|\,\mathbf{X}^t\right] $$
$$ U_t(\mathbf{X}^t)=\mathbb{E}\!\left[u\Big(w(\mathbf{X}^T)-\sum_{\tau=t+1}^T c(\phi_\tau)\Big)\,\Big|\,\mathbf{X}^t\right] $$
再把 \(U_t(\mathbf{X}^t)\) 的确定性等价记为 CE\(=\underline w_t(\mathbf{X}^t)\)，即 \(u(\underline w_t(\mathbf{X}^t))=U_t(\mathbf{X}^t)\)，其中 \(u(\underline w_t(\mathbf{X}^t))\) 是第 \(t\) 期延续值的货币化。记第 \(t\) 期外部机会为 \(\underline w_t\)，则工人 IR 约束为 \(U_t(\mathbf{X}^t)=u(\underline w_t(\mathbf{X}^t))\ge u(\underline w_t)\)（对 \(\forall t\)），即 \(\underline w_t(\mathbf{X}^t)\ge\underline w_t\)（对 \(\forall t\)）。IC 约束可改写为下式 (19.2)：
Then denote the certainty equivalence of \(U_t(\mathbf{X}^t)\) by CE\(=\underline w_t(\mathbf{X}^t)\), i.e. \(u(\underline w_t(\mathbf{X}^t))=U_t(\mathbf{X}^t)\), where \(u(\underline w_t(\mathbf{X}^t))\) is the monetized value of the continuation value in period \(t\). Denote the outside opportunity in period \(t\) by \(\underline w_t\); then the worker's IR constraint becomes \(U_t(\mathbf{X}^t)=u(\underline w_t(\mathbf{X}^t))\ge u(\underline w_t)\) for \(\forall t\), i.e. \(\underline w_t(\mathbf{X}^t)\ge\underline w_t\) for \(\forall t\). The IC constraint can be rewritten as (19.2):
$$ \phi_t(\mathbf{X}^{t-1})\in\arg\max_{\phi'}\sum_{i=0}^n\phi'_i\,u\big(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\big) \tag{19.2} $$
逐期满足 IC / IC period-by-period 确定性等价 \(u(\underline w_t(\mathbf{X}^{t-1},x_t=X_i))\) 由 \(\mathbb{E}[u(w(\mathbf{X}^T)-\sum_{\tau=t+1}^T c(\phi_\tau))\mid\mathbf{X}^{t-1},x_t=X_i]\) 定义，它也涉及 \((\phi_{t+1},\phi_{t+2},\dots,\phi_T)\)，但不涉及 \(\phi_t\)。故委托人若想激励某特定逐期分布序列 \(\{\phi_t(\mathbf{X}^{t-1})\}\)，需逐期满足 IC (19.2)。The certainty equivalence \(u(\underline w_t(\mathbf{X}^{t-1},x_t=X_i))\) is defined by \(\mathbb{E}[u(w(\mathbf{X}^T)-\sum_{\tau=t+1}^T c(\phi_\tau))\mid\mathbf{X}^{t-1},x_t=X_i]\), which also involves \((\phi_{t+1},\phi_{t+2},\dots,\phi_T)\) but does not involve \(\phi_t\). So if the principal wants to incentivize a particular sequence \(\{\phi_t(\mathbf{X}^{t-1})\}\), he needs to satisfy IC (19.2) period-by-period.
命题 19.3 / Proposition 19.3 要实施某特定 \(\{\phi_t(\mathbf{X}^{t-1})\}_{t=1,\dots,T}^{\mathbf{X}^{t-1}\in\mathcal{H}^{t-1}}\)，确定性等价序列 \(\{\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\}\) 在每期被唯一钉定。To implement a particular \(\{\phi_t(\mathbf{X}^{t-1})\}_{t=1,\dots,T}^{\mathbf{X}^{t-1}\in\mathcal{H}^{t-1}}\), the sequence of certainty equivalence \(\{\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\}\) is uniquely pinned down each period.
证明 / Proof 逐期考虑由 IC (19.2) 导出的工人问题（\(\underline w_t\) 为第 \(t\) 期外部机会；之所以扣 \(-c(\phi)\) 是因 \(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\) 只涉及 \((\phi_{t+1},\dots,\phi_T)\)，而我们正在为工人选 \(\phi\)（即 \(\phi_t\)），故须从确定性等价中扣其成本）：Consider the worker's problem from IC (19.2) period-by-period (\(\underline w_t\) is the outside opportunity in period \(t\); the reason we include \(-c(\phi)\) is that \(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\) only involves \((\phi_{t+1},\dots,\phi_T)\), and since we are choosing \(\phi\) (i.e. \(\phi_t\)) for the worker, the cost of \(\phi\) must be deducted from the certainty equivalence):
$$\max_{\phi}\sum_{i=0}^n\phi_i\,u\big(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)-c(\phi)\big)\quad\text{s.t.}\quad \sum_{i=0}^n\phi_i\,u\big(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)-c(\phi)\big)=u(\underline w_t)$$
拉格朗日法、对每个 \(\phi_i\) 求一阶条件，得 (19.3)：By the Lagrangian and the f.o.c. w.r.t. each \(\phi_i\), we get (19.3):
$$u\big(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)-c(\phi)\big)=\sum_{i=0}^n\phi_i\,u'\big(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)-c(\phi)\big)\frac{\partial c(\phi)}{\partial\phi_i} \tag{19.3}$$
与之前同样逻辑：\(u'>0\)、\(u''<0\)，故 (19.3) 左端随 \(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\) 递增、右端递减，至多一个交点。对所有 \(i=0,1,\dots,n\) 与 \(t=1,2,\dots,T\) 重复，便钉死唯一序列 \(\{\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\}\)（若都存在）。\(\blacksquare\)By the same logic as before: \(u'>0\), \(u''<0\), so the LHS of (19.3) is increasing in \(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\) and the RHS decreasing, with at most one intersection point. Repeat for all \(i=0,1,\dots,n\) and \(t=1,2,\dots,T\) to pin down a unique sequence \(\{\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\}\) if every one exists. \(\blacksquare\)
由于各期分布 i.i.d.，对工人而言，\(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\) 与 \(\underline w_t(\mathbf{X}^{t-1})\) 的唯一区别是委托人就给定 \(\mathbf{X}^{t-1}\) 下的 \(x_t\) 所发的工资。故可把第 \(t\) 期态依存工资记为 \(\tilde w_t(x_t=X_i\mid\phi_t(\mathbf{X}^{t-1}))\)，它满足 (19.4)：
Since the distributions for each period are i.i.d., for the worker the only difference between \(\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\) and \(\underline w_t(\mathbf{X}^{t-1})\) is the wage awarded by the principal for \(x_t\) given \(\mathbf{X}^{t-1}\). So we can denote the state-contingent wage in period \(t\) by \(\tilde w_t(x_t=X_i\mid\phi_t(\mathbf{X}^{t-1}))\), which satisfies (19.4):
$$ \tilde w_t\big(x_t=X_i\mid\phi_t(\mathbf{X}^{t-1})\big)=\underline w_t(\mathbf{X}^{t-1},x_t=X_i)-\underline w_{t-1}(\mathbf{X}^{t-1}) \tag{19.4} $$
之所以记为依赖 \(\phi_t(\mathbf{X}^{t-1})\) 的工资，是因为我们先令 \(\{\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\}\) 为命题 19.3 所给、诱导 \(\{\phi_t(\mathbf{X}^{t-1})\}\) 的唯一序列；故发放 \(\{\tilde w_t(x_t=X_i\mid\phi_t(\mathbf{X}^{t-1}))\}\) 便诱导 \(\{\phi_t(\mathbf{X}^{t-1})\}\)。最终工资表由 \(\{\tilde w_t(\cdot)\}\) 生成，对 \(\mathbf{X}^T=(x_1,\dots,x_T)\) 唯一对应序列 \(\{\phi_t(\mathbf{X}^{t-1})\}\)：
The reason we denote the wage as conditional on \(\phi_t(\mathbf{X}^{t-1})\) is that we start by letting \(\{\underline w_t(\mathbf{X}^{t-1},x_t=X_i)\}\) be the unique sequence given by Proposition 19.3 that induces \(\{\phi_t(\mathbf{X}^{t-1})\}\); so awarding \(\{\tilde w_t(x_t=X_i\mid\phi_t(\mathbf{X}^{t-1}))\}\) induces \(\{\phi_t(\mathbf{X}^{t-1})\}\). The final wage schedule is generated by \(\{\tilde w_t(\cdot)\}\) and uniquely corresponds to the sequence \(\{\phi_t(\mathbf{X}^{t-1})\}\) as follows: for \(\mathbf{X}^T=(x_1,\dots,x_T)\):
$$ w(\mathbf{X}^T)=\sum_{t=1}^T\tilde w_t\big(x_t\mid\phi_t(\mathbf{X}^{t-1})\big)+\underline w_0 $$
注 19.2、注 19.3 / Remarks 19.2, 19.3 注 19.2：我们逐期、逐态地选工资 \(\tilde w_t(x_t=X_i\mid\phi_t(\mathbf{X}^{t-1}))\) 来诱导特定序列 \(\{\phi_t(\mathbf{X}^{t-1})\}\)；加 \(\underline w_0\) 保证代理人在第 0 期不离开合约（满足第 0 期 IR）。注 19.3：确定性等价的所有期望都是关于 \(\{\phi_t(\mathbf{X}^{t-1})\}\) 取的，因为采用工资方案 \(\{\tilde w_t(\cdot)\}\) 是最优的，代理人会基于 \(\{\phi_t(\mathbf{X}^{t-1})\}\) 做理性预期。Remark 19.2: we choose the wage \(\tilde w_t(x_t=X_i\mid\phi_t(\mathbf{X}^{t-1}))\) period by period and state by state to induce the particular sequence \(\{\phi_t(\mathbf{X}^{t-1})\}\); adding \(\underline w_0\) guarantees the agent won't leave the contract in period 0 (satisfying IR in period 0). Remark 19.3: all the expectations for certainty equivalence are taken w.r.t. \(\{\phi_t(\mathbf{X}^{t-1})\}\), because using the wage scheme \(\{\tilde w_t(\cdot)\}\) is optimal and the agent will make rational expectations based on \(\{\phi_t(\mathbf{X}^{t-1})\}\).
命题 19.4 / Proposition 19.4 最优合约是 \(\{\phi_t(\mathbf{X}^{t-1})\}_{t=1,\dots,T}^{\mathbf{X}^{t-1}\in\mathcal{H}^{t-1}}=(\phi_0^\star,\dots,\phi_n^\star)^T\)，对应工资表为 \(w(\mathbf{X}^T)=w^\star\cdot A^T=\sum_{i=0}^n w_i^\star A_i^T\)，其中 \(w^\star=(w_0^\star,\dots,w_n^\star)\)、\(A^T=(A_0^T,A_1^T,\dots,A_n^T)\)，\(A_i^T\) 是产出 \(i\) 至第 \(T\) 期的出现次数。The optimal contract is \(\{\phi_t(\mathbf{X}^{t-1})\}_{t=1,\dots,T}^{\mathbf{X}^{t-1}\in\mathcal{H}^{t-1}}=(\phi_0^\star,\dots,\phi_n^\star)^T\), with corresponding wage schedule \(w(\mathbf{X}^T)=w^\star\cdot A^T=\sum_{i=0}^n w_i^\star A_i^T\), where \(w^\star=(w_0^\star,\dots,w_n^\star)\), \(A^T=(A_0^T,A_1^T,\dots,A_n^T)\), and \(A_i^T\) is the number of occurrences of output \(i\) up to period \(T\).
证明 / Proof 先证每期对委托人的最优分布相同。若同一工资表 \(w^\star=(w_0^\star,\dots,w_n^\star)\) 在每期都能诱导同一 \((\phi_0^\star,\dots,\phi_n^\star)\)，则证毕——因为逐期可重复的最优性带来 \(T\) 期的最优性。现在只需说明为何同一工资表在每期诱导同一分布。考虑工人问题的一阶条件 (19.3)，由 (19.4) \(\tilde w_t(x_t=X_i\mid\phi_t(\mathbf{X}^{t-1}))=\underline w_t(\mathbf{X}^{t-1},x_t=X_i)-\underline w_{t-1}(\mathbf{X}^{t-1})\) 改写一阶条件，记第 \(t\) 期对结果 \(i\) 的工资为 \(w_t(x_t=X_i)\)，并用 CARA \(u(y)=-e^{-ry}\) 代入，得 (19.5) 化为：First prove each period's optimal distribution for the principal is the same. If the same wage schedule \(w^\star=(w_0^\star,\dots,w_n^\star)\) in every period can induce the same \((\phi_0^\star,\dots,\phi_n^\star)\), the proof is done — because repeatable optimality over periods leads to optimality for \(T\) periods. Now it remains to show why the same wage schedule induces the same distribution in every period. Consider the worker's f.o.c. (19.3); using (19.4) \(\tilde w_t(x_t=X_i\mid\phi_t(\mathbf{X}^{t-1}))=\underline w_t(\mathbf{X}^{t-1},x_t=X_i)-\underline w_{t-1}(\mathbf{X}^{t-1})\) to rewrite the f.o.c., denoting period \(t\)'s wage for outcome \(i\) by \(w_t(x_t=X_i)\), and plugging in CARA \(u(y)=-e^{-ry}\), the f.o.c. (19.5) becomes:
$$\exp\{-r[w_t(x_t=X_i)-c(\phi)]\}=\sum_{i=0}^n\phi_i\exp\{-r[w_t(x_t=X_i)-c(\phi)]\}\frac{\partial c(\phi)}{\partial\phi_i}$$
（推导中公共因子 \(\exp\{-r\,\underline w_{t-1}(\mathbf{X}^{t-1})\}\) 在两端约去。）此式不涉及先前历史，故同一工资方案在每期都诱导工人所选的同一期分布。既然最优工资方案在所有期相同，便可写出对应最优合约 \(w(\mathbf{X}^t)=w^\star\cdot A^T=\sum_{i=0}^n w_i^\star A_i^T\)，其中 \(w^\star=(w_0^\star,\dots,w_n^\star)\)、\(A^T=(A_0^T,A_1^T,\dots,A_n^T)\)。\(\blacksquare\)（脚注 19.2：若效用非 CARA，一般不必有此相同结果。）(In the derivation the common factor \(\exp\{-r\,\underline w_{t-1}(\mathbf{X}^{t-1})\}\) cancels on both sides.) This does not involve previous history, so the same wage scheme in each period induces the same period distribution chosen by the worker. Since the optimal wage scheme is the same in all periods, we can write the corresponding optimal contract \(w(\mathbf{X}^t)=w^\star\cdot A^T=\sum_{i=0}^n w_i^\star A_i^T\), where \(w^\star=(w_0^\star,\dots,w_n^\star)\), \(A^T=(A_0^T,A_1^T,\dots,A_n^T)\). \(\blacksquare\) (footnote 19.2: if utility is not CARA, we won't necessarily have the same result in general.)
参考文献 / References
- Holmstrom, B., & Milgrom, P. (1987). Aggregation and Linearity in the Provision of Intertemporal Incentives.（CARA 下最优合约为产出聚合的线性合约）
- Holmstrom, B. (1979). Moral Hazard and Observability.（隐藏行动的基础框架，见 [[moral-hazard]]）
References
- Holmstrom, B., & Milgrom, P. (1987). Aggregation and Linearity in the Provision of Intertemporal Incentives. (under CARA the optimal contract is linear in aggregate output)
- Holmstrom, B. (1979). Moral Hazard and Observability. (the foundational hidden-action framework; see [[moral-hazard]])