19. Asset Pricing in Production Economy

Jun He May 30, 2026

资产定价Asset Pricing 生产经济Production Economy AK模型AK Model 调整成本Adjustment Cost 不完全市场Incomplete Market 学习笔记Study Note

Note

本章把生产加入经济：消费由内生选择决定，冲击不再直接作用于消费，而是直接影响投资调整成本（§19.1）或劳动收入（§19.2），再通过经济结构内生地传导到消费。§19.1 在带调整成本的 AK 模型（资本线性、$C_t+I_t=AK_t$）中，用社会计划者问题求最优价值-资本比 $v_t-k_t=f(Z_t)$，在 $\rho=1$ 特例下"猜验"出线性解 $f_1+f_2 Z_t$（$f_2=\frac{\beta}{1-\beta\psi_z}$ 与 $\gamma$ 无关，$f_1$ 依赖 $\gamma$）；$\rho\neq1$ 时用小噪声 $q$ 近似数值求解。§19.2 在不完全市场（不能用状态或有索取权对冲劳动收入冲击、无借贷约束）中，单一代理人通过储蓄平滑消费；用变分法导出欧拉方程与 SDF $\beta\frac{C_t}{C_{t+1}}e^a$，再在稳态附近做一阶近似，证明内生消费 $C_t$ 恰好选择使一系列随机性约束（如 (19.40)）成立。最后给出递归（$\rho=1$）效用下的两种刻画，得到 $\boldsymbol\sigma_c=\boldsymbol\sigma_v$ 与 $\delta$ 的限制。

Note

This chapter adds production: consumption is endogenously chosen, and shocks no longer hit consumption directly but instead hit the investment adjustment cost (§19.1) or labor income (§19.2), transmitting to consumption endogenously through the economy. §19.1 uses an AK model with adjustment cost (linear capital, $C_t+I_t=AK_t$); the social planner's problem gives the optimal value-to-capital ratio $v_t-k_t=f(Z_t)$, which in the $\rho=1$ special case is "guess-and-verified" as linear $f_1+f_2 Z_t$ ($f_2=\frac{\beta}{1-\beta\psi_z}$ independent of $\gamma$, $f_1$ depending on $\gamma$); for $\rho\neq1$ a small-noise $q$-approximation solves it numerically. §19.2 uses an incomplete market (cannot hedge labor-income shocks with state-contingent claims, no borrowing constraint); a single agent smooths consumption by saving; a variational argument gives the Euler equation and SDF $\beta\frac{C_t}{C_{t+1}}e^a$, then a steady-state first-order approximation shows endogenous $C_t$ is chosen exactly so a series of randomness restrictions (e.g. (19.40)) hold. Finally two characterizations under recursive ($\rho=1$) utility yield $\boldsymbol\sigma_c=\boldsymbol\sigma_v$ and a restriction on $\delta$.

19.1 AK Model with Adjustment Cost

19.1.1 Setup

单一（代表性）代理人，单一类型可完全互换的资本与消费品。EZ 偏好 (19.1)：$V_t=[(1-\beta)C_t^{1-\rho}+\beta(R_t^{1-\rho})]^{\frac1{1-\rho}}$，$R_t=\mathbb E[V_{t+1}^{1-\gamma}\mid\mathcal F_t]^{\frac1{1-\gamma}}$ (19.2)。资本演化 (19.3)：

A single (representative) agent, a single perfectly interchangeable capital-consumption good. EZ preference (19.1): $V_t=[(1-\beta)C_t^{1-\rho}+\beta(R_t^{1-\rho})]^{\frac1{1-\rho}}$, $R_t=\mathbb E[V_{t+1}^{1-\gamma}\mid\mathcal F_t]^{\frac1{1-\gamma}}$ (19.2). Capital evolution (19.3):

$$K_{t+1}=K_t\left(1+\Phi\!\left(\frac{I_t}{K_t}\right)\right)\cdot e^{-\alpha_k+Z_t+\boldsymbol\sigma_k\cdot\mathbf W_{t+1}-\frac12|\boldsymbol\sigma_k|^2},\tag{19.3}$$

$I_t$ 为投资，$\mathbf W_{t+1}\sim\mathcal N(\mathbf 0,\mathbf I)$。$\Phi(\cdot)$ 递增凹（$\Phi'>0$、$\Phi''<0$）：投资增加下期资本，但边际效率递减——这正是凸调整成本模型的味道（投资比率越高，转化为资本增量的比例越低）。$Z_t$ 服从 (19.4)：$Z_{t+1}=\psi_z Z_t+\boldsymbol\sigma_z\cdot\mathbf W_{t+1}$。生产技术为 AK 模型（资本线性）(19.5)：$C_t+I_t=AK_t$（$A>0$ 常数）。可选 $(I_t,K_t)$ 为决策变量（$C_t$ 由 (19.5) 定）。对数对象 $v_t\equiv\ln V_t$、$c_t\equiv\ln C_t$、$r_t\equiv\ln R_t$、$i_t\equiv\ln I_t$、$k_t\equiv\ln K_t$。

19.1.2 Useful Facts

(19.3) 除以 $K_t$ 取对数得资本增长 (19.6)：$k_{t+1}-k_t=\ln(1+\Phi(e^{i_t-k_t}))-\alpha_k+Z_t+\boldsymbol\sigma_k\cdot\mathbf W_{t+1}-\frac12|\boldsymbol\sigma_k|^2$。(19.5) 除以 $K_t$ 取对数得 (19.7)：$c_t-k_t=\ln(A-e^{i_t-k_t})$。(19.2) 一次齐次，除以 $K_t$ 取对数得 (19.8)：$r_t-k_t=\frac1{1-\gamma}\ln(\mathbb E[e^{(1-\gamma)(v_{t+1}-k_{t+1}+k_{t+1}-k_t)}\mid\mathcal F_t])$。

19.1.3 The Planner's Problem

社会计划者（单一代理人）求解 (19.10)：$\max_{i_t-k_t}v_t-k_t$。(19.1) 一次齐次，除以 $K_t$ 取对数并代入 (19.7)、(19.8) 得 (19.9)，最优 $i_t-k_t$ 与 $Z_t$ 无关地被最优选定（在最优处 $i_t-k_t$ 抵消），故最优 $v_t-k_t$ 只是 $Z_t$ 的函数，记 $v_t-k_t=f(Z_t)$。

Note

Remark 19.1. 由 (19.10) 可见最优投资依赖 $\rho$，故最优消费/储蓄（投资）比率依赖 $\rho$。

19.1.4 General Case: Approximation

一般 $\rho\neq1$ 时 $f(\cdot)$ 难解析求解。可用 §17.2.1 的小噪声法：先把所有 $\boldsymbol\sigma_k,\boldsymbol\sigma_z$ 标度为 $q\boldsymbol\sigma_k,q\boldsymbol\sigma_z$，在 $q=0$ 附近求一阶、二阶导，再数值估计 $i_t-k_t$ 与 $f$。

19.1.5 Special Case: ρ=1

把 $\rho=1$ 代入 (19.10) 并对 $i_t-k_t$ 取一阶条件 (19.11)，整理得 (19.12)：

$I_t$ is investment, $\mathbf W_{t+1}\sim\mathcal N(\mathbf 0,\mathbf I)$. $\Phi(\cdot)$ is increasing and concave ($\Phi'>0$, $\Phi''<0$): investment raises next period's capital but with diminishing marginal efficiency — the flavor of a convex adjustment-cost model (higher investment ratios convert a smaller proportion into capital increments). $Z_t$ follows (19.4): $Z_{t+1}=\psi_z Z_t+\boldsymbol\sigma_z\cdot\mathbf W_{t+1}$. The production technology is an AK model (linear in capital) (19.5): $C_t+I_t=AK_t$ ($A>0$ constant). Choose $(I_t,K_t)$ as decision variables ($C_t$ pinned by (19.5)). Log objects $v_t\equiv\ln V_t$, $c_t\equiv\ln C_t$, $r_t\equiv\ln R_t$, $i_t\equiv\ln I_t$, $k_t\equiv\ln K_t$.

19.1.2 Useful Facts

Dividing (19.3) by $K_t$ and taking logs gives capital growth (19.6): $k_{t+1}-k_t=\ln(1+\Phi(e^{i_t-k_t}))-\alpha_k+Z_t+\boldsymbol\sigma_k\cdot\mathbf W_{t+1}-\frac12|\boldsymbol\sigma_k|^2$. Dividing (19.5) by $K_t$ and taking logs gives (19.7): $c_t-k_t=\ln(A-e^{i_t-k_t})$. (19.2) is homogeneous of degree 1; dividing by $K_t$ and taking logs gives (19.8): $r_t-k_t=\frac1{1-\gamma}\ln(\mathbb E[e^{(1-\gamma)(v_{t+1}-k_{t+1}+k_{t+1}-k_t)}\mid\mathcal F_t])$.

19.1.3 The Planner's Problem

The social planner (single agent) solves (19.10): $\max_{i_t-k_t}v_t-k_t$. (19.1) is homogeneous of degree 1; dividing by $K_t$, taking logs, and substituting (19.7), (19.8) gives (19.9), and the optimal $i_t-k_t$ is chosen independently of $Z_t$ (at the optimum $i_t-k_t$ cancels out), so the optimal $v_t-k_t$ is only a function of $Z_t$, written $v_t-k_t=f(Z_t)$.

Note

Remark 19.1. From (19.10), optimal investment depends on $\rho$, so the optimal consumption/saving (investment) ratio depends on $\rho$.

19.1.4 General Case: Approximation

For general $\rho\neq1$, $f(\cdot)$ is hard to solve analytically. Use the §17.2.1 small-noise method: scale all $\boldsymbol\sigma_k,\boldsymbol\sigma_z$ to $q\boldsymbol\sigma_k,q\boldsymbol\sigma_z$, take first/second derivatives around $q=0$, then numerically estimate $i_t-k_t$ and $f$.

19.1.5 Special Case: ρ=1

Plugging $\rho=1$ into (19.10) and taking the f.o.c. w.r.t. $i_t-k_t$ (19.11), rearranging gives (19.12):

$$\frac{1-\beta}{\beta}\left(1+\Phi(e^{i_t-k_t})\right)=(A-e^{i_t-k_t})\,\Phi'(e^{i_t-k_t}).\tag{19.12}$$

(19.12) 左端随 $i_t-k_t$ 递增、右端递减，故给定 $\Phi(\cdot)$ 的具体形式，$i_t-k_t$ 被唯一钉住，记为 $i_t^\star-k_t^\star$（与 $Z_t$ 无关，常数）。把 $i_t^\star-k_t^\star$ 代入 (19.9)，猜验 $f(\cdot)$ 线性 (19.13)：

The LHS of (19.12) increases in $i_t-k_t$ and the RHS decreases, so given the specific form of $\Phi(\cdot)$, $i_t-k_t$ is uniquely pinned down, written $i_t^\star-k_t^\star$ (independent of $Z_t$, a constant). Substituting $i_t^\star-k_t^\star$ into (19.9), guess-and-verify $f(\cdot)$ linear (19.13):

$$\underbrace{v_t-k_t}_{f(Z_t)}=f_1+f_2 Z_t.\tag{19.13}$$

用 L'Hôpital 把 (19.9) 在 $\rho=1$ 重写 (19.14)，代入 (19.13)，匹配 $Z_t$ 系数得 (19.16)：$f_2=\frac{\beta}{1-\beta\psi_z}$；匹配常数项得 (19.17) $f_1$。验证成功，$f_1$ 依赖风险厌恶 $\gamma$，而 $f_2$ 完全不依赖 $\gamma$。

Note

Remark 19.2. 有了 (19.17)、(19.16)，可数值算出 $i_t,k_t,v_t$ 序列，进而 $c_t$（或 $C_t$）(19.7)。与第 18 章假设 $C_t$ 外生不同，这里反过来：给定观测到的消费数据，调整模型参数来校准实际消费数据。

19.2 Production Model with Income Shock

此为不完全市场模型：代理人无法用状态或有索取权对冲劳动收入冲击；也无借贷约束。尽管既不能买状态或有索取权、也不能买风险资产，仍能通过储蓄相当好地平滑消费。

19.2.1 Setup

单一（代表性）代理人，单一可互换资本与消费品。状态变量 $\mathbf X_t(q)$ 满足 $\mathbf X_{t+1}(q)=\boldsymbol\psi(\mathbf X_t(q),q\mathbf W_{t+1},q)$（$q$ 度量对不确定性的暴露，$\mathbf W_{t+1}\sim\mathcal N(\mathbf 0,\mathbf I)$）。对数效用、无限期、贴现 $\beta$：$V_t=\sum_{\tau=0}^\infty\beta^\tau\ln C_{t+\tau}$。资本演化 (19.18)：$K_{t+1}+C_t=e^a K_t+Y_t$（$a$ 常数，$\{Y_t\}$ 外生劳动收入）。对数 $y_t(q)\equiv\ln Y_t(q)$、$c_t(q)\equiv\ln C_t(q)$（$\frac{K_{t+1}}{Y_{t+1}}$ 设平稳但不必正，故不取对数）。$Y_t$ 为劳动收入，外生 (19.19 型)：$y_{t+1}(q)-y_t(q)=\kappa(\mathbf X_t(q),\mathbf W_{t+1})$（设 $\kappa$ 对 $\mathbf X_t,\mathbf W_{t+1}$ 可加分离，不影响结果）。$C_t$ 内生选择，故 $C_t(q),K_t(q)$ 也是 $q$ 的函数，$c_{t+1}(q)-c_t(q)=\phi(\mathbf X_t(q),\mathbf W_{t+1})$。由 (17.11) 得一阶导动态 (19.19)：$\mathbf X_{t+1}'(0)=\boldsymbol\psi_{\mathbf X^T}\mathbf X_t'(0)+\boldsymbol\psi_{\mathbf W^T}\mathbf W_{t+1}+\boldsymbol\psi_h$，及 (19.20)、(19.21)：$y_{t+1}'(0)-y_t'(0)=\boldsymbol\kappa_{\mathbf X^T}\mathbf X_t'(0)$、$c_{t+1}'(0)-c_t'(0)=\boldsymbol\phi_{\mathbf X^T}\mathbf X_t'(0)$。

19.2.2 Utility Maximization and Euler Equation

代理人解 (19.22)：$\max_{C_t}\{\ln C_t(q)+\mathbb E[\sum_{\tau=1}^\infty\beta^\tau\ln C_{t+\tau}(q)\mid\mathcal F_t]\}$ s.t. $K_{t+1}(q)+C_t(q)=e^a K_t(q)+Y_t(q)$ (19.23)。用变分法：最优处代理人对"今天多消费 $h$"与否无差异，即 $(C_t,C_{t+1})$ 与 $(C_t-h,C_{t+1}+e^a h)$ 无差异（今天少消费 $h$、明天多 $e^a h$）。对 $h$ 取一阶条件于 $h=0$ 得欧拉方程 (19.24)：

Rewriting (19.9) at $\rho=1$ via L'Hôpital (19.14), substituting (19.13), and matching the $Z_t$ coefficient gives (19.16): $f_2=\frac{\beta}{1-\beta\psi_z}$; matching the constant gives (19.17) $f_1$. Verification succeeds; $f_1$ depends on risk aversion $\gamma$, while $f_2$ is completely independent of $\gamma$.

Note

Remark 19.2. With (19.17), (19.16) one can numerically compute the sequences $i_t,k_t,v_t$ and hence $c_t$ (or $C_t$) via (19.7). Unlike Chapter 18 where $C_t$ was exogenous, here it is reversed: given observed consumption data, adjust model parameters to calibrate the actual consumption data.

19.2 Production Model with Income Shock

This is an incomplete-market model: the agent cannot hedge labor-income shocks with state-contingent claims, and there is no borrowing constraint. Although the agent can buy neither state-contingent claims nor risky assets, they can still smooth consumption quite well by saving.

19.2.1 Setup

A single (representative) agent, a single interchangeable capital-consumption good. The state variable $\mathbf X_t(q)$ satisfies $\mathbf X_{t+1}(q)=\boldsymbol\psi(\mathbf X_t(q),q\mathbf W_{t+1},q)$ ($q$ measures exposure to uncertainty, $\mathbf W_{t+1}\sim\mathcal N(\mathbf 0,\mathbf I)$). Log utility, infinite horizon, discount $\beta$: $V_t=\sum_{\tau=0}^\infty\beta^\tau\ln C_{t+\tau}$. Capital evolution (19.18): $K_{t+1}+C_t=e^a K_t+Y_t$ ($a$ constant, $\{Y_t\}$ exogenous labor income). Logs $y_t(q)\equiv\ln Y_t(q)$, $c_t(q)\equiv\ln C_t(q)$ ($\frac{K_{t+1}}{Y_{t+1}}$ assumed stationary but not necessarily positive, so not logged). $Y_t$ is labor income, exogenous (form 19.19): $y_{t+1}(q)-y_t(q)=\kappa(\mathbf X_t(q),\mathbf W_{t+1})$ ($\kappa$ assumed additively separable in $\mathbf X_t,\mathbf W_{t+1}$, which does not affect results). $C_t$ is endogenously chosen, so $C_t(q),K_t(q)$ are also functions of $q$, with $c_{t+1}(q)-c_t(q)=\phi(\mathbf X_t(q),\mathbf W_{t+1})$. By (17.11) the first-derivative dynamics (19.19): $\mathbf X_{t+1}'(0)=\boldsymbol\psi_{\mathbf X^T}\mathbf X_t'(0)+\boldsymbol\psi_{\mathbf W^T}\mathbf W_{t+1}+\boldsymbol\psi_h$, and (19.20), (19.21): $y_{t+1}'(0)-y_t'(0)=\boldsymbol\kappa_{\mathbf X^T}\mathbf X_t'(0)$, $c_{t+1}'(0)-c_t'(0)=\boldsymbol\phi_{\mathbf X^T}\mathbf X_t'(0)$.

19.2.2 Utility Maximization and Euler Equation

The agent solves (19.22): $\max_{C_t}\{\ln C_t(q)+\mathbb E[\sum_{\tau=1}^\infty\beta^\tau\ln C_{t+\tau}(q)\mid\mathcal F_t]\}$ s.t. $K_{t+1}(q)+C_t(q)=e^a K_t(q)+Y_t(q)$ (19.23). By a variational argument: at the optimum the agent is indifferent between consuming $h$ more today or not, i.e. between $(C_t,C_{t+1})$ and $(C_t-h,C_{t+1}+e^a h)$ (consume $h$ less today, $e^a h$ more tomorrow). The f.o.c. w.r.t. $h$ at $h=0$ gives the Euler equation (19.24):

$$e^a\beta\,\mathbb E\!\left[\frac{C_t(q)}{C_{t+1}(q)}\mid\mathcal F_t\right]=1.\tag{19.24}$$

(19.24) 即定价方程，可重写提取 SDF：$\mathbb E[\underbrace{\beta\frac{C_t(q)}{C_{t+1}(q)}}_{\text{SDF}}e^a\mid\mathcal F_t]=1$——因为今天一单位商品是明天 $e^a$ 单位商品的价格，$\beta\frac{C_t}{C_{t+1}}$ 即 SDF。进一步重写 (19.25)：$e^a\beta\,\mathbb E[\frac{C_t/Y_t}{C_{t+1}/Y_{t+1}}\frac{Y_t}{Y_{t+1}}\mid\mathcal F_t]=1$。

19.2.3 Steady State and First-Order Approximation

稳态设 $q=0$。设 $\frac{Y_{t+1}(0)}{Y_t(0)}=e^{\kappa(\mathbf X_t(0),\mathbf W_{t+1})}\equiv e^\eta$ (19.26)，即 $y_{t+1}(0)-y_t(0)=\eta$ (19.27)。(19.24) 在 $q=0$（无不确定性）必成立：$e^a\beta\frac{C_t(0)}{C_{t+1}(0)}=1$，即 $\frac{C_{t+1}(0)}{C_t(0)}=e^a\beta$ (19.28)。结合 (19.26)、(19.28)，设稳态 $\frac{Y_t(0)}{C_t(0)}$ 为常数得 (19.29)：$e^{a-\eta}\beta=1$。$\frac{K_t(0)}{Y_t(0)}$ 也是常数 (19.30)。

资源约束 (19.18) 一阶近似（围绕 $q=0$）。 除以 $Y_t(q)$ 得 (19.30)，对各比率 $\frac{K_{t+1}(q)}{Y_{t+1}(q)},\frac{Y_{t+1}(q)}{Y_t(q)},\frac{C_t(q)}{Y_t(q)},\frac{K_t(q)}{Y_t(q)}$ 在 $q=0$ 处一阶展开 (19.31)–(19.35)（用 $dx=x\,d\ln x$），代入 (19.30) 得 (19.36)。结合稳态恒等式 (19.37)，整理出 (19.38)：

(19.24) is the pricing equation; rewriting extracts the SDF: $\mathbb E[\underbrace{\beta\frac{C_t(q)}{C_{t+1}(q)}}_{\text{SDF}}e^a\mid\mathcal F_t]=1$ — since one unit of good today is the price of $e^a$ units tomorrow, $\beta\frac{C_t}{C_{t+1}}$ is the SDF. Rewriting further (19.25): $e^a\beta\,\mathbb E[\frac{C_t/Y_t}{C_{t+1}/Y_{t+1}}\frac{Y_t}{Y_{t+1}}\mid\mathcal F_t]=1$.

19.2.3 Steady State and First-Order Approximation

Steady state at $q=0$. Suppose $\frac{Y_{t+1}(0)}{Y_t(0)}=e^{\kappa(\mathbf X_t(0),\mathbf W_{t+1})}\equiv e^\eta$ (19.26), i.e. $y_{t+1}(0)-y_t(0)=\eta$ (19.27). (19.24) must hold at $q=0$ (no uncertainty): $e^a\beta\frac{C_t(0)}{C_{t+1}(0)}=1$, i.e. $\frac{C_{t+1}(0)}{C_t(0)}=e^a\beta$ (19.28). Combining (19.26), (19.28) and assuming steady-state $\frac{Y_t(0)}{C_t(0)}$ constant gives (19.29): $e^{a-\eta}\beta=1$. $\frac{K_t(0)}{Y_t(0)}$ is also constant (19.30).

First-order approximation of the resource constraint (19.18) around $q=0$. Dividing by $Y_t(q)$ gives (19.30); first-order expansions of the ratios $\frac{K_{t+1}(q)}{Y_{t+1}(q)},\frac{Y_{t+1}(q)}{Y_t(q)},\frac{C_t(q)}{Y_t(q)},\frac{K_t(q)}{Y_t(q)}$ at $q=0$ (19.31)–(19.35) (using $dx=x\,d\ln x$), substituting into (19.30) gives (19.36). With the steady-state identity (19.37), this yields (19.38):

$$\underbrace{\left(\frac{K_{t+1}(0)}{Y_{t+1}(0)}\right)_1}_{\equiv\kappa_{t+1}}=\underbrace{e^{a-\eta}}_{\equiv\lambda^{-1}}\underbrace{\left(\frac{K_t(0)}{Y_t(0)}\right)_1}_{\equiv\kappa_t}-\underbrace{\left[\overline{\left(\frac{K}{Y}\right)}(y_{t+1}'(0)-y_t'(0))+e^{-\eta}\overline{\left(\frac{C}{Y}\right)}(c_t'(0)-y_t'(0))\right]}_{\equiv Z_{t+1}}.\tag{19.38}$$

(19.38) 给出 $\kappa_{t+1}=\lambda^{-1}\kappa_t-Z_{t+1}$，递归求解得 (19.39)：$\kappa_t=\sum_{j=1}^\infty\lambda^j Z_{t+j}$。

观察 (19.39)： LHS $\kappa_t$ 仅基于 $t$ 时信息；RHS 基于无穷远未来信息。两者相等的唯一可能是 RHS 上所有未来期随机性的系数恰好抵消。由 (19.20)、(19.21)、(19.19)，$Z_{t+1}$ 是从无穷过去到 $t+1$ 所有冲击的线性组合加常数 $b$：$Z_{t+1}=b+\sum_{j=0}^\infty\boldsymbol\zeta_j\cdot\mathbf W_{t+1-j}$。$\mathbf W_{t+1}$ 不应影响 $\kappa_t$，故 RHS 上 $\mathbf W_{t+1}$ 的系数应为零 (19.40)：

(19.38) gives $\kappa_{t+1}=\lambda^{-1}\kappa_t-Z_{t+1}$, recursively solved as (19.39): $\kappa_t=\sum_{j=1}^\infty\lambda^j Z_{t+j}$.

Observations of (19.39): the LHS $\kappa_t$ rests purely on time-$t$ information; the RHS on infinite-future information. The only way they are equal is that the coefficients of randomness across all future periods on the RHS exactly cancel. By (19.20), (19.21), (19.19), $Z_{t+1}$ is a linear combination of all shocks from the infinite past to $t+1$ plus a constant $b$: $Z_{t+1}=b+\sum_{j=0}^\infty\boldsymbol\zeta_j\cdot\mathbf W_{t+1-j}$. $\mathbf W_{t+1}$ should not affect $\kappa_t$, so its coefficient on the RHS must be zero (19.40):

$$\sum_{j=1}^\infty\lambda^j\boldsymbol\zeta_{j-1}=\mathbf 0,\tag{19.40}$$

(19.40) 必成立、也能成立，因为内生消费 $C_t$ 恰好被选择使该限制满足。

欧拉方程 (19.25) 的近似。 重写 (19.25) 为 (19.41)：$e^a\beta\,\mathbb E[e^{[c_t(q)-y_t(q)]-[c_{t+1}(q)-y_{t+1}(q)]-[y_{t+1}(q)-y_t(q)]}\mid\mathcal F_t]=1$。对期望内对象在 $q=0$ 处一阶展开 (19.42)、(19.43)（用 $dx=x\,d\ln x$），代入 (19.41) 得（在 $q$ 一阶）：$\mathbb E[(c_t'(0)-y_t'(0))-(c_{t+1}'(0)-y_{t+1}'(0))-(y_{t+1}'(0)-y_t'(0))\mid\mathcal F_t]=0$。

递归效用 $\rho=1$. 由 Corollary 9.1，EZ 的 SDF (19.44)：$m_{t+1}^{EZ}=\beta(\frac{C_t}{C_{t+1}})(\frac{V_{t+1}}{\mathbb E[V_{t+1}^{1-\gamma}\mid\mathcal F_t]^{1/(1-\gamma)}})^{1-\gamma}$，设 $\gamma$ 固定。

第一种刻画. 定义 $M_{t+1}(q)\equiv\frac{(V_{t+1}(q))^{1-\gamma}}{\mathbb E[(V_{t+1}(q))^{1-\gamma}\mid\mathcal F_t]}=\frac{e^{(1-\gamma)v_{t+1}(q)}}{\mathbb E[e^{(1-\gamma)v_{t+1}(q)}\mid\mathcal F_t]}$ (19.46)，它满足 §38.2 的两个条件（$\geq0$、期望为 1），可作改变测度。一阶展开 (19.47)（$v_{t+1}'(0)$ 是 $\mathbf W_{t+1}$ 的线性函数 (19.48)：$v_{t+1}'(0)-v_t'(0)=\alpha_v+\boldsymbol\sigma_v\cdot\mathbf W_{t+1}$），故 $M_{t+1}(q)=\alpha_M+(1-\gamma)\boldsymbol\sigma_v\cdot\mathbf W_{t+1}$。在 tilde 测度下 (19.51)：$\tilde{\mathbb E}[\mathbf W_{t+1}\mid\mathcal F_t]=(1-\gamma)\boldsymbol\sigma_v$。代入欧拉方程 (19.49)、(19.50) 整理得 $\tilde{\mathbb E}[(c_t'(0)-y_t'(0))-(c_{t+1}'(0)-y_{t+1}'(0))-(y_{t+1}'(0)-y_t'(0))\mid\mathcal F_t]=0$。

第二种刻画. 由 (9.3)，$\rho=1$ 下 $V_t(q)=(C_t(q))^{1-\beta}(R_t(q))^\beta$，SDF $\frac{S_{t+1}(q)}{S_t(q)}=\beta\frac{C_t(q)}{C_{t+1}(q)}(\frac{V_{t+1}(q)}{R_t(q)})^{1-\gamma}$。对数化 (19.52)：$v_t(q)=(1-\beta)c_t(q)+\beta r_t(q)$、(19.53)：$r_t(q)=\frac1{1-\gamma}\ln(\mathbb E_t[e^{(1-\gamma)v_{t+1}(q)}])$，SDF (19.54)：$s_{t+1}(q)-s_t(q)=\underbrace{\ln\beta}_{=-\delta}+c_t(q)-c_{t+1}(q)+(1-\gamma)(v_{t+1}(q)-r_t(q))$。零阶（稳态）$s_{t+1}(0)-s_t(0)=-\delta-\eta$（因 $v_{t+1}(0)-r_t(0)$，且稳态 $c_t(0)-c_{t+1}(0)=y_t(0)-y_{t+1}(0)=-\eta$）。一阶 (19.55)：$s_{t+1}'(0)-s_t'(0)=c_t'(0)-c_{t+1}'(0)+(1-\gamma)(v_{t+1}'(0)-r_t'(0))$，用 $M_{t+1}(q)$ 作变测度并设 $c_{t+1}'(0)-c_t'(0)=\alpha_c+\boldsymbol\sigma_c\cdot\mathbf W_{t+1}$ (19.56)，在 $q=1$ 处评估，整理得 (19.57)：$-\delta+a-\eta+\mathbb E[c_t'(0)-c_{t+1}'(0)\mid\mathcal F_t]-(1-\gamma)\boldsymbol\sigma_v\cdot\boldsymbol\sigma_v+\frac{|\boldsymbol\sigma_c|^2}2=0$。

设 $c_{t+1}'(0)$ 满足 $\mathbb E[c_t'(0)-c_{t+1}'(0)\mid\mathcal F_t]=0$ (19.58)，即 $\{c_t'(0)\}$ 是鞅（由 (19.56)、(19.58) 等价于 $\alpha_c=0$）。由 (19.52) 得 (19.59)：$v_t'(0)=(1-\beta)c_t'(0)+\beta r_t'(0)$；由 (19.53) 得 (19.60)：$r_t'(0)=\mathbb E_t[v_{t+1}'(0)]$。代入并迭代得 (19.61)：$v_t'(0)=c_t'(0)+\text{constant}$，故 $\boldsymbol\sigma_c=\boldsymbol\sigma_v$（由 (19.48)、(19.56)、(19.61)）。最后把 $\boldsymbol\sigma_c=\boldsymbol\sigma_v$ 与 (19.58) 代入 (19.57) 得 $\delta$ 的限制：

(19.40) must and can hold, because endogenous consumption $C_t$ is chosen exactly so this restriction holds.

Approximation of the Euler equation (19.25). Rewrite (19.25) as (19.41): $e^a\beta\,\mathbb E[e^{[c_t(q)-y_t(q)]-[c_{t+1}(q)-y_{t+1}(q)]-[y_{t+1}(q)-y_t(q)]}\mid\mathcal F_t]=1$. First-order-expand the object inside the expectation at $q=0$ (19.42), (19.43) (using $dx=x\,d\ln x$), substituting into (19.41) gives (to first order in $q$): $\mathbb E[(c_t'(0)-y_t'(0))-(c_{t+1}'(0)-y_{t+1}'(0))-(y_{t+1}'(0)-y_t'(0))\mid\mathcal F_t]=0$.

Recursive utility $\rho=1$. By Corollary 9.1, the EZ SDF (19.44): $m_{t+1}^{EZ}=\beta(\frac{C_t}{C_{t+1}})(\frac{V_{t+1}}{\mathbb E[V_{t+1}^{1-\gamma}\mid\mathcal F_t]^{1/(1-\gamma)}})^{1-\gamma}$, with $\gamma$ fixed.

First characterization. Define $M_{t+1}(q)\equiv\frac{(V_{t+1}(q))^{1-\gamma}}{\mathbb E[(V_{t+1}(q))^{1-\gamma}\mid\mathcal F_t]}=\frac{e^{(1-\gamma)v_{t+1}(q)}}{\mathbb E[e^{(1-\gamma)v_{t+1}(q)}\mid\mathcal F_t]}$ (19.46), satisfying the two §38.2 conditions ($\geq0$, expectation 1), usable as a change of measure. First-order expansion (19.47) ($v_{t+1}'(0)$ is a linear function of $\mathbf W_{t+1}$ (19.48): $v_{t+1}'(0)-v_t'(0)=\alpha_v+\boldsymbol\sigma_v\cdot\mathbf W_{t+1}$), so $M_{t+1}(q)=\alpha_M+(1-\gamma)\boldsymbol\sigma_v\cdot\mathbf W_{t+1}$. Under the tilde measure (19.51): $\tilde{\mathbb E}[\mathbf W_{t+1}\mid\mathcal F_t]=(1-\gamma)\boldsymbol\sigma_v$. Substituting into the Euler equation (19.49), (19.50) yields $\tilde{\mathbb E}[(c_t'(0)-y_t'(0))-(c_{t+1}'(0)-y_{t+1}'(0))-(y_{t+1}'(0)-y_t'(0))\mid\mathcal F_t]=0$.

Second characterization. By (9.3), at $\rho=1$, $V_t(q)=(C_t(q))^{1-\beta}(R_t(q))^\beta$, SDF $\frac{S_{t+1}(q)}{S_t(q)}=\beta\frac{C_t(q)}{C_{t+1}(q)}(\frac{V_{t+1}(q)}{R_t(q)})^{1-\gamma}$. In logs (19.52): $v_t(q)=(1-\beta)c_t(q)+\beta r_t(q)$, (19.53): $r_t(q)=\frac1{1-\gamma}\ln(\mathbb E_t[e^{(1-\gamma)v_{t+1}(q)}])$, SDF (19.54): $s_{t+1}(q)-s_t(q)=\underbrace{\ln\beta}_{=-\delta}+c_t(q)-c_{t+1}(q)+(1-\gamma)(v_{t+1}(q)-r_t(q))$. Zeroth order (steady state) $s_{t+1}(0)-s_t(0)=-\delta-\eta$ (since $v_{t+1}(0)-r_t(0)$, and in steady state $c_t(0)-c_{t+1}(0)=y_t(0)-y_{t+1}(0)=-\eta$). First order (19.55): $s_{t+1}'(0)-s_t'(0)=c_t'(0)-c_{t+1}'(0)+(1-\gamma)(v_{t+1}'(0)-r_t'(0))$; using $M_{t+1}(q)$ as a change of measure and setting $c_{t+1}'(0)-c_t'(0)=\alpha_c+\boldsymbol\sigma_c\cdot\mathbf W_{t+1}$ (19.56), evaluating at $q=1$ gives (19.57): $-\delta+a-\eta+\mathbb E[c_t'(0)-c_{t+1}'(0)\mid\mathcal F_t]-(1-\gamma)\boldsymbol\sigma_v\cdot\boldsymbol\sigma_v+\frac{|\boldsymbol\sigma_c|^2}2=0$.

Assume $c_{t+1}'(0)$ satisfies $\mathbb E[c_t'(0)-c_{t+1}'(0)\mid\mathcal F_t]=0$ (19.58), i.e. $\{c_t'(0)\}$ is a martingale (by (19.56), (19.58) equivalent to $\alpha_c=0$). By (19.52), (19.59): $v_t'(0)=(1-\beta)c_t'(0)+\beta r_t'(0)$; by (19.53), (19.60): $r_t'(0)=\mathbb E_t[v_{t+1}'(0)]$. Substituting and iterating gives (19.61): $v_t'(0)=c_t'(0)+\text{constant}$, so $\boldsymbol\sigma_c=\boldsymbol\sigma_v$ (by (19.48), (19.56), (19.61)). Finally substituting $\boldsymbol\sigma_c=\boldsymbol\sigma_v$ and (19.58) into (19.57) gives the restriction on $\delta$:

$$\delta=a-\eta-\left(\frac12-\gamma\right)|\boldsymbol\sigma_c|^2,$$

即 (19.58) 假设（$\{c_t'(0)\}$ 为鞅）成立所需的 $\delta$ 的必要限制。

the necessary restriction on $\delta$ for the assumption (19.58) ($\{c_t'(0)\}$ a martingale) to hold.

References

Deaton, A. (1991). Saving and Liquidity Constraints. Econometrica 59(5), 1221–1248.