21. Other Specifications with Recursive Utility

Jun He May 31, 2026

资产定价Asset Pricing 递归效用Recursive Utility 习惯形成Habit Persistence 连续时间Continuous Time 随机贴现因子SDF 学习笔记Study Note

Note

本章把 Epstein-Zin 递归效用和两类常见设定结合起来。第一部分在递归效用中嵌入习惯形成 (habit persistence)：关键差别在于即期效用不是定义在消费 $C_t$ 上，而是定义在由当期消费与习惯水平复合而成的有效消费 (effective consumption) $U_t$ 上 (21.3)。通过取极限可得两个特例——耐用品模型 ($\varepsilon=0$，$U_t$ 退化为全部历史消费的加权平均) 与消费上限模型 ($\varepsilon\to\infty$，$U_t=\min\{C_t,H_t\}$)。在 $\rho=\gamma$ 时推出 SDF，并区分内部习惯（$H_t$ 受自身消费影响，边际效用含额外项）与外部习惯（"赶超琼斯"，$H_t$ 外生）。第二部分把递归效用搬到连续时间：通过对欧拉方程关于时间间隔 $\varepsilon$ 求导并用 Ito 引理，得到价值函数漂移 $\mu_{V,t}$ 与扩散 $\boldsymbol\sigma_{V,t}$ 之间的约束关系 (21.25)，并以一个带随机波动的 AK 生产经济作为应用。

Note

This chapter combines Epstein-Zin recursive utility with two common specifications. Part one embeds habit persistence into recursive utility: the key difference is that instantaneous utility is defined not over consumption $C_t$ but over effective consumption $U_t$, a composite of current consumption and the habit level (21.3). Taking limits gives two special cases — the durable goods model ($\varepsilon=0$, where $U_t$ collapses to a weighted average of all past consumption) and the upper-bound consumption model ($\varepsilon\to\infty$, where $U_t=\min\{C_t,H_t\}$). Setting $\rho=\gamma$ delivers the SDF, distinguishing internal habit (where $H_t$ responds to own consumption, adding an extra marginal-utility term) from external habit ("catching up with the Joneses", $H_t$ exogenous). Part two moves recursive utility into continuous time: differentiating the Euler equation with respect to the time increment $\varepsilon$ and applying Ito's lemma yields a constraint linking the value-function drift $\mu_{V,t}$ to the diffusion $\boldsymbol\sigma_{V,t}$ (21.25), illustrated with an AK production economy with stochastic volatility.

21.1 Habit Persistence with Recursive Utility

21.1.1 Setup

代理人的价值函数 $V_t$ 由递归式 (21.1)–(21.2) 给出：

The agent's value function $V_t$ is given recursively by (21.1)–(21.2):

$$V_t=\left[(1-\beta)(U_t)^{1-\rho}+\beta(R_t)^{1-\rho}\right]^{\frac1{1-\rho}}\tag{21.1}$$

$$R_t=\left(\mathbb E\!\left[(V_{t+1})^{1-\gamma}\mid\mathcal F_t\right]\right)^{\frac1{1-\gamma}}\tag{21.2}$$

其中有效消费 (effective consumption) $U_t$ 是当期消费 $C_t$ 与习惯 $H_t$ 的复合 (21.3)，而习惯 $H_t$ 是全部历史消费 $\{C_{t-1},C_{t-2},\dots\}$ 的加权和 (21.4)：

where the effective consumption $U_t$ is a composite of current consumption $C_t$ and habit $H_t$ (21.3), and the habit $H_t$ is a weighted sum of all past consumption $\{C_{t-1},C_{t-2},\dots\}$ (21.4):

$$U_t=\left[(1-\alpha)(C_t)^{1-\varepsilon}+\alpha(H_t)^{1-\varepsilon}\right]^{\frac1{1-\varepsilon}}\quad\text{for }\varepsilon\ge0,\ \alpha\in[0,1]\tag{21.3}$$

$$H_{t+1}=\chi H_t+(1-\chi)C_t\quad\text{for }\chi\in[0,1]\tag{21.4}$$

$\mathcal F_t$ 为 $t$ 时信息集，$V_t,R_t,U_t,H_{t+1}$ 都是一阶齐次的。

Tip

Remark 21.1 (21.1)–(21.2) 与 Epstein-Zin 递归效用 (9.1) 完全相同，唯一区别是即期效用不再定义在消费 $C_t$ 上，而是定义在有效消费 $U_t$ (21.3) 上——$U_t$ 是当期消费 $C_t$ 与习惯 $H_t$（全部历史消费的加权和，见 21.4）的复合。

$\mathcal F_t$ is the time-$t$ information set, and $V_t,R_t,U_t,H_{t+1}$ are all homogeneous of degree 1.

Tip

Remark 21.1 (21.1)–(21.2) are exactly the same as Epstein-Zin recursive utility (9.1), with the only difference being that instantaneous utility is no longer defined over consumption $C_t$ but over effective consumption $U_t$ (21.3) — itself a composite of current consumption $C_t$ and habit $H_t$ (the weighted sum of all past consumption, see 21.4).

21.1.2 Special Cases

(i) 耐用品模型 (durable goods model)：令 $\varepsilon=0$ 且 $\alpha=\chi$，则 (21.3) 退化为线性形式，$U_t=(1-\chi)C_t+\chi H_t=H_{t+1}$。于是有效消费就是下一期习惯，即全部历史消费的加权平均 (21.5)：

(i) Durable goods model: set $\varepsilon=0$ and $\alpha=\chi$; then (21.3) collapses to a linear form, $U_t=(1-\chi)C_t+\chi H_t=H_{t+1}$. So effective consumption equals next-period habit, i.e. a weighted average of all past consumption (21.5):

$$U_t=H_{t+1}=(1-\chi)\sum_{j=0}^{\infty}\chi^{j}C_{t-j}\tag{21.5}$$

由于 $(1-\chi)\sum_{j=0}^{\infty}\chi^{j}=1$，$U_t$ 确实是全部历史消费的加权平均。称其为"耐用品"模型，是因为本期效用来自全部历史消费流，正如一件耐用品在过去持续产生消费流。

(ii) 消费上限模型 (upper-bound consumption model)：令 $\varepsilon\to\infty$，由 L'Hôpital 法则可证 (21.6) 的极限为 $U_t=\min\{C_t,H_t\}$（推导见下）。

Since $(1-\chi)\sum_{j=0}^{\infty}\chi^{j}=1$, $U_t$ is indeed a weighted average of all past consumption. It is called a "durable goods" model because this period's utility comes from the entire stream of past consumption, just as a durable good keeps generating consumption flow from the past.

(ii) Upper-bound consumption model: let $\varepsilon\to\infty$; by L'Hôpital's rule the limit of (21.6) is $U_t=\min\{C_t,H_t\}$ (derivation below).

证明 / Proof：$\varepsilon\to\infty\Rightarrow U_t=\min\{C_t,H_t\}$

取对数并用 L'Hôpital 法则 (21.6)：

Take logs and apply L'Hôpital's rule (21.6):

$$\lim_{\varepsilon\to\infty}U_t=\lim_{\varepsilon\to\infty}\exp\!\left[\frac{(1-\alpha)(C_t)^{1-\varepsilon}\ln C_t+\alpha(H_t)^{1-\varepsilon}\ln H_t}{(1-\alpha)(C_t)^{1-\varepsilon}+\alpha(H_t)^{1-\varepsilon}}\right]\tag{21.6}$$

若 $H_t>C_t$：分子分母同除以 $(H_t)^{1-\varepsilon}$，因 $1-\varepsilon\to-\infty$，$(C_t/H_t)^{1-\varepsilon}\to0$，极限为 $\exp\!\left[\frac{\alpha\ln H_t}{\alpha}\right]$… 经整理得 $U_t=C_t$（取较小者）。
若 $H_t=C_t$：得 $U_t=C_t=H_t$。
若 \(H_t

三种情形合并即 $U_t=\min\{C_t,H_t\}$。这意味着若代理人有储蓄技术，他永远不会消费超过习惯水平 $H_t$。$\blacksquare$

If $H_t>C_t$: divide top and bottom by $(H_t)^{1-\varepsilon}$; since $1-\varepsilon\to-\infty$, $(C_t/H_t)^{1-\varepsilon}\to0$, and after simplification $U_t=C_t$ (the smaller one).
If $H_t=C_t$: $U_t=C_t=H_t$.
If \(H_t

Combining the three cases gives $U_t=\min\{C_t,H_t\}$. This means that if the agent has a saving technology, he will never consume more than the habit level $H_t$. $\blacksquare$

21.1.3 Stochastic Discount Factor

在 $\rho=\gamma$ 的情形下推导 SDF $\frac{S_{t+1}}{S_t}$。代入 $\rho=\gamma$ 后 (21.1)–(21.2) 给出价值的递归 (21.7)：

We derive the SDF $\frac{S_{t+1}}{S_t}$ in the case $\rho=\gamma$. Plugging $\rho=\gamma$ into (21.1)–(21.2) gives the value recursion (21.7):

$$(V_t)^{1-\rho}=(1-\beta)\sum_{j=0}^{\infty}\beta^{j}\,\mathbb E\!\left[(U_{t+j})^{1-\rho}\mid\mathcal F_t\right]\tag{21.7}$$

定义两个边际量：对 $C_t$ 求导（忽略 $C_t$ 通过 $H_{t+1}$ 的间接效应）记为 $\widetilde{MC}_t$ (21.8)；对 $H_t$ 求导（忽略 $H_t$ 通过 $H_{t+1}$ 的间接效应）记为 $\widetilde{MH}_t$ (21.9)：

Define two marginal objects: the derivative w.r.t. $C_t$ (ignoring $C_t$'s indirect effect through $H_{t+1}$), denoted $\widetilde{MC}_t$ (21.8); and the derivative w.r.t. $H_t$ (ignoring $H_t$'s indirect effect through $H_{t+1}$), denoted $\widetilde{MH}_t$ (21.9):

$$\widetilde{MC}_t=(1-\beta)(U_t)^{\varepsilon-\rho}(1-\alpha)(C_t)^{-\varepsilon}\tag{21.8}$$

$$\widetilde{MH}_t=(1-\beta)(U_t)^{\varepsilon-\rho}\,\alpha\,(H_t)^{-\varepsilon}\tag{21.9}$$

全边际消费效用 $MC_t$ 需把 $C_t$ 通过 $H_{t+1},H_{t+2},\dots$ 的间接效应加回。利用 (21.4) 知 $C_t$ 对 $H_{t+j}$ 的系数为 $(1-\chi)\chi^{j-1}$，链式法则给出 (21.10)：

The full marginal utility of consumption $MC_t$ must add back $C_t$'s indirect effects through $H_{t+1},H_{t+2},\dots$. Using (21.4), $C_t$'s coefficient on $H_{t+j}$ is $(1-\chi)\chi^{j-1}$, and the chain rule gives (21.10):

$$MC_t=\widetilde{MC}_t+\sum_{j=1}^{\infty}\beta^{j}\,\mathbb E\!\left[\widetilde{MH}_{t+j}\mid\mathcal F_t\right](1-\chi)\chi^{j-1}\tag{21.10}$$

代回 (21.8)、(21.9) 得显式形式 (21.11)：

Substituting (21.8) and (21.9) gives the explicit form (21.11):

$$MC_t=(1-\beta)(U_t)^{\varepsilon-\rho}(1-\alpha)(C_t)^{-\varepsilon}+\beta(1-\chi)\sum_{j=1}^{\infty}(\beta\chi)^{j-1}\,\mathbb E\!\left[\widetilde{MH}_{t+j}\mid\mathcal F_t\right]\tag{21.11}$$

Tip

内部习惯 vs 外部习惯 - 内部习惯 (internal habit)：$MC_t$ 由 (21.11) 完整给出——代理人内部化了今天消费对未来习惯的提升，故第二项不为零。 - 外部习惯 (external habit，"赶超琼斯")：$H_{t+1}$ 不受单个代理人控制，故 $MC_t$ 就等于 $\widetilde{MC}_t$，第二项为零。

由欧拉方程 (1.20)，SDF 满足 $\mathbb E_t\!\left[\frac{S_{t+1}}{S_t}\right]=\frac{MC_{t+1}}{MC_t}$，更一般地 $\mathbb E_t\!\left[\frac{S_{t+\tau}}{S_t}\right]=\frac{MC_{t+\tau}}{MC_t}$。注意 $MC_{t+\tau}$ 是对价值函数 $V_{t+\tau}$（而非跨期效用）在该期内求导——欧拉方程只对每期价值函数求导才良定义，不能跨期。

SDF 的鞅成分 (martingale component)：把 (21.11) 改写为 (21.12)，再用 (21.9) 把 $\widetilde{MH}_{t+j}$ 写成 (21.13)：

Tip

Internal vs external habit - Internal habit: $MC_t$ is given fully by (21.11) — the agent internalizes how today's consumption raises future habit, so the second term is nonzero. - External habit ("catching up with the Joneses"): $H_{t+1}$ is out of the individual agent's control, so $MC_t$ equals $\widetilde{MC}_t$ and the second term vanishes.

By the Euler equation (1.20), the SDF satisfies $\mathbb E_t\!\left[\frac{S_{t+1}}{S_t}\right]=\frac{MC_{t+1}}{MC_t}$, and more generally $\mathbb E_t\!\left[\frac{S_{t+\tau}}{S_t}\right]=\frac{MC_{t+\tau}}{MC_t}$. Note $MC_{t+\tau}$ takes the derivative of the value function $V_{t+\tau}$ (not the period utility) within that period — the Euler equation is only well-defined period-by-period, not across periods.

Martingale component of the SDF: rewrite (21.11) as (21.12), then use (21.9) to express $\widetilde{MH}_{t+j}$ as in (21.13):

$$MC_t=(1-\beta)\left(\frac{U_t}{C_t}\right)^{\varepsilon}(1-\alpha)(U_t)^{-\rho}+\beta(1-\chi)\sum_{j=1}^{\infty}(\beta\chi)^{j-1}\,\mathbb E\!\left[\widetilde{MH}_{t+j}\mid\mathcal F_t\right]\tag{21.12}$$

$$MC_t=(1-\beta)\left(\frac{U_t}{C_t}\right)^{\varepsilon}(1-\alpha)(U_t)^{-\rho}+(1-\beta)\beta(1-\chi)\sum_{j=1}^{\infty}(\beta\chi)^{j-1}\,\mathbb E\!\left[\frac{\partial U_{t+j}'}{\partial H_{t+j}}\mid\mathcal F_t\right](U_{t+j})^{-\rho}\tag{21.13}$$

由于 $U_t$、$H_t$ 关于 $\{C_t,C_{t-1},\dots\}$ 一阶齐次，$\frac{U_t}{C_t}$ 与 $\frac{\partial U_t}{\partial H_t}$ 关于消费零阶齐次。所以 (21.13) 表明 $MC_t$ 的鞅成分与 $(U_t)^{-\rho}$ 相同（仅差一个零阶齐次标量）。进一步：$S_{t+\tau}$ 的鞅成分与 $(U_{t+\tau})^{-\rho}$ 相同，亦与 $(C_{t+\tau})^{-\rho}$ 相同（因 $\frac{C_{t+\tau}}{U_{t+\tau}}$ 零阶齐次）。于是 $\mathbb E_t[S_{t+\tau}]$ 的鞅成分与 $(C_{t+\tau})^{-\rho}$ 相同，从而 $\frac{S_{t+\tau}}{S_t}$ 的鞅成分为 $\left(\frac{C_{t+\tau}}{C_t}\right)^{-\rho}$——与标准幂效用 SDF 的鞅成分一致。

Since $U_t$ and $H_t$ are h.o.d. 1 in $\{C_t,C_{t-1},\dots\}$, the ratios $\frac{U_t}{C_t}$ and $\frac{\partial U_t}{\partial H_t}$ are h.o.d. 0 in consumption. So (21.13) shows that $MC_t$ has the same martingale component as $(U_t)^{-\rho}$ (up to a h.o.d.-0 scalar). Going further: $S_{t+\tau}$ has the same martingale component as $(U_{t+\tau})^{-\rho}$, which also equals that of $(C_{t+\tau})^{-\rho}$ (since $\frac{C_{t+\tau}}{U_{t+\tau}}$ is h.o.d. 0). Hence $\mathbb E_t[S_{t+\tau}]$ shares the martingale component of $(C_{t+\tau})^{-\rho}$, so the martingale component of $\frac{S_{t+\tau}}{S_t}$ is $\left(\frac{C_{t+\tau}}{C_t}\right)^{-\rho}$ — the same as the standard power-utility SDF.

21.2 Recursive Utility in Continuous Time

21.2.1 Setup

设代理人有连续时间递归效用 (21.14)，其中连续价值算子由 (21.15) 给出，$\delta$ 为贴现率，$\varepsilon$ 为趋于零的小时间区间：

Suppose the agent has continuous-time recursive utility (21.14), with the continuation-value operator given by (21.15), where $\delta$ is the discount rate and $\varepsilon$ is a small time increment going to zero:

$$V_t=\left[(1-e^{-\delta\varepsilon})(C_t)^{1-\rho}+e^{-\delta\varepsilon}(\mathbb R(V_{t+\varepsilon}\mid\mathcal F_t))^{1-\rho}\right]^{\frac1{1-\rho}}\tag{21.14}$$

$$\mathbb R(V_{t+\varepsilon}\mid\mathcal F_t)=\left(\mathbb E\!\left[(V_{t+\varepsilon})^{1-\gamma}\mid\mathcal F_t\right]\right)^{\frac1{1-\gamma}}\tag{21.15}$$

设所有基本随机源都在 $d\mathbf B_t$ 中，则 $V_t$ 必满足扩散 (21.16)，其漂移为标量 $\mu_{V,t}$、扩散为向量 $\boldsymbol\sigma_{V,t}$：

If all fundamental sources of randomness are in $d\mathbf B_t$, then $V_t$ must follow the diffusion (21.16), with scalar drift $\mu_{V,t}$ and vector diffusion $\boldsymbol\sigma_{V,t}$:

$$dV_t=V_t\mu_{V,t}\,dt+V_t\boldsymbol\sigma_{V,t}\cdot d\mathbf B_t\tag{21.16}$$

21.2.2 Characterize the Dynamics of the Recursive Value Function

对 (21.14) 重排得 (21.17)，分为 Part A（即期消费项）与 Part B（连续价值项）：

Rearranging (21.14) gives (21.17), split into Part A (the consumption term) and Part B (the continuation-value term):

$$0=\underbrace{(1-e^{-\delta\varepsilon})\left[(C_t)^{1-\rho}-(V_t)^{1-\rho}\right]}_{\text{Part A}}+\underbrace{e^{-\delta\varepsilon}\left[(\mathbb R(V_{t+\varepsilon}\mid\mathcal F_t))^{1-\rho}-(V_t)^{1-\rho}\right]}_{\text{Part B}}\tag{21.17}$$

对 (21.17) 两边关于 $\varepsilon$ 求导并在 $\varepsilon=0$ 处取值。Part A 导数为 $\delta\left[(C_t)^{1-\rho}-(V_t)^{1-\rho}\right]$ (21.19)；Part B 需用 Ito 引理（详见下方折叠推导），其在 $\varepsilon=0$ 的导数为 (21.25)。结合 $\frac{\partial 0}{\partial\varepsilon}=0$ (21.18) 与 (21.19)、(21.25)，解出漂移与扩散的约束关系：

Differentiate both sides of (21.17) w.r.t. $\varepsilon$, evaluated at $\varepsilon=0$. Part A's derivative is $\delta\left[(C_t)^{1-\rho}-(V_t)^{1-\rho}\right]$ (21.19); Part B requires Ito's lemma (see the collapsible derivation below), with its derivative at $\varepsilon=0$ being (21.25). Combining $\frac{\partial 0}{\partial\varepsilon}=0$ (21.18) with (21.19) and (21.25) solves for the constraint between drift and diffusion:

$$\mu_{V,t}=\frac{\delta}{1-\rho}\left[\left(\frac{C_t}{V_t}\right)^{1-\rho}-1\right]+\frac12\gamma\,|\boldsymbol\sigma_{V,t}|^2\tag{21.25}$$

证明 / Proof：用 Ito 引理推导 Part B 的导数 (21.20)–(21.25)

幂函数的一、二阶导 (21.21)、(21.22)：

First and second derivatives of the power function (21.21), (21.22):

$$\frac{\partial (V_t)^{1-\gamma}}{\partial V_t}=(1-\gamma)(V_t)^{-\gamma}\tag{21.21}$$

$$\frac{\partial^2 (V_t)^{1-\gamma}}{(\partial V_t)^2}=-(1-\gamma)\gamma(V_t)^{-\gamma-1}\tag{21.22}$$

由 Ito 引理 (21.23)，并代入 (21.16) 的扩散：

By Ito's lemma (21.23), substituting the diffusion (21.16):

$$d(V_t)^{1-\gamma}=\frac{\partial (V_t)^{1-\gamma}}{\partial V_t}\,dV_t+\frac12\frac{\partial^2 (V_t)^{1-\gamma}}{(\partial V_t)^2}\,(dV_t)^2\tag{21.23}$$

整理后 $\mathbb E[(V_{t+\varepsilon})^{1-\gamma}\mid\mathcal F_t]$ 对 $\varepsilon$ 的导数为 (21.24)：

After simplification, the derivative of $\mathbb E[(V_{t+\varepsilon})^{1-\gamma}\mid\mathcal F_t]$ w.r.t. $\varepsilon$ is (21.24):

$$\frac{\partial (V_{t+\varepsilon})^{1-\gamma}}{\partial\varepsilon}=(1-\gamma)(V_t)^{1-\gamma}\mu_{V,t}-\frac12(1-\gamma)\gamma(V_t)^{1-\gamma}|\boldsymbol\sigma_{V,t}|^2\tag{21.24}$$

把 (21.24) 代回 Part B（经 (21.15) 与链式法则）得 (21.25) 的右侧：

Substituting (21.24) back into Part B (via (21.15) and the chain rule) gives the right side of (21.25):

$$\left.\frac{\partial\,\text{Part B}}{\partial\varepsilon}\right|_{\varepsilon=0}=(1-\rho)(V_t)^{1-\rho}\left[\mu_{V,t}-\frac12\gamma|\boldsymbol\sigma_{V,t}|^2\right]\tag{21.25*}$$

令 (21.19) 与 (21.25*) 之和为零，即解出 (21.25)。$\blacksquare$

Setting the sum of (21.19) and (21.25*) to zero solves for (21.25). $\blacksquare$

21.2.3 Application

考虑连续时间经济，资本 $K_t$ 按 (21.26) 演化（$I_t$ 为投资，$\{\mathbf B_t\}$ 为三维标准布朗运动，$\alpha_K$ 为折旧）：

Consider a continuous-time economy where capital $K_t$ evolves per (21.26) ($I_t$ is investment, $\{\mathbf B_t\}$ a trivariate standard Brownian motion, $\alpha_K$ depreciation):

$$dK_t=K_t\left(\Phi\!\left(\frac{I_t}{K_t}\right)+Y_t-\alpha_K\right)dt+K_t\sqrt{Z_t}\,\boldsymbol\sigma_K\cdot d\mathbf B_t\tag{21.26}$$

其中：$Y_t$ 是影响资本积累的外生变量，$dY_t=\lambda_Y Y_t\,dt+Y_t\sqrt{Z_t}\,\boldsymbol\sigma_Y\cdot d\mathbf B_t$；$Z_t$ 负责所有过程的随机波动，$dZ_t=-\lambda_Z(Z_t-1)\,dt+\sqrt{Z_t}\,\boldsymbol\sigma_Z\cdot d\mathbf B_t$，且 $\mathbb E[Z_t]=1$、$Z_t\ge0$。设调整成本函数 $\Phi\!\left(\frac{I_t}{K_t}\right)=\phi_1\ln\!\left(1+\phi_2\frac{I_t}{K_t}\right)$。$K_t$ 为内生状态变量，$Z_t,Y_t$ 为外生状态变量。用 Ito 引理对 $\ln K_t$ 得：

where: $Y_t$ is an exogenous variable affecting capital accumulation, $dY_t=\lambda_Y Y_t\,dt+Y_t\sqrt{Z_t}\,\boldsymbol\sigma_Y\cdot d\mathbf B_t$; $Z_t$ drives stochastic volatility of all processes, $dZ_t=-\lambda_Z(Z_t-1)\,dt+\sqrt{Z_t}\,\boldsymbol\sigma_Z\cdot d\mathbf B_t$, with $\mathbb E[Z_t]=1$, $Z_t\ge0$. The adjustment-cost function is $\Phi\!\left(\frac{I_t}{K_t}\right)=\phi_1\ln\!\left(1+\phi_2\frac{I_t}{K_t}\right)$. $K_t$ is the endogenous state, $Z_t,Y_t$ the exogenous states. Applying Ito's lemma to $\ln K_t$:

$$d\ln K_t=\left(\Phi\!\left(\frac{I_t}{K_t}\right)+Y_t-\alpha_K-\frac{|\boldsymbol\sigma_K|^2}{2}Z_t\right)dt+\sqrt{Z_t}\,\boldsymbol\sigma_K\cdot d\mathbf B_t\tag{21.27}$$

若经济为 AK 经济，即资源约束 $C_t+I_t=aK_t$（$a>0$ 为常数），则消费与投资都与资本成比例，模型可解析求解。

Tip

Remark 21.2 此经济可用于研究连续时间的中介资产定价 (intermediary asset pricing)。例如可设两类代理人——专家（金融中介）与家庭——并引入异质性：(1) 差异化风险厌恶 $\gamma_e$（专家）与 $\gamma_h$（家庭）；(2) 差异化生产率 $a_e$、$a_h$；(3) 差异化主观贴现 $\delta_e$、$\delta_h$。在此类模型中仍保有齐次性（如对 $Z_t,K_t,Y_t$）。若两类代理人都用 §21.2.1 的连续时间递归效用，便可研究 SDF、收益率、冲击价格弹性等动态与定价含义。

If the economy is an AK economy, i.e. the resource constraint $C_t+I_t=aK_t$ ($a>0$ constant), then both consumption and investment are proportional to capital and the model is analytically tractable.

Tip

Remark 21.2 This economy can be used to study intermediary asset pricing in continuous time. For example, posit two agent types — experts (financial intermediaries) and households — with heterogeneity in: (1) risk aversion $\gamma_e$ (expert) vs $\gamma_h$ (household); (2) productivity $a_e$, $a_h$; (3) subjective discounting $\delta_e$, $\delta_h$. Homogeneity (e.g. in $Z_t,K_t,Y_t$) is preserved in such models. If both agent types have the continuous-time recursive utility of §21.2.1, one can study the dynamics and pricing implications — SDF, returns, shock-price elasticities.

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