10. Suggested Explanation: Incomplete Market
本章用不完全市场 + 异质投资者解释双重之谜。核心思路：当投资者面对不可保险的特质收入冲击 (idiosyncratic income shocks) 时，个体消费增长在横截面上离散且负偏 (negatively skewed)；由 Jensen 不等式，横截面消费增长的 \(-\alpha\) 次幂平均（即聚合 SDF）会被这种离散和偏度放大，从而用合理的风险厌恶就能产生高股权溢价——不必像 CRRA 代表性代理人那样需要 \(\gamma\approx20\)。三篇文献：(i) Constantinides–Duffie (1996) 反向构造特质收入，使每个投资者的欧拉方程都成立且无交易 (no-trade)，证明不完全市场下定价可行；(ii) Brav, Constantinides and Géczy (2002) 用 CEX 家庭消费数据验证——加入横截面偏度项后，校准的 \(\alpha\approx3\) 不被拒绝；(iii) Constantinides–Ghosh (2017) 进一步让特质消费风险逆周期 (countercyclical)，匹配股权溢价、无风险利率、横截面等多个矩。
This chapter explains the dual puzzle with incomplete markets + heterogeneous investors. The core idea: when investors face uninsurable idiosyncratic income shocks, individual consumption growth is dispersed and negatively skewed in the cross-section; by Jensen's inequality the cross-sectional average of the \(-\alpha\) power of consumption growth (the aggregate SDF) is amplified by this dispersion and skewness, so a reasonable risk aversion can generate a high equity premium — without needing \(\gamma\approx20\) as in the CRRA representative agent. Three papers: (i) Constantinides–Duffie (1996) reverse-engineer idiosyncratic income so every investor's Euler equation holds with no trade, proving pricing works under incomplete markets; (ii) Brav, Constantinides and Géczy (2002) verify with CEX household consumption data — adding the cross-sectional skewness term, a calibrated \(\alpha\approx3\) is not rejected; (iii) Constantinides–Ghosh (2017) further make idiosyncratic consumption risk countercyclical, matching the equity premium, risk-free rate, and the cross-section.
为什么要不完全市场？在完全市场下，异质性其实帮不上忙：分散均衡的资产价格与聚合代表性代理人的效用最大化给出的价格相同（除非让财富分布通过时变的 Lagrange 乘子改变代表性代理人的偏好，或让投资者对状态持有异质信念，使 Arrow–Debreu 状态价格失效）。只有在不完全市场中——特质风险不可对冲、不能用代表性代理人、需用聚合欧拉方程——异质性才变得不可忽略。
Why incomplete markets? Under complete markets, heterogeneity does not help: the decentralized equilibrium prices coincide with those from a representative agent's utility maximization (unless one lets the wealth distribution change the representative agent's preferences through time-varying Lagrange multipliers, or lets investors hold heterogeneous beliefs over states so the Arrow–Debreu state prices break down). Only under incomplete markets — where idiosyncratic risk cannot be hedged, a representative agent cannot be used, and one must use the aggregate Euler equation — does heterogeneity become non-trivial.
10.1 Incomplete Market with Heterogeneous Investors: Constantinides and Duffie (1996)
10.1.1 Setup
证券 \(j\) 除权价格 \(P_{jt}\)、红利 \(d_{jt}\)，向量 \(\mathbf d_t=(d_{1t},\dots,d_{Jt})'\)、\(\mathbf P_t=(P_{1t},\dots,P_{Jt})'\)；无风险债券价格 1、支付 \(1+r_f\)。投资者 \(i\in\mathcal A\)，\(N=\mathrm{card}(\mathcal A)\to\infty\)（无限多投资者，用大数定律）。消费者 \(i\) 收入 \(I_{it}\)、消费 \(C_{it}\)；聚合 \(C_t=\sum_{i\in\mathcal A}C_{it}\)，\(C_t=I_t+D_t\)。投资者用 CRRA 的 vN-M 偏好
Security \(j\) has ex-dividend price \(P_{jt}\) and dividend \(d_{jt}\), with vectors \(\mathbf d_t=(d_{1t},\dots,d_{Jt})'\), \(\mathbf P_t=(P_{1t},\dots,P_{Jt})'\); the risk-free bond has price 1 and payoff \(1+r_f\). Investors \(i\in\mathcal A\) with \(N=\mathrm{card}(\mathcal A)\to\infty\) (infinitely many, to use the law of large numbers). Consumer \(i\) has income \(I_{it}\) and consumption \(C_{it}\); aggregate \(C_t=\sum_{i\in\mathcal A}C_{it}\), \(C_t=I_t+D_t\). Investors have CRRA vN-M preferences
$$U_{it}=\mathbb E\!\left[\sum_{s=0}^\infty e^{-\rho s}\frac{C_{it+s}^{1-\alpha}}{1-\alpha}\ \Big|\ \mathcal F_t\right],\qquad u(C)=\frac{C^{1-\alpha}}{1-\alpha},$$
\(\alpha>0\) 是各投资者共同的相对风险厌恶。信息流 \(\{\mathcal F_t\}\)，初始对称持有 \(\boldsymbol\theta_{i,-1}=\boldsymbol\theta_{k,-1}\)、零债券初始 \(\mathbf b_{i,-1}=\mathbf 0\)。
with \(\alpha>0\) the common relative risk aversion. Filtration \(\{\mathcal F_t\}\), symmetric initial holdings \(\boldsymbol\theta_{i,-1}=\boldsymbol\theta_{k,-1}\) and zero initial bonds \(\mathbf b_{i,-1}=\mathbf 0\).
10.1.2 Decentralized Equilibrium
每个投资者求解 (10.1)，受预算约束 (10.2)：
Each investor solves (10.1) subject to the budget constraint (10.2):
$$\max_{\boldsymbol\theta_{it},\,\mathbf b_{it}}\ \mathbb E\!\left[\sum_{s=t}^\infty e^{-\rho s}\frac{C_{is}^{1-\alpha}}{1-\alpha}\ \Big|\ \mathcal F_t\right]\tag{10.1}$$
$$\text{s.t.}\quad C_{it}=I_{it}+\boldsymbol\theta_{it-1}'(\mathbf P_t+\mathbf d_t)+\mathbf b_{it-1}-\boldsymbol\theta_{it}'\mathbf P_t-\mathbf b_{it}\cdot\mathbf B_t.\tag{10.2}$$
给定聚合红利过程 \(\{\mathbf d_t\}\)、聚合收入过程 \(\{I_t\}\) 与 \(\alpha,\rho\)，竞争均衡是价格过程 \(\{\mathbf P_t,\mathbf B_t\}\) 与策略 \(\{\boldsymbol\theta_{it},\mathbf b_{it}\}\)，使每个投资者解 (10.1)–(10.2)，且市场出清 \(\sum_i\theta_{ijt}=1\)、\(\sum_i b_{ijt}=0\)。
Given the aggregate dividend process \(\{\mathbf d_t\}\), aggregate income process \(\{I_t\}\), and \(\alpha,\rho\), a competitive equilibrium is a price process \(\{\mathbf P_t,\mathbf B_t\}\) and strategies \(\{\boldsymbol\theta_{it},\mathbf b_{it}\}\) such that each investor solves (10.1)–(10.2) and markets clear: \(\sum_i\theta_{ijt}=1\), \(\sum_i b_{ijt}=0\).
10.1.3 Conditions for the Existence of Decentralized Equilibrium
无套利（定理 4.3）保证存在严格为正的 SDF \(m_{j,s+1}>0\)（w.p.1）为所有证券定价。关键技巧——反向构造特质收入，使一个无交易 (no-trade) 均衡成立。构造个体收入
No arbitrage (Theorem 4.3) guarantees a strictly positive SDF \(m_{j,s+1}>0\) (w.p.1) pricing all securities. The key trick — reverse-engineer idiosyncratic income so that a no-trade equilibrium holds. Construct individual income
$$I_{it}=\frac1N\big(\delta_{it}C_t-D_t\big),\qquad \delta_{it}=e^{\sum_{s=1}^t\left(\eta_{is}y_s-\frac{y_s^2}{2}\right)},\qquad y_t=\sqrt{\frac{2}{\alpha^2+\alpha}}\left(\ln m_{t-1,t}+\rho+\alpha\ln\frac{C_t}{C_{t-1}}\right)^{\frac12},$$
其中 \(\eta_{it}\overset{i.i.d.}{\sim}\mathcal N(0,1)\)。\(y_t\) 是横截面特质冲击的标准差，被刻意设成恰好"撑起"目标 SDF。个体消费
with \(\eta_{it}\overset{i.i.d.}{\sim}\mathcal N(0,1)\). Here \(y_t\) is the standard deviation of the cross-sectional idiosyncratic shock, deliberately set so as to exactly "support" the target SDF. Individual consumption is
$$C_{it}=I_{it}+\frac1N D_t=\frac1N\delta_{it}C_t.$$
证明 / Proof：\(\mathbb E[\delta_{it}]=1\)、聚合一致、且私人估值 $=$ 市场估值
(1) \(\mathbb E[\delta_{it}]=1\)。 由对数正态，条件于 \(\{y_s\}\)：
(1) \(\mathbb E[\delta_{it}]=1\). By log-normality, conditioning on \(\{y_s\}\):
$$\mathbb E[\delta_{it}]=\mathbb E\!\left[e^{-\sum_s\frac{y_s^2}{2}+\frac12\sum_s y_s^2}\right]=1.$$
(2) 聚合一致。 由大数定律 \(\sum_i I_{it}=\mathbb E[\delta_{it}]C_t-D_t=C_t-D_t=I_t\) ✓。 (3) 无交易均衡。 个体边际替代率与聚合 SDF 一致，故私人估值等于市场估值 (10.3)、(10.4)：
(2) Aggregation consistent. By the LLN, \(\sum_i I_{it}=\mathbb E[\delta_{it}]C_t-D_t=C_t-D_t=I_t\) ✓. (3) No-trade equilibrium. The individual marginal rate of substitution coincides with the aggregate SDF, so the private valuation equals the market valuation (10.3), (10.4):
$$\hat p_{jt}=\mathbb E\!\left[\frac{u'(C_{i,t+1})}{u'(C_{it})}(P_{j,t+1}+d_{j,t+1})\ \Big|\ \mathcal F_t\right]=P_{jt},\qquad \hat B_{i,s}=\mathbb E[m_{t,s}\mid\mathcal F_t]=B_{i,s}.\quad\blacksquare\tag{10.3–10.4}$$
10.1.4 Summary
放弃完全市场假设，改用单条欧拉方程定出聚合 SDF；通过反向构造收入过程，让每个投资者的欧拉方程都成立、且谁都不想交易。结论：不完全市场 + 异质投资者下定价可行，特质风险通过横截面消费增长的分布（尤其偏度）进入定价。
Abandon the complete-market assumption and use a single Euler equation to pin down the aggregate SDF; by reverse-engineering the income process, every investor's Euler equation holds and no one wants to trade. Bottom line: pricing works under incomplete markets with heterogeneous investors, and idiosyncratic risk enters pricing through the distribution (especially the skewness) of cross-sectional consumption growth.
10.2 Cross-Sectional Skewness Matters: Brav, Constantinides and Géczy (2002)
核心论点：若把横截面消费增长的偏度纳入，校准的相对风险厌恶可以很小。投资者各有特质消费冲击，自己的欧拉方程 (10.5)、(10.6)：
Main argument: once the skewness of cross-sectional consumption growth is included, the calibrated relative risk aversion can be small. Each investor has idiosyncratic consumption shocks and their own Euler equation (10.5), (10.6):
$$\mathbb E\!\left[\frac{u'(C_{i,t+1})}{u'(C_{it})}R_{j,t+1}\right]=1\ \overset{\text{CRRA}}{\Longrightarrow}\ \mathbb E\!\left[\beta\,g_{i,t+1}^{-\alpha}\,R_{j,t+1}\right]=1,\qquad g_{i,t+1}=\frac{C_{i,t+1}}{C_{it}}.\tag{10.5–10.6}$$
对 \(I\) 个投资者取横截面平均，得聚合 SDF \(m^I_{t+1}=\beta\frac1I\sum_{i=1}^I g_{i,t+1}^{-\alpha}\) (10.7)。记样本平均增长 \(g_{t+1}=\frac1I\sum_i g_{i,t+1}\)，对 \(\big(\frac{g_{i,t+1}}{g_{t+1}}-1\big)\) 在 0 附近 Taylor 展开 (10.8)：
Averaging across the \(I\) investors gives the aggregate SDF \(m^I_{t+1}=\beta\frac1I\sum_{i=1}^I g_{i,t+1}^{-\alpha}\) (10.7). With the sample average growth \(g_{t+1}=\frac1I\sum_i g_{i,t+1}\), Taylor-expand in \(\big(\frac{g_{i,t+1}}{g_{t+1}}-1\big)\) around 0 (10.8):
(10.8) 的第三项与横截面消费增长的偏度有关。
- 校准： Brav 等 (2002) 用 CEX 家庭消费数据（1982–1996）。把无风险与市场收益代入 \(\mathbb E[m^I_{t+1}(R_{m,t+1}-R_{f,t+1})]=0\) 校准 \(\alpha\)。加入第三项（偏度）后，校准的 \(\alpha\approx3\) 不被拒绝，在经济上合理；去掉第三项则该结论不成立。
- 贡献： 实证检验假定不完全市场、消费风险不可保险。
- 批评： 论文设 \(\beta=1\)，更稳妥应用两个组合收益、两条方程同时解 \(\alpha,\beta\)；该校准不能同时拟合无风险利率（需用 §10.3 的 Constantinides–Ghosh）；样本量小、标准误大，不易拒绝原假设。
The third term of (10.8) is related to the skewness of cross-sectional consumption growth.
- Calibration: Brav et al. (2002) use CEX household consumption data (1982–1996). Plugging the risk-free and market returns into \(\mathbb E[m^I_{t+1}(R_{m,t+1}-R_{f,t+1})]=0\) calibrates \(\alpha\). With the third (skewness) term, the calibrated \(\alpha\approx3\) is not rejected and is economically plausible; without it the result fails.
- Contribution: the empirical test assumes incomplete markets and uninsurable consumption risk.
- Critiques: the paper sets \(\beta=1\); more robustly one should use two portfolio returns and two equations to solve \(\alpha,\beta\) jointly; the calibration cannot simultaneously fit the risk-free rate (use Constantinides–Ghosh in §10.3); the sample is small with large standard errors, so the null is hard to reject.
10.3 Countercyclical Household Consumption Risk: Constantinides and Ghosh (2017)
在不完全市场中，投资者用 Epstein–Zin 偏好，且无法对冲特质收入冲击。巧妙地构造特质冲击，使异质性只在事前 (ex-ante) 出现——事前各异质代理人的个体 SDF 退化为一个共同 SDF，均衡中无需逐个追踪，从而出现自给自足 (autarchy) 的无交易均衡。
In an incomplete market, investors have Epstein–Zin preferences and cannot hedge idiosyncratic income shocks. The idiosyncratic shocks are cleverly constructed so heterogeneity appears only ex-ante — ex-ante the individual SDFs of heterogeneous agents degenerate to a common SDF that need not be tracked agent-by-agent, yielding an autarchy (no-trade) equilibrium.
10.3.1–10.3.2 Setup and the Model
聚合消费增长 \(\Delta c_{t+1}\) 为 i.i.d. 正态（无自相关）——所以时变必须来自有清晰经济故事的特质收入。特质收入冲击 \(\delta_{i,t}\) 用 Poisson 跳跃构造 (10.9)：
Aggregate consumption growth \(\Delta c_{t+1}\) is i.i.d. normal (no autocorrelation) — so the time variation must come from idiosyncratic income with a clean economic story. The idiosyncratic income shock \(\delta_{i,t}\) is built from Poisson jumps (10.9):
$$\delta_{i,t}=\exp\!\left[\sum_{s=1}^t\Big(\big(j_{i,s}\sigma\eta_{u,s}-j_{i,s}\tfrac{\sigma^2}{2}\big)+\big(\tilde j_{i,s}\eta_{v,s}-\tilde j_{i,s}\tfrac{\sigma^2}{2}\big)\Big)\right],\qquad j_{i,s}\sim\mathrm{Poisson}(\omega_s),\ \tilde j_{i,s}\sim\mathrm{Poisson}(\tilde\omega_s).\tag{10.9}$$
其中 \(\eta\sim\mathcal N(0,1)\)，Poisson 满足 \(\mathbb P\{x=n\}=e^{-\omega}\frac{\omega^n}{n!}\)、\(\mathbb E[x]=\lambda\)。
with \(\eta\sim\mathcal N(0,1)\) and the Poisson satisfying \(\mathbb P\{x=n\}=e^{-\omega}\frac{\omega^n}{n!}\), \(\mathbb E[x]=\lambda\).
10.3.3 Autarchy Equilibrium
由大数定律可证聚合收入 \(I_t=\sum_i\delta_{it}I_d\) 良好定义（同 §10.1.3 的对数正态/Poisson 期望计算）。由于个体消费增长在条件上跨代理人同分布，个体 SDF (10.10) 退化为共同形式，并为市场定价 (10.11)：
By the LLN, aggregate income \(I_t=\sum_i\delta_{it}I_d\) is well-defined (same log-normal/Poisson expectation computation as §10.1.3). Since individual consumption growth is conditionally identically distributed across agents, the individual SDF (10.10) degenerates to a common form and prices the market (10.11):
$$P_{j,t}=\mathbb E_t\big[m_{i,t+1}D_{t+1}\big].\tag{10.11}$$
10.3.4 Countercyclical Household Consumption Risk
定义状态变量 \(x_t=\big(e^{\gamma(\gamma-1)\sigma^2/2}-1\big)\omega_t\)。Constantinides–Ghosh (2017) 限定 \(\gamma>1\)，故 \(e^{\gamma(\gamma-1)\sigma^2/2}-1>0\)，\(x_t\) 与 \(\omega_t\) 正相关；而 \(\omega_t\)（Poisson 强度）控制 \(\delta_{i,t}\) 的横截面离散度——\(x_t\) 越大，家庭消费风险越高。设红利过程
Define the state variable \(x_t=\big(e^{\gamma(\gamma-1)\sigma^2/2}-1\big)\omega_t\). Constantinides–Ghosh (2017) restrict to \(\gamma>1\), so \(e^{\gamma(\gamma-1)\sigma^2/2}-1>0\) and \(x_t\) is positively correlated with \(\omega_t\); and \(\omega_t\) (the Poisson intensity) controls the cross-sectional dispersion of \(\delta_{i,t}\) — higher \(x_t\) means higher household consumption risk. Assume the dividend process
$$\Delta d_{t+1}=\alpha_d+\beta_d\,x_t+\sigma_d\,e_{d,t+1},\qquad e_{d,t+1}\overset{i.i.d.}{\sim}\mathcal N(0,1).$$
正的股权溢价（合理 RRA 下）要求 \(\beta_d<0\)。于是高红利增长（繁荣）\(\Rightarrow\) 低 \(x_t\) \(\Rightarrow\) 低消费风险；低红利增长（衰退）\(\Rightarrow\) 高消费风险——即逆周期的家庭消费风险。
A positive equity premium (with reasonable RRA) requires \(\beta_d<0\). So high dividend growth (boom) \(\Rightarrow\) low \(x_t\) \(\Rightarrow\) low consumption risk; low dividend growth (recession) \(\Rightarrow\) high consumption risk — i.e. countercyclical household consumption risk.
10.3.5 Calibration and Discussion
用 Bansal–Yaron (2004) 的思路，把无风险利率与股权溢价的矩写成可估计参数与待校准参数的函数。结果：
- 带负偏、逆周期、持久的个体消费增长冲击的校准模型，能很好匹配无风险利率、股权溢价、市场价格-红利比、以及聚合红利与消费增长的矩。
- 模型生成顺周期的无风险利率与价格-红利比，以及逆周期的风险溢价与市场收益方差——都接近现实。
- 模型解释了规模、账面市值比、行业分组组合的横截面超额收益。
批评。 (10.9) 中 \(\delta_{i,t}\) 的设定是特设 (ad hoc)、反向工程出来的，只为得到事前同质性以保证可解，缺乏经验或理论支撑。
Using the Bansal–Yaron (2004) approach, write the moments of the risk-free rate and equity premium as functions of estimable and calibrated parameters. Results:
- The calibrated model with negatively skewed, countercyclical, persistent individual consumption-growth shocks fits well the moments of the risk-free rate, equity premium, market price-dividend ratio, and aggregate dividend and consumption growth.
- The model generates a procyclical risk-free rate and price-dividend ratio, and a countercyclical risk premium and market-return variance — all close to reality.
- The model explains the cross-section of excess returns of size-sorted, book-to-market-sorted, and industry-sorted portfolios.
Critique. The specification of \(\delta_{i,t}\) in (10.9) is ad hoc and reverse-engineered, only to obtain ex-ante homogeneity for tractability, and is not well-founded by empirical evidence or theory.
References
- Brav, A., G. M. Constantinides and C. C. Géczy (2002). Asset Pricing with Heterogeneous Consumers and Limited Participation: Empirical Evidence. Journal of Political Economy 110(4), 793–824.
- Constantinides, G. M. (1982). Intertemporal Asset Pricing with Heterogeneous Consumers and without Demand Aggregation. The Journal of Business 55(2), 253–267.
- Constantinides, G. M. and D. Duffie (1996). Asset Pricing with Heterogeneous Consumers. Journal of Political Economy 104(2), 219–240.
- Constantinides, G. M. and A. Ghosh (2017). Asset Pricing with Countercyclical Household Consumption Risk. The Journal of Finance 72(1), 415–460.