33. Aggregation and Risk Sharing
本附录章证明去中心化均衡 (decentralized equilibrium) 与代表性代理人均衡 (representative agent equilibrium) 的等价性，并推导聚合 (aggregation) 与风险分担 (risk sharing) 两个总量概念。代表性代理人解两个问题：(1) 组内配置 (within-allocation)——给定总消费计划与权重 \(\{\lambda_j\}\)，把每种商品分给所有代理人以最大化代表性效用 \(v\) (33.3)；(2) 总消费计划 (aggregate plan)——给定价格选总消费最大化 \(v\) (33.6)。由包络定理证 \(\lambda_j^{-1}u_n^j=\gamma p_n\) (33.9)，与去中心化的 \(u_n^j=\lambda_j p_n\) (33.2) 对比可知：只要代表性代理人的权重 \(\lambda_j\) 取为去中心化均衡中代理人 \(j\) 财富的影子价值 \(\lambda_j^\star\)、且价格相同，二均衡完全重合。风险分担（单一商品、跨期跨状态）结论：\(c_t^j=g_t(c_t)\) 且 \(g_t\) 递增（everyone 的消费与总消费同向，与状态/财富分布无关）；且 \(g_t'\) 随风险容忍度递增——风险容忍度越高的代理人承担越多风险（消费随总消费波动越剧烈）。完全市场下所有（各期各状态）消费决策都在 \(t=0\) 做出，财富分布仅在 \(t=0\) 进入问题（Remark 33.3）。
This appendix chapter proves the equivalence of the decentralized equilibrium and the representative agent equilibrium, and derives the two aggregate notions of aggregation and risk sharing. The representative agent solves two problems: (1) within-allocation — given the aggregate consumption plan and weights \(\{\lambda_j\}\), allocate each good to all agents to maximize the representative utility \(v\) (33.3); (2) aggregate plan — given prices, choose aggregate consumption to maximize \(v\) (33.6). By the envelope theorem, \(\lambda_j^{-1}u_n^j=\gamma p_n\) (33.9); comparing with the decentralized \(u_n^j=\lambda_j p_n\) (33.2) shows: as long as the representative agent's weight \(\lambda_j\) is the shadow value \(\lambda_j^\star\) of agent \(j\)'s wealth in the decentralized equilibrium and prices are the same, the two equilibria coincide exactly. The risk sharing result (single good, across periods and states): \(c_t^j=g_t(c_t)\) with \(g_t\) increasing (everyone's consumption moves with aggregate consumption, regardless of state/wealth distribution); and \(g_t'\) is increasing in risk tolerance — the higher-risk-tolerance agent bears more risk (consumption fluctuates more severely with aggregate consumption). Under complete markets all consumption decisions (across periods and states) are made at \(t=0\), so the wealth distribution enters the problem only at \(t=0\) (Remark 33.3).
33.1 Setup
设 \(N\) 个消费品 \(c_1,\dots,c_N\)、价格 \(p_1,\dots,p_N\)。代理人 \(j\in\mathcal J\) 有效用 \(u^j\)、消费 \(c_1^j,\dots,c_N^j\)、禀赋 \(e_1^j,\dots,e_N^j\)。设各 \(u^j\) 递增凹。线性组合 \(\sum_{j\in\mathcal J}\lambda_j^{-1}u^j(c_1^j,\dots,c_N^j)\) 仍凹，其最大化结果 \(v(c_1,\dots,c_N)\equiv\max_{\{c_1^j,\dots,c_N^j\}_{j\in\mathcal J}}\sum_{j\in\mathcal J}\lambda_j^{-1}u^j(c_1^j,\dots,c_N^j)\) 仍凹。效用凹 + 预算约束凸 → 一阶条件充分、解唯一（对下文所有最大化问题）。
Remark 33.1 此一般设定下效用 \(u^j(c_1^j,\dots,c_N^j)\) 不要求消费品可分（separability）。故可把这些商品视为时间序列商品——代表性代理人存在性无需假设效用的时间可分性。
33.2 Decentralized Equilibrium
考虑代理人 \(j\) 的去中心化决策（\(j\) 非代表性代理人）。其问题 (33.1)：\(\max_{c_1^j,\dots,c_N^j}u^j(c_1^j,\dots,c_N^j)\) s.t. \(\sum_{n=1}^N p_n c_n^j=\sum_{n=1}^N p_n e_n^j\)。拉格朗日（乘子 \(\lambda_j\)）\(\mathcal L=u^j+\lambda_j(\sum p_n e_n^j-\sum p_n c_n^j)\)，f.o.c. 对 \(c_n^j\) (33.2)：
Let there be \(N\) consumption goods \(c_1,\dots,c_N\), prices \(p_1,\dots,p_N\). Agent \(j\in\mathcal J\) has utility \(u^j\), consumption \(c_1^j,\dots,c_N^j\), endowment \(e_1^j,\dots,e_N^j\). Assume each \(u^j\) is increasing and concave. The linear combination \(\sum_{j\in\mathcal J}\lambda_j^{-1}u^j(c_1^j,\dots,c_N^j)\) is still concave, and its maximized result \(v(c_1,\dots,c_N)\equiv\max_{\{c_1^j,\dots,c_N^j\}_{j\in\mathcal J}}\sum_{j\in\mathcal J}\lambda_j^{-1}u^j(c_1^j,\dots,c_N^j)\) is still concave. The concavity of utility and convexity of the budget constraint imply the sufficiency of the f.o.c. and uniqueness of the solution (for all maximization problems below).
Remark 33.1 This general setting with utility \(u^j(c_1^j,\dots,c_N^j)\) does not require separability of consumption goods. So we can think of these goods as time-series goods, and we don't need to assume time separability of utility for the existence of a representative agent.
33.2 Decentralized Equilibrium
Consider agent \(j\)'s decentralized decision-making (\(j\) is not the representative agent). The problem (33.1): \(\max_{c_1^j,\dots,c_N^j}u^j(c_1^j,\dots,c_N^j)\) s.t. \(\sum_{n=1}^N p_n c_n^j=\sum_{n=1}^N p_n e_n^j\). The Lagrangian (multiplier \(\lambda_j\)) \(\mathcal L=u^j+\lambda_j(\sum p_n e_n^j-\sum p_n c_n^j)\), with f.o.c. w.r.t. \(c_n^j\) (33.2):
$$u_n^j(c_1^{j\star},\dots,c_N^{j\star})=\lambda_j p_n,\qquad u_n^j\equiv\frac{\partial u^j(c_1^j,\dots,c_N^j)}{\partial c_n^j}\tag{33.2}$$
33.3 Representative Agent's Two Problems
代表性代理人分两部分解：(1) 总消费计划问题；(2) 组内配置问题。
组内配置问题：给定总消费计划 \(\{c_1,\dots,c_N\}\) 与权重 \(\{\lambda_j\}_{j\in\mathcal J}\)，把每种商品分给所有代理人 \(j\) 以最大化代表性代理人效用。定义新效用 \(v\) (33.3) s.t. (33.4)：
The representative agent solves in two parts: (1) the aggregate consumption plan problem; (2) the within-allocation problem.
Within-allocation problem: given the aggregate consumption plan \(\{c_1,\dots,c_N\}\) and weights \(\{\lambda_j\}_{j\in\mathcal J}\), allocate each good to all agents \(j\) to maximize the representative agent's utility. Define the new utility \(v\) (33.3) s.t. (33.4):
$$v(c_1,\dots,c_N)\equiv\max_{\{c_1^j,\dots,c_N^j\}_{j\in\mathcal J}}\sum_{j\in\mathcal J}\lambda_j^{-1}u^j(c_1^j,\dots,c_N^j)\quad\text{s.t. }\sum_{j\in\mathcal J}c_n^j=c_n,\ n=1,\dots,N\tag{33.3–4}$$
拉格朗日（乘子 \(\mu_n\)）\(\mathcal L=\sum_{j}\lambda_j^{-1}u^j+\sum_{n=1}^N\mu_n(c_n-\sum_j c_n^j)\)，f.o.c. 对 \(c_n^j\) (33.5)：\(\lambda_j^{-1}u_n^j(c_1^j,\dots,c_N^j)=\mu_n\)。
总消费计划问题：给定价格 \(\{p_1,\dots,p_N\}\) 与组内配置定义的 \(v(c_1,\dots,c_N)\)，代表性代理人选总消费计划 \(\{c_1,\dots,c_N\}\) 最大化效用 (33.6)：\(\max_{c_1,\dots,c_N}v(c_1,\dots,c_N)\) s.t. \(\sum_{n=1}^N p_n c_n=\sum_{n=1}^N p_n e_n\)。拉格朗日（乘子 \(\gamma\)），f.o.c. 对 \(c_n\) (33.7)：\(v_n(c_1,\dots,c_N)=\gamma p_n\)，\(v_n\equiv\frac{\partial v}{\partial c_n}\)。
33.4 Equivalence of Decentralized and Representative Agent's Equilibrium
欲证去中心化均衡 \(\{c_1^{j\star},\dots,c_N^{j\star}\}_{j\in\mathcal J}\)（给定 \(\{p_1,\dots,p_N\}\)）与代表性代理人均衡（由其两问题给定 \(\{\lambda_j^\star\}\)、\(\{p_n\}\)、(33.2) 钉定）重合。
33.4.1 Integrate the Two Problems
由 \(v\) 的定义 (33.3) 把约束 (33.4) 代入目标，使 \(v\) 无约束；由包络定理 (33.8)：\(v_n(c_1,\dots,c_N)=\lambda_j^{-1}u_n^j(c_1^j,\dots,c_N^j)\)（此重排不针对个体 \(j\)，故对任意 \(j\) 成立；亦可由 \(v\)=拉格朗日、对 \(c_n\) 求导 + 包络得 \(v_n=\mu_n\)，再由 (33.5) 得 (33.8)）。结合 (33.8) 与 (33.7) 得 (33.9)：
The Lagrangian (multiplier \(\mu_n\)) \(\mathcal L=\sum_{j}\lambda_j^{-1}u^j+\sum_{n=1}^N\mu_n(c_n-\sum_j c_n^j)\), with f.o.c. w.r.t. \(c_n^j\) (33.5): \(\lambda_j^{-1}u_n^j(c_1^j,\dots,c_N^j)=\mu_n\).
Aggregate consumption plan problem: given prices \(\{p_1,\dots,p_N\}\) and the \(v(c_1,\dots,c_N)\) defined by within-allocation, the representative agent chooses the aggregate plan \(\{c_1,\dots,c_N\}\) to maximize utility (33.6): \(\max_{c_1,\dots,c_N}v(c_1,\dots,c_N)\) s.t. \(\sum_{n=1}^N p_n c_n=\sum_{n=1}^N p_n e_n\). The Lagrangian (multiplier \(\gamma\)), with f.o.c. w.r.t. \(c_n\) (33.7): \(v_n(c_1,\dots,c_N)=\gamma p_n\), \(v_n\equiv\frac{\partial v}{\partial c_n}\).
33.4 Equivalence of Decentralized and Representative Agent's Equilibrium
We want to show the decentralized equilibrium \(\{c_1^{j\star},\dots,c_N^{j\star}\}_{j\in\mathcal J}\) (given \(\{p_1,\dots,p_N\}\)) coincides with the representative agent's equilibrium (pinned down by his two problems given \(\{\lambda_j^\star\}\), \(\{p_n\}\), and (33.2)).
33.4.1 Integrate the Two Problems
By the definition of \(v\) (33.3), substitute the constraints (33.4) into the objective so \(v\) is unconstrained; by the envelope theorem (33.8): \(v_n(c_1,\dots,c_N)=\lambda_j^{-1}u_n^j(c_1^j,\dots,c_N^j)\) (this rearrangement is not specific to individual \(j\), so it holds for any \(j\); alternatively, with \(v\) = the Lagrangian, differentiating w.r.t. \(c_n\) + envelope gives \(v_n=\mu_n\), then (33.5) gives (33.8)). Combining (33.8) with (33.7) gives (33.9):
$$\lambda_j^{-1}u_n^j(c_1^j,\dots,c_N^j)=\gamma p_n\tag{33.9}$$
(33.9) 是 \(N\times J\) 个条件，钉定代表性代理人两问题的解 \(\{\tilde c_1,\dots,\tilde c_N\}_j\)（给定 \(\{\lambda_j\}_{j\in\mathcal J}\)、\(\{p_1,\dots,p_N\}\)）。
33.4.2 Compare the Two Outcomes
对比 (33.9) 与 (33.2) 立即可见： - 若代表性代理人的权重取 \(\lambda_j=\lambda_j^\star\)（\(\lambda_j^\star\) 为去中心化问题中代理人 \(j\) 最优处的拉格朗日乘子，即 \(j\) 财富的影子价值）； - 且价格向量 \(\{p_1,\dots,p_N\}\) 在两问题中相同； - 则去中心化问题各 \(j\) 的解 \(\{c_1^{j\star},\dots,c_N^{j\star}\}\) 恰解代表性代理人问题中 \(\gamma=1\) 的 \(N\times J\) 条件 (33.9)； - 且去中心化预算约束 (33.1) 与代表性代理人两问题的预算约束 (33.4)/(33.6) 一致（(33.1) 与 (33.4) 合起来蕴含 (33.6)）。
直觉：组内配置使代表性代理人先想好给定总消费如何最优分配各商品；给定该分配规则（一阶优化），代表性代理人再优化各商品总消费水平、最终钉定整体解。去中心化中各代理人面对价格各自优化使市场对各商品出清；只要价格在两世界相同、且权重 \(\lambda_j\) 取等于影子价值 \(\lambda_j^\star\)，则去中心化与中心化在同一世界以同一目标优化，结果必等价。
Remark 33.2 / 33.3 33.2：构造上，若各个体皆有 von Neumann-Morgenstern (vN-M) 效用，则代表性代理人也有 vN-M 效用。 33.3：市场完全时，所有（各期各状态）消费决策都在 \(t=0\) 做出，故财富分布（\(\lambda_j\)）只在 \(t=0\) 进入问题；条件财富分布（\(t>0\)）不作为状态变量进入问题，因其已在 \(t=0\) 的计算中。
33.5 Aggregate Notions
33.5.1 Aggregation
若所有代理人有同位 (homothetic) 效用、且其禀赋与总禀赋成比例，则相对价格不依赖权重 \(\{\lambda_j\}\)，即相对价格不依赖财富（禀赋）分布。这称聚合 (aggregation)。（详见 He 2019b §2.2.5。）
33.5.2 Risk Sharing
把一般记号推广到只有一个商品、但处于不同期不同状态的情形，即 \(\{c_1,\dots,c_N\}\) 现对应 \(\{c_t(s^t):s^t\in\mathcal S\}_{t=1}^T\)，\(s^t\) 为 \(t\) 期状态空间 \(\mathcal S\) 中的实现状态。个体 \(j\) 的 \(\{c_1^j,\dots,c_N^j\}\) 对应 \(\{c_t^j(s^t):s^t\in\mathcal S\}_{t=1}^T\)。状态的概率分布 \(\mathbf P_t(s^t)\in[0,1]\)、\(\sum_{s^t\in\mathcal S}\mathbf P_t(s^t)=1\) \(\forall t\)。额外假设：所有代理人满足期望效用；所有代理人同贴现因子 \(\beta\)、同信念 \(\{\mathbf P_t(s^t)\}\)。于是把不同期不同状态的同一商品视为静态一期问题中的不同商品，由 (33.8) 得 (33.10)：
(33.9) is a set of \(N\times J\) conditions, pinning down the solution \(\{\tilde c_1,\dots,\tilde c_N\}_j\) of the representative agent's two problems (given \(\{\lambda_j\}_{j\in\mathcal J}\), \(\{p_1,\dots,p_N\}\)).
33.4.2 Compare the Two Outcomes
Comparing (33.9) with (33.2), we immediately see: - if the representative agent's weights are \(\lambda_j=\lambda_j^\star\) (\(\lambda_j^\star\) being the Lagrange multiplier at agent \(j\)'s optimality in the decentralized problem, i.e. the shadow value of \(j\)'s wealth); - and the price vector \(\{p_1,\dots,p_N\}\) is the same in both problems; - then the decentralized solutions \(\{c_1^{j\star},\dots,c_N^{j\star}\}\) for all \(j\) exactly solve the \(N\times J\) conditions (33.9) in the representative agent's problem with \(\gamma=1\); - and the decentralized budget constraint (33.1) is consistent with the representative agent's budget constraints (33.4)/(33.6) (in the sense that (33.1) and (33.4) together imply (33.6)).
Intuition: within-allocation has the representative agent first figure out how to optimally distribute each good given total consumption; given that distribution rule (first-stage optimization), the representative agent then optimizes the aggregate consumption level of each good to finally pin down the entire solution. In the decentralized world each agent solves his own optimization given prices so the market clears for each good. As long as prices are the same in both worlds and the weights \(\lambda_j\) equal the shadow values \(\lambda_j^\star\), the decentralized and centralized worlds optimize the same objective, so the results must be equivalent.
Remark 33.2 / 33.3 33.2: By construction, if individual agents each have von Neumann-Morgenstern (vN-M) utility, then the representative agent has vN-M utility. 33.3: When the market is complete, all consumption decisions (across periods and states) are made at \(t=0\), so the wealth distribution (\(\lambda_j\)) enters the problem only at \(t=0\); the conditional wealth distribution (\(t>0\)) does not enter as a state variable, because it's already in the calculation at \(t=0\).
33.5 Aggregate Notions
33.5.1 Aggregation
If all agents have homothetic utility and their endowments are proportional to the aggregate endowment, then relative prices don't depend on the weights \(\{\lambda_j\}\), i.e. relative prices don't depend on the wealth (endowment) distribution. This is called aggregation. (See He 2019b §2.2.5.)
33.5.2 Risk Sharing
Extend the general notation to one good in different states of different periods, so \(\{c_1,\dots,c_N\}\) now corresponds to \(\{c_t(s^t):s^t\in\mathcal S\}_{t=1}^T\), \(s^t\) the realized state from state space \(\mathcal S\) in period \(t\). Individual \(j\)'s \(\{c_1^j,\dots,c_N^j\}\) corresponds to \(\{c_t^j(s^t):s^t\in\mathcal S\}_{t=1}^T\). The probability distribution of states \(\mathbf P_t(s^t)\in[0,1]\), \(\sum_{s^t\in\mathcal S}\mathbf P_t(s^t)=1\) \(\forall t\). Additional assumptions: all agents satisfy expected utility; all agents have the same discount factor \(\beta\) and the same beliefs \(\{\mathbf P_t(s^t)\}\). So we treat the same good in different states of different periods as different goods in a static one-period problem, and by (33.8) get (33.10):
$$\lambda_j^{-1}u_t^j(c_t^j(s^t))=v_t(c_t(s^t))\tag{33.10}$$
有效价格 \(p_n(s^t)\equiv\frac{p_n}{\mathbf P_t(s^t)}\)（替换 (33.2)/(33.7)/(33.9) 中的 \(p_n\)；此一致性对此前用共同价格的论证关键）。定义函数 \(g_t(c_t(s^t))\equiv(u_t^j)^{-1}(\lambda_j v_t(c_t(s^t)))\) (33.11)，则 \(c_t^j(s^t)=g_t(c_t(s^t))\)。\(g_t\) 递增 → 每个人的消费都与总消费同向，与状态及财富分布无关——称风险分担 (risk sharing)。
33.5.3 Higher Risk Tolerance Agent Bears Higher Risk
把 (33.11) 代入 (33.10) 得 (33.12)：\(\lambda_j^{-1}u_t^j(g_t(c_t(s^t)))=v_t(c_t(s^t))\)。对 \(c_t(s^t)\) 求导得 (33.13)：\(\lambda_j^{-1}u_{t,t}^j(g_t(c_t(s^t)))g_t'(c_t(s^t))=v_{t,t}(c_t(s^t))\)。(33.13) 除以 (33.12) 得 (33.14)：\(\frac{u_{t,t}^j(c_t^j(s^t))}{u_t^j(g_t(c_t(s^t)))}g_t'(c_t(s^t))=\frac{v_{t,t}(c_t(s^t))}{v_t(c_t(s^t))}\)，其 LHS 不依赖 \(j\)（因 RHS 不依赖 \(j\)）。定义个体 \(j\) 关于消费 \(c_t^j(s^t)\) 的风险容忍度（如 (32.4)）(33.15)：\(\tau_j(g_t(c_t(s^t)))\equiv\frac{-u_t^j(c_t^j(s^t))}{u_{t,t}^j(c_t^j(s^t))}\)。代入 (33.14) 重写为：
The effective price \(p_n(s^t)\equiv\frac{p_n}{\mathbf P_t(s^t)}\) (replacing \(p_n\) in (33.2)/(33.7)/(33.9); this consistency is crucial for the earlier argument that used a common price). Define the function \(g_t(c_t(s^t))\equiv(u_t^j)^{-1}(\lambda_j v_t(c_t(s^t)))\) (33.11), then \(c_t^j(s^t)=g_t(c_t(s^t))\). \(g_t\) is increasing → everyone's consumption moves in the same direction as aggregate consumption, regardless of state and wealth distribution — called risk sharing.
33.5.3 Higher Risk Tolerance Agent Bears Higher Risk
Plugging (33.11) into (33.10) gives (33.12): \(\lambda_j^{-1}u_t^j(g_t(c_t(s^t)))=v_t(c_t(s^t))\). Differentiating w.r.t. \(c_t(s^t)\) gives (33.13): \(\lambda_j^{-1}u_{t,t}^j(g_t(c_t(s^t)))g_t'(c_t(s^t))=v_{t,t}(c_t(s^t))\). Dividing (33.13) by (33.12) gives (33.14): \(\frac{u_{t,t}^j(c_t^j(s^t))}{u_t^j(g_t(c_t(s^t)))}g_t'(c_t(s^t))=\frac{v_{t,t}(c_t(s^t))}{v_t(c_t(s^t))}\), whose LHS is independent of \(j\) (since the RHS is independent of \(j\)). Define individual \(j\)'s risk tolerance w.r.t. consumption \(c_t^j(s^t)\) (as in (32.4)) (33.15): \(\tau_j(g_t(c_t(s^t)))\equiv\frac{-u_t^j(c_t^j(s^t))}{u_{t,t}^j(c_t^j(s^t))}\). Substituting into (33.14) rewrites it as:
$$g_t'(c_t(s^t))=\underbrace{\frac{-v_{t,t}(c_t(s^t))}{v_t(c_t(s^t))}}_{>0}\tau_j(g_t(c_t(s^t)))\tag{}$$
故（正）导数 \(g_t'\) 随风险容忍度 \(\tau_j\) 递增。这意味着：在最优配置下，风险容忍度越高的个体，其消费随总消费波动越剧烈，即承担越多风险。此结果非常直观。
So the (positive) derivative \(g_t'\) is increasing in risk tolerance \(\tau_j\). This means: at the optimal allocation, the individual with higher risk tolerance has consumption fluctuate more severely with aggregate consumption, i.e. bears more risk. This result is very intuitive.
References
- He, X. (2019b). Microeconomic Theory Notes by Xindi He.
- Constantinides, G. M. (1982). Intertemporal asset pricing with heterogeneous consumers and without demand aggregation. Journal of Business 55(2), 253–267.
- Rubinstein, M. (1974). An aggregation theorem for securities markets. Journal of Financial Economics 1(3), 225–244.