3. General Equilibrium Framework: Complete Market
本章仍是静态的：在一个两期、有限状态的完全市场 (complete market) 中，用基础物品（如消费品）作计价单位给状态或有证券定价。核心对象是状态价格 \(\Pi(s)\) 与状态价格密度 \(\pi(s)=\Pi(s)/\mathbf P\{s\}\)——后者正是随机贴现因子 (SDF)。由此得到三种等价定价表示（状态价格、期望、风险中性测度），再在均衡中讨论加总 (aggregation) 与风险分担 (risk sharing)：当所有人都是 CRRA 偏好时，存在与财富分布无关的代表性投资者，且任意两人的消费比在所有状态下恒定。
This chapter is still static: in a two-period, finite-state complete market, we price state-contingent securities in units of a numeraire (e.g. the consumption good). The central objects are the state price \(\Pi(s)\) and the state-price density \(\pi(s)=\Pi(s)/\mathbf P\{s\}\) — the latter being exactly the stochastic discount factor (SDF). This yields three equivalent pricing representations (state prices, expectation, risk-neutral measure); in equilibrium we then study aggregation and risk sharing: when everyone has CRRA preferences, a representative agent independent of the wealth distribution exists, and any two agents' consumption ratio is constant across all states.
3.1 Setup
考虑两期模型，\(t=0\) 与 \(t=1\)：
- \(t=0\) 没有不确定性；\(t=1\) 有有限个可能状态 \(\mathcal S=\{s_1,s_2,\dots,s_N\}\)。
- 状态 \(s\) 的（客观）概率记为 \(\mathbf P\{s\}\)，满足 \(\mathbf P\{s\}\ge 0\) 且 \(\sum_{s\in\mathcal S}\mathbf P\{s\}=1\)。
- 记 \(\Pi(s)\) 为 \(t=0\) 时、对 \(t=1\) 状态 \(s\) 支付一单位消费品的或有索取权 (state-contingent claim) 的价格，即 Arrow–Debreu 状态价格。
市场完全：每个状态都有对应的 Arrow–Debreu 证券，因此任意支付都可被复制和定价。
Consider a two-period model, \(t=0\) and \(t=1\):
- At \(t=0\) there is no uncertainty; at \(t=1\) there is a finite set of possible states \(\mathcal S=\{s_1,s_2,\dots,s_N\}\).
- The (objective) probability of state \(s\) is \(\mathbf P\{s\}\), with \(\mathbf P\{s\}\ge 0\) and \(\sum_{s\in\mathcal S}\mathbf P\{s\}=1\).
- Let \(\Pi(s)\) be the \(t=0\) price of a state-contingent claim that pays one unit of the consumption good in state \(s\) at \(t=1\) — i.e. the Arrow–Debreu state price.
The market is complete: every state has its Arrow–Debreu security, so any payoff can be replicated and priced.
3.2 Expectation Representation and Risk Neutral Pricing
设证券 \(X\) 在状态 \(s\) 支付 \(X(s)\)，其 \(t=0\) 价格记为 \(p(X)\)。按状态价格逐状态加总：
Let a security \(X\) generate payoff \(X(s)\) in state \(s\), with \(t=0\) price \(p(X)\). Summing over states with the state prices:
$$p(X)=\sum_{s\in\mathcal S}\Pi(s)\,X(s). \tag{3.1}$$
3.2.1 Expectation Representation: State Price Density Is Pricing Kernel
定义状态价格密度 (state-price density) \(\pi(s)\equiv\dfrac{\Pi(s)}{\mathbf P\{s\}}\)，则 (3.1) 可等价改写为期望形式：
Define the state-price density \(\pi(s)\equiv\dfrac{\Pi(s)}{\mathbf P\{s\}}\). Then (3.1) can be equivalently rewritten as an expectation:
也就是 (3.2)–(3.3)：
That is, (3.2)–(3.3):
$$p(X)=\sum_{s\in\mathcal S}\mathbf P\{s\}\,\pi(s)\,X(s) \tag{3.2}$$
$$\phantom{p(X)}=\mathbb E\big[\pi(s)\,X(s)\big]. \tag{3.3}$$
其中 \(\pi(s)\) 可解释为 \(t=1\) 状态 \(s\) 下一单位消费品折算到 \(t=0\) 的价格（\(s\) 表示它依赖状态），这正是随机贴现因子 (SDF)。
注 3.1（Remark 3.1）。 概率测度 \(\mathbf P\{\cdot\}\) 是状态的物理 / 客观 (physical / objective) 概率分布。
注 3.2（Remark 3.2）。 状态价格密度 \(\pi(s)\) 是每单位概率、每个状态下一单位计价物的价格；它是随机的，即定价核 (pricing kernel)。
Here \(\pi(s)\) can be interpreted as the \(t=0\) price of one unit of the consumption good delivered in state \(s\) at \(t=1\) (\(s\) indicates state dependence) — exactly the stochastic discount factor (SDF).
Remark 3.1. The probability measure \(\mathbf P\{\cdot\}\) is the physical / objective probability distribution of states.
Remark 3.2. The state-price density \(\pi(s)\) is the price of one unit of numeraire per unit of probability in each state; it is stochastic — the pricing kernel.
3.2.2 Risk Neutral Pricing
风险中性概率测度 (risk-neutral measure) \(\mathbf P^\star\{\cdot\}\) 的定义是：用它计算的"期望"（记为 \(\mathbb E^\star[\cdot]\)）再按无风险利率贴现，就给出公平价格：
The risk-neutral measure \(\mathbf P^\star\{\cdot\}\) is defined so that the "expectation" under it (denoted \(\mathbb E^\star[\cdot]\)), discounted at the risk-free rate, yields the fair price:
$$p(X)=\frac{\mathbb E^\star[X(s)]}{1+r_f}\qquad\text{or}\qquad p(X)=\frac{\sum_{s\in\mathcal S}\mathbf P^\star\{s\}\,X(s)}{1+r_f}.$$
与 (3.2) 比较即可读出物理测度、状态价格密度与风险中性测度三者的关系，并得到无风险贴现因子等于全部状态价格之和 (3.4)。
Comparing with (3.2) pins down the relation among the physical measure, the state-price density, and the risk-neutral measure, and yields the riskless discount factor as the sum of all state prices (3.4).
证明 / Proof：风险中性测度与 \(1/(1+r_f)=\sum_s\Pi(s)\)
逐状态比较风险中性定价 \(p(X)=\sum_s\mathbf P^\star\{s\}X(s)/(1+r_f)\) 与 (3.2) \(p(X)=\sum_s\mathbf P\{s\}\pi(s)X(s)\)，对任意支付 \(X\) 成立要求每个状态的系数相等：
Matching the risk-neutral price \(p(X)=\sum_s\mathbf P^\star\{s\}X(s)/(1+r_f)\) with (3.2) \(p(X)=\sum_s\mathbf P\{s\}\pi(s)X(s)\) for every payoff \(X\) requires the state-by-state coefficients to agree:
$$\frac{\mathbf P^\star\{s\}}{1+r_f}=\mathbf P\{s\}\,\pi(s) \;\Longrightarrow\; \frac{\sum_{s\in\mathcal S}\mathbf P^\star\{s\}}{1+r_f}=\sum_{s\in\mathcal S}\mathbf P\{s\}\,\pi(s) \;\Longrightarrow\; \frac{1}{1+r_f}=\sum_{s\in\mathcal S}\Pi(s). \tag{3.4}$$
最后一步用了 \(\sum_s\mathbf P^\star\{s\}=1\) 以及 \(\mathbf P\{s\}\pi(s)=\Pi(s)\)。(3.4) 显然正确：\(\sum_s\Pi(s)\) 保证 \(t=1\) 无论何种状态都得到一单位消费品，因此在 \(t=0\) 的公平价格就是 \(1/(1+r_f)\)。\(\blacksquare\)
The last step uses \(\sum_s\mathbf P^\star\{s\}=1\) and \(\mathbf P\{s\}\pi(s)=\Pi(s)\). Equation (3.4) is exactly right: \(\sum_s\Pi(s)\) buys one unit of the good at \(t=1\) in every state, so it should be fairly priced at \(1/(1+r_f)\) at \(t=0\). \(\blacksquare\)
三种定价表示是同一回事。 状态价格 (3.1)、SDF 期望 (3.3)、风险中性贴现，本质上都是同一个无套利定价，只是把"概率 × 折现"这件事在物理测度 \(\mathbf P\) 和风险中性测度 \(\mathbf P^\star\) 之间重新打包。完全市场保证了 \(\Pi(s)\)、\(\pi(s)\)、\(\mathbf P^\star\) 都唯一存在。The three pricing representations are the same thing. State prices (3.1), the SDF expectation (3.3), and risk-neutral discounting are all one no-arbitrage pricing rule, merely repackaging "probability × discounting" between the physical measure \(\mathbf P\) and the risk-neutral measure \(\mathbf P^\star\). Market completeness guarantees that \(\Pi(s)\), \(\pi(s)\), and \(\mathbf P^\star\) all exist and are unique.
3.3 Aggregation and Risk Sharing
现在引入一个有限的异质投资者 (heterogeneous agents) 集合 \(\mathcal J\)：
- 投资者 \(j\in\mathcal J\) 的 \(t=0\) 消费记为 \(c_0^j\)，\(t=1\) 状态 \(s\) 消费记为 \(c_1^j(s)\)。
- 每人有严格递增、凹的效用函数 \(u^j(\cdot)\)（两期之间不变，但可因人而异）。
- \(\beta\) 为所有人共同的效用贴现因子。
- 投资者 \(j\) 拥有禀赋 \(e_0^j\)（\(t=0\)）与 \(e_1^j(s)\)（\(t=1\) 状态 \(s\)）。
每个投资者求解（预算约束用状态价格给跨期消费定价）：
Now introduce a finite set \(\mathcal J\) of heterogeneous agents:
- Agent \(j\in\mathcal J\) has \(t=0\) consumption \(c_0^j\) and \(t=1\) state-\(s\) consumption \(c_1^j(s)\).
- Each has a strictly increasing, concave utility \(u^j(\cdot)\) (constant across the two periods, but heterogeneous across agents).
- \(\beta\) is the utility discount factor common to all agents.
- Agent \(j\) has endowment \(e_0^j\) (at \(t=0\)) and \(e_1^j(s)\) (state \(s\) at \(t=1\)).
Each agent solves (the budget constraint prices intertemporal consumption with the state prices):
$$\max_{c_0^j,\;\{c_1^j(s)\}_{s\in\mathcal S}}\; u^j\!\big(c_0^j\big)+\mathbb E\big[\beta\,u^j\!\big(c_1^j(s)\big)\big] \quad\text{s.t.}\quad c_0^j+\mathbb E\big[\pi(s)\,c_1^j(s)\big]\le e_0^j+\mathbb E\big[\pi(s)\,e_1^j(s)\big].$$
对整个经济，可行性约束 (feasibility) 要求各期消费加总等于总禀赋：
For the whole economy, feasibility requires aggregate consumption to equal the aggregate endowment in each period:
$$\sum_{j\in\mathcal J}e_0^j=e_0^{\text{agg}},\qquad \sum_{j\in\mathcal J}e_1^j(s)=e_1^{\text{agg}}(s)\ \ \text{for }\forall s\in\mathcal S. \tag{3.5}$$
First-order conditions
推导 / Derivation：拉格朗日与一阶条件 (3.6)–(3.7)
对投资者 \(j\) 写出拉格朗日函数（乘子 \(\lambda^j\) 对应其预算约束）：
Write agent \(j\)'s Lagrangian (multiplier \(\lambda^j\) on the budget constraint):
$$\mathcal L=u^j\!\big(c_0^j\big)+\mathbb E\big[\beta\,u^j\!\big(c_1^j(s)\big)\big]+\lambda^j\Big[e_0^j+\mathbb E\big[\pi(s)\,e_1^j(s)\big]-\big(c_0^j+\mathbb E\big[\pi(s)\,c_1^j(s)\big]\big)\Big].$$
对 \(c_0^j\) 求一阶条件：
The FOC w.r.t. \(c_0^j\):
$$u'\!\big(c_0^j\big)=\lambda^j. \tag{3.6}$$
对每个状态的 \(c_1^j(s)\) 求一阶条件 \(\mathbb E[\beta u^{j\prime}(c_1^j(s))]=\lambda^j\,\mathbb E[\pi(s)]\)，逐状态展开（用 \(\pi(s)\equiv\Pi(s)/\mathbf P\{s\}\)）：
The FOC w.r.t. \(c_1^j(s)\) for each state, \(\mathbb E[\beta u^{j\prime}(c_1^j(s))]=\lambda^j\,\mathbb E[\pi(s)]\), written state by state (using \(\pi(s)\equiv\Pi(s)/\mathbf P\{s\}\)):
$$\beta\,\mathbf P\{s\}\,u^{j\prime}\!\big(c_1^j(s)\big)=\lambda^j\,\Pi(s) \;\Longrightarrow\; u^{j\prime}\!\big(c_1^j(s)\big)=\lambda^j\,\frac{\Pi(s)}{\beta\,\mathbf P\{s\}}. \tag{3.7}$$
(3.7) 只通过 \(\lambda^j\) 依赖于个人 \(j\)，而 \(\lambda^j\) 代表其财富分布 (wealth distribution)。\(\blacksquare\)
Equation (3.7) depends on the individual \(j\) only through \(\lambda^j\), which represents the wealth distribution. \(\blacksquare\)
Aggregation
设 \(u^{j\prime}(\cdot)\) 可逆、逆函数为 \((u^{j\prime})^{-1}(\cdot)\)。用总禀赋条件 \(\sum_j c_1^j(s)=e_1^{\text{agg}}(s)\) 并代入 (3.7) 反解：
Suppose \(u^{j\prime}(\cdot)\) is invertible with inverse \((u^{j\prime})^{-1}(\cdot)\). Use the aggregate condition \(\sum_j c_1^j(s)=e_1^{\text{agg}}(s)\) and invert (3.7):
$$e_1^{\text{agg}}(s)=\sum_{j\in\mathcal J}c_1^j(s)=\sum_{j\in\mathcal J}\big(u^{j\prime}\big)^{-1}\!\Big(\lambda^j\,\frac{\Pi(s)}{\beta\,\mathbf P\{s\}}\Big). \tag{3.8}$$
由 (3.8) 可论证：当 \(u^{j\prime\prime}(\cdot)<0\)（凹）时，总存在一个总效用 / 代表性效用 (aggregate utility) 函数 \(u^{\text{agg}}(\cdot)\)，满足
From (3.8) one can argue that whenever \(u^{j\prime\prime}(\cdot)<0\) (concavity) there always exists an aggregate (representative) utility \(u^{\text{agg}}(\cdot)\) such that
$$u^{\text{agg}\prime}\!\big(e_1^{\text{agg}}(s)\big)=\lambda^{\text{agg}}\,\Pi(s)\quad\text{for }\forall s\in\mathcal S.$$
理由：
- 因为对每个 \(j\) 都有 \(u^{j\prime\prime}(\cdot)<0\)，(3.8) 右边关于 \(\Pi(s)\) 单调递减（对任意 \(s\)）。
- 单调性 \(\Rightarrow\) 存在这样的函数 \(u^{\text{agg}}(\cdot)\)。
- 注意 \(u^{\text{agg}}(\cdot)\) 仍可能依赖于偏好 \(u^j\) 与财富分布 \(\lambda_j\) 的异质性。
当 \(u^{\text{agg}}(\cdot)\) 不依赖于偏好 \(u^j\) 与财富分布 \(\lambda^j\) 时，我们说该经济具有加总性 (aggregation)：可以用单一代表性投资者刻画市场。
Reasoning:
- Since \(u^{j\prime\prime}(\cdot)<0\) for every \(j\), the RHS of (3.8) is decreasing in \(\Pi(s)\) for any \(s\).
- Monotonicity \(\Rightarrow\) existence of such a function \(u^{\text{agg}}(\cdot)\).
- Note \(u^{\text{agg}}(\cdot)\) may still depend on the heterogeneity in \(u^j\) or \(\lambda_j\).
When \(u^{\text{agg}}(\cdot)\) is independent of preferences \(u^j\) and the wealth distribution \(\lambda^j\), we say the economy has aggregation: a single representative agent describes the market.
为什么要加总？ 如果存在代表性投资者，那么市场价格只取决于总禀赋（总消费），而与"财富如何在人群中分配"无关。这让我们能把多投资者经济简化成单投资者的消费基础定价（回到第 1 章的 C-CAPM 框架）。一般情况下做不到，但下面的 CRRA 偏好可以。Why aggregation matters. If a representative agent exists, market prices depend only on the aggregate endowment (aggregate consumption), not on how wealth is distributed across agents. This collapses a multi-agent economy into single-agent consumption-based pricing (back to the C-CAPM of Chapter 1). In general it fails, but the CRRA preferences below deliver it.
Aggregation under CRRA preferences
用特定的效用函数形式即可"消掉"个人依赖。设所有人都是 CRRA：\(u^j(c)=\dfrac{c^{1-\gamma}}{1-\gamma}\)，于是 \(u^{j\prime}(c)=c^{-\gamma}\)。代入 (3.7)：
A special functional form removes the individual dependence. Let everyone be CRRA: \(u^j(c)=\dfrac{c^{1-\gamma}}{1-\gamma}\), so \(u^{j\prime}(c)=c^{-\gamma}\). Substituting into (3.7):
$$c_1^j(s)^{-\gamma}=\lambda^j\,\frac{\Pi(s)}{\beta\,\mathbf P\{s\}}\;\Longrightarrow\; c_1^j(s)=\big(\lambda^j\big)^{-1/\gamma}\Big(\tfrac{1}{\beta\,\mathbf P\{s\}}\Big)^{-1/\gamma}\Pi(s)^{-1/\gamma}.$$
推导 / Derivation：CRRA 下的代表性投资者 (3.9)–(3.10)
把上式代回 (3.8)，并提出与 \(j\) 无关的公因子：
Substitute back into (3.8) and factor out the \(j\)-independent terms:
$$e_1^{\text{agg}}(s)=\sum_{j\in\mathcal J}c_1^j(s)=\Big(\tfrac{1}{\beta\,\mathbf P\{s\}}\Big)^{-1/\gamma}\Pi(s)^{-1/\gamma}\Big[\sum_{j\in\mathcal J}\big(\lambda^j\big)^{-1/\gamma}\Big],$$
$$\Longrightarrow\quad e_1^{\text{agg}}(s)^{-\gamma}=\Big(\tfrac{1}{\beta\,\mathbf P\{s\}}\Big)\Pi(s)\Big[\sum_{j\in\mathcal J}\big(\lambda^j\big)^{-1/\gamma}\Big]^{-\gamma}. \tag{3.9}$$
再用 \(t=0\) 的一阶条件 (3.6)：\(u'(c_0^j)=\lambda^j\Rightarrow (c_0^j)^{-\gamma}=\lambda^j\Rightarrow c_0^j=(\lambda^j)^{-1/\gamma}\)。结合可行性 (3.5)：
Use the \(t=0\) FOC (3.6): \(u'(c_0^j)=\lambda^j\Rightarrow (c_0^j)^{-\gamma}=\lambda^j\Rightarrow c_0^j=(\lambda^j)^{-1/\gamma}\). With feasibility (3.5):
$$e_0^{\text{agg}}=\sum_{j\in\mathcal J}e_0^j=\sum_{j\in\mathcal J}c_0^j\;\Longrightarrow\; e_0^{\text{agg}}=\sum_{j\in\mathcal J}\big(\lambda^j\big)^{-1/\gamma}. \tag{3.10}$$
把 (3.10) 代入 (3.9)，方括号项正好被 \((e_0^{\text{agg}})^{-\gamma}\) 替换：
Plug (3.10) into (3.9); the bracketed term becomes exactly \((e_0^{\text{agg}})^{-\gamma}\):
$$\underbrace{e_1^{\text{agg}}(s)^{-\gamma}}_{\equiv\,u^{\text{agg}\prime}(s)}=\Big(\tfrac{1}{\beta\,\mathbf P\{s\}}\Big)\Pi(s)\,\big(e_0^{\text{agg}}\big)^{-\gamma},$$
即 \(u^{\text{agg}\prime}(s)=\big(\tfrac{1}{\beta\mathbf P\{s\}}\big)\Pi(s)(e_0^{\text{agg}})^{-\gamma}\)，不依赖 \(\lambda_j\)。所以 CRRA 偏好下确实得到了加总结果——代表性投资者也是 CRRA，其"消费"就是总禀赋。\(\blacksquare\)
i.e. \(u^{\text{agg}\prime}(s)=\big(\tfrac{1}{\beta\mathbf P\{s\}}\big)\Pi(s)(e_0^{\text{agg}})^{-\gamma}\), which does not depend on \(\lambda_j\). So with CRRA we do get aggregation — the representative agent is also CRRA, and its "consumption" is the aggregate endowment. \(\blacksquare\)
Risk sharing
从 CRRA 也能立即看到风险分担 (risk sharing)。对两个投资者 \(i\) 与 \(j\) 分别应用 (3.7)：
CRRA also immediately reveals risk sharing. Apply (3.7) to two agents \(i\) and \(j\):
$$c_1^i(s)^{-\gamma}=\lambda^i\,\frac{\Pi(s)}{\beta\,\mathbf P\{s\}}, \tag{3.11}$$
$$c_1^j(s)^{-\gamma}=\lambda^j\,\frac{\Pi(s)}{\beta\,\mathbf P\{s\}}. \tag{3.12}$$
推导 / Derivation：消费比恒定 (3.13)
(3.11) 除以 (3.12)，状态价格项 \(\Pi(s)/(\beta\mathbf P\{s\})\) 完全抵消：
Divide (3.11) by (3.12); the state-price term \(\Pi(s)/(\beta\mathbf P\{s\})\) cancels entirely:
$$\frac{c_1^i(s)^{-\gamma}}{c_1^j(s)^{-\gamma}}=\frac{\lambda^i}{\lambda^j}\;\Longrightarrow\; \frac{c_1^i(s)}{c_1^j(s)}=\Big(\frac{\lambda^i}{\lambda^j}\Big)^{-1/\gamma}. \tag{3.13}$$
(3.13) 左边依赖状态 \(s\)，右边却与状态无关。要让等式对所有 \(s\) 成立，左边的消费比就必须在 \(\forall s\in\mathcal S\) 上恒定——这正是风险分担。\(\blacksquare\)
The LHS of (3.13) is state-dependent, while the RHS is state-independent. For the equality to hold for all \(s\), the consumption ratio on the LHS must be constant across \(\forall s\in\mathcal S\) — exactly risk sharing. \(\blacksquare\)
风险分担的含义。 不论实现哪个状态，投资者 \(j\) 的消费相对投资者 \(i\) 的比值都一样；这对所有 \(j\in\mathcal J\) 都成立。于是唯一改变每个人消费的，是总禀赋的水平：总禀赋不变，则无论实现哪个状态，任何人的消费计划都不变。换言之，可分散的个体风险被完全分担掉，大家只共同承担总量风险 (aggregate risk)。What risk sharing means. Regardless of the realized state, agent \(j\)'s consumption keeps the same ratio to agent \(i\)'s — and this holds for all \(j\in\mathcal J\). So the only thing that moves anyone's consumption is the level of the aggregate endowment: if the aggregate endowment is unchanged, no one's consumption plan changes regardless of the realized state. Diversifiable idiosyncratic risk is fully shared away; agents jointly bear only aggregate risk.
References
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