本讲导读 本讲（Toda 第 2 讲）讲经典的 Samuelson (1969) 最优消费-组合问题（另见 Hakansson (1970)、Toda (2014)）。§1 最优组合问题：1.1 模型（CRRA 可加效用、收益 i.i.d.、预算约束 \(w_{t+1}=R_{t+1}(\theta_t)(w_t-c_t)\)）；1.2 有限期解（Bellman 方程；齐次性 Lemma 1 \(V_T(w)=a_T\frac{w^{1-\gamma}}{1-\gamma}\)；Prop 2 最优组合 \(\theta^*=\arg\max\frac1{1-\gamma}\mathbb E[R(\theta)^{1-\gamma}]\)；消费率与一阶线性差分方程 \(b_T\)；期限越长消费比例越小、组合不变）；1.3 无限期解（一般 DP 问题；Lemma 3 \(V_t^*\le V_t^\infty\)；Prop 4 横截性条件 \(\limsup_T\mathbb E_t[V_T^\infty(x_T)]\le0\) 充要；应用于组合问题验证）。§2 收入波动问题：2.1 模型（CARA \(u(c)=-e^{-\gamma c}/\gamma\)、AR(1) 收入、可负消费）；2.2 解（Prop 5 = Wang (2003) 闭式解 \(c(w,y)=aw+b+dy\)，猜测验证；附注与出处）。含每讲参考文献。

1.1 模型 / 1.1 Model 时间 \(t=0,1,\dots,T\)（可能 \(T=\infty\)）。资产 \(j\in J=\{1,\dots,J\}\)（Samuelson (1969) 中 \(J=2\)，一风险一无风险）。初始财富 \(w_0>0\)，投资者最大化终身期望效用 \(\mathbb E_0\sum_{t=0}^T\beta^t\dfrac{c_t^{1-\gamma}}{1-\gamma}\)，\(\beta>0\) 贴现因子、\(\gamma>0\) 相对风险厌恶、\(c_t\) 消费（可加 CRRA 偏好）。设 \(P_t^j\) 为资产 \(j\) 每股价格、\(D_t^j\) 红利，毛收益 \(R_{t+1}^j=\dfrac{P_{t+1}^j+D_{t+1}^j}{P_t^j}\)（红利期初付、价格期末报）。收益向量 \(\mathbf R_{t+1}=[R_{t+1}^1,\dots,R_{t+1}^J]\)，简化假设 \(\{\mathbf R_t\}\) i.i.d.（但截面联合分布任意）。\(\theta_t^j\) 为投资于资产 \(j\) 的财富比例（多/空头），\(\sum_{j=1}^J\theta_t^j=1\)。组合毛收益 \(R_{t+1}(\theta)=\mathbf R_{t+1}'\theta=\sum_{j=1}^J R_{t+1}^j\theta^j\)。时序：期初收益 \(\mathbf R_t\) 实现并定下财富 \(w_t\)，再选 \(c_t,\theta_t\)。预算约束Time \(t=0,1,\dots,T\) (maybe \(T=\infty\)). Assets \(j\in J=\{1,\dots,J\}\) (in Samuelson (1969), \(J=2\), one risky and one riskless). From initial wealth \(w_0>0\), the investor maximizes lifetime expected utility \(\mathbb E_0\sum_{t=0}^T\beta^t\dfrac{c_t^{1-\gamma}}{1-\gamma}\), with \(\beta>0\) the discount factor, \(\gamma>0\) relative risk aversion, \(c_t\) consumption (additive CRRA preference). Let \(P_t^j\) be the per-share price of asset \(j\), \(D_t^j\) its dividend; the gross return is \(R_{t+1}^j=\dfrac{P_{t+1}^j+D_{t+1}^j}{P_t^j}\) (dividends paid at the beginning, prices quoted at the end). The return vector \(\mathbf R_{t+1}=[R_{t+1}^1,\dots,R_{t+1}^J]\), with the simplifying assumption \(\{\mathbf R_t\}\) i.i.d. (but the cross-sectional joint distribution is arbitrary). \(\theta_t^j\) is the fraction of wealth in asset \(j\) (long/short), \(\sum_{j=1}^J\theta_t^j=1\). The portfolio gross return is \(R_{t+1}(\theta)=\mathbf R_{t+1}'\theta=\sum_{j=1}^J R_{t+1}^j\theta^j\). Timing: at the start of \(t\), returns \(\mathbf R_t\) realize and determine wealth \(w_t\); then \(c_t,\theta_t\) are chosen. The budget constraint is

$$w_{t+1}=R_{t+1}(\theta_t)(w_t-c_t)\ge0.$$

Bellman 方程与齐次性（引理 1）/ Bellman equation and homotheticity (Lemma 1) 因收益 i.i.d.，唯一状态变量是财富。记 \(V_T(w)\) 为还剩 \(T\) 期（含当期共 \(T+1\) 期）时的值函数。\(T\ge1\) 时 Bellman 方程为Since returns are i.i.d., the only state variable is wealth. Let \(V_T(w)\) be the value function when \(T\) periods remain (the investor has \(T+1\) periods to live including the current one). For \(T\ge1\) the Bellman equation is

$$V_T(w)=\max_{c,\theta}\left\{\frac{c^{1-\gamma}}{1-\gamma}+\beta\mathbb E[V_{T-1}(w')]\,\middle|\,w'=R(\theta)(w-c)\right\}.$$

\(T=0\) 时只能消费全部财富，\(V_0(w)=\dfrac{w^{1-\gamma}}{1-\gamma}\)。引理 1：对每个 \(T\)，存在 \(a_T\) 使 \(V_T(w)=a_T\dfrac{w^{1-\gamma}}{1-\gamma}\)。For \(T=0\) the investor can only consume his wealth, \(V_0(w)=\dfrac{w^{1-\gamma}}{1-\gamma}\). Lemma 1: for each \(T\), there exists \(a_T\) such that \(V_T(w)=a_T\dfrac{w^{1-\gamma}}{1-\gamma}\).

引理 1 证明（齐次性）/ Proof of Lemma 1 (homotheticity) 设 \(c_0,\dots,c_T\) 是从财富 \(w\) 出发的最优消费、值 \(V_T(w)\)。由预算约束线性，若初始财富为 \(\lambda w\)（\(\lambda>0\)），则 \(\lambda c_0,\dots,\lambda c_T\) 可行；由效用齐次性，相应终身效用为 \(\lambda^{1-\gamma}V_T(w)\)。又从 \(\lambda w\) 出发的最优值是 \(V_T(\lambda w)\)，故Let \(c_0,\dots,c_T\) be the optimal consumption from wealth \(w\) with value \(V_T(w)\). By the linearity of the budget constraint, if initial wealth is \(\lambda w\) (\(\lambda>0\)), then \(\lambda c_0,\dots,\lambda c_T\) is feasible; by the homotheticity of utility, its lifetime utility is \(\lambda^{1-\gamma}V_T(w)\). Since the optimal value from \(\lambda w\) is \(V_T(\lambda w)\),

$$\lambda^{1-\gamma}V_T(w)\le V_T(\lambda w).\tag{1}$$

在 (1) 中令 \(w'=\lambda w\)、\(\lambda'=1/\lambda\) 得 \((\lambda')^{1-\gamma}V_T(w')\ge V_T(\lambda'w')\) (2)；去撇并结合 (1) 得 \(\lambda^{1-\gamma}V_T(w)\ge V_T(\lambda w)\) (3)。故等号成立。取 \(\lambda=1/w\)：Setting \(w'=\lambda w\), \(\lambda'=1/\lambda\) in (1) gives \((\lambda')^{1-\gamma}V_T(w')\ge V_T(\lambda'w')\) (2); dropping primes and combining with (1) gives \(\lambda^{1-\gamma}V_T(w)\ge V_T(\lambda w)\) (3). So equality holds. Taking \(\lambda=1/w\):

$$V_T(w)=V_T(1)w^{1-\gamma}\equiv a_T\frac{w^{1-\gamma}}{1-\gamma}.\quad\blacksquare$$

命题 2（最优组合）与定义 ρ / Proposition 2 (optimal portfolio) and \(\rho\) 代入引理 1 与预算约束，Bellman 化为 \(a_T\dfrac{w^{1-\gamma}}{1-\gamma}=\max_c\left\{\dfrac{c^{1-\gamma}}{1-\gamma}+\beta a_{T-1}(w-c)^{1-\gamma}\max_\theta\dfrac1{1-\gamma}\mathbb E[R(\theta)^{1-\gamma}]\right\}\) (4)。命题 2：最优组合 \(\theta^*\in\arg\max_\theta\dfrac1{1-\gamma}\mathbb E[R(\theta)^{1-\gamma}]\)，与财富、期限无关。定义 \(\rho=\mathbb E[R(\theta^*)^{1-\gamma}]^{\frac1{1-\gamma}}=\max_\theta\mathbb E[R(\theta)^{1-\gamma}]^{\frac1{1-\gamma}}\)（第二个等号用 \(x\mapsto\frac{x^{1-\gamma}}{1-\gamma}\) 单调）。Substituting Lemma 1 and the budget constraint, the Bellman equation becomes \(a_T\dfrac{w^{1-\gamma}}{1-\gamma}=\max_c\left\{\dfrac{c^{1-\gamma}}{1-\gamma}+\beta a_{T-1}(w-c)^{1-\gamma}\max_\theta\dfrac1{1-\gamma}\mathbb E[R(\theta)^{1-\gamma}]\right\}\) (4). Proposition 2: the optimal portfolio is \(\theta^*\in\arg\max_\theta\dfrac1{1-\gamma}\mathbb E[R(\theta)^{1-\gamma}]\), independent of wealth and horizon. Define \(\rho=\mathbb E[R(\theta^*)^{1-\gamma}]^{\frac1{1-\gamma}}=\max_\theta\mathbb E[R(\theta)^{1-\gamma}]^{\frac1{1-\gamma}}\) (the second equality uses that \(x\mapsto\frac{x^{1-\gamma}}{1-\gamma}\) is monotone).

消费率与 \(b_T\) 差分方程 / The consumption rate and the \(b_T\) difference equation 代入 \(\rho\)，(4) 变为 \(a_T\dfrac{w^{1-\gamma}}{1-\gamma}=\max_c\left\{\dfrac{c^{1-\gamma}}{1-\gamma}+\beta a_{T-1}(w-c)^{1-\gamma}\dfrac{\rho^{1-\gamma}}{1-\gamma}\right\}\) (5)。这是关于 \(c\) 的一元凹最大化，一阶条件 \(c^{-\gamma}-\beta a_{T-1}\rho^{1-\gamma}(w-c)^{-\gamma}=0\) (6)，解得 \(c=\dfrac{w}{1+(\beta a_{T-1}\rho^{1-\gamma})^{\frac1\gamma}}\) (7)。代回 (5) 并整理：\(a_T^{1/\gamma}=1+(\beta\rho^{1-\gamma})^{1/\gamma}a_{T-1}^{1/\gamma}\)。令 \(b_T=a_T^{1/\gamma}\) 得一阶线性差分方程 \(b_T=1+(\beta\rho^{1-\gamma})^{1/\gamma}b_{T-1}\)，\(b_0=1\)，故 \(b_T=\sum_{k=0}^T(\beta\rho^{1-\gamma})^{k/\gamma}=\dfrac{1-(\beta\rho^{1-\gamma})^{\frac{T+1}\gamma}}{1-(\beta\rho^{1-\gamma})^{\frac1\gamma}}\)。由 (7) 最优消费规则 \(c=\dfrac w{b_T}=\dfrac{1-(\beta\rho^{1-\gamma})^{\frac1\gamma}}{1-(\beta\rho^{1-\gamma})^{\frac{T+1}\gamma}}w\)。因 \(1=b_1期限越长，应消费的财富比例 \(1/b_T\) 越小；但组合在各期相同（i.i.d. 假设下）。\(\blacksquare\)Substituting \(\rho\), (5) becomes \(a_T\dfrac{w^{1-\gamma}}{1-\gamma}=\max_c\left\{\dfrac{c^{1-\gamma}}{1-\gamma}+\beta a_{T-1}(w-c)^{1-\gamma}\dfrac{\rho^{1-\gamma}}{1-\gamma}\right\}\) (5). This is a one-variable concave maximization in \(c\), with first-order condition \(c^{-\gamma}-\beta a_{T-1}\rho^{1-\gamma}(w-c)^{-\gamma}=0\) (6), giving \(c=\dfrac{w}{1+(\beta a_{T-1}\rho^{1-\gamma})^{\frac1\gamma}}\) (7). Back-substituting into (5) and simplifying: \(a_T^{1/\gamma}=1+(\beta\rho^{1-\gamma})^{1/\gamma}a_{T-1}^{1/\gamma}\). Letting \(b_T=a_T^{1/\gamma}\) gives the first-order linear difference equation \(b_T=1+(\beta\rho^{1-\gamma})^{1/\gamma}b_{T-1}\), \(b_0=1\), so \(b_T=\sum_{k=0}^T(\beta\rho^{1-\gamma})^{k/\gamma}=\dfrac{1-(\beta\rho^{1-\gamma})^{\frac{T+1}\gamma}}{1-(\beta\rho^{1-\gamma})^{\frac1\gamma}}\). By (7) the optimal consumption rule is \(c=\dfrac w{b_T}=\dfrac{1-(\beta\rho^{1-\gamma})^{\frac1\gamma}}{1-(\beta\rho^{1-\gamma})^{\frac{T+1}\gamma}}w\). Since \(1=b_1the longer the horizon, the smaller the consumed fraction \(1/b_T\); but the portfolio is the same over time (under i.i.d.). \(\blacksquare\)

一般动态规划问题 / A general dynamic programming problem 无限期解大体相同：令有限期解中 \(T\to\infty\)，若 \(\beta\rho^{1-\gamma}<1\)，则系数 \(b=\dfrac1{1-(\beta\rho^{1-\gamma})^{1/\gamma}}\)、消费率 \(c/w=1-(\beta\rho^{1-\gamma})^{1/\gamma}\)。此猜测正确，但有技术细节。考虑更一般问题 \(\max_{\{c_t\}}\mathbb E_0\sum_{t=0}^\infty f_t(c_t,x_t)\) s.t. \(c_t\in\Gamma_t(x_t)\)、\(x_{t+1}=g_{t+1}(c_t,x_t)\)（\(x_t\) 状态、\(c_t\) 控制、\(f_t\) 流效用）。定义 \(T\) 期值函数 \(V_t^T(x)=\sup_{\{c_{t+s}\}_{s=0}^{T-1}}\mathbb E_t\sum_{s=0}^{T-1}f_{t+s}(c_{t+s},x_{t+s})\)，无限期值函数 \(V_t^\infty(x)=\limsup_{T\to\infty}V_t^T(x)\)，真值函数 \(V_t^*(x)=\sup_{\{c_{t+s}\}_{s=0}^\infty}\mathbb E_t\sum_{s=0}^\infty f_{t+s}(c_{t+s},x_{t+s})\)。The infinite-horizon solution is basically the same: letting \(T\to\infty\) in the finite-horizon answer, if \(\beta\rho^{1-\gamma}<1\) the coefficient is \(b=\dfrac1{1-(\beta\rho^{1-\gamma})^{1/\gamma}}\) and the consumption rate \(c/w=1-(\beta\rho^{1-\gamma})^{1/\gamma}\). This guess is correct but has technical subtleties. Consider the more general problem \(\max_{\{c_t\}}\mathbb E_0\sum_{t=0}^\infty f_t(c_t,x_t)\) s.t. \(c_t\in\Gamma_t(x_t)\), \(x_{t+1}=g_{t+1}(c_t,x_t)\) (\(x_t\) state, \(c_t\) control, \(f_t\) flow utility). Define the \(T\)-period value function \(V_t^T(x)=\sup_{\{c_{t+s}\}_{s=0}^{T-1}}\mathbb E_t\sum_{s=0}^{T-1}f_{t+s}(c_{t+s},x_{t+s})\), the infinite-horizon value function \(V_t^\infty(x)=\limsup_{T\to\infty}V_t^T(x)\), and the true value function \(V_t^*(x)=\sup_{\{c_{t+s}\}_{s=0}^\infty}\mathbb E_t\sum_{s=0}^\infty f_{t+s}(c_{t+s},x_{t+s})\).

引理 3 与命题 4（横截性条件）/ Lemma 3 and Proposition 4 (transversality) 引理 3：恒有 \(V_t^*(x)\le V_t^\infty(x)\)。称计划 \(\{c_{t+s}\}\) 递归最优，若它解 \(V_{t+s}^\infty(x_{t+s})=\max_{c\in\Gamma_{t+s}(x_{t+s})}\{f_{t+s}(c,x_{t+s})+\mathbb E_{t+s}V_{t+s+1}^\infty(g_{t+s+1}(c,x_{t+s}))\}\)。命题 4：\(V_t^*(x)=V_t^\infty(x)\) 当且仅当横截性条件 \(\limsup_{T\to\infty}\mathbb E_t[V_T^\infty(x_T)]\le0\) 成立（\(x_T\) 来自递归最优策略）。Lemma 3: always \(V_t^*(x)\le V_t^\infty(x)\). A plan \(\{c_{t+s}\}\) is recursively optimal if it solves \(V_{t+s}^\infty(x_{t+s})=\max_{c\in\Gamma_{t+s}(x_{t+s})}\{f_{t+s}(c,x_{t+s})+\mathbb E_{t+s}V_{t+s+1}^\infty(g_{t+s+1}(c,x_{t+s}))\}\). Proposition 4: \(V_t^*(x)=V_t^\infty(x)\) if and only if the transversality condition \(\limsup_{T\to\infty}\mathbb E_t[V_T^\infty(x_T)]\le0\) holds (\(x_T\) obtained from a recursively optimal policy).

引理 3、命题 4 证明 / Proof of Lemma 3 and Proposition 4 引理 3：任取可行计划 \(\{c_{t+s}\}\)，对任意 \(T\)，\(\mathbb E_t\sum_{s=0}^{T-1}f_{t+s}(c_{t+s},x_{t+s})\le V_t^T(x)\)。令 \(T\to\infty\)：\(\mathbb E_t\sum_{s=0}^\infty f_{t+s}=\lim_T\mathbb E_t\sum_{s=0}^{T-1}f_{t+s}\le\limsup_T V_t^T(x)=V_t^\infty(x)\)。对左端取 \(\sup\) 得 \(V_t^*(x)\le V_t^\infty(x)\)。\(\blacksquare\) 命题 4：取递归最优策略，由定义 \(V_t^\infty(x)=\mathbb E_t\sum_{s=0}^{T-1}f_{t+s}(c_{t+s},x_{t+s})+\mathbb E_t[V_T^\infty(x_T)]\) (8)。令 \(T\to\infty\) 并用引理得 \(V_t^*(x)\ge V_t^\infty(x)-\limsup_T\mathbb E_t[V_T^\infty(x_T)]\)，故 \(\limsup_T\mathbb E_t[V_T^\infty(x_T)]\ge0\) 恒成立；若 \(\le0\) 则 $=0$，上述不等式皆成等式，得 \(V_t^*=V_t^\infty\)。反之若 \(V_t^*=V_t^\infty\)，递归最优策略也最优，在 (8) 中令 \(T\to\infty\) 得 \(\limsup_T\mathbb E_t[V_T^\infty(x_T)]=\lim_T\mathbb E_t[V_T^*(x_T)]=0\le0\)。\(\blacksquare\)Lemma 3: take any feasible plan \(\{c_{t+s}\}\); for any \(T\), \(\mathbb E_t\sum_{s=0}^{T-1}f_{t+s}(c_{t+s},x_{t+s})\le V_t^T(x)\). Letting \(T\to\infty\): \(\mathbb E_t\sum_{s=0}^\infty f_{t+s}=\lim_T\mathbb E_t\sum_{s=0}^{T-1}f_{t+s}\le\limsup_T V_t^T(x)=V_t^\infty(x)\). Taking the sup of the LHS gives \(V_t^*(x)\le V_t^\infty(x)\). \(\blacksquare\) Proposition 4: take a recursively optimal policy; by definition \(V_t^\infty(x)=\mathbb E_t\sum_{s=0}^{T-1}f_{t+s}(c_{t+s},x_{t+s})+\mathbb E_t[V_T^\infty(x_T)]\) (8). Letting \(T\to\infty\) and using the Lemma, \(V_t^*(x)\ge V_t^\infty(x)-\limsup_T\mathbb E_t[V_T^\infty(x_T)]\), so \(\limsup_T\mathbb E_t[V_T^\infty(x_T)]\ge0\) always; if \(\le0\) then $=0$ and all inequalities become equalities, so \(V_t^*=V_t^\infty\). Conversely, if \(V_t^*=V_t^\infty\), the recursively optimal policy is optimal, and letting \(T\to\infty\) in (8) gives \(\limsup_T\mathbb E_t[V_T^\infty(x_T)]=\lim_T\mathbb E_t[V_T^*(x_T)]=0\le0\). \(\blacksquare\)

应用于组合问题：验证横截性 / Application: verifying transversality for the portfolio problem 设 \(\beta\rho^{1-\gamma}<1\)，无限期值函数 \(V^\infty(w)=a\dfrac{w^{1-\gamma}}{1-\gamma}\)，\(a^{1/\gamma}=b=\dfrac1{1-(\beta\rho^{1-\gamma})^{1/\gamma}}>0\)。贴现横截性条件 \(\limsup_T\mathbb E_0[\beta^T V_T^\infty(w_T)]\le0\)：\(\gamma>1\) 时 \(V_T^\infty(w_T)\le0\) 平凡成立；\(0<\gamma<1\) 时由预算约束 \(w_{t+1}=R_{t+1}(\theta^*)(w_t-c_t)\le R_{t+1}(\theta^*)w_t\)，故 \(w_T\le w_0\prod_{t=1}^T R_t(\theta^*)\)，取 \((1-\gamma)\) 次幂取期望 \(\mathbb E_0[w_T^{1-\gamma}]\le w_0^{1-\gamma}\mathbb E[R(\theta^*)^{1-\gamma}]^T=w_0^{1-\gamma}\rho^{(1-\gamma)T}\)，于是 \(\mathbb E_0[\beta^T V_T^\infty(w_T)]\le\dfrac{aw_0^{1-\gamma}}{1-\gamma}(\beta\rho^{1-\gamma})^T\to0\)（因 \(\beta\rho^{1-\gamma}<1\)）。\(\blacksquare\)Suppose \(\beta\rho^{1-\gamma}<1\); the infinite-horizon value function is \(V^\infty(w)=a\dfrac{w^{1-\gamma}}{1-\gamma}\) with \(a^{1/\gamma}=b=\dfrac1{1-(\beta\rho^{1-\gamma})^{1/\gamma}}>0\). The discounted transversality condition \(\limsup_T\mathbb E_0[\beta^T V_T^\infty(w_T)]\le0\): for \(\gamma>1\), \(V_T^\infty(w_T)\le0\) trivially; for \(0<\gamma<1\), the budget constraint \(w_{t+1}=R_{t+1}(\theta^*)(w_t-c_t)\le R_{t+1}(\theta^*)w_t\) gives \(w_T\le w_0\prod_{t=1}^T R_t(\theta^*)\), and taking the \((1-\gamma)\)-th power and expectations, \(\mathbb E_0[w_T^{1-\gamma}]\le w_0^{1-\gamma}\mathbb E[R(\theta^*)^{1-\gamma}]^T=w_0^{1-\gamma}\rho^{(1-\gamma)T}\), so \(\mathbb E_0[\beta^T V_T^\infty(w_T)]\le\dfrac{aw_0^{1-\gamma}}{1-\gamma}(\beta\rho^{1-\gamma})^T\to0\) (since \(\beta\rho^{1-\gamma}<1\)). \(\blacksquare\)

2.1 模型 / 2.1 Model 乘性风险（如随机资产收益）下用 CRRA 效用方便；可加风险（如随机劳动收入）下 CARA 效用更方便。考虑可加 CARA 效用 \(\mathbb E_0\sum_{t=0}^\infty\beta^t u(c_t)\) (9)，\(u(c)=-e^{-\gamma c}/\gamma\)（绝对风险厌恶 \(\gamma>0\)）。可按毛无风险利率 \(R>1\) 借贷。收入风险 AR(1)：\(y_{t+1}=\rho y_t+\varepsilon_{t+1}\) (10)，\(0\le\rho<1\)、\(\varepsilon\) i.i.d.（无失一般性不含常数项）。记 \(w_t\) 为期初金融财富（不含当期收入），预算约束 \(w_{t+1}=R(w_t-c_t+y_t)\)。Bellman 方程 \(V(w,y)=\max_c\{u(c)+\beta\mathbb E[V(R(w-c+y),y')]\mid y'=\rho y+\varepsilon\}\) (11)。CARA 定义在全实轴，故假设消费可为负。Under multiplicative risk (e.g. random asset returns) CRRA is convenient; under additive risk (e.g. random labor income) CARA is more convenient. Consider additive CARA utility \(\mathbb E_0\sum_{t=0}^\infty\beta^t u(c_t)\) (9), with \(u(c)=-e^{-\gamma c}/\gamma\) (absolute risk aversion \(\gamma>0\)). The agent can borrow/save at gross risk-free rate \(R>1\). Income risk is AR(1): \(y_{t+1}=\rho y_t+\varepsilon_{t+1}\) (10), \(0\le\rho<1\), \(\varepsilon\) i.i.d. (WLOG no constant term). Let \(w_t\) be financial wealth at the start of \(t\) (excluding current income); the budget constraint is \(w_{t+1}=R(w_t-c_t+y_t)\). The Bellman equation is \(V(w,y)=\max_c\{u(c)+\beta\mathbb E[V(R(w-c+y),y')]\mid y'=\rho y+\varepsilon\}\) (11). CARA is defined on the whole real line, so consumption may be negative.

命题 5（Wang 2003）/ Proposition 5 (Wang, 2003) 值函数与最优消费规则为 \(V(w,y)=-\dfrac1{\gamma a}e^{-\gamma(aw+b+dy)}\) (12a)，\(c(w,y)=aw+b+dy\) (12b)，其中 \(a=1-1/R\)、\(b=\dfrac1{\gamma(1-R)}\log\beta R\,\mathbb E[e^{-\gamma\frac{R-1}{R-\rho}\varepsilon}]\)、\(d=\dfrac{R-1}{R-\rho}\)。The value function and optimal consumption rule are \(V(w,y)=-\dfrac1{\gamma a}e^{-\gamma(aw+b+dy)}\) (12a), \(c(w,y)=aw+b+dy\) (12b), where \(a=1-1/R\), \(b=\dfrac1{\gamma(1-R)}\log\beta R\,\mathbb E[e^{-\gamma\frac{R-1}{R-\rho}\varepsilon}]\), \(d=\dfrac{R-1}{R-\rho}\).

命题 5 证明（猜测验证）/ Proof of Proposition 5 (guess-and-verify) 代 (12a) 入 Bellman (11)：Substitute (12a) into Bellman (11):

$$-\frac1{\gamma a}e^{-\gamma(aw+b+dy)}=\max_c\left\{-\frac1\gamma e^{-\gamma c}-\frac\beta{\gamma a}\mathbb E[e^{-\gamma(aR(w-c+y)+b+dy')}]\right\}.\tag{13}$$

对 \(c\) 的一阶条件为The first-order condition in \(c\) is

$$e^{-\gamma c}-\beta R\,\mathbb E[e^{-\gamma(aR(w-c+y)+b+dy')}]=0.\tag{14}$$

代 (14) 入 (13) 得Substituting (14) into (13) gives

$$-\frac1{\gamma a}e^{-\gamma(aw+b+dy)}=-\frac1{\gamma a}(a+1/R)e^{-\gamma c}.\tag{15}$$

比较系数知 \(a=1-1/R\)、\(c=aw+b+dy\) 时平凡成立。此时 \(aR(w-c+y)=aw+(1-R)b+(1-R)(d-1)y\)，(14) 变为Comparing coefficients, this holds trivially when \(a=1-1/R\) and \(c=aw+b+dy\). Then \(aR(w-c+y)=aw+(1-R)b+(1-R)(d-1)y\), and (14) becomes

$$e^{-\gamma dy}=\beta R\,\mathbb E[e^{-\gamma((1-R)b+(1-R)(d-1)y+d(\rho y+\varepsilon))}].\tag{16}$$

比较 \(y\) 的系数：\(d=(1-R)(d-1)+\rho d\iff d=\dfrac{R-1}{R-\rho}\)。代回 (16)：Comparing the coefficients of \(y\): \(d=(1-R)(d-1)+\rho d\iff d=\dfrac{R-1}{R-\rho}\). Back-substituting into (16):

$$1=\beta R\,\mathbb E[e^{-\gamma((1-R)b+\frac{R-1}{R-\rho}\varepsilon)}]\iff b=\frac1{\gamma(1-R)}\log\beta R\,\mathbb E[e^{-\gamma\frac{R-1}{R-\rho}\varepsilon}].\quad\blacksquare$$

附注 / Remarks (i) \(\mathbb E[e^{-\gamma\frac{R-1}{R-\rho}\varepsilon}]\) 是 \(\varepsilon\) 的矩母函数 \(M_\varepsilon(s)=\mathbb E[e^{s\varepsilon}]\) 在 \(s=-\gamma\frac{R-1}{R-\rho}\) 处取值。(ii) 该收入波动问题可嵌入一般均衡（Huggett (1993) 模型的一个版本）；Toda (2017) 在 VAR(1) 收入下证明可能有多重均衡（AR(1) 情形唯一）。(iii) Toda (2017) 给出一个收入风险上升反而改善福利的例子。(iv) 横截性条件的证明见 Toda (2017) 附录。(i) \(\mathbb E[e^{-\gamma\frac{R-1}{R-\rho}\varepsilon}]\) is the moment generating function \(M_\varepsilon(s)=\mathbb E[e^{s\varepsilon}]\) of \(\varepsilon\) evaluated at \(s=-\gamma\frac{R-1}{R-\rho}\). (ii) This income fluctuation problem can be embedded into general equilibrium (a version of the Huggett (1993) model); Toda (2017) shows multiple equilibria are possible under VAR(1) income (unique in the AR(1) case). (iii) Toda (2017) gives an example where increasing income risk is welfare-improving. (iv) For a proof of the transversality condition, see the appendix of Toda (2017).