本讲导读 本讲（Toda 第 6 讲）讲不完全市场动态一般均衡模型中的横截面分布。§1 引入（Saito (1998) 模型：均衡 \(r=\mu-\gamma\sigma^2\)、财富满足 Gibrat 比例增长律 \(dw/w=g\,dt+v\,dB\)、对数财富为带漂移布朗运动 ⟹ 横截面对数正态）。§2 Fokker-Planck 方程（Kolmogorov 前向方程）：由扩散 \(dX=g\,dt+v\,dB\) 导出密度 PDE \(\frac{\partial p}{\partial t}=-\frac\partial{\partial x}(gp)+\frac12\frac{\partial^2}{\partial x^2}(v^2p)\)；含生灭项；平稳密度 \(p(x)=\frac1{v^2}\exp(\int\frac{2g}{v^2}dx)\)。§3 例：Ornstein-Uhlenbeck（正态）、\(\mathrm{sgn}\) 漂移（Laplace）、指数时间布朗运动（非对称 Laplace）、带漂移几何布朗运动（逆 Gamma ⟹ 幂律 \(P(X>x)\sim x^{-\alpha}\)，\(\alpha=1-2g/v^2\)）。§4 幂律与 Laplace 分布：幂律/双幂律、矩存在性、双 Pareto 与双 Pareto-对数正态 (dPlN)、Laplace/normal-Laplace、定理 1（几何和的极限是 Laplace）。含每讲参考文献。

Gibrat 比例增长律与对数正态财富 / Gibrat's law and lognormal wealth 考虑 Saito (1998) 模型，为简化设无股市、无总量冲击（仅含特质冲击的私人股权）。均衡无风险利率 \(r=\mu-\gamma\sigma^2\)（\(\mu\) 私人股权期望收益、\(\gamma\) 相对风险厌恶、\(\sigma\) 特质波动），最优消费率 \(m=\beta\varepsilon+(1-\varepsilon)(\mu-\frac{\gamma\sigma^2}2)\)（\(\varepsilon\) 跨期替代弹性）。代入预算约束得Consider the Saito (1998) model; for simplicity assume no stock market and no aggregate shock (only private equity with idiosyncratic shocks). The equilibrium risk-free rate is \(r=\mu-\gamma\sigma^2\) (\(\mu\) expected return on private equity, \(\gamma\) relative risk aversion, \(\sigma\) idiosyncratic volatility), and the optimal consumption rate is \(m=\beta\varepsilon+(1-\varepsilon)(\mu-\frac{\gamma\sigma^2}2)\) (\(\varepsilon\) the elasticity of intertemporal substitution). Substituting into the budget constraint,

$$\frac{dw_{it}}{w_{it}}=g\,dt+v\,dB_{it},\qquad g=\mu-\beta\varepsilon-(1-\varepsilon)\left(\mu-\frac{\gamma\sigma^2}2\right),\ v=\sigma,$$

即个体财富满足 Gibrat (1931) 比例增长律。对 \(f(x)=\log x\) 用 Ito 公式：so individual wealth obeys Gibrat's (1931) law of proportionate growth. Applying Ito's formula to \(f(x)=\log x\):

$$d(\log w_{it})=\frac1{w_{it}}dw_{it}-\frac12\frac1{w_{it}^2}(dw_{it})^2=\left(g-\frac12 v^2\right)dt+v\,dB_{it}.$$

故对数财富是漂移 \(g-\frac12 v^2\)、波动 \(v\) 的布朗运动。设各主体初始财富 \(w_0\)，则 \(t\) 时横截面财富分布为对数正态 \(\log w_{it}\sim N(\log w_0+(g-v^2/2)t,\ v^2 t)\)：对数均值与方差都随时间线性增长。消费与财富成比例故同理。与 Deaton and Paxson (1994) 的实证一致。So log wealth is a Brownian motion with drift \(g-\frac12 v^2\) and volatility \(v\). With every agent starting at \(w_0\), the cross-sectional wealth distribution at \(t\) is lognormal, \(\log w_{it}\sim N(\log w_0+(g-v^2/2)t,\ v^2 t)\): both log mean and variance grow linearly in time. Consumption is proportional to wealth, so the same holds. This is consistent with Deaton and Paxson (1994).

推导（2.1）/ Derivation (2.1) 上例中对数财富恰是带漂移布朗运动故易算分布；一般情形需 Fokker-Planck 方程（Kolmogorov 前向方程）。考虑扩散 \(dX_t=g(t,X_t)dt+v(t,X_t)dB_t\) (1)，\(p(x,t)\) 为 \(X(t)\) 密度。取光滑测试函数 \(F(t,x)\)（\(F(t_1,x)=F(t_2,x)=0\)，\(F,F_x\to0\) 当 \(x\to\pm\infty\)）。由 Ito \(dF=(F_t+F_x g+\frac12 F_{xx}v^2)dt+F_x v\,dB\)，取期望（用鞅性）、从 \(t_1\) 到 \(t_2\) 积分并分部积分，因 \(F\) 几乎任意，被积为零，得In the example log wealth was exactly a Brownian motion with drift, so the distribution was easy; in general we need the Fokker-Planck equation (Kolmogorov forward equation). Consider the diffusion \(dX_t=g(t,X_t)dt+v(t,X_t)dB_t\) (1) with \(p(x,t)\) the density of \(X(t)\). Take a smooth test function \(F(t,x)\) (\(F(t_1,x)=F(t_2,x)=0\), \(F,F_x\to0\) as \(x\to\pm\infty\)). By Ito \(dF=(F_t+F_x g+\frac12 F_{xx}v^2)dt+F_x v\,dB\); taking expectations (martingale property), integrating from \(t_1\) to \(t_2\) and by parts, since \(F\) is (nearly) arbitrary the integrand must vanish, giving

$$\frac{\partial p}{\partial t}=-\frac\partial{\partial x}(gp)+\frac12\frac{\partial^2}{\partial x^2}(v^2 p).\tag{2}$$

若过程偶尔重置（如个体死亡），设单位时间在 \(x\) 处流入 \(j_+\)、流出 \(j_-\)，则 (2) 加 \(+j_+-j_-\)。若以 Poisson 率 \(d\) 死亡并在 \(x_0\) 重生，则 \(\frac{\partial p}{\partial t}=-\frac\partial{\partial x}(gp)+\frac12\frac{\partial^2}{\partial x^2}(v^2 p)+d\delta(x-x_0)-dp\)（\(\delta\) 为 Dirac delta）。If the process is occasionally reset (e.g. individuals die), with influx \(j_+\) and outflux \(j_-\) per unit time at \(x\), add \(+j_+-j_-\) to (2). If a unit dies at Poisson rate \(d\) and is reborn at \(x_0\), then \(\frac{\partial p}{\partial t}=-\frac\partial{\partial x}(gp)+\frac12\frac{\partial^2}{\partial x^2}(v^2 p)+d\delta(x-x_0)-dp\) (\(\delta\) the Dirac delta).

平稳密度（2.2）/ Stationary density (2.2) 若漂移 \(g(x)\)、方差 \(v(x)\) 时不变且存在平稳分布 \(p(x)\)，则 \(0=-\frac{d}{dx}(gp)+\frac12\frac{d^2}{dx^2}(v^2p)\)。关于 \(x\) 积分并用边界 \(p,p'\to0\) 得 \(0=-g(x)p(x)+\frac12(v^2p)'\)。令 \(q=v^2 p\) 解 ODE \(\frac{q'}q=\frac{2g}{v^2}\) 得 \(q(x)=\exp(\int\frac{2g}{v^2}dx)\)，故平稳密度If the drift \(g(x)\) and variance \(v(x)\) are time-independent and a stationary distribution \(p(x)\) exists, then \(0=-\frac{d}{dx}(gp)+\frac12\frac{d^2}{dx^2}(v^2p)\). Integrating in \(x\) with boundary \(p,p'\to0\) gives \(0=-g(x)p(x)+\frac12(v^2p)'\). Letting \(q=v^2 p\) and solving \(\frac{q'}q=\frac{2g}{v^2}\) gives \(q(x)=\exp(\int\frac{2g}{v^2}dx)\), so the stationary density is

$$p(x)=\frac{q(x)}{v(x)^2}=\frac1{v(x)^2}\exp\left(\int\frac{2g(x)}{v(x)^2}\,dx\right),\tag{3}$$

积分常数由 \(\int p=1\) 定。（含死亡率 \(d\) 时为二阶 ODE \(0=-\frac{d}{dx}(gp)+\frac12\frac{d^2}{dx^2}(v^2p)-dp\)，除 \(x_0\) 外成立。\(p(x,t)\) 在一定条件下总收敛到平稳密度，证见 Gardiner 2009。）the integration constant fixed by \(\int p=1\). (With a death rate \(d\), it is a second-order ODE \(0=-\frac{d}{dx}(gp)+\frac12\frac{d^2}{dx^2}(v^2p)-dp\) holding except at \(x_0\). Under conditions \(p(x,t)\) always converges to the stationary density, see Gardiner 2009.)

例 1–3：OU、sgn 漂移、指数时间布朗运动 / Examples 1–3 例 1（Ornstein-Uhlenbeck）：\(dX=-\kappa(X-\mu)dt+v\,dB\)（\(\kappa>0\)，连续时间 AR(1)）。由 (3)，\(g=-\kappa(x-\mu)\)、\(v\) 常数，平稳密度 \(\propto\exp(\int\frac{-2\kappa(x-\mu)}{v^2}dx)=\exp(-\frac{\kappa(x-\mu)^2}{v^2})\)，即 \(N(\mu,\frac{v^2}{2\kappa})\)。例 2（\(\mathrm{sgn}\) 漂移）：\(dX=-\kappa\,\mathrm{sgn}(X-\mu)dt+v\,dB\)。平稳密度 \(\propto\exp(-\frac{2\kappa|x-\mu|}{v^2})\)，即（对称）Laplace 分布（Alfarano et al. 2012；Toda 2012）。例 3（指数时间布朗运动）：\(dX=g\,dt+v\,dB\)，单位以 Poisson 率 \(\delta>0\) 死亡、在 \(x_0\) 重生。稳态 FPE（除 \(x_0\) 外）\(0=-gp'+\frac12 v^2 p''-\delta p\)，通解 \(p(x)=C_1 e^{\lambda_1 x}+C_2 e^{\lambda_2 x}\)，\(\lambda\) 为 \(\frac12 v^2\xi^2-g\xi-\delta=0\) (4) 的根；平稳密度为非对称 Laplace（众数 \(x_0\)、指数 \(\alpha,\beta\)，\(-\alpha,\beta>0\) 为该二次方程的根）。Example 1 (Ornstein-Uhlenbeck): \(dX=-\kappa(X-\mu)dt+v\,dB\) (\(\kappa>0\), the continuous-time AR(1)). By (3), with \(g=-\kappa(x-\mu)\) and constant \(v\), the stationary density \(\propto\exp(\int\frac{-2\kappa(x-\mu)}{v^2}dx)=\exp(-\frac{\kappa(x-\mu)^2}{v^2})\), i.e. \(N(\mu,\frac{v^2}{2\kappa})\). Example 2 (\(\mathrm{sgn}\) drift): \(dX=-\kappa\,\mathrm{sgn}(X-\mu)dt+v\,dB\). The stationary density \(\propto\exp(-\frac{2\kappa|x-\mu|}{v^2})\), the (symmetric) Laplace distribution (Alfarano et al. 2012; Toda 2012). Example 3 (Brownian motion at exponential time): \(dX=g\,dt+v\,dB\), units die at Poisson rate \(\delta>0\) and are reborn at \(x_0\). The steady-state FPE (except at \(x_0\)) is \(0=-gp'+\frac12 v^2 p''-\delta p\), with general solution \(p(x)=C_1 e^{\lambda_1 x}+C_2 e^{\lambda_2 x}\), \(\lambda\) the roots of \(\frac12 v^2\xi^2-g\xi-\delta=0\) (4); the stationary density is asymmetric Laplace (mode \(x_0\), exponents \(\alpha,\beta\) with \(-\alpha,\beta>0\) the roots of this quadratic).

例 4：带漂移几何布朗运动 ⟹ 逆 Gamma ⟹ 幂律 / Example 4: GBM with drift ⟹ inverse gamma ⟹ power law \(dX=(gX+q)dt+vX\,dB\)（带漂移 \(q>0\) 的几何布朗运动）。稳态 FPE \(0=-\frac{d}{dx}[(gx+q)p]+\frac12\frac{d^2}{dx^2}[v^2x^2p]\)，解得逆 Gamma 分布\(dX=(gX+q)dt+vX\,dB\) (geometric Brownian motion with drift \(q>0\)). The steady-state FPE \(0=-\frac{d}{dx}[(gx+q)p]+\frac12\frac{d^2}{dx^2}[v^2x^2p]\) solves to the inverse gamma distribution

$$p(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{-\alpha-1}e^{-\beta/x},\qquad \alpha=1-\frac{2g}{v^2},\ \beta=\frac{2q}{v^2}>0.$$

经济解释：若主体无限寿命、赚常数劳动收入、把财富投入受特质乘性冲击的私人股权，则平稳财富分布是逆 Gamma。因 \(\beta/x\to0\)（\(x\to\infty\)），\(P(X>x)\sim\int_x^\infty y^{-\alpha-1}dy\sim x^{-\alpha}\)，即横截面财富服从指数 \(\alpha=1-2g/v^2\) 的幂律（Benhabib et al. 2011 研究其离散时间优化版本）。Economic interpretation: if agents have infinite lives, earn constant labor income, and invest wealth in private equity subject to idiosyncratic multiplicative shocks, the stationary wealth distribution is inverse gamma. Since \(\beta/x\to0\) as \(x\to\infty\), \(P(X>x)\sim\int_x^\infty y^{-\alpha-1}dy\sim x^{-\alpha}\), so cross-sectional wealth obeys a power law with exponent \(\alpha=1-2g/v^2\) (Benhabib et al. 2011 study a discrete-time optimizing version).

4.1 幂律与双幂律 / 4.1 Power law and double power law 非负随机变量 \(X\) 在上尾服从指数 \(\alpha>0\) 的幂律，若 \(\lim_{x\to\infty}x^\alpha P(X>x)>0\) 存在（Pareto 1896/1897；Mandelbrot 1960/1961）。许多经济变量在下尾也服从幂律：\(\lim_{x\to0}x^{-\beta}P(X0\)。上下尾都服从即双幂律（城市规模、收入、消费及消费增长等）。若 \(X\) 服从指数 \((\alpha,\beta)\) 的双幂律，则 \(X^\eta\) 服从 \((\alpha/\eta,\beta/\eta)\)（\(\eta>0\)）或 \((-\beta/\eta,-\alpha/\eta)\)（\(\eta<0\)）。重要推论：\(\eta\) 阶矩 \(\mathbb E[X^\eta]\) 存在当且仅当 \(-\beta<\eta<\alpha\)——许多计量方法依赖矩存在，故识别幂律很重要。A nonnegative random variable \(X\) obeys a power law in the upper tail with exponent \(\alpha>0\) if \(\lim_{x\to\infty}x^\alpha P(X>x)>0\) exists (Pareto 1896/1897; Mandelbrot 1960/1961). Many economic variables also obey a power law in the lower tail: \(\lim_{x\to0}x^{-\beta}P(X0\). Both tails ⟹ the double power law (city size, income, consumption and consumption growth, etc.). If \(X\) obeys the double power law with exponents \((\alpha,\beta)\), then \(X^\eta\) obeys it with \((\alpha/\eta,\beta/\eta)\) (\(\eta>0\)) or \((-\beta/\eta,-\alpha/\eta)\) (\(\eta<0\)). Key implication: the \(\eta\)-th moment \(\mathbb E[X^\eta]\) exists iff \(-\beta<\eta<\alpha\) — many econometric techniques rely on moment existence, so recognizing a power law matters.

4.2 双 Pareto 与双 Pareto-对数正态 (dPlN) / 4.2 Double Pareto and dPlN 服从双幂律的典型分布是双 Pareto 分布（Reed 2001），密度A canonical double-power-law distribution is the double Pareto distribution (Reed 2001), with density

$$f_{dP}(x)=\begin{cases}\dfrac{\alpha\beta}{\alpha+\beta}M^\alpha x^{-\alpha-1},&(x\ge M)\\[2mm]\dfrac{\alpha\beta}{\alpha+\beta}M^{-\beta}x^{\beta-1},&(0\le x

\(M>0\) 为尺度参数（\(\beta>1\) 时为众数），\(\alpha,\beta>0\) 为形状参数（幂律指数）；\(\beta\to\infty\) 退化为经典 Pareto。其密度在 \(M\) 处有尖点；密度光滑且服从双幂律的例子是双 Pareto-对数正态 (dPlN) 分布（Reed 2003）——独立双 Pareto 与对数正态变量之积。其对数为 normal-Laplace 分布（Reed-Jorgensen 2004，正态与 Laplace 的卷积），密度with \(M>0\) a scale parameter (the mode if \(\beta>1\)) and \(\alpha,\beta>0\) shape parameters (power-law exponents); \(\beta\to\infty\) degenerates to the classical Pareto. Its density has a cusp at \(M\); a smooth-density example obeying the double power law is the double Pareto-lognormal (dPlN) distribution (Reed 2003) — the product of independent double Pareto and lognormal variables. Its logarithm is the normal-Laplace distribution (Reed-Jorgensen 2004, a convolution of normal and Laplace), with density

$$f_{NL}(x)=\frac{\alpha\beta}{\alpha+\beta}\left[e^{\frac{\alpha^2\sigma^2}2-\alpha(x-\mu)}\Phi\!\left(\frac{x-\mu}\sigma-\alpha\sigma\right)+e^{\frac{\beta^2\sigma^2}2+\beta(x-\mu)}\Phi\!\left(-\frac{x-\mu}\sigma-\beta\sigma\right)\right].\tag{7}$$

附录：Laplace 分布与定理 1 / Appendix: the Laplace distribution and Theorem 1 Laplace 分布密度 \(f(x)=\frac{\alpha\beta}{\alpha+\beta}e^{-\alpha(x-m)}\)（\(x\ge m\)）或 \(\frac{\alpha\beta}{\alpha+\beta}e^{-\beta(m-x)}\)（\(x0\) 尺度；\(\alpha\neq\beta\) 即非对称。其特征函数 \(\varphi_X(t)=\frac{e^{imt}}{1-i(\frac1\alpha-\frac1\beta)t+t^2/(\alpha\beta)}\) (9)，均值 \(m+\frac1\alpha-\frac1\beta\)、方差 \(\frac1{\alpha^2}+\frac1{\beta^2}\)。以 \(a=\frac1\alpha-\frac1\beta\)（非对称参数）、\(\sigma=\sqrt{2/(\alpha\beta)}\)（尺度）记 \(X\sim AL(m,a,\sigma)\)，则 \(\varphi_X(t)=\frac{e^{imt}}{1-iat+\sigma^2 t^2/2}\) (10)，\(-\alpha,\beta\) 为 \(\frac{\sigma^2}2\zeta^2-a\zeta-1=0\) (11) 的根。定理 1：设 \(\{X_n\}\) 零均值、满足中心极限定理 \(N^{-1/2}\sum X_n\xrightarrow{d}N(0,\sigma^2)\)，\(N^{-1}\sum a_n\to a\)，\(\nu_p\) 是均值 \(1/p\) 的几何随机变量（独立于 \(\{X_n\}\)）。则 \(p\to0\) 时 \(p^{1/2}\sum_{n=1}^{\nu_p}(X_n+p^{1/2}a_n)\xrightarrow{d}AL(0,a,\sigma)\)。（即 Laplace 是几何和的唯一极限分布——一种稳健性质。）The Laplace distribution has density \(f(x)=\frac{\alpha\beta}{\alpha+\beta}e^{-\alpha(x-m)}\) (\(x\ge m\)) or \(\frac{\alpha\beta}{\alpha+\beta}e^{-\beta(m-x)}\) (\(x0\) scale; \(\alpha\neq\beta\) is asymmetric. Its characteristic function \(\varphi_X(t)=\frac{e^{imt}}{1-i(\frac1\alpha-\frac1\beta)t+t^2/(\alpha\beta)}\) (9), mean \(m+\frac1\alpha-\frac1\beta\), variance \(\frac1{\alpha^2}+\frac1{\beta^2}\). Writing \(a=\frac1\alpha-\frac1\beta\) (asymmetry) and \(\sigma=\sqrt{2/(\alpha\beta)}\) (scale), \(X\sim AL(m,a,\sigma)\) means \(\varphi_X(t)=\frac{e^{imt}}{1-iat+\sigma^2 t^2/2}\) (10), and \(-\alpha,\beta\) are the roots of \(\frac{\sigma^2}2\zeta^2-a\zeta-1=0\) (11). Theorem 1: let \(\{X_n\}\) have zero mean and satisfy the CLT \(N^{-1/2}\sum X_n\xrightarrow{d}N(0,\sigma^2)\), \(N^{-1}\sum a_n\to a\), and \(\nu_p\) a geometric random variable with mean \(1/p\) (independent of \(\{X_n\}\)). Then as \(p\to0\), \(p^{1/2}\sum_{n=1}^{\nu_p}(X_n+p^{1/2}a_n)\xrightarrow{d}AL(0,a,\sigma)\). (So the Laplace is the unique limit distribution of a geometric sum — a robust property.)