效用最大化

#需求理论

模型

消费者最优化问题为

\begin{aligned} max_{{x_{i}}} & u (x_{1}, \dots, x_{n}) \\ s.t. & \sum_{i = 1}^{n} p_{i} x_{i} = m \\ x_{i} \geq 0 \end{aligned}

Lagrangian 函数为

L = u (x_{1}, \dots, x_{n}) + λ (m - \sum_{i = 1}^{n} p_{i} x_{i})

F.O.C.

{\begin{cases} \frac{\partial L}{\partial x_{i}} = \frac{\partial u}{\partial x_{i}} - λ p_{i} = 0 & , x_{i} > 0 \\ \frac{\partial L}{\partial x_{i}} = \frac{\partial u}{\partial x_{i}} - λ p_{i} \leq 0 & , x_{i} = 0 \end{cases}

假设 $x_{a}, x_{b} > 0; x_{c} = 0$ ，其中 $a, b, c \in {1, \dots, n}$ ，可以得到：

替代平衡（substitution balance）等式/不等式

\frac{\partial u / \partial x_{a}}{\partial u / \partial x_{b}} = \frac{p_{a}}{p_{b}} and \frac{\partial u / \partial x_{c}}{\partial u / \partial x_{b}} \leq \frac{p_{c}}{p_{b}}

消费平衡（consumption balance）等式/不等式

\frac{\partial u / \partial x_{a}}{p_{a}} = \frac{\partial u / \partial x_{b}}{p_{b}} = λ \geq \frac{\partial u / \partial x_{c}}{p_{c}}

最优解记作

x_{i}^{M} (p_{1}, \dots, p_{n}, \bar{m}) or x^{M} (p, m)

称为马歇尔需求函数（Marshallian demand function）。