替代-互补函数习题

#数理经济学

习题 1

※求解成本函数

\begin{aligned} min_{z_{i}} & \sum_{i = 1}^{3} w_{i} z_{i} \\ s.t. & a z_{1} + min {b z_{1}, c z_{3}} = q \\ a, b, c > 0 \end{aligned}

令 $x = min {b z_{1}, c z_{3}}$ ，一阶段问题定义为

\begin{aligned} max_{z_{2}, z_{3}} & w_{x} x \equiv w_{2} z_{2} + w_{3} z_{3} \\ s.t. & min {b z_{1}, c z_{3}} = x \\ b, c > 0 \end{aligned}

生产一单位 $x$ 需要 $z_{1} = \frac{1}{b} \land z_{2} = \frac{1}{c}$ ，对应成本为 $\frac{w_{2}}{b} + \frac{w_{3}}{c} = w_{x}$

二阶段问题定义为

\begin{aligned} min_{z_{1}, x} & w_{1} z_{1} + w_{x} x \\ s.t. & a z_{1} + x = q \\ a > 0 \end{aligned}

生产一单位 $q$ 需要 $z_{1} = \frac{1}{a} \lor x = 1$ ，对应成本为 $min {\frac{w_{1}}{a}, w_{x}}$

综上所述，成本函数为 $C (q) = min {\frac{w_{1}}{a}, \frac{w_{2}}{b} + \frac{w_{3}}{c}} q$

※求解成本函数

\begin{aligned} min_{z_{i}} & \sum_{i = 1}^{3} w_{i} z_{i} \\ s.t. & min {a z_{1} + b z_{2}, c z_{3}} = q \\ a, b, c > 0 \end{aligned}

令 $y = a z_{1} + b z_{2}$ ，一阶段问题定义为

\begin{aligned} min_{z_{1}, z_{2}} & w_{y} y \equiv w_{1} z_{1} + w_{2} z_{2} \\ s.t. & a z_{1} + b z_{2} = y \\ a, b > 0 \end{aligned}

生产一单位 $y$ 需要 $z_{1} = \frac{1}{a} \lor z_{2} = \frac{1}{b}$ ，对应成本为 $min {\frac{w_{1}}{a}, \frac{w_{2}}{b}} = w_{y}$

二阶段问题定义为

\begin{aligned} min & w_{y} y + w_{3} z_{3} \\ s.t. & min {y, c z_{3}} = q \\ c > 0 \end{aligned}

生产一单位 $q$ 需要 $y = 1 \land z_{3} = \frac{1}{c}$ ，对应成本为 $w_{y} + \frac{w_{3}}{c}$

综上所述，成本函数为 $C (q) = (min {\frac{w_{1}}{a}, \frac{w_{2}}{b}} + \frac{w_{3}}{3}) q$