罗伊恒等式 罗伊恒等式(Roy's identity)是微观经济学中的一项重要结果,在生产者理论和消费者理论中都有应用。 目录 1 具体表述 1.1 证明 2 参见 3 参考文献 具体表述 设消费者的间接效用函数为 v ( p , w ) {\displaystyle v(\mathbf {p} ,w)} ,则商品 i {\displaystyle i} 的马歇尔需求函数即为 x i m = − ∂ v / ∂ p i ∂ v / ∂ w {\displaystyle x_{i}^{m}=-{\frac {\partial v/\partial p_{i}}{\partial v/\partial w}}} ,其中 p {\displaystyle \mathbf {p} } 为各商品的价格向量, w {\displaystyle w} 为收入。[1] 证明 根据定义,间接效用函数满足约束条件 p ⋅ x = w {\displaystyle \mathbf {p} \cdot \mathbf {x} =w} 下的最大值 v ( p , w ) = max x u ( x ) {\displaystyle v(\mathbf {p} ,w)=\max _{\mathbf {x} }u(\mathbf {x} )} 。因此由带约束的包络定理立即得到 ∂ v ∂ p = ∂ L ∂ p = − λ x {\displaystyle {\frac {\partial v}{\partial \mathbf {p} }}={\frac {\partial {\mathcal {L}}}{\partial \mathbf {p} }}=-\lambda \mathbf {x} } ,其中 L = u ( x ) − λ ( p ⋅ x − w ) {\displaystyle {\mathcal {L}}=u(\mathbf {x} )-\lambda (\mathbf {p} \cdot \mathbf {x} -w)} 为拉格朗日乘数,由其表达式可得 λ = ∂ L ∂ w = ∂ v ∂ w {\displaystyle \lambda ={\frac {\partial {\mathcal {L}}}{\partial w}}={\frac {\partial v}{\partial w}}} ,代入上式即得证。[2] 参见 谢泼德引理参考文献 ^ Varian, Hal. Microeconomic Analysis Third. New York: Norton. 1992: 106–108. ^ Cornes, Richard. Duality and Modern Economics. New York: Cambridge University Press. 1992: 45–47 [2018-11-11]. ISBN 0-521-33291-5. (原始内容存档于2022-04-07).