查看“︁罗伊恒等式”︁的源代码

{{noteTA
|1= zh-cn:谢泼德引理; zh-tw:薛福輔理
|2=zh-hans:希克斯;zh-tw:Hicks;
|G1=Economics
}}

'''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。

== 具体表述 ==
设消费者的[[间接效用函数]]为<math>v(\mathbf{p}, w)</math>，则商品<math>i</math>的马歇尔需求函数即为
:<math>x_i^m = -\frac{\partial v / \partial p_i}{\partial v / \partial w}</math>，
其中<math>\mathbf{p}</math>为各商品的价格向量，<math>w</math>为[[收入]]。<ref>{{cite book |first=Hal |last=Varian |authorlink=哈尔·范里安|title=Microeconomic Analysis |location=New York |publisher=Norton |edition=Third |year=1992 |isbn= |pages=106–108 |url=https://books.google.com/books?id=m20iQAAACAAJ&pg=PA106 }}</ref>

=== 证明 ===
根据定义，间接效用函数满足约束条件<math>\mathbf{p} \cdot \mathbf{x} = w</math>下的最大值<math>v(\mathbf{p}, w) = \max_{\mathbf{x}}u(\mathbf{x})</math>。因此由带约束的[[包络定理]]立即得到
:<math>\frac{\partial v}{\partial \mathbf{p}} = \frac{\partial \mathcal{L}}{\partial \mathbf{p}} = -\lambda \mathbf{x}</math>，
其中<math>\mathcal{L} = u(\mathbf{x}) - \lambda(\mathbf{p} \cdot \mathbf{x} - w)</math>为[[拉格朗日乘数|拉格朗日乘數]]，由其表达式可得<math>\lambda = \frac{\partial \mathcal{L}}{\partial w} = \frac{\partial v}{\partial w}</math>，代入上式即得证。<ref>{{cite book |first=Richard |last=Cornes |title=Duality and Modern Economics |location=New York |publisher=Cambridge University Press |year=1992 |isbn=0-521-33291-5 |pages=45–47 |url=https://books.google.com/books?id=HO8zAAAAIAAJ&pg=PA45 |access-date=2018-11-11 |archive-date=2022-04-07 |archive-url=https://web.archive.org/web/20220407010258/https://books.google.com/books?id=HO8zAAAAIAAJ&pg=PA45 }}</ref>

==參見==
*[[谢泼德引理|薛福輔理]]

== 参考文献 ==
{{reflist}}

[[Category:微观经济学]]