霍特林引理

设共有${\displaystyle n}$种商品，第${\displaystyle i}$种商品的价格为${\displaystyle p_{i}}$，净供给为${\displaystyle y_{i}(\mathbf {p} )}$。厂商的利润函数${\displaystyle \pi (\mathbf {p} )=\max _{\mathbf {y} (\mathbf {p} )}\mathbf {p} \cdot \mathbf {y} (\mathbf {p} )}$，根据包络定理

${\displaystyle {\frac {\partial \pi }{\partial \mathbf {p} }}=\left.{\frac {\partial (\mathbf {p} \cdot \mathbf {y} (\mathbf {p} ))}{\partial \mathbf {p} }}\right\vert _{\mathbf {y} (\mathbf {p} )=\mathbf {y} ^{*}(\mathbf {p} )}=\mathbf {y} ^{*}(\mathbf {p} )}$。