\[\sum_{i=1}^{n}P_{i}X_{i} = I\]
\[
Q = \prod_{i=1}^{n}X_{i}^{\alpha_{i}}
\]

\[MP_{i} = \frac{\partial Q}{\partial X_{i}} = \alpha_{i}\frac{Q}{X_{i}}\]
\[\frac{\partial MP_{i}}{\partial P_{i}} = \frac{\alpha_{i}}{X_{i}}\frac{\partial X_{i}}{\partial P_{i}}(MP_{i}X_{i}-Q) < 0\]

\[= \frac{\alpha_{i}}{X_{i}}\frac{\partial X_{i}}{\partial P_{i}}(\alpha_{i}-1)Q\]

因此，当
\[
\alpha_{i}<1
\]
时，
\[\frac{\partial MP_{i}}{\partial P_{i}} > 0\]
当
\[
\alpha_{i}>1
\]
时，
\[
\frac{\partial MP_{i}}{\partial P_{i}} < 0
\]