罗伊恒等式有
\[V = \frac{\lambda P_1 X_1}{a_1} = \frac{\lambda P_2 X_2}{a_2} = \frac{\lambda P_3 X_3}{a_3}\]
将其代入收入约束
\[
Y = P_1 X_1 + P_2 X_2 + P_3 X_3 = V \frac{a_1 + a_2 + a_3}{\lambda}
\]
得
\[
\lambda = \frac{A(a_1 + a_2 + a_3)}{P_1^{a_1} P_2^{a_2} P_3^{a_3}}
\]
回代入罗伊恒等式得
\[
(a_1 + a_2 + a_3)P_2 X_2 = Y
\]
\[
P_2 = \frac{Y}{(a_1 + a_2 + a_3)X_2}
\]
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