経済学で出る数学
ワークブックでじっくり攻める：応用問題
CES関数の正規化
【問】 CES関数，
\[
f(x_1,x_2)=({\alpha}_1x_1^{\rho}+{\alpha}_2x_2^{\rho})^{\frac{1}{{\rho}}}
\]
は
\[
f(x_1,x_2)=A({\rho})({\beta}x_1^{\rho}+(1-{\beta})x_2^{\rho})^{\frac{1}{{\rho}}}
\]
と書けることを示しなさい．
【解答】
\begin{align*}
f(x_1,x_2)&=({\alpha}_1x_1^{\rho}+{\alpha}_2x_2^{\rho})^{\frac{1}{{\rho}}}\\
&=\Bigl[({\alpha}_1+{\alpha}_2)(\dfrac{{\alpha}_1}{{\alpha}_1+{\alpha}_2}x_1^{\rho}+\dfrac{{\alpha}_2}{{\alpha}_1+{\alpha}_2}x_2^{\rho})\bigr]^{\frac{1}{{\rho}}}\\
&=({\alpha}_1+{\alpha}_2)^{\frac{1}{{\rho}}}\Bigl[(\dfrac{{\alpha}_1}{{\alpha}_1+{\alpha}_2}x_1^{\rho}+
(1-\dfrac{{\alpha}_1}{{\alpha}_1+{\alpha}_2})x_2^{\rho})\bigr]^{\frac{1}{{\rho}}}\\
\end{align*}
なので，
$A({\rho})=({\alpha}_1+{\alpha}_2)^{\frac{1}{{\rho}}}$, ${\beta}=\dfrac{{\alpha}_1}{{\alpha}_1+{\alpha}_2}$とすればよい．
【解答終】
【Further Reading】
Hal R. Varian ‘Microeconomic Analysis; Third Edition,’ W.W.Norton & Company (1992)
ふろく（２）応用問題 一覧へ