経済学で出る数学

ワークブックでじっくり攻める：問7.24解答例

1. $f(tL, tK)=\left(tL\right)^{\frac{1}{3}}\left(tK\right)^{\frac{2}{3}} =t^{\frac{1}{3}+\frac{2}{3}} \left(L\right)^{\frac{1}{3}}\left(K\right)^{\frac{2}{3}} =t\left(L\right)^{\frac{1}{3}}\left(K\right)^{\frac{2}{3}} =tf(L,K)$ より，$1$次同次


2. $f(tL, tK)=\left(tL\right)^{\frac{3}{4}}\left(tK\right)^{\frac{5}{4}} =t^{\frac{3}{4}+\frac{5}{4}} \left(L\right)^{\frac{3}{4}}\left(K\right)^{\frac{5}{4}} =t^2\left(L\right)^{\frac{3}{4}}\left(K\right)^{\frac{5}{4}} =t^2f(L,K)$ より，$2$次同次


3. $f(tL, tK)=\left(tL\right)^{\alpha}\left(tK\right)^{\beta} =t^{\alpha+\beta} \left(L\right)^{\alpha}\left(K\right)^{\beta} =t^{\alpha +\beta}f(L,K)$ より，${\alpha +\beta}$次同次


4. $f(tL, tK)=(tx)^3+(tx)^2(ty)+2(tx)(ty)^2 =t^3x^3+t^3x^2y+t^3xy^2 =t^3\left(x^3+x^2y+xy^2\right)=t^3f(x,y)$ より，$3$次同次