在國際貿易理論中,你會看著世界(國內加國外)的商品相對產量,會介於國內kap國外相對產量之間。以數學表示,若
\frac{x_1}{y_1} < \frac{x_2}{y_2},煞
\frac{x_1}{y_1} < \frac{x_1+x_2}{y_1+y_2} < \frac{x_2}{y_2}。
In international trade theory, you will see that the world's (home plus foreign) relative goods quantity is between homes and foreigns. Mathematically speaking, if
\frac{x_1}{y_1} < \frac{x_2}{y_2}, then
\frac{x_1}{y_1} < \frac{x_1+x_2}{y_1+y_2} < \frac{x_2}{y_2}.
#PaulKrugman #MauriceObstfeld #MarcMelitz
\frac{x_1}{y_1} < \frac{x_2}{y_2} \Rightarrow \frac{x_1+x_2}{y_1+y_2} < \frac{x_2}{y_2} :
\frac{x_1}{y_1} < \frac{x_2}{y_2}
\Rightarrow \frac{x_1+x_2}{y_1} = \frac{x_1}{y_1} + \frac{x_2}{y_1} < \frac{x_2}{y_2}+ \frac{x_2}{y_1}
\Rightarrow \frac{x_1+x_2}{y_1} < \frac{x_2 y_1 + x_2 y_2}{y_2 y_1}
\Rightarrow x_1+x_2 < \frac{x_2 (y_1 + y_2)}{y_2}
\Rightarrow \frac{x_1+x_2}{y_1+y_2} < \frac{x_2}{y_2}
\frac{x_1}{y_1} < \frac{x_2}{y_2} \Rightarrow \frac{x_1}{y_1} < \frac{x_1+x_2}{y_1+y_2} :
\frac{x_1}{y_1} < \frac{x_2}{y_2}
\Rightarrow \frac{x_1}{y_1} + \frac{x_1}{y_2} < \frac{x_2}{y_2} + \frac{x_1}{y_2} = \frac{x_1+x_2}{y_2}
\Rightarrow \frac{x_1 y_2 + x_1 y_1}{y_1 y_2} < \frac{x_1+x_2}{y_2}
\Rightarrow \frac{x_1 (y_1 + y_2)}{y_1} < x_1+x_2
\Rightarrow \frac{x_1}{y_1} < \frac{x_1+x_2}{y_1+y_2}
kā頂面兩个證出來的結果合做伙,就故得證囉。
Combining the results of the above two proofs, QED.