Time is one of the two elements that Americans save carefully,the other being labor
这句话=Time is one of the two elements that Americans save carefully, while the other is labor.
"while the other is labor" 在这里是比较状语从句.当把连接词去掉的时候,the other 和 time是两个不同的主语, 所以要用独立主格表示.
这里是独立主格：名词 / 代词 + -ing分词 的用法.如：
The bus coming here soon,（=Since the bus is coming soon） we should get everything ready. 汽车很快就要来了,我们应该把一切事情准备好.