Suppose k, l, and m grow at constant rates given by $\bar { g } _ { k }$ And $\bar { g } _ { m }$
What is the growth rate of y if $y = m k ^ { 1 / 3 } l ^ { 2 / 3 } ?$
A) $g _ { y } = \bar { g } _ { m } + \bar { g } _ { k } + \bar { g } _ { t }$
B) $g _ { y } = \bar { g } _ { m } \times ( 1 / 3 ) \bar { g } _ { k } \times ( 2 / 3 ) \bar { g } _ { l }$
C) $g _ { y } = \bar { g } _ { m } + ( 1 / 3 ) \left( \bar { g } _ { k } + \bar { g } _ { t } \right)$
D) $g _ { y } = \bar { g } _ { m } + ( 1 / 3 ) \bar { g } _ { k } + ( 2 / 3 ) \bar { g } _ { t }$
E) $g _ { y } = \bar { g } _ { m } + \bar { g } _ { \ell } 1 / 3 + \bar { g } _ { t } 2 / 3$

Question

Suppose k, l, and m grow at constant rates given by $\bar { g } _ { k }$  And $\bar { g } _ { m }$
What is the growth rate of y if $y = m k ^ { 1 / 3 } l ^ { 2 / 3 } ?$
A) $g _ { y } = \bar { g } _ { m } + \bar { g } _ { k } + \bar { g } _ { t }$ 
B) $g _ { y } = \bar { g } _ { m } \times ( 1 / 3 ) \bar { g } _ { k } \times ( 2 / 3 ) \bar { g } _ { l }$ 
C) $g _ { y } = \bar { g } _ { m } + ( 1 / 3 ) \left( \bar { g } _ { k } + \bar { g } _ { t } \right)$ 
D) $g _ { y } = \bar { g } _ { m } + ( 1 / 3 ) \bar { g } _ { k } + ( 2 / 3 ) \bar { g } _ { t }$ 
E) $g _ { y } = \bar { g } _ { m } + \bar { g } _ { \ell } 1 / 3 + \bar { g } _ { t } 2 / 3$

Quizplus · Accepted Answer

The growth rate of a product of variables is the sum of the growth rates of each variable, weighted by their respective exponents. Here, $y = m k^{1/3} l^{2/3}$, so the growth rate $g_y$ is $\bar{g}_m + (1/3)\bar{g}_k + (2/3)\bar{g}_l$.

Suppose K, L, and M Grow at Constant Rates Given gˉk\bar { g } _ { k }gˉ​k​

Suppose K, L, and M Grow at Constant Rates Given $\bar { g } _ { k }$