If the utilization rates of capital (uK)and labor (uN)are procyclical,then the Solow residual,as conventionally measured,is
A) $ Y\left[\left(u_{K} K\right)^{a}\left(u_{N} N\right)^{1-a}\right] $
B) $ Y /\left[\left(u_{K} K\right)^{a}\left(u_{N} N\right)^{1-a}\right] $
C) $ A u_{K}^{a} u_{N}^{1-a} $
D) $ 1 /\left\{Y\left[\left(u_{K} K\right)^{a}\left(u_{N} N\right)^{1-a}\right]\right\} $

Question

Quizplus · Accepted Answer

The Solow residual is conventionally measured as:
Y = F(K,AL)
where L is labor, A is total factor productivity, and uK and uN are the utilization rates of capital and labor respectively. 
Taking logs, we have:
ln Y = ln A + a ln (uK K) + (1-a) ln (uN N)
Therefore, the Solow residual is conventionally measured as the portion of output growth that cannot be accounted for by changes in inputs such as capital and labor. 
Since uK and uN are procyclical, an increase in these variables corresponds to an increase in output (Y). Thus, we need to select the answer choice that includes both uK and uN in a procyclical manner. 
Option C is the correct answer because it has the form A * uk^a * un^(1-a), where A is total factor productivity and uk and un are procyclical. Options A, B, and D do not contain the correct combination of uk and un.

If the Utilization Rates of Capital (UK)and Labor (UN)are Procyclical,then Y[(uKK)a(uNN)1−a] Y\left[\left(u_{K} K\right)^{a}\left(u_{N} N\right)^{1-a}\right] Y[(uK​K)a(uN​N)1−a]

If the Utilization Rates of Capital (UK)and Labor (UN)are Procyclical,then $Y\left[\left(u_{K} K\right)^{a}\left(u_{N} N\right)^{1-a}\right]$