Suppose $Q = \sqrt { \bar { K } L }$ in the short-run. The firm's short-run variable cost curve is
A) $S F C = \frac { Q ^ { 2 } } { \bar { K } }$
B) $S F C = \frac { w Q ^ { 2 } } { \bar { K } } + r \bar { K }$
C) $S F C = w Q ^ { 2 } /{ \bar { K } }$
D) $S F C = r \bar { K }$

Question

Suppose $Q = \sqrt { \bar { K } L }$ in the short-run. The firm's short-run variable cost curve is
A) $S F C = \frac { Q ^ { 2 } } { \bar { K } }$
B) $S F C = \frac { w Q ^ { 2 } } { \bar { K } } + r \bar { K }$
C) $S F C = w Q ^ { 2 } /{ \bar { K } }$ 
D) $S F C = r \bar { K }$

Quizplus · Accepted Answer

The short-run cost curve for variable costs can be expressed as $S F C = wL + rK$, where w is the wage rate, L is labor input, r is the rental rate of capital, and K is capital input. However, since we have Q in terms of K and L, we can substitute $\sqrt{\bar{K}L}$ in for Q, giving us
$$S F C = w\left(\frac{Q^2}{\bar{K}}\right) + r\bar{K}.$$
Simplifying this expression using the given equation for Q, we get
$$SFC = \frac{wQ^2}{\bar{K}}.$$
Therefore, the correct answer is C.

Suppose Q=KˉLQ = \sqrt { \bar { K } L }Q=KˉL​

Suppose $Q = \sqrt { \bar { K } L }$