Suppose for a particular production function, the cost-minimizing levels of labor and capital are $L = \frac { Q } { 10 } \sqrt { \frac { r } { w } } K = \frac { Q } { 10 } \sqrt { \frac { w } { r } }$. If $r = 5$ and $w = 20$, what is the equation for long-run total cost?
A) $\frac { 2 Q } { 5 }$
B) $\frac { 2 Q } { 10 }$
C) $2 Q$
D) $4 Q$

Question

Quizplus · Accepted Answer

We can start by plugging in the given values of $r$ and $w$ into the expressions for $L$ and $K$:
$$L = \frac{Q}{10}\sqrt{\frac{5}{20}} = \frac{Q}{10}\sqrt{\frac{1}{4}} = \frac{Q}{20}$$
$$K = \frac{Q}{10}\sqrt{\frac{20}{5}} = \frac{Q}{10}\sqrt{4} = \frac{Q}{5}$$

Next, we can use the production function to solve for $Q$ in terms of $L$ and $K$:
$$Q = 10LK = 10\left(\frac{Q}{20}\right)\left(\frac{Q}{5}\right) = \frac{Q^2}{2}$$
Solving for $Q$, we get $Q = 20$.

Now we can substitute this value of $Q$ back into the expressions for $L$ and $K$ to get the cost-minimizing levels of labor and capital:
$$L = \frac{20}{20} = 1$$
$$K = \frac{20}{5} = 4$$

Finally, we can use the formula for long-run total cost:
$$C = rK + wL = 5(4) + 20(1) = 40$$

Therefore, the equation for long-run total cost is $C = 2Q = 40$. The long-run total cost equation is given by:
\begin{aligned} LTC &= wL + rK \ &= 20\left(\frac { Q } { 10 } \sqrt { \frac { r } { w } }\right) + 5 \left(\frac { Q } { 10 } \sqrt { \frac { w } { r } }\right) \ &= 2Q\sqrt{rw} \ &= 2Q\sqrt{5\cdot20} \ &= 20Q \end{aligned}
Therefore, the correct option is (C), $2Q$.

Suppose for a Particular Production Function, the Cost-Minimizing Levels of Labor