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

Question

Quizplus · Accepted Answer

The long-run total cost can be calculated by substituting the given values of $L$ and $K$ in the production function:

$$Q = f(K,L) = K^{1/2} L^{1/2} = \frac {Q} {10} \sqrt { \frac {w } {r } } ^ {1/2} \frac {Q } {10} \sqrt { \frac { r } {w } }  ^ {1/2} = \frac {Q^2} {100}$$
Solving for Q, we get:
$$Q = 10\sqrt{Kr}$$
Substituting the given values of $r$ and $w$, we get:
$$Q = 10\sqrt{5K}$$
$$Q = 10\sqrt{\frac{1}{4}L}$$
$$Q = 5\sqrt{L}$$
Now, substitute this value of Q in the cost-minimizing equations for $K$ and $L$:
$$K = \frac {5} {10} \sqrt { \frac {5L} {20} } = \frac {1} {2} \sqrt {L}$$
$$L = \frac {5} {10} \sqrt { \frac {5K} {5} } = \frac {1} {2} \sqrt {K}$$

Next, substitute these values of $K$ and $L$ in the average cost function:
$$AC = \frac {wK + rL} Q = \frac {20(\frac {1} {2} \sqrt {L}) + 5(\frac {1} {2} \sqrt {K})} {5\sqrt {L}} = \frac {2 \sqrt {L} + \sqrt {K} } {\sqrt {L}} = 2\sqrt{\frac{L}{K}} + 1$$
Solving for $\frac {L} {K} $ using the cost-minimizing levels of $L$ and $K$, we get:
$$\frac {L} {K} = \frac {(Q/10)(\sqrt{\frac {r} {w}}) } {(Q/10)(\sqrt{\frac {w} {r}})} = \frac {r}{w} = \frac {5}{20} = \frac {1} {4}$$
Substituting this value in the AC equation, we get:
$$AC = 2\sqrt {\frac {1} {4}} + 1 = 2 \times \frac {1} {2} + 1 = 2$$
Therefore, the long-run total cost equation is:
$$LRAC = ATC = AC \times Q = 2Q$$

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