Please explain how to do this problem and how K=5L? after you find TC and ATC. T

Question

Please explain how to do this problem and how K=5L? after you find TC and ATC.

Thanks

Suppose a firm with a production function Q = KL (where MPL K and MPK- L) is producing 125 units of output by using 5 workers and 25 units of capital. The wage rate (W) per worker is $10 and the rental per unit of capital (R) is $2. If it decreases output to 45 units, long-run average total cost_.__at 125 units of output to at 45 units of output. O falls from $0.80; $0.60 Orises from $1.80; $2 O rises from $0.80; $1.33 falls from $1.80; $1.15

Explanation / Answer

C. Rises from 0.80, 1.33

A profit maximising firm uses input mix such that the ratio of marginal utlity of labor and capital is equal to the ratio of wage and rent.

MU l / MUk = w/r

K/L = 10/2

K = 5L

Long run average cost at 125 units = long run total cost / output

LTC = wL + rK = 5*10 + 25*2 = 100

LAC = 100/125 = 0.8.

Q = 45 and 45 = KL where K =5L

45 = 5L^2

L = 3 and K = 5*3= 15

LTC = 3 *10 + 15*2 = 60

LAc = 66/45 = 1.33

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