A firm has the production function . The wage rate is $10 per unit of labor and the rental rate is $5 per unit of capital and the firm is going to spend $1000 on production.

i. Assuming that the firm is free to choose any level of K and L to emply, how much of each should it emply? How much output will they produce?
ii. Now assume that once the firm has chosen its level of L and K, the level of K becomes fixed. If the price of K increases to $8 per unit, how many units of output can the firm now produce if it spends the same amount?
iii. Once the firm reaches the long run again and is able to vary its level of K, how much should L and K should it employ in order to achieve its original level of output? How much will that level of production cost?

i. The firm should set K = 100, L = 50. This will result in approximately 71 units of output.
ii. The firm can now only emply 20 units of labor and thus can only produce approximately 45 units of output.
iii. The firm should now employ 50 units of labor and only 62.5 units of capital which will give approximately 56 units of output.