This question will ask you to consider a firm with the following CES

production function:

q(K, L) = 

2

√

K + 0.5

√

L

2

This question will ask you to determine the optimal choice of labour and capital as

well as determine what the cost curves look like for a given set of prices. Suppose the

market price for the good being sold is P = $16, the wage rate is w = $12, and the

rental rate is r = $24.

(a) (4 point) What is the optimal choice of labour in the short run if the firm currently

has 144 machines (i.e. K¯ = 144)? Given this choice of capital, does the firm

produce in the short run? If yes, what quantity?

(b) (3 points) What is the short run cost function? Show the quantity the firm would

want to produce the same as in part (a).

(c) (6 points) In the long run both capital and labour can be adjusted. What is the

optimal amount of capital and labour hired? Does the firm make a profit here?

How does profit compare with part (a)?

(d) (4 points) Find the long run cost curve. Show that you get the same outcome as

in part (a) and part (c) for the optimal choice of output.