Question

4. Suppose a small oil drill has the following production function F(K,L) = min(4K,L)

where every drill (captial unit) takes 4 people to operate. Output is measured in barrels.

(a) Suppose there are 10 drills in the oil field. How many workers are needed to produce 40 barrels of oil (q=40)?

(b) Graph the isoquant curves that represent q=20, q=40, and q=60.

(c) Setup the cost minimization problem where labor and capital are flexible. Then find the cost function if the oil company wants to minimize costs?
• L(w,r,q) =

• K(w,r,q) =

•C(w,r,q)=

(d) Suppose the oil company wants to produce 100 barrels. What are the minimum amounts of capital and labor required? What is the cost of producing 100 barrels?

Answer 1

Q = F(K, L) = min(4K,L)

(a) For a fixed proportion production function, cost is minimized when 4K = L.

When q = 40, L = 4K = 4 x 10 = 40

(b) For a fixed proportion production function, isoquants are L-shaped as shown below.

(c) Total cost (C) = wL + rK

Substituting 4K = L in cost function,

C = w x 4K + rK

C = K x (4w + r)

K = C / (4w + r)

L = 4K = 4C / (4w + r)

q = min[{4C / (4w + r)}, {4C / (4w + r)}]

q = 4C / (4w + r)

C = q(4w + r) / 4

(d) When q = 100, we have

100 = min(4K, L)

4K = 100

K = 25

L = 100

Total cost = 100 x (4w + r) / 4 = 25 x (4w + r) = 100w + 25r