Phil and Bob run a paper company. Each week they need to produce 2,000 reams of
ID: 1189430 • Letter: P
Question
Phil and Bob run a paper company. Each week they need to produce 2,000 reams of paper to ship to their customers. The paper plant’s long-run production function is Q= 2K 0.25 L 0.75, where Q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. The rental rate of capital is $5 and the price of labor is $10.
a. What ratio of capital to labor minimizes Phil and Bob’s total costs?
b. How much capital and labor will Phil and Bob need to rent and hire in order to produce 2,000 reams of paper each week?
c. How much will hiring these inputs cost them?
Explanation / Answer
Q = 2K0.25L0.75
w = price of labor = 10
r = price of capital = 5
Total cost, TC = wL + rK
= 10L + 5K
(a) To minimize total costs, the following condition should be satisfied:
MRS = MPL / MPK = w / r
Where MPL = dQ / dL = 2 x 0.75 x (K / L)0.25
MPK = dQ / dK = 2 x 0.25 x (L / K)0.75
So, MRS = MPL / MPK = 3 x (K / L)
Equating MRS with price ratio:
3 x (K / L) = 10 / 5 = 2
3K = 2L
Or, (K / L) = (2/3) [Required cost-minimizing ratio]
(b)
Q = 2K0.25L0.75 = 2,000
And 3K = 2L, or K = (2L / 3) & L = (3 / 2)K
So
2000 = 2 x (2/3)0.25 L0.25L0.75 = 1.81L
L = 2,000 / 1.81 = 1,105 [Rounded off]
So, K = 2L / 3 = 2 x 1105 / 3 = 737 [Rounded off]
(c) With these K & L combination,
TC = 10L + 5K
= (10 x 1105) + (5 x 737)
= 14,735
So, input costs = $14,735