Miguel and Jake run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's long-run production function is Q=4K⁰.⁷⁵L ⁰.²⁵, where Q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. For this production function, the MPL=K⁰.⁷⁵/L⁰.⁷⁵and the MPK =3L⁰.²⁵/K⁰.²⁵. The weekly cost function for the paper plant is C=10K+2L, where C is the total weekly cost.
a) What ratio of capital to labor minimizes Miguel and Jake's total costs?
b) What is the optimal amount of capital and labor Miguel and Jake should rent and hire if they plan to spend $12,000 on producing reams of paper each week?
c) How many reams of paper will they be able to produce?