Question

T uule DelldIUILUU UI Clapiei IIUDICII Michael, Dwight, and Jim 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 = 4K0.75 0.25, where 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 marginal product of labor is MPL = U, and the marginal product of capital is MPx = 30$. The weekly cost function for the paper plant is C= 10K + 2L, where is the total weekly cost. a. What ratio of capital to labor minimizes Michael, Dwight, and Jim's total costs? b. How much capital and labor will Michael, Dwight, and Jim need to rent and hire in order to produce 1,000 reams of paper each week? Round to the nearest whole number. Quantity of capital: units Quantity of labor: workers workers

Answer 1

a)

Cost is minimized at that input combinations where

MPL Wage Rate MPK Rental Rate

Wage Rate = 2

Rental Rate = 10

Solving this fraction by putting in values of MPL , MPK , wage rate and rental rate gives us

K/3L = 1/5

K = 3L/5

This is the cost minimising ratio of capital and labor

b) Q = 1000

Put K = 3L/5 in prouction function

Q = 1000 = 4 x (3/5)^0.75 x L

Solving this gives us

L = 367.64 and K = 220.58

Rounding to nearest whole numbers

L = 368 , K = 221

c) Cost = 10K + 2L = 10(221)+ 2(368) = $2946