3 1 point Firm A produces plastic keychains at its factory B

3. 1 point Firm A produces plastic key-chains at its factory. Both its labor and capital markets are competitive. Wages are S10 per hours and key-chain making equipment (a computer- controlled machine) rents for $5 per hour. The production function is 20K 0.50.5, where q = boxes of key-chains per week, K = hours of key-chain equipment used, and L = hours of labor. Therefore, MPL = 10K0.50.5 , and MPK = 10K 0.5 L0.5 a) Determine the cost-minimizing capital-labor ratio at this firm. b) How much does it cost to produce 16,000 boxes of key-chains? c) Suppose the rental cost of equipment increases to $6 per hour. What is the new cost- used, and L.- hours of labor,MIPL minimizing capital labor ratio?

Solution

(a) W= $10 per hour, R = $5 hour.

MPL =10 K^0.5L^-0.5 and MPK = 10 K^-0.5 L^0.5

Now, Cost minimising condition is MPL/MPK = w/r

10 K^0.5L^-0.5 / 10 K^-0.5 L^0.5 = 10/5

K^0.5 K^0.5 / L^0.5 L^0.5 = 2

K/L =2 [ Cost minimising capital-labor ratio].

(b) Firstly, put q= 16,000 and K =2L in the production function.

q = 20 K^0.5L^0.5

16000 = 20 (2L)^0.5 L^0.5

16000 = 20 (2)^0.5 L

16000/ 20 (2)^0.5 = L

L = 800/2^0.5

Put value of L , we get K = 2 L = 1600 /2^0.5

Now, put value of K and L in cost function, we get

Cost = wL + rK

= (10)(800/2^0.5) + (5)(1600/2^0.5)

= 8000/2^0.5 + 8000/2^0.5

= 16000/2^0.5

= $ 11,315

(c) Suppose rent , r = $6 per hour. Then , new capital labor ratio = K/L = 10/6 =5/3=  1.66.