Question

1. A firm uses labor and machines to produce output according to the production function f(L, M) 2L2 M , where L is the number of units of labor used and M is the number of machines. The cost of labor is $20 per unit and the cost of using a machine is $5. a. Suppose that the firm wants to produce its output in the cheapest possible way. Find the input demand functions for machines and workers. Please show your work. What are the cost-minimizing numbers of machines and workers if the firm wants to produce 20 units of outputs? What is the cost of producing 30 units at these factor prices? Please show your work. b. c. Now suppose that the firm considering using a new perfect substitute technology with the production function f(L, M)=3L+ M (input prices are still the same). Under the new technology; what are the cost-minimizing numbers of machines and workers if the firm wants to produce 30 units of outputs? Should the firm start using the new production technology? Explain your answer.

Answer 1

1 - q = 2 M ; w= $20 ; g = $5 can least cost combination is determined where MRTS = 0. - MRT5 MB 2(1) M MPM aglam 264), + ('z pike So, MRTS = = = 20 =4 =>{M=42] . So, q= 262 m k = 2² (4273 = 22² (2) 1² = UL = L = q) .M=46= 4(72q => m=q) (6239 = 20 = 1 = e = 5 =)[T=5) M= 20) 1- q = 30 =) = 30 = 7,95 = L = 1.857 and [M=30] Cost = wl +M = 20 (7.5) + $(30) = 150 + 150 = ? (coal = 3007 co 9-31 +M This is a substitutes function where one or two inputs will be used depending on the relationship between MRTS and MRTS = MP - Qq laL 3. MPi Sqram = Ž = 3 So, MRTS (W . Thus, only M will be used (T=0] ; 9 = 30 = 3L +M = 360) +M -31 M- 36 | Cost = WL + OM = 90260) + (55)30) = 0 + 150 - Lost & 150 ost is cower under this new technology. firma should start using the new production technology anner Thus,