Question

Suppose that output q represents the number of products produced,K is the number of machines used, and L the number of workers used. To produce one unit of q, one must can use 1 machine OR 4 workers. The cost of a unit of K is r= 100, and the cost of a unit of L is w= 20:First, draw the isoquant for q= 10. Next, draw the isocost line for C= $2000.Using your isocost-isoquant diagram, identify the cost-minimizing number of machines and workers to produce q= 10products per hour.(b) What is the MRTS_K,L for the following production function:Q=KL^2? Is this technology CRS, IRS or DRS? How do you know? (C) What is the MRTS_K,L for the following production function:Q= 2K+L?Is this technologyCRS, IRS or DRS? How do you know?

Answer 1

(a) Isoquant line for q = 10 = K = 10, L/4 or L = 40 (see graph A)

     Isocost line for C = $ 2000 = 100K + 20L = 2000 or, 5K + L = 100 (see graph B)

            For q =10, Minimum cost combination of K and L is as follows: (see graph C)

From the graph, it can be seen that, minimum cost combinations of L and K for q =10 is

(L, K) = (40,0) = C(40,0) = 40

(a) MRTS (K, L) = -dL/dK = MPK/MPL

                     For production function: Q = KL2

                     MPK = L2

                     MPL = 2KL

                     Or, MRTS (K, L) = L2/2KL = L/2K

Now, If K and L increases by m

Q = mk (mL)2 = m2kL2

Thus, Q increases by more than m, thus there is Increasing Return to Scale

(b) For production function: Q = 2K + L

MPK = 2

MPL = 1

Or, MRTS (K, L) = 2/1 = 2

Now, If K and L increases by m

than Q = 2mK + ml

Q = 2m(2K + L)

Thus, Q increases by more than m, thus there is Increasing Return to Scale