Question

(15 points) If input prices are w = 3, and r 2, and q 10KL, what is the least cost input combination required to produce 60 units of output? How would input usage change if output is increased to 240 units? Sketch the solutions on a graph.

Answer 1

If output is 60 units

q = 10KL or for 60 units of output we write 60 = 10KL, KL = 6

MPk = dq/dK = 10L

MPl = dq/dL = 10K

Equilibrium or output max condition

MPl/MPk = w/r

10K/10L = 3/2

K/L = 3/2

K = 3/2L

we also have KL = 6 or L = 6/K (substitute value of L in the equation above)

K = 3/2*6/K

K^2 = 18/2 = 9

K = 3 (K cannot be negative so negative value of K is ignored)

L = 6/3 = 2

If output is 240 units

q = 10KL or for 60 units of output we write 240 = 10KL, KL = 24

MPk = dq/dK = 10L

MPl = dq/dL = 10K

Equilibrium or output max condition

MPl/MPk = w/r

10K/10L = 3/2

K/L = 3/2

K = 3/2L

we also have KL = 24 or L = 24/K (substitute value of L in the equation above)

K = 3/2*24/K

K^2 = 72/2 = 36

K = 6 units (K cannot be negative so negative value of K is ignored)

L = 24/6 = 4 units

3 2 4 Labor Q=240 Q=60 capital K Le The graph is not to seale. This is for representation of figures only.