Question

Let the production function be Q=L1/2 K1/2. Assume Capital, K=1 and the firm pays workers W

. a. Find the marginal product of labor.

b. Show the production function exhibits diminishing marginal productivity.

c. Show the relationship between marginal product and marginal cost

d. Show marginal cost increases as output increases/

Answer 1

Given: Q = L^1/2K^1/2 , K = 1 and wage rate = W

a) Q = L^1/2(1)^1/2

Q = L^1/2

MP_L = dQ/dL = 1/2*L^-1/2

MP_L = 0.5/L^0.5

b) MP_L = 1/2*L^-1/2

dMP_L/dL = (-1/4) * L^-3/2 = -0.25/L^1.5

Since L^1.5 cannot be negative, dMP_L/dL < 0

Thus, the production function exhibits diminishing marginal productivity.

c) MP_L = 0.5/L^1/2

TC = WL + RK

TC = WL + R(1)

TC = WL + R where let R = rent of captital = constant

TC = WQ² + R (Q = L^1/2 i.e. Q² = L)

Now, MC = dTC/dQ = 2WQ

or MC = 2WL^1/2

L^1/2 = MC/2W

Putting in MP_L, we get

MP_L = W / MC

So, it can be seen that marginal productitivity and marginal cost are inversely related.

d) From part c),

MC = 2WQ

From the above equation it can be observed that marginal cost and output Q are directly related. So, as output increases, marginal cost increases.