Question

Show how to derive a relation between the marginal products of labour and capital, MPL and MPK, andf MRTS

Answer 1

Suppose a production function is as below:

Y = AL^a K^b

Where,

Y = total production in units

L = Labor input in units

K = Capital input in units

A = Total constant factor productivity

a = Output elasticity of labor

b = Output elasticity of capital

Now,

MPL = Derivatives of Y with respect to L

            = (d/dY) (AL^a K^b)

            = aAK^b L^(a – 1) [Answer]

MPK = Derivatives of Y with respect to K

            = (d/dY) (AL^a K^b)

            = bAL^a K^(b – 1) [Answer]

MRTS = MPL / MPK

            = aAK^b L^(a – 1) / bAL^a K^(b – 1)

            = (a/b) {(L^(a – 1) ÷ L^a) × (K^b ÷ K^(b – 1))}

            = (a/b) {L^(a – 1 – a) × K^(b – b + 1)}

            = (a/b) {L^(-1) × K^1}

            = (a/b) × (K/L)

            = aK / bL (Answer)