Question

Consider the production function Q = 2(KL)^0.5

a)     What is the marginal product of labour and capital

b)     What is the marginal rate of technical substitution of labor for capital

 c)     What is the elasticity of substitution at a point K = 1, L = 1 if we increase K by one unit?

 

Answer 1

(a) $Q = 2(KL)^{0.5}$

$Q = 2 K^{0.5} L^{0.5}$

Marginal product of labor = Differentiate the above production function with respect to L

$\frac{dQ}{dL}= 0.5 × 2 K^{0.5 }L^{0.5-1}$

 $\frac{dQ}{dL}= 0.5 × 2 K^{0.5 }L^{-0.5}$

$\frac{dQ}{dL}=\frac {K^{0.5 }}{L^{0.5}}$

Marginal product of labor = K^0.5 / L^0.5

$Q = 2 K^{0.5 }L^{0.5}$

Marginal product of capital = Differentiate the above production function with respect to K

$\frac{dQ}{dK} = 0.5 × 2 × K^{0.5-1} L^{0.5}$

$\frac{dQ}{dK }= 0.5 ×2 × K^{-0.5 }L^{0.5}$

$dQ/dK =\frac{ L^{0.5}} { K^{0.5}}$

Marginal product of capital = L^0.5 / K^0.5

 

(b)marginal rate of technical substitution of labor for capital =change in capital/chang in labor

=$\frac{∆K}{∆L}$

$\frac{L^{0.5} K^{0.5}}{K^{0.5} L^{0.5}}$

$=\frac{K}{L}$

(c) Elasticity of substitution σ is given by

$σ=\frac{d ln(k/l)}{d ln(|MRTS|)}$

$=\frac{d ln(k/l)}{d ln(|MRTS|)}=1$

$MRTS=\frac{k}{l}$

Taking log on both sides

$ln\frac{k}{l}=ln({|MRTS|})$

Taking derivative both sides we get

$d ln\frac{k}{l}=d ln(|MRTS|)$

$σ=\frac{d ln(\frac{k}{l})}{d ln(|MRTS|)}=1$