Answer to Question #193149 in Microeconomics for Leilua

Question

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

a)   What is the marginal product of labour and capital (1 marks)

b)   What is the marginal rate of technical substitution of labor for capital (2 marks)

c)   What is the elasticity of substitution at a point K = 1, L = 1 if we increase K by one unit? (2 marks)

Answer 1

(a) $Q = 2(KL)^{0.5}$

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

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

 

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

Marginal product of labor $= \dfrac{K^{0.5} }{ L^{0.5}}$

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

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

$\Rightarrow \dfrac{dQ}{dK} =\dfrac{ L^{0.5}} { K^{0.5}}$

Marginal product of capital $= \dfrac{L^{0.5}} { K^{0.5}}$

 

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

=$\dfrac{∆K}{∆L}$

$=\dfrac{L^{0.5} K^{0.5}}{K^{0.5} L^{0.5}}$

$=\dfrac{K}{L}$

(c) Elasticity of substitution σ is given by

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

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

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

Taking log on both sides

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

Taking derivative both sides we get

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

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