Question

Consider the production function Q = 2(KL)0.5

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

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

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

Answer 1

With the given function of production $Q=2(KL)^{0.5}$

1) To find the products (MPK and also MPL), we use this formula. 

$Q=2(KL)^{0.5}$

$MPK=\frac{∂Q}{∂K}=L^{0.5}$

$MPK=L^{0.5}$

$MPL=\frac{\delta Q}{\delta L}=(KL)^{-0.5}$

2) To find the value of substitution (technical) of labor for capital

$MRTS=\frac {MPL}{MPK}$

$MPTS=\frac{KL^{-0.5}}{L^{0.5}}$

$MRTS=\frac {1}{KL^{0.5}}$

3) To find how much the substitution is elastic

$ε=(\frac{\Delta \frac{1}{k}}{\Delta MRTS})\frac{MRTS}{\frac{1}{k}}$

we know that $MRTS=\frac {1}{KL^{0.5}}$

taking the derivative

$MRTS=\frac {1}{KL^{0.5}}$

$MRTS=0.5(KL)^{-1.5}$

$MRTS=\frac {1}{0.5(KL)^{1.5}}$

$MRTS=\frac {1}{0.5(KL)^{1.5}}$

$\frac{\Delta \frac{1}{k}}{\Delta MRTS}=\frac {1}{0.5(KL)^{1.5}}$

$ε=\frac{1}{0.5KL}$

$ε=\frac{1}{0.5(1\times 1)}$

$ε=2$

the elasticity of substitution would be 2

but, if K is raised by 1, then

$ε=\frac{1}{0.5KL}$

$ε=\frac{1}{0.5(2 \times 1)}$

$ε=1$

Now the elasticity will fall to 1