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

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