Answer to Question #309198 in Microeconomics for Dilsha

Question

1)  A firm’s production function is given by the equation, Q = 25 K^0.5L ^0.5 where ‘Q’ represents units of output, ‘K’ units of capital and ‘L’ units of labor.

a)              Does this production function exhibit increasing, decreasing or constant returns to scale?

b)              Determine MRTSLK

c)              Calculate APL and APK when firm utilizes 25 units of capital and 100 units of labor.

d)              The firm utilizes 40 units of labor and 80 units of capital, if the price of labor is $ 50.00, price of capital $ 100.00 and total capital $ 10,000.00.

Is this factor combination optimum? Give reason. 

Answer 1

Given that $Q=25K^{0.5}L^{0.5}$

a) to determine whether the function is decreasing, constant or increasing, multiply all the inputs by a constant $k$

$Q(kK,kL)=25(kK)^{0.5}(kL)^{0.5}$

$\space=k^{0.5+0.5}(25K^{0.5}L^{0.5})$

Since $0.5+0.5=1$ , the function exhibits a constant returns to scale.

b) $MRTS_{LK}=\frac{dK}{dL}=\frac{-F_L}{F_K}$

$\frac{dK}{dL}=\frac{-F_L}{F_K}=\frac{-12.5K^{0.5}L^{-0.5}}{12.5K^{-0.5}{L^{0.5}}}$

$MRTS_{LK}=\frac{-K}{L}$

c) $AP_L=\frac{Q}{L}=25K^{0.5}L^{-0.5}$

$AP_K=\frac{Q}{K}=25K^{-0.5}L^{0.5}$

At K=25 units and L=100 units

$AP_K=25(25)^{-0.5}(100)^{0.5}=50$

$AP_L=25(25)^{0.5}(100)^{-0.5}=12.5$

d) At the optimum factor combination,

$MRTS_{LK}=\frac{w}{r}$

Where $MRTS_{LK}$ is $\frac{-K}{L}$ w is price of labor $\$50$ and r is the price of capital $\$100$

At the 40 units of labor and 80 units of capital

$\frac{-80}{40}\mathrlap{\,/}{=}\frac{-50}{100}$

This factor combination is not optimum.