Answer to Question #193160 in Microeconomics for Addo Benjamin

Question

Assume the following production function for maize: Q=3K^1.2 L^0.5, where Q is the number of bags of maize produced, K is the units of capital used and L the units of the labour employed.

(a) Derive the MRTS of K and L.

(b) Estimate the factor intensity of the production function and interpret results.

(c) Is the above production function characterized by decreasing returns of scale? Explain

Answer 1

$Q=3K^{1.2} L^{0.5}$

Q is the number of bags of maize produced

K is the units of capital used 

L the units of the labor employed

(a) MRTS

$MRTS_{KL}=\frac{MPL}{MPK}$

MPL=Chnage in output/Change in Labor

$MPL=\frac{∆Q}{∆L}$

$Q=3K^{1.2} L^{0.5}$

$\frac{∆Q}{dL}=3×0.5K^{1.2}L^{-0.5}$

$MPL=1.5K^{1.2}L^{-0.5}$

MPK=Change in Output/Change in capital

$MPK=\frac{∆Q}{∆K}$

$Q=3K^{1.2} L^{0.5}$

$\frac {∆Q}{∆K}=3×1.2K^{0.2}L^{0.5}$

$MPK=3.6K^{0.2}L^{0.5}$

$MRTS=\frac{1.5}{3.6}×\frac{K}{L}$

(b) Factor Intensities

Factor intensities are measured by using the share of each input factor in the production function.

$Q=3K^{1.2} L^{0.5}$

Factor intensity=Share of the labor/Share of the capital

Factor intensity$=\frac{0.5}{1.2}=0.416$

Larger the value of this ratio, the technique will be labor-intensive

Lower the value of this ratio, the techniques will be capital intensive.

So, we can see that the value is lower, hence, the technique used in this production function is capital intensive.

 

(c) Returns to scale

To check for decreasing returns for scale, we will increase the input by some constant 't'. If the new output increases by less than 't', then the production function is decreasing returns to scale.

If the new output is greater than t-Increasing returns to scale

If the new output is less than t-Decreasing returns to scale

If the new output is equal to t-Constant returns to scale.

$Q=3K^{1.2} L^{0.5}$

$Q'=3(tK)^{1.2}(tL)^{0.5}$        [Q' is the new output]

$Q'=t^{1.2}+0.5[3K^{1.2}L^{0.5}]$

$Q'=t^{1.7}Q$

So, the new output is greater than t.

Hence, increasing returns to scale and not decreasing returns to scale.