Answer to Question #221299 in Microeconomics for Riswan

Question

The Long-run production function is given by; Y = 180 L1.2 K1.8

Where, Y = Output (mt/day), L = Labour (hours/mt) K = Capita (Rs/mt)

a) Calculate Marginal Product of Labour (MPL) and Marginal Product of Capital (MPK), if L=12 and K=20

b) Derive the equation for Isoquent and graphically show it by assuming L= 10, 15, 20 25 and 30.

c) Determine factor intensity and returns to scale of this production function.

d) Prove that the elasticity of labour is 1.2 and elasticity of capital is 1.8

Answer 1

a) The long-run production function is given as,

$Y=180\times L^{0.8}K^{1.8}..............................................(i)$

where, Y = units of output produced

L = units of labor employed

K = units of capital employed

 

The marginal physical (MPL) productivity of labor states the change in units of output produced due to employment of an additional unit of labor. It is derived as,

$MPL=\frac{\delta Y}{\delta L}$

$=0.8\times L^{-0.2}$

Now if L = 12 units then MPL,

$=0.8\times (12)^{-0.2}\\=0.487$

The marginal physical (MPK) productivity of capital states the change in units of output produced due to employment of an additional unit of capital. It is derived as,

$MPK=\frac{\delta Y}{\delta K}$

$= 1.8\times K^{0.8}$

Now if K=20 then MPK,

$=1.8\times(20)^{0.8}\\ = 19.77$

b) The isoquant shows various combinations of inputs that manufactures same level of output.

The equation of isoquant is obtained in the following manner –

$Y=180\times L^{0.8}\times K^{1.8}$

$K^{1.8}=\frac{Y}{180\times L^{0.8}}$

$K=[\frac{Y}{180\times L^{0.8}}]^{\frac{1}{1.8}}$

$K=[\frac{1}{180\times L^{0.8}}]^{\frac{1}{1.8}}..............................................................(1)$

The equation (1) is the equation of unit isoquant (since Y (output) = 1).

The following figure depicts the graph of a unit isoquant –

c) The factor intensity shows the relative importance of inputs in the production process. In others words, it measures the amount of each type of input required to produce a certain amount of output. Here, MPK > MPL, this means that in order to produce 1 unit of the good the requirement of capital is higher than the requirement of labor. Therefore, he production process is capital-intensive.

The returns to scale shows the amount by which the output changes as all the factors are altered by same proportion. The returns to scale is calculated in the following manner,

$Y=F(180\times L^{0,8}\times K^{1.8}$

Now, if$\space L=α\times L$ and $K = α\times K,$ then

$Y'=F[180\times (αL)^{0.8}\times (αK)^{1.8}]$

$=\alpha^{0.8+1.8}\times F(180\times K^{1.8})$

$=\alpha^{2.6\times Y}.......................................................(where, \alpha>0)$

So, as the inputs are changed by changed by α, the output increases by more than α times.

Hence, the production function expresses increasing (IRS) returns to scale.