Answer to Question #212476 in Microeconomics for Clarise

Question

A firm faces the production function below and can buy L at Php240 a unit and K at Php4 a unit.

A. If it has a budget of Php 16,000 what combination of K and L should it use to maximize output?

Q = 2K 0.2 L 0.6

Answer 1

$Q= 2K^{0.2}L^{0.6} ............ (1)$

Wage w = Php 240

rent = Php 4 

Budegt = 16000

Budegt line 

$wL+rK = budegt$  

Putting values 

$240L+4K = 16000 ....... (2)$

Slope of the isocost line $= -\frac{240}{4}$  

Marginal product of labour $MPL = 2(0.6)K^{0.2}L^{-0.4}$   

Marginal product of labour $MPK = 2(0.2)K^{-0.8}L^{0.6}$

Slope of the isoquants $=-MRTS =\frac {-MPL}{MPK}$

$MRTS =\frac {-2(0.6)K^{0.2}L^{-0.4}}{ 2(0.2)K^{-0.8}L^{0.6} }\\=-\frac{3K}{L}$

Condition of optimality 

The slope of isoquants = slope of the isocost line 

$-\frac{3K}{L}=-\frac{240}{4}\\\frac{3K}{L}=60\\K=20L$

Putting value of K in isocost equation 

$240L+4(20L) = 16000$

$i.e. 320L=16000$

$i.e. L =50$

So $K = 20(50) = 1000$

If K = 1000 and L =50, then we get maximum output.