Question

The Cobb-Douglas function for a new product is given by:                                                            

N (L,K) = 1.19L^0.72K^0.18                  

Where    C* =Rs. 750,    w = Rs. 30,     r = Rs. 40

Determine the amount of labor and capital that the firm should use in order to maximize output. What is this level of output? Also interpret the value of Lagrange multiplier, λ.  

Answer 1

given Cobb-Douglas production function as

$N (L,K) = 1.19L^{0.72}K^{0.18}$

the cost function will be

$C=wL+rK\\750=30L+40K$

The Lagrangian optimization equation is then given by:

$\delta(L,K)=1.19L^{0.72}K^{0.18}+\lambda(750-30L-40K)$

$\frac{\delta(L,K)}{\delta L}=0.8568\frac{K^{0.18}}{L^{0.18}}-30\lambda=0$

$\frac{\delta(L,K)}{\delta K}=0.2142\frac{L^{0.72}}{K^{0.72}}-40\lambda=0$

Dividing the first equation with the second, we get:

$\frac{4K}{L}=\frac{3}{4}\\K=\frac{3L}{16}$

Now, taking the derivative of the Lagrangian with respect to lambda, we get

$\frac{\delta(L,K)}{\delta \lambda}=750-30L-40K=0$

Using $K = \frac{3L}{16}$ we get

$750-30L-40\frac{3L}{16}=0\\70-30L-7.5L=0$

From this, we get L = 20

Therefore, $K = \frac{3L}{16} = \frac{(3 x 20)}{16} = 3.75$

Therefore, the level of output is given by $N(L, K) = 1.19 \times (20)^{0.72} \times (3.75)^{0.18} = 1.19 \times 8.644 \times 1.268 = 13.043$

The Lagrangian multiplier's value can be calculated from the calculated values of K and L and putting them in the first derivative equation.

$0.8568\frac{3.75^{0.18}}{20^{0.18}}=30\lambda\\\frac{1.086}{1.714}=30\lambda\\\frac{0.6336}{30}=\lambda=0.021$

Here, the Lagrangian multiplier tells us the rate of change in the total output with respect to a change in the inputs (K and L).