Question

A firm has a production function y = X1X2. If the minimum cost of production at w1 = w2 = 1 is equal to 4, what is y equal to?

Answer 1

As per the given question,

$y=x_1 x_2$

Now,

$L=w_1x_1+w_2x_2+\lambda(y-x_1x_2)$

$0=\frac{dy}{d\lambda}=y-x_1x_2$

$0=\frac{dy}{d\lambda}=w_1-\lambda x_2$

$0=\frac{dy}{d\lambda}=w_2-\lambda x_2$

So, $\lambda = \frac{w_1}{x_2}=\frac{w_2}{x_1}$

We know that $4=w_1x_1+w_2x_2$

$4=w_1x_1+\frac{w_1\times x_1}{w_2}w_2$

$x_1=\frac{4}{2w_1}=\frac{2}{w_1}$

$x_2=\frac{2}{w_2}$

Hence, $y=x_1 x_2$

$y=\frac{2\times 2}{w_1\times w_2}$

$y=4$