Cobb-Douglas production function

Question

This question really boggled my mind. Could you please solve and explain to me?

Answer 1

What is $w$ and what is $r$, are both values $>0$? Do we have complete competition, and hence fimrs are price-takers? Based on those assumptions and that we have an economy which is at equilibrium one would simply plug in the partial derivatives into the given equation:

We have $ wL= \frac{\delta Q}{\delta L}L$ and $rK= \frac{\delta Q}{\delta K}K$ so if you plug those terms into $ wL+rK$ and use the given information that $Q=wL+rK$, you get:

$50 *K^{0.4} *L^{0.6} = 50*0,6*L^{-0,4}* K^{0,4}*L+50*0,4*K^{-0,6}*L^{0,6}*K$.

Now manipulate the right hand side and you see immediately that both side are equal. This means that the income of the households equals the amount of goods produced at equilibrium.