Struggling to put two equations together effectively again. I have my income equation:
$$ y=zk^\alpha $$
And I'm trying to plug it into my marginal product of capital
$$ MPK=\alpha z k^{\alpha-1} $$
I know the end product is
$$\alpha (zy^{\alpha -1})^\frac{1}{\alpha} $$
But I can't see how to convert income into an effective equation to plug into my problem. I always end up with funny powers that don't seem to fit.
Thanks!
You have $y=zk^\alpha$, so you can express $k$ in terms of $y$ as follows:- $$k^\alpha=\left(\frac{y}{z}\right)\Rightarrow k= \left(\frac{y}{z}\right)^{\frac{1}{\alpha}} $$ Substituting the value of $k$ into the $MPK$ results in:- $$MPK=\alpha z k^{\alpha-1}=\alpha z\left(\frac{y}{z}\right)^{\frac{\alpha-1}{\alpha}}=\alpha z^{(1-\frac{\alpha-1}{\alpha})}y^{\frac{\alpha-1}{\alpha}}=\alpha z^{\frac{1}{\alpha}}y^{\frac{\alpha-1}{\alpha}}=\alpha (zy^{\alpha-1})^{\frac{1}{\alpha}}$$