In the Keynesian model,
to make as much profit as possible, firms will choose the level of the real wage that gets the most effort from workers for each dollar of real wages paid
- Source: Macroeconomics, 10e Andrew B. Abel, Ben S. Bernanke, Dean Croushore Copyright © 2020 Book ID: 1FR37TIEM0I
I think this is assuming that worker effort is equivalent to output, which is equivalent to revenue.
But why does maximizing revenue/cost maximize profit? I get why maximizing revenue-cost would maximize profit (because revenue-cost is the definition of profit).
Assuming revenue is r(w) and wage is w, we are setting $\frac{\partial}{\partial w}\frac{r(w)}{w}=0$, which is $\frac{r'(w)w-r(w)}{w^2}=0$, so $r'(w)w-r(w)=0$. But how is that the same as $\frac{\partial}{\partial w}(r(w)-w)=0$ or $r'(w)-1=0$ which is profit maximization?
Not sure if I am thinking about this the right way.