I am not able even after a whole hour spent on figuring this out to understand the following passage. Please, can you help me out somehow? It is part of the explanation of how a tax affect a monopoly. We start by the assumption that marginal revenue equals margina cost (plus the tax):
$$a-2by = c+t$$
and we solve for y which leads to:
$$y=\frac{a-c-t}{2b}$$
Than I don't understand how I can get to this: $$\frac{\Delta y}{\Delta t}=- \frac {1}{2b}$$
and finally I don't understand why the demand is: $$p(y)=a-by$$
Thanks for the help in advance!
By definition $y'=\lim_{\Delta t\to 0}\frac{\Delta y}{\Delta t}$ so when $\Delta t$ is so small , $y'\thickapprox\frac{\Delta y}{\Delta t} $ and then you have : $$y'=\frac{-1}{2b}$$ Moreover, you know that $$\int MR=TR$$ This means that: $$TR=\int (a-2bt)dt=at-bt^2$$ Regarding the definition of Total Revenue, $TR=p(y)y$ wherein $p$ is your demand function, so: $$at-bt^2=(a-bt)t\longrightarrow p(y)=a-by$$