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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!

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    I can explain why $\Delta y/\Delta t=-1/2b$. First, notice that $y$ depends on $t$, so one can write $y=y(t)=(a-c-t)/2b$. Hence $$\frac{\Delta y}{\Delta t}=\frac{y(t+\Delta t)-y(t)}{\Delta t}=\frac{(a-c-t-\Delta t)-(a-c-t)}{2b\Delta t}=-\frac{1}{2b}.$$2012-09-26
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    I suspect demand being $a-by$ is an assumption, making revenue $(a-by) \times y = ay - by^2$ and marginal revenue its derivative with respect to price $a-2by$. If not, then you can do the calculation in reverse (assuming there is no revenue when there is no demand).2012-09-26
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    sorry @Merci but I still don't fully understand, is this a partial derivative? are a and c variables (t is a variable right?). thanks so much so far!2012-09-26

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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: $$at-bt^2=(a-bt)t\longrightarrow p(y)=a-by$$

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    Nice work! I hope you awaken refreshed (as at the time of my posting this comment, I hope you are in a heavenly slumber! :-)2013-03-22
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    Thanks for nice charming words, Amy.2013-03-22
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    Hello! I've been "waiting" for you, dear friend!2013-03-22