This is what the lecturer drew in my class. $S_T$ is supply curve after a tax of $t$ dollars per unit is imposed .But I don't understand several things.
1. He said the tax amount is from $P_T^* $ to $P_T^*-t$. But shouldn't it be from $p^*$ to $p^*_T$
2.Also please explain how the consumer share of tax and producer share of tax is obtained.
3.Also how is producer surplus be that given area.
I really don't understand what is going on.So all the help is appreciated. Also please tell me some good books where I can understand these concepts and also good online courses available.
The yellow rectangle on the picture is inaccurately drawn: its lower right vertex should be higher, on the pre-tax supply curve. In present form, it appears to be based on the value of $p$ where the post-tax supply curve meets $q=0$, which isn't a relevant quantity here. Here is my version:
The tax amount is $t$, which is the amount by which $S_T$ is higher than $S$. The idea is that the price at which the producer is willing to sell now includes $t$. As a result of this raise in the supply curve, the equilibrium point shifts not only up but also to the left: the equilibrium price goes up and the demand goes down. It is a purely geometric fact that the intersection point does not move up as much as the supply curve itself. Thus, the quantity $p^*_T-p^*$, i.e., the amount by which equilibrium price goes up, is less than the amount of tax.
That quantity $p^*_T-p^*$ is the consumer share of tax, since it describes how much more consumers will pay per unit. The rest of tax is, therefore, producer's share.
Example
Before tax, the supply equation is $p=15q+20$ and demand equation is $p=100-5q$. The equilibrium is found from $15q+20 = 100 - 5q$, namely $q=4$, $p=80$.
Add the tax of \$10 per unit. The new supply equation is $p=15q+30$; the supply curve (line) went up by $10$. The new equilibrium is found from $15q+30 = 100-5q$, namely $q=3.5$, $p=82.5$.
In this example, only \$2.50 of tax is passed to consumers; the rest (\$7.50) is producer's share.
The producer surplus is a bit different topic; the fact that it's equal to red area is basically the definition (again, note that the red area is too small, because the yellow rectangle is too big). The Economic surplus wiki explains it pretty well, I think. See also Deadweight loss.