Definition
Firstly, consider the supply function:
QS = S(PS): output of production is a function of producer's price. Quantity is a function of price.
Interpretation:
It can be regarded as from the profit maximization problem: for a given output price (as well as input price), the producer would solve for the optimal amount of input use to maximize the profit (output price = marginal cost). The optimal amount of input would also gives the optimal output level given current price vector.
Or, it can be regarded as a collection of producers with different marginal cost. When price increases, more producers have their marginal cost <= price, so they start to produce.
Similarly,the demand function QD = D(PM) can be interpreted by either the substitution effect for utility maximization function, or a collection of consumers. When price decreases, more consumer demand that good.
Then a supply-demand graph tell us that given the supply and demand function, when price reach a certain level, amount of demand = amount ot supply, the economy solves out. That is to say: we solve the economy when QD = QS (in perfect market, PS = PM. However, when there are tax or quota, PS may not equal PM, but we still have the same Q for this commodity) That is the key when I should "stop" the analysis.
Without tax, there is only one price in the economy. If there is tax, there are two price in the economy, one faces by producer and the other one faced by consumer .
When there is a tax, it can be interpreted as tax on producer: for given price, producer would produce less goods, or tax to consumer: for a given price, consumer would demand less goods.
In any case, now the economy would end up with quantity Q.If consumer bear the tax, producer gets P2 but consumer pays P1. So the consumer surplus changes from A+B+E to A, producer surplus changes from C+D+F to D. The tax rate = P1 - P2. When the quantity is Q, the government gets tax value Q(P1- P2), or B+C. Then the deadweight loss is E+F
Similarly, consider subsidy:
Hint: Use Q and original supply and demand curve to research each agent's welfare! Suppose subsidy is given to producer. In the analysis of Q, we assume that the supply curve move rightward. However, if we say from producer's side, with the subsidy, its supply curve does not move. So the next step analysis: analysis of welfare should be based on original supply / demand curve!
Consumer surplus:
A + B → A + B + D +F
Producer surplus:
D+G → D+G+B+C
Total subsidy: B+D+C+F+E
Deadweight loss (by government): E.
Note: the tax / subsidy rate is not added to previous equilibrium price, because with the tax / subsidy, the equilibrium price and quantity would change. The tax / subsidy rate is the difference between two agent's price!