Equilibrium price and quantity after tax

Consider the following market with the demand and supply equations

Q_D = 100 - 2P

Q_S = 3P

where Q_D is the quantity demanded and Q_S is the quantity supplied. There is currently no tax applied to this market. We solve for the equilibrium price and quantity by equating demand and supply (Q_D = Q_S) such that:

 100 - 2P = 3P

Solving for P yields:

 P = 20

The equilibrium quantity can be determined by substituting price back into the supply or demand equation. Using the supply equation we see that the equilibrium quantity is:

Q = 3*(20) = 60

Now suppose that the government decides that consumers will pay a tax of $1 per unit. In this case the tax is levied on the demand side of the market. I find it easy to denote the after-tax price paid by consumers as being a new variable P_t which I define as the price + the tax rate, such that:

P_t = P + 1

Substituting that new price into the demand equations yields the new demand equation:

Q_D = 100 - 2P_t

Q_D = 100 - 2(P + 1)

We can now equate the supply and demand equations, giving:

100 - 2(P + 1) = 3P

P = 19.6

And the after-tax price is:

P_t = 20.6

Substituting this back into the supply equation yields the new equilibrium quantity of output:

 Q = 58.8

In this case, the price received by consumers decreases, the price paid by consumers increases and the equilibrium quantity goes down.

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