# 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.