Consider the following market with the demand and supply equations

where is the quantity demanded and 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 () such that:

Solving for yields:

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:

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 which I define as the price + the tax rate, such that:

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

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

And the after-tax price is:

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

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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