A per-unit tax (an excise tax) works like a supply shifter (What Shifts Supply): sellers now need more per unit to be willing to sell. The result is a smaller market, a gain for the government and a deadweight loss (Total Surplus and Deadweight Loss).
Key idea
A tax of $t$ per unit drives a wedge between the price buyers pay, $P_b$, and the price sellers keep, $P_s$: $P_b - P_s = t$. The quantity falls to $Q_t$, where buyers' quantity at $P_b$ equals sellers' quantity at $P_s$. Tax revenue $= t \times Q_t$. Deadweight loss $= \tfrac{1}{2} \times t \times (Q^* - Q_t)$.
Why it works
Collected from sellers, the tax means a seller who needed \$4 to sell a unit now needs \$4 plus the tax: the supply curve shifts up by $t$. The new crossing with demand gives the buyers' price and quantity; sellers keep that price minus $t$. Revenue is a rectangle of height $t$ and width $Q_t$, money that moves from buyers and sellers to the government. The units between $Q_t$ and $Q^*$ would have been worth more to buyers than they cost sellers, but the tax stops them: that triangle is lost.
Worked example: A \$3 tax
$Q_d = 20 - 2P$ and $Q_s = -4 + 2P$. A \$3 per-unit tax is collected from sellers. Find $P_b$, $P_s$, the quantity, the revenue and the deadweight loss.
- Sellers keep $P_b - 3$: set $20 - 2P_b = -4 + 2(P_b - 3)$.
- $20 - 2P_b = -10 + 2P_b$, so $30 = 4P_b$ and $P_b = 7.5$; $P_s = 4.5$.
- $Q_t = 20 - 2(7.5) = 5$ (down from 8).
- Revenue $= \$3 \times 5 = \$15$. $DWL = \tfrac{1}{2} \times 3 \times (8 - 5) = \$4.50$.
Answer: $P_b = \$7.50$, $P_s = \$4.50$, $Q = 5$, revenue \$15, DWL \$4.50.
Worked example: Collected from buyers instead
Suppose the same \$3 tax is collected from buyers at the register. What changes?
- Now demand shifts down by \$3: buyers will pay sellers at most their old willingness minus the tax.
- Solve $20 - 2(P_s + 3) = -4 + 2P_s$: $14 - 2P_s = -4 + 2P_s$, so $P_s = 4.5$ and buyers pay $4.5 + 3 = 7.5$ in total.
Answer: Nothing that matters: the same \$7.50, \$4.50 and 5 units.
Common mistake
Expecting the price to rise by the full tax, or computing revenue with the old quantity. Here the buyers' price rose by \$1.50, not \$3, and revenue is $3 \times 5$, not $3 \times 8 = \$24$. Check: $P_b - P_s$ must equal the tax, and both prices must give the same new quantity.