Three Ways to Tax

Run one market under three taxes: lump-sum, specific, and ad valorem.

A lump-sum tax is a fixed bill, so it never enters the price decision. Trade stays untouched. A specific tax drives a fixed wedge into every unit sold. An ad valorem tax widens the wedge by a share of the price. Only wedged taxes shrink trade and leave a loss triangle. The two wedge formulas are below.

For example, three taxes visit a 🍬 candy shop. The lump-sum tax is 10,000 won a month, sold or not. The specific tax takes 100 won per candy. The ad valorem tax takes 10 percent of the price. Only the per-candy taxes raise the price and shrink how many get bought.

Pb=Ps+tP_b = P_s + t
Pb=(1+τ)PsP_b = (1 + \tau) P_s

Part of these learning paths

Examples & Exercises

A specific tax of t=3t = 3 lands on each pack of cigarettes.

The market has demand P=12−QP = 12 - Q and supply P=2+QP = 2 + Q.

Find the traded quantity, the revenue, and the deadweight loss.

Specific tax 33 — the wedge pushes quantity from 55 down to 3.53.5.
Show proof

  1. Untaxed, equilibrium is 12−Q=2+Q12 - Q = 2 + Q, so Q=5Q = 5 and P=7P = 7. A specific tax cuts quantity by tB+D=32=1.5\frac{t}{B + D} = \frac{3}{2} = 1.5, leaving Q=3.5Q = 3.5.

  2. Buyers pay Pb=12−3.5=8.5P_b = 12 - 3.5 = 8.5 while sellers receive Ps=2+3.5=5.5P_s = 2 + 3.5 = 5.5. Revenue is 3×3.5=10.53 \times 3.5 = 10.5.

  3. Deadweight loss is the lost-trade triangle, t22(B+D)=94=2.25\frac{t^2}{2(B + D)} = \frac{9}{4} = 2.25. With equal slopes the tax is split half and half — who pays is decided by responsiveness, not by the law's wording.

Now tax the same market with an ad valorem tax of 30%30\% — a tax proportional to price, like a VAT.

What are the quantity, the wedge, and the revenue? How does it differ from a specific tax?

Ad valorem 30%30\% — the wedge opens in proportion to the price.
Show proof

  1. Under an ad valorem tax the buyer's price is 1.31.3 times the seller's: Pb=1.3PsP_b = 1.3 P_s. Feeding that into demand and supply, 12−Q=1.3(2+Q)12 - Q = 1.3(2 + Q) gives Q≈4.09Q \approx 4.09.

  2. Sellers receive Ps≈6.09P_s \approx 6.09 and buyers pay Pb≈7.91P_b \approx 7.91, so the wedge is about 1.831.83; revenue is 1.83×4.09≈7.461.83 \times 4.09 \approx 7.46 and deadweight loss is about 0.830.83.

  3. The difference is the wedge's character.

    A specific wedge is a fixed amount per unit, but an ad valorem wedge is proportional to price — pricier goods, bigger wedge.

    That is why ad valorem revenue rides along with prices.

1. Intro to Microeconomics