Supply, Demand, and Equilibrium

The competitive equilibrium is where the demand and supply curves cross. That crossing fixes both price and quantity at once.

Now add a per-unit tax and a wedge opens up. The price buyers pay pulls apart from the price sellers receive. Fewer trades happen, and the lost trades destroy welfare. That destroyed part is the deadweight loss in the formula below. What remains splits into consumer surplus, producer surplus, and tax revenue.

For example, say a 🍬 candy sells for 100 won at the corner 🏪 shop. A tax of 20 won per candy makes shoppers pay 110 won while the shop keeps only 90. The 20 won in the middle is the wedge. Pricier candy means fewer friends buy it.

Q∗=A−CB+D,P∗=A−BQ∗Q^* = \frac{A-C}{B+D}, \qquad P^* = A - BQ^*
DWL=t(Q∗−Qt)2DWL = \frac{t(Q^* - Q_t)}{2}

Part of these learning paths

Examples & Exercises

In a market with demand P=10−QP = 10 - Q and supply P=2+QP = 2 + Q, a per-unit tax t=2t = 2 is imposed.

Compute by hand the equilibrium quantity, the buyer and seller prices, and the deadweight loss.

Q∗=A−CB+D,Qt=A−C−tB+DQ^* = \frac{A - C}{B + D}, \qquad Q_t = \frac{A - C - t}{B + D}
DWL=t(Q∗−Qt)2DWL = \frac{t (Q^* - Q_t)}{2}
A=10A = 10, C=2C = 2, t=2t = 2 — the tax wedge and the Harberger triangle.
Show proof

  1. The no-tax equilibrium is Q∗=(10−2)/2=4Q^* = (10-2)/2 = 4, P∗=10−4=6P^* = 10 - 4 = 6.

  2. After the tax the quantity is Qt=(10−2−2)/2=3Q_t = (10-2-2)/2 = 3. The seller price on the supply curve is Ps=2+3=5P_s = 2 + 3 = 5, the buyer price on the demand curve is Pb=10−3=7P_b = 10 - 3 = 7, and the gap is exactly t=2t = 2.

  3. Consumer surplus is (10−7)⋅3/2=4.5(10-7)\cdot 3/2 = 4.5, producer surplus is (5−2)⋅3/2=4.5(5-2)\cdot 3/2 = 4.5, revenue is 2⋅3=62 \cdot 3 = 6.

    Deadweight loss is 2⋅(4−3)2=1\frac{2 \cdot (4-3)}{2} = 1 — the triangle between value and cost on the lost unit.

    They sum to 4.5+4.5+6+1=164.5+4.5+6+1 = 16, the no-tax total surplus.

    4.5+4.5+6+1⏟DWL=164.5 + 4.5 + 6 + \underbrace{1}_{DWL} = 16

In the same example, compute how the tax burden falls on buyers versus sellers, and explain it through the slopes of demand and supply.

A=10A = 10, C=2C = 2, t=2t = 2 — incidence follows the slope ratio.
Show proof

  1. The buyer burden is Pb−P∗=7−6=1P_b - P^* = 7 - 6 = 1 and the seller burden is P∗−Ps=6−5=1P^* - P_s = 6 - 5 = 1. They sum to t=2t = 2.

  2. Here demand and supply have equal slope magnitudes (11), so the burden splits evenly. In general the buyer share is DB+D\frac{D}{B+D} and the seller share is BB+D\frac{B}{B+D} — the relatively inelastic (steeper) side bears more.

    buyer share=DB+D,seller share=BB+D\text{buyer share} = \frac{D}{B+D}, \qquad \text{seller share} = \frac{B}{B+D}

9. Competitive Markets