Tax vs Quota: Deadweight Loss

A tax and a quota are different policy tools. Yet both shrink the amount traded.

Switch the tool and the lost welfare keeps the same shape. The deadweight-loss triangle holds its area either way. Only the destination of the transfer changes. With a tax it becomes government revenue. With a quota it becomes rent for whoever holds the quota.

For example, say the corner 🏪 shop sells 10 🍬 candies a day. The government caps sales at 6 a day, so 4 trades vanish. Some friends who wanted candy cannot buy it. That frustration over vanished trades is the deadweight loss. The shop holding the selling permit pockets the rent instead.

DWLtax=t22(B+D),DWLquota=(B+D)(Q∗−Qˉ)22DWL_{tax} = \frac{t^2}{2(B+D)}, \qquad DWL_{quota} = \frac{(B+D)(Q^* - \bar{Q})^2}{2}

Part of these learning paths

Examples & Exercises

With demand P=10−QP = 10 - Q and supply P=2+QP = 2 + Q, compute the deadweight loss of a tax t=2t = 2 directly as the area of the Harberger triangle, and check it against the formula DWL=t2/(2(B+D))DWL = t^2/(2(B+D)).

DWL=t22(B+D)DWL = \frac{t^2}{2(B+D)}
Tax t=2t = 2 — base and height of the DWL triangle.
Show proof

  1. The triangle's height (the price wedge) is t=2t = 2 and its base (the lost quantity) is ΔQ=Q∗−Qt=4−3=1\Delta Q = Q^* - Q_t = 4 - 3 = 1. The area is 2⋅1/2=12 \cdot 1 / 2 = 1.

  2. The formula gives DWL=22/(2⋅2)=1DWL = 2^2/(2 \cdot 2) = 1 — it matches. Deadweight loss grows with the square of the tax rate, so doubling the rate quadruples the loss.

    t→2t  ⇒  DWL→4⋅DWLt \to 2t \;\Rightarrow\; DWL \to 4 \cdot DWL

In the same market, find the deadweight loss of a quota fixing quantity at Qˉ=3\bar{Q} = 3, and compare it with the tax rate that produces the same quantity.

DWLquota=(B+D)(Q∗−Qˉ)22DWL_{quota} = \frac{(B+D)(Q^* - \bar{Q})^2}{2}
Quota Qˉ=3\bar{Q} = 3 — the equivalent tax rate is t=2t = 2.
Show proof

  1. Under the quota the demand price is Pd(3)=7P_d(3) = 7 and the supply price is Ps(3)=5P_s(3) = 5, so the price gap is 22. Deadweight loss is (7−5)⋅(4−3)/2=1(7-5)\cdot(4-3)/2 = 1.

  2. The tax producing the same quantity Qt=3Q_t = 3 is t=(B+D)(Q∗−Qˉ)=2⋅1=2t = (B+D)(Q^* - \bar{Q}) = 2 \cdot 1 = 2, with the same DWL of 11. In the linear model a quota and a tax give the same DWL whenever they give the same quantity — the difference is whether the transfer is tax revenue or quota rent.

9. Competitive Markets