A tax raises revenue, but it also stops some trades that would have helped both sides. The value lost on those trades is the deadweight loss. With Tax Incidence with Elasticities and the surplus integrals of Consumer Surplus as an Integral and Producer Surplus as an Integral, we can measure it and see how it grows with the tax.
Key idea
The deadweight loss of a per-unit tax is the value of the lost trades: $$DWL = \int_{Q_t}^{Q^*}\big(P_D(Q) - P_S(Q)\big)\,dQ \approx \tfrac{1}{2}\,t\,\Delta Q.$$ With straight lines, $\Delta Q = \dfrac{bd}{b + d}\,t$, so $DWL = \dfrac{1}{2}\cdot\dfrac{bd}{b + d}\,t^2$: it grows with the square of the tax.
Why it works
Each unit between $Q_t$ and $Q^*$ was worth $P_D(Q)$ to some buyer and cost only $P_S(Q)$ to make, so it would have added $P_D - P_S$ to total surplus. The tax prevents it. The gap is $t$ at $Q_t$ (that is what the tax wedge means) and 0 at $Q^*$, so with straight lines the lost area is a triangle, $\tfrac{1}{2}t\,\Delta Q$. Doubling the tax doubles both the height and the base: four times the area. In elasticity form (the Harberger approximation), $DWL \approx \dfrac{1}{2}\,t^2\,\dfrac{Q}{P}\cdot\dfrac{\varepsilon_S|\varepsilon_D|}{\varepsilon_S + |\varepsilon_D|}$: taxes on goods with very inelastic demand or supply waste less.
Worked example: Measuring the loss
$Q_d = 100 - 2P$, $Q_s = 3P - 20$, $t = 5$.
- From the last lesson: $Q^* = 52$, and the tax gives $P_b = 27$, $Q_t = 100 - 54 = 46$, so $\Delta Q = 6$.
- Formula check: $\dfrac{2\cdot 3}{2 + 3}\cdot 5 = 6$.
- $DWL = \tfrac{1}{2}(5)(6) = 15$. Revenue: $5 \times 46 = 230$.
Answer: $DWL = 15$; revenue 230.
Worked example: Doubling the tax
Same market, $t = 10$.
- $\Delta Q = \tfrac{6}{5}(10) = 12$, so $Q_t = 40$.
- $DWL = \tfrac{1}{2}(10)(12) = 60$, four times as much. Revenue: $10 \times 40 = 400$, less than double.
Answer: $DWL = 60$.
Common mistake
Expecting a tax twice as large to cause twice the deadweight loss. With straight lines it is four times as large, because both the wedge and the number of lost trades double. Check: write DWL as a function of $t$. If it is linear in $t$, you have left out the change in quantity.
A practical consequence that economists broadly agree on: raising a given amount of revenue with a low rate on a broad base wastes less than a high rate on a narrow base, other things equal. Which taxes are fair, and how to weigh efficiency against fairness, are separate questions on which people disagree.