1. Formula for the Excess Burden
Appendix to Lecture on Efficient taxation 
Course of Applied Economics 4

2. Excess Burden Measurement with Demand Curves
Excess burden = ½ ηPbq1tb2
Price per pound of barley
Tax revenues
Excess burden of tax
(1 + tb)Pb
S’b g f h d i Pb Sb Db q2 q1
Pounds of barley per year
Course of Applied Economics 4

3. Course of Applied Economics 4
Step 1: base and height
Area fdi in figure in terms of compensated demand elasticity. The triangle area is given by the formula
𝐴= 1 2 ×𝑏𝑎𝑠𝑒×ℎ𝑒𝑖𝑔ℎ𝑡= 1 2 ×𝑑𝑖×𝑓𝑑 (1)
fd is just the difference between the after tax and the before tax P (ΔPo)
𝑓𝑑=∆ 𝑃 𝑜 = 1+ 𝑡 𝑜 𝑃 𝑜 − 𝑃 𝑜 = 𝑡 𝑜 ×𝑃 𝑜 (2)
di is the change in quantity Δq induced by the price rise
𝑑𝑖=∆𝑞= 𝑞 1 − 𝑞 2
Course of Applied Economics 4

4. Step 2: use of elasticity formula
The definition of price elasticity applied to this case is
𝜀= ∆𝑞 𝑃 𝑜 ∆ 𝑃 𝑜 𝑞
So that
∆𝑞=𝜀 𝑞 𝑃 𝑜 ∆ 𝑃 𝑜 (4)
From (2) we know that ∆ 𝑃 𝑜 = 𝑡 𝑜 ×𝑃 𝑜 , so that (4) yields
∆𝑞=𝜀 𝑞 𝑃 𝑜 𝑡 𝑜 𝑃 𝑜 =𝜀×𝑞× 𝑡 𝑜
Course of Applied Economics 4

5. Step 3: derivation of the formula for excess burden
Finally, recall that 𝑑𝑖=∆𝑞 and substitute both (5) and (2) into (1) to obtain
𝐴= 1 2 𝑑𝑖 𝑓𝑑
= 1 2 𝜀𝑞 𝑡 𝑜 𝑡 𝑜 𝑃 𝑜
= 1 2 ×𝜀×𝑞× 𝑃 𝑜 × 𝑡 𝑜 2
Q.E.D.
Course of Applied Economics 4