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

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

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

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

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