Download Formula for the Excess Burden

Formula for the Excess Burden
Appendix to Lecture 2
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Course of Public Finance - Southeast University - Nanjing, China
Diagram
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Course of Public Finance - Southeast University - Nanjing, China
Step 1: base and height
 Area fdi in figure in terms of compensated demand elasticity.
The triangle area is given by the formula
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2
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𝐴 = × 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 = × 𝑑𝑖 × 𝑓𝑑 (1)
 fd is just the difference between the after tax and the before
tax P (ΔPo)
𝑓𝑑 = ∆𝑃𝑜 = 1 − 𝑡𝑜 𝑃𝑜 − 𝑃𝑜 = 𝑡𝑜 × 𝑃𝑜 (2)
 di is the change in quantity Δq induced by the price rise
𝑑𝑖 = ∆𝑞 = 𝑞1 − 𝑞2
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Course of Public Finance - Southeast University - Nanjing, China
Step 2: use of elasticity formula
 The definition of price elasticity applied to this case is
∆𝑞𝑃𝑜
𝜀=
∆𝑃𝑜 𝑞
So that
∆𝑞 = 𝜀
𝑞
𝑃𝑜
∆𝑃𝑜
(4)
 From (2) we know that ∆𝑃𝑜 = 𝑡𝑜 × 𝑃𝑜 , so that (4) yields
∆𝑞 = 𝜀
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𝑞
𝑃𝑜
𝑡𝑜 𝑃𝑜 = 𝜀 × 𝑞 × 𝑡𝑜
Course of Public Finance - Southeast University - Nanjing, China
Step 3: derivation of the formula for
excess burden
 Finally, recall that 𝑑𝑖 = ∆𝑞 and substitute both (5) and (2)
into (1) to obtain
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𝐴 = 𝑑𝑖 𝑓𝑑
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= 𝜀𝑞𝑡𝑜 𝑡𝑜 𝑃𝑜
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1
= × 𝜀 × 𝑞 × 𝑃𝑜 × 𝑡𝑜
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 Q.E.D.
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Course of Public Finance - Southeast University - Nanjing, China
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