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# ch11_appen - Chapter 11 Appendix 1 Derivation of the...

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1 Chapter 11 Appendix

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2 Derivation of the Formula for Excess Burden of a Unit Tax W = 1/2 T × Q Where: W = excess burden (Deadweight loss) T = tax Q = change in equilibrium quantity as a result of the tax
3 Solving for Q Step 1: Some Definitions T = P G P N P G = P G P * P N = P N P *

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4 Solving for Q Step 2: Elasticity comes into play E D =( Q / Q *)/( P G / P *) =( Q / Q *) × ( P */ P G ) =( Q / Q *) × [ P */( P G P *)] E S =( Q / Q *)/( P N / P *) =( Q / Q *) × ( P */ P N ) = ( Q / Q *) × [ P */( P N P *)]
5 Solving for Q Step 3: Solving for P G E D = ( Q / Q *) × [ P */( P G P *)] ( P G P *) = ( Q / Q *) × ( P */ E D ) P G = ( Q / Q *) × ( P */ E D ) + P *

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6 Solving for Q Step 4: Solving for P N E S = ( Q / Q *) × ( P */( P N P *) ( P N P *) = ( Q / Q *) × ( P */ E S ) P N = ( Q / Q *) × ( P */ E S ) + P *
7 Solving for Q Step 5: Using the T = P G P N definition T = P G P N = ( Q / Q *) × ( P */ E D ) +P* – [( Q / Q *) × ( P */ E S ) + P *] = ( Q / Q *) × ( P */ E D ) – ( Q / Q *) × ( P */ E S ) = ( Q / Q *) × ( P *) × [(1/ E D ) – (1/ E S )] = ( Q / Q *) × ( P *) × [( E S E D )/( E D E S )]

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