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Chapter 11 Appendix

Chapter 11 Appendix. Derivation of the Formula for Excess Burden of a Unit Tax. W = 1/2 T × D Q. Where: W = excess burden (Deadweight loss) T = tax D Q = change in equilibrium quantity as a result of the tax. Solving for D Q . Step 1: Some Definitions T = P G – P N

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Chapter 11 Appendix

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  1. Chapter 11Appendix

  2. Derivation of the Formula for Excess Burden of a Unit Tax W = 1/2T × DQ Where: W = excess burden (Deadweight loss) T = tax DQ = change in equilibrium quantity as a result of the tax

  3. Solving for DQ • Step 1: Some Definitions • T = PG– PN • DPG = PG – P* • DPN = PN– P*

  4. Solving for DQ • Step 2: Elasticity comes into play • ED=(DQ/Q*)/(DPG/P*) =(DQ/Q*) × (P*/DPG) =(DQ/Q*) × [P*/(PG –P*)] • ES=(DQ/Q*)/(DPN/P*) =(DQ/Q*) × (P*/DPN) = (DQ/Q*) × [P*/(PN–P*)]

  5. Solving for DQ • Step 3: Solving for PG ED= (DQ/Q*) × [P*/(PG –P*)] (PG–P*) = (DQ/Q*) × (P*/ED) PG = (DQ/Q*) × (P*/ED) + P*

  6. Solving for DQ • Step 4: Solving for PN • ES= (DQ/Q*) × (P*/(PN–P*) • (PN–P*) = (DQ/Q*) × (P*/ES) • PN = (DQ/Q*) × (P*/ES) +P*

  7. Solving for DQ • Step 5: Using the T = PG – PN definition  T = PG – PN = (DQ/Q*) × (P*/ED) +P* – [(DQ/Q*) × (P*/ES) +P*] = (DQ/Q*) × (P*/ED) – (DQ/Q*) × (P*/ES) = (DQ/Q*) × (P*) × [(1/ED) – (1/ES)] = (DQ/Q*) × (P*) × [(ES–ED)/(EDES)]

  8. Solving for DQ • Step 6: Solving T = (DQ/Q*) × (P*) [(ES–ED)/(EDES)] for DQ T = (DQ/Q*) × (P*) [(ES–ED)/(EDES)]  • So  DQ = T(P*/Q*) × [(EDES)/(ES–ED)]  • Plugging back into W = 1/2TDQ • W = 1/2T2(P*/Q*) × [(EDES)/(ES–ED)]

  9. Derivation for the Ad-Valorem Tax • If the pre- and post-tax prices are close to one another, then • W = 1/2t2(P*Q*) × [(EDES) / (ES–ED)] • If LRAC is perfectly inelastic, then • W = 1/2t2 (P*Q*) × (ED) × [(ES)/(ES–ED)] = 1/2t2 (P*Q*) × (ED) • because [(ES)/(ES–ED)] approaches 1.

  10. Individual Losses in Welfare Under Perfect Competition • If there is perfect competition, then ED is infinite from the firm owner’s perspective. • This implies that • DWL = 1/2t2(P*Q*)ES

  11. Compensated Demand Curves • Recall that compensated demand curves show the relationship between price and quantity demanded, excluding the income effect. It only looks at the substitution away from the taxed good.

  12. Figure 11A.1 Regular and Compensated Demand Curves For a Normal Good Compensated Demand Curve P1 Price Regular Demand Curve Q1 0 Gasoline per Year

  13. Compensated Supply Curves • Recall that compensated supply curves show the relationship between price and quantity supplied excluding the income effect. It only looks at the substitution away from the taxed good.

  14. Figure 11A.2 Using A Compensated Demand Curve to Isolate The Substitution Effect of a Tax-Induced Price Increase ST S E2 PG E1 1.00 Price (Dollars) A PN DR DQS DC Q2 Q1 Gasoline per Year (Gallons) 0 DQ

  15. Figure 11A.3 A Compensated Supply Curve for an Input Compensated Labor Supply Curve W1 Wages Regular Labor Supply Curve Q1 0 Input Services per Year

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