1. 1 Chapter 11 Appendix

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

3. 3 Solving for  Q Step 1: Some Definitions T = P G – P N  P G = P G – P*  P N = P N – P*

4. 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. 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*

6. 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. 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 )]

8. 8 Solving for  Q Step 6: Solving T = (  Q/Q*) × (P*) [(E S – E D )/(E D E S )] for  Q T = (  Q/Q*) × (P*) [(E S – E D )/(E D E S )] So  Q = T(P*/Q*) × [(E D E S )/(E S – E D )] Plugging back into W = 1/2T  Q W = 1/2T 2 (P*/Q*) × [(E D E S )/(E S – E D )]

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

10. 10 Individual Losses in Welfare Under Perfect Competition If there is perfect competition, then E D is infinite from the firm owner’s perspective. This implies that DWL = 1/2t 2 (P*Q*)E S

11. 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. 12 Figure 11A.1 Regular and Compensated Demand Curves For a Normal Good Price Gasoline per Year 0 Curve Regular Demand Q1Q1 P1P1 Compensated Demand Curve

13. 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. 14 Figure 11A.2 Using A Compensated Demand Curve to Isolate The Substitution Effect of a Tax-Induced Price Increase DCDC STST Price (Dollars) Gasoline per Year (Gallons) 0 Q1Q1 1.00 DRDR S E1E1 QSQS PGPG PNPN Q2Q2 QQ A E2E2

15. 15 Figure 11A.3 A Compensated Supply Curve for an Input Wages Input Services per Year 0 Q1Q1 W1W1 Regular Labor Supply Curve Compensated Labor Supply Curve