Chapter 11
Appendix
1
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
2
Solving for DQ

Step 1: Some Definitions



T = PG – PN
DPG = PG – P*
DPN = PN – P*
3
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*)]
4
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*
5
Solving for DQ

Step 4: Solving for PN
= (DQ/Q*) × (P*/(PN – P*)

ES

(PN – P*) = (DQ/Q*) × (P*/ES)

PN
= (DQ/Q*) × (P*/ES) +P*
6
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)]
7
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)]
8
Derivation for the Ad-Valorem Tax

If the pre- and post-tax prices are close to one
another, then


If LRAC is perfectly inelastic, then


W = 1/2t2(P*Q*) × [(EDES) / (ES – ED)]
W = 1/2t2 (P*Q*) × (ED) × [(ES)/(ES – ED)]
= 1/2t2 (P*Q*) × (ED)
because [(ES)/(ES – ED)] approaches 1.
9
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
10
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.
11
Figure 11A.1 Regular and Compensated Demand
Curves For a Normal Good
Price
Compensated Demand Curve
P1
Regular Demand
Curve
0
Q1
Gasoline per Year
12
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.
13
Figure 11A.2 Using A Compensated Demand Curve to
Isolate The Substitution Effect of a Tax-Induced Price
Increase
S
T
Price (Dollars)
S
E2
PG
1.00
PN
E1
A
DQS
0
Q2 Q1
DQ
DC
DR
Gasoline per
Year (Gallons) 14
Wages
Figure 11A.3 A Compensated Supply Curve
for an Input
Compensated Labor
Supply Curve
W1
Regular Labor Supply Curve
0
Q1
Input Services per Year
15