Sales and Purchase Taxes: Who
Bears the Burden?
1
1. Introduction
Imagine that a government wishes to raise
some tax revenue


Two schemes are being considered:
(i) Sales Tax;
(ii) Purchase Tax
2
3.

Demand
Tax

Consider unit purchase tax

Consumer liable for £t per unit purchased

Thus, imposition of tax will reduce consumer’s
reservation price for the good
3
Figure 1: (Unit) Purchase Tax
p
pd
0
q
4
Figure 1: (Unit) Purchase Tax
p
tax
pd
ptd
0
q
5
Figure 1: (Unit) Purchase Tax
p
t = £2
5
pd
3
ptd
0
10
q
6
4. Supply

The Supply Curve Shows the relationship between
price and quantity supplied ceteris paribus

That is:

qs at particular price per unit

(minimum) price per unit suppliers willing to accept
for particular quantity.
7
Figure 8: (Inverse) Supply Function ; ps = ps(q)
p
ps
0
q
8
Figure 2: (Inverse) Supply Function ; ps = ps(q)
p
… quantity supplied at a particular price
ps
5
0
10
q
9
Figure 2: (Inverse) Supply Function ; ps = ps(q)
p
… seller’s reservation price (i.e. minimum
price seller wiling to accept per unit)
ps
5
0
10
q
10
4.

Supply
Tax

Consider unit sales tax

Seller liable for £t per unit purchased

Thus, imposition of tax will increase seller’s
reservation price for the good
11
Figure 3: (Unit) Sales Tax
p
ps
0
q
12
Figure 4: (Unit) Sales Tax
p
pts
tax
0
ps
q
13
Figure 4: (Unit) Sales Tax
p
pts
11
t = £2
ps
9
0
10
q
14
5. Comparison

How do the two types of tax impact upon
buyers and sellers?

Assume first a sales tax – i.e. a tax is imposed
upon sellers per unit sold

How does this affect market equilibrium?
15
Figure 5: (Unit) Sales Tax
p
ps
p*
pd
0
q
*
q
16
Figure 5: (Unit) Sales Tax
p
pts
*
t
p
ps
t
p*
pd
0
*
t
q
q
*
q
17
5. Comparison

Thus, a unit sales tax:
(i) Reduces the quantity traded;
(ii) Raises the equilibrium price
18
5. Comparison

Now, consider a unit purchase tax …
19
Figure 6: (Unit) Purchase Tax
p
ps
p*
pd
0
q
*
q
20
Figure 6: (Unit) Purchase Tax
p
ps
p*
pt*
pd
ptd
0
*
t
q
q
*
q
21
5. Comparison

Thus, a unit purchase tax:
(i)
Reduces the quantity traded
(ii)
Reduces the equilibrium price
22
5. Comparison

So, which alternative, as a buyer, would you
prefer?

Must consider gross and net price

Unit tax drives a wedge between price paid
and received
23
5. Comparison

Unit Sales Tax …

Seller is responsible for paying the tax

Net price seller receives is equilibrium price less tax
Seller
pNet
= pt* - t
24
5. Comparison

Unit Sales Tax …

Seller is responsible for paying the tax

Net price seller receives is equilibrium price less tax
Seller
pNet
= pt* - t
25
Figure 7: (Unit) Sales Tax
p
pts
*
t
p
ps
t
p*
pd
0
*
t
q
q
*
q
26
Figure 7: (Unit) Sales Tax
p
pts
Buyer Pays
*
t
p
ps
t
p*
Seller Receives
pt* - t
pd
0
*
t
q
q
*
q
27
6. Comparison

Unit Purchase Tax

Buyers is responsible for tax

Net price buyer pays is equilibrium price plus tax
p
Buyer
Net
*
t
= p +t
28
Figure 15: (Unit) Purchase Tax
p
ps
p*
pd
0
q
*
q
29
Figure 15: (Unit) Purchase Tax
p
ps
p*
pt*
t
pd
ptd
0
*
t
q
q
*
q
30
Figure 7: (Unit) Purchase Tax
p
Buyer Pays
ps
pt* + t
p*
pt*
t
Seller Receives
pd
ptd
0
*
t
q
q
*
q
31
5. Comparison

It can be shown that the burden of the tax does not
depend upon whom it is imposed

The buyer and seller will share the burden depending
upon the slopes of their demand and supply curves

These slopes affect the ability of buyers and seller to
‘pass on’ the burden of the tax to one another
32
5. Comparison

Consider first a unit sales tax …
33
Figure 8: (Unit) Sales Tax
p
ps
p*
pd
0
q
*
q
34
Figure 8: (Unit) Sales Tax
p
pts
*
t
p
ps
t
p*
pt* - t
pd
0
*
t
q
q
*
q
35
Figure 8: (Unit) Sales Tax
p
pts
Buyer Pays
*
t
p
ps
t
p*
pt* - t
pd
Seller Receives
0
*
t
q
q
*
q
36
Figure 8: (Unit) Sales Tax
p
pts
Buyer Pays
*
t
p
ps
t
A
p*
B
pt* - t
pd
Seller Receives
0
*
t
q
q
*
q
37
Figure 8: (Unit) Sales Tax
p
Buyer’s Burden
pts
Buyer Pays
*
t
p
ps
t
A
p*
B
pt* - t
0
Seller’s Burden
pd
Seller Receives
*
t
q
q
*
q
38
5. Comparison

And now a unit purchase tax …
39
Figure 9: (Unit) Purchase Tax
p
ps
p*
pd
0
q
*
q
40
Figure 9: (Unit) Purchase Tax
p
ps
pt* + t
p*
pt*
t
pd
ptd
0
*
t
q
q
*
q
41
Figure 9: (Unit) Purchase Tax
p
Buyer Pays
ps
pt* + t
p*
pt*
t
Seller Receives
pd
ptd
0
*
t
q
q
*
q
42
Figure 9: (Unit) Purchase Tax
p
Buyer Pays
ps
pt* + t
C
p*
D
pt*
t
Seller Receives
pd
ptd
0
*
t
q
q
*
q
43
Figure 9: (Unit) Purchase Tax
p
Buyer’s Burden
ps
pt* + t
C
p*
D
pt*
t
Seller’s Burden
pd
ptd
0
*
t
q
q
*
q
44
5. Comparison

Thus:
A+B=t=C+D
A = Buyer’s Burden = C
B = Seller’s Burden = D

The relative tax burden does not depend upon whom
the tax is imposed

The buyer and seller will share the burden depending
upon the slopes of their demand and supply curves
45
5. Comparison

Try to prove this using the following linear (normal)
demand and supply equations:
( )
q ( p ) = c + dp
q d p = a - bp
s

Solve for the pre- and post-tax equilibria under both a
sales and purchase tax and show that the relative
burdens are the same
46
5. Comparison

It can be shown that …
b
Seller 's Burden =
b+ d
d
Buyer 's Burden =
b+ d

… under both a unit sales tax and a unit purchase tax
47
5. Comparison

It can be shown, for example, that a seller is
able to pass on more of the burden of a sales
tax the steeper (i.e. less elastic) is the buyer’s
demand curve …
48
Figure 10: (Unit) Sales Tax
p
p
pts
d
A = Buyer’s Burden
B = Seller’s Burden
*
t
p
ps
t
A
p*
B
pt* - t
0
*
t
q
q
*
q
49
Figure 10: (Unit) Sales Tax
p1d
p
p
pts
d
A = Buyer’s Burden
B = Seller’s Burden
*
t
p
ps
t
A1
A
p*
B1
B
pt* - t
0
*
t
q
q
*
q
50
5. Comparison

In the limit, if the demand curve is vertical (i.e.
perfectly inelastic) then the seller is able to
pass on all of the burden of a sales tax to the
buyer …
51
Figure 10: (Unit) Sales Tax
d
2
p
p
p
pts
d
A = Buyer’s Burden
B = Sellers Burden
t
*
t
p
ps
A2
A
p*
B
pt* - t
0
*
t
q
q
*
q
52
5. Comparison

Note, vertical demand curve implies b = 0 such that:
b
0
Seller 's Burden =
=
=0
b+ d 0+ d
d
d
Buyer 's Burden =
=
=1
b+ d 0+ d

Buyer (Seller) bears all (none) of the burden
53
6. Conclusion

The relative burden a unit tax is determined by the
relative slopes of the demand and supply curves

These slopes determine the extent to which buyers
and sellers can ‘pass on’ the burden of the tax to one
another

Who is legally liable for the tax is not important
54
7. Measuring Welfare

Demand and supply curves – reservation price
schedules of buyers and sellers

That is, the maximum (minimum) price buyers
(sellers) are prepared to pay (accept)

If we know the prices that buyers (sellers) actually
pay (receive), then we can derive a measure of
aggregate surplus and, thus, social welfare
55
Figure 11: Consumer Surplus (CS)
p
10
8
6
4
p* = 2
pd
0
1
2
3
4
q*
=5
q
56
Figure 11: Consumer Surplus (CS)
p
TWP = 10 + 8 + 6 + 4 + 2 = 30
p*q* = 10
10
8
6
4
p* = 2
CS = 20
pd
0
1
2
3
4
q*
=5
q
57
Figure 11: Consumer Surplus (CS)
p
CS
p*
Expenditure =p*q*
Demand
0
q*
q
58
Figure 12: Producer Surplus (PS)
p
Revenue =
Supply
p*q*
q*
p*
PS
p*
0
q*
q
59
Figure 13: Social Welfare (W)
p
Supply
W = CS + PS
CS
p*
PS
Demand
0
q*
q
60
Figure 14: Social Welfare and Tax
p
pts
Buyer Pays
pt*
CS
t
ps
p*
pt* - t
PS
pd
Seller Receives
0
*
t
q
q
*
q
61
Figure 14: Social Welfare and Tax
p
pts
CS
pt*
p
t
ps
T = tq
*
pt* - t
PS
0
pd
*
t
q
q
*
q
62
Figure 14: Social Welfare and Tax
p
pts
CS
pt*
t
T
pt* - t
ps
DWL
PS
0
pd
*
t
q
q
*
q
63
8. Final Comments

The relative burden of a unit tax is determined by the
relative slopes of the demand and supply curves

Who is legally liable for the tax does not affect the
relative burden

But, both sales and purchase unit taxes lead to the
same deadweight loss in social welfare.
64
p
p d = ptd
ps
p* = pt*
0
q
*
q
65
p
t
*
ps
*
t
p =p
pd
d
t
p
0
*
t
q
q
*
q
66
p
pd
pts
pt* = p* + t
t
ps
p*
0
q
*
q
67
p
*
t
pts
*
p = p +t
t
p
ps
*
pd
0
*
t
q
q
*
q
68
p
ps
p*
pd
t
ptd
pt* = p* - t
0
*
t
q
q
*
q
69
p
ps
pd
p*
t
ptd
pt* = p* - t
0
q
*
q
70
p
p s = pts
p* = pt*
pd
0
q
*
q
71
p
pts
t
*
pd
*
t
p =p
0
ps
*
t
q
q
*
q
72