This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Proof of Tax Equivalence
We made the claim in class that the eﬀects of a tax on equilibrium price do not depend on whether
you collect the tax from consumers or suppliers. The following is a proof of this claim. 1 Collect from producers
D(p(τ )) = S (p(τ ), τ )
dD
dS
=
dτ
dτ
∂D ∂p
∂S ∂p ∂S
=
+
∂p ∂τ
∂p ∂τ
∂τ
∂S
∂τ ∂p
=
∂τ ∂D
∂p − ∂S
∂p Because producers remit the tax they care directly only about the aftertax price, p − τ . Therefore,
∂S
∂S
∂τ = − ∂p . In words, the change in quantity supplied from a unit increase in tax τ is the same as
the change from a unit decrease in transaction price p. Using this fact,
∂p
=
∂τ ∂S
∂p
∂S
∂p − ∂D
∂p = ν
ν− What we found above is the change in the transaction price p. To calculate tax incidence, let’s
consider what happens to producer price P S and to the consumer price P D :
P S = p(τ ) − τ
dP S
∂p
=
−1
dτ
∂τ
ν
−1
=
ν−
= ν− <0 The consumer does not remit the tax directly. Therefore,
P D = p(τ )
∂p
dP D
=
dτ
∂τ
ν
=
ν− 1 >0 2 Collect from consumers
D(p(τ ), τ ) = S (p(τ ))
dD
dS
=
dτ
dτ
∂D ∂p ∂D
∂S ∂p
+
=
∂p ∂τ
∂τ
∂p ∂τ
∂p
=
∂τ ∂D
∂τ
∂S
∂p − ∂D
∂p Now consumers remit the tax so they only care about the aftertax price. To them, a unit increase
in tax τ is equivalent to a unit increase in transaction price p. Therefore, ∂D = ∂D . Using this
∂τ
∂p
fact,
∂p
=
∂τ ∂D
∂p
∂S
∂p − ∂D
∂p = ν− Let’s again examine what happens to the producer price P S and consumer price P D now that the
consumers remit the tax instead of the producers. For producers:
P S = p(τ )
dP S
∂p
=
dτ
∂τ
=
ν− <0 For consumers:
P D = p(τ ) − τ
dP D
∂p
=
−1
dτ
∂τ
=
−1
ν−
ν
=
>0
ν−
Note that dP S
dτ and dP D
dτ don’t depend on who remits the tax. This concludes the proof. 2 ...
View
Full
Document
This note was uploaded on 12/10/2011 for the course ECON 401 taught by Professor Burbidge,john during the Winter '08 term at Waterloo.
 Winter '08
 Burbidge,John

Click to edit the document details