Chapter Sixteen
Equilibrium
均衡
Structure
Market equilibrium
Quantity tax and equilibrium
Tax incidence （ 税收分担）
Deadweight loss （ 额外净损失）
Market Equilibrium
A market is in equilibrium when total
quantity demanded by buyers equals
total quantity supplied by sellers.
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
p*
q*
D(p*) = S(p*); the market
is in equilibrium.
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
p*
S(p’)
D(p’) < S(p’); an excess
of quantity supplied over
quantity demanded.
p’
D(p’)
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
p*
S(p’)
D(p’) < S(p’); an excess
of quantity supplied over
quantity demanded.
p’
D(p’)
Market price must fall towards p*.
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
p*
D(p”)
D(p”) > S(p”); an excess
of quantity demanded
over quantity supplied.
p”
S(p”)
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
p*
D(p”)
D(p”) > S(p”); an excess
of quantity demanded
over quantity supplied.
p”
S(p”)
Market price must rise towards p*.
Market Equilibrium – Linear D & S
D p a bp( )
S p c dp( )
At the equilibrium price p*,D(p*) = S(p*).
That is,a bp c dp* *
which gives p a cb d*
and q D p S p
ad bc
b d
* * *( ) ( ),
Market Equilibrium
p
D(p),S(p)
D(p) = a-bp
Market
demand
Market
supply
S(p) = c+dp
p
a c
b d
*?
db
bcadq *
Market Equilibrium
Can we calculate the market
equilibrium using the inverse market
demand and supply curves?
Yes,it is the same calculation.
Market Equilibrium
q D p a bp p a qb D q( ) ( ),1
q S p c dp p c qd S q( ) ( ),1
the equation of the inverse market
demand curve,And
the equation of the inverse market
supply curve.
Market Equilibrium
q
D-1(q),
S-1(q)
D-1(q) = (a-q)/b
Market
inverse
demand S-1(q) = (-c+q)/d
p*
q*
At equilibrium,
D-1(q*) = S-1(q*).
Market inverse supply
Market Equilibrium
p D q a qb 1 ( )p S q c qd 1 ( ),and
At the equilibrium quantity q*,D-1(p*) = S-1(p*).
That is,a q
b
c q
d
* *
which gives q
ad bc
b d
*
andp D q S q
a c
b d
* * *( ) ( ),
1 1
Market Equilibrium
q
D-1(q),
S-1(q)
D-1(q) = (a-q)/b
Market
demand
Market
supply
S-1(q) = (-c+q)/d
p
a c
b d
*?
db
bcadq *
Market Equilibrium
Two special cases:
quantity supplied is fixed,
independent of the market price,
and
quantity supplied is extremely
sensitive to the market price.
Market Equilibrium
S(p) = c+dp,so d=0
and S(p)? c.
p
qq* = c
Market quantity supplied is
fixed,independent of price.
Market Equilibrium
S(p) = c+dp,so d=0
and S(p)? c.
p
q
p*
D-1(q) = (a-q)/b
Market
demand
q* = c
Market quantity supplied is
fixed,independent of price.
Market Equilibrium
S(p) = c+dp,so d=0
and S(p)? c.
p
q
p* =
(a-c)/b
D-1(q) = (a-q)/b
Market
demand
q* = c
p* = D-1(q*); that is,
p* = (a-c)/b.
Market quantity supplied is
fixed,independent of price.
Market Equilibrium
S(p) = c+dp,so d=0
and S(p)? c.
p
q
D-1(q) = (a-q)/b
Market
demand
q* = c
p* = D-1(q*); that is,
p* = (a-c)/b.
p a cb d*
q ad bcb d*with d = 0 give
p a cb*
q c*,?
p* =
(a-c)/b
Market quantity supplied is
fixed,independent of price.
Market Equilibrium
Market quantity supplied is
extremely sensitive to price.
S-1(q) = p*.
p
q
p*
Market Equilibrium
Market quantity supplied is
extremely sensitive to price.
S-1(q) = p*.
p
q
p*
D-1(q) = (a-q)/b
Market
demand
q*
Market Equilibrium
Market quantity supplied is
extremely sensitive to price.
S-1(q) = p*.
p
q
p*
D-1(q) = (a-q)/b
Market
demand
q* =
a-bp*
p* = D-1(q*) = (a-q*)/b so
q* = a-bp*
Quantity Taxes
A quantity tax levied at a rate of $t is
a tax of $t paid on each unit traded.
If the tax is levied on sellers then it is
an excise tax.
If the tax is levied on buyers then it is
a sales tax.
Quantity Taxes
What is the effect of a quantity tax on
a market’s equilibrium?
How are prices affected?
How is the quantity traded affected?
Who pays the tax?
How are gains-to-trade altered?
Quantity Taxes
A tax rate t makes the price paid by
buyers,pb,higher by t from the price
received by sellers,ps.
p p tb s
Quantity Taxes
Even with a tax the market must
clear.
I.e,quantity demanded by buyers at
price pb must equal quantity supplied
by sellers at price ps.
D p S pb s( ) ( )?
Quantity Taxes
p p tb sD p S pb s( ) ( )?and
describe the market’s equilibrium.
Notice that these two conditions apply no
matter if the tax is levied on sellers or on
buyers.
Hence,a sales tax rate $t has the
same effect as an excise tax rate $t.
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$t
An excise tax
raises the market
supply curve by $t
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
An excise tax
raises the market
supply curve by $t,
raises the buyers’
price and lowers the
quantity traded.
$tpb
qt
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
An excise tax
raises the market
supply curve by $t,
raises the buyers’
price and lowers the
quantity traded.
$tpb
qt
And sellers receive only ps = pb - t.
ps
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
An sales tax lowers
the market demand
curve by $t
$t
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
An sales tax lowers
the market demand
curve by $t,lowers
the sellers’ price and
reduces the quantity
traded.$t
qt
ps
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
An sales tax lowers
the market demand
curve by $t,lowers
the sellers’ price and
reduces the quantity
traded.$t
pb
qt
And buyers pay pb = ps + t.
ps
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
A sales tax levied at
rate $t has the same
effects on the
market’s equilibrium
as does an excise tax
levied at rate $t.$t
pb
qt
ps
$t
Quantity Taxes & Market Equilibrium
Who pays the tax of $t per unit
traded?
The division of the $t between buyers
and sellers is the incidence of the tax
(税收分担 ).
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps
Tax paid by
buyers
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps Tax paid bysellers
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps
Tax paid by
buyers
Tax paid by
sellers
Quantity Taxes & Market Equilibrium
E.g,suppose the market demand and
supply curves are linear.
D p a bpb b( )
S p c dps s( )
Quantity Taxes & Market Equilibrium
andD p a bpb b( )S p c dps s( ),
Quantity Taxes & Market Equilibrium
and
With the tax,the market equilibrium satisfies
and so
and
D p a bpb b( )S p c dps s( ),
p p tb sD p S pb s( ) ( )?
p p tb sa bp c dpb s,
Quantity Taxes & Market Equilibrium
D p a bpb b( )S p c dps s( ),and
With the tax,the market equilibrium satisfies
p p tb sD p S pb s( ) ( )?and so
p p tb sa bp c dpb s,and
Substituting for pb gives
a b p t c dp p a c btb ds s s( ),
Quantity Taxes & Market Equilibrium
p a c btb dsand p p tb sgive
The quantity traded at equilibrium is
q D p S p
a bp
ad bc bdt
b d
t
b s
b
( ) ( )
.
p a c dtb db
Quantity Taxes & Market Equilibrium
p a c btb ds
p a c dtb db
q ad bc bdtb dt
As t? 0,ps and pb? the
equilibrium price if
there is no tax (t = 0) and qt
the quantity traded at equilibrium
when there is no tax.
ad bc
b d
,
*,pdb ca
Quantity Taxes & Market Equilibrium
p a c btb ds
p a c dtb db
q ad bc bdtb dt
As t increases,ps falls,
pb rises,
and qt falls.
Quantity Taxes & Market Equilibrium
p a c btb ds
p a c dtb db
q ad bc bdtb dt
The tax paid per unit by the buyer isp p a c dt
b d
a c
b d
dt
b db
*,
Quantity Taxes & Market Equilibrium
p a c btb ds
p a c dtb db
q ad bc bdtb dt
The tax paid per unit by the buyer isp p a c dt
b d
a c
b d
dt
b db
*,
The tax paid per unit by the seller isp p a c
b d
a c bt
b d
bt
b ds
*,
Quantity Taxes & Market Equilibrium
p a c btb ds
p a c dtb db
q ad bc bdtb dt
The total tax paid (by buyers and sellers
combined) is
T tq t ad bc bdtb dt,
Tax Incidence and Own-Price
Elasticities
The incidence of a quantity tax
depends upon the own-price
elasticities of demand and supply.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
Change to buyers’
price is pb - p*.
Change to quantity
demanded is Dq.
Dq
Tax Incidence and Own-Price
Elasticities
Around p = p* the own-price elasticity
of demand is approximately
D
b
q
q
p p
p
D
*
*
*
Tax Incidence and Own-Price
Elasticities
Around p = p* the own-price elasticity
of demand is approximately
D
b
b
D
q
q
p p
p
p p
q p
q
D
D
*
*
*
*
*
*
.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
Change to sellers’
price is ps - p*.
Change to quantity
demanded is Dq.
Dq
Tax Incidence and Own-Price
Elasticities
Around p = p* the own-price elasticity
of supply is approximately
S
s
q
q
p p
p
D
*
*
*
Tax Incidence and Own-Price
Elasticities
Around p = p* the own-price elasticity
of supply is approximately
S
s
s
S
q
q
p p
p
p p
q p
q
D
D
*
*
*
*
*
*
.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps
Tax paid by
buyers
Tax paid by
sellers
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps
Tax paid by
buyers
Tax paid by
sellers
Tax incidence =
p p
p p
b
s
*
*
.
Tax Incidence and Own-Price
Elasticities
Tax incidence =
p p
p p
b
s
*
*
.
p p
q p
q
b
D
*
*
*
.
D
p p
q p
q
s
S
*
*
*
.
D
Tax Incidence and Own-Price
Elasticities
Tax incidence =
p p
p p
b
s
*
*
.
p p
q p
q
b
D
*
*
*
.
D
p p
q p
q
s
S
*
*
*
.
D
So
p p
p p
b
s
S
D
*
*
.
Tax Incidence and Own-Price
Elasticitiesp p
p p
b
s
S
D
*
*
.
Tax incidence is
The fraction of a $t quantity tax paid
by buyers rises as supply becomes more
own-price elastic or as demand becomes
less own-price elastic.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
As market demand
becomes less own-
price elastic,tax
incidence shifts more
to the buyers.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
As market demand
becomes less own-
price elastic,tax
incidence shifts more
to the buyers.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
ps= p*
$tpb
qt = q*
As market demand
becomes less own-
price elastic,tax
incidence shifts more
to the buyers.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
ps= p*
$tpb
qt = q*
As market demand
becomes less own-
price elastic,tax
incidence shifts more
to the buyers.
When?D = 0,buyers pay the entire tax,even
though it is levied on the sellers.
Tax Incidence and Own-Price
Elasticitiesp p
p p
b
s
S
D
*
*
.
Tax incidence is
Similarly,the fraction of a $t quantity
tax paid by sellers rises as supply
becomes less own-price elastic or as
demand becomes more own-price elastic.
Deadweight Loss and Own-Price
Elasticities
A quantity tax imposed on a
competitive market reduces the
quantity traded and so reduces
gains-to-trade (i.e,the sum of
Consumers’ and Producers’
Surpluses).
The lost total surplus is the tax’s
deadweight loss（ 额外净损失）,or
excess burden.
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
CS
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
PS
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
CS
PS
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
CS
PS
The tax reduces
both CS and PS
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
CS
PS
The tax reduces
both CS and PS,
transfers surplus
to government
Tax
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
CS
PS
The tax reduces
both CS and PS,
transfers surplus
to government
Tax
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
CS
PS
The tax reduces
both CS and PS,
transfers surplus
to government
Tax
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
CS
PS
The tax reduces
both CS and PS,
transfers surplus
to government,
and lowers total
surplus.
Tax
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps Deadweight loss
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
Deadweight loss falls
as market demand
becomes less own-
price elastic.
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
ps= p*
$tpb
qt = q*
Deadweight loss falls
as market demand
becomes less own-
price elastic.
When?D = 0,the tax causes no deadweight
loss.
Deadweight Loss and Own-Price
Elasticities
Deadweight loss due to a quantity
tax rises as either market demand or
market supply becomes more own-
price elastic.
If either?D = 0 or?S = 0 then the
deadweight loss is zero.
Equilibrium
均衡
Structure
Market equilibrium
Quantity tax and equilibrium
Tax incidence （ 税收分担）
Deadweight loss （ 额外净损失）
Market Equilibrium
A market is in equilibrium when total
quantity demanded by buyers equals
total quantity supplied by sellers.
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
p*
q*
D(p*) = S(p*); the market
is in equilibrium.
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
p*
S(p’)
D(p’) < S(p’); an excess
of quantity supplied over
quantity demanded.
p’
D(p’)
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
p*
S(p’)
D(p’) < S(p’); an excess
of quantity supplied over
quantity demanded.
p’
D(p’)
Market price must fall towards p*.
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
p*
D(p”)
D(p”) > S(p”); an excess
of quantity demanded
over quantity supplied.
p”
S(p”)
Market Equilibrium
p
D(p),S(p)
q=D(p)
Market
demand
Market
supply
q=S(p)
p*
D(p”)
D(p”) > S(p”); an excess
of quantity demanded
over quantity supplied.
p”
S(p”)
Market price must rise towards p*.
Market Equilibrium – Linear D & S
D p a bp( )
S p c dp( )
At the equilibrium price p*,D(p*) = S(p*).
That is,a bp c dp* *
which gives p a cb d*
and q D p S p
ad bc
b d
* * *( ) ( ),
Market Equilibrium
p
D(p),S(p)
D(p) = a-bp
Market
demand
Market
supply
S(p) = c+dp
p
a c
b d
*?
db
bcadq *
Market Equilibrium
Can we calculate the market
equilibrium using the inverse market
demand and supply curves?
Yes,it is the same calculation.
Market Equilibrium
q D p a bp p a qb D q( ) ( ),1
q S p c dp p c qd S q( ) ( ),1
the equation of the inverse market
demand curve,And
the equation of the inverse market
supply curve.
Market Equilibrium
q
D-1(q),
S-1(q)
D-1(q) = (a-q)/b
Market
inverse
demand S-1(q) = (-c+q)/d
p*
q*
At equilibrium,
D-1(q*) = S-1(q*).
Market inverse supply
Market Equilibrium
p D q a qb 1 ( )p S q c qd 1 ( ),and
At the equilibrium quantity q*,D-1(p*) = S-1(p*).
That is,a q
b
c q
d
* *
which gives q
ad bc
b d
*
andp D q S q
a c
b d
* * *( ) ( ),
1 1
Market Equilibrium
q
D-1(q),
S-1(q)
D-1(q) = (a-q)/b
Market
demand
Market
supply
S-1(q) = (-c+q)/d
p
a c
b d
*?
db
bcadq *
Market Equilibrium
Two special cases:
quantity supplied is fixed,
independent of the market price,
and
quantity supplied is extremely
sensitive to the market price.
Market Equilibrium
S(p) = c+dp,so d=0
and S(p)? c.
p
qq* = c
Market quantity supplied is
fixed,independent of price.
Market Equilibrium
S(p) = c+dp,so d=0
and S(p)? c.
p
q
p*
D-1(q) = (a-q)/b
Market
demand
q* = c
Market quantity supplied is
fixed,independent of price.
Market Equilibrium
S(p) = c+dp,so d=0
and S(p)? c.
p
q
p* =
(a-c)/b
D-1(q) = (a-q)/b
Market
demand
q* = c
p* = D-1(q*); that is,
p* = (a-c)/b.
Market quantity supplied is
fixed,independent of price.
Market Equilibrium
S(p) = c+dp,so d=0
and S(p)? c.
p
q
D-1(q) = (a-q)/b
Market
demand
q* = c
p* = D-1(q*); that is,
p* = (a-c)/b.
p a cb d*
q ad bcb d*with d = 0 give
p a cb*
q c*,?
p* =
(a-c)/b
Market quantity supplied is
fixed,independent of price.
Market Equilibrium
Market quantity supplied is
extremely sensitive to price.
S-1(q) = p*.
p
q
p*
Market Equilibrium
Market quantity supplied is
extremely sensitive to price.
S-1(q) = p*.
p
q
p*
D-1(q) = (a-q)/b
Market
demand
q*
Market Equilibrium
Market quantity supplied is
extremely sensitive to price.
S-1(q) = p*.
p
q
p*
D-1(q) = (a-q)/b
Market
demand
q* =
a-bp*
p* = D-1(q*) = (a-q*)/b so
q* = a-bp*
Quantity Taxes
A quantity tax levied at a rate of $t is
a tax of $t paid on each unit traded.
If the tax is levied on sellers then it is
an excise tax.
If the tax is levied on buyers then it is
a sales tax.
Quantity Taxes
What is the effect of a quantity tax on
a market’s equilibrium?
How are prices affected?
How is the quantity traded affected?
Who pays the tax?
How are gains-to-trade altered?
Quantity Taxes
A tax rate t makes the price paid by
buyers,pb,higher by t from the price
received by sellers,ps.
p p tb s
Quantity Taxes
Even with a tax the market must
clear.
I.e,quantity demanded by buyers at
price pb must equal quantity supplied
by sellers at price ps.
D p S pb s( ) ( )?
Quantity Taxes
p p tb sD p S pb s( ) ( )?and
describe the market’s equilibrium.
Notice that these two conditions apply no
matter if the tax is levied on sellers or on
buyers.
Hence,a sales tax rate $t has the
same effect as an excise tax rate $t.
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$t
An excise tax
raises the market
supply curve by $t
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
An excise tax
raises the market
supply curve by $t,
raises the buyers’
price and lowers the
quantity traded.
$tpb
qt
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
An excise tax
raises the market
supply curve by $t,
raises the buyers’
price and lowers the
quantity traded.
$tpb
qt
And sellers receive only ps = pb - t.
ps
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
An sales tax lowers
the market demand
curve by $t
$t
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
An sales tax lowers
the market demand
curve by $t,lowers
the sellers’ price and
reduces the quantity
traded.$t
qt
ps
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
An sales tax lowers
the market demand
curve by $t,lowers
the sellers’ price and
reduces the quantity
traded.$t
pb
qt
And buyers pay pb = ps + t.
ps
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
A sales tax levied at
rate $t has the same
effects on the
market’s equilibrium
as does an excise tax
levied at rate $t.$t
pb
qt
ps
$t
Quantity Taxes & Market Equilibrium
Who pays the tax of $t per unit
traded?
The division of the $t between buyers
and sellers is the incidence of the tax
(税收分担 ).
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps
Tax paid by
buyers
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps Tax paid bysellers
Quantity Taxes & Market Equilibrium
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps
Tax paid by
buyers
Tax paid by
sellers
Quantity Taxes & Market Equilibrium
E.g,suppose the market demand and
supply curves are linear.
D p a bpb b( )
S p c dps s( )
Quantity Taxes & Market Equilibrium
andD p a bpb b( )S p c dps s( ),
Quantity Taxes & Market Equilibrium
and
With the tax,the market equilibrium satisfies
and so
and
D p a bpb b( )S p c dps s( ),
p p tb sD p S pb s( ) ( )?
p p tb sa bp c dpb s,
Quantity Taxes & Market Equilibrium
D p a bpb b( )S p c dps s( ),and
With the tax,the market equilibrium satisfies
p p tb sD p S pb s( ) ( )?and so
p p tb sa bp c dpb s,and
Substituting for pb gives
a b p t c dp p a c btb ds s s( ),
Quantity Taxes & Market Equilibrium
p a c btb dsand p p tb sgive
The quantity traded at equilibrium is
q D p S p
a bp
ad bc bdt
b d
t
b s
b
( ) ( )
.
p a c dtb db
Quantity Taxes & Market Equilibrium
p a c btb ds
p a c dtb db
q ad bc bdtb dt
As t? 0,ps and pb? the
equilibrium price if
there is no tax (t = 0) and qt
the quantity traded at equilibrium
when there is no tax.
ad bc
b d
,
*,pdb ca
Quantity Taxes & Market Equilibrium
p a c btb ds
p a c dtb db
q ad bc bdtb dt
As t increases,ps falls,
pb rises,
and qt falls.
Quantity Taxes & Market Equilibrium
p a c btb ds
p a c dtb db
q ad bc bdtb dt
The tax paid per unit by the buyer isp p a c dt
b d
a c
b d
dt
b db
*,
Quantity Taxes & Market Equilibrium
p a c btb ds
p a c dtb db
q ad bc bdtb dt
The tax paid per unit by the buyer isp p a c dt
b d
a c
b d
dt
b db
*,
The tax paid per unit by the seller isp p a c
b d
a c bt
b d
bt
b ds
*,
Quantity Taxes & Market Equilibrium
p a c btb ds
p a c dtb db
q ad bc bdtb dt
The total tax paid (by buyers and sellers
combined) is
T tq t ad bc bdtb dt,
Tax Incidence and Own-Price
Elasticities
The incidence of a quantity tax
depends upon the own-price
elasticities of demand and supply.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
Change to buyers’
price is pb - p*.
Change to quantity
demanded is Dq.
Dq
Tax Incidence and Own-Price
Elasticities
Around p = p* the own-price elasticity
of demand is approximately
D
b
q
q
p p
p
D
*
*
*
Tax Incidence and Own-Price
Elasticities
Around p = p* the own-price elasticity
of demand is approximately
D
b
b
D
q
q
p p
p
p p
q p
q
D
D
*
*
*
*
*
*
.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
Change to sellers’
price is ps - p*.
Change to quantity
demanded is Dq.
Dq
Tax Incidence and Own-Price
Elasticities
Around p = p* the own-price elasticity
of supply is approximately
S
s
q
q
p p
p
D
*
*
*
Tax Incidence and Own-Price
Elasticities
Around p = p* the own-price elasticity
of supply is approximately
S
s
s
S
q
q
p p
p
p p
q p
q
D
D
*
*
*
*
*
*
.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps
Tax paid by
buyers
Tax paid by
sellers
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
pb
qt
ps
Tax paid by
buyers
Tax paid by
sellers
Tax incidence =
p p
p p
b
s
*
*
.
Tax Incidence and Own-Price
Elasticities
Tax incidence =
p p
p p
b
s
*
*
.
p p
q p
q
b
D
*
*
*
.
D
p p
q p
q
s
S
*
*
*
.
D
Tax Incidence and Own-Price
Elasticities
Tax incidence =
p p
p p
b
s
*
*
.
p p
q p
q
b
D
*
*
*
.
D
p p
q p
q
s
S
*
*
*
.
D
So
p p
p p
b
s
S
D
*
*
.
Tax Incidence and Own-Price
Elasticitiesp p
p p
b
s
S
D
*
*
.
Tax incidence is
The fraction of a $t quantity tax paid
by buyers rises as supply becomes more
own-price elastic or as demand becomes
less own-price elastic.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
As market demand
becomes less own-
price elastic,tax
incidence shifts more
to the buyers.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
As market demand
becomes less own-
price elastic,tax
incidence shifts more
to the buyers.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
ps= p*
$tpb
qt = q*
As market demand
becomes less own-
price elastic,tax
incidence shifts more
to the buyers.
Tax Incidence and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
ps= p*
$tpb
qt = q*
As market demand
becomes less own-
price elastic,tax
incidence shifts more
to the buyers.
When?D = 0,buyers pay the entire tax,even
though it is levied on the sellers.
Tax Incidence and Own-Price
Elasticitiesp p
p p
b
s
S
D
*
*
.
Tax incidence is
Similarly,the fraction of a $t quantity
tax paid by sellers rises as supply
becomes less own-price elastic or as
demand becomes more own-price elastic.
Deadweight Loss and Own-Price
Elasticities
A quantity tax imposed on a
competitive market reduces the
quantity traded and so reduces
gains-to-trade (i.e,the sum of
Consumers’ and Producers’
Surpluses).
The lost total surplus is the tax’s
deadweight loss（ 额外净损失）,or
excess burden.
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
CS
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
PS
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
No tax
CS
PS
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
CS
PS
The tax reduces
both CS and PS
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
CS
PS
The tax reduces
both CS and PS,
transfers surplus
to government
Tax
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
CS
PS
The tax reduces
both CS and PS,
transfers surplus
to government
Tax
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
CS
PS
The tax reduces
both CS and PS,
transfers surplus
to government
Tax
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
CS
PS
The tax reduces
both CS and PS,
transfers surplus
to government,
and lowers total
surplus.
Tax
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps Deadweight loss
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
p*
q*
$tpb
qt
ps
Deadweight loss falls
as market demand
becomes less own-
price elastic.
Deadweight Loss and Own-Price
Elasticities
p
D(p),S(p)
Market
demand
Market
supply
ps= p*
$tpb
qt = q*
Deadweight loss falls
as market demand
becomes less own-
price elastic.
When?D = 0,the tax causes no deadweight
loss.
Deadweight Loss and Own-Price
Elasticities
Deadweight loss due to a quantity
tax rises as either market demand or
market supply becomes more own-
price elastic.
If either?D = 0 or?S = 0 then the
deadweight loss is zero.
