Download (p*).

Equilibrium

A market is in equilibrium when total quantity
demanded by buyers equals total quantity supplied
by sellers.
Market
p
demand
q=D(p)
D(p)
p
Market
supply
q=S(p)
S(p)
Market
p
demand
Market
supply
q=S(p)
q=D(p)
D(p), S(p)
Market
p
demand
Market
supply
q=S(p)
p*
q=D(p)
q*
D(p), S(p)
Market
p
demand
Market
supply
q=S(p)
D(p*) = S(p*); the market
is in equilibrium.
p*
q=D(p)
q*
D(p), S(p)
Market
p
demand
Market
supply
q=S(p)
D(p’) < S(p’); an excess
of quantity supplied over
quantity demanded.
q=D(p)
p’
p*
D(p’)
S(p’)
D(p), S(p)
Market
p
demand
Market
supply
q=S(p)
D(p’) < S(p’); an excess
of quantity supplied over
quantity demanded.
q=D(p)
p’
p*
D(p’)
S(p’)
D(p), S(p)
Market price must fall towards p*.
Market
p
demand
Market
supply
q=S(p)
D(p”) > S(p”); an excess
of quantity demanded
over quantity supplied.
q=D(p)
p*
p”
S(p”)
D(p”)
D(p), S(p)
Market
p
demand
Market
supply
q=S(p)
D(p”) > S(p”); an excess
of quantity demanded
over quantity supplied.
q=D(p)
p*
p”
S(p”)
D(p”)
D(p), S(p)
Market price must rise towards p*.

An example of calculating a market equilibrium
when the market demand and supply curves are
linear.
D(p)  a  bp
S(p)  c  dp
Market
p
demand
Market
supply
S(p) = c+dp
p*
D(p) = a-bp
q*
D(p), S(p)
Market
p
demand
Market
supply
S(p) = c+dp
What are the values
of p* and q*?
p*
D(p) = a-bp
q*
D(p), S(p)
D(p)  a  bp
S(p)  c  dp
At the equilibrium price p*, D(p*) = S(p*).
D(p)  a  bp
S(p)  c  dp
At the equilibrium price p*, D(p*) = S(p*).
That is,
a  bp*  c  dp*
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
ac
p 
bd
*
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
ac
p 
bd
*
ad  bc
and q  D(p )  S(p ) 
.
bd
*
*
*
Market
p
demand
*
Market
supply
S(p) = c+dp
p 
ac
bd
D(p) = a-bp
ad  bc
q 
bd
*
D(p), S(p)

Can we calculate the market equilibrium using the
inverse market demand and supply curves?


Can we calculate the market equilibrium using the
inverse market demand and supply curves?
Yes, it is the same calculation.
aq
1
q  D(p)  a  bp  p 
 D ( q),
b
the equation of the inverse market
demand curve. And
cq
q  S(p)  c  dp  p 
 S 1 ( q),
d
the equation of the inverse market
supply curve.
D-1(q),
S-1(q)
Market
inverse
demand
Market inverse supply
S-1(q) = (-c+q)/d
p*
D-1(q) = (a-q)/b
q*
q
D-1(q), Market
S-1(q) demand
Market inverse supply
S-1(q) = (-c+q)/d
At equilibrium,
D-1(q*) = S-1(q*).
p*
D-1(q) = (a-q)/b
q*
q
pD
1
aq
cq
1
( q) 
.
and p  S ( q) 
d
b
At the equilibrium quantity q*, D-1(p*) = S-1(p*).
pD
1
aq
cq
1
( q) 
.
and p  S ( q) 
d
b
At the equilibrium quantity q*, D-1(p*) = S-1(p*).
That is,
a  q*  c  q*

b
d
pD
1
aq
cq
1
( q) 
.
and p  S ( q) 
d
b
At the equilibrium quantity q*, D-1(p*) = S-1(p*).
That is,
a  q*  c  q*

b
d
* ad  bc
which gives q 
bd
pD
1
aq
cq
1
( q) 
.
and p  S ( q) 
d
b
At the equilibrium quantity q*, D-1(p*) = S-1(p*).
That is,
a  q*  c  q*

b
d
* ad  bc
which gives q 
bd
*
and p  D
1
*
1
(q )  S
ac
(q ) 
.
bd
*
D-1(q), Market
S-1(q) demand
*
Market
supply
S-1(q) = (-c+q)/d
p 
ac
bd
D-1(q) = (a-q)/b
ad  bc
q 
bd
*
q

Two special cases:
quantity supplied is fixed, independent of the
market price, and
 quantity supplied is extremely sensitive to the
market price.

Market quantity supplied is
fixed, independent of price.
p
q*
q
Market quantity supplied is
fixed, independent of price.
S(p) = c+dp, so d=0
and S(p)  c.
p
q* = c
q
Market
p
demand
Market quantity supplied is
fixed, independent of price.
S(p) = c+dp, so d=0
and S(p)  c.
D-1(q) = (a-q)/b
q* = c
q
Market
p
demand
Market quantity supplied is
fixed, independent of price.
S(p) = c+dp, so d=0
and S(p)  c.
p*
D-1(q) = (a-q)/b
q* = c
q
Market
p
demand
p* =
(a-c)/b
Market quantity supplied is
fixed, independent of price.
S(p) = c+dp, so d=0
and S(p)  c.
p* = D-1(q*); that is,
p* = (a-c)/b.
D-1(q) = (a-q)/b
q* = c
q
Market
p
demand
p* =
(a-c)/b
Market quantity supplied is
fixed, independent of price.
S(p) = c+dp, so d=0
and S(p)  c.
p* = D-1(q*); that is,
p* = (a-c)/b.
D-1(q) = (a-q)/b
a  c q* = c
p 
bd
* ad  bc
q 
bd
*
q
Market
p
demand
p* =
(a-c)/b
Market quantity supplied is
fixed, independent of price.
S(p) = c+dp, so d=0
and S(p)  c.
p* = D-1(q*); that is,
p* = (a-c)/b.
D-1(q) = (a-q)/b
q
a  c q* = c
p 
bd
* ad  bc with d = 0 give
q 
bd
*
ac
p 
b
*
q*  c.

Two special cases are
when quantity supplied is fixed, independent of the
market price, and
 when quantity supplied is extremely sensitive to the
market price.


p
Market quantity supplied is
extremely sensitive to price.
q
p
Market quantity supplied is
extremely sensitive to price.
S-1(q) = p*.
p*
q
Market
p
demand
Market quantity supplied is
extremely sensitive to price.
S-1(q) = p*.
p*
D-1(q) = (a-q)/b
q
Market
p
demand
Market quantity supplied is
extremely sensitive to price.
S-1(q) = p*.
p*
D-1(q) = (a-q)/b
q*
q
Market
p
demand
Market quantity supplied is
extremely sensitive to price.
S-1(q) = p*.
p* = D-1(q*) = (a-q*)/b so
q* = a-bp*
p*
D-1(q) = (a-q)/b
q* =
a-bp*
q



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.





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?

A tax rate t makes the price paid by buyers, pb,
higher by t from the price received by sellers, ps.
pb  ps  t


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(pb )  S(ps )
D(pb )  S(ps )
pb  ps  t
and
describe the market’s equilibrium.
Notice these conditions apply no
matter if the tax is levied on sellers or on
buyers.
D(pb )  S(ps )
pb  ps  t
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.
Market
p
demand
Market
supply
No tax
p*
q*
D(p), S(p)
Market
p
demand
Market
supply
$t
An excise tax
raises the market
supply curve by $t
p*
q*
D(p), S(p)
Market
p
demand
Market
supply
$t
pb
p*
qt q*
An excise tax
raises the market
supply curve by $t,
raises the buyers’
price and lowers the
quantity traded.
D(p), S(p)
Market
p
demand
Market
supply
$t
pb
p*
ps
qt q*
An excise tax
raises the market
supply curve by $t,
raises the buyers’
price and lowers the
quantity traded.
D(p), S(p)
And sellers receive only ps = pb - t.
Market
p
demand
Market
supply
No tax
p*
q*
D(p), S(p)
Market
p
demand
Market
supply
An sales tax lowers
the market demand
curve by $t
p*
$t
q*
D(p), S(p)
Market
p
demand
p*
ps
Market
supply
$t
qt q*
An sales tax lowers
the market demand
curve by $t, lowers
the sellers’ price and
reduces the quantity
traded.
D(p), S(p)
Market
p
demand
pb
p*
ps
Market
supply
$t
qt q*
An sales tax lowers
the market demand
curve by $t, lowers
the sellers’ price and
reduces the quantity
traded.
D(p), S(p)
And buyers pay pb = ps + t.
Market
p
demand
Market
supply
$t
pb
p*
ps
$t
qt 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.
D(p), S(p)


Who pays the tax of $t per unit traded?
The division of the $t between buyers and sellers is
the incidence of the tax.
Market
p
demand
Market
supply
pb
p*
ps
qt q*
D(p), S(p)
Market
Market
p
demand
supply
Tax paid by
buyers
pb
p*
ps
qt q*
D(p), S(p)
Market
p
demand
pb
p*
ps
Market
supply
Tax paid by
sellers
qt q*
D(p), S(p)
Market
Market
p
demand
supply
Tax paid by
buyers
pb
p*
ps
Tax paid by
sellers
qt q*
D(p), S(p)

E.g. suppose the market demand and supply curves
are linear.
D(pb )  a  bpb
S(ps )  c  dps
D(pb )  a  bpb and S(ps )  c  dps .
D(pb )  a  bpb and S(ps )  c  dps .
With the tax, the market equilibrium satisfies
pb  ps  t and D(pb )  S(ps ) so
pb  ps  t and a  bpb  c  dps .
D(pb )  a  bpb and S(ps )  c  dps .
With the tax, the market equilibrium satisfies
pb  ps  t and D(pb )  S(ps ) so
pb  ps  t and a  bpb  c  dps .
Substituting for pb gives
a  c  bt
a  b(ps  t )  c  dps  ps 
.
bd
a  c  bt
ps 
and
bd
pb  ps  t give
a  c  dt
pb 
bd
The quantity traded at equilibrium is
qt  D( pb )  S( ps )
ad  bc  bdt
 a  bpb 
.
bd
a  c  bt
ps 
bd
a  c  dt
pb 
bd
ad  bc  bdt
q 
bd
t
ac
As t  0, ps and pb 
 p*, the
bd
equilibrium price if
ad  bc
t
,
there is no tax (t = 0) and q 
bd
the quantity traded at equilibrium
when there is no tax.
a  c  bt
ps 
bd
a  c  dt
pb 
bd
As t increases,
and
ad  bc  bdt
q 
bd
t
ps falls,
pb rises,
qt falls.
a  c  bt
ps 
bd
a  c  dt
pb 
bd
ad  bc  bdt
q 
bd
t
The tax paid per unit by the buyer is
a  c  dt a  c
dt
pb  p 


.
bd
bd bd
*
a  c  bt
ps 
bd
a  c  dt
pb 
bd
ad  bc  bdt
q 
bd
t
The tax paid per unit by the buyer is
a  c  dt a  c
dt
pb  p 


.
bd
bd bd
*
The tax paid per unit by the seller is
a  c a  c  bt
bt
p  ps 


.
bd
bd
bd
*
a  c  bt
ps 
bd
a  c  dt
pb 
bd
ad  bc  bdt
q 
bd
t
The total tax paid (by buyers and sellers
combined) is
ad  bc  bdt
T  tq  t
.
bd
t

What was the impact of
tax?
Taxes discourage market
activity.
 When a good is taxed, the
quantity sold is smaller.
 Buyers and sellers share
the tax burden.


The incidence of a quantity tax depends upon the
own-price elasticities of demand and supply.
p
Market
demand
Market
supply
$t
pb
p*
ps
qt q*
D(p), S(p)
p
Market
demand
Market
supply
$t
pb
p*
ps
qt q*
Dq
Change to buyers’
price is pb - p*.
Change to quantity
demanded is Dq.
D(p), S(p)
Around p = p* the own-price elasticity
of demand is approximately
Dq
*
q
D 
pb  p*
p*
Around p = p* the own-price elasticity
of demand is approximately
Dq
*
q
D 
pb  p*
p*
 pb  p* 
Dq  p*
 D  q*
.
p
Market
demand
Market
supply
$t
pb
p*
ps
qt q*
D(p), S(p)
p
Market
demand
Market
supply
$t
pb
p*
ps
qt q*
Dq
Change to sellers’
price is ps - p*.
Change to quantity
demanded is Dq.
D(p), S(p)
Around p = p* the own-price elasticity
of supply is approximately
Dq
*
q
S 
*
ps  p
*
p
Around p = p* the own-price elasticity
of supply is approximately
Dq
*
q
S 
ps  p*
p*
 ps  p* 
Dq  p*
 S  q*
.
p
pb
p*
ps
Market
Market
demand
supply
Tax paid by
buyers
Tax paid by
sellers
qt q*
D(p), S(p)
Market
Market
p
demand
supply
Tax paid by
buyers
pb
p*
ps
Tax paid by
sellers
qt q*
Tax incidence =
* D(p), S(p)
pb  p
*
p  ps
.
*
Tax incidence =
*
pb  p 
*
Dq  p
D  q
*
.
pb  p
*
p  ps
.
*
ps  p 
*
Dq  p
 S q
*
.
*
Tax incidence =
*
pb  p 
So
*
Dq  p
D  q
*
*
pb  p
*
p  ps
.
pb  p
*
p  ps
.
*
ps  p 
S
 
.
D
*
Dq  p
 S q
*
.
*
Tax incidence is
pb  p
*
p  ps
S
 
.
D
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.
p
Market
demand
Market
supply
$t
pb
p*
ps
qt q*
As market demand
becomes less ownprice elastic, tax
incidence shifts more
to the buyers.
D(p), S(p)
p
Market
demand
Market
supply
$t
pb
p*
ps
qt q*
As market demand
becomes less ownprice elastic, tax
incidence shifts more
to the buyers.
D(p), S(p)
Market
p
demand
pb
ps= p*
Market
supply
$t
qt = q*
As market demand
becomes less ownprice elastic, tax
incidence shifts more
to the buyers.
D(p), S(p)
p
pb
ps= p*
Market
demand
Market
supply
$t
As market demand
becomes less ownprice elastic, tax
incidence shifts more
to the buyers.
D(p), S(p)
qt = q*
When D = 0, buyers pay the entire tax, even
though it is levied on the sellers.
*
Tax incidence is
pb  p
*
p  ps
S
 
.
D
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.
So, how is the burden of the tax divided?
The burden of a tax falls more
heavily on the side of the
market that is less elastic.


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, or excess burden.
p
Market
demand
Market
supply
No tax
p*
q*
D(p), S(p)
p
Market
demand
Market
supply
No tax
p*
CS
q*
D(p), S(p)
p
Market
demand
Market
supply
No tax
p*
PS
q*
D(p), S(p)
p
Market
demand
Market
supply
No tax
p*
CS
PS
q*
D(p), S(p)
p
Market
demand
Market
supply
No tax
p*
CS
PS
q*
D(p), S(p)
p
Market
demand
Market
supply
$t
pb CS
p*
ps PS
qt q*
The tax reduces
both CS and PS
D(p), S(p)
p
Market
demand
Market
supply
$t
pb CS
p* Tax
ps PS
qt q*
The tax reduces
both CS and PS,
transfers surplus
to government
D(p), S(p)
p
Market
demand
Market
supply
$t
pb CS
p* Tax
ps PS
qt q*
The tax reduces
both CS and PS,
transfers surplus
to government
D(p), S(p)
p
Market
demand
Market
supply
$t
pb CS
p* Tax
ps PS
qt q*
The tax reduces
both CS and PS,
transfers surplus
to government
D(p), S(p)
Market
p
demand
Market
supply
$t
pb CS
p* Tax
ps PS
qt q*
The tax reduces
both CS and PS,
transfers surplus
to government,
and lowers total
surplus.
D(p), S(p)
Market
p
demand
Market
supply
$t
pb CS
p* Tax
ps PS
Deadweight loss
qt q*
D(p), S(p)
Market
p
demand
Market
supply
$t
pb
p*
ps
Deadweight loss
qt q*
D(p), S(p)
Market
p
demand
Market
supply
$t
pb
p*
ps
qt q*
Deadweight loss falls
as market demand
becomes less ownprice elastic.
D(p), S(p)
Market
p
demand
Market
supply
$t
pb
p*
ps
qt q*
Deadweight loss falls
as market demand
becomes less ownprice elastic.
D(p), S(p)
p
pb
ps= p*
Market
demand
Market
supply
$t
Deadweight loss falls
as market demand
becomes less ownprice elastic.
D(p), S(p)
qt = q*
When D = 0, the tax causes no deadweight
loss.


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.