© 2010 w. w. norton & company, inc. 16 equilibrium

© 2010 W. W. Norton & Company, Inc.

16 Equilibrium

© 2010 W. W. Norton & Company, Inc. 2

Market Equilibrium

A market is in equilibrium when total quantity demanded by buyers equals total quantity supplied by sellers.


Market Equilibriump

D(p)

q=D(p)

Marketdemand


Market Equilibriump

S(p)

Marketsupply

q=S(p)


Market Equilibriump

D(p), S(p)

q=D(p)

Marketdemand

Marketsupply

q=S(p)


Market Equilibriump

D(p), S(p)

q=D(p)

Marketdemand

Marketsupply

q=S(p)

p*

q*


Market Equilibriump

D(p), S(p)

q=D(p)

Marketdemand

Marketsupply

q=S(p)

p*

q*

D(p*) = S(p*); the marketis in equilibrium.


Market Equilibriump

D(p), S(p)

q=D(p)

Marketdemand

Marketsupply

q=S(p)

p*

S(p’)

D(p’) < S(p’); an excessof quantity supplied overquantity demanded.

p’

D(p’)


Market Equilibriump

D(p), S(p)

q=D(p)

Marketdemand

Marketsupply

q=S(p)

p*

S(p’)

D(p’) < S(p’); an excessof quantity supplied overquantity demanded.

p’

D(p’)

Market price must fall towards p*.


Market Equilibriump

D(p), S(p)

q=D(p)

Marketdemand

Marketsupply

q=S(p)

p*

D(p”)

D(p”) > S(p”); an excessof quantity demandedover quantity supplied.

p”

S(p”)


Market Equilibriump

D(p), S(p)

q=D(p)

Marketdemand

Marketsupply

q=S(p)

p*

D(p”)

D(p”) > S(p”); an excessof quantity demandedover quantity supplied.

p”

S(p”)

Market price must rise towards p*.


Market Equilibrium

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 Equilibriump

D(p), S(p)

D(p) = a-bp

Marketdemand

Marketsupply

S(p) = c+dp

p*

q*


Market Equilibriump

D(p), S(p)

D(p) = a-bp

Marketdemand

Marketsupply

S(p) = c+dp

p*

q*

What are the valuesof p* and q*?


Market EquilibriumD p a bp( ) S p c dp( )

At the equilibrium price p*, D(p*) = S(p*).



At the equilibrium price p*, D(p*) = S(p*).That is, a bp c dp * *




which gives pa cb d

*




which gives pa cb d

*

and q D p S pad bcb d

* * *( ) ( ) .


Market Equilibriump

D(p), S(p)

D(p) = a-bp

Marketdemand

Marketsupply

S(p) = c+dpp

a cb d

*

dbbcad

q*


Market Equilibrium

Can we calculate the market equilibrium using the inverse market demand and supply curves?


Market Equilibrium

Can we calculate the market equilibrium using the inverse market demand and supply curves?

Yes, it is the same calculation.


Market Equilibriumq D p a bp p

a qb

D q ( ) ( ),1

q S p c dp pc qd

S q ( ) ( ),1

the equation of the inverse marketdemand curve. And

the equation of the inverse marketsupply curve.


Market Equilibrium

q

D-1(q),S-1(q)

D-1(q) = (a-q)/b

Marketinversedemand

Market inverse supplyS-1(q) = (-c+q)/d

p*

q*


Market Equilibrium

q

D-1(q),S-1(q)

D-1(q) = (a-q)/b

Marketdemand

S-1(q) = (-c+q)/d

p*

q*

At equilibrium,D-1(q*) = S-1(q*).

Market inverse supply


Market Equilibriump D q

a qb

1( ) p S qc qd

1( ) .and

At the equilibrium quantity q*, D-1(p*) = S-1(p*).



a qb

1( ) p S qc qd

1( ) .and

At the equilibrium quantity q*, D-1(p*) = S-1(p*).That is, a q

bc q

d * *



a qb

1( ) p S qc qd

1( ) .and


bc q

d * *

which gives qad bcb d

*



a qb

1( ) p S qc qd

1( ) .and


bc q

d * *

which gives qad bcb d

*

and p D q S qa cb d

* * *( ) ( ) .

1 1


Market Equilibrium

q

D-1(q),S-1(q)

D-1(q) = (a-q)/b

Marketdemand

Marketsupply

S-1(q) = (-c+q)/dp

a cb d

*

dbbcad

q*


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 EquilibriumMarket quantity supplied isfixed, independent of price.

p

qq*


Market Equilibrium

S(p) = c+dp, so d=0and S(p) c.

p

qq* = c

Market quantity supplied isfixed, independent of price.


Market Equilibrium


p

qq* = c

D-1(q) = (a-q)/b

Marketdemand



Market Equilibrium


p

q

p*

D-1(q) = (a-q)/b

Marketdemand

q* = c



Market Equilibrium


p

q

p* =(a-c)/b

D-1(q) = (a-q)/b

Marketdemand

q* = c

p* = D-1(q*); that is,p* = (a-c)/b.



Market Equilibrium


p

q

D-1(q) = (a-q)/b

Marketdemand

q* = c

p* = D-1(q*); that is,p* = (a-c)/b.

p* =(a-c)/b


pa cb d

*

qad bcb d

*


Market Equilibrium


p

q

D-1(q) = (a-q)/b

Marketdemand

q* = c

p* = D-1(q*); that is,p* = (a-c)/b.

pa cb d

*

qad bcb d

*

with d = 0 give

pa c

b*

q c* .

p* =(a-c)/b



Market Equilibrium

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.


Market EquilibriumMarket quantity supplied isextremely sensitive to price.

p

q



S-1(q) = p*.

p

q

p*



S-1(q) = p*.

p

q

p*

D-1(q) = (a-q)/b

Marketdemand



S-1(q) = p*.

p

q

p*

D-1(q) = (a-q)/b

Marketdemand

q*



S-1(q) = p*.

p

q

p*

D-1(q) = (a-q)/b

Marketdemand

q* =a-bp*

p* = D-1(q*) = (a-q*)/b soq* = 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 s D p S pb s( ) ( )and

describe the market’s equilibrium.Notice these conditions apply nomatter if the tax is levied on sellers or onbuyers.


Quantity Taxes

p p tb s D p S pb s( ) ( )and

describe the market’s equilibrium.Notice that these two conditions apply nomatter if the tax is levied on sellers or onbuyers.Hence, a sales tax rate $t has thesame effect as an excise tax rate $t.


Quantity Taxes & Market Equilibrium

p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

No tax



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$t

An excise taxraises the marketsupply curve by $t



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

An excise taxraises the marketsupply curve by $t,raises the buyers’price and lowers thequantity traded.

$tpb

qt



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

An excise taxraises the marketsupply curve by $t,raises the buyers’price and lowers thequantity traded.

$tpb

qt

And sellers receive only ps = pb - t.

ps



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

No tax



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

An sales tax lowersthe market demandcurve by $t

$t



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

An sales tax lowersthe market demandcurve by $t, lowersthe sellers’ price andreduces the quantitytraded.$t

qt

ps



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

An sales tax lowersthe market demandcurve by $t, lowersthe sellers’ price andreduces the quantitytraded.$t

pbpb

qt

pb

And buyers pay pb = ps + t.

ps



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

A sales tax levied atrate $t has the sameeffects on themarket’s equilibriumas does an excise taxlevied at rate $t.$t

pbpb

qt

pb

ps

$t



Who pays the tax of $t per unit traded?

The division of the $t between buyers and sellers is the incidence of the tax.



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

pbpb

qt

pb

ps



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

pbpb

qt

pb

ps

Tax paid by buyers



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

pbpb

qt

pb

psTax paid by sellers



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

pbpb

qt

pb

ps

Tax paid by buyers

Tax paid by sellers



E.g. suppose the market demand and supply curves are linear.

D p a bpb b( ) S p c dps s( )



andD p a bpb b( ) S p c dps s( ) .



and

With the tax, the market equilibrium satisfies

and so

and

D p a bpb b( ) S p c dps s( ) .

p p tb s D p S pb s( ) ( )

p p tb s a bp c dpb s .



D p a bpb b( ) S p c dps s( ) . and

With the tax, the market equilibrium satisfies

p p tb s D p S pb s( ) ( )and so

p p tb s a bp c dpb s .and

Substituting for pb gives

a b p t c dp pa c bt

b ds s s

( ) .



pa c bt

b ds and p p tb s give

The quantity traded at equilibrium is

q D p S p

a bpad bc bdt

b d

tb s

b

( ) ( )

.

pa c dt

b db



pa c bt

b ds

pa c dt

b db

qad bc bdt

b dt

As t 0, ps and pb theequilibrium price ifthere is no tax (t = 0) and qt the quantity traded at equilibriumwhen there is no tax.

ad bcb d

,

*,pdbca



pa c bt

b ds

pa c dt

b db

qad bc bdt

b dt

As t increases, ps falls,

pb rises,

and qt falls.



pa c bt

b ds

pa c dt

b db

qad bc bdt

b dt

The tax paid per unit by the buyer isp p

a c dtb d

a cb d

dtb db

* .



pa c bt

b ds

pa c dt

b db

qad bc bdt

b dt

The tax paid per unit by the buyer isp p

a c dtb d

a cb d

dtb db

* .

The tax paid per unit by the seller isp p

a cb d

a c btb d

btb ds

* .



pa c bt

b ds

pa c dt

b db

qad bc bdt

b dt

The total tax paid (by buyers and sellerscombined) is

T tq tad bc bdt

b dt

.


Tax Incidence and Own-Price Elasticities

The incidence of a quantity tax depends upon the own-price elasticities of demand and supply.



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps

Change to buyers’price is pb - p*.Change to quantitydemanded is q.

q



Around p = p* the own-price elasticityof demand is approximately

Db

q

q

p p

p

*

*

*



Around p = p* the own-price elasticityof demand is approximately

Db

bD

q

q

p p

p

p pq p

q

*

*

*

**

*.



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps

Change to sellers’price is ps - p*.Change to quantitydemanded is q.

q



Around p = p* the own-price elasticityof supply is approximately

Ss

q

q

p p

p

*

*

*



Around p = p* the own-price elasticityof supply is approximately

Ss

sS

q

q

p p

p

p pq p

q

*

*

*

**

*.



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

pbpb

qt

pb

ps

Tax paid by buyers

Tax paid by sellers



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

pbpb

qt

pb

ps

Tax paid by buyers

Tax paid by sellers

Tax incidence = p p

p pb

s

*

*.



Tax incidence = p p

p pb

s

*

*.

p pq p

qb

D

*

*

*.

p p

q p

qs

S

*

*

*.



Tax incidence = p p

p pb

s

*

*.

p pq p

qb

D

*

*

*.

p p

q p

qs

S

*

*

*.

So p p

p pb

s

S

D

*

*.



p p

p pb

s

S

D

*

*.

Tax incidence is

The fraction of a $t quantity tax paidby buyers rises as supply becomes moreown-price elastic or as demand becomesless own-price elastic.



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps

As market demandbecomes less own-price elastic, taxincidence shifts moreto the buyers.



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps




p

D(p), S(p)

Marketdemand

Marketsupply

ps= p*

$tpb

qt = q*




p

D(p), S(p)

Marketdemand

Marketsupply

ps= p*

$tpb

qt = q*


When D = 0, buyers pay the entire tax, even though it is levied on the sellers.



p p

p pb

s

S

D

*

*.

Tax incidence is

Similarly, the fraction of a $t quantitytax paid by sellers rises as supplybecomes less own-price elastic or asdemand 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, or excess burden.



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

No tax



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

No taxCS



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

No tax

PS



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

No taxCS

PS



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps

CS

PS

The tax reducesboth CS and PS



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps

CS

PS

The tax reducesboth CS and PS,transfers surplusto government

Tax



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps

CS

PS


Tax



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps

CS

PS

The tax reducesboth CS and PS,transfers surplusto government,and lowers total surplus.

Tax



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps

CS

PSTax

Deadweight loss



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps Deadweight loss



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps

Deadweight loss fallsas market demandbecomes less own-price elastic.



p

D(p), S(p)

Marketdemand

Marketsupply

p*

q*

$tpb

qt

ps




p

D(p), S(p)

Marketdemand

Marketsupply

ps= p*

$tpb

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.

© 2010 w. w. norton & company, inc. 16 equilibrium

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