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Demand, elasticities, market equilibrium, revenue, deadweight
loss
(Course Micro-economics)
Ch 15-16 Varian
Teachers: Jongeneel/Van Mouche
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Chapter Fifteen
Market Demand
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From Individual to Market Demand Functions
Think of an economy containing n consumers, denoted by i = 1, … ,n.
Consumer i’s ordinary demand function for commodity j is
x p p mji i* ( , , )1 2
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From Individual to Market Demand Functions
When all consumers are price-takers, the market demand function for commodity j is
If all consumers are identical then
where M = nm.
X p p m m x p p mjn
ji i
i
n( , , , , ) ( , , ).*1 2
11 2
1
X p p M n x p p mj j( , , ) ( , , )*1 2 1 2
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From Individual to Market Demand Functions
The market demand curve is the “horizontal sum” of the individual consumers’ demand curves.
E.g. suppose there are only two consumers; i = A,B.
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From Individual to Market Demand Functions
p1 p1
x A1* x B
1*
x xA B1 1*
p1
20 15
35
p1’
p1”
p1’
p1”
p1’
p1”
The “horizontal sum”of the demand curvesof individuals A and B.
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Elasticities
Elasticity measures the “sensitivity” of one variable with respect to another.
The elasticity of variable X with respect to variable Y is
x yxy,
%%
.
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Economic Applications of Elasticity
Economists use elasticities to measure the sensitivity ofquantity demanded of commodity
i with respect to the price of commodity i (own-price elasticity of demand)demand for commodity i with
respect to the price of commodity j (cross-price elasticity of demand).
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Economic Applications of Elasticity
demand for commodity i with respect to income (income elasticity of demand)quantity supplied of commodity i
with respect to the price of commodity i (own-price elasticity of supply)
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Economic Applications of Elasticity
quantity supplied of commodity i with respect to the wage rate (elasticity of supply with respect to the price of labor)and many, many others.
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Own-Price Elasticity of Demand
Q: Why not use a demand curve’s slope to measure the sensitivity of quantity demanded to a change in a commodity’s own price?
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Own-Price Elasticity of Demand
5 50
10 10slope= - 2
slope= - 0.2
p1 p1
10-packs Single Units
X1* X1
*
In which case is the quantity demandedX1* more sensitive to changes to p1?It is the same in both cases.
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Own-Price Elasticity of Demand
Q: Why not just use the slope of a demand curve to measure the sensitivity of quantity demanded to a change in a commodity’s own price?
A: Because the value of sensitivity then depends upon the (arbitrary) units of measurement used for quantity demanded.
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Own-Price Elasticity of Demand
x p
xp1 1
1
1* ,
*%%
is a ratio of percentages and so has nounits of measurement. Hence own-price elasticity of demand is a sensitivity measure that is independent of units of measurement (unit free).
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Arc and Point Elasticities
An “average” own-price elasticity of demand for commodity i over an interval of values for pi is an arc-elasticity, usually computed by a mid-point formula.
Elasticity computed for a single value of pi is a point elasticity.
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Arc Own-Price Elasticity
pi
Xi*
pi’pi’+h
pi’-h
What is the “average” own-priceelasticity of demand for pricesin an interval centered on pi’?
X p
i
ii i
Xp
* ,
*%%
%'
ph
pii
1002
%( " '")( " '") /
*XX X
X Xii i
i i
100
2
Xi'"Xi"
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Arc Own-Price Elasticity
X p
i
ii i
Xp
* ,
*%%
%'
ph
pii
1002
%( " '")( " '") /
*XX X
X Xii i
i i
100
2
So
is the arc own-price elasticity of demand.
.h2
)"'X"X(2/)"'X"X(
'pp%X% ii
ii
i
i
*i
p,X i*i
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Point Own-Price Elasticity
pi
Xi*
pi’
What is the own-price elasticityof demand in a very small intervalof prices centered on pi’?
Xi'
X p
i
i
i
ii i
pX
dXdp
* ,
*''
is the elasticity at the point ( ', ' ).X pi i
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Point Own-Price Elasticity
E.g. Suppose pi = a - bXi. Then Xi = (a-pi)/b and
X p
i
i
i
ii i
p
X
dXdp
* , *
*
.b1
dpdX
i
*i Therefore,
X p
i
i
i
ii i
pa p b b
pa p
* , ( ) /.
1
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Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
X p
i
ii i
pa p
* ,
p aa
a a
1
0
a/2
a/2b
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Point Own-Price Elasticity
pi
Xi*
a
pi = a - bXi*
a/b
X p
i
ii i
pa p
* ,
1
0
a/2
a/2b
own-price elastic
own-price inelastic(own-price unit elastic)
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Point Own-Price Elasticity
pi
Xi*
X kp kpk
pi i
ai
i
* 22
2 everywhere alongthe demand curve.
Constant elasticity demand curve
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Revenue and Own-Price Elasticity of Demand
If raising a commodity’s price causes little decrease in quantity demanded, then sellers’ revenues rise.
Hence own-price inelastic demand causes sellers’ revenues to rise as price rises.
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Revenue and Own-Price Elasticity of Demand
If raising a commodity’s price causes a large decrease in quantity demanded, then sellers’ revenues fall.
Hence own-price elastic demand causes sellers’ revenues to fall as price rises.
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Revenue and Own-Price Elasticity of Demand
R p p X p( ) ( ).* Sellers’ revenue is
So
X p*( ) .1
dpdX
)p(X
p1)p(X
*
**
dRdp
X p pdXdp
**
( )
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Revenue and Own-Price Elasticity of Demand
dRdp
X p *( ) 1
so if 1 thendRdp0
and a change to price does not altersellers’ revenue.
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Revenue and Own-Price Elasticity of Demand
dRdp
X p *( ) 1
but if 1 0 thendRdp 0
and a price increase raises sellers’revenue.
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Revenue and Own-Price Elasticity of Demand
dRdp
X p *( ) 1
And if 1 thendRdp 0
and a price increase reduces sellers’revenue.
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Revenue and Own-Price Elasticity of Demand
In summary:
1 0Own-price inelastic demand;price rise causes rise in sellers’ revenue.
Own-price unit elastic demand;price rise causes no change in sellers’revenue.
1
Own-price elastic demand;price rise causes fall in sellers’ revenue.
1
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Marginal Revenue and Own-Price Elasticity of Demand
A seller’s marginal revenue is the rate at which revenue changes with the number of units sold by the seller.
MR qdR q
dq( )
( ).
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Marginal Revenue and Own-Price Elasticity of Demand
p(q) denotes the seller’s inverse demand function; i.e. the price at which the seller can sell q units. Then
MR qdR q
dqdp q
dqq p q( )
( ) ( )( )
R q p q q( ) ( ) so
p qq
p qdp q
dq( )
( )( )
.1
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Marginal Revenue and Own-Price Elasticity of Demand
MR q p qq
p qdp q
dq( ) ( )
( )( )
.
1
dqdp
pq
and
so MR q p q( ) ( ) .
11
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Marginal Revenue and Own-Price Elasticity of Demand
MR q p q( ) ( )
11 says that the rate
at which a seller’s revenue changeswith the number of units it sellsdepends on the sensitivity of quantitydemanded to price; i.e., upon theof the own-price elasticity of demand.
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Marginal Revenue and Own-Price Elasticity of Demand
1
1)q(p)q(MR
If 1 then MR q( ) .0
If 1 0 then MR q( ) . 0
If 1 then MR q( ) . 0
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Selling onemore unit raises the seller’s revenue.
Selling onemore unit reduces the seller’s revenue.
Selling onemore unit does not change the seller’srevenue.
Marginal Revenue and Own-Price Elasticity of Demand
If 1 then MR q( ) .0
If 1 0 then MR q( ) . 0
If 1 then MR q( ) . 0
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ExampleVegetables AuctionExample with linear inverse demand for tomatoes.
p q a bq( ) .
Then R q p q q a bq q( ) ( ) ( )
and MR q a bq( ) . 2
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Example Vegetables Auction
p q a bq( )
MR q a bq( ) 2
a
a/b
ptomatoes
qtomatoesa/2b
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Example Vegetables Auction
p q a bq( ) MR q a bq( ) 2
a
a/b
p
qa/2b
q
$
a/ba/2b
R(q)
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Example Vegetables Auction
p q a bq( )
MR q a bq( ) 2a
a/b
p
qa/2b
q
$
a/ba/2b
R(q)
S(0)
S(1)=S(0)-INT
Auction’s market
intervention
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Chapter Sixteen
Equilibrium
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Market Equilibrium
A market is in equilibrium when total quantity demanded by buyers equals total quantity supplied by sellers.
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Market Equilibrium
p
D(p), S(p)
q=D(p)
Marketdemand
Marketsupply
q=S(p)
p*
q*
D(p*) = S(p*); the marketis in equilibrium.
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Market Equilibrium
p
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*.
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Market Equilibrium
p
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*.
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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( )
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Market Equilibrium
p
D(p), S(p)
D(p) = a-bp
Marketdemand
Marketsupply
S(p) = c+dp
p*
q*
What are the valuesof p* and q*?
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Market EquilibriumD p a bp( ) S p c dp( )
At the equilibrium price p*, D(p*) = S(p*).
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Market EquilibriumD p a bp( ) S p c dp( )
At the equilibrium price p*, D(p*) = S(p*).That is, a bp c dp * *
which gives pa cb d
*
and q D p S pad bcb d
* * *( ) ( ) .
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Market Equilibrium
p
D(p), S(p)
D(p) = a-bp
Marketdemand
Marketsupply
S(p) = c+dpp
a cb d
*
dbbcad
q*
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Market Equilibrium
Can we calculate the market equilibrium using the inverse market demand and supply curves?
Yes, it is the same calculation.
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Market Equilibrium
Two special cases:quantity supplied is fixed,
independent of the market price, andquantity supplied is extremely
sensitive to the market price.
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Market Equilibrium
S(p) = c+dp, so d=0and S(p) c.
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 quantity supplied isfixed, independent of price.
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Market Equilibrium
Two special cases arewhen quantity supplied is fixed,
independent of the market price, andwhen quantity supplied is
extremely sensitive to the market price.
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Market Equilibrium
Market quantity supplied isextremely sensitive to price.
p
q
Market quantity supplied is fixed (cf milk quota)
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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.
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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?
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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
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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( ) ( )
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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.
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Quantity Taxes & Market Equilibrium
p
D(p), S(p)
Marketdemand
Marketsupply
p*
q*
No tax
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Quantity Taxes & Market Equilibrium
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
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Quantity Taxes & Market Equilibrium
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
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Quantity Taxes & Market Equilibrium
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
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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.
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Quantity Taxes & Market Equilibrium
p
D(p), S(p)
Marketdemand
Marketsupply
p*
q*
pbpb
qt
pb
ps
Tax paid by buyers
Tax paid by sellers
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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 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
( ) .
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Quantity Taxes & Market Equilibrium
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
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Quantity Taxes & Market Equilibrium
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
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Quantity Taxes & Market Equilibrium
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.
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Quantity Taxes & Market Equilibrium
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
* .
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Quantity Taxes & Market Equilibrium
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
.
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Tax Incidence and Own-Price Elasticities
The incidence of a quantity tax depends upon the own-price elasticities of demand and supply.
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Tax Incidence and Own-Price Elasticities
p
D(p), S(p)
Marketdemand
Marketsupply
p*
q*
pbpb
qt
pb
ps
Tax paid by buyers
Tax paid by sellers
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Tax Incidence and Own-Price Elasticities
p
D(p), S(p)
Marketdemand
Marketsupply
p*
q*
€tpb
qt
ps
As market demandbecomes less own-price elastic, taxincidence shifts moreto the buyers.
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Tax Incidence and Own-Price Elasticities
p
D(p), S(p)
Marketdemand
Marketsupply
ps= p*
€tpb
qt = q*
As market demandbecomes less own-price elastic, taxincidence shifts moreto the buyers.
When D = 0, buyers pay the entire tax, even though it is levied on the sellers.
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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.
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Deadweight Loss and Own-Price Elasticities
p
D(p), S(p)
Marketdemand
Marketsupply
p*
q*
€tpb
qt
ps
CS
PSTax
Deadweight loss
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Deadweight Loss and Own-Price Elasticities
p
D(p), S(p)
Marketdemand
Marketsupply
p*
q*
$tpb
qt
ps Deadweight loss
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Deadweight Loss and Own-Price Elasticities
p
D(p), S(p)
Marketdemand
Marketsupply
p*
q*
€tpb
qt
ps
Deadweight loss fallsas market demandbecomes less own-price elastic.
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Deadweight Loss and Own-Price Elasticities
p
D(p), S(p)
Marketdemand
Marketsupply
ps= p*
€tpb
qt = q*
Deadweight loss fallsas market demandbecomes less own-price elastic.
When D = 0, the tax causes no deadweight loss.
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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.