What is the Optimal Incidence of Taxation inCoupled Markets?
Stephen Kinsella David Ramsey
University of Limerick
Irish Economic Association, April 25–27, 2008
Today
Idea
Model
Derivation
Numerical Examples
Further Work
This Paper in one slide
Question What happens to coupled markets when one getstaxed?
Method Study imposition of the tax under more and morecomplex market arrangements
Results As markets become more coupled, firm relationshipsreally matter for the imposition of the tax
Practical application Environmental taxes; tourism subsidies.
Further Work Econometric testing of cross price elasticities: neednatural/computational experiment
This Paper in one slide
Question What happens to coupled markets when one getstaxed?
Method Study imposition of the tax under more and morecomplex market arrangements
Results As markets become more coupled, firm relationshipsreally matter for the imposition of the tax
Practical application Environmental taxes; tourism subsidies.
Further Work Econometric testing of cross price elasticities: neednatural/computational experiment
This Paper in one slide
Question What happens to coupled markets when one getstaxed?
Method Study imposition of the tax under more and morecomplex market arrangements
Results As markets become more coupled, firm relationshipsreally matter for the imposition of the tax
Practical application Environmental taxes; tourism subsidies.
Further Work Econometric testing of cross price elasticities: neednatural/computational experiment
This Paper in one slide
Question What happens to coupled markets when one getstaxed?
Method Study imposition of the tax under more and morecomplex market arrangements
Results As markets become more coupled, firm relationshipsreally matter for the imposition of the tax
Practical application Environmental taxes; tourism subsidies.
Further Work Econometric testing of cross price elasticities: neednatural/computational experiment
Markets we study
i) the classical case of a monopoly producing one good,
ii) the case of two companies each producing a good inwhich one company is a Stackelberg leader. Weconsider the e!ect of placing a tax on a) the leaderand b) the follower,
iii) the case of two companies each producing a good inwhich there is no such hierarchy,
iv) the case of a monopoly producing two goods.
Notation
!i ,j % ! in demand for good i for 1% change in price of good jW (p) ProfitpD(p) Revenuem Marginal cost of productionE fixed costs i! m = m̄p total pricet levy/tax
Methodology
1. Linearise around a demand curve
2. Assume tax is small relative to total price
3. Solve Stackelberg game
4. Derive analytical expressions for e!ect on e"ciency of theimposition of tax
Methodology
1. Linearise around a demand curve
2. Assume tax is small relative to total price
3. Solve Stackelberg game
4. Derive analytical expressions for e!ect on e"ciency of theimposition of tax
Methodology
1. Linearise around a demand curve
2. Assume tax is small relative to total price
3. Solve Stackelberg game
4. Derive analytical expressions for e!ect on e"ciency of theimposition of tax
Classical Model
W (p) = (A" Bp)(p "m)" E , (1)
p! =m
2+
A
2B. (2)
B ="!D(p!)
p!; A = (1" !)D(p!).
m =p!(1 + !)
!# p! =
!m
1 + !.
This is the monopoly (non-discrimination) pricing rule from Tirole(1993, pg. 76.)
Classical Model
W (p) = (A" Bp)(p "m)" E , (1)
p! =m
2+
A
2B. (2)
B ="!D(p!)
p!; A = (1" !)D(p!).
m =p!(1 + !)
!# p! =
!m
1 + !.
This is the monopoly (non-discrimination) pricing rule from Tirole(1993, pg. 76.)
Classical Model
W (p) = (A" Bp)(p "m)" E , (1)
p! =m
2+
A
2B. (2)
B ="!D(p!)
p!; A = (1" !)D(p!).
m =p!(1 + !)
!# p! =
!m
1 + !.
This is the monopoly (non-discrimination) pricing rule from Tirole(1993, pg. 76.)
Classical Model
W (p) = (A" Bp)(p "m)" E , (1)
p! =m
2+
A
2B. (2)
B ="!D(p!)
p!; A = (1" !)D(p!).
m =p!(1 + !)
!# p! =
!m
1 + !.
This is the monopoly (non-discrimination) pricing rule from Tirole(1993, pg. 76.)
Now impose a tax, t
W (p) = (p "m " t)(A" Bp)" E .
The marginal tax revenue will be D(p!)
#W = (m " p!)#D + (1"#p)D(p!) = D(p!).
#S = #pD(p!) =D(p!)
2
i.e. half of the marginal tax revenue. It follows that the marginale"ciency of this tax is 2
3 .
Now impose a tax, t
W (p) = (p "m " t)(A" Bp)" E .
The marginal tax revenue will be D(p!)
#W = (m " p!)#D + (1"#p)D(p!) = D(p!).
#S = #pD(p!) =D(p!)
2
i.e. half of the marginal tax revenue. It follows that the marginale"ciency of this tax is 2
3 .
Now impose a tax, t
W (p) = (p "m " t)(A" Bp)" E .
The marginal tax revenue will be D(p!)
#W = (m " p!)#D + (1"#p)D(p!) = D(p!).
#S = #pD(p!) =D(p!)
2
i.e. half of the marginal tax revenue. It follows that the marginale"ciency of this tax is 2
3 .
Now impose a tax, t
W (p) = (p "m " t)(A" Bp)" E .
The marginal tax revenue will be D(p!)
#W = (m " p!)#D + (1"#p)D(p!) = D(p!).
#S = #pD(p!) =D(p!)
2
i.e. half of the marginal tax revenue. It follows that the marginale"ciency of this tax is 2
3 .
Now impose a tax, t
W (p) = (p "m " t)(A" Bp)" E .
The marginal tax revenue will be D(p!)
#W = (m " p!)#D + (1"#p)D(p!) = D(p!).
#S = #pD(p!) =D(p!)
2
i.e. half of the marginal tax revenue. It follows that the marginale"ciency of this tax is 2
3 .
Hierarchical Model, 2 firms, one good each
Two goods, two firms: p1, p2, Firm 1 is Stackelberg leader.Linearised Demand Functions look like
W1(p1, p2) = (p1 "m1)[A1 " B1p1 + C1p2]" E1
W2(p1, p2) = (p2 "m2)[A2 " B2p2 + C2p1]" E2.
Hierarchical Model, 2 firms, one good each
Two goods, two firms: p1, p2, Firm 1 is Stackelberg leader.Linearised Demand Functions look like
W1(p1, p2) = (p1 "m1)[A1 " B1p1 + C1p2]" E1
W2(p1, p2) = (p2 "m2)[A2 " B2p2 + C2p1]" E2.
Hierarchical Model, contdFirst Eqm condition
"W2
"p2 |p=(p1,p!2 (p1))= 0.
Which leads to
p!2(p1) =m
2+
A2 + C2p1
2B2.
And the second eqm condition is
"W1
"p1 |p=(p!1 ,p!2 (p!1 ))= 0.
Which leads to
p!1 =2A1B2 + C1A2 + C1B2m2 + 2B1B2m1 " C1C2m1
4B1B2 " 2C1C2(3)
Hierarchical Model, contdFirst Eqm condition
"W2
"p2 |p=(p1,p!2 (p1))= 0.
Which leads to
p!2(p1) =m
2+
A2 + C2p1
2B2.
And the second eqm condition is
"W1
"p1 |p=(p!1 ,p!2 (p!1 ))= 0.
Which leads to
p!1 =2A1B2 + C1A2 + C1B2m2 + 2B1B2m1 " C1C2m1
4B1B2 " 2C1C2(3)
Hierarchical Model, contdFirst Eqm condition
"W2
"p2 |p=(p1,p!2 (p1))= 0.
Which leads to
p!2(p1) =m
2+
A2 + C2p1
2B2.
And the second eqm condition is
"W1
"p1 |p=(p!1 ,p!2 (p!1 ))= 0.
Which leads to
p!1 =2A1B2 + C1A2 + C1B2m2 + 2B1B2m1 " C1C2m1
4B1B2 " 2C1C2(3)
Hierarchical Model, contdFirst Eqm condition
"W2
"p2 |p=(p1,p!2 (p1))= 0.
Which leads to
p!2(p1) =m
2+
A2 + C2p1
2B2.
And the second eqm condition is
"W1
"p1 |p=(p!1 ,p!2 (p!1 ))= 0.
Which leads to
p!1 =2A1B2 + C1A2 + C1B2m2 + 2B1B2m1 " C1C2m1
4B1B2 " 2C1C2(3)
Hierarchical Model, contd.
p!2 =4A2B1B2 " A2C1C2 + 2C2A1B2 " C1C2B2m2 + 2C2m1B1B2 "m1C1C 2
2 + 4B1B22m2
4B2(2B1B2 " C1C2).
(4)From the definition of the cross-price elasticities, we have
B1 = "!11D1(p!1 , p!2)
p!1; B2 = "!22D2(p!1 , p
!2)
p!2(5)
C1 =!12D1(p!1 , p
!2)
p!2; C2 =
!21D2(p!1 , p!2)
p!1(6)
A1 = D1(p!1 , p
!2)(1" !11 " !12); A2 = D2(p
!1 , p
!2)(1" !22 " !21). (7)
Hierarchical Model, contd.
p!2 =4A2B1B2 " A2C1C2 + 2C2A1B2 " C1C2B2m2 + 2C2m1B1B2 "m1C1C 2
2 + 4B1B22m2
4B2(2B1B2 " C1C2).
(4)From the definition of the cross-price elasticities, we have
B1 = "!11D1(p!1 , p!2)
p!1; B2 = "!22D2(p!1 , p
!2)
p!2(5)
C1 =!12D1(p!1 , p
!2)
p!2; C2 =
!21D2(p!1 , p!2)
p!1(6)
A1 = D1(p!1 , p
!2)(1" !11 " !12); A2 = D2(p
!1 , p
!2)(1" !22 " !21). (7)
Hierarchical Model, contd.Using these relationships, together with Equations (3) and (4), we obtain
p!1 =m1(2!11!22 " !12!21)
2!22(!11 + 1)" !12!21; p!2 =
!22m2
1 + !22.
Message
Although the structure of the market a!ects the price set by thefollower, this is not apparent when the price is written in terms ofthe cross-price elasticities and marginal costs (the formula isanalogous to the classical model). This is due to the fact that thecross-price elasticity depends on the equilibrium price.
Numerical Examples
! When goods are substitutes
! When goods are complements
Other Results
Market Structure Marginal E"ciency of Taxation1 firm, 1 good 2
3
2 firms, 1 good (Stackelberg leader) 23 "
2!2,1p!2 D2(p!1 ,p!2 )18!2,2p!1 D1(p!1 ,p!2 )+3!2,1p!2 D2(p!1 ,p!2 )
2 firms, 1 good (Stackelberg follower) 23 "
2!1,2[2p!1 !2,2D1(p!1 ,p!2 )"p!2 !2,1D2(p!1 ,p!2 )]p!2 D2(p!1 ,p!2 )[36!2,2!1,1"21!1,2!2,1]+6p!1 !2,2!1,2D1(p!1 ,p!2 )
2 firms, 1 good, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
1 firm, 2 goods, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
2 firms, 1 good, hierarchy 23 "
H1H2
Table: Summary of results on marginal e"ciency of taxation by market structure.
Other Results
Market Structure Marginal E"ciency of Taxation1 firm, 1 good 2
3
2 firms, 1 good (Stackelberg leader) 23 "
2!2,1p!2 D2(p!1 ,p!2 )18!2,2p!1 D1(p!1 ,p!2 )+3!2,1p!2 D2(p!1 ,p!2 )
2 firms, 1 good (Stackelberg follower) 23 "
2!1,2[2p!1 !2,2D1(p!1 ,p!2 )"p!2 !2,1D2(p!1 ,p!2 )]p!2 D2(p!1 ,p!2 )[36!2,2!1,1"21!1,2!2,1]+6p!1 !2,2!1,2D1(p!1 ,p!2 )
2 firms, 1 good, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
1 firm, 2 goods, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
2 firms, 1 good, hierarchy 23 "
H1H2
Table: Summary of results on marginal e"ciency of taxation by market structure.
Other Results
Market Structure Marginal E"ciency of Taxation1 firm, 1 good 2
3
2 firms, 1 good (Stackelberg leader) 23 "
2!2,1p!2 D2(p!1 ,p!2 )18!2,2p!1 D1(p!1 ,p!2 )+3!2,1p!2 D2(p!1 ,p!2 )
2 firms, 1 good (Stackelberg follower) 23 "
2!1,2[2p!1 !2,2D1(p!1 ,p!2 )"p!2 !2,1D2(p!1 ,p!2 )]p!2 D2(p!1 ,p!2 )[36!2,2!1,1"21!1,2!2,1]+6p!1 !2,2!1,2D1(p!1 ,p!2 )
2 firms, 1 good, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
1 firm, 2 goods, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
2 firms, 1 good, hierarchy 23 "
H1H2
Table: Summary of results on marginal e"ciency of taxation by market structure.
Other Results
Market Structure Marginal E"ciency of Taxation1 firm, 1 good 2
3
2 firms, 1 good (Stackelberg leader) 23 "
2!2,1p!2 D2(p!1 ,p!2 )18!2,2p!1 D1(p!1 ,p!2 )+3!2,1p!2 D2(p!1 ,p!2 )
2 firms, 1 good (Stackelberg follower) 23 "
2!1,2[2p!1 !2,2D1(p!1 ,p!2 )"p!2 !2,1D2(p!1 ,p!2 )]p!2 D2(p!1 ,p!2 )[36!2,2!1,1"21!1,2!2,1]+6p!1 !2,2!1,2D1(p!1 ,p!2 )
2 firms, 1 good, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
1 firm, 2 goods, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
2 firms, 1 good, hierarchy 23 "
H1H2
Table: Summary of results on marginal e"ciency of taxation by market structure.
Other Results
Market Structure Marginal E"ciency of Taxation1 firm, 1 good 2
3
2 firms, 1 good (Stackelberg leader) 23 "
2!2,1p!2 D2(p!1 ,p!2 )18!2,2p!1 D1(p!1 ,p!2 )+3!2,1p!2 D2(p!1 ,p!2 )
2 firms, 1 good (Stackelberg follower) 23 "
2!1,2[2p!1 !2,2D1(p!1 ,p!2 )"p!2 !2,1D2(p!1 ,p!2 )]p!2 D2(p!1 ,p!2 )[36!2,2!1,1"21!1,2!2,1]+6p!1 !2,2!1,2D1(p!1 ,p!2 )
2 firms, 1 good, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
1 firm, 2 goods, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
2 firms, 1 good, hierarchy 23 "
H1H2
Table: Summary of results on marginal e"ciency of taxation by market structure.
Other Results
Market Structure Marginal E"ciency of Taxation1 firm, 1 good 2
3
2 firms, 1 good (Stackelberg leader) 23 "
2!2,1p!2 D2(p!1 ,p!2 )18!2,2p!1 D1(p!1 ,p!2 )+3!2,1p!2 D2(p!1 ,p!2 )
2 firms, 1 good (Stackelberg follower) 23 "
2!1,2[2p!1 !2,2D1(p!1 ,p!2 )"p!2 !2,1D2(p!1 ,p!2 )]p!2 D2(p!1 ,p!2 )[36!2,2!1,1"21!1,2!2,1]+6p!1 !2,2!1,2D1(p!1 ,p!2 )
2 firms, 1 good, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
1 firm, 2 goods, no hierarchy 23 "
!2,1[2p!2 !1,1D2(p!1 ,p!2 )"p!1 !1,2D1(p!1 ,p!2 )]p!1 D1(p!1 ,p!2 )[18!1,1!2,2"6!1,2!2,1]+3p!2 !1,1!2,1D2(p!1 ,p!2 )
2 firms, 1 good, hierarchy 23 "
H1H2
Table: Summary of results on marginal e"ciency of taxation by market structure.
Further Work
Testing We’d like to test the magnitudes of theseine"ciencies econometrically and/or experimentally.
Comments? Any comments/questions are welcome
References
Jean Tirole. The Theory of Industrial Organization. MIT Press,1993.