What is the Optimal Incidence of Taxation in Coupled Markets

What is the Optimal Incidence of Taxation inCoupled Markets?

Stephen Kinsella David Ramsey

University of Limerick

Irish Economic Association, April 25–27, 2008

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

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

Classical Model

W (p) = (A" Bp)(p "m)" E , (1)

2B. (2)

B ="!D(p!)

p!; A = (1" !)D(p!).

m =p!(1 + !)

!# p! =

1 + !.

This is the monopoly (non-discrimination) pricing rule from Tirole(1993, pg. 76.)

Classical Model

W (p) = (A" Bp)(p "m)" E , (1)

2B. (2)

B ="!D(p!)

p!; A = (1" !)D(p!).

m =p!(1 + !)

!# p! =

1 + !.

Classical Model

W (p) = (A" Bp)(p "m)" E , (1)

2B. (2)

B ="!D(p!)

p!; A = (1" !)D(p!).

m =p!(1 + !)

!# p! =

1 + !.

Classical Model

W (p) = (A" Bp)(p "m)" E , (1)

2B. (2)

B ="!D(p!)

p!; A = (1" !)D(p!).

m =p!(1 + !)

!# p! =

1 + !.

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!)

i.e. half of the marginal tax revenue. It follows that the marginale"ciency of this tax is 2

Now impose a tax, t

W (p) = (p "m " t)(A" Bp)" E .

#W = (m " p!)#D + (1"#p)D(p!) = D(p!).

#S = #pD(p!) =D(p!)

Now impose a tax, t

W (p) = (p "m " t)(A" Bp)" E .

#W = (m " p!)#D + (1"#p)D(p!) = D(p!).

#S = #pD(p!) =D(p!)

Now impose a tax, t

W (p) = (p "m " t)(A" Bp)" E .

#W = (m " p!)#D + (1"#p)D(p!) = D(p!).

#S = #pD(p!) =D(p!)

Now impose a tax, t

W (p) = (p "m " t)(A" Bp)" E .

#W = (m " p!)#D + (1"#p)D(p!) = D(p!).

#S = #pD(p!) =D(p!)

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

"p2 |p=(p1,p!2 (p1))= 0.

Which leads to

p!2(p1) =m

A2 + C2p1

And the second eqm condition is

"p1 |p=(p!1 ,p!2 (p!1 ))= 0.

Which leads to

p!1 =2A1B2 + C1A2 + C1B2m2 + 2B1B2m1 " C1C2m1

4B1B2 " 2C1C2(3)

"p2 |p=(p1,p!2 (p1))= 0.

Which leads to

p!2(p1) =m

A2 + C2p1

"p1 |p=(p!1 ,p!2 (p!1 ))= 0.

Which leads to

p!1 =2A1B2 + C1A2 + C1B2m2 + 2B1B2m1 " C1C2m1

4B1B2 " 2C1C2(3)

"p2 |p=(p1,p!2 (p1))= 0.

Which leads to

p!2(p1) =m

A2 + C2p1

"p1 |p=(p!1 ,p!2 (p!1 ))= 0.

Which leads to

p!1 =2A1B2 + C1A2 + C1B2m2 + 2B1B2m1 " C1C2m1

4B1B2 " 2C1C2(3)

"p2 |p=(p1,p!2 (p1))= 0.

Which leads to

p!2(p1) =m

A2 + C2p1

"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

p!2(5)

C1 =!12D1(p!1 , p

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

p!2(5)

C1 =!12D1(p!1 , p

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 =

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

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 firms, 1 good, hierarchy 23 "

Table: Summary of results on marginal e"ciency of taxation by market structure.

Other Results

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 )

Other Results

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 )

Other Results

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 )

Other Results

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 )

Other Results

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 )

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.

What is the Optimal Incidence of Taxation in Coupled Markets

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