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What is the Optimal Incidence of Taxation in
             Coupled Markets?

         Stephen Kinsella       David Ramsey

...
Today


  Idea


  Model


  Derivation


  Numerical Examples


  Further Work
This Paper in one slide



      Question What happens to coupled markets when one gets
               taxed?
       Metho...
This Paper in one slide



      Question What happens to coupled markets when one gets
               taxed?
       Metho...
This Paper in one slide



      Question What happens to coupled markets when one gets
               taxed?
       Metho...
This Paper in one slide



      Question What happens to coupled markets when one gets
               taxed?
       Metho...
Markets we study



           i) the classical case of a monopoly producing one good,
          ii) the case of two compa...
Notation




    i,j     % ↑ in demand for good i for 1% change in price of good j
    W (p)   Profit
    pD(p)   Revenue
 ...
Methodology




    1. Linearise around a demand curve
    2. Assume tax is small relative to total price
    3. Solve Sta...
Methodology




    1. Linearise around a demand curve
    2. Assume tax is small relative to total price
    3. Solve Sta...
Methodology




    1. Linearise around a demand curve
    2. Assume tax is small relative to total price
    3. Solve Sta...
Classical Model



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

                                ...
Classical Model



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

                                ...
Classical Model



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

                                ...
Classical Model



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

                                ...
Now impose a tax, t



                   W (p) = (p − m − t)(A − Bp) − E .
   The marginal tax revenue will be D(p ∗ )

 ...
Now impose a tax, t



                   W (p) = (p − m − t)(A − Bp) − E .
   The marginal tax revenue will be D(p ∗ )

 ...
Now impose a tax, t



                   W (p) = (p − m − t)(A − Bp) − E .
   The marginal tax revenue will be D(p ∗ )

 ...
Now impose a tax, t



                   W (p) = (p − m − t)(A − Bp) − E .
   The marginal tax revenue will be D(p ∗ )

 ...
Now impose a tax, t



                   W (p) = (p − m − t)(A − Bp) − E .
   The marginal tax revenue will be D(p ∗ )

 ...
Hierarchical Model, 2 firms, one good each




   Two goods, two firms: p1 , p2 , Firm 1 is Stackelberg leader.
   Linearise...
Hierarchical Model, 2 firms, one good each




   Two goods, two firms: p1 , p2 , Firm 1 is Stackelberg leader.
   Linearise...
Hierarchical Model, contd
   First Eqm condition
                                   δW2
                                  ...
Hierarchical Model, contd
   First Eqm condition
                                   δW2
                                  ...
Hierarchical Model, contd
   First Eqm condition
                                   δW2
                                  ...
Hierarchical Model, contd
   First Eqm condition
                                   δW2
                                  ...
Hierarchical Model, contd.
                                                                               2        2
     ...
Hierarchical Model, contd.
                                                                               2        2
     ...
Hierarchical Model, contd.
   Using these relationships, together with Equations (3) and (4), we obtain

                 ...
Message




  Although the structure of the market affects the price set by the
  follower, this is not apparent when the p...
Numerical Examples




      When goods are substitutes
      When goods are complements
Other Results
    Market Structure                                       Marginal Efficiency of Taxation
                   ...
Other Results
    Market Structure                                       Marginal Efficiency of Taxation
                   ...
Other Results
    Market Structure                                       Marginal Efficiency of Taxation
                   ...
Other Results
    Market Structure                                       Marginal Efficiency of Taxation
                   ...
Other Results
    Market Structure                                       Marginal Efficiency of Taxation
                   ...
Other Results
    Market Structure                                       Marginal Efficiency of Taxation
                   ...
Further Work




      Testing We’d like to test the magnitudes of these
              inefficiencies econometrically and/or...
References




   Jean Tirole. The Theory of Industrial Organization. MIT Press,
     1993.
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What is the Optimal Incidence of Taxation in Coupled Markets

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What is the Optimal Incidence of Taxation in Coupled Markets

  1. 1. What is the Optimal Incidence of Taxation in Coupled Markets? Stephen Kinsella David Ramsey University of Limerick Irish Economic Association, April 25–27, 2008
  2. 2. Today Idea Model Derivation Numerical Examples Further Work
  3. 3. This Paper in one slide Question What happens to coupled markets when one gets taxed? Method Study imposition of the tax under more and more complex market arrangements Results As markets become more coupled, firm relationships really matter for the imposition of the tax Practical application Environmental taxes; tourism subsidies. Further Work Econometric testing of cross price elasticities: need natural/computational experiment
  4. 4. This Paper in one slide Question What happens to coupled markets when one gets taxed? Method Study imposition of the tax under more and more complex market arrangements Results As markets become more coupled, firm relationships really matter for the imposition of the tax Practical application Environmental taxes; tourism subsidies. Further Work Econometric testing of cross price elasticities: need natural/computational experiment
  5. 5. This Paper in one slide Question What happens to coupled markets when one gets taxed? Method Study imposition of the tax under more and more complex market arrangements Results As markets become more coupled, firm relationships really matter for the imposition of the tax Practical application Environmental taxes; tourism subsidies. Further Work Econometric testing of cross price elasticities: need natural/computational experiment
  6. 6. This Paper in one slide Question What happens to coupled markets when one gets taxed? Method Study imposition of the tax under more and more complex market arrangements Results As markets become more coupled, firm relationships really matter for the imposition of the tax Practical application Environmental taxes; tourism subsidies. Further Work Econometric testing of cross price elasticities: need natural/computational experiment
  7. 7. Markets we study i) the classical case of a monopoly producing one good, ii) the case of two companies each producing a good in which one company is a Stackelberg leader. We consider the effect of placing a tax on a) the leader and b) the follower, iii) the case of two companies each producing a good in which there is no such hierarchy, iv) the case of a monopoly producing two goods.
  8. 8. Notation i,j % ↑ in demand for good i for 1% change in price of good j W (p) Profit pD(p) Revenue m Marginal cost of production E fixed costs iff m = m¯ p total price t levy/tax
  9. 9. 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 effect on efficiency of the imposition of tax
  10. 10. 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 effect on efficiency of the imposition of tax
  11. 11. 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 effect on efficiency of the imposition of tax
  12. 12. Classical Model W (p) = (A − Bp)(p − m) − E , (1) m A p∗ = + . (2) 2 2B − D(p ∗ ) B= ; A = (1 − )D(p ∗ ). p∗ p ∗ (1 + ) m m= ⇒ p∗ = . 1+ This is the monopoly (non-discrimination) pricing rule from Tirole (1993, pg. 76.)
  13. 13. Classical Model W (p) = (A − Bp)(p − m) − E , (1) m A p∗ = + . (2) 2 2B − D(p ∗ ) B= ; A = (1 − )D(p ∗ ). p∗ p ∗ (1 + ) m m= ⇒ p∗ = . 1+ This is the monopoly (non-discrimination) pricing rule from Tirole (1993, pg. 76.)
  14. 14. Classical Model W (p) = (A − Bp)(p − m) − E , (1) m A p∗ = + . (2) 2 2B − D(p ∗ ) B= ; A = (1 − )D(p ∗ ). p∗ p ∗ (1 + ) m m= ⇒ p∗ = . 1+ This is the monopoly (non-discrimination) pricing rule from Tirole (1993, pg. 76.)
  15. 15. Classical Model W (p) = (A − Bp)(p − m) − E , (1) m A p∗ = + . (2) 2 2B − D(p ∗ ) B= ; A = (1 − )D(p ∗ ). p∗ p ∗ (1 + ) m m= ⇒ p∗ = . 1+ This is the monopoly (non-discrimination) pricing rule from Tirole (1993, pg. 76.)
  16. 16. 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 ∗ ). D(p ∗ ) ∆S = ∆pD(p ∗ ) = 2 i.e. half of the marginal tax revenue. It follows that the marginal efficiency of this tax is 2 . 3
  17. 17. 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 ∗ ). D(p ∗ ) ∆S = ∆pD(p ∗ ) = 2 i.e. half of the marginal tax revenue. It follows that the marginal efficiency of this tax is 2 . 3
  18. 18. 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 ∗ ). D(p ∗ ) ∆S = ∆pD(p ∗ ) = 2 i.e. half of the marginal tax revenue. It follows that the marginal efficiency of this tax is 2 . 3
  19. 19. 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 ∗ ). D(p ∗ ) ∆S = ∆pD(p ∗ ) = 2 i.e. half of the marginal tax revenue. It follows that the marginal efficiency of this tax is 2 . 3
  20. 20. 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 ∗ ). D(p ∗ ) ∆S = ∆pD(p ∗ ) = 2 i.e. half of the marginal tax revenue. It follows that the marginal efficiency of this tax is 2 . 3
  21. 21. 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 − B1 p1 + C1 p2 ] − E1 W2 (p1 , p2 ) = (p2 − m2 )[A2 − B2 p2 + C2 p1 ] − E2 .
  22. 22. 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 − B1 p1 + C1 p2 ] − E1 W2 (p1 , p2 ) = (p2 − m2 )[A2 − B2 p2 + C2 p1 ] − E2 .
  23. 23. Hierarchical Model, contd First Eqm condition δW2 = 0. δp2 |p=(p1 ,p2 (p1 )) ∗ Which leads to ∗ m A2 + C2 p1 p2 (p1 ) = + . 2 2B2 And the second eqm condition is δW1 = 0. δp1 |p=(p1 ,p2 (p1 )) ∗ ∗ ∗ Which leads to ∗ 2A1 B2 + C1 A2 + C1 B2 m2 + 2B1 B2 m1 − C1 C2 m1 p1 = (3) 4B1 B2 − 2C1 C2
  24. 24. Hierarchical Model, contd First Eqm condition δW2 = 0. δp2 |p=(p1 ,p2 (p1 )) ∗ Which leads to ∗ m A2 + C2 p1 p2 (p1 ) = + . 2 2B2 And the second eqm condition is δW1 = 0. δp1 |p=(p1 ,p2 (p1 )) ∗ ∗ ∗ Which leads to ∗ 2A1 B2 + C1 A2 + C1 B2 m2 + 2B1 B2 m1 − C1 C2 m1 p1 = (3) 4B1 B2 − 2C1 C2
  25. 25. Hierarchical Model, contd First Eqm condition δW2 = 0. δp2 |p=(p1 ,p2 (p1 )) ∗ Which leads to ∗ m A2 + C2 p1 p2 (p1 ) = + . 2 2B2 And the second eqm condition is δW1 = 0. δp1 |p=(p1 ,p2 (p1 )) ∗ ∗ ∗ Which leads to ∗ 2A1 B2 + C1 A2 + C1 B2 m2 + 2B1 B2 m1 − C1 C2 m1 p1 = (3) 4B1 B2 − 2C1 C2
  26. 26. Hierarchical Model, contd First Eqm condition δW2 = 0. δp2 |p=(p1 ,p2 (p1 )) ∗ Which leads to ∗ m A2 + C2 p1 p2 (p1 ) = + . 2 2B2 And the second eqm condition is δW1 = 0. δp1 |p=(p1 ,p2 (p1 )) ∗ ∗ ∗ Which leads to ∗ 2A1 B2 + C1 A2 + C1 B2 m2 + 2B1 B2 m1 − C1 C2 m1 p1 = (3) 4B1 B2 − 2C1 C2
  27. 27. Hierarchical Model, contd. 2 2 4A2 B1 B2 − A2 C1 C2 + 2C2 A1 B2 − C1 C2 B2 m2 + 2C2 m1 B1 B2 − m1 C1 C2 + 4B1 B2 m2 ∗ p2 = . 4B2 (2B1 B2 − C1 C2 ) (4) From the definition of the cross-price elasticities, we have ∗ ∗ ∗ ∗ 11 D1 (p1 , p2 ) 22 D2 (p1 , p2 ) B1 = − ∗ ; B2 = − ∗ (5) p1 p2 ∗ ∗ ∗ ∗ 12 D1 (p1 , p2 ) 21 D2 (p1 , p2 ) C1 = ∗ ; C2 = ∗ (6) p2 p1 ∗ ∗ ∗ ∗ A1 = D1 (p1 , p2 )(1 − 11 − 12 ); A2 = D2 (p1 , p2 )(1 − 22 − 21 ). (7)
  28. 28. Hierarchical Model, contd. 2 2 4A2 B1 B2 − A2 C1 C2 + 2C2 A1 B2 − C1 C2 B2 m2 + 2C2 m1 B1 B2 − m1 C1 C2 + 4B1 B2 m2 ∗ p2 = . 4B2 (2B1 B2 − C1 C2 ) (4) From the definition of the cross-price elasticities, we have ∗ ∗ ∗ ∗ 11 D1 (p1 , p2 ) 22 D2 (p1 , p2 ) B1 = − ∗ ; B2 = − ∗ (5) p1 p2 ∗ ∗ ∗ ∗ 12 D1 (p1 , p2 ) 21 D2 (p1 , p2 ) C1 = ∗ ; C2 = ∗ (6) p2 p1 ∗ ∗ ∗ ∗ A1 = D1 (p1 , p2 )(1 − 11 − 12 ); A2 = D2 (p1 , p2 )(1 − 22 − 21 ). (7)
  29. 29. Hierarchical Model, contd. Using these relationships, together with Equations (3) and (4), we obtain ∗ m1 (2 11 22 − 12 21 ) ∗ 22 m2 p1 = ; p2 = . 2 22 ( 11 + 1) − 12 21 1 + 22
  30. 30. Message Although the structure of the market affects the price set by the follower, this is not apparent when the price is written in terms of the cross-price elasticities and marginal costs (the formula is analogous to the classical model). This is due to the fact that the cross-price elasticity depends on the equilibrium price.
  31. 31. Numerical Examples When goods are substitutes When goods are complements
  32. 32. Other Results Market Structure Marginal Efficiency of Taxation 2 1 firm, 1 good 3∗ ∗ ∗ 2 2,1 p2 D2 (p1 ,p2 ) 2 2 firms, 1 good (Stackelberg leader) 3 − 18 2,2 p1 D1 (p1 ,p2 )+3 2,1 p2 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ [2p 2,2 D (p1 2∗ ,p ∗ )−p ∗ D (p ∗ ,p ∗ )] 2 2 firms, 1 good (Stackelberg follower) 3 − p∗ D2 (p∗ ,p1,2 1 2,2 1,11−21 1,2 2,12 2,1 ∗ 22,21 1,22D1 (p∗ ,p∗ ) ∗ )[36 ]+6p1 2 1 2 1 2 ∗ ∗ ∗ ∗ D (p ∗ ,p ∗ )] 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p 2 firms, 1 good, no hierarchy 3 − p∗ D1 (p∗ ,p∗ )[18 1,1 2,2 −6 1,2 2,11 1,2 ∗ 11,11 2,12D2 (p∗ ,p∗ ) ]+3p2 1 1 2 1 2 ∗ ∗ ∗ ∗ ∗ ∗ 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p1 1,2 D1 (p1 ,p2 )] 1 firm, 2 goods, no hierarchy 3 − p1 D1 (p1 ,p2 )[18 1,1 2,2 −6 1,2 2,1 ]+3p2 1,1 2,1 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 H1 2 firms, 1 good, hierarchy 3 − H2 Table: Summary of results on marginal efficiency of taxation by market structure.
  33. 33. Other Results Market Structure Marginal Efficiency of Taxation 2 1 firm, 1 good 3∗ ∗ ∗ 2 2,1 p2 D2 (p1 ,p2 ) 2 2 firms, 1 good (Stackelberg leader) 3 − 18 2,2 p1 D1 (p1 ,p2 )+3 2,1 p2 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ [2p 2,2 D (p1 2∗ ,p ∗ )−p ∗ D (p ∗ ,p ∗ )] 2 2 firms, 1 good (Stackelberg follower) 3 − p∗ D2 (p∗ ,p1,2 1 2,2 1,11−21 1,2 2,12 2,1 ∗ 22,21 1,22D1 (p∗ ,p∗ ) ∗ )[36 ]+6p1 2 1 2 1 2 ∗ ∗ ∗ ∗ D (p ∗ ,p ∗ )] 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p 2 firms, 1 good, no hierarchy 3 − p∗ D1 (p∗ ,p∗ )[18 1,1 2,2 −6 1,2 2,11 1,2 ∗ 11,11 2,12D2 (p∗ ,p∗ ) ]+3p2 1 1 2 1 2 ∗ ∗ ∗ ∗ ∗ ∗ 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p1 1,2 D1 (p1 ,p2 )] 1 firm, 2 goods, no hierarchy 3 − p1 D1 (p1 ,p2 )[18 1,1 2,2 −6 1,2 2,1 ]+3p2 1,1 2,1 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 H1 2 firms, 1 good, hierarchy 3 − H2 Table: Summary of results on marginal efficiency of taxation by market structure.
  34. 34. Other Results Market Structure Marginal Efficiency of Taxation 2 1 firm, 1 good 3∗ ∗ ∗ 2 2,1 p2 D2 (p1 ,p2 ) 2 2 firms, 1 good (Stackelberg leader) 3 − 18 2,2 p1 D1 (p1 ,p2 )+3 2,1 p2 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ [2p 2,2 D (p1 2∗ ,p ∗ )−p ∗ D (p ∗ ,p ∗ )] 2 2 firms, 1 good (Stackelberg follower) 3 − p∗ D2 (p∗ ,p1,2 1 2,2 1,11−21 1,2 2,12 2,1 ∗ 22,21 1,22D1 (p∗ ,p∗ ) ∗ )[36 ]+6p1 2 1 2 1 2 ∗ ∗ ∗ ∗ D (p ∗ ,p ∗ )] 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p 2 firms, 1 good, no hierarchy 3 − p∗ D1 (p∗ ,p∗ )[18 1,1 2,2 −6 1,2 2,11 1,2 ∗ 11,11 2,12D2 (p∗ ,p∗ ) ]+3p2 1 1 2 1 2 ∗ ∗ ∗ ∗ ∗ ∗ 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p1 1,2 D1 (p1 ,p2 )] 1 firm, 2 goods, no hierarchy 3 − p1 D1 (p1 ,p2 )[18 1,1 2,2 −6 1,2 2,1 ]+3p2 1,1 2,1 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 H1 2 firms, 1 good, hierarchy 3 − H2 Table: Summary of results on marginal efficiency of taxation by market structure.
  35. 35. Other Results Market Structure Marginal Efficiency of Taxation 2 1 firm, 1 good 3∗ ∗ ∗ 2 2,1 p2 D2 (p1 ,p2 ) 2 2 firms, 1 good (Stackelberg leader) 3 − 18 2,2 p1 D1 (p1 ,p2 )+3 2,1 p2 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ [2p 2,2 D (p1 2∗ ,p ∗ )−p ∗ D (p ∗ ,p ∗ )] 2 2 firms, 1 good (Stackelberg follower) 3 − p∗ D2 (p∗ ,p1,2 1 2,2 1,11−21 1,2 2,12 2,1 ∗ 22,21 1,22D1 (p∗ ,p∗ ) ∗ )[36 ]+6p1 2 1 2 1 2 ∗ ∗ ∗ ∗ D (p ∗ ,p ∗ )] 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p 2 firms, 1 good, no hierarchy 3 − p∗ D1 (p∗ ,p∗ )[18 1,1 2,2 −6 1,2 2,11 1,2 ∗ 11,11 2,12D2 (p∗ ,p∗ ) ]+3p2 1 1 2 1 2 ∗ ∗ ∗ ∗ ∗ ∗ 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p1 1,2 D1 (p1 ,p2 )] 1 firm, 2 goods, no hierarchy 3 − p1 D1 (p1 ,p2 )[18 1,1 2,2 −6 1,2 2,1 ]+3p2 1,1 2,1 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 H1 2 firms, 1 good, hierarchy 3 − H2 Table: Summary of results on marginal efficiency of taxation by market structure.
  36. 36. Other Results Market Structure Marginal Efficiency of Taxation 2 1 firm, 1 good 3∗ ∗ ∗ 2 2,1 p2 D2 (p1 ,p2 ) 2 2 firms, 1 good (Stackelberg leader) 3 − 18 2,2 p1 D1 (p1 ,p2 )+3 2,1 p2 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ [2p 2,2 D (p1 2∗ ,p ∗ )−p ∗ D (p ∗ ,p ∗ )] 2 2 firms, 1 good (Stackelberg follower) 3 − p∗ D2 (p∗ ,p1,2 1 2,2 1,11−21 1,2 2,12 2,1 ∗ 22,21 1,22D1 (p∗ ,p∗ ) ∗ )[36 ]+6p1 2 1 2 1 2 ∗ ∗ ∗ ∗ D (p ∗ ,p ∗ )] 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p 2 firms, 1 good, no hierarchy 3 − p∗ D1 (p∗ ,p∗ )[18 1,1 2,2 −6 1,2 2,11 1,2 ∗ 11,11 2,12D2 (p∗ ,p∗ ) ]+3p2 1 1 2 1 2 ∗ ∗ ∗ ∗ ∗ ∗ 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p1 1,2 D1 (p1 ,p2 )] 1 firm, 2 goods, no hierarchy 3 − p1 D1 (p1 ,p2 )[18 1,1 2,2 −6 1,2 2,1 ]+3p2 1,1 2,1 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 H1 2 firms, 1 good, hierarchy 3 − H2 Table: Summary of results on marginal efficiency of taxation by market structure.
  37. 37. Other Results Market Structure Marginal Efficiency of Taxation 2 1 firm, 1 good 3∗ ∗ ∗ 2 2,1 p2 D2 (p1 ,p2 ) 2 2 firms, 1 good (Stackelberg leader) 3 − 18 2,2 p1 D1 (p1 ,p2 )+3 2,1 p2 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ [2p 2,2 D (p1 2∗ ,p ∗ )−p ∗ D (p ∗ ,p ∗ )] 2 2 firms, 1 good (Stackelberg follower) 3 − p∗ D2 (p∗ ,p1,2 1 2,2 1,11−21 1,2 2,12 2,1 ∗ 22,21 1,22D1 (p∗ ,p∗ ) ∗ )[36 ]+6p1 2 1 2 1 2 ∗ ∗ ∗ ∗ D (p ∗ ,p ∗ )] 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p 2 firms, 1 good, no hierarchy 3 − p∗ D1 (p∗ ,p∗ )[18 1,1 2,2 −6 1,2 2,11 1,2 ∗ 11,11 2,12D2 (p∗ ,p∗ ) ]+3p2 1 1 2 1 2 ∗ ∗ ∗ ∗ ∗ ∗ 2 2,1 [2p2 1,1 D2 (p1 ,p2 )−p1 1,2 D1 (p1 ,p2 )] 1 firm, 2 goods, no hierarchy 3 − p1 D1 (p1 ,p2 )[18 1,1 2,2 −6 1,2 2,1 ]+3p2 1,1 2,1 D2 (p1 ,p2 ) ∗ ∗ ∗ ∗ ∗ ∗ 2 H1 2 firms, 1 good, hierarchy 3 − H2 Table: Summary of results on marginal efficiency of taxation by market structure.
  38. 38. Further Work Testing We’d like to test the magnitudes of these inefficiencies econometrically and/or experimentally. Comments? Any comments/questions are welcome
  39. 39. References Jean Tirole. The Theory of Industrial Organization. MIT Press, 1993.

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