A Tax revenue maximization

A.1 A necessary conditions for revenue maximizing tax rates When we maximize the tax revenue associated with spirits, it can be shown that thefirst order condition can be written as

t^N

p^N_R =− 1 El_p^N

RS_R^N. (36)

Since 0 < t^N/p^N_R < 1 we must have that El_pN

RS_R^N < −1. In words: A necessary condition for the existence of a tax revenue maximizing tax rate on spirits is that the price elasticity of registered spirit purchase must take on values greater than one in absolute value, i.e. the demand must be sensitive enough. Likewise we have for Sweden

t^S

p^S_R =− 1 El_pS

R S^S_R+S_S^N . (37)

Andfinally for Denmark

t^D

p^D_R =− 1

El_pD

R S_R^D+S_D^N +S_D^S . (38) The only difference from (36) being that we must include foreigners cross-border shopping in the formula for Sweden and Denmark, i.e., it is the change in total taxed demand which matters.

A.2 Governments’ objective: tax revenue maximization In this section we presents the results when the governments maximize tax revenue from spirits. For Norway we have

max

t^N

R^N(t^N, t^S, t^D) =t^NS_R^N, (39) whereR^N is the tax revenue from spirits in Norway which in general depend on own and other countries’ tax rates on spirits. Likewise, the Sweden maximizes tax revenue

max

t^S R^S(t^N, t^S, t^D) =t^S S^S_R+S_S^N . (40) Note that Norwegian (outward) cross-border shopping, i.e., spirits purchased in Sweden, is included in the Swedish revenue implying that Sweden gains tax revenue at the expense of Norway. Finally the objective of the Danish government is

max

t^S

R^D(t^N, t^S, t^D) =t^D S_R^D+S_D^S +S_D^N (41) Note that both Swedish and Norwegian cross-border shopping is included in the tax revenue for Denmark.

The first order condition associated with(34) can be written as

∂R^N

∂t^N =S_R^N+t^N∂S_R^N

∂t^N = 0. (42)

Likewise thefirst order condition for(40) is

∂R^S

∂t^S =S_R^S+S_S^N+t^S ∂S_R^S

∂t^S +∂S_S^N

∂t^S = 0. (43) Andfinally thefirst order condition for(41)is

∂R^D

∂t^D =S_R^D+S^N_D+S^S_D+t^D ∂S_R^D

∂t^D +∂S_D^N

∂t^D +∂S_D^S

∂t^D = 0. (44) (42)−(43) implicitly define a reaction function for Norway, Sweden and Denmark respectively, i.e., the maximizing tax rate for a given country as a function of the other countries’ tax rates. Let these functions be writ-ten as t^N(t^S, t^D), t^S(t^N, t^D) and t^D(t^N, t^S). Two types of game are then considered, a simultaneous and a sequential game.

The equilibrium tax rates for a simultaneous game solves the following equation

∂R^N

∂t^N = ∂R^S

∂t^S = ∂R^D

∂t^D = 0, (45)

i.e., the equilibrium is defined as the intersection of all the reaction functions.

For the sequential game we will assume that Denmark sets their tax

rate first, then followed by Sweden and finally Norway. We use backward

induction and start with Norway max

t^N R^N(t^N, t^S, t^D), (46) i.e., Norway maximizes the tax revenue taking the Swedish and Danish tax rate as given. The solution to this problem gives the reaction function for Norway, i.e., the maximizing Norwegian tax rate as a function of the Swedish and Danish tax rates,t^N(t^S, t^D). Sweden maximizes tax revenue subject to the reaction function of Norway, taking the Danish tax rate as given, hence

max

t^S

R^S(t^N, t^S, t^D) (47) s.t. t^N(t^S, t^D).

The solution to this problem implicitly gives us the Swedish maximizing tax rate as a function of the Danish tax rate set in the first period of the game, i.e. t^S t^D . Denmark maximizes tax revenue subject to this and the reaction function for Norway:

max

t^D R^D(t^N, t^S, t^D) (48) s.t. t^N t^S, t^D and t^S(t^D) .

The maximizing Danish tax rate in equilibrium is given by the solution to this problem, and can be written as:

∂R^D

∂t^D + ∂R^D

∂t^N

∂t^N

∂t^S

∂t^S

∂t^D +∂t^N

∂t^D +∂R^D

∂t^S

∂t^S

∂t^D = 0. (49) A.3 Simulation results

The tax revenue from spirits first increases monotonically when tax rate on spirits increases, and then, after reaching a maximum point, decreases monotonically with the tax rate. This is in line with earlierfindings. ²³

Tables A.3.1 and A.3.2 show that there exists an equilibrium under each type of game, and present the simulation results. Table A.3.1 gives us the results for the simultaneous game. In thefirst row we have the equilibrium tax rates for each country. We see that the equilibrium tax rates is highest in Norway, then followed by Sweden and Denmark. Compared to the situation

2 3In Aasness and Nygård (2009) we simulated Dupuit-curves (better known as Laffer-curves) for different cross-border exposed goods in Norway. For some goods this showed that atfirst the tax revenue increases monotonically with the tax rate, then, after reaching a maximum, decreases monotonically with the tax rate.

in 2004, this implies a price increase on spirits for all countries. The price increase is by far highest for Norway (30.1%) and Denmark (22.6%), followed by Sweden (2.2%). The ordering of the elasticities is consistent with the ordering of the own price elasticity of registered spirits in tables 1, 3 and 5.

A country with a low price sensitivity with respect to registered spirits, gets the highest tax rate. Note that the equilibrium price differential between Norway and the two other countries is higher than in 2004. Especially the price differential between Norway and Sweden has increased substantially.

On the other hand, the price differential between Sweden and Denmark is lower compared to the situation in 2004.

If we turn to the sequential game in Table A.3.2 we see that the sequen-tial game consists of higher tax rates and higher tax revenue for all countries, compared to the simultaneous game. Note that the equilibrium tax rates change somewhat in the tables, although not much. This suggests that the maximizing tax rate in each country is rather insensitive with respect to the two other countries tax rate. This is confirmed by simulations of reaction curves.

Table A.3.1. Equilibrium tax rates, prices and tax revenue - simultaneous game, revenue maximization

Norway Sweden Denmark

Tax rate (t/q) 3,169 2,879 1,131

Price change (%) 1) 30,1 2,1 22,6

Price level 2) 101,1 52,4 38,1

Tax revenue 3263,9 4692,7 1888,6

1) Price change compared to initial situation, i.e. situation in 2004.

2) Norwegian equilibrium price in simultaneous game for MITC with utility maximization =100.

Table A.3.2. Equilibrium tax rates, prices and tax revenue - sequential game, revenue maximization

Norway Sweden Denmark

Tax rate (t/q) 3,172 2,902 1,140

Price change (%) 1) 30,2 2,7 23,1

Price level 2) 101,2 52,7 38,3

Tax revenue 3266,4 4694,7 1890,5

1) Price change compared to initial situation, i.e. situation in 2004.

2) Norwegian equilibrium price in simultaneous game for MITC with utility maximization =100.

B An alternative model with explicit

In document Commodity tax competition – purchases of spirits in the Scandinavian countries (sider 21-25)