Consider an economic federation comprising two countries, which dier with respect to their ability of commitment. Country 1 - to be called the horizontal Stackelberg leader - acts as rst mover vis-à-vis both the federal government and the other country. In contrast, country 2 - to be called the horizontal follower - acts as rst mover only vis-à-vis the federal government.8
The decision making structure within the federation is as follows. First, the horizontal Stackelberg leader chooses its policy while anticipating, and in- corporating into its decision problem, how the horizontal follower, the federal
7This approach is in line with Fuest and Huber (1997) and Aronsson and Sjögren (2004a,
2004b).
8The number of member countries has been normalized to two - additional horizontal
government and the domestic private sector respond to its policy. Second, the horizontal follower chooses its policy, taking as given the policies decided by the horizontal Stackelberg leader, while anticipating the federal government’s and the domestic private sectors’ response to its policy. Third, the federal government chooses maximum allowable emissions (environmental targets) for the countries. These targets are conditioned on the national tax policy, while anticipating the behavioral responses by the private sector in each country. Fi- nally, the private sector makes its choices conditional on the domestic public policy. However, when solving for the optimal tax policy, it is convenient to start backwards, i.e. with the private sector, the federal government and so on.
2.1
Consumers
Each member country is made up of identical consumers, the number of which is normalized to one. The preferences of the consumer in country l (l = 1> 2) are dened by the utility function
Xl= x¡fl> {l> }l¢ ! (H) (1)
where fl is the consumption of a good produced by a clean technology,{l the consumption of a good produced by a dirty technology and }l leisure. Leisure is dened as }l = K ol, where K is a xed time endowment and ol the hours of work. The variableH is the environmental damage and is treated as exogenous by the consumer. The function x (·) is assumed to be increasing in each argument and strictly concave, while! (·) is increasing and strictly convex in the environmental damage.
The consumer’s budget constraint is given by
zlol Wl¡zlol¢= fl+ s{l (2)
where zl is the gross wage rate and Wl¡zlol¢ is the income tax paid to the national government. The price of good {l is denoted by s, while good fl is a numeraire good. Good { is assumed to be traded on a world market, which is meant to imply that its price is treated as exogenous by the federal and national governments as well as by the private agents.
The optimal tax problem will be dened in terms of a conditional indirect utility function and conditional demand functions. Therefore, it is convenient
to solve each consumer’s optimization problem in two stages (following Chris- tiansen (1984)). In the rst stage, fl and {l are chosen conditional on}l and the budget constraintLl= fl+s{l, whereLlis the consumer’s net income. This gives the conditional demand functions for the two goods,{l= {¡Ll> }l> s¢and fl= f¡Ll> }l> s¢, as well as the conditional indirect utility function
Yl= Y¡Ll> }l> s> H¢= y¡Ll> }l> s¢ ! (H) = (3)
In the second stage, the hours of work are chosen to maximize the conditional indirect utility function subject to the budget constraintLl = zlol Wl¡zlol¢. The rst order condition is written
Cyl
CLlzl
¡
1 l¢= Cyl
C}l (4)
whereyl= y¡Ll> }l¢, and where l = CWl¡zlol¢@C¡zlol¢is the marginal income tax rate.
2.2
Production
Turning to the production part of the model, each country is made up of two competitive production sectors which produce goodsf and {, respectively. Within each sector, all rms are identical and their number normalized to one. In both sectors, labor and a sector specic input are used in the production, while the production of good {l also requires the use of an environmentally bad input, el.9 The use of bad input causes emissions that are transboundary.
Therefore, the aggregate environmental damage in each country, H, is dened as
H = e1+ e2= (5)
The prots in sectorsf and { are given by l
f = ifl¡olf¢ zlofl (6)
l
{ = si{l¡ol{> el¢ zlo{l wlel (7)
9The sector specic production factor - introduced to avoid xed wages - will be suppressed
for notational convenience throughout the paper. It is implicitly assumed that the sector specic factor is owned by the domestic government.
wherewlis the production tax on the ‘bad’ input, and where subindices indicate production sectors.10>11 The production functions,ifl(·) and i{l(·), are increas- ing and strictly concave in their respective arguments, and characterized by decreasing returns to scale. Moreover, it is assumed that the supply of the bad input is completely elastic, meaning that the price of the bad input is indepen- dent of changes in the demand. Therefore, it will be convenient to normalize the producer price of the bad input to zero. This assumption follows related literature (see e.g. Aronsson et al. (2006a)) and implies that the unit cost of the bad input in sector{ equals the production tax. It is also assumed that labor is immobile between countries but completely mobile between sectors, meaning that the wage in each country will be the same in both sectors. The rst order conditions for prot maximization can be written as
Cil f¡olf¢ Col f z l = 0 (8) sCi{l ¡ ol {> el¢ Col { z l = 0 (9) sCi{l ¡ ol {> el¢ Cel wl = 0= (10)
By combining equations (8)-(10) with the labor market condition ofol ol{+olf, the equilibrium levels of zl, ol{, olf andel can be written as functions ofwl and ol, i.e. zl = zl¡wl> ol¢ (11) ol { = ol{¡wl> ol¢ (12) ol f = olf¡wl> ol¢ (13) el = el¡wl> ol¢ (14)
where the xed consumer price, s, is suppressed for notational convenience12.
1 0Since there are no consumption taxes, the producer price equals the consumer price. 1 1See Cremer and Gahvari (2001) for a discussion about the choice between a consumption
tax and a production tax, and its implications for the results.