Optimal Tax Policy and Environmental Externality: A General Equilibrium Analysis
4. The optimal taxation
4.1 Marginal tax rates for households
In this model the tax system consists of a direct income tax and an indirect commodity tax. The total tax paid by a worker of type h household is the sum of these two taxes τh
( )
Y =T( )
Yh +tTXh. The marginal effective tax rate, referred to from now on as METR, can be found by differentiating total taxes with respect to income, Yh: into Eq. (13) gives us the METR for type h1 .
The conditional demand functions = h
d
condition can be written in the form 1 0
∂ =
q X . Taking these properties into account Eq. (14) becomes
63
The marginal effective tax rates can be determined from this form by solving
term h
p from the first order conditions (4)-(7) for both household types respectively.
Consider first the high ability type. The METR for type 2 is
ˆ .
The effective marginal tax rate of type 2 consists of a positive coefficient 12 w and of two terms A2 and B2, where subscripts refer to the type of worker. Thus, the traditional ‘no distortion at the top’ result does not hold in this framework, resulting from two effects A2 and B2.
The term A2 describes the effect of the externality: if there were no externality, the shadow price would be zero and the whole term would vanish.8 When the externality is harmful, we notice that term A2 is positive when the dirty good is a normal good and a substitute for leisure. Intuitively this can be interpreted such that discouraging the labour supply of the high-skilled type becomes desirable, when an increase in income raises the demand for the dirty good thereby worsening the environmental situation.
8 A similar term is also found in Pirttilä and Tuomala (1997).
The term B2 captures the effect of endogenous wages. Since 2 >0
As the two terms have opposite signs (when the externality is socially harmful and the dirty good is a normal good and a substitute for leisure), the overall level of the marginal effective tax rate of the high ability type remains ambiguous as long as utility functions and demands are not specified.
Likewise, the effective marginal tax rate for the low ability type is given by
ˆ . positive coefficient 11
w and three terms: one referring to the externality, another resulting from the self-selection constraint and the third reflecting the effect of endogenous wages.9
The first term, A1 corresponds to that of the high ability person and is positive if the dirty good is a normal good and a substitute for leisure. The same intuition of the desirability of taxing labour supply as a means to discourage the consumption of the dirty good also holds here. Term B1 defines the effect of the
9 Note that this distinction is only terminological; the endogenousness of the wage ratio also originates from the self-selection constraint as the changes in relative wages affect the incentives to mimic.
65 self-selection constraint. The term can be shown to be positive as a result of the single-crossing property10.
Term C1 is an outcome of the endogenous wages. It has a positive effect on the METR, since 1
∂L Ω
∂ is negative. This implies that the effect of the wage adjustment increases the marginal effective tax rate for the low ability person. Intuitively, discouraging the labour supply of type 1 decreases the wage ratio which worsens the incentives for mimicking. As all terms, A1, B1 and C1, are positive, the marginal effective tax rate of the low ability type is strictly positive when the dirty good is normal and a substitute for leisure.
The conclusions from Eq. 16 and 17 are that (under the assumptions made) the effective marginal tax rate for the low ability type is strictly positive and the sign of the effective marginal tax rate for the high ability person is ambiguous. The effect of endogenous wages decreases the METR of the high productivity worker (term B2) and raises the METR of the low productivity worker (term C1). If there were no externality, terms A1 and A2 would disappear from the equations of marginal effective tax rates. When the dirty good is a normal good and a substitute for leisure and the externality is socially harmful, the presence of the externality increases METR for both household types. This result is in accordance with Pirttilä and Tuomala (1997).
An effect similar to terms B2 and C1including the effect of labour supply Lh on the wage ratio has also been reported in other papers on optimal taxes with
10 This property states that the indifference curve of the high ability person is flatter than that of the low ability person at a given point. Usually this is presented in income-consumption space in the form VˆY2 VˆB2 −VY1 VB1 >0. Taking a partial derivative of the utility functions with respect to income Y and substituting these derivatives into the usual form of single-crossing condition gives us
(
VˆL2 VˆB2)
Ω−VL1 VB1 >0.endogenous wages. Aronsson and Sjögren (2003) consider optimal taxation in a case where wages are determined by labour unions. The corresponding effect is captured in their elasticity type term l, but resulting from the imperfect labour market, the sign of the term is not obvious in their framework. Micheletto (2004) also considers marginal effective tax rates with endogenous factor prices and finds that it is desirable to distort the labour supply decisions when, by so doing, government can increase the wage ratio and thus decrease the desirability of mimicking. However, as in his model the wage ratio is a function of factor prices, and there are differences in the intensity of using each labour type, the overall effect of the wage adjustment on the METRs depends on the degree of the complementarity between private good and labour.