20
(i) Suppose, that λ1 ,λ2 ,λc,λf,λt, λm, λx and λy are the values attained by the Lagrange
multipliers in the solution to (20) and that the partial derivatives Xe and Rhj for j = 1, 2 and
h = £,m are also evaluated at that solution. Set prices pt = λf>,pm = λm,py = λy and con-
sider the alternative tax schemes A and B:
-Aisdefinedby: tA = λ1, t ; = λ1, tcA = λc, tA = λxXe and tA = tAA = 0 ;
-Bisdefinedby: tB = tB2 = 0, tCB = λc, teB = λxXe +∑ j λjRe , t£ = Σ j λ-R and
tmB =∑jλjRmj .
With these prices and either tax scheme A or tax scheme B all markets clear and the equilib-
rium allocation is efficient.
(ii) When the tax scheme A is implemented, the efficient tax rates satisfy
tA ʃ > 1 Y
te {<} y
Xr > 0 and/or Xr > 0 and Xr = 0,
ra1ra2rc


=Xr=0,
rc
(28)
X=X=0 and X>0.
ra1 ra2 rc
To prove Proposition 4i insert the prices and tax rates as assigned in Proposition 4 into (27)
and verify that this substitution makes (27) coincide with (21). The tax scheme A in Proposi-
tion 4i is a pure emissions tax scheme in the sense that a tax is levied on the emission of each
polluting residual, while non-emission items like the inputs labor and material are not taxed.
We infer from (21) that (in equilibrium) with the tax scheme A all tax rates are set equal to the
marginal environmental damage of the respective residuals15:
(29)
tA = MD X : = -∑ x . v for v = e,ra,ra ,c
vxv i i a1a2
Uy
We also know that if abatement takes place, optimal emissions tax rates need to equal mar-
ginal abatement costs. In our model production residuals are the only residuals subject to
abatement. Therefore we will now focus on those residuals. Invoking (27b) we obtain, after
some rearrangement of terms,
teA=Ye-∑ taAjRej-tcAYe. (30)
ee aece
15 To simplify the comparison of marginal conditions characterizing either efficient allocations or market alloca-
tions we measure all prices and tax rates in terms of the consumer good by setting py ≡ 1 .
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