On the Optimal Taxation of Capital Income

112 JONES, MANUELLI, AND ROSSI
heterogeneity. What we show here is that this finding holds in some cases
even with a representative household that supplies two different types of
labor if the planner is constrained to choose the tax rates on the two types
of labor equally. (An assumption implicit in Stiglitz's work.)
Here, the period utility function of the household is given by u(c, 1&n 1,
1&n 2) and the production function is F(k, n 1, n 2).
It can be shown that the W function for this example is given by:
W(c, n 1, n 2)=u(c,1&n 1,1&n 2)+*[u cc&u n1 n 1&u n2 n 2]
and that the constraints of the planner's problem are
ct+x t+g t F(kt, n1t, n2t), kt+1=(1&$)kt+xt, ,(t)#u n1 (t) F n1 (t)&u n2 (t) F n2 (t)=0.
The first two constraints describe feasibility while the third uses the first
order conditions of the household to impose the constraint that the two
labor tax rates should be equal, { n1 ={ n2 for all t. Letting the Lagrange
multipliers on the first and third constraints in the steady state be given by
+* and '* respectively, the relevant steady state conditions are
W c*=+*&'*, c*
W* n1 =&+*F* n1 +'* n1
1=;[F k*+1&$]+;'*, k* +*.
In section (c), below, we provide a numerical example in which { k {0.
The equations above indicate when the limiting tax will be zero. If { k =0,
it must be the case that either '* or, n* is zero. From the definition of ,,
it follows that , k is given by
, k= u n1
Fn1\ Fn1k Fn1 It follows that , k=0 if and only if
F n1k
F n1
= Fn2k .
Fn2 & F n2k
F n2+ .
This condition defines a special class of production functions which
include Cobb-Douglas but not the general CES case. To get our