On the Optimal Taxation of Capital Income

98 JONES, MANUELLI, AND ROSSI
The necessary conditions for an interior solution of the consumer's maximization
problem are given by
p t=; t u c(t)
1+{ c
t
(1+{ c
0 )
u c(0)
t=0, 1, ... (1.a)
ul(t)=(1&{ n
t ) wtMn(t) uc(t) 1+{ c
t
t=0, 1, ... (1.b)
ul(t)= uc(t) 1+{ c
Gn(t) t Gx(t) t=0, 1, ... (1.c)
pt=p t+1[1&$ k+(1&{ k
t+1 )rt+1] t=0, 1, ... (1.d)
(1+{ m
t )=(1&{n t )wtMx(t) t=0, 1, ... (1.e)
pt Gx(t)=pt+1{ (1&$ h+Gh(t+1)) +(1&{
Gx(t+1) n
t+1 )wt+1Mh(t+1) =
t=0, 1, ..., (1.f )
in addition to the constraints on problem (P.1). Using the first order conditions
and the assumption that G and M are homogeneous of degree one it
is possible to show that in ``equilibrium'' the consumer's budget constraint
can be greatly simplified.
Specifically, consider the term t=0 p t[x kt&(1&{ k
t )r tk t]. Using the law
of motion for k t we can rewrite this sum as
:
t=0
p t[x kt&(1&{ k
t )r tk t]=p 0[(1&$ k)+(1&{ k
0 )r 0]k 0
+ :
t=1
k t[p t[1&$ k+(1&{ k
t )r t]&p t&1].
However, the second term on the right hand side is zero given (1.d). Next,
consider the term t=0 pt[xht+(1+{ m
t )xmt&(1&{ n
t ) wtM(xmt, ht, nmt)]. Given the law of motion for the stock of human capital (which holds as an
equality) and the assumption that G and M are linearly homogeneous in
(x, h), it follows that this term is given by
:
t=0
p t_ ((1+{m
t )&(1&{n
t ) w tM x(t))x mt
+ h t+1&(1&$ h+G h(t))
G x(t)
h t&(1&{ n
t ) w tM h(t)h t& .