4 J
.B. Davies et al. Journal of Public Economics 77 2000 1 –28
The main results in this paper are as follows. The optimal tax solution depends on the government’s ability to distinguish private human capital investment from
private consumption. If it has some ability in this regard, only a consumption tax should be used to finance public provision of goods and services. If subsidies are
feasible for observed education investment, the consumption tax rate is in- dependent of the degree of observability but the subsidy rate is higher the lower is
the observability. If subsidies are not feasible, the consumption tax rate is lower the more limited is the observability. Optimal tax rates for goods that provide
consumption and education investment simultaneously are below normal rates for observed pure consumption. Growth and welfare are positively related to in-
dependent of the degree of observability without with subsidies. The growth effect of a consumption tax for public goods is also discussed.
The remainder of this paper is organized as follows. The next section introduces the model where private investments in education are fully observed. Section 3
investigates the model with limited observability of education investment. Section 4 makes some extensions. The last section gives some concluding remarks.
2. The model with full observability
To facilitate the comparison of this model with previous ones, we start with the conventional assumption that governments can fully observe the uses of private
goods expenditures for education and consumption. This model has an infinite number of periods and a constant population with measure one. Agents are
identical, infinitely lived, and each is endowed with one unit of time per period. They allocate their time among leisure z , production l , and education h . There is
t t
t
a private good that can be either consumed or invested in human capital, and two publicly provided goods: public consumption and public education investment.
The representative agent’s preferences are defined over private consumption c ,
t
leisure z , and public consumption g as
t t
` s 2t
U 5
O
r ln c 1 ln z 1
b ln g , b . 0, 0 , r , 1,
t s
s s
s 5t
where t and s refer to periods in time, b measures the taste for public
consumption, and r is the discount factor. Correspondingly, the recursive value
function is V 5 ln c 1 ln z 1
bln g 1 rV .
1
t t
t t
t 11
As in Glomm and Ravikumar 1992, the production of the private good is proportional to the use of effective labor units according to:
y 5 H l 2
t t t
J .B. Davies et al. Journal of Public Economics 77 2000 1 –28
5
where y is output, l the units of labor, and H human capital skill.
t t
t
In contrast to optimal taxation models or public goods models that use a neoclassical framework e.g. Krusell et al., 1996, here human capital or skills
accumulate through education:
u 12u a 12
a
H 5 A q e
h H , A . 0, 0 ,
u , 1, 0 , a , 1. 3
t 11 t
t t
t
In Eq. 3, q and e are, respectively, private and public investments of goods in
t t
education, h the units of time used in learning, A a productivity parameter, a the
t
importance of physical inputs, and u the importance of private investment relative
to public investment. Note that private and public investments in education are not perfect substitutes.
We assume that government expenditures on public consumption g and public
t
education investment e are financed by flat-rate taxes. The available tax
t
instruments include a tax on the purchase of the private good at a rate t , and a
c
labor income tax at a rate t . With full observability of private investment in
l
education, it is important to see if there is any deductibility for educational
4
expenditures under the optimal tax structure. Denote the deduction rate as x and let d 5 1 2 x. Individuals’ budget and time constraints are given by
1 1 t c 5 1 2 t H l 2 1 1 t dq ,
4
c t
l t t
c t
z 5 1 2 h 2 l . 5
t t
t
When d 5 0 or x 5 1, there is full deductibility for education expenditures, and t
c
becomes a consumption tax. When d 5 1 or x 5 0, there is no deductibility and hence
t is a commodity indirect tax. A negative d or x . 1 stands for a subsidy.
c
The government budget constraint is given by ¯ ¯
¯ ¯
g 1 e 5 R 5 t c 1 t H l 1 t dq
6
t t
t c t
l t t
c t
where R is the total tax revenue. Eq. 6 means that the tax revenue per household
t
¯ ¯ ¯
equals taxes from average consumption c , average labor income H l , and average
t t t
¯ ¯ ¯
¯ private education investment q . With identical agents, c 5 c , H l 5 H l , and
t t
t t t
t t
¯q 5 q . We allow an optimal allocation of R between g and e with e 5 cR and
t t
t t
t t
t
g 5 1 2 cR for 0 , c , 1.
t t
Given the tax rates ht ,t ,dj, the publicly provided goods hg ,e j and initial
c l
t t
¯ human capital
hH ,H j, the representative agent solves the following concave
t t
programming problem:
4
Deductibility in principle is feasible under an income tax, but in practice has been observed mainly under consumption taxes.
6 J
.B. Davies et al. Journal of Public Economics 77 2000 1 –28
¯ VH , H ,
t , t ,d
t t
c l
1 au
a 21 au u 2 1 u
H l 1 2 t 2 1 1 d t H
A h H
e
t t
l c
t 11 t
t t
]]]]]]]]]]]]]
5 max ln
H
1 1 t
c
h ,l ,H
t t
t 11
¯ 1 ln1 2 h 2 l 1
b ln g 1 rVH ,H
, t ,t ,d .
7
J
t t
t t 11
t 11 c
l
¯ Observe that average human capital H affects individuals’ welfare via e and g by
t t
t
6 and 7. The first-order conditions for the above optimization problem are h :1 1 2 h 2 l 5 q 1 1 d
t 1 2 a [1 1 t c h au ], 8
t t
t t
c c
t t
l :1 1 2 h 2 l 5 H 1 2 t [1 1 t c ],
9
t t
t t
l c
t
H :1 1 d
t q [1 1 t c auH ] 5
r≠V ≠H
. 10
t 11 c
t c
t t 11
t 11 t 11
The envelope condition is ≠V ≠H 5 [l 1 2 t 1 1 2 a1 1 dt q auH ]
t t
t l
c t
t
[1 1 t c ].
c t
Eqs. 8 and 10 and the envelope condition state that the loss in utility from investing in human capital now will be compensated by the gain in utility from
higher productivity later. Eq. 9 equates the loss in utility from working for an additional unit of time less leisure to the gain in utility from earning more
income for private consumption.
Solving the above first-order conditions, we have r1 2 a
]]]]]] h 5 h 5
, 11
t
2 2 r[1 2 a1 2 u ]
1 2 r1 2 a
]]]]]] l 5 l 5
, 12
t
2 2 r[1 2 a1 2 u ]
h1 2 r[1 2 a1 2 u ]j1 2 t
l
]]]]]]]]] c 5
H ; g H ,
13
t t
c t
h2 2 r[1 2 a1 2 u ]j1 1 t
c
aur1 2 t
l
]]]]]]]]] q 5
H ; g H .
14
t t
q t
h2 2 r[1 2 a1 2 u ]j1 1 dt
c
From the above solution, the time allocation among h , l and z is the same in
t t
t
each period, and independent of all the tax rates due to our specification of the
J .B. Davies et al. Journal of Public Economics 77 2000 1 –28
7
5
preferences and production technology. However, private consumption c and
t
private education investment q are proportional to the agent’s own human capital
t
stock, changing over time, and responsive to these taxes. An increase in the labor income tax rate, or in the indirect tax rate for d . 0 partial deductibility, reduces
both private consumption and private education investment. Using the above results, our logarithmic preferences give the following value
function: ¯
V 5 B 1 Dln H 1 Eln H ,
t t
t
1 r1 1 b
]] ]]]
B 5 ln[
g 1 2 h 2 l] 1 bln[1 2 cg ] 1 ln
g ,
H J
c R
u
1 2 r
1 2 r
1 ]]]]]]
D 5 ,
1 2 r[1 2 a1 2 u ]
b h1 2 r[1 2 a1 2 u ]j 1 ar1 2 u
]]]]]]]]]] E 5
15 1 2
r h1 2 r[1 2 a1 2 u ]j
12 a
au a 12u
with g 5 Ah
g cg
5 H H and
g 5 t g 1 t l 1 t dg . In 15, B
u q
R t 11
t R
c c l
c q
depends on the taxes while D and E are independent of the taxes. The growth rate of the representative agent’s income is given by
m 5 g 2 1. 16
u
Here, m is positive if the productivity parameter A is large enough. We now
determine the optimal allocation of tax revenue R between public investment and
t
public consumption.
Proposition 1. The optimal share of public education investment is
c 5 ar1 1 b 1 2 u [b1 2 r 1 ar1 1 b 1 2 u ].
Proof. Maximizing B hence V with respect to c requires B 5 [2b1 2 rc 1
t c
2
ar1 2 u 1 1 b 1 2 c] [1 2 r c1 2 c] 5 0. The value of c follows. h The division of tax revenue between public consumption and investment
depends on their relative importance in preferences and technology. The more
5
With other specifications of preferences and technologies, the time allocation may be responsive to changes in tax rates, in particular when tax rates are chosen to generate very different ratios of tax
revenue to income. We expect that changes in the time allocation are negligible when different mixes of consumption and labor income tax rates are chosen to yield a constant ratio of tax revenue to income,
as in our model the ratio of tax revenue to income is determined for optimal provision of public goods. With a CES utility function specification and a similar time allocation problem, the numerical
simulations in Devereux and Love 1994; Parts A and B in Table 1 support our conjectures here. The empirical evidence also suggests that the response in hours of work to changes in the net wage is very
small for prime age male earners. See, e.g., Pencavel 1986.
8 J
.B. Davies et al. Journal of Public Economics 77 2000 1 –28
important the public consumption larger b , the larger the fraction of tax revenue
devoted to public consumption [larger 1 2 c]. Similarly, the more important the
public investment in education [larger a1 2 u ], the larger the fraction of tax
revenue devoted to public investment larger c. Obviously, if b 5 0, then
c 5 1; if u 5 1, then c 5 0. Define I
; aru1 1 b 1 2 r and J ; b 1 ar1 2 u 1 1 b 1 2 r. Opti- mal taxation is characterized by:
Proposition 2. With full observability of private investment in education, all
combinations of t . 0 and t 0 such that d 5 2 t t and g 5 Jl 1 1 I 1 J
c l
l c
R
are optimal. In particular, t 5 d 5 0 and t 5 J is an optimal scheme.
l c
Proof. See Appendix A.
The intuition for the optimal tax is to remove dynamic inefficiency and to optimize the size of tax revenue for public goods. The factor 1 2
t 1 1 t d in
l c
14 represents the dynamic inefficiency of the flat-rate taxes. The inefficiency is eliminated if d 5 0 and
t 5 0 or if we allow a subsidy such that d 5 2 t t . With
l l
c
this subsidy, it is easy to see from 4 the equivalence between proportional consumption taxes and labor income taxes: 1 1
t c 5 1 2 t H l 2 q . Conse-
c t
l t t
t
quently, all combinations of taxes that correct the dynamic inefficiency and generate the optimal tax revenue for public goods will be optimal. The optimal tax
revenue relative to income g
depends positively on the taste for public
R
consumption b and the role of public investment in education a1 2 u .
The full deductibility for educational expenditures under the optimal tax without subsidies d 5 0 in Proposition 2 arises under the conventional assumption that
6
private investments in education are fully observable. Also, under full observabili- ty the optimal tax without subsidies and without labor income taxes is found to be
equivalent to the ones with both subsidies and labor income taxes. However, what happens if there is limited observability of private investment in education?
3. The model with limited observability
