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# Oxford University Press 2007 Oxford Economic Papers 60 (2008), 5787 57All rights reserved doi:10.1093/oep/gpm018
Fiscal policy and endogenous growth with
public infrastructure
By Pierre-Richard Agenor
School of Social Sciences, University of Manchester, Manchester, M13 9PL;
and Centre for Growth and Business Cycle Research;
e-mail: [email protected]
Optimal tax and spending allocation rules are derived in an endogenous growth modelwhere raw labor must be educated to become productive and infrastructure services
affect the schooling technology. Growth- and welfare-maximizing solutions are com-
pared under nonseparable utility in consumption and government-provided services.
Among other results, it is shown that the optimal share of spending on infrastructure
depends not only on production elasticities but also on the quality of schooling and the
degree to which infrastructure services affect the production of educated labor.
Congestion costs in education raise the optimal share of spending on infrastructure.
JEL classifications: O41, H54, I28.
1. Introduction
The impact of public investment on growth has been the subject of much attention
in recent academic research and policy debates. As documented by Zagler and
Durnecker (2003), much academic research (both empirical and analytical) has
focused, in particular, on the effects of public infrastructure. At the empirical
level, Easterly and Rebelo (1993) found in an early contribution a positive associa-tion across countries between public investment in infrastructure (namely,
transportation and communications) and growth. In a re-run of the Easterly-
Rebelo regressions, Miller and Tsoukis (2001) confirmed this effect.1 Bose et al.
(2003), using panel data for 30 developing countries and an econometric method-
ology that explicitly accounts for the government budget constraint and possible
biases arising from omitted variables, found that the share of government capital
expenditure in GDP is positively and significantly related to income growth
per capita, whereas the share of current expenditure is not. Calderon and Serven
(2004), using also a large sample of countries and panel data covering the period
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economy is endowed only with raw labor, which must be educated to become
productive. Knowledge is thus embodied in (educated) workers. Unlike Uzawa-
Lucas type models, where human capital is disembodied and can expand without
bounds, in the present framework knowledge is defined as a fixed set of skills and it
is the number of (educated) workers that can grow without bounds (as long as the
growth rate of the raw labor force is, of course, positive). Moreover, and unlike the
case considered in Agenor (2005a), in the present setting government-provided
services have an impact on household utility, and infrastructure has a positive
effect on the rate of accumulation of educated labor.
The latter assumption captures the view that infrastructure services (better roads,
reliable access to electricity, etc.) may enhance the ability of individuals to study
and acquire skills. This is a particularly important consideration for low-income
developing countries. In many of these countries, the lack of an adequate networkof roads makes access to schools (particularly in rural areas) difficult; dropout rates
tend to be higher when children must walk long distances to get to school (see Levy,
2004). The lack of access to electricity hampers the ability to study, both in the
classroom and at home. In some countries, the lack of adequate toilet facilities for
girls in rural area schools has led many parents to deny an education to their
daughters.4 As it turns out, accounting for the impact of infrastructure on the
schooling technology has important implications for the determination of the
optimal allocation of government expenditure.
Another issue that I address in the paper is the existence of congestion costs ineducation, which have proved to be a particularly important factor in assessing the
quality of schooling in low-income countries. According to recent data from
UNESCO and the World Bank, student-teacher ratios in these countries dramati-
cally exceed average ratios observed in industrial countries. For instance, at 44 to 1,
the pupil-teacher ratio in Sub-Saharan Africa is on average three times higher than
that of developed countries; moreover, one in four countries in the region has
ratios above 55 to 1 (see UNESCO, 2005). Much research in the endogenous
growth literature has examined the issue of congestion costs in infrastructure, aswell as their implications for private capital formation and the optimal allocation of
public expenditure (see, for instance, Turnovsky, 1997; Fisher and Turnovsky, 1998;
and Eicher and Turnovsky, 2000). But almost none has focused on congestion costs
in education and their implications for human capital accumulation, the optimal
composition of spending, and growth.5
The remainder of the paper proceeds as follows. Section 2 presents the basic
framework, which assumes that utility is separable in consumption and
..........................................................................................................................................................................
4 See Agenor and Moreno-Dodson (2007) for a detailed discussion of the impact of infrastructure on
education outcomes, as well as a review of the evidence.
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government-provided services. Section 3 solves for the steady-state growth rate in
the decentralized equilibrium, and examines its dynamic properties. Section 4
illustrates the functioning of the model by considering a revenue-neutral change
in the composition of public spending, namely a switch from spending on educa-
tion to infrastructure services. Section 5 derives the growth-maximizing fiscalstructure. Section 6 considers the case where utility is nonseparable in consumption
and government services, and compares the growth- and welfare-maximizing
solutions. Section 7 introduces congestion costs in education in the model, and
examines the effect of these costs on the optimal allocation of government expen-
diture. The final section summarizes the main results of the paper and offers some
concluding remarks.
2. The basic frameworkConsider an economy populated by a single, infinitely-lived household. It produces
a single traded good, whose price is fixed on world markets. The good can be used
for either consumption or investment. The economys endowment consists of raw
labor, which must be educated (through a publicly-funded schooling system) to be
used in market production. The government provides infrastructure, education,
and utility-enhancing services, at no charge. It levies a flat tax on marketed output
to finance its expenditure.
2.1 Market activity
Aggregate marketed output, Y, is produced with private physical capital, public
infrastructure services, educated labor, using a Cobb-Douglas technology:6
Y G I EK
1P AP
GI
KP
E
KP
KP, 1
whereKPis the stock of private capital,GIgovernment infrastructure services,Ethestock of educated labor, 2 0, 1 the proportion of the educated labor force
employed in goods production, AP 40, and , 2 0, 1. Thus, production
exhibits constant returns to scale in all factors. As long as GI=KP and E=KP are
constant (which turns out to be the case in the steady state), output will exhibit
linearity in the stock of private capital. The production function is then essentially
an AK-type technology.7
..........................................................................................................................................................................
6 Throughout the paper, the time subscript tis omitted. A dot over a variable is used to denote its time
derivative.
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2.2 Household preferences
The household-producer maximizes the discounted stream of future utility
V 1
0
C1
1 ln GH
exptdt, 15 1, 2
where C is aggregate consumption, GH utility-enhancing government services,
4 0 the discount rate, 1= the intertemporal elasticity of substitution
(with 1 corresponding to the logarithmic utility function), and 4 0 a coeffi-
cient that measures the impact of GH on the households instantaneous utility.
Thus, as for instance in Turnovsky (1996, 2000, 2004), Chang (1999), and Baier
and Glomm (2001), publicly-provided services affect the households utility
directly. However, unlike these authors, and in line with Cassou and Lansing(2003) and Piras (2005), private consumption and public services are assumed to
be additively separable in this initial specification. This is in line with the empirical
evidence provided by Karras (1994), McGrattan et al. (1997), Chiu (2001), and
Okubo (2003), among others.
Output is taxed at the rate 2 0, 1. The household spends on consumption
and accumulates capital. It receives teachers pay from the government,
1 wGE, where wG is the real wage and 1 the proportion of teachers
in the educated labor force. As noted earlier, education is provided free of
charge and there are no user fees for infrastructure services used in private produc-
tion. Thus, the household benefits from an implicit rent associated with public
spending.
Abstracting from capital depreciation, the households resource constraint is
C _KP 1 Y 1 wGE: 3
2.3 Education technology
Raw labor, which grows at a constant rate, must be educated before it can be
used in market production. In turn, the production of educated labor
requires the combination of teachers, students, and government infrastructure
services:
_E Q1 EGIL1
, 4
where _E is the flow of newly-educated workers, L the number of students, Q a
variable that measures the quality of schooling, and 2 0 1 Thus, the educa-
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Using (7), eq. (6) can be rewritten as
_E
E AE
GE
E
GI
E
, 8
where AE a11 1. Thus, the growth rate of the stock of educated
labor depends on public spending on both education and infrastructure services per
teacher (or educated worker). Note that, in light of (7), it does not matter whether
the quality of schooling is defined as a function of the ratio of government spending
per teacher (as in (5)) or per student. In addition, to ensure that the number of
teachers has a positive effect on the flow supply of educated labor, I assume that
51.10 Note also that a form similar to (8) can be obtained despite
Dropping (5) if spending on education is assumed to affect directly the production
of educated labor (rather than through quality), that is, if for instance (5) takes theform _E Ga1E 1 E
a2 Ga3I L
1ai , where ai 2 0, 1. Using (7), it can then be
shown that _E=E would again be given by (8), with a1 and a3 . In that
case, no additional restrictions on a1and a2 are necessary.
2.4 Government
The government provides infrastructure, education, and utility-enhancing services
to the household, pays salaries to teachers, and collects a proportional tax on
marketed output. It cannot issue debt and must maintain a balanced budget con-
tinuously.11
Thus, the governments flow budget constraint is given byXhI, E, H
Gh 1 wGE Y: 9
Expenditure on all categories of services are taken to be determined as fractions
of tax revenue, so that Gh hY, with h 2 0, 1, and h I, E, H. Teachers
salaries are also fixed as a fraction of tax revenue. The government budget
constraint therefore implies that
XhI, E, H
h
1,10
which determines residually one of the spending shares.12
..........................................................................................................................................................................10 Common empirical estimates of are in the range 0.10.2. Estimates of are more difficult to come
by; in the numerical exercises performed in Agenor (2005b), values of 0.1 and 0.2 are used. In either
case, however, the condition 51 appears not to be very restrictive.11 See Turnovsky (1996, 1997) for the case of debt financing in a similar context. Because Ricardian
equivalence holds in the present framework, excluding borrowing is simply a matter of convenience.
With government borrowing, agents would foresee that higher taxation is required later in order to
repay the accumulated debt, and this would need to be accounted for.12 To avoid a corner solution (in which 1), I assume implicitly that educated workers seek employ-
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From (3) and (9), the market-clearing condition for the goods market is
Y C _KP
XhI, E, HGh: 11
3. The decentralized equilibrium
In the present setting, a decentralized equilibrium can be defined as follows:
Definition 1 A decentralized equilibrium is a set of infinite sequences for the
quantities fC, KP, Eg1t0, such that fC, KPg
1t0 maximize eq. (2) subject to (3), and
fKP, Eg1t0 satisfy eq. (8), (10), and (11), for given values of the tax rate, , the ratio
of government wages to output, , and the spending shares on public services, h,with h I, E, H.
This equilibrium can be characterized as follows. The household-producer
maximizes (2) subject to (3), taking the tax rate, , government-provided services
and wage payments as given. The current-value Hamiltonian for this problem can
be written as
C1
1 ln G
H 1 Y 1 w
GE C,
whereis the costate variable associated with constraint (3). Using (1), first-order
optimality conditions for this problem can be written as, setting s 1
1 , so that s 2 0, 1:
d
dC 0 ) C , 12
_= d
dKP sAP
GI
KP
E
KP
, 13
together with the budget constraint (3), the initial condition KP0 K0P, and the
transversality condition
limt!1
KPexpt 0: 14
Equation (12) equates the marginal utility of consumption to the shadow value
of private capital, Equation (13) is the standard Keynes-Ramsey consumption
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Combining eqs (12) and (13) yields
_C
C sAP
GI
KP
E
KP
" #, 15
whereas substituting (1) in (3) yields
_KP 1 APGI
KP
E
KP
KP 1 wGE C: 16
As shown in Appendix 1, eqs (8), (10), (15), and (16) can be further manipulated
to lead to a system of two nonlinear differential equations (see eqs (A7) and (A8))
inc C=KPande E=KP. These two equations, together with the initial condition
e0 E0=K0P40, and the transversality condition (13), rewritten as
limt!1
c1 expt 0, 17
determine the dynamics of the decentralized economy. The balanced-growth path
(BGP) can therefore be defined as follows:
Definition 2 The BGP is a set of sequences fc, eg1t0, spending shares and tax rate
satisfying Definition 1, such that for an initial condition e0eqs (8), (15), and (16)and the transversality condition (17) are satisfied, and consumption, the stock of
educated labor, as well as the stock of private capital, all grow at the same constant
rate _C=C _E=E _KP=KP.
The transversality condition (17) is always satisfied along any interior BGP
because consumption and the stock of private capital grow at the same constant
rate, implying that the ratio c C=KP is also constant.13 Given the form of the
utility function used here, a necessary condition to get bounded utility (that is, for
the integral in (2) to converge) is 41 . This condition imposes no restric-
tion when41, that is,5 1. When5 1, it imposes an upper bound on admis-
sible values for the rate of growth. Equivalently, it requires the rate of time
preference to be sufficiently large.
From eqs (A4) and (A6) in Appendix 1, is given by the equivalent forms14
sAPI=1
~e=1 , 18
AE
=1I
=1~e!, 19
..........................................................................................................................................................................13 _ _ _ _ _
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where ~e denotes the steady-state value ofe, A a constant term, and
! 1
1 50:
As shown in Appendix 1, the following proposition can be established:
Proposition 1 Along an equilibrium path with a strictly positive growth rate, the
BGP is unique. There is only one stable path converging to this equilibrium.
This proposition implies therefore that the model is locally determinate.
Its dynamics can be analyzed using phase diagrams, as illustrated in Fig. 1.
The _e 0 curve (denoted EE in the figure) always has a positive slope in c-e
space, whereas the slope of the _
c
0 curve (denoted CC) can be either upward-or downward-sloping, depending on the size of the elasticity of intertemporal
substitution, . The upper (lower) panel corresponds to the case where is rela-
tively high (low), in a sense made precise in Appendix 1. The saddlepath, denoted
SS, may therefore have either a positive or a negative slope. Following a jump in c
(as a result, for instance, of a change in the tax rate or one of the spending shares
parameters),candemay or may not move in the same direction. The reason is that
the transitional dynamics are driven by the ratio of educated labor to private
capital, and as this ratio increases (falls), the marginal productivity of private
capital increases (falls) as well. In turn, this tends to raise (reduce) consumptionand investment over time.
4. Revenue-neutral spending shift
The steady-state effects and transitional dynamics associated with revenue-neutral
changes in spending shares are straightforward to analyse in the present setting.
In particular, Appendix 1 establishes that an increase in I, offset by a reduction
in E(that is, with dI dE 0, holding and constant), has an ambiguous
effect on the steady-state growth rate. This result is summarized in the followingproposition:
Proposition 2 With the tax rate and the share of government spending on wages
held constant, a switch in the composition of public expenditure from education
to infrastructure services has an ambiguous effect on the steady-state growth rate.
If infrastructure services do not affect the education technology ( 0), the net
effect depends only on =. With 4 0, the net effect depends also on =.
To understand the intuition behind these results, consider first the case where
0. Increasing the fraction of government spending on infrastructure (for a
given stock of educated labor) increases the marginal product of physical capital,
66 fiscal policy and endogenous growth
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Thus, the positive effect of the increase in the share of spending on infrastructure is
accompanied by a lower supply of educated workers, which tends to lower private
production and reduce the growth rate. The net effect on output and the growth
rate depends on how productive the two inputs are in relative terms, that is, on
the ratio =. As shown in Appendix 1, if= exceeds the elasticity of the steady-
state value of the educated labor-capital ratio with respect to the share of spending
in infrastructure, the growth effect (as well as the effect on the consumption-capital
S
c
S
e
E
E
C
C
A
c~
e~
0
S
c
S
e
E
E
C
C
A
c~
e~0
Fig. 1. The balanced growth equilibrium.
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types of services are in the production of goods, but also to how productive they are
in the production of educated labor, as measured by the ratio =. Even if infra-
structure services have a small impact on the production of goods, a high value of
= may still lead to an increase in ~e, ~c, and . In the particular case where 0,
that is, if education services do not affect the acquisition of skills, the impact on thesteady-state growth rate is unambiguously positive.
Figure 2 illustrates two possible outcomes, under the assumptions that the
elasticity of intertemporal substitution is relatively high and the ratio = is not
too large. In both cases the educated labor-capital ratio falls in the steady state. But
the consumption-capital ratio and the growth rate may either increase or fall,
depending on the magnitude of =, as noted earlier. In both panels curve CC
shifts to the left, but curve EEcan shift either to the right (upper panel) or the left
(lower panel). In the upper (lower) panel,CCshifts by more (less) thanEEand the
consumption-capital ratio falls (increases). The steady-state consumption-capital
ratio falls in the first case and increases in the second.15 As noted earlier, during the
transition, the fall in the educated labor-capital ratio lowers the marginal produc-
tivity of capital, leading to a gradual reduction in the stock of physical capital.
This reduction is large enough to ensure that the consumption-capital ratio
increases over time.
5. The growth-maximizing fiscal structure
To determine the growth-maximizing fiscal structure in the decentralized equili-
brium involves setting simultaneously the tax rates and expenditure shares so that
@=@ @=@h 0, 8h. Given the structure of the model, this problem can be
addressed in two stages, the first of which involves solving for the growth-
maximizing tax rate, keeping spending shares constant. From (18) and (19), the
following proposition can be readily established:
Proposition 3 With all government spending shares held constant, the growth-
maximizing value of the tax rate is .
This result generalizes the rule , which was first established by Barro
(1990) in a flow model, and subsequently in stock models by Futagami et al.
(1993), Turnovsky (1997), and Baier and Glomm (2001). The Barro rule holds
only in the particular case where educated labor (or, more generally, human
capital) has no effect on production, that is, 0. Note also that the growth-
maximizing tax rate depends only on parameters characterizing the goods
production technology; it is independent not only of how the revenue is spent
(that is, the spending shares h and ), but also of the parameters characterizing
the education technology. As discussed later, this is not generally the case with thewelfare-maximizing solution.
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The second stage involves allocating public expenditure between infrastructure,
education and utility-enhancing services, with the tax rate and the share of govern-
ment spending on wages both taken as given. The following proposition can be
shown to hold:
Proposition 4 With the tax rate and the share of government spending on wagesheld constant, the growth-maximizing composition of public expenditure is
S
c
S
e
E
E
C
C
Ac~
e~
0
S
c
S
e
E
EC
C
A
c~
e~
0
A
B
A
B
Fig. 2. Shift in the Composition of Government Spending from
Education to Infrastructure
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That the optimal value ofH is zero is of course a direct implication of assuming
separability between utility-enhancing government services and private
consumption. As a consequence, spending on this type of services has no effect
on growth. To provide a more explicit interpretation of the second result in
Proposition 4, it is convenient to consider the particular case where 0. Giventhe budget constraint (10), Proposition 4 implies therefore that
I
51, 21
with E 1 I from (10).
Equation (20) implies that if the production function for educated labor does
not depend on infrastructure services, that is, with 0, the optimal allocation
of spending between infrastructure and education would depend only on the
parameters characterizing the goods production technology not on the education
technology, as I have shown elsewhere (see Agenor, 2005a). Specifically, from (21)
the growth-maximizing shares would be I = and E = .
Combined with Proposition 3, the optimal tax structure implies therefore that
the shares of spending in output would be I and
E .
16 With
4 0, however, the education technology does matter for the allocation of govern-
ment spending; the optimal share of spending on infrastructure (education) is
higher (lower) than otherwise. As can be inferred from (21), the more productiveinfrastructure services are in fostering the acquisition of skills (the higher is), or
the lower the quality of education (the lower is), the higher should be the optimal
share of spending on infrastructure services. Note also that an improvement in the
quality of education has no effect on the optimal allocation of public expenditure if
infrastructure services do not affect the production of educated labor (that is,
@I=@ 0 if 0). And in the limit case where ! 0 (so that public spending
on education has no effect on the schooling technology), I! 1, as could be
expected.
Given that the growth-maximizing tax rate is independent of spending shares,and conversely that the growth-maximizing shares are independent of the tax rate,
the results in Propositions 3 and 4 describe completely the optimal fiscal structure
in a decentralized economy. The issue to which I turn next is how robust these
results are if utility is nonseparable in consumption and government-provided
services.
6. Nonseparable utility and welfare maximizing rules
In the foregoing discussion, the utility function of the household was assumed to beadditively separable between private consumption and public services, as suggested
70 fiscal policy and endogenous growth
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by some of the empirical evidence. This specification implies that with the tax rate
and the share of government spending on wages held constant, a switch in the
composition of public expenditure from utility-enhancing services to the provision
of either infrastructure or education services is unambiguously growth-enhancing.
This result is of course due to the fact that the share of spending on utility-enhancing services has no effect on the economys steady-state growth rate, neither
directly nor indirectlythrough changes in the steady-state value of the ratio of
educated labor to physical capital.
However, while separability may be intuitively obvious for some categories of
utility-enhancing public services (such as security or national defense), this is not
quite the case for others, such as health or environment services. Better health care,
for instance, may have a direct impact on the ability of individuals to consume and
enjoy their free time.17 Greater access to public parks in busy cities may have the
same effect.
Accordingly, I now consider a more general utility function, which I take to be
nonseparable in CandGH:
maxC
V
10
CGH
1
1 exptdt, 22
where again 4 0 measures the contribution of government-provided services to
utility and 1= is, as before, the intertemporal elasticity of substitution. This
specification is similar to the one used by Turnovsky (1996), among others.18
Consider first the growth-maximizing policies. Following the results in Agenor
(2005c), Agenor and Neanidis (2006), as well as in Appendix 2, the following
proposition can be established:
Proposition 5With nonseparable utility as given in (22), the growth-maximizing tax
rate is the same as given in Proposition 1. If changes in the share of spending on
infrastructure are offset by an adjustment in the share of spending on education,
formula (21) remains valid, whereas if the offsetting change results from an adjust-
ment in spending on utility-enhancing services, the growth-maximizing share of
spending on infrastructure is equal to unity.
The first result in this proposition is fairly intuitive; the allocation between
productive components of public expenditure does not depend on whether
nonproductive services affect utility if the policymakers objective is to maximize
growth. The second result is also easy to understand, given that (as can be inferred
from the results in Appendix 2), neither the share of spending on utility-enhancing
..........................................................................................................................................................................17
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services, H, nor the parameter , appear explicitly in the solution of the steady-
state growth rate.
Consider now the welfare-maximizing solution, which involves a planner choos-
ing optimally all quantities and policy instruments so as to maximize (subject to
appropriate constraints) the households discounted lifetime utility, as given
in (22).19 Because the government budget constraint (10) must hold at all times,
one of the spending shares must again be chosen residually. In what follows, I will
assume thatEis determined through (10), in order to focus on potential trade-offs
between spending on infrastructure and utility-enhancing services.
By using eqs (A27), (A29), and (A30) in Appendix 2, and setting 0 for
simplicity, the social planners problem is to maximize, with respect to
C, I, H, , KP, and E,
fCHAPI
E
KP 1=1g1
1
K 1 APIE
KP 1=1 C
n o EA
EI
1 2 E1!K!P ,
where A AEA2P, 1 =1 , 2 =1 , 1 ,
and Kand Edenote the costate variables associated with eqs (A29) and (A30),
respectively.Optimality conditions are given by
GH CG
H
K, 23
CGH 1
1
K 1 Y
1
EA
EI1
2
E
1!
K
!
P
I
1
E ,
24
CGH
1H
EAE
1I
2 E1!K!P
E
; 25
CGH
1KY 26
EA
E
I1 2 E1!K!
P
,
72 fiscal policy and endogenous growth
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_K
K
1 1
Y
KP
E
K
1 27
AEI1 2 E1!K!P 1KP
1K
CGH
1
1 1
KP,
_E
E
1
1 A
EI
1 2 E1!K!P1
E28
K
E
1
Y
E
1
E CG
H
11E
,
together with the transversality conditions
limt!1
KKPexpt limt!1
EEexpt 0: 29
Rewriting (23), taking logs, and differentiating with respect to time yields
_C
C
_K
K
_GH
GH
, 30
where 1 1= and _GH=GH _Y=Y. Using (27) and (A22) in Appendix 2,
eq (30) becomes
_C
C
1 1
Y
KP
E
K
1 31
AEI
1 2 E1!K!P1
KP
1
K CG
H
1 1
1
KP
1
_E
E
1
_KP
KP
:
Appendix 2 shows how eqs (31), (A29), and (A30), can be further manipulated
to produce a system of two nonlinear differential equations in c and e. These
equations, together with the initial condition e040 and the transversality condi-
tions (29), characterize the dynamics of the centrally-planned economy.
The steady-state growth rate is given by eq (19) and its equivalent form
1 1 f 1
E
H
~C~Y
!" #32
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Appendix 2 discusses the uniqueness and stability of the BGP. The dynamics of
the model are qualitatively similar to those derived for the decentralized equili-
brium and can therefore be analyzed in a diagram similar to Fig. 1.
More importantly for the purpose at hand, it can be established from (23)(26)
that
1 E
H
~C
~Y
! 0;
~C
~Y
! 1
" #
~C
~Y
!
" #
I 1
E
0;
1 EH
I 1 E
~C~Y
! 0;
where E 1 I H. These equations represent a system in ; I, and H.
Along the BGP, because Cand Ygrow at the same rate, ~C= ~Y is constant and so
are all fiscal policy instruments. In general, an explicit solution cannot be derived,
given the high degree of nonlinearity of this system. However, direct inspection
leads to the following proposition:
Proposition 6Under non-separable utility as given in (22), and given that from (11)~C= ~Y 1 h = 1
=~c , the welfare-maximizing values of the tax rate and
the spending shares along the BGP are not independent of each other and depend
in general on the sensitivity of household utility to government-provided services.
In principle, the welfare-maximizing problem of the central planner could also
be specified in terms of the instantaneous utility function shown in (2), that is, with
separability between consumption and health services. However, the first-order
optimality conditions (in the particular case where 1) also give a nonlinear
system in ; 1 and H for a constant consumption-output ratio, and an explicitanalytical solution cannot be derived either. Exact results are therefore omitted to
save space, given that their qualitative features are similar to those summarized in
Proposition 6.
7. Congestion costs
As noted earlier, much of the endogenous growth literature on congestion costs
has focused on two types of congestion costs: those affecting the use of
infrastructure services (or capital) in the production of goods, and those affecting
utility-enhancing services 20 In the context of the present paper, a more novel issue
74 fiscal policy and endogenous growth
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taking the form of threshold effects. This section examines these issues, with a focus
on the growth-maximizing solutions.
Specifically, two alternative ways of modeling congestion costs in education are
considered. The first approach consists of assuming that the productivity of infra-
structure services falls with an increase in the flow number of students (or newly-educated individuals), _L:
_E Q1 E GI_L
L1, 33
where 0 measures the degree of congestion (with 1 denoting proportional
congestion). For instance, too many students using publicly-provided internet
services may slow the speed of access for everybody, thereby diminishing theusefulness of these services for creating educated labor.
Given that from (7), _E _L, it is straightforward to show that using equation (33)
instead of (4) does not affect any of the previous results regarding the
growth-maximizing fiscal structure (Propositions 3 and 4). The reason is clear
eq. (33) boils down to a geometric transformation (with all coefficients
multiplied by 1 1) of the original expression for educated labor flows.
Although this changes the stability properties of the model and its transitional
dynamics (and thus the speed of convergence to the BGP), it does not alter
the optimal allocation rule because it is the ratio = that appears inProposition 4.21
The second approach to account for congestion in the production of educated
labor is to assume that quality is subject to threshold effects. In the foregoing
analysis, it was assumed that the ratio of students to teachers,L=1 E(denoted
byxbelow) is kept fixed at a by the government. However, whether a is high or
low does have implications for the quality of schooling. Specifically, suppose that
the quality of education, Q, has a parameter that can be high or low (Hor L,
respectively), depending on whether the student-teacher ratio is lower or greater
than a threshold value, aC:
Q QH x
H,
QL xL
,
if a aC
if a4aC
:
..........................................................................................................................................................................20 For examples of the former literature, see Turnovsky (1997), Fisher and Turnovsky (1998), and
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This specification implies that the model can generate multiple BGPs, a full
characterization of which is beyond the scope of this paper. For the purpose at
hand, it is sufficient to observe that, from Proposition 4 (or more specifically
eq. (21)), the two regimes would be characterized by different growth-maximizing
allocations between infrastructure and education services: in the L-regime, theoptimal share E would indeed be lower than in the H-regime. These results
can be summarized in the following proposition:
Proposition 7Congestion costs in education (in the form of threshold effects on the
quality of schooling) raise the growth-maximizing share of public spending on
infrastructure.
Given that the growth-maximizing tax rate does not depend on the parameters
characterizing the education technology, Proposition 3 is not affected by the intro-
duction of threshold effects on the quality of schooling. But because the welfare-
maximizing tax rate and spending shares depend on the steady-state ratio of
consumption to output, they also depend indirectly on the parameters of the
education technology. As a result, they would both be affected by a threshold
effect on . However, establishing analytically the effect of a shift in on the
optimal tax structure is more difficult in this case.
8. Concluding remarksThe purpose of this paper was to study the determination of optimal taxation andallocation of public resources between utility-enhancing, infrastructure, and educa-
tion services. The analysis was based on an endogenous growth model with two key
features: raw labor must be educated to become productive and infrastructure
services affect the schooling technology. Government spending is financed by a
tax on output. The balanced growth path was derived, and the transitional
dynamics associated with a revenue-neutral shift in the composition of public
spending from education to infrastructure services were analyzed. It was shown
that, in general, such as shift has an ambiguous effect on the growth rate and thesteady-state values of consumption- and the supply of educated labor (both mea-
sured in proportion of the private capital stock).
The third and fourth parts of the paper focused on the determination of the
optimal tax rate and spending shares, under both separable and nonseparable
household utility in consumption and government-provided services. In both
cases, the growth-maximizing tax rate was found to be equal to the sum of the
elasticities of output with respect to infrastructure services and educated labor,
and thus to be independent of the schooling technology. With separable utility,
the growth-maximizing composition of public spending was shown to be a zero
share for utility-enhancing services, whereas the optimal allocation to infrastruc-
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are not education-enhancing, the growth-maximizing allocation between infra-
structure and education would depend solely on the parameters characterizing
the goods production technology. With nonseparable utility, however, it was
shown that the growth-maximizing solution coincides with the one obtained
under separable utility only if changes in the share of spending on infrastructureare offset by an adjustment in the share of spending on education. If, on the
contrary, these changes are offset by an adjustment in spending on utility-enhan-
cing services, the growth-maximizing share of spending on infrastructure is equal
to unityeven though spending on education is productive.
The welfare-maximizing solutions under nonseparable utility were shown to be
such that the optimal values of the tax rate and the spending shares are constant
along the balance growth path, and dependent on each other. In addition, it was
shown that they depend in general on the sensitivity of household utility to gov-
ernment-provided services. Finally, the last section focused on congestion costs in
education. It was shown that if these costs take the form of threshold externalities
associated with the quality of schooling, the optimal share of spending on infra-
structure (education) is positively (negatively) related to the degree of congestion in
education.
The analysis in this paper could be extended in several directions. One area of
investigation would be to consider the case where it is the stock of public capital in
infrastructure, rather than the flow of spending on infrastructure services, that
affects the production technology for goods and educated labor. I have pursuedthis line of research in a companion paper (see Agenor, 2005b), and the results
indicate that the optimal rules remain qualitatively similar to those derived in the
present paper. However, transitional dynamics and welfare-maximizing solutions
cannot be explicitly analysed in that context, and recourse to numerical techniques
becomes necessary. A second direction would be to consider other forms of dis-
tortionary taxation, such as a tax on consumption, wages, or capital income, as for
instance in Turnovsky (1996, 2000). With an endogenous supply of (raw) labor,
a tax on wages would affect private decisions to allocate time between labor and
leisure. Similarly, a tax on the return to private capital would affect decisionsbetween consumption and investment. Both taxes are likely to have ambiguous
effects on growth, because their adverse effect on private investment and labor
supply would be at least in part offset by the increase in the stock of public capital
induced by higher revenues. An interesting issue in this context is the extent to
which collection costs (which can be particularly large for direct taxes in developing
countries) affect the optimal tax structure.
Finally, as noted in the introduction, an important difference between the pre-
sent paper and others in the Uzawa-Lucas tradition is the assumption that knowl-
edge is embodied in workers. Moreover, it was assumed that all workers (once
educated) share the same amount of knowledge In practice, it is hard to distinguish
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rate, for instance, may have a significant effect on growth either because the
number of educated workers acts as a constraint on the expansion of physical
capital to begin with (as one would expect in low-income countries), or because
greater literacy increases the economys ability to create knowledge (a more likely
scenario in high-income countries). It is therefore difficultexcept perhaps byrefining standard growth accounting techniquesto determine whether it is the
number of educated workers, or changes in the average level of education, that
affects growth, based on these indicators. A worthwhile extension of the present
analysis would be of course to combine these two assumptions and assume that
although knowledge is embodied, the (average) level of skills per worker changes
over time. Doing so would allow a distinction between two types of education
expenditurethose aimed at running schools and those aimed directly at creating
knowledge. But the degree of complexity that would then be added could compli-
cate the analysis significantly and could require the use of numerical simulation
methods.
Acknowledgements
I am grateful to Kyriakos Neanidis, Devrim Yilmaz, the Editor, and three anonymous
referees for helpful discussions and comments on a previous version. I bear sole responsi-
bility, however, for the views expressed here.
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From the definition ofGI, and using (1),
GI IY IAPGI
KP
E
KP
KP,
that is, with e E=KP ,GI
KP IAP
GI
KP
e,
or equivalently
GI
KP IAPe
1=1
: A2
Substituting (A2) in (A1) yields
_KP
KP q API
e
1=1c
,
A3
wherec C=KP .
Similarly, using (A2), equation (15) can be rewritten as
_C
C s API
e 1=1
h i
: A4
From the definition ofGE, and using (1),
GE EY EAPGI
KP
E
KP
KP,
so that, using (A2),
GE
KP E API
e 1=1
E APIe
1=1
: A5
Equation (8) gives
_E
E AE
GE
KP
KP
E
GI
KP
KP
E
AE
GE
KP
GI
KP
e,
so that, using (A2) and (A5),
_E
E AE
EAPIe
1=1
IAP=1e=1
e:
This expression can be rewritten as
_E
E A
E
1I
2 e!, A6
whereA AEA
2
P , 1 =1 , 2 =1 , and
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Combining eqs (A3), (A4), and (A6) yields
_c
c I
e 1=1
c, A7
_ee
AE1I
2 e! q APIe
1=1c, A8
where
A1=1P s q: A9
This expression is negative (positive) if is lower (higher) than the ratio q / s:
sg sg s1q: A10
To investigate the dynamics in the vicinity of the steady state, the system ( A7)-(A8) can
be linearized to give_c
_e
a11 a12
a21 a22
c ~c
e ~e
, A11
where the aijare given by
a11 ~c, a21 ~e,
a22 !AE
1I
2 ~e! q
1 API
~e
1=150,
a12 ~c
1 I
=1~e!=,
where ~e and ~cdenote the stationary values ofe and c , and sga12 sg22
From (A7), setting _c 0 yields
~c I~e
1=1: A12
Similarly, from (A8), setting _e 0 yields
~c q API~e 1=1AE1I 2 ~e!: A13
From these two equations, it can be seen that the slopes of the CC and EE curves are
given by
d~c
d~e
CC
a12
a11
1 I
=1~e!=,
d~c
d~e
EE
a22
a2140:
Thus, whereas EEalways has a positive slope, the slope ofCC is upward- (downward-)
sloping if is negative (positive).c is a jump variable, whereas e is predetermined over time. Saddlepath stability requires
bl ( i i ) T h hi di i h ld h d i f h
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If~e=I50 (which is always the case if 0 or more generally if=is small), the effect
on growth is positive if =4~e=I. If ~e=I40 the effect on growth is always positive.
Graphically, it can be verified from (A12) and (A14) that a rise in Ialways leads to a shift in
CCto the left (given that 4 0), whereas the shift in EEcan be either to the right (upper
panel of Fig. 2) or the left (lower panel).The impact effect of a rise in I on the consumption-private capital ratio, given that
de0=d 0, is
@c0
@I
@~c
@I
a12
~c
@~e
@I
, A16
which is also ambiguous in general, given that @ ~c=@Iis ambiguous. If@ ~e=@I50 , given that
a1240 , then @c0=@I50 if@~c=@I50 , as shown in the upper panel of Fig. 2.
Appendix 2Dynamic form with nonseparable utility
Suppose that the households discounted utility is given by (22), which is repeated here for
convenience:
maxC
V
10
CGH1
1 exptdt, 40: A17
In the decentralized economy, the household solves problem (A17) subject to (1) and (3),taking the tax rate, , and government-provided services, GH, as given. The current-value
Hamiltonian for this problem can be written as
CG
H
1
1 1 AP
GI
KP
E
KP
KP S C
( ),
where is the costate variable associated with (3) and S 1 wGE .
From the first-order condition d=dC 0 and the costate condition_
d=dKP , optimality conditions for this problem can be written, withs 1 , where 1 , as
GHCGH
, A18
_= sAPGI
KP
E
KP
, A19
together with the budget constraint (3) and the transversality condition
limt!1
KPexpt 0: A20
84 fiscal policy and endogenous growth
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Equation (A18) can be rewritten as
C G11=H :
Taking logs of this expression and differentiating with respect to time yields
_C
C
_
_GH
GH
, A21
where 1 1=.
Given that the tax rate is constant, _GH=GH _GI=GI _Y=Y. From (1),
_Y
Y
_GI
GI
_E
E
1
_KP
KP ,
that is,
_Y
Y
1
_E
E
1
_KP
KP
: A22
Substituting this expression, together with (A19), in (A21) yields
_C
C sAP
GI
KP
E
KP
( )
1
_E
E
_KP
KP
: A23
The next step is to eliminate _KP=KP and _E=E in (A23). From (A3) in Appendix 1,
_KP
KP q API
e
h i1=1c, A24
wherec C=KP and q 1 I E H. Similarly, from (A6),
_EE
AE1I
2 e!, A25
where again A AEA2P, 1 =1 , 2 =1 , and !
1=1 250.
Substituting (A24), and (A25) in (A23) and using (A2) yields
_C
C s API
e
h i1=1 A26
1
AE1I
2 e! q APIe
h i
1=1c
:
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Combining (A24), (A25), and (A26) yields a dynamic system in c and e, which can be
analyzed along the lines of Appendix 1. With 51, the qualitative properties of the model
(uniqueness and stability) are the same as before.23 The growth-maximizing policies
(as summarized in Proposition 5) can be derived directly from (A25) and (A26), setting
c ~cand e
~e .Consider now the welfare-maximizing policies. To specify the social planners problem,
note first that, from (1), given that GI IY,
Y APIE
KP
h i1=1: A27
From (11), the economys consolidated budget constraint can be written as
Y C _KP X
hI, E, H
Gh, A28
that is, using (A27),
_KP q APIE
KP
h i1=1C: A29
Using Gh hYand (A27), eq. (8) becomes
_E AEI1 2 E1!K!P: A30
To get an expression for E=K, use (23) and (25) to obtain
E
K
EY
HA
E
1I
2 E1!K!P 1 C
Y
: A31
Substituting (A31) in (31) yields, using (23), (A29), (A30), and setting 0;
_C
C API
e
h i1=1
A32
1 AE
1I
2 e!
1 c ,
where
1 1
E
Hg
C
Y
40:
Divide (A29) byKPyields
_KP
KP q API
e
h i1=1c: A33
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Subtracting (A33) from (A32) yields
_c
c API
e
1=1
A34
1 A
E
1I
2 e!
1 1
c ,
where q. The sign of this expression depends on the size of ; as before, it is
negative for sufficiently small.
Now, dividing (A30) byEyields
_E
E A
EI
1 2 e!: A35
Subtracting (A33) from (A35) implies
_e
e AE
1I
2 e! q APIe
h i1=1
c, A36
which is identical to (A8). Equations (A34) and (A36) represent a system of two nonlinear
differential equations inc C=KPande E=KP. Uniqueness and stability properties of this
system can be established along the lines of Appendix 1 (see also Agenor and Neanidis,
(2006, Appendix C). It can readily be shown that Fig. 1 represents again the phase diagram of
the centrally-planned economy.
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