Fiscal Policy and Endoge

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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,



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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



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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

,



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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



19/32

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

p.-r. agnor 75


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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-



21/32

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

p.-r. agnor 77


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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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Agenor, P.-R. (2005b) Schooling and public capital in a model of endogenous growth,

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p.-r. agnor 79


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25/32

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

p.-r. agnor 81


26/32

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



27/32


28/32

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



29/32

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

:

p.-r. agnor 85


30/32

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



31/32

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.

p.-r. agnor 87


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Fiscal Policy and Endoge