WPS4405
Policy ReseaRch WoRking PaPeR 4405
Fiscal Policy, Public Expenditure
Composition, and Growth
Theory and Empirics
Willi Semmler
Alfred Greiner
Bobo Diallo
Armon Rezai
Anand Rajaram
The World Bank
Africa Region
Public Sector Governance Department
November 2007
Policy ReseaRch WoRking PaPeR 4405
Abstract
This paper responds to the development policy debate on the modeling strategy of Greiner, Semmler and
involving the World Bank and the IMF on the use of Gong (2005) we develop a general model that features
fiscal policy not only for economic stabilization but also a government that undertakes public expenditure on (a)
to promote economic growth and increase per capita education and health facilities which enhance human
income. A key issue in this debate relates to the effect capital, (b) public infrastructure such as roads and bridges
of the composition of public expenditure on economic necessary for market activity, (c) public administration to
growth. Policy makers and some researchers have argued support government functions, (d) transfers and public
that expenditure on growthenhancing functions could consumption facilities, and (e) debt service. The proposed
enhance future revenue and justify the provision of model is numerically solved, calibrated and the impact of
"fiscal space" in the budget. But there are no simple the composition of public expenditure on the longrun
ways to identify the growthmaximizing composition per capita income explored for low, lowermiddle and
of public expenditure. The current paper lays out a uppermiddleincome countries. Policy implications and
research strategy to explore the effects of fiscal policy, practical policy rules are spelled out, the extension to
including the composition of public expenditure, on an estimable model indicated, a debt sustainability test
economic growth, using a time series approach. Based proposed, and the outofsteadystate dynamics studied.
This papera product of the Public Sector Governance Departmentis part of a larger effort in the Poverty Reduction
and Economic Management (PREM) VicePresidency to encourage active policy research on fiscal policy and growth
issues, with a particular focus on expenditure composition choices and their growth and sustainability implications. Policy
Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at
arajaram@worldbank.org and SemmlerW@newschool.edu.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Fiscal Policy, Public Expenditure Composition, and Growth
Theory and Empirics1
Willi Semmler,2 Alfred Greiner,3 Bobo Diallo,4 Armon Rezai,5 and
Anand Rajaram6
1We want to thank Lars Gruene (Univ. of Bielefeld) for extensive help in the numerical part of the paper, David Bevan
(Oxford Univ.) and to Shanta Devarajan for valuable guidance, and the audience at the World Bank workshop on
"Growth and Fiscal Policy," June 2006, for useful comments. Financial support from the BankNetherlands Partnership
Program is gratefully acknowledged.
2Schwartz Center for Economic Policy Analysis, New School, New York and CEM, Bielefeld
3Dept. of Economics and CEM, Bielefeld University
4Schwartz Center for Economic Policy Analysis, New School, New York
5Schwartz Center for Economic Policy Analysis, New School, New York
6World Bank, Washington, D.C
1 Introduction
Starting from a "production function" view of the growth and development process with a
narrow focus on physical capital, labor force, and exogenous technical progress, growth theory has
come to acknowledge the role of an economy's endowment and continuing investment in human
capital, physical infrastructure, research and knowledge development, institutions, and innovation
in explaining its growth trajectory.
The implications for policy design are somewhat harder to divine. From the perspective of a
government, which policies enhance an economy's growth prospects and how large are policy
effects on growth? In part, the answer depends on the empirical approach taken. Studies based on
crosscountry regressions have explored a large number of forces of growth, but they face
methodological perils such as the huge heterogeneity of countries (different technologies,
institutions, and preferences across countries), and the uncertainty of presumed underlying models
and parameters. Consequently, nonrobustness of the outcomes and ambiguous policy implications
are a common criticism of studies based on crosscountry regressions.7
Greiner, Semmler and Gong (2005) argue that a time series perspective on economic growth
may be more useful to pursue in designing growth and development strategies. Growth models, as
advocated by Greiner and colleagues, (1) allow for better specification of microeconomic behavior
of economic agents, (2) enable time series study of the forces of growth for a country or a group of
countries at particular stages of economic growth, (3) permit analysis using econometric time series
methods, and (4) allow us to spell out (though mostly in the context of smallscale models)
important implications for growth and development policies.
Since such models, which lend themselves to some time series tests, are often difficult to treat
when a larger number of forces of growth are introduced, Greiner and colleagues (2005) have
suggested identifying the specific forces that are important in a particular country consistent with its
particular stage of development. They mainly consider externalities and learning from others,
education and human capital, the creation and accumulation of knowledge and public infrastructure
as forces of economic growth in a time series context. Overall, this approach allows analysts to
better derive the public policy implications for economic growth.
The primary interest of this paper is to explore, as the World Bank's papers to the
Development Committee (2006, 2007) suggest, whether countries could better use fiscal policy
(and in particular, the level and composition of public expenditure) to promote sustainable increases
in growth and welfare for low and middleincome countries. We are encouraged by some recent
work by Glomm and Rioja (2006), Agenor and Neanidis (2006), and country studies undertaken by
World Bank research.8
Section 2 provides a selective discussion of recent literature and empirical evidence on public
investment in infrastructure, health, and education and their effect on growth. In section 3 we sketch
a general social planner type model that includes choices regarding the accumulation of physical
capital as well as the composition of investment in infrastructure, health, and education. In section
4, the model is calibrated by matching actual differences in per capita income for three groups of
countries (for lowincome, lowermiddleincome and uppermiddleincome countries). In addition
the effects of a change in foreign aid and other factors that have an impact on growth are explored.
Section 5 undertakes a similar calibration exercise; here, however, by considering the growth and
welfare effects arising from a change in the composition of the public expenditure. Section 6
discusses sustainability tests of fiscal policy. Section 7 concludes the paper and spells out some
practical policy rules. In the appendix the data sources are discussed and the detailed results for the
lowermiddleincome and uppermiddleincome countries are reported. Also, in the appendix the
7See Rodrik (2005) for a methodological criticism of crosscountry regressions in identifying growth effects of policies.
See also Easterly (2005) in the Handbook of Economic Growth for a skeptical review of the role of policies in
explaining crosscountry growth.
8See the papers by Arestoff and Hurlin (2006), Balonos (2005), Ferreira and Arajo (2006), and Suescun (2005).
2
outofsteadystate dynamics of a simplified version of the model are explored, and it is shown how
that the model can be turned into an estimable model using time series data.
2 Literature and Evidence
The scope for policy to influence economic growth depends on the underlying model of
growth. So long as the Solow model dominated economists' view of growth, there was little role for
fiscal policy to influence the long term rate of growth, which depended on exogenous technical
progress.9 Lucas (1988) and Barro (1990) opened the door to a rich literature on endogenous growth
theory and a corresponding attempt to develop our understanding of the implications for fiscal
policy. Tanzi and Zee (1997) rovide a relatively early review of the resulting literature on fiscal
policy and concluded that despite the lack of robust empirical results, endogenous growth theory
provided the basis for confidence that fiscal policy could affect long run growth performance of
countries.
The theoretical literature on fiscal policy has studied the effect of `productive' and
`unproductive' spending and distortionary and nondistortionary taxation on long term growth. This
literature generally predicts that productive spending financed by nondistortionary taxes will have
a positive effect on long term growth whereas the opposite combination (unproductive spending
financed by distortionary taxation) will have a negative effect. The algebraic sign of other spendtax
combinations is more ambiguous since there are opposing effects at work. Barro (1990) suggests
that an inverted Ucurve relationship would exist between productive government expenditure and
economic growth, as the rising cost of distortionary taxation necessary to finance spending
overtakes the declining benefits of productive spending.
Early models of growth that featured government expenditure used fairly simple
characterizations of productive and unproductive spending  public investment was viewed to be
productive whereas public consumption was unproductive. Devarajan et al. (1996) developed a
model with public investment and consumption expenditure to show that the growth impact of
public investment could be negative if there was excessive investment. Glomm and Ravikumar
(1997) considered the implications of government expenditure on infrastructure (which influences
private production) as well as on education which results in human capital accumulation. More
recent literature (Zagler and Durnecker (2003), Glomm and Rioja (2006), Blankenau and Simpson
(2004), Agenor and eanidis (2006)) provide a more disaggregated discussion of government
expenditure, typically including spending on public infrastructure, health, and education, which are
described as providing inputs for private production. Zagler and Durnecker (2003) define an
economy where output is produced using labor, private capital and public infrastructure expenditure
and consider the effects of government spending and taxation on long term growth rates. Glomm
and Rioja (2006) consider the implications of shifting expenditure from transfers to infrastructure or
education and conclude, based on empirical evidence from Brazil, that at the margin the growth
implications are small. Blankenau and Simpson (2004) focus on education expenditure and growth.
Some of the papers take account of the interdependence among these expenditures, with the
productivity of health spending depending on education and infrastructure expenditure or stocks,
and vice versa. The intuition behind such complementarities is well known  good sanitation and
water supply infrastructure has large health benefits, including a reduction in incidence of malaria
and gastrointestinal deseases. This in turn has a positive effect on school attendance rates and on
learning outcomes (see Bundy and et al. (2005)) as well as on labor productivity in market
activities.
While the theoretical models have become more nuanced about public expenditure and their
financing, the empirical evidence is often perceived to be ambiguous, reflecting problems of data,
differing research methodologies and econometric techniques. Gemmell (2007) provides a useful
9Temple (2003) argues that the scope for policy to have an influence on the level of output should merit the attention of
policy makers and analysts but has been neglected because of a misguided focus on effects on the long term growth rate
and an undervaluation of level effects.
3
review of the evidence, and concludes that more recent literature uses more reliable methods
(including a clearer specification of the government budget constraint) to derive robust evidence, at
least for OECD countries, of long run impacts of fiscal policy on economic growth. Even for
developing countries he finds that, consistent with theory, recent studies show a positive medium to
longrun growth effect of certain categories of expenditure, such as transport and communication
infrastructure, education and health. However, the complementarities between health, education,
infrastructure and growth involve tradeoffs in the actual development process. Public resources are
limited and given the constraints governments often need to weigh the benefits of expenditure on
one against the benefits of spending on the other. In the following section a model is formulated to
represent both the complementarities and the tradeoffs of public investment.
3 A Model of Fiscal Policy and Economic Growth
The model to be presented is based on the work by Greiner et al. (2005) but takes into account
some of the generalizations that have been put forward by Corsetti and Roubini (1996) and others.
The current paper develops a model of growth with a private and a public sector, the latter described
by a government that can choose to raise resources through taxation, borrowing, or foreign aid. In
our model we assume that the tax rate is chosen optimally and there is a welldefined intertemporal
budget constraint. We allow for foreign debt and spell out fiscal policy rules that ensure
sustainability of debt. Besides the accumulation of physical capital by the private sector, the model
includes public investment in infrastructure, health and education. In contrast to Agenor and Yilmaz
(2006) who work with expenditure flows, we develop a model with stocks: private physical capital,
human capital and a broad notion of public capital that includes public infrastructure to support
market production as well as facilities for health and education services. The paper attempts to
derive some policy insights using empirical techniques to estimate the model. Given that time series
models that study forces of growth in a historical context are rather complex and require high
quality time series data to be estimated, we adopt a calibration technique to assess the growth
impact of fiscal policy on countries identified by levels of income/development.
3.1 The Model
The Economy
Production in this stylized economy is characterized by a Cobb Douglas function
( Ak (u1h) (v1g) for the production of market goods and ((1u1)h)1(v2g)2 (v3g)3 for the
production of human capital) with private capital k , public capital g , and human capital h . Note
that only a part of public capital (v1g ) is used to support private market production, while other
parts are used to help to build human capital in the form of schools (v2g ) and medical centers
(v3g ) and do not directly influence production.10 This realistically describes the longer gestation lag
in creating human capital relative to typical physical infrastructure.
The Government
Government (represented by a social planner) raises taxes optimally without distortion and
can choose four types of public expenditure: (i) public investment to enhance education and health
services that increase the stock of human capital, (ii) investment on public infrastructure assets
(transport and communication system as well as energy, water supply and sanitation), to support
market production and the creation of human capital, (iii) transfers and public consumption
representing expenditure with public goods' characteristics which may enter into households'
preferences (public parks, civic facilities and consumption transfers), and (iv) public administration
necessary for the functioning of the government (including justice, security and tax collection).
These four expenditure streams allow us to consider the effect of public expenditure composition on
10Since the v 's represent shares, v1 + v2 + v3 = 1.
4
growth.
The stylized model developed in this paper focuses on deriving the implications for growth
and welfare due to alternative government choices regarding the allocation of the stock and flow of
public capital. The government can choose to allocate public investment in infrastructure assets
(such as roads) that directly influence market production or in assets relevant for human capital
accumulation (such as schools and clinics) that may have a lagged effect on growth. The growth
and welfare effects are worth studying since they help us highlight the intertemporal consequences
and tradeoffs of policy choices confronting many governments.11
The model enables government to acquire foreign debt and recognizes debt service as a part of
the government budget. Foreign transfers to lowincome countries are assumed to reflect foreign aid
that is earmarked for public investment. The social planner is assumed to maximize a logarithmic
welfare function, a form that simplifies preferences.
When we define the variables, all variables are in per capita form and we define public capital
as nonexcludable but subject to congestion in the production of market goods and the creation of
human capital, because it is per capita public capital ( g = G/L ) which affects per capita output.
Table 1 Variable Overview (all in per capita terms)
Variables Description
b Debt
c Consumption
d Debt service
ep Amount of resources absorbed by the public sector
k Private apital
g Public capital
h Human capital
y GDP
Table 2 Parameter Overview (all in per capita terms)
Parameters Description
A Productivity factor
ip
f Foreign aid
n Population growth rate
r Borrowing rate
u1 Fraction of h used to support the production of k
1u1 Fraction of h used to build new h
v1 Fraction of g used to support the production of k
v2 Fraction of g used to build new h in the form of schools
v3 Fraction of g used to build new h in the form of medical centers
1 Share of public resources used for the building of new g
2 Share of public resources used for transfers and public consumption
3 Share of public resources used for the functioning of the administration
11To focus on the effects of public capital allocation, the model assumes a stylized world where the government decides
on the allocation of public capital and the private sector provides the labor to staff the schools and clinics as well as the
labor employed in market production. One could imagine an economy where private sector volunteers provide the
human capital services but rely on complementary government facilities. Human capital allocation therefore has no
direct impact on the government budget. An extension to the current paper could consider the implication of
government employment of teachers and doctors and then consider the compositional choice regarding capital and
recurrent wage expenditure.
5
4 Share of public resources used for debt services
Elasticity of GDP w.r.t. k
Elasticity of GDP w.r.t. h
Elasticity of GDP w.r.t. g
k Depreciation of private capital (k )
g Depreciation of public capital ( g )
h Depreciation of human capital (h )
1 Elasticity of h w.r.t. existing h
2 Elasticity of h w.r.t. g in the form of schools
2 Elasticity of h w.r.t. g in the form of medical centers
Weight of transfers and public consumption in utility
Discount rate
U Utility/welfare
The amount of resources absorbed by the public sector, ep , is used for public infrastructure
(ip = 1ep ), transfers and public consumption (cp = 2ep ), and public administration (tr = 3ep ).
The latter has neither utility nor productive effects, but possesses public goods features and is
necessary for the functioning of the state. Finally, for the debt service we have
d = (11 2 3)ep = 4ep.12
When public investment is turned into public capital, we can think of three uses of the public
capital: First, there exists public capital which raises productivity of market production, such as
transport systems (roads, bridges, harbors) and utilities (for example water supply). In the model,
that part is v1g . The other fraction of public capital is used for facilitating the formation of human
capital via health services, v2g , and education, v3g . In our view, the formulation with one public
capital stock, which is divided (and subdivided) between use in production and human capital
formation, is sufficient to study the composition effect of public expenditure.
Note that for the government budget constraint we can write G = ip + (1 +2 +3 +4)ep ,
f
with ip +1ep earmarked for public investment and 4ep for debt service. In the steady state then
f
we have ip +1ep = (g + n)g . The flow of expenditure should be properly subdivided in spending
f
for infrastructure, health, and education, according to the fractions v1, v2 and v3.13 Moreover, note
that the government revenue is also T = ip + ep , but what can be used for domestic spending is only
f
T 4ep , since 4ep flows out for debt service. The primary surplus of the government budget is
4ep.
We presume a planner solves the system of equations below to derive the social optimum:
12Note that we can allow for net borrowing. If we define net borrowing as ibp , earmarked for public investment, we
will, then, have to add ibp in eqns. (4) and (5) as additional borrowing. We can think of ip in eqn. (4) as representing
f
both foreign aid and net borrowing. In the steady state debt, b , and the debttoincome ratio, b, will correspondingly
y
fall, since the maximal sustainable debt level will decrease as more debt is taken on.
13We might think that the annual revenue, earmarked for ip +1ep , is handed over to a trust fund that manages the
f
public current and capital expenditure related to infrastructure, health and education.
6
(c(2ep) )1 
max (1)
c,ep,u1 0e(n)t 1 1dt
subject to
k& = Ak (u1h) (v1g)  c ep  (k + n)k (2)
h& = ((1u1)h)1(v2g)2 (v3g)3 hh (3)
g& = ip +1ep  (g + n)g
f (4)
b& = (r  n)b  (11 2 3)ep (5)
Note that for simplicity we assume an additively separable utility function
U() = lnc + ln(2ep) . Note also that equations (2)  (5) describe the constraints to the growth of
private capital, human capital, public capital and debt. Since we only consider an exogenous growth
model in this version, + + 1, e.g. = = = 0.33. One could introduce here some
externalities, as in Greiner et al. (2005, ch.3), to obtain endogenous growth.
In a further extension we could allow for foreign borrowing (for public investment) by adding
a positive term to eqns. (4) and (5). Since this borrowing will be constant and permanent, the level
b
of permissible debt level will fall. However, since income will increase, the debttoincome ratio
y
will also fall. This is exactly what we expect from a numerical exercise.14
Our formulation of public debt implies that public debt is foreign public debt. New borrowing
from abroad can be allowed for.15 Interest payments on existent foreign debt (for example foreign
bonds) is rb (at world interest rate). Repayments and interest payments do not go to the domestic
household sector, but go to foreigners. The usual transversality condition is assumed to hold (see
section 3.2).
Next we explore the stationary state of the model in its form of eqns. (1)  (5). We employ the
Hamiltonian to sketch a solution of the model. The Hamiltonian is
H(c,ep,k,h, g) = ln c +ln(2ep)
+ 1[Ak(u1h)(v1g) cep (k +n)k]
+ 2[((1u1)h)1(v2g)2(v3g)3 hh] (6)
+ 3[ip +1ep (g +n)g]
f
+ 4[(r n)b(11 2 3)ep]
with 1,2,3,4 the costate variables. The first order conditions for the two choice
variables, c and ep , are:
H
c = 0 c1 = 1 (7)
14Note that this term does not have to be constant. It suffices if the term is positive on average for the results to hold.
15See footnote 3.
7
H 1 (8)
ep = 0 ep = 1  31  4(11 2 3)
For the costate variables we have
&1 = 1( n) H
(9)
k = 1(  n)  1[Ak1(u1h) (v1g) (k + n)]
&2 = 2( n) H
h = 2(  n)  1[Ak(u1h)1u1(v1g) ]
(10)
 2[1((1u1)h)11(1u1)((1v1)g)2 h]
&3 = 3( n) H
g = 3(  n)  1[Ak (u1h) (v1g)1 v1]
(11)
 2[((1u1)h)1((1v1)g)212(1v1)]+3(g +n)
&4 = 4( n) H
(12)
b = 4(  n)  4(r  n)
Equations (7)(12), which are derived from the two first order conditions with respect to the
choice variables c and ep and the four equations for the costate variables 1, 2, 3,and 4 , give
us, together with the four state variable equations (2)(5), a system of ten equations in ten variables
{c,ep,k,h, g,b,1,2,3,4}.
Writing eqns. (7) and (8) as
0 = 1  c1 (13)
0 = 1  1 ep 1 (14)
and setting the differential equations (2)(5) as well as (9)(12) equal to zero, we can obtain a
stationary state {c*,e*p,k*,h*, g*,b*1,*2,*3,*4}. Plugging these into the production function
*
y = Ak (u1h) (v1g) , we can also obtain the per capita income, y*, at the stationary state.
Given our specification of human capital and its use, we can study the effect of the allocation
of human capital on per capita income and the other macroeconomic variables. Furthermore,
varying our parameter set v1, v2 and v3, we can explore the impact of different public expenditure
compositions on the steadystate outcomes of per capita income and the other variables. A further
effect on per capita income is predicted to be caused by a change of the parameter A and of foreign
aid, if ip > 0 . Overall, from all of those comparative static studies we expect to obtain information
f
on per capita income, consumption and the other macro variables.
Specifying all parameters, the following numerical stationary solution is obtained from
necessary optimality conditions using the computer software Mathematica:16 17 18
16The parameter values used are: n = 0.015 , = 0.03 , k = 0.075 , h = 0.075 , g = 0.05 , = = = 0.33,
u1 = 0.85 , v1 = 0.8 , v2 = 0.1, v3 = 0.1 , 1 = 2 = 0.2 , 3 = 0.25 , 1 = 0.1, 2 = 0.7 , 3 = 0.1 , ip = 0.05 ,
f
8
k* = 86.59, h* = 21.97 , g* =17.86 , b* = 20.20, y* = 27.55, b*
= 0.73
y
With this solution technique we can undertake the above described comparative static
analyses and some important calibrations, see sects. 4 and 5.19
3.2 Fiscal Policy to Ensure Debt Sustainability20
The model of section 3.1 permits government to borrow from capital markets, in particular to
borrow from abroad in order to undertake public investment. In this section, we want to explore the
effects and implications of government borrowing from capital markets and briefly discuss the
sustainability of fiscal policy when borrowing is allowed for.21 22
Let us assume that the government has borrowed from abroad for undertaking public
investment. The implication of this type of expenditure is that along the transition path, this will
raise the growth rate of public infrastructure and lead to a higher level of public capital. This higher
level brings a distortion into the model by raising the marginal product of private capital. As a
consequence, the investment share is increased and the growth rate of consumption rises implying
higher welfare after a sufficiently long adjustment period. Note that this also leads to higher growth
of physical and human capital. However, in our growth model these growth effects only hold on the
transition path. In the longrun, higher public investment leads to higher levels of output and
consumption but does not affect the growth rate of endogenous variables.
Since here borrowing from abroad equals loans or issuing of bonds to foreigners, the
government must pay it back plus interest payments. Debt repayment is represented by 4ep . More
concretely, the government has to stick to the intertemporal budget constraint which can be written
as

b(0) = e (r()n)ds()d t

0 lime (r()n)db(t)
0 = 0, (15)
0 t
where s denotes the primary surplus which, in our model, is given by 4ep . The first term in
(15) states that percapita government debt at time zero must equal the discounted stream of future
primary surpluses. This implies that the government must run primary surpluses, if it starts with a
positive stock of debt. Equivalent to this formulation is the requirement that discounted debt
converges to zero asymptotically. It should be mentioned that in theoretical exogenous growth
models the intertemporal budget constraint is met, provided the interest rate exceeds the population
growth rate on average, because percapita debt converges to a finite value. Nevertheless, it is
A = 1, = 0.1, r = 0.07 .
17The Mathematica code is available at www.newschool.edu/gf/cem.
18Note that these values are dimensionless and are calibrated in such a way as to fit the data. Economic implications are
derived in later sections, where the effects of comparative changes are explored.
19Allowing for additional borrowing by adding a constant in eqns. (4) and (5) and setting this borrowing at the same
size as foreign aid, ibp = 0.05 , we obtain the following numerical values: k* = 88.67 , h* = 22.36 , g* = 18.43 ,
*
y* = 28.21, b b
= 0.67 . As one can observe, b and fall, indicating a fall in permissible debt.
y y
20Since this section treats empirical matters, its notation is changed in order to comply with readily available data (e.g.
G now refers to primary expenditure and not public capital as in the rest of the paper).
21Again, note that in case of net borrowing a constant needs to be added to eqn. ((5)) and the foreign aid parameter ip f
in eqn. ((4)) is supposed to be corrected for the availability of new investment funds for public capital.
22In Greiner et al. (2005), ch. 6, more particular model versions are developed with government borrowing for specific
government expenditures.
9
important for real world economies to check for rules whether fiscal policies of countries are such
that they fulfill the intertemporal budget constraint or not.
One possible rule for sustainability is to require that the government adjusts the primary
surplus to GDP ratio to variations in the debtGDP ratio. To see that a positive linear dependence of
the primary surplus ratio23 on the debt ratio can guarantee sustainability, we assume that the
primary surplus ratio is given by
s(t) = + (t) b(t), (16)
y(t) y(t)
with y(t) percapita GDP. (t) determines how strongly the primary surplus reacts to
changes in the public debtGDP ratio and , which is assumed to be constant, can be interpreted as
a systematic component determining how the level of the primary surplus reacts to a rise in GDP.
can also be seen as representing other constant variables which affect the primary surplusGDP
ratio.
Using equation ((16)), the differential equation describing the evolution of public debt can be
rewritten as
b&(t) = (r(t)  n)b(t)  s(t) = (r(t)  n  (t))b(t)  y(t). (17)
t
(r ( )n)d
Solving this differential equation and multiplying both sides by e 0 to get the
present value of public debt yields
(
t t
( r( )n)d ( )d ()r()+n+())d
t d ,
e 0 b(t) = e 0 b
(0)  y(0) e 0 (18)
0
with the growth rate of GDP. Equation (18) shows that a positive value for on average,
t
so that ( )d converges to plus infinity asymptotically, is necessary for sustainability. It is also
0
sufficient, if r()  n  () is positive on average.24
It should be noted that a positive value for r  n  characterizes a dynamically efficient
deterministic economy. If the economy is stochastic, however, this need not necessarily hold for the
economy to be dynamically efficient. Nevertheless, testing the reaction of the primary surplus to
variations in the debt ratio is reasonable because a positive reaction is necessary for sustainability as
mentioned above. Further, this test yields insight as to how governments deal with public debt and,
thus, shows how important the goal of stabilizing debt is for the government.
One advantage of the sustainability test just presented is to be seen in the fact that it does not
depend on the interest rate, in contrast to other tests where the assumed interest rate may be crucial
as to the outcome whether a given policy is sustainable or not. In addition, the proposed rule is
intuitively plausible from an economic point of view. If a government runs a deficit, it has to run a
primary surplus in the future to pay back the deficit. Otherwise, sustainability is not given. In our
model economy, this implies a withdrawal of resources from the economy. Therefore, a deficit
financed increase in public investment is beneficial for the economy, if the gain in productivity is
sufficiently high to cover the interest payments and the loan. A detailed discussion of different
types of sustainability tests and how the above suggested sustainability rule can be implemented is
undertaken in section 6 of the paper. Next we explore the change of the basic structure of the model
and its impact on the macroeconomic variables such as output, consumption, capital stock and debt.
23Note in the subsequent part we solely focus on the sustainability issue and ignore the feedback effects of a changing
surplus on the other spending categories.
t

24 ( )d
This holds because l'H o^ pital gives the limit of the second term in (18) multiplied by e 0 as
t
(r( )n())d
e 0 /.
10
4 Exploring the Basic Structure of the Model
We want to explore the effect of foreign aid, if , the productivity factor, A , and the fraction of
p
human capital used in market goods production, u1, on the steadystate variables of three group of
countries classified by income level. We follow the World Bank's 2006 paper to the Development
Committee in classifying countries in lowincome, lowermiddleincome, and uppermiddleincome
groups. Due to the quality of the data, however, only a reduced list of countries is used for the
calibration exercise.25 We employ only data for the time period 1994 to 2004. First, we will
calibrate the effect of foreign aid, ip .26 By looking at actual data of foreign aid per capita, we are
f
able to determine a range in which our parameters can vary. Before we can evaluate the effect of
foreign aid, productivity factor, and the fraction of human capital used in market goods production
on per capita income, we need to calibrate our abovementioned model (1) to (5) such that we can
roughly reproduce the differences in per capita income across the three groups of countries. Since,
as discussed above, there are many forces of growth causing the differences in per capita income,
we adjust the productivity level of the three groups, the parameter A , such that we roughly obtain
the actual differences in per capita income, normalizing the parameter, A , of the lowest per capita
income group. Therefore A is set to 1 for the lowest income group. We hereby take the
composition of total public expenditure, 1 , 2 , 3 , and 4 , as given for each group as we find
them in the data. Now we employ our Mathematica program to obtain stationary state solutions for
the three groups of countries.27 This is equivalent to a comparative static analysis for the three
country groups. Our exercise consists in keeping the parameter, A , constant for each group of
countries and equal to its calibrated value. Then we let foreign aid, ip , vary during that period and
f
see how such a change affects the steadystate values k, h, g, c, ep, y , b
, and U . Such an
y
exercise was conducted for the low, lowermiddle, and uppermiddle income groups. In the next
step we vary the productivity parameter, A , for each group of countries, employing for each
country group the A that we used to calibrate the model to obtain the actual differences in per
capita income, and increasing it gradually. The parameter, A , is set equal to 1, 1.18, and 1.45 for
the low, lowermiddle, and uppermiddle income groups respectively. Finally, we vary u1, the
fraction of human capital used in market goods production, from 0.1 to 0.8, keeping foreign aid
equal to its 10year average, and the other parameters constant.
4.1 Effects of Foreign Aid per Capita, ip
f
b
The comparative static results for all income groups show that k , h , g , c , ep , y , , and
y
U are linear in ip and there is clearly a positive relationship between the level of foreign aid and
f
b
all the variables, except for public resources per capita, ep , and debt per GDP, , which show a
y
25See Appendix A.4 for the complete list of countries.
26Note that in the following we presume that there is no new borrowing, so that ip is solely foreign aid.
f
27The Mathematica code is available at www.newschool.edu/gf/cem.
11
negative relationship with respect to foreign aid, ip . f 28 As the foreign aid goes up the public
resources per capita falls slightly. The results suggest that any increase in foreign aid per capita
would increase k, h, g, c and y but would reduce only very slightly the optimally chosen public
resources, ep . Such a negative relationship could be explained by the fact that public resource is a
result of a lumpsum taxation and as more aid flows in, there is less and less incentive to tax, and as
b
a result, a decrease in public resources, ep . A similar argument can be made for debt per GDP, .
y
As more aid flows in, there is less and less incentive to borrow, and as a result, there would be a fall
b
in as ip increases. Actual data for the period 19942004 shows that the per capita foreign aid
f
y
went from 2 to 4. To conduct our comparative static exercises, we increased ip by increments of f
0.2 from 2 to 4, which is the range provided by the actual data.29 The results of our comparative
static exercises for the lowincome group are summarized in Table 4 in appendix A.5. The results
for the remaining two income groups can also be found in the appendix A.5. Figure 1 below shows
b
the relationship between ip and y , U ,
f , and ep respectively for the lowincome group.
y
Similar results are observed for the lowermiddle and uppermiddle income groups.30
Figure 1: Effects of Foreign Aid ip on the Steady State Variables (LowIncome Group)
f
4.2
28This paper does not address the problems of moral hazard and inefficiency that often characterize aid inflows into
poor governance environments. Waste, corruption, and inefficiency would clearly weaken the effects of foreign aid on
growth and welfare.
29Recent literature has studied the effect of foreign aid on per capita income in country by country studies (see Easterly
(2006)) or in crosssection studies (see Reddy and Minoiu (2006)). The latter study finds significant and large effects.
30Note that the effect of an increase of A has rather large effects on output. But also note that an increase of our A by
6% would correspond to an increase in the efficiency of labor by roughly 20%, since we have written A as outside the
production function, namely as exogenous technical change affecting total factor productivity.
12
Effect of A, the productivity factor
b
The comparative static results for all income groups show that k , h , g , c , ep , y , , and
y
U are linear in A and there is clearly a positive relationship between the level of foreign aid and all
b
the variables. The results suggest that any increase in A would increase k , h , g , c , ep , y , ,
y
and U . Because the lowincome group is our reference group, A is increased from 1 to 1.06, and
the effect on the steadystate variables is observed. As one might predict, results show that there is a
positive linear relationship between A and all the steadystate values of the variables, with the
fastest increase in k , y , and g , relative to the other variables. The results of our comparative static
exercises for the lowincome group are summarized in Table 5 in appendix A.5. The results for the
other two income groups can also be found in the appendix. Figure 2 below shows the relationship
between A and y , and k respectively for the lowincome group. Similar results are observed for
the lowermiddle and uppermiddle income groups.
<>
Figure 2: Effects of the Productivity Factor A on the Steady State Variables (LowIncome Group)
4.3 Effects of the Fraction of Human Capital Used in Market Goods
Production, u1
To analyze the effect of the parameter u1, it was increased from 0.05 to 0.95 with increments
of 0.05, and the steadystate values corresponding to the different values of the parameter were
recorded. The comparative static results on u1 for all income groups show a humpshape, nonlinear
b
relationship, with respect to k , g , h , c , ep , y , , and U . With respect to the humpshape
y
relationship, we can make the following observations: First, we observe a positive relationship
b
between u1 and k , g , c , ep , y , , and U for u1 0.8 , and a negative relationship for u1 0.8 .
y
b
We observe increasing returns for k , g , c , ep , y , , when u1 0.3 and decreasing returns when
y
u1 0.3. As for U , and b
, we always observe decreasing returns for all values of u1. Second, as to
y
human capital, h , we observe a positive relationship between 1u1 and h when 1u1 0.3 , and a
negative relationship for 1u1 0.3 . Finally, it was observed in our exercises that the nature, and
form of the humpshape relationship depend on the choice of our parameters. For instance, we
chose 1, 2 , and 3 to sum up to less than 1, indicating decreasing returns. When the above
13
parameters are chosen such that they sum up to more than 1, both the nature of the relationship and
the optimal point for u1 change. The results of our comparative static exercise for the lowincome
group are summarized in Table 6 in appendix . The results for the other two income groups can also
be found in the appendix A.5. Figure 3 above shows the relationship between u1 and y , U , h , and
b , and between 1u1 and h , for the lowincome group. Similar results are observed for the lower
y
middle and uppermiddle income groups.
Figure 3: Effect of the Fraction of Human Capital Used in Market Goods Production u1 on
the Steady State Variables (LowIncome Group)
The observation that u1, has a nonlinear, humpshaped relationship with all the variables suggests
a decreasing return effect so that, beyond a point, any increase in the fraction of resources allocated
to one area takes away resources from other areas that may indeed contribute more to growth.
Furthermore, the humpshaped form of the effect of 1u1 on per capita human capital, h , implies
that any increase in the parameters would first increase h , but beyond the 30% threshold, the effect
becomes negative. Such a reversal of the relationship could signal the fact that as too much human
capital is devoted to human capital production, there is less and less of it available for market goods
production.
5 Exploring the Effect of Public Expenditure Composition
In this section the model is calibrated for different expenditure structures and the effects of
changes in the composition of public investment expenditure are explored for the three country
groups. As above, the classification in low, lowermiddle and uppermiddleincome countries is
14
retained.31 Here, again, only the results of the lowincome groups are reported. For the results of the
other two country groups, see Appendix A.6. The data on public expenditure is obtained from the
IMF's Government Finance Statistics. Since these are too detailed for our analysis, the different
expenditure categories are summarized to match the following model parameters: public investment
in infrastructure, 1v1 , education, 1v2 , and health, 1v3 , public transfers, 2 , public consumption,
3 and debt services, 4 = (11 2 3).32 Note that the values reflect the averages for all
countries of one category.
By plugging the obtained numerical values (1 = 0.4035 , 2 = 0.0623, 3 = 0.4342 ) into the
model, setting = 0.03, k = 0.075, h = 0.075, g = 0.05 , = = = 0.33, u1 = 0.85,
1 =2 = 0.2, 3 = 0.25, r = 0.08, = 0.1, and using the values for A and ip that were found in f
the previous section ( A =1, n = 0.0195 ip = 3.361), one can simulate the effect of changes in the
f
composition of public expenditure on the stationary states by letting v1, v2 and v3 vary.33
In the following this is done for the lowincome country group.34 Although all graphs are
presented in the 3dimensional space such that the interaction of all three investment expenditure
parameters can be observed at the same time, the stationary states for shifts from public investment
in health, 1v2 , to public investment in education, 1v3 , are of particular interest.35
Generally, the results suggest that, in order to maximize the growth of per capita income,
more than half of public investment expenditure should be on infrastructure while the residual
should be roughly equally allocated to investments that support production of human capital.
Figures 4 to 12, all of which are humpshaped, depict the effects of changes in the investment
expenditure composition on the various model parameters. Figures 4 to 9 show all possible
investment expenditure compositions for income, y , capital, k , public capital, g , public debt, b ,
consumption, c and total public resource absorption, ep , are qualitatively the same.
b
Only the graphs for human capital, h , utility, U , and debt/income ratio, , differ in their
y
behavior. In Figure 12 one can see that human capital can be increased beyond its welfare and
growthmaximizing level by setting v2 = 0.30 and v3 = 0.20 (and v1 = 0.5). At this point the
maximal h is reached, however, the benefits of its increase do not offset the costs of higher
depreciation. Figure 10 depicts the changes in utility over the range of v1, v2 and v3. Whilst the
curvature of U is less pronounced than in the previous graphs, it is humpshaped and possesses an
interior maximum which lies at general maximum, i.e. v1 = 0.65, v2 = 0.15 and v3 = 0.20 . The
b
shape of the debt to income ratio, , differs completely from the other variables as it does not
y
b
possess an interior maximum. As one can see in graph 11, the curvature of is strictly concave
y
and attains its maximum value at the origin, i.e. v2 = v3 = 0 (and v1 =1). The fact that is, however,
more interesting is that the ratio is approximately stable through most of the range of v2 and v3.
31See appendix A.4 for the complete list of countries.
32See appendix A.3 for details.
33Note that described the numerical values for , k , h, g , , , , u1, 1, 2, 3, 1 , 2 , 3 and remain
the same for the remainder of this section.
34Similar results hold for the other two groups. Complete sets of graphs for the other groups are listed in Appendix A.6.
35Note that v1 + v2 + v3 = 1 always has to hold which restricts the plane to a triangular space. By fixing v2 or v3 the
effects of changes in v1 are also always observable.
15
Sustainability problems due to shifts in the investment expenditure composition do not occur in the
long run.
The aim of this section was to conduct a comparative static exercise for changes in the
composition of public investment expenditure. The results show that in the long run the welfare
and growth maximum lies at a composition of roughly v1 = 0.65, v2 = 0.15 and v3 = 0.20 . Debt
sustainability problems did not occur throughout the possible range of v1, v2 and v3. Yet, overall to
the issue of debt sustainability we have presumed here a very passive debt policy as indicated in
eqn. ((5)), which assumes that new debt is not issued and the repayment follows a fixed rule,
namely 4ep . If we allow for new debt issue, as discussed in section 3.1, the debt to output ratio, b
y
is likely to increase.
Figure 4: Effects of Changes on Income y Figure 5: Effects of Changes on Capital k
Figure 7: Effects of Changes on Public Figure 6: Effects of Changes on Public
Capital g Debt b
16
Figure 8: Effects of Changes on Figure 9: Effects of Changes on Public
Consumption c Absorption ep
Figure 11: Effects of Changes on Utility U Figure 10: Effects of Changes on Debt to
Income Ratio b
y
Figure 12: Effects of Changes on
Human Capital h
6 Testing Debt Sustainability36
36As in section 3.2 the notation of this section is changed in order to comply with empirically available data.
17
In section 3.2 we modeled the intertemporal budget constraint in the context of our model
but we did not pursue this question further in our calibration studies in section 4 and 5. If one
studies particular countries (or country groups) from a time series perspective, it is an important
empirical question whether certain public policies for growth stay within the bounds of a
sustainable fiscal policy. Here we briefly discuss tests for sustainability of fiscal policies and
suggest a version that has been tested for advanced countries. We will then suggest sustainability
tests that can be undertaken for specific country studies.37.
If one studies the effect of fiscal policy on economic growth, one needs to check, as indicated
in section 3.2, whether the government fulfills the intertemporal budget constraint. As shown in
section 3.2, in economic terms, this constraint states that public (net) debt at time zero must equal
the expected value of future presentvalue primary surpluses. In general terms, this requirement is
also often referred to as the noPonzi game condition. In a general form we can write this as
follows. Neglecting stochastic effects and assuming that the interest rate is constant, the inter
temporal budget constraint can be written as
B(0) = er S ( )d,
p (19)
0
with r the constant interest rate, B(0) public debt at time zero and S the primary surplus.38
p
Equivalent to equation (19) is the following equation
limertB (t) = 0, (20)
t
with B(t) public debt at time t , stating that the present value of public debt converges to
zero for t .
In the economic literature numerous studies exist which explore whether (19) and (20) hold in real
economies.39 For example, Hamilton and Flavin (1984) suggest testing for the presence of a bubble
term in the time series of public net debt which would indicate that a given fiscal policy is not
sustainable. Trehan and Walsh (1991) proposed to test whether the budget deficit is stationary or to
test whether the primary budget deficit and the public debt series are cointegrated and (1L)St p
is stationary, with 0 <1+ rt. Another test, proposed by Wilcox (1989), is to test whether the
series of undiscounted debt displays an unconditional mean of zero. If this holds the intertemporal
government constraint will be fulfilled, because the intertemporal budget constraint requires the
discounted debt to converge to zero.40 This implies that all government debt will be repaid at some
point in time.
Moreover, another aspect of these tests which has given rise to criticism is that those tests
need strong assumptions, because the transversality condition involves an expectation about states
in the future that are difficult to obtain from a single set of time series data and because assumptions
about the discount rate have to be made.41
Alternative tests check solely if in the long run some debttoincome ratio remains stationary.
A test procedure which circumvents the problems associated with the above first type of test
focuses on the time series of the debt ratio, i.e. on the ratio of public debttoGDP. If this series is
constant the intertemporal budget constraint is fulfilled for dynamically efficient economies. To
see this let B/Y = c1 be the constant debt ratio, with Y denoting GDP and c1 a positive constant.
Inserting B/Y = c1 in ((20)) yields
limc1Y0e( r)t = 0, (21)
t
for < r , with > 0 the constant growth rate of GDP. The condition < r characterizes a
37For a detailed discussion of sustainability tests see Greiner et al. (2005).
38For further derivations see e.g. Blanchard and Fischer (1989), ch. 2.
39See e.g. Hamilton and Flavin (1984), Kremers (1988), Wilcox (1989), and Trehan and Walsh (1991).
40As to these tests applied to Germany see Greiner and Semmler (1999).
41See e.g. Bohn (1995,1998).
18
dynamically efficient economy and is likely to hold in real economies. For example, in EU
countries this seems to be obvious if one compares interest rates with GDP growth rates. But even
the US, where growth rates have exceeded interest rates on safe government bonds since the 1990s,
is a dynamically efficient economy.42 Therefore, for advanced countries we may limit our
considerations to the case of dynamic efficient economies and assume that the discount rate of
government debt exceeds the GDP growth rate.43 For developing countries, like the low and lower
middle income countries of this study, this would have to be explored.
However, testing for stationarity of the debttoGDP ratio is characterized by some
shortcomings, too. It is difficult to distinguish between a time series which is stationary about a
positive intercept and one that shows a trend. This holds because standard unit root regressions have
low power against autoregressive alternatives if the AR coefficient is close to one. As a
consequence, the hypothesis that a given fiscal policy is sustainable has been rejected too easily.
Therefore, Bohn (1998) suggests to test whether the primary deficittoGDP ratio is a positive
linear function of the debttoGDP ratio. If this holds, a given fiscal policy will be sustainable. As
discussed in section 3.2 the intuitive reasoning behind this argument is that if a government raises
the primary surplus as the public debt ratio rises, it takes a corrective action which stabilizes the
debt ratio and makes public debt sustainable. Before one can undertake empirical tests which apply
this concept we need to advance some theoretical considerations about the relevance of this test. We
limit our considerations to deterministic economies.
Assuming, as in section 3.2, that the primary surplus to GDP ratio depends on a constant and
linearly on the debttoGDP ratio, this variable can be written as
S (t)
p T(t) G(t) = + Y(t) ,
B(t)
= (22)
Y(t) Y(t)
with T(t) tax revenue at t , G(t) public spending at t, and are constants which can be
negative or positive. is a systematic component which determines how the level of the primary
deficit reacts to variations in GDP. can also be interpreted as other (constant) economic variables
which affect the surplus ratio.
The coefficient can be called a reaction coefficient since it gives the response of the
primary surplus ratio to an increase in the debt ratio. Inserting (22) in the differential equation,
giving the evolution of public debt, the latter equation is then given by
B&(t) = rB(t) + G(t) T(t) = (r  )B(t) Y(t). (23)
Solving this equation we get
B(t) = r )t (24)
  Y(0)et + e(r C1,
with B(0) > 0 debt at time t = 0 which is assumed to be strictly positive and with C1 a
constant given by C1 = B(0) Y(0)/(r   ) . Multiplying both sides of (24) by ert leads to
ertB(t) = t (25)
r   Y(0)e(r)t + e C1.
The first term on the right hand side in (25) converges to zero in dynamically efficient
economies and the second term converges for > 0 and diverges for < 0. These considerations
show that > 0 guarantees that the intertemporal budget constraint of the government holds.
Further, with equation (22) the debttoGDP ratio evolves according to
42For details of such conditions see Abel et al. (1989).
43In dynamically inefficient economies the government budget constraint is irrelevant because in that case the
government can play a Ponzi game.
19
b& = b  = b(r   ).
B& Y&
(26)
B Y
Solving this differential equation, we obtain the debttoGDP ratio b as
b(t) =  )t (27)
(r   ) +e(r C2,
where C2 is a constant given by C2 = b(0) /(r   ), with b(0) B(0)/Y(0) the debt
GDP ratio at time t = 0.
Equation (27) shows that the debttoGDP ratio remains bounded if r   < 0 holds. This
shows that a positive does not assure boundedness of the debtGDP ratio although the inter
temporal budget constraint of the government is fulfilled in this case. Only if is larger than the
difference between the interest rate and the GDP growth rate the debt ratio remains bounded. These
considerations demonstrate that sustainability of debt may be given even if the debttoGDP ratio
rises over time  a situation which seems to hold for some countries.
But it must also be pointed out that for too high a level of public debt the government will
probably not be able to raise the primary surplus any further. Then, our rule formulated in equation
(27) will break down. This holds because, the present value of future surpluses must equal public
debt at any finite point in time. So, if the government is not able to raise the primary surplus as
public debt rises any longer from a certain point of time, say t1 , onwards, is zero or even
negative. Then, a fiscal policy is sustainable only if the government has succeeded to reduce public
debt to zero up to that point of time t1 . This implies that in the longrun the debttoGDP ratio must
be constant, although it may well rise transitorily. However, it is ultimately an empirical question
what the country specific will be.44
In this section therefore we suggest some method related to the Bohn method (Bohn(1998))
which allows one to estimate sustainability, but at the same time works, as section 3.2 has indicated,
with a time varying reaction of governments to the debt to GDP ratio. This allows one to get some
insight using empirical tests about the sustainability of policies. As we have pointed out in section
3.2 our strategy for testing the sustainability of debt has the advantage that the test does not depend
on the interest rate which is used to discount public debt as needed in the first type of the above
discussed tests.
Following our setup in section 3.2, the equation suggested to be tested is as follows:
st = tbt +Zt +t,
y y (28)
where st and bt are the primary surplus to GDP and the debttoGDP ratio, respectively. Zt
y y
is a vector which consists of the number 1 and of other factors related to the primary surplus and t
is an error term which is i.i.d. N(0, ). 2
The idea behind estimating (28) is the tax smoothing hypothesis according to which public
deficits should be used to finance transitory government spending, for example higher public
spending during recessions. Further, the variables contained in Zt may differ depending on which
country is analyzed. In the US, for example, military spending is a variable which exerts a strong
influence on the primary surplus. In European countries social spending plays an important role
affecting the primary surplus. For other countries, for example, lowincome countries, other
spending categories may be relevant.
We want to note, as concerns the primary surplus, one has to distinguish between the primary
surplus including the social surplus and the primary surplus exclusive of the social surplus, where
social surplus means the surplus of social insurance system. This holds because in some countries
deficits of the social insurance system raise the stock of public debt, since the government sector
and the social insurance system are not separated, whereas this does not hold for other countries.
44This could be estimated separately for each country in country studies.
20
So, in some European countries the social insurance systems are autonomous and do not borrow in
capital markets. For example, if the social insurance system runs deficits, these deficits are either
transitory because they must be paid back in the next period or the deficits are covered by reserves
from earlier years.
If, on the other hand, the social insurance system has a surplus this does not raise the surplus
of the government, but is used as reserve. However, it happens that the government subsidizes the
social insurance which leads to a decline in the surplus of the government. This amount, however, is
included in the regular surplus or deficit of the government so that taking it into account in the
surplus or deficit of the government and adding that of the social insurance would lead to double
counting.
These considerations demonstrate that institutional regulations determine which concept for
the primary surplus should be used. As we noted above, social security and transfer arrangements in
countries of different income groups seem to be different. Our suggested test should therefore be
adjusted to each type of income group  or country if country specific fiscal policy studies are
undertaken.
7 Conclusion
The World Bank has argued in two recent policy papers that fiscal policy design should seek
to ensure macroeconomic stability as well as promote growth and the longrun welfare of a country.
The growth impact of the composition of public expenditure is an important aspect of the design of
fiscal policy that is consequently the focus of research interest. In this paper we suggest a general
model of fiscal policy and growth in an economy with a government that taxes optimally and
undertakes public expenditure on (a) education and health facilities, (b) public infrastructure such as
roads and bridges, (c) public administration, (d) transfers and public consumption facilities, and (e)
debt service. We use this model to explore the impact of shifts in the composition of expenditure
on longrun per capita income and other macroeconomic variables. We also set up the model in a
way that allows us to study whether fiscal policy is sustainable. This model is solved, the impact of
foreign aid, the allocation of human and public capital and fiscal expenditure on per capita income
and other macro variables explored.
Foreign aid per capita and the productivity factor both have a positive and linear effect on per
capita GDP and welfare. Such a result is clearly what would have been predicted by the theory as
inflows of foreign aid are assumed to be used for investment in roads, schools, hospitals, or any
other infrastructure that plays an important role in raising productivity. Foreign aid has either no
effect, or a slightly negative effect, on the stock of resources absorbed by the public sector.. Such a
result could be due to some "crowdingin" effect, reflecting the fact that as foreign aid flows in,
more investment and production opportunities open up and resources are used more privately and
less publicly.
Second, the model suggests that the choice of allocating human capital to market production
or to human capital production poses an important tradeoff. As the share of human capital that is
devoted to human capital production rises (starting from a very low share) income, as well as
welfare, initially rises but then falls. In addition, beyond a certain point, the ability to expand the
stock of human capital itself is actually diminished. Thus, there exists a growthmaximizing
allocation of human capital between final production and the creation of additional human capital.
Overall, our model suggests that the larger fraction of human capital should be used for market
production.45
The calibration exercise was also undertaken for different compositions of public investment
expenditure for the three country groups. The results of this exercise show that the composition of
public investment expenditure matters, as the gains of moving to the optimal allocation between
45Of course, this depends on the assumption in our parameter constellations, in particular on the elasticity parameters in
the creation of human capital, which add up to 0.6 , a number far below 1.
21
public infrastructure, and education and health facilities are significant. Based on the model and the
calibration exercise, a practical rule of thumb seems to be that about twothirds of public investment
should be directed towards public infrastructure that facilitates market production. The remaining
third should be split more or less evenly between public investments in facilities that support the
provision of health and education. Such a division of resources would maximize (per capita) income
and welfare. This is due to the fact that the facilitation of market good production directly increases
the availability of public resources, while the other two expenditure categories first have to
permeate the economic system before affecting the availability of public resources and thus growth
and utility. This result has relevance for policy debate on the composition of expenditure necessary
to achieve the Millenium Development Goals (MDGs), which include an income target (halve the
number of people below the poverty line) as well as specific targets related to the achievement of
health status and access to educational services. The stylized model in this paper suggests that
greater emphasis on the health and educational targets relative to investments that may contribute to
expansion of market production may result in slower progress on reducing poverty.
During this numerical exercise, attention was also paid to the evolution of public debt. Quite
surprisingly, the debttoincome ratio was almost invariant to all possible investment expenditure
compositions. In the considered case the ratio stabilized below 70%. Here the rule of thumb seems
to emerge that in the long run debt sustainability does not pose a problem as long as resources for
public investments are used in a growthmaximizing way.
We conclude with some observations regarding possible extensions of the model described in
this paper. One possibility (described in the appendix) is to estimate the model using time series
methods. However, this will only be possible for countries where highquality time series data are
available. An important extension of the model would be to develop the revenue side of fiscal
policy in order to study the effects of various forms of taxation on per capita income. Another
extension would be to consider the growth implications of variations in the balance between
recurrent and capital expenditure by allowing government employment of teachers and doctors.
Finally, the debt sustainability tests proposed in this paper could be applied to countryspecific
studies.
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24
Appendix
The appendix begins with the outofsteadystate solution for a simplified version of the
model (A.1) and a suggestion of how to estimate the simpler model using time series analysis (A.2).
Appendix A.3 explains how the data used in the calibration was constructed and appendix A.4 lists
the countries used. Appendix A.5 lists the data used to compute the graphs shown in section 4 and
additional data on the two other country groups which have not been discussed in the paper.
Appendix A.6 does the same with graphs on the long run effects of public expenditure composition
that have been omitted in the main part of the paper.
A.1 OutofSteadyState Solutions
Since economies are rarely at their steady states, it is, for practical purposes, of great
importance to explore what decisions should be taken out of the steady state. There is, however, no
analytical solution for our decision variables out of the steady state. Yet, once the parameters have
been set and the stationary state of the variables {c,ep,k,h, g,1,2,3} have been computed, the
dynamic programming algorithm developed by Gruene and Semmler (2004) can be used to study
the outofsteadystate dynamics not only of consumption, c , and the use of resources by the
public, ep , but also the state variables: physical capital stock, human capital and public capital
stock.46 Using our parameter set above, next we will compute the outofsteadystate solutions using
a numerical algorithm. This way we can study the following: When, for example, the capital stock
of a country is k < k* (and h < h* , g < g*) the dynamic programming solution allows one to judge
whether c should be high or low, how ep behaves, and how the control variables relate to each
other out of the steady state.
Using the algorithm of Gruene and Semmler (2004) we have solved the model (1)(4), with u1
fixed, and thus solely with c and ep as control variables and k,h and g as state variables.47 Yet,
since the value function (1) is a function of the decision variables c and ep and the state variables
k, h and g (the former again depending on k, h and g ) one can only obtain proper graphs by
appropriate projections and study the value function and behavior of decision variables in a two
dimensional subspace.
We have solved the model (1)(4) in the vicinity of the steady state k* = 65.47, h* =18.16 ,
g* =11.47 which was obtained by solving the equation system arising from the Hamiltonian in
section 3.1.48 Next, in the vicinity of the positive steady state we have taken a cube
[60,70]×[16,20]×[9,13] for k , u and g , with a subspace for the choice variables [7,8]×[7,8], for
c and ep .
Figure 13 shows four sample trajectories in a three dimensional space, with initial conditions
for trajectory 1 (T1): (66.2, 18.9, 11.8) which are above the steady state k*,h*, g* . Trajectory 2 (T2)
has initial conditions: (67.5, 18.7, 11.4) and is, thus, far to the right of the steady state k*,h*, g* .
Trajectory 3 (T3): (63.2, 16.8, 11.1) starts below the steady state k*,h*, g* . Finally, Trajectory 4
(T4) starts with k = 68.2 above, with h =16.5 below and with g at the steady state.
46The out of steady state dynamics are analyzed for a simplified version of the model which omits public debt.
47Note that this simplified version omits the debt equation (5) of the model.
48There is, however, a second steady state which was neglected because of negative values of some state variables.
25
Figure 13: Out of Steady State Dynamics
As one can observe in Figure 13, the correctly taken optimal decisions c and ep make the
initial states of state variables converging toward the steady state k*,h*, g* for all four sample
trajectories. Note that trajectories (T1), (T2), (T3) and (T4) do not only exhibit some irregular
features but also cross each other. This comes from the fact that we follow the trajectories in a 2
dim subspace state variables only, namely in the k  h space. For graphical purposes, the third state
variable g is fixed in Figure 13.
Figure 14: Decision Variablec in the k  h space
If one stores the decision variables c and ep at each grid point of the 3dim cube of k,h and
g one can then plot them in a 3dim space where the height represents the numerical value of the
decision variable. In Figure 14, for example, the height stands for the value of c , and the other two
axes represent k and h .
In Figure 14 it is clearly visible that if k > k* and h > h* the optimal consumption c is
required to be high, above its steady state. The reverse holds for the space k < k* and h < h* . Here
c has to be below its steady state. This behavior of the optimal c is a result that one would also
expect from economic intuition.
26
Figure 15: Decision Variableep in the k  h space
On the other hand as Figure 15 shows the decision variable ep is not so monotonically
dependent on k and h . In some regions ep is higher than in others. Note that we have neglected
here to study the behavior of c depending on government capital, g . This also can be easily done,
but this might not be so relevant in the context of our study.
Next, we want to study the dependence of the total use of public resources, ep , on private
capital, k , and public capital, g .
Figure 16: Decision Variableep in the k  g space
Figure 16 shows how the decision variable ep behaves in the k  g space. We can observe
that ep will be high above the steady state, for g > g* , and low below the steady state, for g < g* .
The decision variable ep does not seem to depend much on the level of the capital stock, k .
Finally, we want to make a comment on the constraints that we have put on the control
variables, c and ep . For both the lower constraint is 7 and the upper is 8 so that all optimal c and
ep are staying within those constraints. But note that with c* = 7.04 and e*p = 7.53 the decision
variables are above the lower constraints of 7. Putting such constraints on the decision variables just
means that the decision variables cannot go fast enough down and thus the state variables that are
affected by this are not going up fast enough. This is the reason why the trajectory (T4) in Figure 13
27
first moves far down (to the left) in the state variable k and then rises again moving toward its
steadystate value.
The above exercise with constraints for the decision variables is nevertheless of great
importance for practical economic policy. It basically means that the decision variables do not have
to be exactly at their optimal values in order to exhibit convergence dynamics. For actual economies
an exact optimal control is indeed hard to achieve, since for actual economies, there is model
uncertainty  which model fits the economy  as well as data uncertainty (see Brock et al. (2003)).
Our computations with constraints for the decision variables then means that the decision variables
need to be only above or below their steadystate values in the appropriate state space in order to
fulfill the requirement to be roughly optimal. This aspect appears to us of great practical
importance, since decisions for c and ep have to be only approximately correct.
A.2 The Estimable Form of the Model
Next we want to spell out some implications of how to estimate our model with time series
methods. Even for our simplified social planner model of eqns. (1) to (4) it could be too ambitious
to estimate the model employing Euler equations, derived from the first order conditions for the two
control variables for consumption, c , and total use of resources, ep , and the decision variable u1.
As the model is written, it gives us the outcome of the social optimum. We propose to reduce the
model further, and treat, for the purpose of a time series estimation, only consumption as decision
variable. The other choice variables, ep and u1 are just treated as historical variables, as time series
observations. This is also likely to inject some realistic features into the model.
For a time series study we thus propose to include only one Euler equation in the time series
estimation procedure. We suggest to only use the equations (2)(4) and the equation for c& , which
can be derived from the first order condition of the decision variable c as
c& &1 ( ) (29)
c =  c& = c Ak1(u1h) (v1g)  k
1
The model reflects now the fact that it is only optimized with respect to private consumption.
The decision making process for all the public variables might be considered too complex to be
presumed as the result of some optimization process. Yet, in the normative sense, as we discuss
below, we still might consider the other public decision variables as choice variables so as to give
us a guidance to welfare improving policies. Above, we have undertaken some calibration and
comparative static study in order to explore the impact of fiscal policy decisions on per capita
income and welfare.
A.3 Data
Two different sources were used for gathering the data necessary for the calibration and
comparative static exercises. First, information on GDP, population size and foreign aid flows was
taken from the World Bank's World Development Indicators (WDI). Second, information on the
public expenditure composition was taken from the Government Finance Statistics (GFS). In order
to make the data compatible with the model, only category 7 (and its subcategories) of the GFS
were used. Public infrastructure investment, v11ep , consists of "Economic affairs" (704),
"Environmental protection" (705) and "Housing and community amenities" (706). Public
expenditure on health, v21ep , was taken from "Health" (707). Public expenditure on education,
v31ep , consists of "Education" (709). "Recreation, culture, and religion" (708) and "Social
28
protection" (710) was subsumed under public transfers, 2ep . Public consumption and debt
payments, 3ep and 4ep , were defined as the sum of "General public services" (701), "Defense"
(702) and "Public order and safety" (703). Since the information on debt service is very spotty, we
chose 4 = 0.10 in a rough approximation and set the accordingly decreased above sum equal to
3.
We follow the IMF Development Committee (2006) in its classification of countries in low,
lowermiddle and uppermiddle income countries. However, due to the bad quality of the data in the
GFS, we dropped some countries and reduced the list of countries to the ones listed in appendix
A.4.
A.4 Country Groups
Table 3: Reduced List of Countries
LowIncome LowerMiddle Income UpperMiddle Income
Bhutan Belarus Argentina
Burundi Bolivia Croatia
Ethiopia Bulgaria Czech Republic
India Dominican Republic Estonia
Moldova Egypt Hungary
Myanmar El Salvador Lithuania
Pakistan Iran Mauritius
Yemen Jamaica Mongolia
Zambia Maldives Panama
Morocco Poland
Philippines Uruguay
Romania
Sri Lanka
Thailand
Tunisia
A.5 Appendix to Section 4
Table 4: Effect of Foreign Aid, ip , on the SteadyState Variables (LowIncome Group)
f
if
p ep c k h g b y U
2.0 1539.3 890.4 10848 1152.6 9005 2531.8 3451.6 89.70
2.2 1539.1 891.0 10850 1152.8 9006 2531.5 3452.2 89.71
2.4 1538.9 891.6 10851 1152.9 9008 2531.2 3452.7 89.72
2.6 1538.8 892.1 10853 1153.0 9010 2530.8 3453.2 89.73
2.8 1538.6 892.7 10855 1153.2 9012 2530.5 3453.8 89.73
3.0 1538.4 893.2 10856 1153.3 9014 2530.2 3454.3 89.74
3.2 1538.2 893.8 10858 1153.4 9015 2529.9 3454.8 89.75
3.4 1538.0 894.4 10860 1153.6 9017 2529.6 3455.4 89.76
3.6 1537.8 894.9 10861 1153.7 9019 2529.3 3455.9 89.80
3.8 1537.6 895.5 10863 1153.8 9021 25290. 3456.4 89.77
4.0 1537.5 896.0 10865 1153.9 9023 2528.7 3457.0 89.78
29
Table 5: Effect of the Productivity Factor, A , on the SteadyState Variables (LowIncome Group)
A ep c k h g b y U
1.000 1538 894 10859 1153 9016 2530 3455 89.8
1.005 1589 923 11214 1175 9311 2613 3568 90.2
1.010 1641 953 11580 1196 9614 2699 3684 90.6
1.015 1694 984 11955 1218 9926 2786 3804 91.1
1.020 1749 1015 12341 1240 10246 2877 3927 91.5
1.025 1805 1048 12737 1262 10575 2969 4053 91.9
1.030 1863 1081 13144 1284 10913 3065 4182 92.3
1.035 1923 1115 13562 1307 11259 3163 4315 92.8
1.040 1984 1150 13992 1330 11615 3263 4452 93.2
1.045 2047 1186 14432 1354 11981 3366 4592 93.6
1.050 2111 1223 14885 1377 12356 3472 4736 94.0
1.055 2177 1261 15349 1401 12742 3581 4884 94.4
1.060 2245 1300 15825 1425 13137 3692 5035 94.9
Table 6: Effect of the Fraction of Human Capital Used in Market Goods Production u1
on the SteadyState Variables (LowIncome Group)
u1 ep c k h g b y U
0.05 9 15 104 144 98 14 33 32.4
0.10 39 33 321 253 274 64 102 44.4
0.15 93 64 700 383 588 152 223 53.7
0.20 168 107 1230 517 1028 276 391 60.8
0.25 263 162 1899 648 1582 433 604 66.4
0.30 376 227 2691 774 2239 618 856 71.0
0.35 503 300 3586 893 2982 828 1141 74.9
0.40 642 380 4563 1002 3793 1056 1452 78.1
0.45 789 464 5597 1100 4650 1298 1781 80.8
0.50 940 551 6657 1184 5530 1547 2118 83.1
0.55 1090 637 7710 1252 6404 1793 2453 85.1
0.60 1233 719 8716 1303 7239 2028 2773 86.8
0.65 1363 793 9628 1332 7995 2242 3063 88.1
0.70 1471 855 10386 1338 8624 2419 3305 89.2
0.75 1547 899 10918 1315 9065 2544 3474 89.8
0.80 1576 916 11123 1256 9236 2592 3539 90.1
0.85 1538 894 10859 1153 9016 2530 3455 89.8
0.90 1399 814 9882 989 8206 2301 3144 88.5
0.95 1084 633 7665 721 6366 1782 2439 85.0
Table 7: Effects of Foreign Aid, ip , on SteadyState Variables (LowerMiddleIncome Group)
f
if
p ep c k h g b y U
3.00 5824.8 4659.7 45470 2742.7 42050 8642.2 14467.8 104.38
3.20 5824.7 4660.3 45472 2742.8 42052 8642.0 14468.4 104.38
3.40 5824.6 4660.9 45474 2742.9 42055 8641.8 14469.0 104.38
3.60 5824.5 4661.4 45476 2743.0 42057 8641.6 14469.6 104.38
3.80 5824.3 4662.0 45478 2743.0 42059 8641.4 14470.2 104.38
4.00 5824.2 4662.6 45480 2743.1 42061 8641.3 14470.8 104.38
4.20 5824.1 4663.1 45481 2743.2 42064 8641.1 14471.4 104.38
4.40 5823.9 4663.7 45483 2743.3 42066 8640.9 14472.0 104.38
4.60 5823.8 4664.3 45485 2743.4 42068 8640.7 14472.6 104.40
4.80 5823.7 4664.8 45487 2743.5 42071 8640.5 14473.2 104.39
5.00 5823.6 4665.4 45489 2743.5 42073 8640.3 14473.8 104.39
30
Table 8: Effects of the Productivity Factor, A , on the SteadyState Variables (LowerMiddleIncome Group)
A ep c k h g b y U
1.180 5824 4663 45481 2743 42063 8641 14471 104.4
1.185 5986 4792 46742 2786 43229 8881 14873 104.7
1.190 6152 4924 48034 2829 44423 9127 15283 105.1
1.195 6321 5059 49355 2872 45645 9378 15704 105.4
1.200 6494 5198 50707 2916 46895 9636 16134 105.7
1.205 6672 5339 52091 2961 48174 9899 16574 106.1
1.210 6853 5484 53506 3006 49483 10168 17025 106.4
1.215 7039 5632 54953 3051 50821 10443 17485 106.8
1.220 7229 5784 56434 3097 52190 10725 17956 107.1
1.225 7423 5939 57948 3144 53590 11013 18438 107.4
1.230 7621 6097 59496 3191 55021 11308 18931 107.8
1.235 7824 6259 61079 3238 56485 11609 19434 108.1
1.240 8032 6425 62698 3286 57982 11917 19949 108.4
Table 9: Effect of u1, the Fraction of Human Capital Used in Market Goods Production
on the SteadyState Variables (LowerMiddleIncome Group)
u1 ep c k h g b y U
0.05 35 41 328 278 316 52 104 44.3
0.10 154 137 1260 574 1176 229 401 59.8
0.15 359 300 2856 894 2651 532 909 69.8
0.20 644 528 5082 1217 4710 956 1617 76.9
0.25 1004 815 7889 1532 7305 1490 2510 82.4
0.30 1430 1155 11211 1835 10376 2122 3567 86.8
0.35 1912 1540 14968 2119 13850 2836 4763 90.4
0.40 2437 1959 19068 2380 17641 3616 6067 93.5
0.45 2993 2403 23404 2613 21651 4441 7447 96.0
0.50 3564 2859 27853 2813 25764 5288 8862 98.2
0.55 4130 3311 32272 2977 29849 6128 10268 100.1
0.60 4672 3743 36492 3097 33752 6931 11611 101.6
0.65 5162 4134 40316 3168 37287 7659 12828 102.9
0.70 5570 4460 43498 3181 40229 8264 13840 103.8
0.75 5856 4688 45728 3126 42292 8688 14550 104.5
0.80 5967 4777 46591 2988 43089 8852 14824 104.7
0.85 5824 4663 45481 2743 42063 8641 14471 104.4
0.90 5299 4243 41383 2351 38274 7862 13167 103.2
0.95 4106 3291 32082 1713 29674 6092 10208 100.0
Table 10: Effects of Foreign Aid, ip , on SteadyState Variables (UpperMiddleIncome Group)
f
ifp ep c k h g b y U
3.0 10068.2 11326.4 89637 3597.5 68113 13335.4 28520.8 106.26
3.2 10068.1 11327.3 89640 3597.6 68116 13335.3 28521.8 106.26
3.4 10068.1 11328.1 89643 3597.7 68119 13335.2 28522.8 106.27
3.6 10068.0 11328.9 89646 3597.8 68122 13335.1 28523.8 106.27
3.8 10067.9 11329.8 89649 3597.9 68125 13334.9 28524.8 106.27
4.0 10067.8 11330.6 89652 3598.0 68128 13334.8 28525.8 106.27
4.2 10067.7 11331.4 89656 3598.0 68131 13334.7 28526.8 106.27
4.4 10067.6 11332.3 89659 3598.1 68134 13334.6 28527.7 106.27
4.6 10067.5 11333.1 89662 3598.2 68137 13334.5 28528.7 106.30
4.8 10067.4 11333.9 89665 3598.3 68140 13334.4 28529.7 106.27
5.0 10067.4 11334.8 89668 3598.4 68143 13334.3 28530.7 106.27
31
Table 11: Effects of the Productivity Factor, A , on the SteadyState Variables (UpperMiddleIncome Group)
A ep c k h g b y U
1.450 10068 11331 89654 3598 68130 13335 28526 106.3
1.455 10295 11586 91674 3643 69664 13636 29169 106.5
1.460 10526 11846 93733 3689 71228 13942 29824 106.8
1.465 10762 12111 95830 3735 72822 14254 30491 107.0
1.470 11002 12381 97967 3782 74446 14572 31171 107.3
1.475 11247 12656 100145 3829 76100 14897 31864 107.5
1.480 11496 12936 102363 3876 77785 15227 32570 107.8
1.485 11750 13221 104622 3924 79502 15563 33289 108.0
1.490 12009 13512 106924 3973 81251 15906 34021 108.3
1.495 12272 13808 109268 4021 83032 16255 34767 108.5
1.500 12541 14110 111656 4071 84846 16610 35527 108.8
1.505 12814 14417 114088 4120 86693 16972 36301 109.0
1.510 13092 14729 116564 4170 88574 17341 37089 109.3
Table 12: Effect of u1, the Fraction of Human Capital Used in Market Goods Production
on the SteadyState Variables (UpperMiddleIncome Group)
u1 ep c k h g b y U
0.05 62 87 623 355 488 82 198 49.9
0.10 269 319 2465 749 1886 356 784 65.1
0.15 623 717 5610 1170 4276 825 1785 74.4
0.20 1116 1271 10000 1594 7611 1478 3182 81.1
0.25 1738 1970 15534 2008 11815 2302 4943 86.1
0.30 2474 2797 22083 2405 16791 3277 7027 90.1
0.35 3307 3732 29491 2778 22419 4380 9383 93.5
0.40 4215 4753 37575 3120 28561 5583 11956 96.3
0.45 5176 5833 46124 3426 35056 6856 14676 98.6
0.50 6162 6940 54894 3690 41719 8161 17466 100.6
0.55 7141 8041 63606 3904 48338 9458 20238 102.3
0.60 8076 9091 71927 4062 54660 10697 22886 103.7
0.65 8923 10043 79465 4155 60387 11819 25284 104.9
0.70 9629 10836 85739 4173 65153 12753 27280 105.8
0.75 10123 11391 90136 4100 68494 13408 28680 106.3
0.80 10314 11606 91837 3919 69787 13661 29221 106.5
0.85 10068 11329 89648 3598 68123 13335 28524 106.3
0.90 9160 10309 81569 3083 61985 12132 25954 105.2
0.95 7099 7993 63232 2246 48054 9402 20119 102.2
32
A.6 Appendix to Section 5
Since the calibration exercise for the composition of public expenditure involves too many
parameters to list all data in tables, the depiction of the lowermiddle and uppermiddleincome
cases is limited to graphs. As one can see, these graphs show the same qualitative effects as the
lowincome group's that were discussed in the main part of the paper.
Figure 17: Effects of Expenditure Composition Changes on the
per capita Parameters in the LowerMiddleIncome Case
33
Figure 18: Effects of Expenditure Composition Changes on the
per capita Parameters in the UpperMiddleIncome Case
34