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WPS4405
P OLICY R ESEARCH W ORKING P APER
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
involving the World Bank and the IMF on the use of
fiscal policy not only for economic stabilization but also
to promote economic growth and increase per capita
income. A key issue in this debate relates to the effect
of the composition of public expenditure on economic
growth. Policy makers and some researchers have argued
that expenditure on growth-enhancing functions could
enhance future revenue and justify the provision of
"fiscal space" in the budget. But there are no simple
ways to identify the growth-maximizing composition
of public expenditure. The current paper lays out a
research strategy to explore the effects of fiscal policy,
including the composition of public expenditure, on
economic growth, using a time series approach. Based
on the modeling strategy of Greiner, Semmler and
Gong (2005) we develop a general model that features
a government that undertakes public expenditure on (a)
education and health facilities which enhance human
capital, (b) public infrastructure such as roads and bridges
necessary for market activity, (c) public administration to
support government functions, (d) transfers and public
consumption facilities, and (e) debt service. The proposed
model is numerically solved, calibrated and the impact of
the composition of public expenditure on the long-run
per capita income explored for low-, lower-middle- and
upper-middle-income countries. Policy implications and
practical policy rules are spelled out, the extension to
an estimable model indicated, a debt sustainability test
proposed, and the out-of-steady-state dynamics studied.
This paper—a product of the Public Sector Governance Department—is part of a larger effort in the Poverty Reduction
and Economic Management (PREM) Vice-Presidency 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
1
We 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 Bank-Netherlands Partnership
Program is gratefully acknowledged.
2
Schwartz Center for Economic Policy Analysis, New School, New York and CEM, Bielefeld
3
Dept. of Economics and CEM, Bielefeld University
4
Schwartz Center for Economic Policy Analysis, New School, New York
5
Schwartz Center for Economic Policy Analysis, New School, New York
6
World 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
cross-country 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 cross-country 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 small-scale 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 middle-income 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 low-income, lower-middle-income and upper-middle-income 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
lower-middle-income and upper-middle-income countries are reported. Also, in the appendix the
7
See Rodrik (2005) for a methodological criticism of cross-country 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 cross-country growth.
8
See the papers by Arestoff and Hurlin (2006), Balonos (2005), Ferreira and Arajo (2006), and Suescun (2005).
-2-
out-of-steady-state 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 non-distortionary taxation on long term growth. This
literature generally predicts that productive spending financed by non-distortionary 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 spend-tax
combinations is more ambiguous since there are opposing effects at work. Barro (1990) suggests
that an inverted U-curve 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 gastro-intestinal 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
9
Temple (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
long-run 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 trade-offs 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 trade-offs 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 well-defined inter-temporal
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
ε
ε
ε
α
( Ak (u1h) β (v1 g )γ for the production of market goods and ((1 − u1 )h) 1 (v2 g ) 2 (v3 g ) 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 ( v1 g ) is used to support private market production, while other
parts are used to help to build human capital in the form of schools ( v2 g ) and medical centers
( v3 g ) 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
10
Since 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 inter-temporal consequences
and trade-offs 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 low-income 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 non-excludable 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
b
c
d
ep
k
g
h
y
Description
Debt
Consumption
Debt service
Amount of resources absorbed by the public sector
Private apital
Public capital
Human capital
GDP
Table 2 Parameter Overview (all in per capita terms)
Parameters
A
i pf
n
r
u1
1 − u1
v1
v2
v3
α1
α2
α3
Description
Productivity factor
Foreign aid
Population growth rate
Borrowing rate
Fraction of h used to support the production of k
Fraction of h used to build new h
Fraction of g used to support the production of k
Fraction of g used to build new h in the form of schools
Fraction of g used to build new h in the form of medical centers
Share of public resources used for the building of new g
Share of public resources used for transfers and public consumption
Share of public resources used for the functioning of the administration
11
To 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-
α4
α
β
γ
δk
δg
δh
ε1
ε2
ε2
η
ρ
U
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
Depreciation of private capital ( k )
Depreciation of public capital ( g )
Depreciation of human capital ( h )
Elasticity of h w.r.t. existing h
Elasticity of h w.r.t. g in the form of schools
Elasticity of h w.r.t. g in the form of medical centers
Weight of transfers and public consumption in utility
Discount rate
Utility/welfare
The amount of resources absorbed by the public sector, e p , is used for public infrastructure
( i p = α1e p ), transfers and public consumption ( c p = α 2 e p ), and public administration ( tr = α 3e p ).
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 = (1 − α1 − α 2 − α 3 )e p = α 4 e p . 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 v1 g . The other fraction of public capital is used for facilitating the formation of human
capital via health services, v2 g , and education, v3 g . 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 = i pf + (α1 + α 2 + α 3 + α 4 )e p ,
with i pf + α1e p earmarked for public investment and α 4 e p for debt service. In the steady state then
we have i pf + α1e p = (δ g + n) g . The flow of expenditure should be properly subdivided in spending
for infrastructure, health, and education, according to the fractions v1 , v2 and v3 .13 Moreover, note
that the government revenue is also T = i pf + e p , but what can be used for domestic spending is only
T − α 4 e p , since α 4 e p flows out for debt service. The primary surplus of the government budget is
α 4e p .
We presume a planner solves the system of equations below to derive the social optimum:
12
b
Note that we can allow for net borrowing. If we define net borrowing as i p , earmarked for public investment, we
b
f
will, then, have to add i p in eqns. (4) and (5) as additional borrowing. We can think of i p in eqn. (4) as representing
both foreign aid and net borrowing. In the steady state debt, b , and the debt-to-income ratio, b , will correspondingly
y
fall, since the maximal sustainable debt level will decrease as more debt is taken on.
13
We might think that the annual revenue, earmarked for i p + α1e p , is handed over to a trust fund that manages the
f
public current and capital expenditure related to infrastructure, health and education.
-6-
⎞
⎛ (c (α 2 e p )η )1−σ
−( ρ − n ) t ⎜
e
− 1⎟dt
max ∫0
⎟
⎜
c ,e p ,u1
1−σ
⎠
⎝
∞
(1)
subject to
k& =
Ak α (u1h) β (v1 g )γ − c − e p − (δ k + n)k
h& = ((1 − u1 )h) (v2 g ) (v3 g ) − δ h h
(2)
(3)
g&
(4)
ε1
ε2
= i pf + α1e p − (δ g + n) g
ε3
b& = (r − n)b − (1 − α1 − α 2 − α 3 )e p
(5)
Note that for simplicity we assume an additively separable utility function
U (⋅) = ln c + η ln (α 2 e p ) . 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, α + β + γ ≤ 1, e.g. α = β = γ = 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 debt-to-income ratio
y
14
will also fall. This is exactly what we expect from a numerical exercise.
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, e p , k , h, g ) = ln c + ηln(α 2 e p )
+ λ1[ Ak α (u1h) β (v1 g )γ − c − e p − (δ k + n)k ]
+ λ2 [((1 − u1 )h) 1 (v2 g ) 2 (v3 g ) 3 − δ h h]
ε
ε
+ λ3 [i pf + α1e p − (δ g + n) g ]
ε
(6)
+ λ4 [(r − n)b − (1 − α1 − α 2 − α 3 )e p ]
with λ1 , λ2 , λ3 , λ4 the co-state variables. The first order conditions for the two choice
variables, c and e p , are:
∂H
= 0 ⇒ c −1 = λ1
∂c
14
15
Note that this term does not have to be constant. It suffices if the term is positive on average for the results to hold.
See footnote 3.
-7-
(7)
∂H
= 0 ⇒ ηe −p1 = λ1 − λ3α1 − λ4 (1 − α1 − α 2 − α 3 )
∂e p
(8)
For the co-state variables we have
λ&1 = λ1 ( ρ − n) −
∂H
= λ1 ( ρ − n) − λ1[ Aαk α −1 (u1h) β (v1 g )γ − (δ k + n)]
∂k
∂H
= λ2 ( ρ − n) − λ1[ Ak α β (u1h) β −1 u1 (v1 g )γ ]
∂h
ε −1
ε
− λ2 [ε 1 ((1 − u1 )h) 1 (1 − u1 )((1 − v1 ) g ) 2 − δ h ]
λ&2 = λ2 ( ρ − n) −
∂H
= λ3 ( ρ − n) − λ1[ Ak α (u1h) β γ (v1 g )γ −1 ⋅ v1 ]
∂g
ε
ε −1
− λ2 [((1 − u1 )h) 1 ((1 − v1 ) g ) 2 ε 2 (1 − v1 )] + λ3 (δ g + n)
λ&3 = λ3 ( ρ − n) −
λ&4 = λ4 ( ρ − n) −
∂H
= λ4 ( ρ − n ) − λ4 ( r − n )
∂b
(9)
(10)
(11)
(12)
Equations (7)-(12), which are derived from the two first order conditions with respect to the
choice variables c and e p and the four equations for the co-state variables λ1 , λ2 , λ3 , and λ4 , give
us, together with the four state variable equations (2)-(5), a system of ten equations in ten variables
{c, e p , k , h, g , b, λ1 , λ2 , λ3 , λ4 } .
Writing eqns. (7) and (8) as
0 = λ1 − c −1
(13)
0 = λ1 − λα 1 − ηe −p1
(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*λ1* , λ*2 , λ*3 , λ*4 } . Plugging these into the production function
y = Ak α (u1h) β (v1 g )γ , 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 steady-state 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 i pf > 0 . Overall, from all of those comparative static studies we expect to obtain information
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
The parameter values used are: n = 0.015 , ρ = 0.03 , δ k = 0.075 , δ h = 0.075 , δ g = 0.05 , α = β = γ = 0.33 ,
f
u1 = 0.85 , v1 = 0.8 , v2 = 0.1 , v3 = 0.1 , ε1 = ε 2 = 0.2 , ε 3 = 0.25 , α1 = 0.1 , α 2 = 0.7 , α 3 = 0.1 , i p = 0.05 ,
16
-8-
*
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
k * = 86.59, h* = 21.97 , g * = 17.86 , b* = 20.20 , y * = 27.55 ,
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 long-run, 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 α 4 e p . More
concretely, the government has to stick to the inter-temporal budget constraint which can be written
as
b(0) = ∫ e
∞ −
∫
τ
0
(r ( μ )−n)dμ
s (τ )dτ ↔ lim e
t →∞
−
∫ (r (τ )−n)dτ b(t ) = 0,
t
0
(15)
where s denotes the primary surplus which, in our model, is given by α 4 e p . The first term in
0
(15) states that per-capita 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 inter-temporal budget constraint is met, provided the interest rate exceeds the population
growth rate on average, because per-capita debt converges to a finite value. Nevertheless, it is
A = 1 , η = 0.1 , r = 0.07 .
17
The Mathematica code is available at www.newschool.edu/gf/cem.
Note 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.
19
Allowing for additional borrowing by adding a constant in eqns. (4) and (5) and setting this borrowing at the same
*
*
*
b
size as foreign aid, i p = 0.05 , we obtain the following numerical values: k = 88.67 , h = 22.36 , g = 18.43 ,
18
*
y = 28.21 ,
b
*
= 0.67
y
. As one can observe, b and
b
fall, indicating a fall in permissible debt.
y
20
Since 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).
21
f
Again, note that in case of net borrowing a constant needs to be added to eqn. ((5)) and the foreign aid parameter i p
in eqn. ((4)) is supposed to be corrected for the availability of new investment funds for public capital.
In Greiner et al. (2005), ch. 6, more particular model versions are developed with government borrowing for specific
government expenditures.
22
-9-
important for real world economies to check for rules whether fiscal policies of countries are such
that they fulfill the inter-temporal 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 debt-GDP 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
b(t )
s (t )
,
(16)
= α + β (t )
y (t )
y (t )
with y (t ) per-capita GDP. β (t ) determines how strongly the primary surplus reacts to
changes in the public debt-GDP ratio and α , 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.
α can also be seen as representing other constant variables which affect the primary surplus-GDP
ratio.
Using equation ((16)), the differential equation describing the evolution of public debt can be
rewritten as
(17)
b&(t ) = (r (t ) − n)b(t ) − s (t ) = (r (t ) − n − β (t ))b(t ) − α y (t ).
∫ (r (τ )−n)dτ
e
−
t
0
Solving this differential equation and multiplying both sides by
to get the
present value of public debt yields
t
t
τ
⎛
− ( r (τ )−n) dτ
− β (τ ) dτ
( β ( μ )−r ( μ )+ n+γ ( μ ))dμ ⎞⎟
t
⎜
∫
∫
∫
0
0
0
(18)
e
b(t ) = e
dτ ⎟,
⎜ b(0) − α y (0) ∫0e
⎟
⎜
⎠
⎝
with γ the growth rate of GDP. Equation (18) shows that a positive value for β on average,
so that
∫ β (τ )dτ
t
converges to plus infinity asymptotically, is necessary for sustainability. It is also
sufficient, if r ( μ ) − n − γ ( μ ) is positive on average.24
It should be noted that a positive value for r − n − γ 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.
0
23
Note 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.
24
∫ β (τ )dτ
This holds because l'H ô pital gives the limit of the second term in (18) multiplied by e
−
e
−
∫ (r (μ )−n−γ (μ ))dμ /β .
t
0
t
0
-10-
as
4
Exploring the Basic Structure of the Model
We want to explore the effect of foreign aid, i fp , the productivity factor, A , and the fraction of
human capital used in market goods production, u1 , on the steady-state 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 low-income, lower-middle-income, and upper-middle-income
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, i pf .26 By looking at actual data of foreign aid per capita, we are
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 above-mentioned 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, α1 , α 2 , α 3 , and α 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, i pf , vary during that period and
see how such a change affects the steady-state values k , h, g , c, e p , y , b , and U . Such an
y
exercise was conducted for the low-, lower-middle-, and upper-middle 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-, lower-middle-, and upper-middle 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 10-year average, and the other parameters constant.
f
4.1 Effects of Foreign Aid per Capita, i p
b
, and
y
and there is clearly a positive relationship between the level of foreign aid and
The comparative static results for all income groups show that k , h , g , c , e p , y ,
U are linear in i pf
all the variables, except for public resources per capita, e p , and debt per GDP,
25
26
27
b
, which show a
y
See Appendix A.4 for the complete list of countries.
f
Note that in the following we presume that there is no new borrowing, so that i p is solely foreign aid.
The Mathematica code is available at www.newschool.edu/gf/cem.
-11-
negative relationship with respect to foreign aid, i pf .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, e p . Such a negative relationship could be explained by the fact that public resource is a
result of a lump-sum 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, e p . 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 i pf increases. Actual data for the period 1994-2004 shows that the per capita foreign aid
y
went from 2 to 4. To conduct our comparative static exercises, we increased i pf by increments of
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 low-income 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 i pf and y , U , , and e p respectively for the low-income group.
y
Similar results are observed for the lower-middle and upper-middle income groups.30
Figure 1: Effects of Foreign Aid ipf on the Steady State Variables (Low-Income Group)
4.2
28
This 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.
29
Recent literature has studied the effect of foreign aid on per capita income in country by country studies (see Easterly
(2006)) or in cross-section studies (see Reddy and Minoiu (2006)). The latter study finds significant and large effects.
30
Note 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
, 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 , e p , y , ,
y
and U . Because the low-income group is our reference group, A is increased from 1 to 1.06, and
the effect on the steady-state variables is observed. As one might predict, results show that there is a
positive linear relationship between A and all the steady-state 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 low-income 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 low-income group. Similar results are observed for
the lower-middle and upper-middle income groups.
The comparative static results for all income groups show that k , h , g , c , e p , y ,
<<AU: IN FIG. 2, THE RIGHT-HAND GRAPH IS NOT SHOWING UP CORRECTLY:>>
Figure 2: Effects of the Productivity Factor A on the Steady State Variables (Low-Income 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 steady-state values corresponding to the different values of the parameter were
recorded. The comparative static results on u1 for all income groups show a hump-shape, non-linear
b
relationship, with respect to k , g , h , c , e p , y , , and U . With respect to the hump-shape
y
relationship, we can make the following observations: First, we observe a positive relationship
b
between u1 and k , g , c , e p , 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 , e p , y , , when u1 ≤ 0.3 and decreasing returns when
y
b
u1 ≥ 0.3 . As for U , and , we always observe decreasing returns for all values of u1 . Second, as to
y
human capital, h , we observe a positive relationship between 1 − u1 and h when 1 − u1 ≤ 0.3 , and a
negative relationship for 1 − u1 ≥ 0.3 . Finally, it was observed in our exercises that the nature, and
form of the hump-shape relationship depend on the choice of our parameters. For instance, we
chose ε 1 , ε 2 , and ε 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 low-income
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 1 − u1 and h , for the low-income group. Similar results are observed for the lowery
middle and upper-middle income groups.
Figure 3: Effect of the Fraction of Human Capital Used in Market Goods Production u1 on
the Steady State Variables (Low-Income Group)
The observation that u1 , has a non-linear, hump-shaped 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 hump-shaped form of the effect of 1 − u1 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-, lower-middle- and upper-middle-income countries is
-14-
retained.31 Here, again, only the results of the low-income 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, α1v1 , education, α1v2 , and health, α1v3 , public transfers, α 2 , public consumption,
α 3 and debt services, α 4 = (1 − α1 − α 2 − α 3 ) .32 Note that the values reflect the averages for all
countries of one category.
By plugging the obtained numerical values ( α1 = 0.4035 , α 2 = 0.0623 , α 3 = 0.4342 ) into the
model, setting ρ = 0.03 , δ k = 0.075 , δ h = 0.075 , δ g = 0.05 , α = β = γ = 0.33 , u1 = 0.85 ,
ε 1 = ε 2 = 0.2 , ε 3 = 0.25 , r = 0.08 , η = 0.1 , and using the values for A and i pf that were found in
the previous section ( A = 1 , n = 0.0195 i pf = 3.361 ), one can simulate the effect of changes in the
composition of public expenditure on the stationary states by letting v1 , v2 and v3 vary.33
In the following this is done for the low-income country group.34 Although all graphs are
presented in the 3-dimensional 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, α1v2 , to public investment in education, α1v3 , 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 hump-shaped, 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, e p , are qualitatively the same.
b
, differ in their
y
behavior. In Figure 12 one can see that human capital can be increased beyond its welfare- and
growth-maximizing 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 hump-shaped 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 .
Only the graphs for human capital, h , utility, U , and debt/income ratio,
31
See appendix A.4 for the complete list of countries.
See appendix A.3 for details.
33
Note that described the numerical values for ρ , δ k , δ h , δ g , α , β , γ , u1 , ε 1 , ε 2 , ε 3 ,
32
α1 , α 2 , α 3
and η remain
the same for the remainder of this section.
Similar results hold for the other two groups. Complete sets of graphs for the other groups are listed in Appendix A.6.
35
Note that v1 + v2 + v3 = 1 always has to hold which restricts the plane to a triangular space. By fixing v2 or v3 the
34
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 welfareand 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,
b
namely α 4 e p . If we allow for new debt issue, as discussed in section 3.1, the debt to output ratio,
y
is likely to increase.
Figure 4: Effects of Changes on Income y
Figure 7: Effects of Changes on Public
Capital g
Figure 5: Effects of Changes on Capital k
Figure 6: Effects of Changes on Public
Debt b
-16-
Figure 8: Effects of Changes on
Consumption c
Figure 9: Effects of Changes on Public
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
36
Testing Debt Sustainability36
As 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 inter-temporal 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 inter-temporal 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 present-value primary surpluses. In general terms, this requirement is
also often referred to as the no-Ponzi game condition. In a general form we can write this as
follows. Neglecting stochastic effects and assuming that the interest rate is constant, the intertemporal budget constraint can be written as
B (0) = ∫ e − rτ S p (τ ) dτ ,
∞
0
(19)
with r the constant interest rate, B(0) public debt at time zero and S p the primary surplus.38
Equivalent to equation (19) is the following equation
− rt
(20)
lim e B(t ) = 0,
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 co-integrated and (1 − λL) Stp
is stationary, with 0 ≤ λ < 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 inter-temporal
government constraint will be fulfilled, because the inter-temporal 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 debt-to-income 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 debt-to-GDP. If this series is
constant the inter-temporal 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
lim c1Y0 e
t →∞
(γ − r )t
= 0,
(21)
for γ < r , with γ > 0 the constant growth rate of GDP. The condition γ < r characterizes a
37
For a detailed discussion of sustainability tests see Greiner et al. (2005).
For further derivations see e.g. Blanchard and Fischer (1989), ch. 2.
39
See e.g. Hamilton and Flavin (1984), Kremers (1988), Wilcox (1989), and Trehan and Walsh (1991).
40
As to these tests applied to Germany see Greiner and Semmler (1999).
41
See e.g. Bohn (1995,1998).
38
-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 lowermiddle income countries of this study, this would have to be explored.
However, testing for stationarity of the debt-to-GDP 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 deficit-to-GDP ratio is a positive
linear function of the debt-to-GDP 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 debt-to-GDP ratio, this variable can be written as
⎛ B (t ) ⎞
S p (t ) T (t ) − G (t )
⎟⎟,
(22)
= α + β ⎜⎜
=
Y (t )
Y (t )
⎝ Y (t ) ⎠
with T (t ) tax revenue at t , G (t ) public spending at t , α and β are constants which can be
negative or positive. α is a systematic component which determines how the level of the primary
deficit reacts to variations in GDP. α can also be interpreted as other (constant) economic variables
which affect the surplus ratio.
The coefficient β 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
(23)
B& (t ) = r B(t ) + G (t ) − T (t ) = (r − β ) B(t ) − α Y (t ).
Solving this equation we get
⎛
⎞
α
⎟⎟Y (0)eγ t + e ( r − β ) t C1 ,
(24)
B (t ) = ⎜⎜
−
−
r
γ
β
⎝
⎠
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)α/(r − γ − β ) . Multiplying both sides of (24) by e − rt leads to
⎛
α
e −rt B(t ) = ⎜⎜
⎝ r −γ − β
⎞
⎟⎟Y (0)e (γ − r ) t + e − β t C1.
⎠
(25)
The first term on the right hand side in (25) converges to zero in dynamically efficient
economies and the second term converges for β > 0 and diverges for β < 0. These considerations
show that β > 0 guarantees that the inter-temporal budget constraint of the government holds.
Further, with equation (22) the debt-to-GDP ratio evolves according to
42
For details of such conditions see Abel et al. (1989).
In dynamically inefficient economies the government budget constraint is irrelevant because in that case the
government can play a Ponzi game.
43
-19-
⎛ B& Y& ⎞
b& = b⎜⎜ − ⎟⎟ = b(r − β − γ ) − α .
⎝B Y⎠
Solving this differential equation, we obtain the debt-to-GDP ratio b as
(26)
α
+ e ( r −γ − β ) t C 2 ,
(27)
(r − β − γ )
where C2 is a constant given by C2 = b(0) − α/(r − β − γ ), with b(0) ≡ B(0)/Y (0) the debtb(t ) =
GDP ratio at time t = 0.
Equation (27) shows that the debt-to-GDP ratio remains bounded if r − γ − β < 0 holds. This
shows that a positive β does not assure boundedness of the debt-GDP ratio although the intertemporal budget constraint of the government is fulfilled in this case. Only if β 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 debt-to-GDP 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, β 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 long-run the debt-to-GDP ratio must
be constant, although it may well rise transitorily. However, it is ultimately an empirical question
what the country specific β 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:
(28)
sty = β t bty + α Τ Z t + ε t ,
where sty and bty are the primary surplus to GDP and the debt-to-GDP ratio, respectively. Z t
is a vector which consists of the number 1 and of other factors related to the primary surplus and ε t
is an error term which is i.i.d. N (0, σ 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 Z t 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, low-income 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.
44
This 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 long-run 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 long-run 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 “crowding-in” 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 trade-off. 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 growth-maximizing
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
45
Of 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 two-thirds 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 debt-to-income 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 growth-maximizing 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 high-quality 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 country-specific
studies.
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-24-
Appendix
The appendix begins with the out-of-steady-state 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 Out-of-Steady-State 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, e p , k , h, g , λ1 , λ2 , λ3 } have been computed, the
dynamic programming algorithm developed by Gruene and Semmler (2004) can be used to study
the out-of-steady-state dynamics not only of consumption, c , and the use of resources by the
public, e p , 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 out-of-steady-state 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 e p 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 e p 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 e p 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 twodimensional 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 e p .
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.
46
The out of steady state dynamics are analyzed for a simplified version of the model which omits public debt.
Note that this simplified version omits the debt equation (5) of the model.
48
There is, however, a second steady state which was neglected because of negative values of some state variables.
47
-25-
Figure 13: Out of Steady State Dynamics
As one can observe in Figure 13, the correctly taken optimal decisions c and e p 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 2dim 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 Variable c in the k − h space
If one stores the decision variables c and e p at each grid point of the 3-dim cube of k , h and
g one can then plot them in a 3-dim 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 Variable e p in the k − h space
On the other hand as Figure 15 shows the decision variable e p is not so monotonically
dependent on k and h . In some regions e p 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, e p , on private
capital, k , and public capital, g .
Figure 16: Decision Variable e p in the
k − g space
Figure 16 shows how the decision variable e p behaves in the k − g space. We can observe
that e p will be high above the steady state, for g > g * , and low below the steady state, for g < g * .
The decision variable e p 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 e p . For both the lower constraint is 7 and the upper is 8 so that all optimal c and
e p 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
steady-state 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 steady-state 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 e p 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, e p , 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, e p 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&
λ&
= − 1 → c& = c Aαk α −1 (u1h) β (v1 g )γ − ρ − δ k
(29)
λ1
c
(
)
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, v1α1e p , consists of “Economic affairs” (704),
“Environmental protection” (705) and “Housing and community amenities” (706). Public
expenditure on health, v2α1e p , was taken from “Health” (707). Public expenditure on education,
v3α1e p , consists of “Education” (709). “Recreation, culture, and religion” (708) and “Social
-28-
protection” (710) was subsumed under public transfers, α 2 e p . Public consumption and debt
payments, α 3e p and α 4 e p , 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 α 4 = 0.10 in a rough approximation and set the accordingly decreased above sum equal to
α3 .
We follow the IMF Development Committee (2006) in its classification of countries in low,
lower-middle and upper-middle 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
Low-Income
Bhutan
Burundi
Ethiopia
India
Moldova
Myanmar
Pakistan
Yemen
Zambia
Lower-Middle Income
Belarus
Bolivia
Bulgaria
Dominican Republic
Egypt
El Salvador
Iran
Jamaica
Maldives
Morocco
Philippines
Romania
Sri Lanka
Thailand
Tunisia
Upper-Middle Income
Argentina
Croatia
Czech Republic
Estonia
Hungary
Lithuania
Mauritius
Mongolia
Panama
Poland
Uruguay
A.5 Appendix to Section 4
f
Table 4: Effect of Foreign Aid, i p , on the Steady-State Variables (Low-Income Group)
i fp
ep
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
1539.3
1539.1
1538.9
1538.8
1538.6
1538.4
1538.2
1538.0
1537.8
1537.6
1537.5
c
890.4
891.0
891.6
892.1
892.7
893.2
893.8
894.4
894.9
895.5
896.0
k
10848
10850
10851
10853
10855
10856
10858
10860
10861
10863
10865
h
1152.6
1152.8
1152.9
1153.0
1153.2
1153.3
1153.4
1153.6
1153.7
1153.8
1153.9
-29-
g
9005
9006
9008
9010
9012
9014
9015
9017
9019
9021
9023
b
2531.8
2531.5
2531.2
2530.8
2530.5
2530.2
2529.9
2529.6
2529.3
25290.
2528.7
y
3451.6
3452.2
3452.7
3453.2
3453.8
3454.3
3454.8
3455.4
3455.9
3456.4
3457.0
U
89.70
89.71
89.72
89.73
89.73
89.74
89.75
89.76
89.80
89.77
89.78
Table 5: Effect of the Productivity Factor, A , on the Steady-State Variables (Low-Income Group)
A
1.000
1.005
1.010
1.015
1.020
1.025
1.030
1.035
1.040
1.045
1.050
1.055
1.060
ep
c
1538
1589
1641
1694
1749
1805
1863
1923
1984
2047
2111
2177
2245
894
923
953
984
1015
1048
1081
1115
1150
1186
1223
1261
1300
k
10859
11214
11580
11955
12341
12737
13144
13562
13992
14432
14885
15349
15825
h
1153
1175
1196
1218
1240
1262
1284
1307
1330
1354
1377
1401
1425
g
9016
9311
9614
9926
10246
10575
10913
11259
11615
11981
12356
12742
13137
b
y
2530
2613
2699
2786
2877
2969
3065
3163
3263
3366
3472
3581
3692
3455
3568
3684
3804
3927
4053
4182
4315
4452
4592
4736
4884
5035
U
89.8
90.2
90.6
91.1
91.5
91.9
92.3
92.8
93.2
93.6
94.0
94.4
94.9
Table 6: Effect of the Fraction of Human Capital Used in Market Goods Production u1
on the Steady-State Variables (Low-Income Group)
u1
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
ep
9
39
93
168
263
376
503
642
789
940
1090
1233
1363
1471
1547
1576
1538
1399
1084
c
15
33
64
107
162
227
300
380
464
551
637
719
793
855
899
916
894
814
633
k
104
321
700
1230
1899
2691
3586
4563
5597
6657
7710
8716
9628
10386
10918
11123
10859
9882
7665
h
g
b
y
144
253
383
517
648
774
893
1002
1100
1184
1252
1303
1332
1338
1315
1256
1153
989
721
98
274
588
1028
1582
2239
2982
3793
4650
5530
6404
7239
7995
8624
9065
9236
9016
8206
6366
14
64
152
276
433
618
828
1056
1298
1547
1793
2028
2242
2419
2544
2592
2530
2301
1782
33
102
223
391
604
856
1141
1452
1781
2118
2453
2773
3063
3305
3474
3539
3455
3144
2439
U
32.4
44.4
53.7
60.8
66.4
71.0
74.9
78.1
80.8
83.1
85.1
86.8
88.1
89.2
89.8
90.1
89.8
88.5
85.0
f
Table 7: Effects of Foreign Aid, i p , on Steady-State Variables (Lower-Middle-Income Group)
i fp
3.00
3.20
3.40
3.60
3.80
4.00
4.20
4.40
4.60
4.80
5.00
ep
5824.8
5824.7
5824.6
5824.5
5824.3
5824.2
5824.1
5823.9
5823.8
5823.7
5823.6
c
4659.7
4660.3
4660.9
4661.4
4662.0
4662.6
4663.1
4663.7
4664.3
4664.8
4665.4
k
45470
45472
45474
45476
45478
45480
45481
45483
45485
45487
45489
h
2742.7
2742.8
2742.9
2743.0
2743.0
2743.1
2743.2
2743.3
2743.4
2743.5
2743.5
-30-
g
42050
42052
42055
42057
42059
42061
42064
42066
42068
42071
42073
b
8642.2
8642.0
8641.8
8641.6
8641.4
8641.3
8641.1
8640.9
8640.7
8640.5
8640.3
y
14467.8
14468.4
14469.0
14469.6
14470.2
14470.8
14471.4
14472.0
14472.6
14473.2
14473.8
U
104.38
104.38
104.38
104.38
104.38
104.38
104.38
104.38
104.40
104.39
104.39
Table 8: Effects of the Productivity Factor, A , on the Steady-State Variables (Lower-Middle-Income Group)
A
1.180
1.185
1.190
1.195
1.200
1.205
1.210
1.215
1.220
1.225
1.230
1.235
1.240
ep
c
5824
5986
6152
6321
6494
6672
6853
7039
7229
7423
7621
7824
8032
4663
4792
4924
5059
5198
5339
5484
5632
5784
5939
6097
6259
6425
k
45481
46742
48034
49355
50707
52091
53506
54953
56434
57948
59496
61079
62698
h
2743
2786
2829
2872
2916
2961
3006
3051
3097
3144
3191
3238
3286
g
42063
43229
44423
45645
46895
48174
49483
50821
52190
53590
55021
56485
57982
b
8641
8881
9127
9378
9636
9899
10168
10443
10725
11013
11308
11609
11917
y
14471
14873
15283
15704
16134
16574
17025
17485
17956
18438
18931
19434
19949
U
104.4
104.7
105.1
105.4
105.7
106.1
106.4
106.8
107.1
107.4
107.8
108.1
108.4
Table 9: Effect of u1 , the Fraction of Human Capital Used in Market Goods Production
on the Steady-State Variables (Lower-Middle-Income Group)
u1
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
ep
c
35
154
359
644
1004
1430
1912
2437
2993
3564
4130
4672
5162
5570
5856
5967
5824
5299
4106
41
137
300
528
815
1155
1540
1959
2403
2859
3311
3743
4134
4460
4688
4777
4663
4243
3291
k
328
1260
2856
5082
7889
11211
14968
19068
23404
27853
32272
36492
40316
43498
45728
46591
45481
41383
32082
h
278
574
894
1217
1532
1835
2119
2380
2613
2813
2977
3097
3168
3181
3126
2988
2743
2351
1713
g
316
1176
2651
4710
7305
10376
13850
17641
21651
25764
29849
33752
37287
40229
42292
43089
42063
38274
29674
b
52
229
532
956
1490
2122
2836
3616
4441
5288
6128
6931
7659
8264
8688
8852
8641
7862
6092
y
104
401
909
1617
2510
3567
4763
6067
7447
8862
10268
11611
12828
13840
14550
14824
14471
13167
10208
U
44.3
59.8
69.8
76.9
82.4
86.8
90.4
93.5
96.0
98.2
100.1
101.6
102.9
103.8
104.5
104.7
104.4
103.2
100.0
f
Table 10: Effects of Foreign Aid, i p , on Steady-State Variables (Upper-Middle-Income Group)
i fp
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
4.8
5.0
ep
c
10068.2
10068.1
10068.1
10068.0
10067.9
10067.8
10067.7
10067.6
10067.5
10067.4
10067.4
11326.4
11327.3
11328.1
11328.9
11329.8
11330.6
11331.4
11332.3
11333.1
11333.9
11334.8
k
89637
89640
89643
89646
89649
89652
89656
89659
89662
89665
89668
h
3597.5
3597.6
3597.7
3597.8
3597.9
3598.0
3598.0
3598.1
3598.2
3598.3
3598.4
-31-
g
68113
68116
68119
68122
68125
68128
68131
68134
68137
68140
68143
b
y
13335.4
13335.3
13335.2
13335.1
13334.9
13334.8
13334.7
13334.6
13334.5
13334.4
13334.3
28520.8
28521.8
28522.8
28523.8
28524.8
28525.8
28526.8
28527.7
28528.7
28529.7
28530.7
U
106.26
106.26
106.27
106.27
106.27
106.27
106.27
106.27
106.30
106.27
106.27
Table 11: Effects of the Productivity Factor, A , on the Steady-State Variables (Upper-Middle-Income Group)
ep
A
1.450
1.455
1.460
1.465
1.470
1.475
1.480
1.485
1.490
1.495
1.500
1.505
1.510
c
10068
10295
10526
10762
11002
11247
11496
11750
12009
12272
12541
12814
13092
k
11331
11586
11846
12111
12381
12656
12936
13221
13512
13808
14110
14417
14729
g
h
89654
91674
93733
95830
97967
100145
102363
104622
106924
109268
111656
114088
116564
3598
3643
3689
3735
3782
3829
3876
3924
3973
4021
4071
4120
4170
y
b
68130
69664
71228
72822
74446
76100
77785
79502
81251
83032
84846
86693
88574
13335
13636
13942
14254
14572
14897
15227
15563
15906
16255
16610
16972
17341
U
28526
29169
29824
30491
31171
31864
32570
33289
34021
34767
35527
36301
37089
106.3
106.5
106.8
107.0
107.3
107.5
107.8
108.0
108.3
108.5
108.8
109.0
109.3
Table 12: Effect of u1 , the Fraction of Human Capital Used in Market Goods Production
on the Steady-State Variables (Upper-Middle-Income Group)
u1
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
ep
62
269
623
1116
1738
2474
3307
4215
5176
6162
7141
8076
8923
9629
10123
10314
10068
9160
7099
c
87
319
717
1271
1970
2797
3732
4753
5833
6940
8041
9091
10043
10836
11391
11606
11329
10309
7993
k
623
2465
5610
10000
15534
22083
29491
37575
46124
54894
63606
71927
79465
85739
90136
91837
89648
81569
63232
g
h
355
749
1170
1594
2008
2405
2778
3120
3426
3690
3904
4062
4155
4173
4100
3919
3598
3083
2246
488
1886
4276
7611
11815
16791
22419
28561
35056
41719
48338
54660
60387
65153
68494
69787
68123
61985
48054
-32-
b
82
356
825
1478
2302
3277
4380
5583
6856
8161
9458
10697
11819
12753
13408
13661
13335
12132
9402
y
198
784
1785
3182
4943
7027
9383
11956
14676
17466
20238
22886
25284
27280
28680
29221
28524
25954
20119
U
49.9
65.1
74.4
81.1
86.1
90.1
93.5
96.3
98.6
100.6
102.3
103.7
104.9
105.8
106.3
106.5
106.3
105.2
102.2
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 lower-middle- and upper-middle-income
cases is limited to graphs. As one can see, these graphs show the same qualitative effects as the
low-income 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 Lower-Middle-Income Case
-33-
Figure 18: Effects of Expenditure Composition Changes on the
per capita Parameters in the Upper-Middle-Income Case
-34-