Download On National Fiscal Policy and Growth

On National Fiscal Policy and Growth: Searching for Optimality
under Externality
ABSTRACT
In this paper, we examine the view of capital fundamentalism claiming that national fiscal policies,
with public investment being subject to adjustment costs, can be considered as the primary determinant
of economic growth. According to our analysis, a country that experiences a low rate of growth with a
relatively low public to private capital ratio can generate and attain a higher long-run rate of economic
growth, equivalent to the growth rate of public capital. It is revealed that the after-tax marginal product
of capital, hence the rate of return, depends positively on the ratio of private to public capital,
something that sharply contradicts the results obtained in the rather traditional strand of research where
the rate of return was invariant with that particular ratio. We also reconsider some properties of optimal
fiscal policy and conclude that, in accordance to conventional priors, maximisation of the private-sector
utility function corresponds to maximisation of the growth rate of the economy.
1. Introduction
It is already well known from the relevant literature that models of economic
growth can generate long-run growth without relying on theories of population
change, as in Becker and Barro (1988), or exogenous changes in technological
progress, due to Romer (1986). A general feature of these models is the presence of
constant or increasing returns in the process of accumulating the factors of
production; Lucas (1988), and Romer (1989). While exogenous technological change
can be ruled out, such models can be viewed as equilibrium models of endogenous
technological change in which long-run growth is primarily motivated by the
accumulation of knowledge by forward-looking and profit-maximising agents. In
contrast to models in which capital exhibits diminishing marginal productivity, the
stock of knowledge can endlessly grow. Even in a situation where all inputs of
production are held constant, there is no reason why knowledge must also be constant
at some steady state and, accordingly, no further research should be undertaken; Barro
(1990), and Angelopoulos et al. (2007). Apparently, it is the co-existence of three
elements, namely, increasing returns in the production of output, externalities, and
decreasing returns in the production of new knowledge that can produce a wellspecified competitive and/or equilibrium model of growth; Rodrik (2005).
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One strand of the literature on endogenous economic growth is concerned with
models in which private and social returns to investment diverge, so that decentralised
choices can lead to suboptimal rates of saving and economic growth; Arrow (1962),
and Acemoglou et al. (2003). In particular, private returns to scale may be
diminishing but social returns reflecting various externalities, such as spillovers of
knowledge, can be either constant or increasing. Another strand of the literature is
concerned with models without externalities, in which the privately determined
choices of saving and growth can be Pareto optimal; Rebelo (1991), and Gale and
Orszag (2003). These models rely on constant returns to private capital that is broadly
defined to encompass human and physical capital. Still, apart from displaying
constant returns to scale technology, these models generate steady-state growth paths,
thus being compatible with the stylised facts of economic growth, as described in
Kaldor (1961), and enhanced by Arestis (2007).
Unambiguously, one of the most interesting aspects of the recent revival of
growth theory is the focus on the long-run effects of economic policy as such are
reflected in the wide cross-country dispersion in average rates of growth. Therefore,
the role of public policy is central in generating long-run growth; Easterly (2005).
King and Rebelo (1993), among others, examine whether national fiscal policies
could explain the observed disparity in growth rates across countries by isolating the
effects of taxation on long-run growth and, yet, by assuming that government
expenditures do not affect private-sector preferences or production technologies.
Furthermore, public policy has the feature that either government spending or tax
rates are exogenous. There is also a large literature on tax-policy issues in the
neoclassical growth model concluding that high income-tax rates end up to lower
growth rates; Sato (1967), Feldstein (1974), Stiglitz (1978), Becker (1985), and
Easterly et al. (2004). However, in the neoclassical model such an effect can explain
the observed cross-country differences in growth rates only during the transition path
towards the steady state since it is established that the steady-state rate is given by the
rate of exogenous technical progress. According to the traditional theory of “capital
fundamentalism”, as surveyed by King and Levine (1994), national fiscal policies can
be considered as the main determinant of growth. In essence, it is argued that
investment rates are crucially important to economic growth when cross-section
estimates are under consideration and, further, that differences in growth rates across
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countries can be explained by differences in the process of capital accumulation;
Levine and Renelt (1992), Mankiw et al (1992), and Rodrik (2005).
Barro and Sala-i-Martin (1992, 1995) have developed a series of models, in
which investment in infrastructure affects output through the production function, as a
factor along with capital and labour, in order to study the influence of the supply of
public goods on growth rates. Clearly, the rate of output growth can be positively
related to the share of government purchases, in the form of public services, while
examining various policy implications under alternative schemes of the production
function. In a similar reasoning, Glomm and Ravikumar (1994) explore the
implications for capital accumulation when investment in infrastructure enters into the
private-production function as an external input, but with the contribution of
infrastructure
to
private-factor
productivity
being
subject
to
congestion.
Consequently, not only does government expenditure in the form of public investment
play a decisive role for the performance of the economy, while strengthening the
dynamic character of policy analysis, but also it provides a rationale for empirical
studies that establish a strong positive link between investment and output growth
rates; Aschauer (1989), Baxter and King (1993), Easterly and Rebelo (1993), Dollar
and Svensson (2000), and Bekaert et al. (2005).
In the present context, we introduce a simple open-economy model of
endogenous growth in which the production function, apart from labour, consolidates
physical and human capital. Following Barro and Sala-i-Martin (1995), and
Alogoskoufis and Kalyvitis (1996), the formation of private capital is subject to costs
of adjustment so that, the economy’s total (private and public) capital ratio adjusts
gradually towards the steady state. Therefore, it remains to see whether, despite any
externalities dovetailed with the formation of private capital, an efficient use of
government spending in public infrastructure can solely determine the steady-state
growth of the economy. Only in this way can the production-enhancing role of
government expenditures be underscored, thereby suggesting that policy makers
should use public capital in a prudent and effective manner, pointing towards the
adoption of “functional finance”, suggested recently by Arestis and Sawyer (2010);
for a rather post-Keynesian variant, see Casares and McCallum (2000). Then, in a
rather closed-economy version, we extend the analysis by focusing on the issue of
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financing such expenditures, especially, when the government pursues a balanced
budget. Contrary to conventional priors, the after-tax marginal product of capital,
therefore the rate of return, depends positively on the ratio of private to public capital.
Finally, following Barro (1990), and motivated by Economides et al. (2007), but also
Arestis and Sawyer (2010), we examine whether such policies can be optimal
regarding both governmental and decentralised choices, in a framework characterised
by the familiar externalities implied by public expenditures and taxation. It is shown
that private-sector utility maximisation corresponds to maximisation of the economy’s
growth rate.
The organisational structure of the paper is the following. Section 2 introduces
the basic model, for a small country like Greece, in the presence of adjustment costs
on private capital. Section 3 discusses the implications of public investment in
infrastructure for the steady-state economic growth. Section 4 examines the issue of
financing government expenditure and its impact on both the decentralised economy
and the social planner, while exploring the optimality dimension of such policies.
Some concluding remarks are offered in the last section, with the stability analysis of
the system being displayed in the Appendix.
2. The Model
Assume a small open economy that embraces a large number of competitive
firms. Without loss of generality and aggregating across firms, the production
function may be given the following expression;
Y  AK a (hL) 1a
(1)
where: Y denotes output, K is the private-sector capital, and L stands for labour, with
α and 1-α being the shares of private capital and labour, respectively.
Parameter A reflects the constant technology level, with A>0.
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The assumption of constant returns becomes more plausible whenever, as in
our case, capital is broadly viewed to encompass both human and physical capital.
Indeed, parameter h represents human capital and we consider it to be a function of
the existing total (private and public) capital of the economy, denoted by K and G
respectively, so that;
hψ
where:
K β G 1β
L
(2)
ψ>0 stands for an efficiency parameter that captures the degree of the
economy’s efficiently used total capital. It becomes evident that, the
representative firm’s output is a function of its private capital and of the
economy’s total capital. However, human and physical capital need not be
perfect substitutes in production.
Therefore, production may exhibit roughly constant returns to scale in the two
types of capital taken together, but diminishing returns in either input separately. In
other words, even with a broad concept of private capital, production involves
decreasing returns to private inputs if the government inputs, acting as a complement,
expand in a different fashion. According to the conventional strands of the literature,
and assuming full depreciation of public capital, we can refer to government inputs
interchangeably as the flow of government purchases or the stock of accumulated
public capital. Alternatively, we could think of government spending as the quantity
of public services provided to all firms, as a non-rival and non-excludable good.
Obviously, the model abstracts from externalities associated with the use of public
services such as, various congestion effects which might arise for highways or some
other publicly provided services. Furthermore, in our setting, government expenditure
in the form of investment in infrastructure leaves the household utility unaltered.
The change in the capital stock of the representative firm is given by;
  I  δK
K
5
(3)
where: I clearly shows gross investment, and the dot expresses a derivative with
respect to time.
In the presence of adjustment costs during the formation of private capital, the
cost in units of output for each unit of investment is an increasing function of I in
relation to K, so that;
Cost of investment = I[1 
φ I
( )]
2 K
(4)
with φ > 0 standing for the sensitivity of the adjustment costs to the total amount
invested. It becomes evident that, the costs of adjustment depend on gross rather than
net investment.
Therefore, the infinite horizon problem of the representative firm is to
maximise the present discounted value of its net cash flow, that is output minus the
labour expenses and the cost of investment, taking h as given in order to have;
t
max  e  rt [Y  wL  [1 
0
φ I 
 ]I]dt
2K
(5)
subject to (1) and (3), with r being the world real interest rate assuming uncovered
parity, w is the real wage rate, and δ shows the rate of depreciation. Now, we can
analyse the optimisation problem by setting up the current value Hamiltonian as
follows;
φ I


H  e  rt Y  wL  [1  ( )]I  q(I  δK)
2 K


(6)
The maximisation entails the standard first-order conditions so that,
H/L  H/I  0 and q  H/ K , along with the usual transversality condition of
lim (qKe  rt )  0
t 
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(7)
Hence, the first-order conditions can be expressed as;
w  A(1  α)[
K α 1α
] h
L
(8)
 I  q 1
 
φ
K
(9)
K 
q  (r  δ)q  Aα  
L
α 1
h
1 a
φ I 
  
2K
2
(10)
If we now substitute (2) into (1), the modified aggregate production function
can be presented as;
K
Y  Aψ1 a  
G
β(1 α)
Κ α G 1 a
(11)
Substituting (9) into (10) while rearranging terms, we can get an expression for the
modified first-order conditions, so that;
w  A(1   )ψ
1-α
K 
 G 
 β(1 a )
G 1α
 I  q 1
 
φ
K
q  (r  δ)q  Aαψ
1 α
K 
 G 
(α 1)(1β)
(12)
(13)
(q  1) 2

2φ
(14)
Equation (12) is the usual equation of the marginal product of labour to the
wage rate, something that holds because there are no adjustment costs attached to
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changes in labour input. Equation (13) indicates that the relation between the shadow
value of private capital and private investment is monotonically increasing, in terms
of contemporaneous output, while the relation between the adjustment-cost parameter
and private investment is clearly negative. In addition, equation (13) states that the
shadow value of installed capital exceeds unity due to the presence of adjustments
costs. Equation (14) offers the change in the shadow value of capital as a positive
function of the market rate of return and the rate of depreciation of private capital,
minus the return on private capital along with the marginal reduction of the costs of
adjustment as private capital increases. Of course, if the costs of adjustment were
absent so that the shadow value of capital would equal one, the market rate of return
would be given by the difference between the rate of return on private capital and the
rate of depreciation.
3. Public Investment in Infrastructure
Let us consider now a situation where the government chooses a growth rate
of investment in infrastructure equal to π. It becomes obvious that the change in the
capital stock is now turned into;
  I  δK  πK
K
(15)
By defining the ratio of private to public capital K/G =κ, the equation of motion for κ
is given by;
κ 
I  δK  πK
G
whereas from equation (9), it can be shown as;
κ  κ
q 1  φ(δ  π)

κ
φ
φ
Accordingly, we can write equation (14) as;
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(16)
q  (r  δ)q  Aαψ 1-a κ (α 1)(1β) 
(q  1) 2
2φ
(17)
while the transversality condition in (7) can be now expressed in terms of the state
variable κ as;
lim (qκe (r π)t )  0
(18)
t 
which implies that if q and κ are constant in the steady state, the real interest rate must
exceed the growth rate of public investment in infrastructure.
Using equations (9) and (15) we obtain the steady-state value of q in terms of
the policy parameter π, so that;
q*  1  φ(δ  π)
(19)
while replacing (19) into (17) and rearranging terms, we obtain the steady-state value
of κ as follows;
1

 (1 α)(1β)
Ααψ1 α
κ*  
2 
 (r  δ)[1  φ(δ  π)  φ/2(δ  π) ] 
(20)
Of course, in order to have a positive steady-state value of κ the following must be
true;
(r  δ)[1  φ(δ  π)]  φ/2(δ  π)2
The inequality holds if π<δ+2r, which is always satisfied by the transversality
condition in (18). Clearly, equations (16) and (17) form a system of two differential
equations, the transitional dynamics of which can be characterised using a phase
diagram in terms of q and κ, as depicted in Figure A. Yet, we can better understand
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the relation between the two state variables by setting equation (17) equal to zero and
then differentiating q with respect to κ, so that;
(q  1) 2  2φ(r  δ)q  2φAαψ 1α κ (α 1)(1β)  0
(21)
dq
φAαψ 1α (1   )(1  β)κ (α 2)β(1α)

dκ
(q  1)  φ(r  δ)
(22)
The numerator is positive, and the denominator is negative if q<1+φ(r+δ), and
consequently, the q  0 locus is downward sloping. Obviously, the slope is positive if
q>1+φ(r+δ). Both inequalities hold due to the steady-state value of q, and the fact that
r>π. The system described in Figure A exhibits saddlepath stability, with the stable
arm being downward sloping throughout. In particular, if the economy begins at low
values of κ, then q>q* applies, implying that for a low private to public ratio the initial
value of q exceeds its corresponding steady-state value. That is to say, the high market
value of installed capital stimulates a great deal, but not an infinite amount, of
investment. Besides, such a relation can be verified by looking at equation (9). Over
time, the increase in private capital leads to decreases in q and, hence, to reductions in
i/κ until the steady-state growth rate of π+δ is finally reached. According to theory,
therefore, a poor economy with low initial levels of capital will experience high
values of installed capital, given higher costs of adjustment and high growth rates of
the capital stock.
Overall our analysis suggests that, a country that experiences a low rate of
economic growth can obtain higher growth rates by raising the long-run share of its
government expenditures in infrastructure. A higher level of public investment in
infrastructure induces a rise in the marginal product of capital, thus further stimulating
private-sector investment. This can be confirmed by recalling equation (17) where,
the marginal product of private capital appears to be a negative function of the ratio of
private to public capital. As a result, both private-sector capital accumulation and
investment unavoidably rise. Eventually, the steady state of the country will be
characterised by a lower ratio of private to public capital and a higher rate of
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economic growth that is equal to the growth rate of governmental spending in
infrastructure.
From a policy perspective, it is interesting to note that our economy apart from
being capable of generating endogenous growth, it also provides a fruitful insight
concerning the role of the efficiency parameter that measures the degree of use of
human capital. Indeed, under such an endogenously determined rate of economic
growth, our economy depends solely and exclusively on the growth rate of
government expenditures in infrastructure rather, than on the prudent or efficient use
of human capital in the steady state. However, we should not forget the productionenhancing role of the efficiency parameter during the transitional dynamics towards
the steady state, since a larger value of ψ, equivalently a more efficient use of
government expenditures in infrastructure, has been positively contributing to a
proportionate increase of the output level and, consequently, to the overall rate of
economic growth.
4. Government Financing and Pareto Optimality
In this section, we examine the issue of financing government expenditure
through taxation, along with its impact on the growth rates of the decentralised
economy and that of the social planner, respectively, and, yet, we explore the issue of
whether such policies can be Pareto optimal. In order to keep the analysis simple, we
neglect the existence of adjustment costs in the formation of private capital, as the
introduction of taxes entails by definition a distorting externality. Noticeably, in such
a closed-type version the government enters into the economy in the form of a
purchases flow of the quantity of public services, provided to all firms as a non-rival
and non-excludable good, but without causing any externalities brought about by
congestion effects which might arise, say for highways, or some other publicly
provided services.
The representative infinite-lived household seeks to maximise overall utility as
follows;
11

U   u(c)e ρt dt
0
where c is consumption, and ρ>0 is the constant rate of time preference.
The utility function is of the form;
u(θ) 
c1θ  1
1 θ
where θ>0 , so that the marginal utility can have a constant elasticity of –θ.
The familiar condition for consumption optimisation is given by;
γ c  (1/θ)(r  ρ)
(23)
and the analogous transversality condition as of;
lim  (κe  rt )  0
t 
clearly, γ stands for the growth rate of consumption at each point in time.
Accordingly, the representative firm’s cash flow or profit at any point in time
can be given by;
Profit  (1   )Y  wL  (r  δ)Κ
(24)
with w being the wage rate, and r+δ showing the rental rate of price.
Next, consider a government that wishes to run a balanced budget that is
financed by a proportional tax at rate τ on the aggregate of gross output, so that;
G  τΥ
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(25)
The production function of our economy is known to be;
Υ  Αψ1α κ β(1α)α G1a
(26)
which comes if we set κ=K/G in equation (11). Assume further that τ and, hence, the
gross ratio of public expenditure, G/Y, is constant over time.
Now the representative firm’s after-tax profit condition can be expressed as a
relation between the after-tax marginal product of labour and the after-tax marginal
product of capital. Obviously, profit maximisation and the zero-profit condition imply
that the wage rate must be equal to the rental price, in order to give;
r  δ  (1  τ)( Υ/ Κ) or equivalently;
r  δ  (1  τ)Αψ 1α G1a [β  α(β  1)]κ (β1)(1α)
(27)
Using (25) and (26), we can get an expression for G as of;
G  (τΑ) 1/α ψ (1α)/α κ β(1α)/α1
(28)
If we then use (28) and substitute for G in (27), we have the following, modified
rental price as;
r  δ  (1  τ)Αψ 1α [(τΑ) 1/α ψ (1α)/α κ β(1α)(α 1) ]1α [β  α(β  1)]κ (β 1)(1α)
(29)
It becomes evident that, the after-tax marginal product therefore the rate of
return r, depends positively on κ though at a decreasing rate, something that
contradicts the results obtained in the conventional strands of the literature where, the
rate of return r was invariant with κ. Moreover we see from equation (29) that, the
after-tax marginal product of capital plays the same role in the growth process with
the constant A in the AK model and, accordingly, with the constant private-sector
marginal product of capital in typical equilibrium models.
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Assuming absence of any transitional dynamics, since the model generates
endogenous growth the terms involving exponential power become asymptotically
negligible, the growth rates of consumption, capital and output respectively, c, κ and
y, all equal the same constant γ of equation (23). Hence, we can determine the growth
rate of the decentralised economy as given below;
γ DP 


1 1/α (1α)/α (1α)/α β(1α)/α
Α ψ
τ
κ
[β  α(β  1)](1  τ)  δ  ρ
θ
(30)
We observe in equation (30) two governmental effects on growth working in
opposite directions. First, we have the negative effect of taxation on the after-tax
marginal product of capital, captured by the 1-τ term. Second, there is a positive effect
of the efficiently used public expenditures, represented by the τ(1-α)/α term. Figure B
depicts the growth rate of the decentralised economy against the share of government
expenditures. In essence, at low values of τ the positive effect dominates and,
consequently, the economic growth rate rises with taxes. At relatively higher values of
τ the adverse impact of distorting taxation becomes more significant, until the rate of
economic growth reaches a peak. But for still higher values of τ, the negative effect
becomes dominant and, hence, the growth rate of the economy declines in accordance
with taxes.
Differentiating γ with respect to τ while setting the derivative to zero from
(30), we can obtain the maximum value of γ given by;
τ  G/Y  1  α
(31)
In order to interpret such a result, we need to calculate the marginal product of
government expenditures given in equation (26), so that;
Υ/ G  (1   )(Y/G)  (1   )/τ
Apparently, the result of (31) corresponds to the natural efficiency condition for the
size of the government, since ∂Y/∂G=1, thus proving the fact that the marginal cost is
equated to the marginal benefit.
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In this kind of models, a benevolent government would seek to maximise the
utility attained by the representative household in a first-best environment. However,
although the condition ∂Y/∂G=1 would be part of this utility maximisation problem,
such a condition might not necessarily hold in a second-best scenario in which taxes
were of distorting nature. Moreover, the utility maximisation might not correspond to
maximisation of the economic growth rate either. Nevertheless, given the structure of
our Cobb-Douglas production function, as we shall next see, maximisation of the
utility leads to maximisation of the growth rate, thus corresponding to the efficiency
condition ∂Y/∂G=1.
Let us now consider how the government would act in order to maximise the
utility attained by the representative household. Remember the utility form of the
representative infinite-lived household, at the beginning of this section. Further recall
that, the economy has no transitional dynamics and is always at a position of steadystate growth in which all quantities evolve at the rate γ shown in equation (30).
Having this in mind and using c(t)=c(0)eγt, while carrying out the integration, we can
evaluate in closed form the utility of the representative household as follows;
U
1  [c(0)]1θ
1
 

(1  θ)  ρ  γ(1  θ) ρ 
(32)
where, it is clear that the denominator should be positive and it is, given the familiar
transversality condition. Still, it can be seen that the values of both γ and c(0) exert a
favourable effect on U. As mentioned above, maximisation of U is equivalent to
maximisation of γ. But, as it becomes evident, a government may not always wish to
maximise the growth rate since such a development may come at expense of a lower
growth in consumption.
The initial level of consumption is given by;
C(0)=Y(0)-G(0)-I(0)
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where: G(0) = τY(0)
I(0) = (γ+δ) Κ(0), and
 +δK
I= K
Given a starting amount of capital κ(0), the levels of all variables are again
determined. Therefore, using these conditions and substituting in equations (26) and
(28), the initial level of consumption can be expressed as;


c(0)  κ(0) Α1/α ψ (1/α)/α τ (1α)/α κ β(1α)/α [β  α(β  1)](1  τ)  γ  δ
(33)
Note that equations (30) and (33) imply that, the initial quantity of consumption per
person can be also written as;
c(0)  κ(0)[ρ  γ(θ  1)]
(34)
So far, we have seen that equations (30) and (33) determine the values of γ and
c(0), respectively, as functions of τ=g/y. Yet, it has been stated that such formulas can
be used in order to also determine the governmental share in gross output that
maximises the representative utility. Hence, by substituting (34) into (32) we can
characterise the connection between the two variables, U and γ, in order to get;
U
1  κ(0)1θ [ρ  γ(θ  1)]1θ 1 
 

(1  θ) 
ρ  γ(1  θ)
ρ
(35)
Clearly, equation (35) reveals that U depends on γ but not separately on τ.
Since we already know that maximisation of γ entails τ=1-α, it is safe to argue that
such a condition also maximises the utility attained by the representative household.
Still, it is evident that U is monotonically increasing in γ, thus proving that
maximisation of U corresponds to maximisation of γ.
It would be then conceptually useful to examine whether the growth rate and
the outcomes, obtained by the decentralised choices of firms and households, can be
Pareto optimal considering the social planner’s problem. In particular, the social
16
planner satisfies the condition ∂Y/∂G=1 and, hence, G/Y=τ=1-α. We already know
that, such an efficiency condition of the public sector would apply in a first-best
environment and, yet, the social planner always attains the first-best solutions. We
also know that, the key distortion in our decentralised economy is that the private
sector is faced with a private marginal product of capital (1-τ)∂Y/∂Κ, which
unavoidably falls short of the corresponding marginal product of capital faced by the
public sector, due to the tax rate τ. Therefore, it is this wedge between private and
social returns responsible for reducing the growth rate γ, as shown as equation (30).
Something that can be found by replacing (1-τ)∂Y/∂G with ∂Y/∂Κ. In turn, the 1-τ
term is replaced by 1. If we further set G/Y=τ=1-α, the social-planner growth rate can
be expressed as;
γS 


1 1/α (1 α)/α
Α ψ
(1  α) (1 α)/α κ β(1 α)/α [β  α(β  1)]  δ  ρ
θ
(36)
On balance, it is possible to generate a rate of economic growth in such a
decentralised framework, thereby obtaining first-best outcomes. Obviously, the
government sets G/Y=τ=1-α in order to get quantity of public goods and, yet, finances
its expenditures with a tax of zero marginal rate concerning the production process.
Finally it would be natural to consider, as an artificial tax experiment, whether the
command optimum could still be implemented if we replaced the income tax with a
lump-sum tax, in the growth rate of the decentralised environment given in equation
(30). Under such a scenario, the private marginal return on capital is ∂Y/∂Κ, rather
than (1-τ)∂Y/∂K. Not surprisingly, but interestingly, the new marginal product of
capital, that the private sector deals with, corresponds to the marginal product of
capital previously adopted by the social planner. Consequently, the lump-sum
modified rate of consumption growth that optimising individuals would choose can be
expressed as;
γ DL 


1 1/α (1α)/α (1-α)/α β(1α)/α
Α ψ
τ
κ
[β  α(β  1)]  δ  ρ
θ
(37)
Obviously, the only difference between equations (37) and (30) is the absence
of the 1-τ term inside the brackets. However, in the operation of a lump-sum tax, the
17
private sector responds to the higher return on capital by choosing a proportionately
higher rate of consumption growth and, perhaps, a higher rate of savings. Overall, in
such a model, a consumption tax would be equivalent to a lump-sum tax, since issues
pertinent to labour-leisure choices have been intentionally ignored.
5. Concluding Remarks and Policy Lessons
One of the most interesting and, at the same time, contentious, for policy
debates, aspects of the recent revival of growth theory is the focus on the long-run
effects of macroeconomic policy, especially in a world of hypothetically high
integration, given the wide cross-country dispersion in average growth rates. In the
present analysis, we introduced a simple model of endogenous economic growth in
which, apart from labour, the production function consolidates physical and human
capital available to both private and public sectors. In our attempt to provide a
theoretical framework for endogenous growth in a small economy like Greece, the
emphasis has been placed upon two main issues. First, we examined the view of
capital fundamentalism claiming that, national policies on capital formation can be
considered as the primary determinant of growth. For that purpose, we studied the
impact of government expenditure in infrastructure on the rate of economic growth,
with the formation of private capital being subject to adjustment costs. Second, we
explored the implications of financing such a spending programme in public
investment, ideally, when a government wishes to run a balanced budget. In
adjunction to that, we investigated whether such policies can be Pareto optimal,
regarding the choices of both the decentralised economy and the social planner, in a
framework characterised by the familiar externalities that public expenditures and
taxation encompass.
With respect to the first issue, we showed that the growth rate of the economy
is positively related to, and solely determined by, the growth rate of government
expenditure in infrastructure at steady state, that is also associated with a lower ratio
of private to public capital. Yet, a higher level of investment in infrastructure induces
a rise in the shadow value of capital, hence an equal rise in the marginal product of
18
private capital that, in turn, brings about a higher level of private investment.
However, as a consequence of the costs of adjustment in the formation of private
capital, the economy does not immediately reach the new steady state. By contrast,
the steady state of the economy is gradually achieved until the growth rates of private
and public investment are eventually equalised.
With respect to the second issue of financing government spending through
taxation, we proceeded by analysing such policies in a rather closed-type economy
while checking the ensuing Pareto-optimality issues. Clearly, and by definition, the
introduction of taxes entails two effects on the growth rates of different-sized
governments. An increase in taxes usually reduces the growth rate of the economy,
but an increase in the share of government to output ratio induces a rise in the
marginal product of capital which, in turn, enhances the growth rate. Typically, the
latter effect dominates when the government is small while the former effect is
dominant in the case of a large government. Moreover, our analysis revealed that the
after-tax marginal product of capital, therefore the rate of return depends positively,
though at a decreasing rate, on the ratio of private to public capital, something which
is in sharp contrast with the results obtained in more traditional literature strands, in
which the rate of return was invariant with that particular ratio. Concerning the
optimality issues we gather that, as the government chooses to always obtain first-best
outcomes, the social planner finances the spending programme at zero marginal rates
as opposed to the decentralised economy of the private sector which is faced with a
positive tax rate, thus a higher marginal product of capital. In effect, we have shown
that maximisation of the private-sector utility corresponds to maximisation of the
growth rate of the economy, as a direct consequence of the specification of our CobbDouglas production function.
We wish to close up by reminding that we abstracted from issues relating to
public debt dynamics and its impact on private investment, or issues pertinent to time
inconsistency and governmental precommitment. However, for a formal and thorough
analysis of such policy targets, see, respectively, Saint-Paul (1992), Krichel and
Levine (1995), and Jafarey et al (1996). Notwithstanding such rather earlier literature,
it would be extremely useful to emphasise on the newer or modern lines of growth
literature capturing various technology shocks, enriched by a more active role of
19
monetary authorities, as in Gali et al. (2003), or idiosyncratic behaviour of
institutional nature, due to Acemoglou (2006), and overall policy governance
dimensions with financial applications, as in Canzoneri et al. (2002), Mukand and
Rodrik (2005), Kaufmann et al. (2007), and Malley et al. (2007), respectively. Indeed,
work along such research thematic would strengthen the role of macro-governance
aspects during the conduction of policy design and certainly enhance its actual
implementation. On balance, in attempting to probe the generality of our findings, we
hope that our “hybrid” in nature macro-modelling approach might prove appealing to
the Transition economies of the South, Central and Eastern parts of the continent,
entering the European Union, but, mainly, economically informative to theoretical
advances and, perhaps, technically conclusive to empirical inquiries.
Appendix
Stability Analysis of the Model
Consider the system of the two differential equations, (16) and (17), respectively. The particular
solutions of the specific system can be obtained, and best understood, by first setting up a generalised
version of its dynamic behaviour over time, such as;
q  q(q, k )
(i)
D
k  k (q, k )
(ii)
20
q
q
q
k
k
q
k
k
In order to have saddle-path stability, one of the two characteristic roots must be positive and the other
negative so that the determinant of the corresponding eigenvalues in the relevant matrix should yield;
D=(ad-bc)<0.
From equation (17), we can get the following expressions;
a).
b).
q
(q  1)
 (r   ) 
q

q
 Aa 1 (1   )(1   ) ( 2) (1 )
k
It can be seen, and thus verified, from equation (13) that both eigenvalues are positive since, at
equilibrium, q is greater than unity.
Accordingly, we can have from equation (16) the analogous relations;
c).
k 1

q 
d).
k
0
k
It becomes evident, that the third eigenvalue is positive while the fourth is, at equilibrium, always zero
which is a straightforward consequence of eq.(20).
Therefore, our matrix-eigenvalues determinant can be presented as follows;
D
a() b()
0
c() d (o)
Apparently, the model experiences a well-defined saddle-path stability.
21
22
23
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