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Endogenous growth, government debt and budgetary regimes
ALFRED GREINER University of Bielefeld Bielefeld, Germany
WILLI SEMMLER University of BieIefeld Bielefeld, Germany and New School University New York City, New York
Endogenous Growth, Government Debt and Budgetary Regimes* In this paper we analyze an endogenous growth model, in which sustained per capita growth results from investment in public capital arid the government is allowed to borrow from the capital market. As to the government behavior we do not suppose that governments optimize but instead stick to some well-defined budgetary regimes. The impact of a deficit financed increase in productive government spending is analyzed. It is shown that the growth effect of this fiscal policy crueially depends on the budgetary regime in use. In particular, a more strict budgetary regime, that is, an economy where the public defieit is primarily used for public investment, does not necessarily imply a lower growth rate. Simulations demonstrate that the growth maximizing ineome tax rate is about in the range of the elasticity of output with respeet to public capital.
1. Introduction In recent years, growth theory has received renewed attention with the rise of a new line of research that explains growth rates endogenously. Within this "new" growth theory, one approach posits that the government supplies productive services that increase the marginal product of private capital and, thus, positively influence economic growth. If the production function is linear homogenous in private and public capital jointly, and government spendings are endogenous, this approach yields an endogenous rate of growth. Arrow and Kurz (1970) were the first to introduce productive public capital in growth models, but in their model growth is still determined by exogenous factors. Barro (1990) takes up this approach and presents a *We thank two referees for detailed comments on an earlier version. This paper was written whde Alfred Greiner was visiting the Department of Economics of the Graduate Faculty of the New School for Social Research in New York. Its hospitality and generous provision of facilities, as well as generous financial support from the Deutsche Forschungsgemeingchaft (DFG) and the German Marshall Fund of the United States is gratefully acknowledged.
Journal of Macroeconomics, Summer 2000, Vol. 22, No. 3, pp. 363-384 Copyright © 2000 by Louisiana State University Press 0164-0704/2000/$1.50
363
Alfred Greiner and Wilh Semmter model with endogenously determined growth rates and a balaJaced budget. He, however, assumes that government spending as a flow variable enters the macroeconomic production function, whereas Arrow and Kurz suppose that the stock of public capital shows productive effects, Futagami, Morita and Shibata (1993) present an endogenous growth model where the stock of public capital has positive effects as concerns the marginal product of private capital thus leading to endogenous growth, The Arrow and Kurz version is also supported by an empirical study by Aschauer (1989) who showed that the stock of public capital is indeed more important than the flow of government spending. His study shows a value of 0.38-0.56 for the elasticity of output with respect to public capital. This, and therefore, Aschauer's study had been very controversial. For a survey of the criticism as well as other empirical studies investigating the link between output and public capital, see Munnell (1992), Gramhch (1994) or Sturm, Kuper and de Haan (1997). In this paper we take up the approach presented by Futagami, Morita and Shibata (1993) and analyze how fiscal policy affects the balanced growth rate of an economy. In contrast to their model, however, we also allow for a budget deficit of the government and take into account government debt. But the government decisions are supposed to be restricted by certain budgetary regimes t ° which a government must stick, These regimes are generally formulated in terms of the instruments (expenditure and tax regimes) or in terms of a well-defined target like the budget deficit for example or the size of public debt. Van Ewijk and van de Klundert (1993) analyze the impact of different budgetary regimes for the dynamics of growth and public debt in a conventional growth model. They consider three different regimes, which they name according to the economists who introduced them in the economics literature (van Ewijk and van de Klundert 1993, 123-124). A general result they derive is that the linkage of public spending for research and education to the budgetary position may be a destabilizing factor. Further, they find that the regime where the government keeps the budget deficit constant is less favorable as concerns productivity growth in comparison to a regime where the government may vary the budget deficit. However, the regimes they consider postulate that some variables which are endogenous in our setting remain constant over time so that they cannot be used within our model of endogenous and sustained per capita growth. Therefore, we define some new regimes taking into account sustained growth within which the impact of fiscal policy is investigated. The remainder of the paper is organized as follows. In the next section, we present the basic model and introduce different budgetary regimes, Section 3 gives analytical results and in Section 4, we present simulations to
364
Endogenous Growth, Government Debt and Budgetary Regimes illustrate the results of the analytical model and to derive some additional outcomes. Section 5, finally, concludes the paper.
2. The Basic Model We consider a closed economy which is composed of three sectors: the household sector, a representative firm and the government. The household is supposed to maximize the discounted stream of utilities arising from consumption subject to its budget constraint:
max c(t)
fO~ e-~Cu(C(t))dt
;
(1)
subject to
C(t) + S(t) = (w(t) + r(t)S(t))(t - r) + Tp(t).
(2)
The term C(t) represents private consumption at time t, S(t) = K(t) + B(t) denotes assets which comprise physical capital K(t) plus government bond or public debt B(t). Tp(t) stands for lumpsum transfer payments, the household takes as given in solving its optimization problem. The variable • gives the income tax rate and w(t) and r(t) denote the wage rate and the interest rate respectively. The variable P gives the constant rate of time preference and the labor supply is assumed to be constant and normalized to one so that all variables give per capita quantities. The depreciation rate of physical capital is set equal to zero. As to the utility function we assume a function of the form (C(t)l-~)/ (1 - a), with - a denoting the elasticity of marginal utility with respect to consumption or the negative of the inverse of the instantaneous elasticity of substitution between consumption at two points in time, which is assumed to be constant. For ~ = 1 the utility function is the logarithmic function In C(t). If the utility function is used to describe attitudes towards risk ~ has an alternative interpretation. In this case ~ is the coefficient of relative risk aversion. A number of empirical studies have been undertaken aiming at estimating this coefficient under the assumption that it is constant, by looking at the consumers' willingness to shift consumption across time in response to changes in interest rates. The estimates of ~ vary to a great degree but are usually at or above unity (see Blanchard and Fischer 1989, 44; Lucas 1990; or Hall 1988). Assuming that the rate of growth, g, of the endogenous variables is bounded by g < p it is shown in an appendix available on request that there
365
Alfred Greiner and Willi Semmler
exists a solution to this optimization problem. To find this solution we formulate the current value Hamiltonian which is given by, 1 H(') = u(c) + 7((w + rS)(1 - r) + Tp - C). The necessary optimality conditions are then obtained as 7
=
c -~
,
= 7(P -
(3)
(1 - T)r),
(4)
+ (rS + w)(1 -
= -C
z) + T p .
(5)
These conditions are also sufficient if the limiting transversality condition l i m t ~ e-etT(t) (K(t) + B(t)) = 0 is fulfilled. 2 Differentiating 7 = C - ° with respect to time and using 2> = 7(P - (1 - r)r), this system can be reduced to a two dimensional differential equation system: p -
C
S
-
(1 - ~)r +
~r
- - , ~r
C + (r + w / S ) ( 1 S
(6)
T -
z) +
-P.
S
(7)
The productive sector is assumed to be represented by one firm which behaves competitively exhibiting a production function of the form, f(K, G) = K 1-~ G ~ .
(S)
The variable G gives the stock of productive public capital which is a nonrival and non-excludable good 3 and 1 - a, ¢x E (001)0 denotes the share of private capital in the production function. Since K denotes per capita capital, the wage rate and the interest rate are determined as w =cLK 1- ~G ~ and r = ( 1 - a ) K - ~ G ~. The budget constraint of the government, finally, is given by/3 + T = rB + Cp + Tp + G. The variable Cp stands for public consumption and T is the tax revenue, which is given by T = r(w + rS). As to public con1In the following we omit the time argument if no ambiguity arises. ~This condition is automatically fulfilled ifg < p holds, with g the long run balanced growth rate. This also guarantees the boundedness of the utility functional. ZFor an analysis where the public good is non-excludable but rival or subject to congestion, see Barro and Sala-i-Martin (1992). In another version of this model we take account of congestion effects and population growth (cf. Greiner and Semmler 1999).
366
Endogenous Growth, Government Debt and Budgetary Regimes sumption we suppose that it is a pure drain on resources and provides no benefits to the economy as it is often assumed in the optimal tax literature. An alternative specification which would not change our results would be to assume that public consumption enters the utility function in an additively separable way (for a discussion and an economic justification of that assumption see Judd 1985, 301 or Turnovsky 1995, 405). We suppose that public consumption and transfer payments to the household constitutes a certain part of the tax revenue, i.e. Cp = (0zT and Tp = (01T, (01, (02 < 1. Moreover, the government is not allowed to play a Ponzi game, that is, fimt-~ B(t)e -Igr~*)d~ = 0 must hold, and we define four alternative budgetary regimes to which the government must stick. 4 The first states that government expenditures for public consumption, transfer payments and interest payments must be smaller than the tax revenue, Cp + Tp + rB = (00T, with (00 < 1. We will refer to this regime as regime (A). This regime is called the "Golden Rule of Public Finance" and postulates that it is not justified to run a deficit to finance non-productive expenditures which do not yield returns in the future, Public deficits then are only feasible to finance productive public investment which increases a public capital stock and raises aggregate production. 5 A slight modification of this regime, and in a way natural extension, is obtained when allowing that only a certain part of the interest payments on public debt must be financed out of the tax revenue and the remaining part may be paid by issuing new bonds. In this ease, the budgetary regime is described by Cp + Tp + (04rB = (POT,with (04 ~ (0, 1), We refer to this regime as regime (B). The third regime, regime (C), states that public consumption plus transfers to individuals must not exceed the tax revenue, but the government runs into debt in order to finance public investment and interest payments, Cp + Tp = (00T, (00 < 1. Further, in all regimes investment in infrastructure is supposed to constitute a certain part of the remaining tax revenue, G = (0a ' (1 - (00)T, (03 >-- 0. In regime (A) for example this implies that government debt only increases if (0a > 1. Table 1 gives a survey of the regimes. Let us in the next section take a closer look at our economy described by this basic model and derive some implications of those different regimes for the growth rate of economies.
4It should also be mentionedthat increases in governmentexpenditurewould lead to proportional decreasesin consumptionif the first did not influenceproductionpossibilitiesand if we had non-distortonarytaxes. This followsfrom the Ricardian equivalencetheorem (cf. Blanchard and Fischer 1989, ch. 2). 5Regime (A) can be found in the German constitutionand is bindingfor the government.It was also used to derivethe deficitcriterionin the treaty of Maastricht.
367
Alfred Greiner and Willi Semmler
TABLE 1. Regim e ...........Target
. . . . . . . . . . . . . . . . . . . . . . Deficit due t o
,
(A)
Cp + Tp + rB < T
public investment
(B)
Cp + Tp + (04rB < T
public investment + (1 - (04)rB
(c) cp + rp. + ~ > T, (Cp + Tp + G) + rB . . . . . . . . . . . % +. r , < r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Analytioal Results Before we analyze our economy we first state that regimes (A), (B) and (C) can be derived from regime (B) for appropriate values of (04 and (03. To see' that more clearly we write down the budgetary regime for regime (B) which is given by
(9)
Cp + Tp + ~4rB = (0oT.
Further, the debt accumulation equation for regime (B) is given by [3 = rB + Cp + Tp + G - T = ((% - 1)(i - ~oa)r + (1 - (04)rB.
(10)
Setting (04 = i in (9) and (10) and (0a > 1 it is immediately seen that we obtain regime (A). In that ease the whole interest payment on outstanding public debt must be financed by the tax revenue. If (04 E (0, 1) only (o4 of the interest payments is financed by the tax revenue and the rest, (1 - (04)rB, is paid by raising public debt, giving regime (B). In the extreme, i.e. for (04 = 0, all interest payments on outstanding debt are financed by issuing new debt and we now have regime (C), for (0a > i. In that case we get Cp + Tp + G - T = (No - 1)(1 - (03)T > 0, To find the differential equation describing the evolution of physical capital, that is the economy-wide resource constraint, we first note that the budget constraint of the individual gives [( + B = - C + (w + rK + rB)(1 - z) + .Tp. Using the definition of B the term K + / ) can be written as [( + B = K + rB + (0zT + .Tp + ~oa(1 - (00)r - T. Combining those two expressions then yields K = - C + ( w + r K ) - ((02+(03(1-(00)) z(w + rK + rB ). Using the equilibrium conditions w + rK = K 1- ~G ~ and r = (t - ot)G~K -~ then gives the economy-wide resource constraint as
K 368
-
K
+
-
~(~
+ ~(i-
~o))
] +
(i
-
~)~
.
(11)
Endogenous Growth, Government Debt and Budgetary Regimes
The description of our economy is completed by Equation (6) giving the evolution of consumption and by the differential equation describing the path of the stock of public capital G. Thus, we get the following four dimensional differen[ial equation systems with appropriate initial conditions and the limiting transversatity condition: K -
~-
K +
(~0p ~
-
C
-G- =
- r((o~ + pa(1 - ~Oo))
1 + (I - cx)~2 ;
1)(1 - (0a)z((t - ~) + ~ ) ( G ) ~
+
(12)
+ (1 - ~04)(1 - ~x)(G);
(13)
(1 - z)(1 - a ) K - ~ G '~ ....... ; a
-
+/1
(14)
\K]
-
GJ
(15)
w e write that system in the rates of growth which tend to zero if the decline in the marginal product of physieat capital, caused by a rising capital stock, is not made up for by an ~ncrease in public capital. However, if the stock of public capital is sufficiently large so that the marginal product of private capital does not converge to zero in the tong run we can observe endogenous growth. In this case the r.h.s, of, our system is always positive, and we first have to perform a change of the system to be able to continue our analysis. Defining c = C/K, b ~- B / K and x = G / K the new system of differentiat equations is given by d/c = C / C - K/K, l~/b = B / B - K/K, and ic/x = G / G - [(/K leading to
_ec = -~r + x~ ((i. -, ~)(i,~r -' ~)
+
r(0,2 +
~,a(1 -
0,o)).(t
+
(1 -
o,)b)), ÷
(16) --
XaT
(((1
--
~)
÷
b-1)(~90
--
1)(1 - foa) + ((02 + ~0a(I - ~00))
(1 + (1 - ~)b)) + x°(1 - ~)(1 - ~4) + c - x=; -
X
=
x ~-1
(I
+
(1
-
ot)b)(oa(1
-
(1 + (1 - ~)b) + c - x".
0,o)~ +
x~(~o~
+
(Oa(1 -
(17) ~Oo))
(is,)
A stationary point of this system then corresponds to a balanced growth path 369'
Alfred Greiner and Willi Semmler of the original system w h e r e all variables grow at the same rate. In the following we will confine our analytical analysis to this path and examine h o w fiscal policy financed t h r o u g h additional debt affects the growth rate on that p a t h P Moreover, we will exclude the economically meaningless stationary point c = x = b = 0 so that we can consider system (16)-(18) in the rates of growth and confine our analysis to an interior stationary point. Doing so we get for c from e/c = O,
C
P- o-
~
--
x ~ (r((02 + (0a(1
(0o))(1 + (1 -
¢~)b) - 1 + (1 - r)(la - a ) ) .
(19)
Further, we know that the constraint Cp + T v + (04"FB = (0oT must be fulfilled on the balanced growth path. This condition makes (00 an endogenous variable which d e p e n d s on b, (0 1, (02, (04, %and a. Using T = r(KI-~G a + rB), it is easily seen that in the steady state (00 is d e t e r m i n e d as (04(1 -- 0¢) (00 = ((01 + (02) + Z((1 -- Or) + b - l ) "
(20)
But it must be noted that (00 < 1 is i m p o s e d as an additional constraint, which must always be fulfilled. Inserting c from d/c = 0 and (00 = (00(b,(0b(02,(04,z',°t) in [9/b and ~/x and setting the 1.h.s. equal to zero completely describes the balanced growth path of our economy. The equations are obtained as 0 = q(') =- g/~ = x~ (z((1 - ca) + b-1)(~Oo( ) - 1)(1 - (0a) (1
0
:
ql(')
~
-
¢04)(1
-
(21)
~)~;
~- = X~-l(1 + (1 -- ec)b)¢%(1 - (%( . ))z - x~ x
(1-T~1-~)) + p__ +a
(1 - r)(1 - a) o"
+ ;--.
(22)
O"
Solving for b and x then determines the balanced growth path for our economy, on which all variables grow at the same constant rate. The ratio G/K = x denotes the ratio of public capital to private capital which determines the marginal p r o d u c t of private capital and, as a consequence, the balanced
SUnder a slight technical assumption it is shown in an appendix available on request that the no-Ponzi game condition is met for this path. 370
Endogenous Growth, Government Debt and Budgetary Regimes growth rate. The balanced growth rate is obtained from (14), which merely depends on x at the steady state and on the exogenously given parameters. In the following, we intend to investigate the impact of fiscal policy on the balanced growth rate of our economy. To do so we implicitly differentiate system (21)-(22) with respect to the parameter under consideration. This shows how the ratio G/K changes and, together with (14), gives the impact on the growth rate of our economy. Using D/b ~ q(x, b, ~o0;...), and yc/x =- ql(x, b, ~Oo;. . .), and implicitly differentiating (21)-(22) with respect to the parameters leads to
Ob/Oz] = - M - 1 [ Oq(')/Oz ] OxlazJ kaqt(-)lazj ' The variable z stands for the parameters, and the matrix M is given by
[ Oq(.)/Ob M = iOqt(.)/Ob
Oq(.)/Ox ] Oq~(.)/OxJ'
The negative of its inverse is calculated as
-M- I = _
1 det M
. [
Oql(.)/Ox [ - Oql(')/Ob
-Oq(')/Ox] Oq(')/Ob] '
with det M denoting the determinant of M. Now, we can analyze how the growth rate in our economy reacts to fiscal policy. A result which holds independent of the budgetary regime under consideration is that an increase in public consumption and transfer payments reduces the balanced growth rate. That result is not too surprising. Therefore, we only mention that outcome but do not go into the details (a formal proof can be found in Greiner 1996, chap. 4.2). A more interesting topic is the question of how a deficit financed increase in public investment and how the choice of the budgetary regime affect economic growth. To begin with, we consider regimes (A) and (B).
Regimes (A) and (B) First, we study regime (A), which is obtained from (21)-(22) by setting ~04 = 1 and (03 > 1. The question we want to answer is whether a rise in public investment financed through additional government debt increases economic growth if we take into account feedback effects of the higher level of public debt. To investigate this case we note that it can be represented by an increase in the parameter (0a. On the one hand, a higher (0a means 371
Alfred Greiner and Willi Semmler more direct investment in public capital but, on the other hand, it also changes the ratio of b, which determines (J/G both directly and indirectly through (%. Proposition 1 gives the condition which must be fulfilled for a positive growth effect of a deficit financed increase in public investment. PROPOSITION 1. A deficit financed increase of investment in public capital raises (reduces) the balanced growth t'ate in regime (A) if Z =- z ( 1 - (00)(1 + (1 - ot)b)[(0a + ((03 - 1)/(b(1 - or))] > (<)1.
Proof
First we note that under a slight additional technical assumption it can be shown that in regime (A) det M > 0 holds. 7 To prove this proposition we derive from (14), 0g
0(03
-- (1 -- T)(1
a
-- O~)O~xa_l
~._.~_._. 0(0a
Implicitly differentiating (21) and (22) again gives an expression for the change in x at the steady state. Since - det M <0, x, and thus the balanced growth rate, rises if -(Oql(')/Ob)(Oq(.)/O(0a) + (Oq(.)/Ob)(Oql(.)/O~oa) < 0 and vice versa. The sign of that expression is negative (positive) if 1 - z(1 - (0o)(1 +(1 - a)b)[~0a + (~03 - 1)/(b(1 - a))]--- - Z + 1 < (>)0. Thus, proposition 1 is proved. It should be noted that it is the feedback effect of government debt that exerts a negative influence on the growth rate if investment in infrastructure is financed through additional debt. This is seen by setting O(0o/Ob = 0. The expression (Oq(.)/Ob)(Oql(.)/O(0a) - (Oql(.)/Ob)(Oq(.)/O~oa) then is always negative and a defcit financed rise in investment in infrastructure raises x and the balanced growth rate. In that case our result would be equivalent to the one derived by Turnovsky (1995, 418). The introduction of budgetary regimes, however, implies that a deficit financed increase in public investment shows a feedback effect which acts through two channels: first, an increase in public debt raises interest payments which must be financed by the tax revenue and consequently reduces the resources available for public investment. Here, we may speak in a way of internal crowding out (as in van Ewijk and van de Klundert 1993, 123). But that effect only holds for regimes (A) and (B). Second, the introduction of the budgetary regimes implies that the interest payments on public debt appear in the economy wide resource constraint (11) and lead to a (external) crowding out of private investment. That effect holds for all three regimes. 7The derivationas well as the detailed proof is containedin an appendixwhich is available on request. 372
Endogenous Growth, Government Debt and Budgetary Regimes
Since governments are not allowed to increase public debt arbitrarily because public debt does have effects (see e.g., Easterly and Rebelo 1993 or Fischer 1993), it is apparent that the introduction of budgetary regimes to model negative feedback effects of higher government debt is justified. That is also the reason why budgetary regimes have frequently been introduced in the economics literature (for a detailed survey of budgetary regimes see van Ewijk 1991). As to the effect concerning the ratio of public debt to private capital, b, no unambiguous result can be derived; that is, dais ratio may rise or fall. Therefore, we will present a numerical example in Section 4 in order to gain further insight into that problem. It can also be shown that, in economies with a high share of nonproductive government spending, like high interest payments, public consumption and/or transfer payments, that is, a large value for (00, a deficit financed rise in public investment is more likely to produce negative growth effects. This is immediately seen by differentiating Z with respect to ~00, which is negative. Another point which we should like to treat is how the tax rate must be chosen to maximize economic growth. Since a certain part of the tax revenue is used for investment in public capital which raises the balanced growth rate, it becomes immediately clear that the growth maximizing income tax rate does not equal zero. Thus, our result is consistent with the one derived by Barro (1990) and similar to the outcome observed by Jones, Manuelli and Rossi (1993), who demonstrate that the optimal income tax rate does not equal zero if government spending has a direct positive effect on private investment. But this outcome clearly is in contrast to the result of the standard Ramsey type growth model and also to endogenous growth models with physical and human capital (see e.g. Lucas 1990, or MilesiFerretti and Roubini 1998). The growth maximizing income tax rate is computed by differentiating (11) with respect to r. This leads to Og _ 1 - OL x~ Or cr
[
-1
+ ~ r
Or
1
This shows that the increase in the income tax rate raises or lowers the balanced growth rate if the elasticity of G/K with respect to r is larger or smaller than the expression z/((l - r)ot). The economic mechanism behind this result becomes immediately clear. For given parameter values an economy with a high elasticity of G/K with respect to r is more likely to experience growth if the income tax rate is increased because the higher investment in public capital, caused by more tax revenue, leads to a relatively strong increase in the ratio G/K and, thus, in the marginal product of private physical 373
Alfred Greiner and WiUi Semmler capital. This positive effect then dominates the negative direct one of higher income taxes leading to a reallocation of private resources from investment to consumption. However, it is not possible to calculate the term Ox/Orfor our analytical model in detail, because this expression becomes extremely complicated. This also motivates the use of numerical examples to gain further insight into our model. Another point of interest is the question of what happens if a less strict budgetary regime is assumed in which only a certain part of the government's interest payments must be financed by the tax revenue. Formally, that is obtained by setting (o4 ~ (0, 1) which gives regime (B). Now, the negative feedback effect of public investment financed by public debt is expected to be lower in magnitude since only (O4rB of the interest payments must be financed by the current tax revenue. Proposition 2 gives the precise result. PROPOSITION 2. Holding the national debt~private capital ratio fixed, a positive growth effect of higher deficit financed investment in public capital is the more likely the lower (O4.
Proof To prove the proposition we proceed as in the proof of Proposition 1. Doing the same steps as in that proof with (o0 now given by (o0 = ((01 -[- (O2) -}- (O4 (1 -- a)br -1 (1 + (1 - a)b) -1 and taking into account the derivatives of this term with--respect to r and b we get -(Oql(.)/Ob)(Oq(.)/ 0(O3) + (Oq(')/Ob)(Oql(')/O(o3) = (O4 - r ( 1 - (o0)(1 + (1 - a)b)[(o3 + ((O3 - 1)/(b(1 - a))]. In analogy to Proposition 1, the balanced growth rate rises if this expression is negative, that is, if Z1 --- z(1 - (o0)(1 + (1 - ~)b)[(oa + ((O3 - 1)/(b(1 - or))]-> (O4.Since (#4 E (0, 1) and (oopositively varies with (O4the second part of the proposition is proved. That proposition states that a positive effect of public investment financed by additional debt is the more likely the smaller the feedback effect of a higher level of government debt imposed by the budgetary regime. With that proposition one could be tempted to come to the conclusion that governments only have to follow less restrictive budgetary polieies and earl thus generate positive growth effects of a deficit financed public investment on the balanced growth rate. But two points must be taken into account. First, that proposition is only valid for a given debt/capital ratio, which is an endogenous variable. That is, if a less strict budgetary regime is in use this regime may cause a higher debt/capital ratio which probably compensates for the positive direct effect of less interest payments to be financed out of the tax revenue so that the economy may end up with less economic growth.
374
Endogenous Growth, Government Debt and Budgetary Regimes The second point we have not yet addressed is the question of whether such a balanced growth path exists at all, on which economic per capita variables grow at a constant rate. Above we mentioned that such a path is only feasible if the decline in the marginal product of private capital is compensated by investment in public capital. However, in our analytical model we unfortunately cannot answer that question more precisely because the resulting expression becomes too complicated. Therefore, we will conduct some numerical simulations to highlight the problem of the existence of a steadystate growth path and try to find whether a less strict budgetary regime probably makes endogenous growth impossible. Further it can be shown (see Greiner 1996, 4.2) that the balanced growth rate of the economy does not necessarily rise if a less strict budgetary regime is imposed. A higher value for (04 may well be compatible with a higher steady-state growth rate. This ambiguity results from the fact that (04 in this regime shows two different effects. On the one side, a lower value for (04 implies that fewer interest payments on public debt must be financed through the tax revenue, leaving more resources for investment in public capital. In our model that direct effect tends to decrease (00, which can be seen by differentiating (00 with respect to (04- On the other hand, there is an indirect effect of (04 by influencing the steady-state value for b. If a lower interest payment must be financed out of the tax revenue, the level of public debt will probably be higher and thus the steady-state value of public debt per private capital, b. That indirect effect tends to increase the amount of tax revenue used for the debt service, increasing (00 in our economy and, thus, tends to lower investment in public capital that will show negative repercussions for economic growth. Whether the direct effect dominates the indirect one, implying that a less strict budgetary regime is accompanied by higher economic growth, or vice versa cannot be determined for the analytical model and depends on the specific conditions of the economy under consideration. In the simulations we will present a numerical example where that case is illustrated, but first we present some analytical results for our model assuming the budgetary regime (C).
Regime (C) In regime (C) the government does not have to finance any part of its interest payments and investment out of the tax revenue. But the latter must be sufficiently high to meet public consumption and transfer payments. The constraint is then written as Cp + Tp = (POT,(00 < 1, which is obtained from (9) by setting (04 = 0. It should be noted that for (04 = 0, (00 is given by (00 = (01 + (02. The evolution of the government debt in that regime is /3
375
Alfred Greiuer and Willi Scrawler = rB + C p + Tp + (7, - T = rB + (1 - ~00)(~0a - t)T0 where(~ = ~0a(1 - ~0o)T and :~0a > t. Formally, regime (C) is obtained from (21)-(22) by setting (o4 = 0 and ta > t. It should be noted that for ¢a < 1 government expenditure exclusive of interest paYments are .smaller fhan the tax revenue, We do not consider fhis regime but refer to Greiner (i996, :chap. 4.2) Where it is analyzed in detail. Let us first consider regime (C) and try to find :some analytical results. To gai~ further insight° we explicitly compute b at the steady state from q ( . ) = 0 as
b ............... -- a~(~00- :t){1 -~03) :x-~p - ( 1 - - z ) : ( t - a ) + ~(i~)(I-~4 + ((eo~l)(t~{03)~)
(z3)
It is immediately seen that for Pa > !, (1 - r) > a ( l - ~04 + (:(o0 - t)(t - ~03)z) must hold ifb is to be positive, This demonstrates that for ~04 = 0, that is, for regime (C), ~ must be smaller than 1 - a, and thus smaller than 1, for a b.alav,ced growth path with a positive level of government :debt to exist. Since a is -the inverse of the instantaneous elasticity of substitution, the economic interpretation of this result is that the representative individual in this economy mnst ~have a utility function w~th an extremely high instantaneous elasticity of substittt-tion for,a balanced gmwth path with government debt to exist. That is, the household must he extremely willing to forgo current consumption and shift it i~to the future. As mentioned in the introduction, most empirical studies suggest that a takes on a value around 1. In this ease a balanced :grawth path can only exist in regime (C) ff the government has a negative level of debt, that is if it is a creditor. We state this result in proposition 3, PrtOPOSITION 3. In regime (C), 1 ~ • > ¢r is a uecessary condition for a balanced g~ow.th path with positi.,ve government debt to exist. If this inequality does not hold, a balanced growth path is necessarily associated with negative ,government debt; that is, the government must be a ,creditor. The proof follows immediately from the expression for b in steady state. To find the effects of fiscal policy on the balanced growth rate we proceed in analogy to the section titled Regime (A) and tB). Since the balanced growth rate which is given by (14) does not directly depend on (P3 the ,zariation in x, induced by a change of this parameter, shows whether the growth rate increases or declines, The growth effect of a deficit financed rise in public investment is given in Proposition 4. 376
Endogenous Growth, Government Debt and Budgetary Regimes P:aOI'OStTION 4, In .regime (C), a deficit financed increase in public invest-
ment raises the balanced growth rate. Pro@ Again det M > 0 holds. Then, the sign of (Oqi(-)/Ob)(Oq(')/O~oa) - (Oq~(.)/Ob) (aq~(-)Z0e3) determines the effect on x and, thus, on the balanced g r o ~ rate. If it is positive [he growth rate rises and vice versa. In .an appendix to the paper, Which is a~ailable on request, we show that ttais expression is ~nambiguous!y positive. This proposition states that a rise in public investment raises the hal_ aneed growth rate, This outcome is probably due to the fact that there is no ir~terrtal crowding out in ~is regime (-with ~0a > 1). But we should also like to ,emphasize that, for ~a < 1, -that is, if only interest payments on public ,debt :are :financed by public deficit and public investment, public consumption a~d transfer payments are financed by the tax revenue, a rise in pubhe investment has no ~nambiguons effect. That is it may raise or lower economic ,gr,o~, although there is no internal crowding out either. This :case is .studied in detail in Greiaer (!996, eh~p. 4,2), Further, ~e .should like to point out that this regime (C) is of less rdevanee for real world economies. That holds because one always has to be aware of the existence problem concerning the balanced growth path. For ¢r :-~ 1 - ~, which seems to be quite realistic, this proposition can only be app~ed if the government is a creditor. In the more .realistic ease of a positive go~ernment debt, the instantaneous ehsticity .of substitution has .to he very high for this proposition to be relevant. As to the gr ~owth maximizing income tax rate we imme~a.tely see that :it is given by the :same expression as in regimes (A) and (B)and cannot be precisely determined in the analytical model. Therefore, we present some numerical simulations to find its magnitude. Before ~e continue the presentation let us briefly summarize ,,~hatwe have achieved. We ,have analyzed how a government may influence the batmaced growth rate by increasing productive investment in public ,capitat, As to the financing of this spending.we ha~e assumed that the government issues new ~ d s , It t,urned out that the results crucially depended on the budgetary regime nnder consideration, For regime (A) we saw that a deficit financed increase ,of investment in infrastructure raises balanced growth if a certain condition is met. In particular, a positive effect is more likely the lower the s'hare of non-productive government :spending, other things , e q ~ . Further, we could show that applying this regime less restrictively, that is, by requiring that only a certain part of,debt payments must be financed out of the tax revenue, .giving regime (B) a positive effect of deficit financed increases in public investment is more likely, but only for a given debffcapital 377
Alfred Greiner and Willi Semmler ratio which, however, is an endogenous variable that is expected to be the higher the less restrictive the budgetary regime is. As to the growth rate itself, no concrete results could be derived, implying that both cases could occur. So, a less strict budgetary regime could be accompanied either by a higher or lower balanced growth rate. In regime (C), where only public consumption and transfer payments are financed by the current tax revenue, we found that only for extremely high values of the instantaneous elasticity of substitution can a balanced growth path with a positive government debt exist. For ¢r -> 1, what most empirical studies consider as realistic, the government must be a creditor if endogenous growth is to hold. Moreover, we could show that in this regime a deficit financed increase in public investment always raises economic growth. In the next section we intend to present some numerical examples to illustrate our analytical results and to shed some light on the question of whether the existence of a balanced growth path is influenced by the choice of the budgetary regime as well as to find the growth maximizing income tax rate, because these questions could not be answered within our analytical framework.
4. Numerical Examples To start with we fix some of our parameter values. As to the elasticity of output with respect to public capital we take the same value as in Barro (1990) and set a = 0.25. The utility function is supposed to be logarithmic, giving ¢r = 1. The discount rate is taken to be p = 0.3. Interpreting one time period as 5 years then implies that the annual discount rate is 6%. These values are kept constant throughout all simulations. First, let us present an example to illustrate regime (A). In this regime all expenditures, with the exception of public investment, had to be financed out of the tax revenue. The parameter values for ~01and ~2, giving that part of the tax revenue which is used for transfer payments and public consumption, are set to ~01 = 0.3 and ,co2 = 0.35. These are about the ratios of transfers to individuals per total government revenue and of public consumption per total government revenue for West Germany for the mid eighties (cf. Sachverst~indigenrat 1993, table 38). The parameter ~a is set to ~a = 1.5. For these parameters, Table 2 reports the steady state values 8 for x, b and ~0 as SFor all simulations, the dynamic behavior of the system is characterized by saddle point stability, The eigenvalues of the Jacobian are given in an appendix available on request.
378
Endogenous Growth, Government Debt and Budgetary Regimes TABLE 2. z
x
b ,,
0.15 0.20 0.22
g ,
0_153 0.0511 0.208 0.0694 0.232 0,0772
,,,
9)0 ,
0,0990 0,1052 0.1058
, , ,
i
r ,
0.896 0.897 0.899
x
b
,,
g i
0.23 0.25 0.30
0.244 0.269 0.337
0.0813 0.0896 0.1124
,
9)0 ,
0.1058 0.1051 0,1001
0.900 0.902 0.909
well as the growth rate for different income tax rates. 9 The values are rounded to the third and fourth decimal point, respectively. Table 2 shows that the maximum growth occurs for income tax rates of about 22-23%, which are a little smaller than the elasticity of output with respect to public capital a. It must be noted that we took one time period to comprise 5 years so that the annual growth rate is about 2%. If we raise the parameter 9)3, that is, if we increase public investment fnanced by additional debt and set 9)s = 1.65, we get the following results, shown in Table 3. TABLE 3.
0,15 0.20 0.22
x
b
g
9)0
0.142 0.192 0.214
0.0559 0,0758 0.0843
0.0912 0.0974 0.0979
0.918 0.919 0.920
r
x
0.23 0.225 0.25 0.248 0.30 0.310
b
g
9)0
0.0887 0.0977 0.1220
0.0978 0.0969 0.0917
0.921 0.923 0.930
This demonstrates that the feedback effect in regime (A) is so high that the positive effect of higher investment in public capital is more than compensated by the additional interest payments generated through deficit financing. In this ease, increasing 9)3 leads to less public investment because the additional interest payments caused by this fiscal policy increase 9)0 in a way so that in the end less resources are available for public investment. In this situation reducing the government debt or public consumption or transfer payments sets flee resources which earl then be used for public investment and may raise economic growth. We also see that the ratio of public debt to private capital, b, rises, That was to be expected because a higher 9)3 means a higher public deficit, other things equal. It should be mentioned that a deficit financed increase in public investment is to be seen as a crowding out of private investment if this policy 9There exist two steady states with these parameters. But the second yields ~0 = 1 and a zero growth rate.
379
Alfred Greiner and Willi Semmler TABLE 4. r
x
b
g
0.15 0.20 0,21
0.131 0,178 0.187
0.0599 0.0813 0.0860
0.0837 0.0895 0.0896
~00
r
x
0.922 0,22 0.196 0.923 0.25 0.224 0.924 0.30 0.265
b
g
~00
0_0908 0.1061 0.1360
0.0894 0.0869 0,0765
0.925 0.930 0.943
reduces the ratio x = G/K. This holds because a lower value for x reduces the return to private investment, that is, lowers the marginal product of private capital r = (1 - o0x~. With the parameter values in Table 2, a rise in ~0a leads to: c~r
3~3 3r
0~3 Or 0~3 ar
0.0819 for r = 0.15 ; 0.0743 for r = 0.2 ; 0.0818 for r = 0.25 ; 0.0910 for r = 0.3 .
&Pa In Table 4 we present the results for ~04 = 0.95, meaning that 95% of the interest payments must be met by the tax revenue, whereas the rest is financed by additional debt. All other parameters are as in the simulation for Table 2. That table illustrates the discussion following Proposition 2. It shows that a less strict budgetary regime does not necessarily lead to a higher balanced growth rate. The reason for that outcome is that, in this case, the parameter ~00is larger than in the economy with regime (A) and, therefore, investment in public capital is lower. This result holds because the direct effect of less interest payments which must be financed out of the tax revenue, which would lead to a lower q0, is more than compensated by the higher level of government debt per private capital, b, which tends to increase ~0, so that the economy ends up with a higher value for ~0. But it must be emphasized that the reverse effect could probably also be observed and the outcome depends on the specification of the economy. This holds because for the analytical model no unambiguous result could be obtained. To find how this economy reacts to increases in public investment financed by a higher debt, we calculate the derivative of x with respect to ~3, yielding 380
Endogenous Growth, Government Debt and Budgetary Regimes &¢
-0.320537
0~3
2.67813
-
0.119687 for ~ = 0.15 ;
0x d{o3
-0.266592 1.65434
0.161147 for r = 0.2 •
#x 0~o3
-0.151911 0.435745
0.348625 for T = 0.30 .
-
These results show that Oxl3~3is still negative for ~94 0.95. This effect is again due to the higher debt/capital ratio b which causes an increase in ¢0, which offsets the positive effect of a lower ¢4. Therefore, this example underlines that proposition 2 is only valid for a given debt/capital ratio. If we further decrease ~04and set P4 = 0.9, the balanced growth path does not exist any longer, meaning that there is no sustained per capita growth for our economy. This demonstrates that a less strict budgetary regime does probably not allow sustained per capita growth at all. Reducing ~4 further, we observe that sustained per capita growth is again possible for ¢4 around ¢4 = 0.05. But it must be emphasized that this regime is only feasible if the government is a creditor, that is, for a negative level of public debt. For ~04 = 0 (~4 = 0.05) the steady-state value for b is b = - 0.0641 (-0.0718). The balanced growth rate associated with this steady state is 0.187 (0.190). Recalling that one time period comprises 5 years, this corresponds to an annual growth rate of about 3.74 (3.8)%. =
5. Conclusions The goal of this paper was to analyze an endogenous growth model with productive government spending and public debt. In our study, the government does not have to run a balanced budget at any moment in time, but it must stick to certain budgetary regimes, which impose constraints on the government's fiscal behavior. It should be noted that we cannot resort to the Ricardian equivalence theorem in our model because it contains distortionary taxes and productive government spending. We demonstrated that less strict budgetary regimes, that is, economies where a greater fraction of interest payments may be financed by raising public debt, make a positive growth effect of a deficit financed increase of public investment more likely. But it must be stressed that this outcome only holds if the public debt to private capital ratio, which is an endogenous variable, is held fixed. Simulations demonstrated that less strict budgetary regimes are associated with a higher government debt which offset the positive direct growth effect of a less strict budgetary regime. Further, we could demonstrate that a less strict budgetary regime possibly does not allow 381
Alfred Greiner and Willi Semmler sustained per capita growth or that it does only if the government owns assets and is a lender. That result became particularly obvious in regime (B), where sustained growth with positive public debt did not exist once the fraction of interest payments paid by raising public debt exceeded a certain threshold level, Those outcomes demonstrate that imposing restrictions as to public deficits may in fact be reasonable from an economic point of view. Our results are of relevance as to the discussion about the budgetary goals of the EMU. Here we have only provided a theoretical model to discuss those issues. For an empirical methodology and estimation strategies, see Greiner and Semmler (1999). Note, however, we do not consider effects of monetary policy in our model and the role of budgetary rules as concerns the effects of inflation (see Woodford 1996). Moreover, we saw that there exists a growth maximizing income tax rate as in the models by Barro (1990) and Futagami, Morita and Shibata (1993). However, we should also like to emphasize that maximizing economic growth need not necessarily be equivalent to maximizing welfare, even if one limits the investigations to the balanced growth path. The reason is that a fiscal policy which reduces the balanced growth rate, like an increase in public transfers or in the tax rate above its growth maximizing value, may raise the level of private consumption. A rise in the level of private consumption has a positive welfare effect which can outweigh the negative one of a lower growth rate. This can be proved rigorously for this model without public debt (cf. Greiner and Hanusch 1998) as well for the Barro (1990) model (cf. Greiner 1998). For the model presented in this paper, however, a formal proof is not possible since public debt makes the analysis too complicated. Another point which should be made clear is that in our model we could only take into account one source of economic growth, namely productive government spending, while others, such as production externalities or efforts to increase human capital and knowledge, were neglected because they would have made the model intractable. Therefore, the result of our simulations should be interpreted with some care if one wants to evaluate the quantitative effect of fiscal policy in real economies. Our numerical examples pertain to the demonstration of the analytical model and its qualitative results which indeed seem to be very robust.
Received: May 1998 Final version: October 1998
382
Endogenous Growth, Government Debt and Budgetary Regimes References Arrow, Kenneth J., and Mordecai Kurz. Public Investment, the Rate of Return, and Optimal Fiscal Policy. Baltimore: The John Hopkins Press, 1970. Aschauer, David A. "Is Public Expenditure Productive?" Journal of Monetary Economics 23 (1989): 177 200. Barro, Robert J. "Government Spending in a Simple Model of Endogenous Growth." Journal of Political Economy 98 (1990): $103-25. Blanchard, Olivier J., and Stanley Fischer. Lectures on Macroeconomics. Cambridge, Massachusetts: MIT Press, 1989. Easterly, William, and Sergio Rebelo. "Fiscal Policy and Economic Growth: An Empirical Investigation." Journal of Monetary Economics 32 (1993): 417-58. Ewijk, Casper van. On the Dynamics of Growth and Debt. Oxford: Oxford University Press, 1991. Ewijk, Caspar van, and Theo van de Klundert. "Endogenous Technology, Budgetary Regimes and Public Policy." In The Political Economy of Government Debt, edited by Hattie A. A. Verbon and Frans A. A. M. Van Winden. Amsterdam: North-Holland, 1993. Fischer, Stanley. "The Role of Macroeconomic Factors in Growth." Journal of Monetary Economics 32 (1993): 485-512. Futagami, Koichi, Yuichi Morita, and Akihisa Shibata. "Dynamic Analysis of an Endogenous Growth Model with Public Capital." Scandinavian Jourhal of Economics 95 (1993): 607-25. Gramlich, Edward M. "Infrastructure Investment: A Review Essay." Journal of Economic Literature 32 (1994): 1176--96. Greiner, Alfred. Fiscal Policy and Economic Growth. Aldershot: Avebury Publishing Company, 1996. --. "Fiscal Policy in an Endogenous-Growth Model with Public Investment: A Note." Journal of Economics 68 (1998): 199-98. Greiner, Alfred, and Horst Hanusch. "Growth and Welfare Effects of Fiscal Policy in an Endogenous Growth Model with Public Investment." International Tax and Public Finance 5 (1998): 249-61. Greiner, Alfred, and Willi Semmler. "An Endogenous Growth Model with Public Borrowing and Government Capital." Annals of Operations Research 88 (1999): 65-79. --. "An Inquiry into the Sustainability of German Fiscal Policy: Some Time Series Tests.'" Public Finance Review (27) (1999): 221-37. Hall, Robert. "Intertemporal Substitution in Consumption." Journal of Political Economy 96 (1988): 339-57. Jones, Larry E., Rodolfo E. Manuelli, and Peter E. Rossi. "Optimal Taxation
383
Alfred Greiner and Willi Semmler in Models of Endogenous Growth." Journal of Political Economy 101 (t993): 485517. Judd, Kenneth L. "Short-Run AnaIysis of Fiscal P01iey in a Simple Perfect Foresight Model." Journal of Political Economy 9~ (1985): 298-319. Lueas, Robert El. "Supply-Side Economics: An AnaIytieal Review." Oxford Economic Papers 49. (1990): 293-316. Milesi-Ferretti, Gian M., and Nourie{ Roubini. "On the Taxatior~ of Human and Physical Capital in Models of Endogenous Growth."Journat of PubliC Economics 70 (1998): 237-54. Munnelt, Alicia H. "Infrastructure Investment and Economic Growth.'Jourhal of Economic Perspectives 6 (1992): 189-98_ Sachverstandigenrat zur Begutaehtung der gesamtwitsehaftlichen Lage. Zeit zum Handeln-Antriebskr~fie stdrken jahresgutachten 1993/94. Stuttgart: Metzler-Poesehet, 1993:. Sturm, Jan Egbert, Gerhard H. Kuper, and Jakob de ttaan. "Modelling Government Investment and Economic Growth on: a Macro Level A Review." CCSO Series No. 29 University of Groningen, 1997. Turnovsley, Stephen. MacroeconomicDynamics. Cambridge, Massachusetts'. MIT Press, 1995. Woodford, Michael "Control of Public Deficits: A Requirement }'or Price Stability?" NBER Working Papers, 5684, i996.
384
WILLI SEMMLER University of BieIefeld Bielefeld, Germany and New School University New York City, New York
Endogenous Growth, Government Debt and Budgetary Regimes* In this paper we analyze an endogenous growth model, in which sustained per capita growth results from investment in public capital arid the government is allowed to borrow from the capital market. As to the government behavior we do not suppose that governments optimize but instead stick to some well-defined budgetary regimes. The impact of a deficit financed increase in productive government spending is analyzed. It is shown that the growth effect of this fiscal policy crueially depends on the budgetary regime in use. In particular, a more strict budgetary regime, that is, an economy where the public defieit is primarily used for public investment, does not necessarily imply a lower growth rate. Simulations demonstrate that the growth maximizing ineome tax rate is about in the range of the elasticity of output with respeet to public capital.
1. Introduction In recent years, growth theory has received renewed attention with the rise of a new line of research that explains growth rates endogenously. Within this "new" growth theory, one approach posits that the government supplies productive services that increase the marginal product of private capital and, thus, positively influence economic growth. If the production function is linear homogenous in private and public capital jointly, and government spendings are endogenous, this approach yields an endogenous rate of growth. Arrow and Kurz (1970) were the first to introduce productive public capital in growth models, but in their model growth is still determined by exogenous factors. Barro (1990) takes up this approach and presents a *We thank two referees for detailed comments on an earlier version. This paper was written whde Alfred Greiner was visiting the Department of Economics of the Graduate Faculty of the New School for Social Research in New York. Its hospitality and generous provision of facilities, as well as generous financial support from the Deutsche Forschungsgemeingchaft (DFG) and the German Marshall Fund of the United States is gratefully acknowledged.
Journal of Macroeconomics, Summer 2000, Vol. 22, No. 3, pp. 363-384 Copyright © 2000 by Louisiana State University Press 0164-0704/2000/$1.50
363
Alfred Greiner and Wilh Semmter model with endogenously determined growth rates and a balaJaced budget. He, however, assumes that government spending as a flow variable enters the macroeconomic production function, whereas Arrow and Kurz suppose that the stock of public capital shows productive effects, Futagami, Morita and Shibata (1993) present an endogenous growth model where the stock of public capital has positive effects as concerns the marginal product of private capital thus leading to endogenous growth, The Arrow and Kurz version is also supported by an empirical study by Aschauer (1989) who showed that the stock of public capital is indeed more important than the flow of government spending. His study shows a value of 0.38-0.56 for the elasticity of output with respect to public capital. This, and therefore, Aschauer's study had been very controversial. For a survey of the criticism as well as other empirical studies investigating the link between output and public capital, see Munnell (1992), Gramhch (1994) or Sturm, Kuper and de Haan (1997). In this paper we take up the approach presented by Futagami, Morita and Shibata (1993) and analyze how fiscal policy affects the balanced growth rate of an economy. In contrast to their model, however, we also allow for a budget deficit of the government and take into account government debt. But the government decisions are supposed to be restricted by certain budgetary regimes t ° which a government must stick, These regimes are generally formulated in terms of the instruments (expenditure and tax regimes) or in terms of a well-defined target like the budget deficit for example or the size of public debt. Van Ewijk and van de Klundert (1993) analyze the impact of different budgetary regimes for the dynamics of growth and public debt in a conventional growth model. They consider three different regimes, which they name according to the economists who introduced them in the economics literature (van Ewijk and van de Klundert 1993, 123-124). A general result they derive is that the linkage of public spending for research and education to the budgetary position may be a destabilizing factor. Further, they find that the regime where the government keeps the budget deficit constant is less favorable as concerns productivity growth in comparison to a regime where the government may vary the budget deficit. However, the regimes they consider postulate that some variables which are endogenous in our setting remain constant over time so that they cannot be used within our model of endogenous and sustained per capita growth. Therefore, we define some new regimes taking into account sustained growth within which the impact of fiscal policy is investigated. The remainder of the paper is organized as follows. In the next section, we present the basic model and introduce different budgetary regimes, Section 3 gives analytical results and in Section 4, we present simulations to
364
Endogenous Growth, Government Debt and Budgetary Regimes illustrate the results of the analytical model and to derive some additional outcomes. Section 5, finally, concludes the paper.
2. The Basic Model We consider a closed economy which is composed of three sectors: the household sector, a representative firm and the government. The household is supposed to maximize the discounted stream of utilities arising from consumption subject to its budget constraint:
max c(t)
fO~ e-~Cu(C(t))dt
;
(1)
subject to
C(t) + S(t) = (w(t) + r(t)S(t))(t - r) + Tp(t).
(2)
The term C(t) represents private consumption at time t, S(t) = K(t) + B(t) denotes assets which comprise physical capital K(t) plus government bond or public debt B(t). Tp(t) stands for lumpsum transfer payments, the household takes as given in solving its optimization problem. The variable • gives the income tax rate and w(t) and r(t) denote the wage rate and the interest rate respectively. The variable P gives the constant rate of time preference and the labor supply is assumed to be constant and normalized to one so that all variables give per capita quantities. The depreciation rate of physical capital is set equal to zero. As to the utility function we assume a function of the form (C(t)l-~)/ (1 - a), with - a denoting the elasticity of marginal utility with respect to consumption or the negative of the inverse of the instantaneous elasticity of substitution between consumption at two points in time, which is assumed to be constant. For ~ = 1 the utility function is the logarithmic function In C(t). If the utility function is used to describe attitudes towards risk ~ has an alternative interpretation. In this case ~ is the coefficient of relative risk aversion. A number of empirical studies have been undertaken aiming at estimating this coefficient under the assumption that it is constant, by looking at the consumers' willingness to shift consumption across time in response to changes in interest rates. The estimates of ~ vary to a great degree but are usually at or above unity (see Blanchard and Fischer 1989, 44; Lucas 1990; or Hall 1988). Assuming that the rate of growth, g, of the endogenous variables is bounded by g < p it is shown in an appendix available on request that there
365
Alfred Greiner and Willi Semmler
exists a solution to this optimization problem. To find this solution we formulate the current value Hamiltonian which is given by, 1 H(') = u(c) + 7((w + rS)(1 - r) + Tp - C). The necessary optimality conditions are then obtained as 7
=
c -~
,
= 7(P -
(3)
(1 - T)r),
(4)
+ (rS + w)(1 -
= -C
z) + T p .
(5)
These conditions are also sufficient if the limiting transversality condition l i m t ~ e-etT(t) (K(t) + B(t)) = 0 is fulfilled. 2 Differentiating 7 = C - ° with respect to time and using 2> = 7(P - (1 - r)r), this system can be reduced to a two dimensional differential equation system: p -
C
S
-
(1 - ~)r +
~r
- - , ~r
C + (r + w / S ) ( 1 S
(6)
T -
z) +
-P.
S
(7)
The productive sector is assumed to be represented by one firm which behaves competitively exhibiting a production function of the form, f(K, G) = K 1-~ G ~ .
(S)
The variable G gives the stock of productive public capital which is a nonrival and non-excludable good 3 and 1 - a, ¢x E (001)0 denotes the share of private capital in the production function. Since K denotes per capita capital, the wage rate and the interest rate are determined as w =cLK 1- ~G ~ and r = ( 1 - a ) K - ~ G ~. The budget constraint of the government, finally, is given by/3 + T = rB + Cp + Tp + G. The variable Cp stands for public consumption and T is the tax revenue, which is given by T = r(w + rS). As to public con1In the following we omit the time argument if no ambiguity arises. ~This condition is automatically fulfilled ifg < p holds, with g the long run balanced growth rate. This also guarantees the boundedness of the utility functional. ZFor an analysis where the public good is non-excludable but rival or subject to congestion, see Barro and Sala-i-Martin (1992). In another version of this model we take account of congestion effects and population growth (cf. Greiner and Semmler 1999).
366
Endogenous Growth, Government Debt and Budgetary Regimes sumption we suppose that it is a pure drain on resources and provides no benefits to the economy as it is often assumed in the optimal tax literature. An alternative specification which would not change our results would be to assume that public consumption enters the utility function in an additively separable way (for a discussion and an economic justification of that assumption see Judd 1985, 301 or Turnovsky 1995, 405). We suppose that public consumption and transfer payments to the household constitutes a certain part of the tax revenue, i.e. Cp = (0zT and Tp = (01T, (01, (02 < 1. Moreover, the government is not allowed to play a Ponzi game, that is, fimt-~ B(t)e -Igr~*)d~ = 0 must hold, and we define four alternative budgetary regimes to which the government must stick. 4 The first states that government expenditures for public consumption, transfer payments and interest payments must be smaller than the tax revenue, Cp + Tp + rB = (00T, with (00 < 1. We will refer to this regime as regime (A). This regime is called the "Golden Rule of Public Finance" and postulates that it is not justified to run a deficit to finance non-productive expenditures which do not yield returns in the future, Public deficits then are only feasible to finance productive public investment which increases a public capital stock and raises aggregate production. 5 A slight modification of this regime, and in a way natural extension, is obtained when allowing that only a certain part of the interest payments on public debt must be financed out of the tax revenue and the remaining part may be paid by issuing new bonds. In this ease, the budgetary regime is described by Cp + Tp + (04rB = (POT,with (04 ~ (0, 1), We refer to this regime as regime (B). The third regime, regime (C), states that public consumption plus transfers to individuals must not exceed the tax revenue, but the government runs into debt in order to finance public investment and interest payments, Cp + Tp = (00T, (00 < 1. Further, in all regimes investment in infrastructure is supposed to constitute a certain part of the remaining tax revenue, G = (0a ' (1 - (00)T, (03 >-- 0. In regime (A) for example this implies that government debt only increases if (0a > 1. Table 1 gives a survey of the regimes. Let us in the next section take a closer look at our economy described by this basic model and derive some implications of those different regimes for the growth rate of economies.
4It should also be mentionedthat increases in governmentexpenditurewould lead to proportional decreasesin consumptionif the first did not influenceproductionpossibilitiesand if we had non-distortonarytaxes. This followsfrom the Ricardian equivalencetheorem (cf. Blanchard and Fischer 1989, ch. 2). 5Regime (A) can be found in the German constitutionand is bindingfor the government.It was also used to derivethe deficitcriterionin the treaty of Maastricht.
367
Alfred Greiner and Willi Semmler
TABLE 1. Regim e ...........Target
. . . . . . . . . . . . . . . . . . . . . . Deficit due t o
,
(A)
Cp + Tp + rB < T
public investment
(B)
Cp + Tp + (04rB < T
public investment + (1 - (04)rB
(c) cp + rp. + ~ > T, (Cp + Tp + G) + rB . . . . . . . . . . . % +. r , < r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Analytioal Results Before we analyze our economy we first state that regimes (A), (B) and (C) can be derived from regime (B) for appropriate values of (04 and (03. To see' that more clearly we write down the budgetary regime for regime (B) which is given by
(9)
Cp + Tp + ~4rB = (0oT.
Further, the debt accumulation equation for regime (B) is given by [3 = rB + Cp + Tp + G - T = ((% - 1)(i - ~oa)r + (1 - (04)rB.
(10)
Setting (04 = i in (9) and (10) and (0a > 1 it is immediately seen that we obtain regime (A). In that ease the whole interest payment on outstanding public debt must be financed by the tax revenue. If (04 E (0, 1) only (o4 of the interest payments is financed by the tax revenue and the rest, (1 - (04)rB, is paid by raising public debt, giving regime (B). In the extreme, i.e. for (04 = 0, all interest payments on outstanding debt are financed by issuing new debt and we now have regime (C), for (0a > i. In that case we get Cp + Tp + G - T = (No - 1)(1 - (03)T > 0, To find the differential equation describing the evolution of physical capital, that is the economy-wide resource constraint, we first note that the budget constraint of the individual gives [( + B = - C + (w + rK + rB)(1 - z) + .Tp. Using the definition of B the term K + / ) can be written as [( + B = K + rB + (0zT + .Tp + ~oa(1 - (00)r - T. Combining those two expressions then yields K = - C + ( w + r K ) - ((02+(03(1-(00)) z(w + rK + rB ). Using the equilibrium conditions w + rK = K 1- ~G ~ and r = (t - ot)G~K -~ then gives the economy-wide resource constraint as
K 368
-
K
+
-
~(~
+ ~(i-
~o))
] +
(i
-
~)~
.
(11)
Endogenous Growth, Government Debt and Budgetary Regimes
The description of our economy is completed by Equation (6) giving the evolution of consumption and by the differential equation describing the path of the stock of public capital G. Thus, we get the following four dimensional differen[ial equation systems with appropriate initial conditions and the limiting transversatity condition: K -
~-
K +
(~0p ~
-
C
-G- =
- r((o~ + pa(1 - ~Oo))
1 + (I - cx)~2 ;
1)(1 - (0a)z((t - ~) + ~ ) ( G ) ~
+
(12)
+ (1 - ~04)(1 - ~x)(G);
(13)
(1 - z)(1 - a ) K - ~ G '~ ....... ; a
-
+/1
(14)
\K]
-
GJ
(15)
w e write that system in the rates of growth which tend to zero if the decline in the marginal product of physieat capital, caused by a rising capital stock, is not made up for by an ~ncrease in public capital. However, if the stock of public capital is sufficiently large so that the marginal product of private capital does not converge to zero in the tong run we can observe endogenous growth. In this case the r.h.s, of, our system is always positive, and we first have to perform a change of the system to be able to continue our analysis. Defining c = C/K, b ~- B / K and x = G / K the new system of differentiat equations is given by d/c = C / C - K/K, l~/b = B / B - K/K, and ic/x = G / G - [(/K leading to
_ec = -~r + x~ ((i. -, ~)(i,~r -' ~)
+
r(0,2 +
~,a(1 -
0,o)).(t
+
(1 -
o,)b)), ÷
(16) --
XaT
(((1
--
~)
÷
b-1)(~90
--
1)(1 - foa) + ((02 + ~0a(I - ~00))
(1 + (1 - ~)b)) + x°(1 - ~)(1 - ~4) + c - x=; -
X
=
x ~-1
(I
+
(1
-
ot)b)(oa(1
-
(1 + (1 - ~)b) + c - x".
0,o)~ +
x~(~o~
+
(Oa(1 -
(17) ~Oo))
(is,)
A stationary point of this system then corresponds to a balanced growth path 369'
Alfred Greiner and Willi Semmler of the original system w h e r e all variables grow at the same rate. In the following we will confine our analytical analysis to this path and examine h o w fiscal policy financed t h r o u g h additional debt affects the growth rate on that p a t h P Moreover, we will exclude the economically meaningless stationary point c = x = b = 0 so that we can consider system (16)-(18) in the rates of growth and confine our analysis to an interior stationary point. Doing so we get for c from e/c = O,
C
P- o-
~
--
x ~ (r((02 + (0a(1
(0o))(1 + (1 -
¢~)b) - 1 + (1 - r)(la - a ) ) .
(19)
Further, we know that the constraint Cp + T v + (04"FB = (0oT must be fulfilled on the balanced growth path. This condition makes (00 an endogenous variable which d e p e n d s on b, (0 1, (02, (04, %and a. Using T = r(KI-~G a + rB), it is easily seen that in the steady state (00 is d e t e r m i n e d as (04(1 -- 0¢) (00 = ((01 + (02) + Z((1 -- Or) + b - l ) "
(20)
But it must be noted that (00 < 1 is i m p o s e d as an additional constraint, which must always be fulfilled. Inserting c from d/c = 0 and (00 = (00(b,(0b(02,(04,z',°t) in [9/b and ~/x and setting the 1.h.s. equal to zero completely describes the balanced growth path of our economy. The equations are obtained as 0 = q(') =- g/~ = x~ (z((1 - ca) + b-1)(~Oo( ) - 1)(1 - (0a) (1
0
:
ql(')
~
-
¢04)(1
-
(21)
~)~;
~- = X~-l(1 + (1 -- ec)b)¢%(1 - (%( . ))z - x~ x
(1-T~1-~)) + p__ +a
(1 - r)(1 - a) o"
+ ;--.
(22)
O"
Solving for b and x then determines the balanced growth path for our economy, on which all variables grow at the same constant rate. The ratio G/K = x denotes the ratio of public capital to private capital which determines the marginal p r o d u c t of private capital and, as a consequence, the balanced
SUnder a slight technical assumption it is shown in an appendix available on request that the no-Ponzi game condition is met for this path. 370
Endogenous Growth, Government Debt and Budgetary Regimes growth rate. The balanced growth rate is obtained from (14), which merely depends on x at the steady state and on the exogenously given parameters. In the following, we intend to investigate the impact of fiscal policy on the balanced growth rate of our economy. To do so we implicitly differentiate system (21)-(22) with respect to the parameter under consideration. This shows how the ratio G/K changes and, together with (14), gives the impact on the growth rate of our economy. Using D/b ~ q(x, b, ~o0;...), and yc/x =- ql(x, b, ~Oo;. . .), and implicitly differentiating (21)-(22) with respect to the parameters leads to
Ob/Oz] = - M - 1 [ Oq(')/Oz ] OxlazJ kaqt(-)lazj ' The variable z stands for the parameters, and the matrix M is given by
[ Oq(.)/Ob M = iOqt(.)/Ob
Oq(.)/Ox ] Oq~(.)/OxJ'
The negative of its inverse is calculated as
-M- I = _
1 det M
. [
Oql(.)/Ox [ - Oql(')/Ob
-Oq(')/Ox] Oq(')/Ob] '
with det M denoting the determinant of M. Now, we can analyze how the growth rate in our economy reacts to fiscal policy. A result which holds independent of the budgetary regime under consideration is that an increase in public consumption and transfer payments reduces the balanced growth rate. That result is not too surprising. Therefore, we only mention that outcome but do not go into the details (a formal proof can be found in Greiner 1996, chap. 4.2). A more interesting topic is the question of how a deficit financed increase in public investment and how the choice of the budgetary regime affect economic growth. To begin with, we consider regimes (A) and (B).
Regimes (A) and (B) First, we study regime (A), which is obtained from (21)-(22) by setting ~04 = 1 and (03 > 1. The question we want to answer is whether a rise in public investment financed through additional government debt increases economic growth if we take into account feedback effects of the higher level of public debt. To investigate this case we note that it can be represented by an increase in the parameter (0a. On the one hand, a higher (0a means 371
Alfred Greiner and Willi Semmler more direct investment in public capital but, on the other hand, it also changes the ratio of b, which determines (J/G both directly and indirectly through (%. Proposition 1 gives the condition which must be fulfilled for a positive growth effect of a deficit financed increase in public investment. PROPOSITION 1. A deficit financed increase of investment in public capital raises (reduces) the balanced growth t'ate in regime (A) if Z =- z ( 1 - (00)(1 + (1 - ot)b)[(0a + ((03 - 1)/(b(1 - or))] > (<)1.
Proof
First we note that under a slight additional technical assumption it can be shown that in regime (A) det M > 0 holds. 7 To prove this proposition we derive from (14), 0g
0(03
-- (1 -- T)(1
a
-- O~)O~xa_l
~._.~_._. 0(0a
Implicitly differentiating (21) and (22) again gives an expression for the change in x at the steady state. Since - det M <0, x, and thus the balanced growth rate, rises if -(Oql(')/Ob)(Oq(.)/O(0a) + (Oq(.)/Ob)(Oql(.)/O~oa) < 0 and vice versa. The sign of that expression is negative (positive) if 1 - z(1 - (0o)(1 +(1 - a)b)[~0a + (~03 - 1)/(b(1 - a))]--- - Z + 1 < (>)0. Thus, proposition 1 is proved. It should be noted that it is the feedback effect of government debt that exerts a negative influence on the growth rate if investment in infrastructure is financed through additional debt. This is seen by setting O(0o/Ob = 0. The expression (Oq(.)/Ob)(Oql(.)/O(0a) - (Oql(.)/Ob)(Oq(.)/O~oa) then is always negative and a defcit financed rise in investment in infrastructure raises x and the balanced growth rate. In that case our result would be equivalent to the one derived by Turnovsky (1995, 418). The introduction of budgetary regimes, however, implies that a deficit financed increase in public investment shows a feedback effect which acts through two channels: first, an increase in public debt raises interest payments which must be financed by the tax revenue and consequently reduces the resources available for public investment. Here, we may speak in a way of internal crowding out (as in van Ewijk and van de Klundert 1993, 123). But that effect only holds for regimes (A) and (B). Second, the introduction of the budgetary regimes implies that the interest payments on public debt appear in the economy wide resource constraint (11) and lead to a (external) crowding out of private investment. That effect holds for all three regimes. 7The derivationas well as the detailed proof is containedin an appendixwhich is available on request. 372
Endogenous Growth, Government Debt and Budgetary Regimes
Since governments are not allowed to increase public debt arbitrarily because public debt does have effects (see e.g., Easterly and Rebelo 1993 or Fischer 1993), it is apparent that the introduction of budgetary regimes to model negative feedback effects of higher government debt is justified. That is also the reason why budgetary regimes have frequently been introduced in the economics literature (for a detailed survey of budgetary regimes see van Ewijk 1991). As to the effect concerning the ratio of public debt to private capital, b, no unambiguous result can be derived; that is, dais ratio may rise or fall. Therefore, we will present a numerical example in Section 4 in order to gain further insight into that problem. It can also be shown that, in economies with a high share of nonproductive government spending, like high interest payments, public consumption and/or transfer payments, that is, a large value for (00, a deficit financed rise in public investment is more likely to produce negative growth effects. This is immediately seen by differentiating Z with respect to ~00, which is negative. Another point which we should like to treat is how the tax rate must be chosen to maximize economic growth. Since a certain part of the tax revenue is used for investment in public capital which raises the balanced growth rate, it becomes immediately clear that the growth maximizing income tax rate does not equal zero. Thus, our result is consistent with the one derived by Barro (1990) and similar to the outcome observed by Jones, Manuelli and Rossi (1993), who demonstrate that the optimal income tax rate does not equal zero if government spending has a direct positive effect on private investment. But this outcome clearly is in contrast to the result of the standard Ramsey type growth model and also to endogenous growth models with physical and human capital (see e.g. Lucas 1990, or MilesiFerretti and Roubini 1998). The growth maximizing income tax rate is computed by differentiating (11) with respect to r. This leads to Og _ 1 - OL x~ Or cr
[
-1
+ ~ r
Or
1
This shows that the increase in the income tax rate raises or lowers the balanced growth rate if the elasticity of G/K with respect to r is larger or smaller than the expression z/((l - r)ot). The economic mechanism behind this result becomes immediately clear. For given parameter values an economy with a high elasticity of G/K with respect to r is more likely to experience growth if the income tax rate is increased because the higher investment in public capital, caused by more tax revenue, leads to a relatively strong increase in the ratio G/K and, thus, in the marginal product of private physical 373
Alfred Greiner and WiUi Semmler capital. This positive effect then dominates the negative direct one of higher income taxes leading to a reallocation of private resources from investment to consumption. However, it is not possible to calculate the term Ox/Orfor our analytical model in detail, because this expression becomes extremely complicated. This also motivates the use of numerical examples to gain further insight into our model. Another point of interest is the question of what happens if a less strict budgetary regime is assumed in which only a certain part of the government's interest payments must be financed by the tax revenue. Formally, that is obtained by setting (o4 ~ (0, 1) which gives regime (B). Now, the negative feedback effect of public investment financed by public debt is expected to be lower in magnitude since only (O4rB of the interest payments must be financed by the current tax revenue. Proposition 2 gives the precise result. PROPOSITION 2. Holding the national debt~private capital ratio fixed, a positive growth effect of higher deficit financed investment in public capital is the more likely the lower (O4.
Proof To prove the proposition we proceed as in the proof of Proposition 1. Doing the same steps as in that proof with (o0 now given by (o0 = ((01 -[- (O2) -}- (O4 (1 -- a)br -1 (1 + (1 - a)b) -1 and taking into account the derivatives of this term with--respect to r and b we get -(Oql(.)/Ob)(Oq(.)/ 0(O3) + (Oq(')/Ob)(Oql(')/O(o3) = (O4 - r ( 1 - (o0)(1 + (1 - a)b)[(o3 + ((O3 - 1)/(b(1 - a))]. In analogy to Proposition 1, the balanced growth rate rises if this expression is negative, that is, if Z1 --- z(1 - (o0)(1 + (1 - ~)b)[(oa + ((O3 - 1)/(b(1 - or))]-> (O4.Since (#4 E (0, 1) and (oopositively varies with (O4the second part of the proposition is proved. That proposition states that a positive effect of public investment financed by additional debt is the more likely the smaller the feedback effect of a higher level of government debt imposed by the budgetary regime. With that proposition one could be tempted to come to the conclusion that governments only have to follow less restrictive budgetary polieies and earl thus generate positive growth effects of a deficit financed public investment on the balanced growth rate. But two points must be taken into account. First, that proposition is only valid for a given debt/capital ratio, which is an endogenous variable. That is, if a less strict budgetary regime is in use this regime may cause a higher debt/capital ratio which probably compensates for the positive direct effect of less interest payments to be financed out of the tax revenue so that the economy may end up with less economic growth.
374
Endogenous Growth, Government Debt and Budgetary Regimes The second point we have not yet addressed is the question of whether such a balanced growth path exists at all, on which economic per capita variables grow at a constant rate. Above we mentioned that such a path is only feasible if the decline in the marginal product of private capital is compensated by investment in public capital. However, in our analytical model we unfortunately cannot answer that question more precisely because the resulting expression becomes too complicated. Therefore, we will conduct some numerical simulations to highlight the problem of the existence of a steadystate growth path and try to find whether a less strict budgetary regime probably makes endogenous growth impossible. Further it can be shown (see Greiner 1996, 4.2) that the balanced growth rate of the economy does not necessarily rise if a less strict budgetary regime is imposed. A higher value for (04 may well be compatible with a higher steady-state growth rate. This ambiguity results from the fact that (04 in this regime shows two different effects. On the one side, a lower value for (04 implies that fewer interest payments on public debt must be financed through the tax revenue, leaving more resources for investment in public capital. In our model that direct effect tends to decrease (00, which can be seen by differentiating (00 with respect to (04- On the other hand, there is an indirect effect of (04 by influencing the steady-state value for b. If a lower interest payment must be financed out of the tax revenue, the level of public debt will probably be higher and thus the steady-state value of public debt per private capital, b. That indirect effect tends to increase the amount of tax revenue used for the debt service, increasing (00 in our economy and, thus, tends to lower investment in public capital that will show negative repercussions for economic growth. Whether the direct effect dominates the indirect one, implying that a less strict budgetary regime is accompanied by higher economic growth, or vice versa cannot be determined for the analytical model and depends on the specific conditions of the economy under consideration. In the simulations we will present a numerical example where that case is illustrated, but first we present some analytical results for our model assuming the budgetary regime (C).
Regime (C) In regime (C) the government does not have to finance any part of its interest payments and investment out of the tax revenue. But the latter must be sufficiently high to meet public consumption and transfer payments. The constraint is then written as Cp + Tp = (POT,(00 < 1, which is obtained from (9) by setting (04 = 0. It should be noted that for (04 = 0, (00 is given by (00 = (01 + (02. The evolution of the government debt in that regime is /3
375
Alfred Greiuer and Willi Scrawler = rB + C p + Tp + (7, - T = rB + (1 - ~00)(~0a - t)T0 where(~ = ~0a(1 - ~0o)T and :~0a > t. Formally, regime (C) is obtained from (21)-(22) by setting (o4 = 0 and ta > t. It should be noted that for ¢a < 1 government expenditure exclusive of interest paYments are .smaller fhan the tax revenue, We do not consider fhis regime but refer to Greiner (i996, :chap. 4.2) Where it is analyzed in detail. Let us first consider regime (C) and try to find :some analytical results. To gai~ further insight° we explicitly compute b at the steady state from q ( . ) = 0 as
b ............... -- a~(~00- :t){1 -~03) :x-~p - ( 1 - - z ) : ( t - a ) + ~(i~)(I-~4 + ((eo~l)(t~{03)~)
(z3)
It is immediately seen that for Pa > !, (1 - r) > a ( l - ~04 + (:(o0 - t)(t - ~03)z) must hold ifb is to be positive, This demonstrates that for ~04 = 0, that is, for regime (C), ~ must be smaller than 1 - a, and thus smaller than 1, for a b.alav,ced growth path with a positive level of government :debt to exist. Since a is -the inverse of the instantaneous elasticity of substitution, the economic interpretation of this result is that the representative individual in this economy mnst ~have a utility function w~th an extremely high instantaneous elasticity of substittt-tion for,a balanced gmwth path with government debt to exist. That is, the household must he extremely willing to forgo current consumption and shift it i~to the future. As mentioned in the introduction, most empirical studies suggest that a takes on a value around 1. In this ease a balanced :grawth path can only exist in regime (C) ff the government has a negative level of debt, that is if it is a creditor. We state this result in proposition 3, PrtOPOSITION 3. In regime (C), 1 ~ • > ¢r is a uecessary condition for a balanced g~ow.th path with positi.,ve government debt to exist. If this inequality does not hold, a balanced growth path is necessarily associated with negative ,government debt; that is, the government must be a ,creditor. The proof follows immediately from the expression for b in steady state. To find the effects of fiscal policy on the balanced growth rate we proceed in analogy to the section titled Regime (A) and tB). Since the balanced growth rate which is given by (14) does not directly depend on (P3 the ,zariation in x, induced by a change of this parameter, shows whether the growth rate increases or declines, The growth effect of a deficit financed rise in public investment is given in Proposition 4. 376
Endogenous Growth, Government Debt and Budgetary Regimes P:aOI'OStTION 4, In .regime (C), a deficit financed increase in public invest-
ment raises the balanced growth rate. Pro@ Again det M > 0 holds. Then, the sign of (Oqi(-)/Ob)(Oq(')/O~oa) - (Oq~(.)/Ob) (aq~(-)Z0e3) determines the effect on x and, thus, on the balanced g r o ~ rate. If it is positive [he growth rate rises and vice versa. In .an appendix to the paper, Which is a~ailable on request, we show that ttais expression is ~nambiguous!y positive. This proposition states that a rise in public investment raises the hal_ aneed growth rate, This outcome is probably due to the fact that there is no ir~terrtal crowding out in ~is regime (-with ~0a > 1). But we should also like to ,emphasize that, for ~a < 1, -that is, if only interest payments on public ,debt :are :financed by public deficit and public investment, public consumption a~d transfer payments are financed by the tax revenue, a rise in pubhe investment has no ~nambiguons effect. That is it may raise or lower economic ,gr,o~, although there is no internal crowding out either. This :case is .studied in detail in Greiaer (!996, eh~p. 4,2), Further, ~e .should like to point out that this regime (C) is of less rdevanee for real world economies. That holds because one always has to be aware of the existence problem concerning the balanced growth path. For ¢r :-~ 1 - ~, which seems to be quite realistic, this proposition can only be app~ed if the government is a creditor. In the more .realistic ease of a positive go~ernment debt, the instantaneous ehsticity .of substitution has .to he very high for this proposition to be relevant. As to the gr ~owth maximizing income tax rate we imme~a.tely see that :it is given by the :same expression as in regimes (A) and (B)and cannot be precisely determined in the analytical model. Therefore, we present some numerical simulations to find its magnitude. Before ~e continue the presentation let us briefly summarize ,,~hatwe have achieved. We ,have analyzed how a government may influence the batmaced growth rate by increasing productive investment in public ,capitat, As to the financing of this spending.we ha~e assumed that the government issues new ~ d s , It t,urned out that the results crucially depended on the budgetary regime nnder consideration, For regime (A) we saw that a deficit financed increase ,of investment in infrastructure raises balanced growth if a certain condition is met. In particular, a positive effect is more likely the lower the s'hare of non-productive government :spending, other things , e q ~ . Further, we could show that applying this regime less restrictively, that is, by requiring that only a certain part of,debt payments must be financed out of the tax revenue, .giving regime (B) a positive effect of deficit financed increases in public investment is more likely, but only for a given debffcapital 377
Alfred Greiner and Willi Semmler ratio which, however, is an endogenous variable that is expected to be the higher the less restrictive the budgetary regime is. As to the growth rate itself, no concrete results could be derived, implying that both cases could occur. So, a less strict budgetary regime could be accompanied either by a higher or lower balanced growth rate. In regime (C), where only public consumption and transfer payments are financed by the current tax revenue, we found that only for extremely high values of the instantaneous elasticity of substitution can a balanced growth path with a positive government debt exist. For ¢r -> 1, what most empirical studies consider as realistic, the government must be a creditor if endogenous growth is to hold. Moreover, we could show that in this regime a deficit financed increase in public investment always raises economic growth. In the next section we intend to present some numerical examples to illustrate our analytical results and to shed some light on the question of whether the existence of a balanced growth path is influenced by the choice of the budgetary regime as well as to find the growth maximizing income tax rate, because these questions could not be answered within our analytical framework.
4. Numerical Examples To start with we fix some of our parameter values. As to the elasticity of output with respect to public capital we take the same value as in Barro (1990) and set a = 0.25. The utility function is supposed to be logarithmic, giving ¢r = 1. The discount rate is taken to be p = 0.3. Interpreting one time period as 5 years then implies that the annual discount rate is 6%. These values are kept constant throughout all simulations. First, let us present an example to illustrate regime (A). In this regime all expenditures, with the exception of public investment, had to be financed out of the tax revenue. The parameter values for ~01and ~2, giving that part of the tax revenue which is used for transfer payments and public consumption, are set to ~01 = 0.3 and ,co2 = 0.35. These are about the ratios of transfers to individuals per total government revenue and of public consumption per total government revenue for West Germany for the mid eighties (cf. Sachverst~indigenrat 1993, table 38). The parameter ~a is set to ~a = 1.5. For these parameters, Table 2 reports the steady state values 8 for x, b and ~0 as SFor all simulations, the dynamic behavior of the system is characterized by saddle point stability, The eigenvalues of the Jacobian are given in an appendix available on request.
378
Endogenous Growth, Government Debt and Budgetary Regimes TABLE 2. z
x
b ,,
0.15 0.20 0.22
g ,
0_153 0.0511 0.208 0.0694 0.232 0,0772
,,,
9)0 ,
0,0990 0,1052 0.1058
, , ,
i
r ,
0.896 0.897 0.899
x
b
,,
g i
0.23 0.25 0.30
0.244 0.269 0.337
0.0813 0.0896 0.1124
,
9)0 ,
0.1058 0.1051 0,1001
0.900 0.902 0.909
well as the growth rate for different income tax rates. 9 The values are rounded to the third and fourth decimal point, respectively. Table 2 shows that the maximum growth occurs for income tax rates of about 22-23%, which are a little smaller than the elasticity of output with respect to public capital a. It must be noted that we took one time period to comprise 5 years so that the annual growth rate is about 2%. If we raise the parameter 9)3, that is, if we increase public investment fnanced by additional debt and set 9)s = 1.65, we get the following results, shown in Table 3. TABLE 3.
0,15 0.20 0.22
x
b
g
9)0
0.142 0.192 0.214
0.0559 0,0758 0.0843
0.0912 0.0974 0.0979
0.918 0.919 0.920
r
x
0.23 0.225 0.25 0.248 0.30 0.310
b
g
9)0
0.0887 0.0977 0.1220
0.0978 0.0969 0.0917
0.921 0.923 0.930
This demonstrates that the feedback effect in regime (A) is so high that the positive effect of higher investment in public capital is more than compensated by the additional interest payments generated through deficit financing. In this ease, increasing 9)3 leads to less public investment because the additional interest payments caused by this fiscal policy increase 9)0 in a way so that in the end less resources are available for public investment. In this situation reducing the government debt or public consumption or transfer payments sets flee resources which earl then be used for public investment and may raise economic growth. We also see that the ratio of public debt to private capital, b, rises, That was to be expected because a higher 9)3 means a higher public deficit, other things equal. It should be mentioned that a deficit financed increase in public investment is to be seen as a crowding out of private investment if this policy 9There exist two steady states with these parameters. But the second yields ~0 = 1 and a zero growth rate.
379
Alfred Greiner and Willi Semmler TABLE 4. r
x
b
g
0.15 0.20 0,21
0.131 0,178 0.187
0.0599 0.0813 0.0860
0.0837 0.0895 0.0896
~00
r
x
0.922 0,22 0.196 0.923 0.25 0.224 0.924 0.30 0.265
b
g
~00
0_0908 0.1061 0.1360
0.0894 0.0869 0,0765
0.925 0.930 0.943
reduces the ratio x = G/K. This holds because a lower value for x reduces the return to private investment, that is, lowers the marginal product of private capital r = (1 - o0x~. With the parameter values in Table 2, a rise in ~0a leads to: c~r
3~3 3r
0~3 Or 0~3 ar
0.0819 for r = 0.15 ; 0.0743 for r = 0.2 ; 0.0818 for r = 0.25 ; 0.0910 for r = 0.3 .
&Pa In Table 4 we present the results for ~04 = 0.95, meaning that 95% of the interest payments must be met by the tax revenue, whereas the rest is financed by additional debt. All other parameters are as in the simulation for Table 2. That table illustrates the discussion following Proposition 2. It shows that a less strict budgetary regime does not necessarily lead to a higher balanced growth rate. The reason for that outcome is that, in this case, the parameter ~00is larger than in the economy with regime (A) and, therefore, investment in public capital is lower. This result holds because the direct effect of less interest payments which must be financed out of the tax revenue, which would lead to a lower q0, is more than compensated by the higher level of government debt per private capital, b, which tends to increase ~0, so that the economy ends up with a higher value for ~0. But it must be emphasized that the reverse effect could probably also be observed and the outcome depends on the specification of the economy. This holds because for the analytical model no unambiguous result could be obtained. To find how this economy reacts to increases in public investment financed by a higher debt, we calculate the derivative of x with respect to ~3, yielding 380
Endogenous Growth, Government Debt and Budgetary Regimes &¢
-0.320537
0~3
2.67813
-
0.119687 for ~ = 0.15 ;
0x d{o3
-0.266592 1.65434
0.161147 for r = 0.2 •
#x 0~o3
-0.151911 0.435745
0.348625 for T = 0.30 .
-
These results show that Oxl3~3is still negative for ~94 0.95. This effect is again due to the higher debt/capital ratio b which causes an increase in ¢0, which offsets the positive effect of a lower ¢4. Therefore, this example underlines that proposition 2 is only valid for a given debt/capital ratio. If we further decrease ~04and set P4 = 0.9, the balanced growth path does not exist any longer, meaning that there is no sustained per capita growth for our economy. This demonstrates that a less strict budgetary regime does probably not allow sustained per capita growth at all. Reducing ~4 further, we observe that sustained per capita growth is again possible for ¢4 around ¢4 = 0.05. But it must be emphasized that this regime is only feasible if the government is a creditor, that is, for a negative level of public debt. For ~04 = 0 (~4 = 0.05) the steady-state value for b is b = - 0.0641 (-0.0718). The balanced growth rate associated with this steady state is 0.187 (0.190). Recalling that one time period comprises 5 years, this corresponds to an annual growth rate of about 3.74 (3.8)%. =
5. Conclusions The goal of this paper was to analyze an endogenous growth model with productive government spending and public debt. In our study, the government does not have to run a balanced budget at any moment in time, but it must stick to certain budgetary regimes, which impose constraints on the government's fiscal behavior. It should be noted that we cannot resort to the Ricardian equivalence theorem in our model because it contains distortionary taxes and productive government spending. We demonstrated that less strict budgetary regimes, that is, economies where a greater fraction of interest payments may be financed by raising public debt, make a positive growth effect of a deficit financed increase of public investment more likely. But it must be stressed that this outcome only holds if the public debt to private capital ratio, which is an endogenous variable, is held fixed. Simulations demonstrated that less strict budgetary regimes are associated with a higher government debt which offset the positive direct growth effect of a less strict budgetary regime. Further, we could demonstrate that a less strict budgetary regime possibly does not allow 381
Alfred Greiner and Willi Semmler sustained per capita growth or that it does only if the government owns assets and is a lender. That result became particularly obvious in regime (B), where sustained growth with positive public debt did not exist once the fraction of interest payments paid by raising public debt exceeded a certain threshold level, Those outcomes demonstrate that imposing restrictions as to public deficits may in fact be reasonable from an economic point of view. Our results are of relevance as to the discussion about the budgetary goals of the EMU. Here we have only provided a theoretical model to discuss those issues. For an empirical methodology and estimation strategies, see Greiner and Semmler (1999). Note, however, we do not consider effects of monetary policy in our model and the role of budgetary rules as concerns the effects of inflation (see Woodford 1996). Moreover, we saw that there exists a growth maximizing income tax rate as in the models by Barro (1990) and Futagami, Morita and Shibata (1993). However, we should also like to emphasize that maximizing economic growth need not necessarily be equivalent to maximizing welfare, even if one limits the investigations to the balanced growth path. The reason is that a fiscal policy which reduces the balanced growth rate, like an increase in public transfers or in the tax rate above its growth maximizing value, may raise the level of private consumption. A rise in the level of private consumption has a positive welfare effect which can outweigh the negative one of a lower growth rate. This can be proved rigorously for this model without public debt (cf. Greiner and Hanusch 1998) as well for the Barro (1990) model (cf. Greiner 1998). For the model presented in this paper, however, a formal proof is not possible since public debt makes the analysis too complicated. Another point which should be made clear is that in our model we could only take into account one source of economic growth, namely productive government spending, while others, such as production externalities or efforts to increase human capital and knowledge, were neglected because they would have made the model intractable. Therefore, the result of our simulations should be interpreted with some care if one wants to evaluate the quantitative effect of fiscal policy in real economies. Our numerical examples pertain to the demonstration of the analytical model and its qualitative results which indeed seem to be very robust.
Received: May 1998 Final version: October 1998
382
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