Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue2.Feb2000:
Journal of Economic Dynamics & Control
24 (2000) 227}246
Public services, increasing returns, and equilibrium
dynamics
Junxi Zhang*
School of Economics and Finance, The University of Hong Kong, Pokfulam Road, Hong Kong
Received 1 January 1997; accepted 30 November 1998
Abstract
This paper is concerned with casting o! the standard assumption } the production
function displays socially constant returns to scale } in a simple growth model with
public inputs. In the model, public services work as a source of utility and as an input to
production. Making use of the global bifurcation technique, we show that the model can
generate a whole spectrum of interesting dynamics, which do not arise with constant
returns. Speci"cally, when increasing returns are not exceedingly strong, not only does an
indeterminate steady state take place, but also there exist divergent perfect foresight
paths in expanding oscillations and economic cycles. The role of history and expectations
in shaping equilibrium is also considered. ( 2000 Elsevier Science B.V. All rights
reserved.
JEL classixcation: H42; O41; C62
Keywords: Public services; Increasing returns to scale; Indeterminacy of equilibrium;
Self-ful"lling expectations; Economic cycles
1. Introduction
In recent years there is growing interest in studying the relationship between
economic growth and public services. This literature builds on Barro's (1990)
seminal model of endogenous growth with a public sector. In the model, the
government uses tax revenue to "nance public expenditures, which include the
provision of infrastructure services, the protection of property rights, and other
* Corresponding author. Tel.: 852 2857 8502; fax: 2548 1152; e-mail: jjzhang@econ.hku.hk.
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 8 8 - 8
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J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
productive activities. Under the assumption that the production function displays socially constant returns to scale, it is found that public services, especially
infrastructure services, are essential for persistent growth. This policy implication has often been proposed to developing countries which face serious problems of scarcity of public goods, in particular infrastructure.
The central assumption embedded in this class of models deserves careful
scrutiny. First, the proviso, albeit helps reduce our e!orts in calculating growth
rates, eliminates some important features which arise in dynamic economies.
There is a wide agreement that it is primarily responsible for the absence of
transitional dynamics in the Barro model, since the model can be essentially
reduced to a version of the AK model.1 By contrast, the transitional dynamics
should allow us to understand the behavior of the system in the short run.
Second, accurate analysis of growth in less developed economies suggests that
these countries actually experience substantial increasing returns to scale during
the process of economic development, particularly at early stages. Thus, it is
necessary to expand the class of production functions for a more adequate
description of these economies. Third, the developments in economic modeling
for the past decade or so have been emphasizing non-standard elements,
including scale economies. Failure to take them into account seems di$cult to
reconcile.
This paper is concerned with casting o! the above standard assumption, that
is, the production function is only a narrow class of socially constant returns to
scale. In the model, government expenditures serve as a source of utility and as
an input to production (see also Abe, 1995). Under rather general functional
forms of utility and production, we show that the model can generate many
interesting dynamics. With mild increasing returns,2 not only does an indeterminate steady state take place, but also there exist divergent perfect foresight paths
in expanding oscillations and economic cycles, two possibilities which are
relatively unknown in the existing literature.
1 There are some attempts which adopt modi"ed versions of the Barro model in order to induce
transitional dynamics. Futagami et al. (1993) introduce public capital and demonstrate this possibility. Notably, the local stability of the transitional path is simple: the steady-growth equilibrium is
a saddle point. In a similar model, Glomm and Ravikumar (1994) reach the same conclusion.
A recent paper by Abe (1995) takes a di!erent approach. He generalizes the assumption of linear
homogeneity in the income identity and allows for increasing returns, and shows that indeterminacy
of equilibria may arise. Although our model is related to this strand of research, we di!er in two
main aspects in addition to the primary concern. First, Abe's analysis is of the local nature, while
ours is of the global nature since we make use of the global bifurcation technique. Second, our model
generates much richer dynamics, including the existence of economic cycles and multiple perfect
foresight paths in the vicinity of an unstable equilibrium, which are absent from the Abe model.
2 In this paper, we say that increasing returns are mild (strong) if the share of public services in
production does not (does) exceed one; see discussions in Sections 3 and 4 for details.
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
229
More speci"cally, there are three distinct cases when increasing returns are
mild. The "rst one involves in that the elasticity parameter of utility to government spending is su$ciently large. We "nd that the system gives rise to a unique
interior equilibrium which turns out to be a sink, and thus it yields a continuum
of equilibrium paths converging to the stationary point. The second case
concerns a moderate elasticity. It is shown that the interior equilibrium is
a source. Taking into account two terminal points, an immediate question arises
as to which state the economy actually gets established. The nature of this
question entails one to carry out an analysis of the global perfect foresight
dynamics. The information on the local dynamics is not enough, because, for
example, the existence of a unique perfect foresight path in a neighborhood of
a stationary state does not necessarily rule out the existence of other perfect
foresight paths in the large. Global dynamics also bring the possibility of
a meaningful and thorough understanding of the history versus expectations
distinction, an issue known in other "elds of economics: industrial organization,
international trade, macroeconomics, etc (e.g., see Krugman, 1991; Matsuyama,
1991). Making use of the global dynamics technique, it is possible to determine
precisely under what circumstances history matters and when expectations may
be decisive.
Perhaps the most interesting case is that when the elasticity parameter is
equal to a critical value, economic cycles emerge. The existence of cycles is
established using the Hopf bifurcation theorem (see Guchenheimer and Holmes,
1990), and a detailed description of the limit cycle, including its explicit expression, bifurcated period and stability conditions, is presented. Since the ultimate
reason for studying nonlinear cycles is to develop an endogenous theory of the
business cycle that can compete with the dominant linear stochastic paradigm,
our "nding is useful in that it provides yet another mechanism for economic
systems to generate periodic equilibria endogenously.
On the other hand, when increasing returns are very strong, the local stability
of the interior equilibrium changes entirely because it becomes a saddle point.
Given an initial condition, there exists a unique equilibrium path which converges monotonically to the stationary state.
The rest of the paper is organized as follows. In Section 2, we cast the model
and derive a system of nonlinear di!erential equations. Section 3 performs the
global analysis for the case of mild increasing returns to scale, while Section 4
brie#y considers the case of strong increasing returns. Finally, Section 5 concludes the paper.
2. The basic model with mild increasing returns to scale
The model economy is populated with a continuum of in"nitely lived agents,
whose measure is normalized to one. The representative agent supplies one unit
230
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
of labor services inelastically, and at any moment she chooses consumption so
as to maximize the discounted sum of utility, given by
P
=
e~ot u(c , g ) dt,
(1)
t t
0
where o is the subjective discount rate, c is private consumption, and g is
t
t
government consumption. Notice that in Eq. (1), government spending directly
a!ects consumer behavior, since she derives utility from public goods and
services, for example, public roads, parks, schools, etc. We assume that although
she chooses the level of consumption by participating in free markets, she takes
government expenditure as given. For simplicity, the instantaneous utility
function presumably possesses the form3
; "
0
(2)
u(c , g )"(1/p) (cpgg),
t t
t t
where p (1 and g 4 1 are parameters. The parametric restriction of p (1
makes utility strictly concave in the choice variable c . If g '0 (g (0), then
t
private consumption and government expenditure are pairwise Edgeworth
complements (substitutes).4 However, when g ( 0, it suggests a minor technical
di$culty. To ensure that the government consumption's marginal utility is
positive, Eq. (1) is modi"ed as
P
=
e~ot [u(c , g )#v(g )] dt,
(1a)
t t
t
0
where v@'0. Nevertheless, under the usual assumption that individuals have no
control over g , the maximization problem can be solved by ignoring g 's
t
t
contribution to ; through the function v. Hence, Eqs. (1) and (1a) are treated as
0
operationally equivalent counterparts throughout this paper.
Following Barro (1990), we assume that public goods and services, such as
infrastructure, also serve as an input to private production.5 But, in addition, we
allow increasing returns to scale:6
; "
0
f (k , g )"kagb,
t t
t t
(3)
3 It should be noted that the functional forms of utility and production in this paper are chosen so
as not only to retain their main features but also to allow for explicit solutions of the equilibria.
4 Two goods are said to be Edgeworth complements (substitutes) if the marginal utility of one
increases (decreases) as the quantity of the other increases.
5 Throughout this paper we regard public services as if they are publicly-provided private goods,
which are rival and excludable. For more general settings, see Barro and Sala-i-Martin (1992).
6 The existence of an equilibrium in this type of models with a general convex production function
but without public goods has been stated in Skiba (1978).
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
231
where a#b'1 but b (1; and k is capital stock. In this section, the case with
t
b (1 is referred to as &mild' increasing returns, while the case of strong
increasing returns with b'1 will be studied in Section 4. Furthermore, to make
the model interesting, we postulate that the economy's total capital stock per
person is kM . Consequently, the domain for the capital stock is restricted to [0, kM ].
To "nance spending on public goods and services, the government has access
to a tax on income, q. The budget is balanced every instant, so that7
(4)
g "q f (k , g )"q kagb
t t
t
t t
The consumption-saving decisions of the representative agent are determined by
solving the following problem: maximizing Eq. (1) subject to
(5)
c #kQ "(1!q) f (k , g ),
t
t t
t
the nonnegativity conditions c 5 0 and k 5 0, and an initial condition on
t
t
k , k '0. The current-value Hamiltonian for this problem is (super#uous time
t 0
subscripts are dropped hereafter):
H"(1/p) (cpgg)#j[(1!q) f (k, g)!c],
where j is costate variable. The "rst-order necessary conditions include
j"cp~1gg,
(6a)
jQ "j[o!a(1!q)qb@(1~b)k(a`b~1)@(1~b)],
(6b)
and the associated transversality condition is
lim j k e~oT"0.
(7)
T T
T?=
Substituting Eqs. (3), (4) and (6a) into the budget constraint (5), it follows that
qg@ckag@c
,
kQ "(1!q)qb@(1~b)ka@(1~b)!
j1@(1~p)
(8)
where c,(1!b)(1!p)'0. Eqs. (6b) and (8) jointly de"ne a planar dynamical
system in (j, k) on [0,R)][0, kM ]. Since j is a &jumping' variable, the initial
value for j, j , must be chosen to make a path consistent with these equilibrium
0
conditions. In this model the notion of history is captured by the capital stock.
3. Global dynamics
In this section, we carry out a global analysis of the dynamical system (6b)
and (8). Before proceeding, "rst the local dynamics are analyzed. An interior
7 The upper bound for k, kM , is then equivalent to an upper bound for g via this equation.
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J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
stationary state (jH, kH) is determined by the following relations: jQ "0 and
kQ "0, leading to the following equilibrium values:
C
kH"
D
(1~b)@(a`b~1)
aqg@(1~b)
o
, jH"
(kH)(ag@(1~b))~(1~p). (9)
o
a(1!q)qb@(1~b)
It can be readily veri"ed that the transversality condition (7) is met at the
equilibrium point. As shown in Appendix A, the equilibrium dynamics depend
on the sign of the trace of the Jacobian coe$cient matrix of the system. If the
trace is negative (positive), called Case A (B) below, the equilibrium is a sink
(source). The borderline case of a zero trace remains important in this model,
suggesting that the solution curve of the dynamical system, Eqs. (6b) and (8), is
a Jordan curve, i.e., a closed curve that does not intersect itself. We discuss these
global dynamical features of the system case by case.
Case A: g '1!p. This case entails the elasticity parameter of utility to
government spending, g, to be su$ciently large. First consider the behavior of
the trajectories in the phase diagram. The curve for which jQ "0 is vertical, while
the curve for which kQ "0 is upward sloping, which is determined by the
following equation:
q*g~b(1~p)+@(1~b)
ka*g~(1~p)+@(1~b).
j"
(1!q)1~p
(10)
There are three possible shapes of this locus. When a[g!(1!p)]'1!b, it is
convex; when a[g!(1!p)](1!b, it is concave. For the knife-edge case of
a[g!(1!p)]"1!b, it is a straight line from the origin. Fig. 1a and b depicts
the phase portraits for the "rst two possibilities, while the transparent borderline
case is not drawn. Notably, the jQ "0 locus is the standard modi"ed golden rule
for capital accumulation in the presence of taxation and public goods, which
says that ultimately the (after-tax) productivity of capital is determined by the
rate of time preference. The unique interior equilibrium point is denoted as E.
The two loci divide each phase portrait into four regions, where the components
of directions of movement are indicated by arrows: jQ '0 in the left-hand side of
the locus jQ "0 and jQ (0 in the right; above the locus of kQ "0, kQ '0, and kQ (0
below it.
Next, this case features that the determinant is positive while the trace is
negative. A negative trace and a positive determinant mean that there is a locally
indeterminate steady state, i.e., there exists an indeterminate steady state with
a continuum of equilibrium trajectories converging to the steady state. In
addition, because of the local indeterminacy, there may be stationary sunspot
equilibria in the neighborhood of E.
The following proposition presents the results for Case A (see Appendix A):
Proposition 1. ;nder the assumption of mild increasing returns of scale, a#b '1
and b (1, the dynamical system has an indeterminate steady state in the interior,
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
233
Fig. 1. The dynamical system for Case A: (a) a[g!(1!p)]'1!b; and (b) a[g!(1!p)](1!b.
E, if g '1!p. More speci,cally, when g (g(1, E is a stable node; when
H
g(g , E is a stable focus.
H
Fig. 2a and b displays these possible kinds of equilibrium dynamics. The two
terminal points, at which k"0 and k"kM , are denoted as E and E (lower and
L
U
234
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
Fig. 2. The equilibrium dynamic paths for Case A: (a) g (g(1; and (b) g(g .
H
H
upper bounds for k), respectively. The upper bound for k, kM , means that the
economy operates at its full potential. The multiple equilibrium paths in Fig. 2b
are of particular importance, as it shows the existence of sunspot equilibria in
the vicinity of E. Suppose that initial capital stocks for two economies were quite
close, like those in South Korea and the Philippines not so long ago. The above
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
235
Fig. 3. The dynamical system for Case B.
analysis makes evident that their subsequent growth experience could dramatically di!er, which may be the result of indeterminacy of equilibria, that is,
di!erent counties follow di!erent equilibrium trajectories toward a stationary
state or a balanced growth path. This result is reminiscent of the recent literature
of indeterminacy without public goods, e.g., Benhabib and Farmer (1994) and
Benhabib and Perli (1994).
Case B: g (1!p. This case is profoundly di!erent from Case A. First, as
shown in Fig. 3, the kQ "0 locus becomes downward sloping and convex
(also see Eq. (10)). Second, the local stability of the unique interior equilibrium E changes drastically: it becomes a source since the trace is positive (see
Eq. (A.3) in Appendix A). Third, the lower terminal point, E , becomes instead
L
(0,#R), which is equivalent to c"0 via Eqs. (6a) and (10). This state with
zero consumption and capital stock can be viewed as a state of poverty
and stagnation, which generically corresponds to the &poverty trap' in the
literature.
Given that the unique interior stationary point is a source, the solution to the
dynamical system will hit either boundary eventually, for any initial values,
k and j , that satisfy k 3(0, kM ). Following Krugman (1991), these two positions
0
0
0
that the economy reaches in the long run can also be loosely classi"ed as
equilibrium points. Then, several interesting questions immediately arise. Which
equilibrium does the economy go to? Does the initial condition matter? When
might self-ful"lling expectations play a role? The answers to these questions
236
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
impinge on the nature of the characteristic roots for E. As shown in Appendix A,
we can state:
Proposition 2. Suppose that the following conditions hold: a#b '1, b (1, and
g (1!p. ¹hen the unique interior equilibrium E of the system is a source. If g
L
( g (1!p, E is an unstable focus; when g(g , E is an unstable node.
L
When g (g(1!p, the linearized system diverges from the stationary state
L
in expanding oscillations. The kind of equilibrium dynamics is delineated in
Fig. 4a, in which perfect foresight paths consist of a pair of interwining, noncrossing spirals around E. As the "gure makes clear, the set of equilibria depends
on the size of the initial capital stock relative to the benchmarks k and k ,
1
2
de"ned by the projections onto the k axis of the left-most point of E 's stable
L
manifold and the right most point of E 's stable manifold, respectively. The
U
interval (k , k ) is generically called the &overlap' region of the stable manifolds
1 2
associated with E and E , where the extent of the overlap region depends in
L
U
a complicated way on parameter values. When initial capital stock falls out of
that region, determinate equilibrium dynamics emerge. If k 3(0, k ), there exists
0
1
an equilibrium path that the economy will be trapped in the lower terminal
equilibrium E . Similarly, if initial level is su$ciently high such that k ' k ,
L
0
2
the economy converges monotonically to E . In both circumstances, history
U
dictates the choice of equilibrium in the long run.
If k 3(k , k ), there are multiple trajectories that satisfy all the equilibrium
0
1 2
conditions, and the initial condition is no longer su$cient to determine the
outcome of the economy in the future. The economy can reach the high
equilibrium E even when the economy is initially located to the left of E, if
U
agents coordinate their expectations on the equilibrium path converging to E .
U
Fig. 4. The equilibrium dynamical paths for Case B: (a) g (g(1!p; and (b) g(g .
L
L
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
237
Given the perfect foresight nature of the model, expectations are entirely
self-ful"lling. In addition, the model reveals that starting from the right of E does
not guarantee the convergence to E . In a word, expectations will matter when
U
the initial capital falls into the overlap region. The policy implication is obvious
like that proposed in other studies: to break the vicious cycle of poverty, it calls
for active state intervention.
The above result accords reasonably well with intuition. In this model, if
b (1, the level of public services is an increasing function of the level of private
capital, as seen from Eq. (4). Suppose some agents increase their levels of capital.
Then the level of public services will increase, which in turn will raise the
marginal return of capital and induce everyone to increase capital accumulation.
In other words, there gives rise to a positive capital spillover. As argued by
Krugman (1991) and Matsuyama (1991), if expectations are coordinated in
a way that everyone believes that the economy will end up in the high equilibrium, then it will.
When g(g , E possesses two distinct, real, positive eigenvalues. Hence, the
L
only compatible portrait belongs to a scenario that determinate perfect foresight
paths emerge in that the linearized system steadily diverges from the stationary
state, no matter where the economy starts. Fig. 4b depicts this case. There is
a &threshold' for capital: when k ( k*, the economy converges to the low
0
terminal point E ; when k ' k*, it converges to the high terminal point E .
L
0
U
Clearly, history plays a decisive role in determining the long-run position of the
economy.
¹he borderline case: g "1!p. Denote this value as g . This case remains the
0
most interesting one, because economic cycles now take place. To show this, we
notice that the determinant is positive, while the trace is equal to zero, thereby
implying that the Jacobian has a pair of pure imaginary eigenvalues. If we de"ne
a real positive number
4o2(a#b!1)
'0,
d ,
0
ac
the two pure imaginary eigenvalues at g are equal to id and !id , respec0
0
0
tively. Moreover, it is straightforward to show that at g"g , the real part
0
of the derivative of eigenvalue with respect to g does not vanish. Thus, the
loss of stability can be guaranteed by small perturbations in the bifurcation
parameter g. According to the Hopf bifurcation theorem (Guchenheimer
and Holmes, 1990), it follows that
Proposition 3. Suppose that a#b '1, b (1, and g "1!p. ¹hen, there exist
economic cycles } the Hopf bifurcation } in the system.
In what follows, we will give a detailed description of the limit cycle, including
its explicit expression, bifurcated period and stability conditions. To do so, we
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J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
write the Jacobian of the system, evaluated at g"g , as (also see Eq. (A.1) in
0
Appendix A):
C
0
J(g ),
0
J (g )
21 0
C
D
!J (g )
12 0
0
D
o(a#b!1) jH(g )
0
!
1!b
kH(g )
0 ,
(11)
"
o kH(g )
0
0
ag jH(g )
0
0
where J (g ) '0, J (g )'0, jH(g )"[q/(1!q)]g0, and k*(g ) is given by
12 0
21 0
0
0
Eq. (9). Introducing two new variables x,k!kH and y,j!jH, then the two
di!erential equations become
0
yR (g )"!J (g ) x(g ),
0
12 0
0
(12a)
xR (g )"!J (g ) y(g ),
0
21 0
0
(12b)
These equations can be transformed into a standard second-order equation for
x(g ):
0
d2x(g )
0 #J (g ) J (g ) x(g )"0.
12 0 21 0
0
dt2
(13)
Similarly, one can obtain the same equation for y(g ). Eq. (13) is the well-known
0
van der Pol equation in the absence of damping. In order to determine a particular solution, we now need two side conditions on the path of each variable.
Suppose that the system initially starts at the point (0, y ). The other condition
0
for each variable is then determined by Eqs. (12a) and (12b). Accordingly, the
system settles into the following oscillations:
y "y cos(u(g )t),
t
0
0
(14a)
x "y JJ (g )/J (g ) sin(u(g )t),
(14b)
t
0
21 0 12 0
0
where u(g ),[J (g )J (g )]1@2 is the oscillation frequency.8 The bifurcated
0
12 0 21 0
cycle has period 2p/u(g ). It is observed that the amplitude of the x oscillation,
0
t
y [J (g )/J (g) )]1@2, plays a crucial role in shaping the cycles. Attention now
0 21 0 12 0
turns to determine the relative magnitude of J (g ) and J (g ). Unfortunately,
12 0
21 0
explicit comparison cannot be made. There appears to exist a critical value for
8 From the initial and subsequent positions, we can see that the points move clockwise.
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
239
the equilibrium ratio of jH(g )/kH(g ):
0
0
C D S
jH(g )
1!b
0 "
.
kH(g )
ag (a#b!1)
0 c
0
Evidently, if jH(g )/kH(g )l[jH(g )/kH(g )] , J (g )mJ (g ). As a result, we
0
0
0
0 c 21 0
12 0
have
Proposition 4. For J (g ) ' (() J (g ), the contour of the limit cycles is
21 0
12 0
a family of ellipses with k (j) as the major axis. For J (g ) " J (g ), it becomes
21 0
12 0
circular.
Fig. 5a}c depicts the three possibilities, in which the kQ "0 locus is horizontal.
Next, the stability conditions are examined. To make the attempted task
manageable, we still work with the linearized system but slightly perturb the
bifurcation parameter g, while all other parameters remain "xed. The resulting
Jacobian as opposed to Eq. (11) is
C
0
D
C
!J
12 "
J,
J
J
21
22
D
o(a#b!1) jH
!
kH
1!b
,
o
kH
o[(1!p)!g]
a(1!p) jH
c
0
(15)
where jH and kH are given by Eq. (9). Notice that the sign of J and J are
12
21
positive, while that of J is indeterminate. When g approaches g from the
22
0
above (below), J is negative (positive). The presence of this term yields the
22
following equations:
yR "!J x,
(16a)
12
xR "J y#J x.
(16b)
21
22
Consequently, we can obtain the more general version of the van der Pol
equation
dx
d2x
#u2x"J
,
22 dt
dt2
(17)
where u,(J J )1@2. J is the so-called damping rate. It is known that
12 21
22
a positive (negative) value for J means that the amplitude of the oscillations
22
grows (decays); also see Appendix B. Thus, we can prove the following result:
Proposition 5. ¹he stability of the bifurcated cycle is determined by the sign of J .
22
If J is positive, the cycle is unstable; while if it is negative, the cycle is
22
supercritically stable.
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J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
Fig. 5. The equilibrium dynamic paths for the Borderline Case: (a) J (g )'J (g ); (b)
21 0
12 0
J (g )"J (g ); and (c) J (g )(J (g ).
21 0
12 0
21 0
12 0
This proposition makes clear that all stable cycles are supercritical. Supercritical bifurcation means that the location of the equilibrium changes smoothly
with parameter value near the bifurcation point. Let us see why it is supercritical, but not subcritical, bifurcation that matters in this model. The preceding
discussion suggests that when the bifurcation parameter g, the elasticity of utility
to government expenditure, equals g , the system is neutral, i.e., it is neither
0
stable or unstable. Hence, the structure of the system is quite sensitive to this
elasticity parameter. Whenever it is increased, the equilibrium becomes stable;
whenever it is decreased, the equilibrium becomes unstable. However, there is no
structural change in the system.
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
241
It should be pointed out that estimates of the preference parameters p and g in
a framework with utility-enhancing public services have not yet been obtained.
As a result, the question of whether there are economic cycles in the borderline
case of g"1!p remains unanswered. At this stage, we are still unable to rule
out such a possibility. Hopefully, our theoretical demonstration of the potential
importance of this case may encourage empirical work on this issue.
4. The case of strong increasing returns to scale
In this section, we brie#y consider a special case of increasing returns: the
share of public services in production, b, exceeds one, i.e., b'1.9 Unlike what
happened in the previous section, the local stability of the unique stationary
state E and the entire equilibrium dynamics completely change. First, the shapes
of the kQ "0 locus alter for both Cases A and B. From Eq. (11), the curve
becomes downward sloping for Case A, while upward sloping for Case B.
Moreover, in Case B, the locus can be either convex, or concave, or a straight
line from the origin, depending on certain parameter values.
Second, the local stability of E changes as well. The condition of b'1 (or
equivalently c(0) makes the determinant negative (see Eq. (A.2) in Appendix A), which yields:
Proposition 6. =hen increasing returns are so strong that b '1, the unique
equilibrium is a saddle point.
This proposition holds because one of the roots of the coe$cient matrix is
positive and the other is negative. Fig. 6a and b displays the phase portraits for
Cases A and B, and similar diagram can be drawn for the borderline case. These
"gures suggest that given an initial capital stock k 3(0, kM ), there is a unique
0
equilibrium path, which converges monotonically to the stationary state. That
path belongs to the stable manifold of the stationary state.
Intuitively, we note from the government budget constraint (4) that if increasing returns are very strong, the level of public goods eventually becomes
a decreasing function of private capital. Even if some individuals choose to
increase their levels of capital, self-ful"lling equilibrium dynamics will not arise.
The reason is as follows: the level of public goods decreases, which inevitably
lowers the marginal return of capital, and thus others have no incentive to
coordinate their expectations by increasing capital accumulation.
9 It can be shown that the case of decreasing returns to scale produces qualitatively the same
implication, except that Cases A and B are simply switched.
242
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
Fig. 6. The equilibrium dynamic paths for b'1: (a) Case A; and (b) Case B.
5. Concluding remarks
In this paper we have examined the implications of relaxing the standard
assumption } the production function exhibits socially constant returns to scale
} in a simple growth model with public goods. It was found that the model is
capable of generating a whole spectrum of interesting dynamics, which do not
arise with constant returns. Two features are worth noting for the case of
moderate increasing returns. The "rst one involves in a possibility that the
dynamic path diverges from an interior equilibrium in expanding oscillations,
suggesting that self-ful"lling expectations play an essential role in determining
the long-run position of the economy. The second one is that economic cycles
take place, implying that the endogenously determined government spending is
also periodic when the system is in the neighborhood of the equilibrium.
Our "ndings are obtained within a decentralized stationary framework. We
believe, however, that the model can be extended to more general settings,
such as incorporating public investment and endogenizing government decisions. Another interesting extension lies in allowing the government budget
to be balanced in a present value sense and analyzing the implication of this
modi"cation.
Acknowledgements
I would like to thank an anonymous referee for helpful comments and
suggestions.
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
243
Appendix A. Dynamic analysis
Linearizing Eqs. (6b) and (8) in the vicinity of the equilibrium de"ned in
Eq. (9) yields
o(a#b!1) jH
0
!
Qj
kH j!jH
1!b
.
(A.1)
"
o
kH
o[(1!p)!g]
k!kH
kQ
a(1!p) jH
c
CD
C
DC
D
The determinant and the trace of this Jacobian can be computed as
o2(a#b!1)
Det"
'0,
ac
(A.2)
o[(1!p)!g]
Trace"
,
c
(A.3)
where use has been made of c(,(1!b)(1!p))'0. Since the determinant of
the Jacobian of Eq. (A.1) measures the product of the roots and the trace
measures the sum, one can use information on the sign of determinant and trace
to check the dimension of the stable manifold of the steady state. From Eq. (A.2)
and (A.3), we can only determine the sign of the determinant, which is positive,
implying that the two characteristic eigenvalues of the linearized system are of
the same sign. In other words, a positive determinant can be associated with
either a stable manifold of dimension zero (an unstable steady state) or a stable
manifold of dimension two (a completely stable steady state), depending on the
sign of the trace. Given that the sign of the trace is indeterminate, we can
distinguish among the following two cases (the borderline case is discussed in
detail in the text and in Appendix B):
Case A: g '1!p. The trace is negative, suggesting that the equilibrium is
a sink. That is, the corresponding steady state is locally indeterminate.
The discriminant of the coe$cient matrix in Eq. (A.1) is
G
H
o2 [(1!p)!g]2 4(a#b!1)
!
.
D"(Trace)2!4 Det"
a
c
c
(A.4)
By "xing all parameters except for g in Eq. (A.4), we "nd that there exists a value
of g, denoted by g :10
H
g "(1!p)#2Jc(a#!1)/a
H
(A.5)
10 g is obtained directly from Eq. (A.4) by setting the right hand side to zero and noting that
H
g '1!p. To yield meaningful discussion, we require that g (1, which is generally warranted
H
under a wide range of parameter values, e.g., the scale parameter b is in the left neighborhood of one.
244
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
such that
(1) if g (g(1, the system has two distinct, real, negative roots. The steady
H
state is a stable node; and
(2) if g(g , the eigenvalues form a complex conjugate pair with a negative
H
real part. The steady state is a stable focus.
Case B: g (1!p. The trace is positive, suggesting that the equilibrium is
a source. There exists a value of g, denoted by g :
L
(A.6)
g "(1!p)!2Jc(a#b!1)/a,
L
such that the stability properties change when g crosses this value. The following
generic con"gurations are possible:
(1) when g (g(1!p, the system has a pair of imaginary roots with
L
positive real parts. The steady state is an unstable focus; and
(2) when g(g , the system possesses two distinct, real, positive eigenvalues.
L
The steady state is an unstable node.
Appendix B. Proof of Proposition 5
The procedure we shall use is called &the method of slowly varying amplitude
and phase' (see, e.g., Sanders and Verhaulst, 1984). Inspection of Eq. (17)
indicates that if the right-hand side were zero, then x would oscillate
sinusoidally in time [see Eq. (14b)]. This leads us to introduce the following
expressions for x and its time derivative [see Eq. (12b)]:
x "a sin(ut#/ ),
(B.1)
t
t
t
xR "a u cos(ut#/ ),
(B.2)
t
t
where a and / are time-varying amplitude and phase, respectively. Notice that
5
5
Eq. (B.2) is not the usual derivative of x , which in fact is equal to
t
xR "aR sin(ut#/)#a(u#/Q ) cos(ut#/).
(B.3)
Equating Eq. (B.3) with (B.2), it produces
aR sin(ut#/)#a/Q cos(ut#/)"0.
(B.4)
The second derivative of x can be computed from Eq. (B.2):
x( "aR u cos(ut#/)!au(u#/Q )sin(ut#/).
(B.5)
Substituting Eqs. (B.5) and (B.2) into Eq. (17) and rearranging yield
aR cos(ut#/)!a/Q sin(ut#/)"J a cos(ut#/).
22
(B.6)
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
245
Next, we derive two equations for aR and /Q . Using the following algebraic
manipulations, Eq. (B.4)]sin(ut#/)#Eq. (B.6)]cos(ut#/) and Eq. (B.4)]
cos(ut#/)!Eq. (B.6)]sin(ut#/), we get
aR "J a cos2(wt#/),
(B.7)
22
/Q "!J a sin(ut#/) cos(ut#/).
(B.8)
22
It should be noted that Eqs. (B.7) and (B.8) are equivalent to Eq. (17). Now, we
invoke the crucial idea of the method of slowly varying amplitude and phase:
when a trajectory gets closer to the limit cycle, its amplitude a and phase / vary
slowly over the time scale of the period of oscillation. Therefore, the time
derivatives of these variables are roughly constant over one period of oscillation.
On the basis, we integrate the right-hand sides of Eqs. (B.7) and (B.8) over one
period and treat the amplitude and phase in these integrations as constants.
Making use of the following integrals:
P
P
1
u 2n@u
cos2(ut#/) dt" ,
2
2n
0
u 2n@u
sin(ut#/) cos(ut#/) dt"0,
2n
0
we have the approximate equations
J
aR " 22 a,
2
/Q "0.
(B.9)
(B.10)
As seen before, when J "0, the limit cycle is reached. To "nd the rate at
22
which a nearby trajectory approaches the cycle, Eq. (B.9) alludes that it invokes
an exponential relation, which depends on the sign of J . If J '(() 0, the
22
22
amplitude of oscillations, a , grows (decays).
t
References
Abe, N., 1995. Poverty trap and growth with public goods. Economics Letters 47, 361}366.
Barro, R., 1990. Government spending in a simple model of endogenous growth. Journal of Political
Economy 98, S103}S125.
Barro, R., Sala-i-Martin, X., 1992. Public "nance in models of economic growth. Review of
Economic Studies 59, 645}661.
Benhabib, J., Farmer, R., 1994. Indeterminacy and increasing returns. Journal of Economic Theory
63, 19}41.
Benhabib, J., Perli, R., 1994. Uniqueness and indeterminacy: on the dynamics of endogenous growth.
Journal of Economic Theory 63, 113}142.
Futagami, K., Morita, Y., Shibata, A., 1993. Dynamic analysis of an endogenous growth model with
public capital. Scandinavian Journal of Economics 95, 607}625.
246
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
Glomm, G., Ravikumar, B., 1994. Public investment in infrastructure in a simple growth model.
Journal of Economic Dynamics and Control 18, 1173}1187.
Guchenheimer, J., Holmes, P., 1990. Nonlinear Oscillations, Dynamical Systems and Bifurcations of
Vector Fields. 3rd ed., Springer, New York.
Krugman, P., 1991. History versus expectations, Quarterly Journal of Economics 106, 651}667.
Matsuyama, K., 1991. Increasing returns, industrialization, and indeterminacy of equilibrium.
Quarterly Journal of Economics 106, 617}650.
Sanders, J., Verhaulst, F., 1984. Averaging Methods in Nonlinear Dynamical Systems. Springer,
New York.
Skiba, A., 1978. Optimal growth with a convex-concave production function. Econometrica 46,
527}539.
24 (2000) 227}246
Public services, increasing returns, and equilibrium
dynamics
Junxi Zhang*
School of Economics and Finance, The University of Hong Kong, Pokfulam Road, Hong Kong
Received 1 January 1997; accepted 30 November 1998
Abstract
This paper is concerned with casting o! the standard assumption } the production
function displays socially constant returns to scale } in a simple growth model with
public inputs. In the model, public services work as a source of utility and as an input to
production. Making use of the global bifurcation technique, we show that the model can
generate a whole spectrum of interesting dynamics, which do not arise with constant
returns. Speci"cally, when increasing returns are not exceedingly strong, not only does an
indeterminate steady state take place, but also there exist divergent perfect foresight
paths in expanding oscillations and economic cycles. The role of history and expectations
in shaping equilibrium is also considered. ( 2000 Elsevier Science B.V. All rights
reserved.
JEL classixcation: H42; O41; C62
Keywords: Public services; Increasing returns to scale; Indeterminacy of equilibrium;
Self-ful"lling expectations; Economic cycles
1. Introduction
In recent years there is growing interest in studying the relationship between
economic growth and public services. This literature builds on Barro's (1990)
seminal model of endogenous growth with a public sector. In the model, the
government uses tax revenue to "nance public expenditures, which include the
provision of infrastructure services, the protection of property rights, and other
* Corresponding author. Tel.: 852 2857 8502; fax: 2548 1152; e-mail: jjzhang@econ.hku.hk.
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 8 8 - 8
228
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
productive activities. Under the assumption that the production function displays socially constant returns to scale, it is found that public services, especially
infrastructure services, are essential for persistent growth. This policy implication has often been proposed to developing countries which face serious problems of scarcity of public goods, in particular infrastructure.
The central assumption embedded in this class of models deserves careful
scrutiny. First, the proviso, albeit helps reduce our e!orts in calculating growth
rates, eliminates some important features which arise in dynamic economies.
There is a wide agreement that it is primarily responsible for the absence of
transitional dynamics in the Barro model, since the model can be essentially
reduced to a version of the AK model.1 By contrast, the transitional dynamics
should allow us to understand the behavior of the system in the short run.
Second, accurate analysis of growth in less developed economies suggests that
these countries actually experience substantial increasing returns to scale during
the process of economic development, particularly at early stages. Thus, it is
necessary to expand the class of production functions for a more adequate
description of these economies. Third, the developments in economic modeling
for the past decade or so have been emphasizing non-standard elements,
including scale economies. Failure to take them into account seems di$cult to
reconcile.
This paper is concerned with casting o! the above standard assumption, that
is, the production function is only a narrow class of socially constant returns to
scale. In the model, government expenditures serve as a source of utility and as
an input to production (see also Abe, 1995). Under rather general functional
forms of utility and production, we show that the model can generate many
interesting dynamics. With mild increasing returns,2 not only does an indeterminate steady state take place, but also there exist divergent perfect foresight paths
in expanding oscillations and economic cycles, two possibilities which are
relatively unknown in the existing literature.
1 There are some attempts which adopt modi"ed versions of the Barro model in order to induce
transitional dynamics. Futagami et al. (1993) introduce public capital and demonstrate this possibility. Notably, the local stability of the transitional path is simple: the steady-growth equilibrium is
a saddle point. In a similar model, Glomm and Ravikumar (1994) reach the same conclusion.
A recent paper by Abe (1995) takes a di!erent approach. He generalizes the assumption of linear
homogeneity in the income identity and allows for increasing returns, and shows that indeterminacy
of equilibria may arise. Although our model is related to this strand of research, we di!er in two
main aspects in addition to the primary concern. First, Abe's analysis is of the local nature, while
ours is of the global nature since we make use of the global bifurcation technique. Second, our model
generates much richer dynamics, including the existence of economic cycles and multiple perfect
foresight paths in the vicinity of an unstable equilibrium, which are absent from the Abe model.
2 In this paper, we say that increasing returns are mild (strong) if the share of public services in
production does not (does) exceed one; see discussions in Sections 3 and 4 for details.
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
229
More speci"cally, there are three distinct cases when increasing returns are
mild. The "rst one involves in that the elasticity parameter of utility to government spending is su$ciently large. We "nd that the system gives rise to a unique
interior equilibrium which turns out to be a sink, and thus it yields a continuum
of equilibrium paths converging to the stationary point. The second case
concerns a moderate elasticity. It is shown that the interior equilibrium is
a source. Taking into account two terminal points, an immediate question arises
as to which state the economy actually gets established. The nature of this
question entails one to carry out an analysis of the global perfect foresight
dynamics. The information on the local dynamics is not enough, because, for
example, the existence of a unique perfect foresight path in a neighborhood of
a stationary state does not necessarily rule out the existence of other perfect
foresight paths in the large. Global dynamics also bring the possibility of
a meaningful and thorough understanding of the history versus expectations
distinction, an issue known in other "elds of economics: industrial organization,
international trade, macroeconomics, etc (e.g., see Krugman, 1991; Matsuyama,
1991). Making use of the global dynamics technique, it is possible to determine
precisely under what circumstances history matters and when expectations may
be decisive.
Perhaps the most interesting case is that when the elasticity parameter is
equal to a critical value, economic cycles emerge. The existence of cycles is
established using the Hopf bifurcation theorem (see Guchenheimer and Holmes,
1990), and a detailed description of the limit cycle, including its explicit expression, bifurcated period and stability conditions, is presented. Since the ultimate
reason for studying nonlinear cycles is to develop an endogenous theory of the
business cycle that can compete with the dominant linear stochastic paradigm,
our "nding is useful in that it provides yet another mechanism for economic
systems to generate periodic equilibria endogenously.
On the other hand, when increasing returns are very strong, the local stability
of the interior equilibrium changes entirely because it becomes a saddle point.
Given an initial condition, there exists a unique equilibrium path which converges monotonically to the stationary state.
The rest of the paper is organized as follows. In Section 2, we cast the model
and derive a system of nonlinear di!erential equations. Section 3 performs the
global analysis for the case of mild increasing returns to scale, while Section 4
brie#y considers the case of strong increasing returns. Finally, Section 5 concludes the paper.
2. The basic model with mild increasing returns to scale
The model economy is populated with a continuum of in"nitely lived agents,
whose measure is normalized to one. The representative agent supplies one unit
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J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
of labor services inelastically, and at any moment she chooses consumption so
as to maximize the discounted sum of utility, given by
P
=
e~ot u(c , g ) dt,
(1)
t t
0
where o is the subjective discount rate, c is private consumption, and g is
t
t
government consumption. Notice that in Eq. (1), government spending directly
a!ects consumer behavior, since she derives utility from public goods and
services, for example, public roads, parks, schools, etc. We assume that although
she chooses the level of consumption by participating in free markets, she takes
government expenditure as given. For simplicity, the instantaneous utility
function presumably possesses the form3
; "
0
(2)
u(c , g )"(1/p) (cpgg),
t t
t t
where p (1 and g 4 1 are parameters. The parametric restriction of p (1
makes utility strictly concave in the choice variable c . If g '0 (g (0), then
t
private consumption and government expenditure are pairwise Edgeworth
complements (substitutes).4 However, when g ( 0, it suggests a minor technical
di$culty. To ensure that the government consumption's marginal utility is
positive, Eq. (1) is modi"ed as
P
=
e~ot [u(c , g )#v(g )] dt,
(1a)
t t
t
0
where v@'0. Nevertheless, under the usual assumption that individuals have no
control over g , the maximization problem can be solved by ignoring g 's
t
t
contribution to ; through the function v. Hence, Eqs. (1) and (1a) are treated as
0
operationally equivalent counterparts throughout this paper.
Following Barro (1990), we assume that public goods and services, such as
infrastructure, also serve as an input to private production.5 But, in addition, we
allow increasing returns to scale:6
; "
0
f (k , g )"kagb,
t t
t t
(3)
3 It should be noted that the functional forms of utility and production in this paper are chosen so
as not only to retain their main features but also to allow for explicit solutions of the equilibria.
4 Two goods are said to be Edgeworth complements (substitutes) if the marginal utility of one
increases (decreases) as the quantity of the other increases.
5 Throughout this paper we regard public services as if they are publicly-provided private goods,
which are rival and excludable. For more general settings, see Barro and Sala-i-Martin (1992).
6 The existence of an equilibrium in this type of models with a general convex production function
but without public goods has been stated in Skiba (1978).
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
231
where a#b'1 but b (1; and k is capital stock. In this section, the case with
t
b (1 is referred to as &mild' increasing returns, while the case of strong
increasing returns with b'1 will be studied in Section 4. Furthermore, to make
the model interesting, we postulate that the economy's total capital stock per
person is kM . Consequently, the domain for the capital stock is restricted to [0, kM ].
To "nance spending on public goods and services, the government has access
to a tax on income, q. The budget is balanced every instant, so that7
(4)
g "q f (k , g )"q kagb
t t
t
t t
The consumption-saving decisions of the representative agent are determined by
solving the following problem: maximizing Eq. (1) subject to
(5)
c #kQ "(1!q) f (k , g ),
t
t t
t
the nonnegativity conditions c 5 0 and k 5 0, and an initial condition on
t
t
k , k '0. The current-value Hamiltonian for this problem is (super#uous time
t 0
subscripts are dropped hereafter):
H"(1/p) (cpgg)#j[(1!q) f (k, g)!c],
where j is costate variable. The "rst-order necessary conditions include
j"cp~1gg,
(6a)
jQ "j[o!a(1!q)qb@(1~b)k(a`b~1)@(1~b)],
(6b)
and the associated transversality condition is
lim j k e~oT"0.
(7)
T T
T?=
Substituting Eqs. (3), (4) and (6a) into the budget constraint (5), it follows that
qg@ckag@c
,
kQ "(1!q)qb@(1~b)ka@(1~b)!
j1@(1~p)
(8)
where c,(1!b)(1!p)'0. Eqs. (6b) and (8) jointly de"ne a planar dynamical
system in (j, k) on [0,R)][0, kM ]. Since j is a &jumping' variable, the initial
value for j, j , must be chosen to make a path consistent with these equilibrium
0
conditions. In this model the notion of history is captured by the capital stock.
3. Global dynamics
In this section, we carry out a global analysis of the dynamical system (6b)
and (8). Before proceeding, "rst the local dynamics are analyzed. An interior
7 The upper bound for k, kM , is then equivalent to an upper bound for g via this equation.
232
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
stationary state (jH, kH) is determined by the following relations: jQ "0 and
kQ "0, leading to the following equilibrium values:
C
kH"
D
(1~b)@(a`b~1)
aqg@(1~b)
o
, jH"
(kH)(ag@(1~b))~(1~p). (9)
o
a(1!q)qb@(1~b)
It can be readily veri"ed that the transversality condition (7) is met at the
equilibrium point. As shown in Appendix A, the equilibrium dynamics depend
on the sign of the trace of the Jacobian coe$cient matrix of the system. If the
trace is negative (positive), called Case A (B) below, the equilibrium is a sink
(source). The borderline case of a zero trace remains important in this model,
suggesting that the solution curve of the dynamical system, Eqs. (6b) and (8), is
a Jordan curve, i.e., a closed curve that does not intersect itself. We discuss these
global dynamical features of the system case by case.
Case A: g '1!p. This case entails the elasticity parameter of utility to
government spending, g, to be su$ciently large. First consider the behavior of
the trajectories in the phase diagram. The curve for which jQ "0 is vertical, while
the curve for which kQ "0 is upward sloping, which is determined by the
following equation:
q*g~b(1~p)+@(1~b)
ka*g~(1~p)+@(1~b).
j"
(1!q)1~p
(10)
There are three possible shapes of this locus. When a[g!(1!p)]'1!b, it is
convex; when a[g!(1!p)](1!b, it is concave. For the knife-edge case of
a[g!(1!p)]"1!b, it is a straight line from the origin. Fig. 1a and b depicts
the phase portraits for the "rst two possibilities, while the transparent borderline
case is not drawn. Notably, the jQ "0 locus is the standard modi"ed golden rule
for capital accumulation in the presence of taxation and public goods, which
says that ultimately the (after-tax) productivity of capital is determined by the
rate of time preference. The unique interior equilibrium point is denoted as E.
The two loci divide each phase portrait into four regions, where the components
of directions of movement are indicated by arrows: jQ '0 in the left-hand side of
the locus jQ "0 and jQ (0 in the right; above the locus of kQ "0, kQ '0, and kQ (0
below it.
Next, this case features that the determinant is positive while the trace is
negative. A negative trace and a positive determinant mean that there is a locally
indeterminate steady state, i.e., there exists an indeterminate steady state with
a continuum of equilibrium trajectories converging to the steady state. In
addition, because of the local indeterminacy, there may be stationary sunspot
equilibria in the neighborhood of E.
The following proposition presents the results for Case A (see Appendix A):
Proposition 1. ;nder the assumption of mild increasing returns of scale, a#b '1
and b (1, the dynamical system has an indeterminate steady state in the interior,
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
233
Fig. 1. The dynamical system for Case A: (a) a[g!(1!p)]'1!b; and (b) a[g!(1!p)](1!b.
E, if g '1!p. More speci,cally, when g (g(1, E is a stable node; when
H
g(g , E is a stable focus.
H
Fig. 2a and b displays these possible kinds of equilibrium dynamics. The two
terminal points, at which k"0 and k"kM , are denoted as E and E (lower and
L
U
234
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
Fig. 2. The equilibrium dynamic paths for Case A: (a) g (g(1; and (b) g(g .
H
H
upper bounds for k), respectively. The upper bound for k, kM , means that the
economy operates at its full potential. The multiple equilibrium paths in Fig. 2b
are of particular importance, as it shows the existence of sunspot equilibria in
the vicinity of E. Suppose that initial capital stocks for two economies were quite
close, like those in South Korea and the Philippines not so long ago. The above
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
235
Fig. 3. The dynamical system for Case B.
analysis makes evident that their subsequent growth experience could dramatically di!er, which may be the result of indeterminacy of equilibria, that is,
di!erent counties follow di!erent equilibrium trajectories toward a stationary
state or a balanced growth path. This result is reminiscent of the recent literature
of indeterminacy without public goods, e.g., Benhabib and Farmer (1994) and
Benhabib and Perli (1994).
Case B: g (1!p. This case is profoundly di!erent from Case A. First, as
shown in Fig. 3, the kQ "0 locus becomes downward sloping and convex
(also see Eq. (10)). Second, the local stability of the unique interior equilibrium E changes drastically: it becomes a source since the trace is positive (see
Eq. (A.3) in Appendix A). Third, the lower terminal point, E , becomes instead
L
(0,#R), which is equivalent to c"0 via Eqs. (6a) and (10). This state with
zero consumption and capital stock can be viewed as a state of poverty
and stagnation, which generically corresponds to the &poverty trap' in the
literature.
Given that the unique interior stationary point is a source, the solution to the
dynamical system will hit either boundary eventually, for any initial values,
k and j , that satisfy k 3(0, kM ). Following Krugman (1991), these two positions
0
0
0
that the economy reaches in the long run can also be loosely classi"ed as
equilibrium points. Then, several interesting questions immediately arise. Which
equilibrium does the economy go to? Does the initial condition matter? When
might self-ful"lling expectations play a role? The answers to these questions
236
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
impinge on the nature of the characteristic roots for E. As shown in Appendix A,
we can state:
Proposition 2. Suppose that the following conditions hold: a#b '1, b (1, and
g (1!p. ¹hen the unique interior equilibrium E of the system is a source. If g
L
( g (1!p, E is an unstable focus; when g(g , E is an unstable node.
L
When g (g(1!p, the linearized system diverges from the stationary state
L
in expanding oscillations. The kind of equilibrium dynamics is delineated in
Fig. 4a, in which perfect foresight paths consist of a pair of interwining, noncrossing spirals around E. As the "gure makes clear, the set of equilibria depends
on the size of the initial capital stock relative to the benchmarks k and k ,
1
2
de"ned by the projections onto the k axis of the left-most point of E 's stable
L
manifold and the right most point of E 's stable manifold, respectively. The
U
interval (k , k ) is generically called the &overlap' region of the stable manifolds
1 2
associated with E and E , where the extent of the overlap region depends in
L
U
a complicated way on parameter values. When initial capital stock falls out of
that region, determinate equilibrium dynamics emerge. If k 3(0, k ), there exists
0
1
an equilibrium path that the economy will be trapped in the lower terminal
equilibrium E . Similarly, if initial level is su$ciently high such that k ' k ,
L
0
2
the economy converges monotonically to E . In both circumstances, history
U
dictates the choice of equilibrium in the long run.
If k 3(k , k ), there are multiple trajectories that satisfy all the equilibrium
0
1 2
conditions, and the initial condition is no longer su$cient to determine the
outcome of the economy in the future. The economy can reach the high
equilibrium E even when the economy is initially located to the left of E, if
U
agents coordinate their expectations on the equilibrium path converging to E .
U
Fig. 4. The equilibrium dynamical paths for Case B: (a) g (g(1!p; and (b) g(g .
L
L
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
237
Given the perfect foresight nature of the model, expectations are entirely
self-ful"lling. In addition, the model reveals that starting from the right of E does
not guarantee the convergence to E . In a word, expectations will matter when
U
the initial capital falls into the overlap region. The policy implication is obvious
like that proposed in other studies: to break the vicious cycle of poverty, it calls
for active state intervention.
The above result accords reasonably well with intuition. In this model, if
b (1, the level of public services is an increasing function of the level of private
capital, as seen from Eq. (4). Suppose some agents increase their levels of capital.
Then the level of public services will increase, which in turn will raise the
marginal return of capital and induce everyone to increase capital accumulation.
In other words, there gives rise to a positive capital spillover. As argued by
Krugman (1991) and Matsuyama (1991), if expectations are coordinated in
a way that everyone believes that the economy will end up in the high equilibrium, then it will.
When g(g , E possesses two distinct, real, positive eigenvalues. Hence, the
L
only compatible portrait belongs to a scenario that determinate perfect foresight
paths emerge in that the linearized system steadily diverges from the stationary
state, no matter where the economy starts. Fig. 4b depicts this case. There is
a &threshold' for capital: when k ( k*, the economy converges to the low
0
terminal point E ; when k ' k*, it converges to the high terminal point E .
L
0
U
Clearly, history plays a decisive role in determining the long-run position of the
economy.
¹he borderline case: g "1!p. Denote this value as g . This case remains the
0
most interesting one, because economic cycles now take place. To show this, we
notice that the determinant is positive, while the trace is equal to zero, thereby
implying that the Jacobian has a pair of pure imaginary eigenvalues. If we de"ne
a real positive number
4o2(a#b!1)
'0,
d ,
0
ac
the two pure imaginary eigenvalues at g are equal to id and !id , respec0
0
0
tively. Moreover, it is straightforward to show that at g"g , the real part
0
of the derivative of eigenvalue with respect to g does not vanish. Thus, the
loss of stability can be guaranteed by small perturbations in the bifurcation
parameter g. According to the Hopf bifurcation theorem (Guchenheimer
and Holmes, 1990), it follows that
Proposition 3. Suppose that a#b '1, b (1, and g "1!p. ¹hen, there exist
economic cycles } the Hopf bifurcation } in the system.
In what follows, we will give a detailed description of the limit cycle, including
its explicit expression, bifurcated period and stability conditions. To do so, we
238
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
write the Jacobian of the system, evaluated at g"g , as (also see Eq. (A.1) in
0
Appendix A):
C
0
J(g ),
0
J (g )
21 0
C
D
!J (g )
12 0
0
D
o(a#b!1) jH(g )
0
!
1!b
kH(g )
0 ,
(11)
"
o kH(g )
0
0
ag jH(g )
0
0
where J (g ) '0, J (g )'0, jH(g )"[q/(1!q)]g0, and k*(g ) is given by
12 0
21 0
0
0
Eq. (9). Introducing two new variables x,k!kH and y,j!jH, then the two
di!erential equations become
0
yR (g )"!J (g ) x(g ),
0
12 0
0
(12a)
xR (g )"!J (g ) y(g ),
0
21 0
0
(12b)
These equations can be transformed into a standard second-order equation for
x(g ):
0
d2x(g )
0 #J (g ) J (g ) x(g )"0.
12 0 21 0
0
dt2
(13)
Similarly, one can obtain the same equation for y(g ). Eq. (13) is the well-known
0
van der Pol equation in the absence of damping. In order to determine a particular solution, we now need two side conditions on the path of each variable.
Suppose that the system initially starts at the point (0, y ). The other condition
0
for each variable is then determined by Eqs. (12a) and (12b). Accordingly, the
system settles into the following oscillations:
y "y cos(u(g )t),
t
0
0
(14a)
x "y JJ (g )/J (g ) sin(u(g )t),
(14b)
t
0
21 0 12 0
0
where u(g ),[J (g )J (g )]1@2 is the oscillation frequency.8 The bifurcated
0
12 0 21 0
cycle has period 2p/u(g ). It is observed that the amplitude of the x oscillation,
0
t
y [J (g )/J (g) )]1@2, plays a crucial role in shaping the cycles. Attention now
0 21 0 12 0
turns to determine the relative magnitude of J (g ) and J (g ). Unfortunately,
12 0
21 0
explicit comparison cannot be made. There appears to exist a critical value for
8 From the initial and subsequent positions, we can see that the points move clockwise.
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
239
the equilibrium ratio of jH(g )/kH(g ):
0
0
C D S
jH(g )
1!b
0 "
.
kH(g )
ag (a#b!1)
0 c
0
Evidently, if jH(g )/kH(g )l[jH(g )/kH(g )] , J (g )mJ (g ). As a result, we
0
0
0
0 c 21 0
12 0
have
Proposition 4. For J (g ) ' (() J (g ), the contour of the limit cycles is
21 0
12 0
a family of ellipses with k (j) as the major axis. For J (g ) " J (g ), it becomes
21 0
12 0
circular.
Fig. 5a}c depicts the three possibilities, in which the kQ "0 locus is horizontal.
Next, the stability conditions are examined. To make the attempted task
manageable, we still work with the linearized system but slightly perturb the
bifurcation parameter g, while all other parameters remain "xed. The resulting
Jacobian as opposed to Eq. (11) is
C
0
D
C
!J
12 "
J,
J
J
21
22
D
o(a#b!1) jH
!
kH
1!b
,
o
kH
o[(1!p)!g]
a(1!p) jH
c
0
(15)
where jH and kH are given by Eq. (9). Notice that the sign of J and J are
12
21
positive, while that of J is indeterminate. When g approaches g from the
22
0
above (below), J is negative (positive). The presence of this term yields the
22
following equations:
yR "!J x,
(16a)
12
xR "J y#J x.
(16b)
21
22
Consequently, we can obtain the more general version of the van der Pol
equation
dx
d2x
#u2x"J
,
22 dt
dt2
(17)
where u,(J J )1@2. J is the so-called damping rate. It is known that
12 21
22
a positive (negative) value for J means that the amplitude of the oscillations
22
grows (decays); also see Appendix B. Thus, we can prove the following result:
Proposition 5. ¹he stability of the bifurcated cycle is determined by the sign of J .
22
If J is positive, the cycle is unstable; while if it is negative, the cycle is
22
supercritically stable.
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J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
Fig. 5. The equilibrium dynamic paths for the Borderline Case: (a) J (g )'J (g ); (b)
21 0
12 0
J (g )"J (g ); and (c) J (g )(J (g ).
21 0
12 0
21 0
12 0
This proposition makes clear that all stable cycles are supercritical. Supercritical bifurcation means that the location of the equilibrium changes smoothly
with parameter value near the bifurcation point. Let us see why it is supercritical, but not subcritical, bifurcation that matters in this model. The preceding
discussion suggests that when the bifurcation parameter g, the elasticity of utility
to government expenditure, equals g , the system is neutral, i.e., it is neither
0
stable or unstable. Hence, the structure of the system is quite sensitive to this
elasticity parameter. Whenever it is increased, the equilibrium becomes stable;
whenever it is decreased, the equilibrium becomes unstable. However, there is no
structural change in the system.
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
241
It should be pointed out that estimates of the preference parameters p and g in
a framework with utility-enhancing public services have not yet been obtained.
As a result, the question of whether there are economic cycles in the borderline
case of g"1!p remains unanswered. At this stage, we are still unable to rule
out such a possibility. Hopefully, our theoretical demonstration of the potential
importance of this case may encourage empirical work on this issue.
4. The case of strong increasing returns to scale
In this section, we brie#y consider a special case of increasing returns: the
share of public services in production, b, exceeds one, i.e., b'1.9 Unlike what
happened in the previous section, the local stability of the unique stationary
state E and the entire equilibrium dynamics completely change. First, the shapes
of the kQ "0 locus alter for both Cases A and B. From Eq. (11), the curve
becomes downward sloping for Case A, while upward sloping for Case B.
Moreover, in Case B, the locus can be either convex, or concave, or a straight
line from the origin, depending on certain parameter values.
Second, the local stability of E changes as well. The condition of b'1 (or
equivalently c(0) makes the determinant negative (see Eq. (A.2) in Appendix A), which yields:
Proposition 6. =hen increasing returns are so strong that b '1, the unique
equilibrium is a saddle point.
This proposition holds because one of the roots of the coe$cient matrix is
positive and the other is negative. Fig. 6a and b displays the phase portraits for
Cases A and B, and similar diagram can be drawn for the borderline case. These
"gures suggest that given an initial capital stock k 3(0, kM ), there is a unique
0
equilibrium path, which converges monotonically to the stationary state. That
path belongs to the stable manifold of the stationary state.
Intuitively, we note from the government budget constraint (4) that if increasing returns are very strong, the level of public goods eventually becomes
a decreasing function of private capital. Even if some individuals choose to
increase their levels of capital, self-ful"lling equilibrium dynamics will not arise.
The reason is as follows: the level of public goods decreases, which inevitably
lowers the marginal return of capital, and thus others have no incentive to
coordinate their expectations by increasing capital accumulation.
9 It can be shown that the case of decreasing returns to scale produces qualitatively the same
implication, except that Cases A and B are simply switched.
242
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
Fig. 6. The equilibrium dynamic paths for b'1: (a) Case A; and (b) Case B.
5. Concluding remarks
In this paper we have examined the implications of relaxing the standard
assumption } the production function exhibits socially constant returns to scale
} in a simple growth model with public goods. It was found that the model is
capable of generating a whole spectrum of interesting dynamics, which do not
arise with constant returns. Two features are worth noting for the case of
moderate increasing returns. The "rst one involves in a possibility that the
dynamic path diverges from an interior equilibrium in expanding oscillations,
suggesting that self-ful"lling expectations play an essential role in determining
the long-run position of the economy. The second one is that economic cycles
take place, implying that the endogenously determined government spending is
also periodic when the system is in the neighborhood of the equilibrium.
Our "ndings are obtained within a decentralized stationary framework. We
believe, however, that the model can be extended to more general settings,
such as incorporating public investment and endogenizing government decisions. Another interesting extension lies in allowing the government budget
to be balanced in a present value sense and analyzing the implication of this
modi"cation.
Acknowledgements
I would like to thank an anonymous referee for helpful comments and
suggestions.
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
243
Appendix A. Dynamic analysis
Linearizing Eqs. (6b) and (8) in the vicinity of the equilibrium de"ned in
Eq. (9) yields
o(a#b!1) jH
0
!
Qj
kH j!jH
1!b
.
(A.1)
"
o
kH
o[(1!p)!g]
k!kH
kQ
a(1!p) jH
c
CD
C
DC
D
The determinant and the trace of this Jacobian can be computed as
o2(a#b!1)
Det"
'0,
ac
(A.2)
o[(1!p)!g]
Trace"
,
c
(A.3)
where use has been made of c(,(1!b)(1!p))'0. Since the determinant of
the Jacobian of Eq. (A.1) measures the product of the roots and the trace
measures the sum, one can use information on the sign of determinant and trace
to check the dimension of the stable manifold of the steady state. From Eq. (A.2)
and (A.3), we can only determine the sign of the determinant, which is positive,
implying that the two characteristic eigenvalues of the linearized system are of
the same sign. In other words, a positive determinant can be associated with
either a stable manifold of dimension zero (an unstable steady state) or a stable
manifold of dimension two (a completely stable steady state), depending on the
sign of the trace. Given that the sign of the trace is indeterminate, we can
distinguish among the following two cases (the borderline case is discussed in
detail in the text and in Appendix B):
Case A: g '1!p. The trace is negative, suggesting that the equilibrium is
a sink. That is, the corresponding steady state is locally indeterminate.
The discriminant of the coe$cient matrix in Eq. (A.1) is
G
H
o2 [(1!p)!g]2 4(a#b!1)
!
.
D"(Trace)2!4 Det"
a
c
c
(A.4)
By "xing all parameters except for g in Eq. (A.4), we "nd that there exists a value
of g, denoted by g :10
H
g "(1!p)#2Jc(a#!1)/a
H
(A.5)
10 g is obtained directly from Eq. (A.4) by setting the right hand side to zero and noting that
H
g '1!p. To yield meaningful discussion, we require that g (1, which is generally warranted
H
under a wide range of parameter values, e.g., the scale parameter b is in the left neighborhood of one.
244
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
such that
(1) if g (g(1, the system has two distinct, real, negative roots. The steady
H
state is a stable node; and
(2) if g(g , the eigenvalues form a complex conjugate pair with a negative
H
real part. The steady state is a stable focus.
Case B: g (1!p. The trace is positive, suggesting that the equilibrium is
a source. There exists a value of g, denoted by g :
L
(A.6)
g "(1!p)!2Jc(a#b!1)/a,
L
such that the stability properties change when g crosses this value. The following
generic con"gurations are possible:
(1) when g (g(1!p, the system has a pair of imaginary roots with
L
positive real parts. The steady state is an unstable focus; and
(2) when g(g , the system possesses two distinct, real, positive eigenvalues.
L
The steady state is an unstable node.
Appendix B. Proof of Proposition 5
The procedure we shall use is called &the method of slowly varying amplitude
and phase' (see, e.g., Sanders and Verhaulst, 1984). Inspection of Eq. (17)
indicates that if the right-hand side were zero, then x would oscillate
sinusoidally in time [see Eq. (14b)]. This leads us to introduce the following
expressions for x and its time derivative [see Eq. (12b)]:
x "a sin(ut#/ ),
(B.1)
t
t
t
xR "a u cos(ut#/ ),
(B.2)
t
t
where a and / are time-varying amplitude and phase, respectively. Notice that
5
5
Eq. (B.2) is not the usual derivative of x , which in fact is equal to
t
xR "aR sin(ut#/)#a(u#/Q ) cos(ut#/).
(B.3)
Equating Eq. (B.3) with (B.2), it produces
aR sin(ut#/)#a/Q cos(ut#/)"0.
(B.4)
The second derivative of x can be computed from Eq. (B.2):
x( "aR u cos(ut#/)!au(u#/Q )sin(ut#/).
(B.5)
Substituting Eqs. (B.5) and (B.2) into Eq. (17) and rearranging yield
aR cos(ut#/)!a/Q sin(ut#/)"J a cos(ut#/).
22
(B.6)
J. Zhang / Journal of Economic Dynamics & Control 24 (2000) 227}246
245
Next, we derive two equations for aR and /Q . Using the following algebraic
manipulations, Eq. (B.4)]sin(ut#/)#Eq. (B.6)]cos(ut#/) and Eq. (B.4)]
cos(ut#/)!Eq. (B.6)]sin(ut#/), we get
aR "J a cos2(wt#/),
(B.7)
22
/Q "!J a sin(ut#/) cos(ut#/).
(B.8)
22
It should be noted that Eqs. (B.7) and (B.8) are equivalent to Eq. (17). Now, we
invoke the crucial idea of the method of slowly varying amplitude and phase:
when a trajectory gets closer to the limit cycle, its amplitude a and phase / vary
slowly over the time scale of the period of oscillation. Therefore, the time
derivatives of these variables are roughly constant over one period of oscillation.
On the basis, we integrate the right-hand sides of Eqs. (B.7) and (B.8) over one
period and treat the amplitude and phase in these integrations as constants.
Making use of the following integrals:
P
P
1
u 2n@u
cos2(ut#/) dt" ,
2
2n
0
u 2n@u
sin(ut#/) cos(ut#/) dt"0,
2n
0
we have the approximate equations
J
aR " 22 a,
2
/Q "0.
(B.9)
(B.10)
As seen before, when J "0, the limit cycle is reached. To "nd the rate at
22
which a nearby trajectory approaches the cycle, Eq. (B.9) alludes that it invokes
an exponential relation, which depends on the sign of J . If J '(() 0, the
22
22
amplitude of oscillations, a , grows (decays).
t
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