Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue5-7.Jul2000:

Journal of Economic Dynamics & Control
24 (2000) 1121}1144

Critical debt and debt dynamics
Willi Semmler!,",*, Malte Sieveking#
!Department of Economics, University of Bielefeld, 33501 Bielefeld, Germany
"Graduate Faculty, New School for Social Research, 65 Fifth Avenue, New York, NY 10003, USA
#Department of Mathematics, University of Frankfurt, Robert-Mayer-Str. 6-10, (6) Frankfurt, Germany

Abstract
We study the debt dynamics and sustainable debt for an open economy which borrows
from abroad in order to "nance consumption. To service the debt the country may
exploit a renewable resource. We show that there is for every resource stock R a critical
level BH(R) of debt above which debt tends to in"nity but below which it may be steered
to zero. We demonstrate how to compute BH(R) using an ODE of steepest descent. If the
economy maximizes a discounted integral of utility depending on consumption and the
resource stock, the critical debt BH(R) may be reached in "nite time. In such a situation
slight perturbations of the optimal consumption lead to insolvency. The maximum
principle ceases to be valid in this case. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: C61; F32; F34; O16
Keywords: Intertemporal model; Resources; Growth; Current account; Foreign debt

1. Introduction
In recent economic literature on open economies it is shown that external
borrowing can speed up growth and lead to a faster convergence of per capita

* Corresponding author. The paper has bene"ted from discussions at a conference on &Computational Economics' Amsterdam, at seminars at the University of Bielefeld, Germany, the Colegio di
Mexico, Mexico City, and Asia University, Tokyo. We are also grateful for communications and
discussions with Graciela Chichilnisky, Geo!rey Heal, Oliver Blanchard, Ken Judd, and Michael
Rauscher. Moreover, we want to thank three referees and Cars Hommes for their helpful comments.

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 9 ) 0 0 0 3 9 - 1

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W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

income between countries.1 On the other hand, it is often demonstrated that an
increase of a country's debt beyond a critical level can lead to unsustainable debt
and insolvency of the country. A country may then lose its creditworthiness

causing a sudden reversal of capital #ows.2 This in turn may lead to a currency
and "nancial crisis and a large output loss. Moreover, as Nishimura and
Ohyama (1995) have shown there appears to exist debt cycles in international
borrowing and lending. They present empirical evidence that a number of
countries went through those debt cycles in the past.
The problem of debt dynamics has usually been addressed in the context of
a Ramsey (1928) type model. Variants of such a model have been studied by
Blanchard and Fischer (1989, Chapter 2) for a closed economy with inside debt
and by Blanchard (1983), Cohen and Sachs (1986) and Barro et al. (1995) for an
open economy with external debt.3 In these models agents are assumed to
maximize a discounted stream of utility from consumption goods.4
For the open economy variant an important ingredient is a current account
determined by intertemporal decisions.5 In an open economy, the decisions to
consume and invest imply a decision to borrow whenever output is less than
investment and consumption.6 Current account de"cits lead to an increase in
the country's debt.7 In order to remain solvent, roughly speaking, the country's
debt should be no greater than the net wealth of the country.
The adoption of the above framework of intertemporal decision making
agents to countries which export natural resources has built on the literature
where agents optimally extract resources for consumption (see Plourde, 1970;

Clark, 1990) and where the stock of a resource is the state variable. Those
models have been extended to a trading economy, see Dasgupta et al. (1978),
there, however, for exhaustible resources. As other studies we simplify the
problem by disregarding investment decisions and we thus eliminate the capital
stock as state variable.8 We consider a renewable resource and substitute the
1 See, for example, Barro et al. (1995).
2 Such reversal of capital #ows have recently been studied empirically in a series of papers by
Milesi-Ferretti and Razin (1996, 1997). Their latest paper studies the Asian "nancial crisis in the light
of above approach to insolvency.
3 In the latter work there are also variants considered with credit constrained open economies.
4 Blanchard (1983), in an extension of his model, also includes the disutility of debt in the utility
function.
5 See Swensson and Razin (1983) and Sachs (1982).
6 We want to note, however, that in empirical literature the hypothesis of the independence of
investment from domestic savings has been questioned; see Feldstein and Horioka (1980).
7 This, of course, presumes a free access to capital markets which does not always exists, see Sen
(1991)
8 See Nishimura and Ohyama (1995) and Sachs (1982) where investment is also neglected. A model
with investment and capital stock is studied in Sieveking and Semmler (1997b).

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

1123

stock of the renewable resource for the capital stock dynamics. Except for
a di!erent formulation of the extraction rate our model resembles the one by
Rauscher (1990) and Feichtinger and Novak (1991).
Often in the study of problems of the above type Pontryagin's maximum
principle and the associated Hamiltonian is employed. Such a procedure,
however, is incorrect because of the bounds on debt that those models imply in
order to be economically meaningful. Rauscher (1990) e.g. requires that debt
eventually falls below a "xed universal bound. Blanchard (1983) requires that
the discounted debt tends to zero. (The rate of discount on utility equals that of
interest on debt in his model.) In the present paper we show that those kind of
conditions on debt are equivalent to an inequality constraint for debt: there is
for every initial resource stock a critical level of debt, below which debt may be
steered to zero but above which debt tends to in"nity, no matter how the rate of
extraction and consumption is chosen. In other words the above conditions on
debt, also called non-explosivenes conditions, can be met if and only if debt stays
below the critical level.

The existence of such a critical level seems to be obvious to economists. Yet
the consequences seem to have passed unnoticed: there are state constraints
which invalidate the maximum principle (in its usual form). More precisely,
suppose an optimally controlled resource debt path becomes critical in "nite
time (the state constraint becomes binding). Up to that time the maximum
principle is applicable. From then on, however, the path will be incompatible
with the maximum principle. Also the economy will be in a precarious situation.
Consumption is not permissible and a slight perturbation leads to debt explosion, i.e. to insolvency.
While this calls for a safety margin to be built into the model if used as
a planning instrument the near insolvency occurring near critical states may
explain problems that real economies encounter when forced to optimize consumption.
We show in Sections 5 and 6 that there is numerical as well as analytical
evidence that all optimally controlled paths become critical in "nite time, if the
discount rate (of utility) is su$ciently small. It is more plausible * and also
ver"ed in a numerical example (in Section 6) * that debt becomes critical in
"nite time if the discount rate of utility is large compared to interest rate on debt.
In general, even a rough estimate of the region of initial states which become
critical in "nite time, as well as an estimate of the time, remains an unresolved
problem. The answer will not be simple, since we know from Rauscher (1990)
and Feichtinger and Novak (1991) and from our own simulations in Section 6

that limit cycles exist for certain parameters and initial values.
To compute the critical level of debt is a nontrivial problem. Our second main
result (Theorem 1) is that the computational issue can be solved using certain
ODE initial value problems. The ODEs (there are two of them) correspond to
steering debt as steeply as possible downwards in the resource-debt space. There

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W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

is one ODE for steepest debt descent with increasing resource and one for
decreasing resource.
We believe that the kind of analysis presented in our paper might be practically useful in risk analysis. For this purpose, however, one would have to take
stochastic in#uences into account, thus increasing the number of state variables
to at least 3. If another state variable for example capital stock is incorporated
we count 4 state variables and there will be corresponding di$culties to deduce
anything from the model.
The remainder of the paper is organized as follows. Section 2 introduces the
model. Sections 3 introduces vector "eld analysis and de"nes critical debt.
Section 4 shows how to compute the critical debt from solutions to certain ODE

initial value problems. Section 5 studies the optimally controlled system, in
particular the limiting behavior of the trajectories of the resource and the
country's debt for large and very small discount rates. In Section 6 we compute
the critical curve and some trajectories in particular one which exhibits a limit
cycle. Section 7 draws some conclusions.

2. The model
We consider a country that is well endowed with a resource, transforms the
optimally extracted resource into tradable goods and makes optimal consumption decisions. We presume that the resource is renewable the growth rate of
which is determined by the Pearl}Verhulst logistic model commonly employed
in resource economics. As in Beltratti et al. (1993,1994) we assume that utility
depends on the #ow of consumption goods as well as a renewable resource. The
current account in our model is determined by the decisions to extract the
resource and by intertemporal consumption decisions.
The model has two control variables, the extraction rate and consumption,
and two state equations, the stock of the resource and foreign debt. We de"ne
the following optimal control problem:

G

Max
c,q
s.t.

(P )
d

:=e~dt;(R, c) dt,
0
RQ "g(R)!qR, 04q4Q,

(1)

BQ "h(B)!pf (qR)#c, 04c4C,

(2)

B4BH(R)"critical debt,

(R(0), B(0))"(R , B ).

0 0
Here R(t) is the stock of the resource at time t, B(t) the country's accumulated
debt at time t, q(t)R(t) the extraction rate of the resource at time t, q(t) the
extraction e!ort (control variable, [0, Q]), c(t) the consumption (control variable,
[0, C]), f the production function for exportable goods, g the reproduction

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

1125

function for the stock of the resource, h(B) the interest payment on debt B,
a convex function of B and BH(R) the critical debt.
The state constraint B4BH(R) needs some explanation, since BH(R) is not an
arbitrary function of R but de"ned in terms of (1) and (2). In fact, BH is the
solution of another optimization problem in terms of g,h and f, which has to be
solved before solving P . Suppose h(B)"dB for some interest rate d. Then the
d
non-explosiveness condition
lim B(t) e~dt"0
t?=

(see Blanchard, 1983) is easily seen to be satis"able, for some control q, if and
only if

P

=
B(0)4sup
e~dtf (qR) dt": BH(R)
q 0
s.t. RQ "g(R)!qR, R(0)"R B(0)"B
0
0
which is the present value of the resource stock R(0) for a discount rate d. Thus,
the present value is the maximal initial debt B , which may be held bounded by
0
an appropriate control q * given the initial stock R . The latter one is our
0
de"nition of BH(R) no matter what h(B) is, see Section 3 for a more formal
de"nition. Note also that we use qR as extraction rate instead of q. This prevents
the resource from becoming negative.

We make the following assumptions: d'0, g, h, f,; continuously di!erentiable cP;(R, c) concave, f concave, ; '0, ; '0, Q'max
g(R)/R,
c
R
0yRyR.!9
C' f (QR ). More speci"cally, we posit
.!9
1. f'0, f (0)"0, f@'0, fA(0, f A continuous,
2. g(R)"Rg (R) with g continuously di!erentiable, g 50 on some interval
1
1
1
(0, R ), g(R )"0,
.!9
.!9
3. h@'0, h@ continuous, h(0)"0,
4. in order to simplify the analysis we shall assume in Section 4 that there is at
most one critical point on the curve h(B)"f (g(R)) where the extremal vector
"eld v is tangent (see Section 3).
The "rst state equation (1) has been widely employed in resource economics,
particularly in the economics of "shery and forestry. The resource dynamics (1),
de"ned by the Pearl}Verhulst reproduction function, g(R), and the extraction rate, qR, is modeled along the lines of Plourde (1970) and Clark (1990).
The current account dynamics (2) follows the aforementioned intertemporal
theory for an open economy along the lines of Swensson and Razin (1983),
Sachs (1981, 1982) and Blanchard (1983). The current account is determined by
the debt service, h(B), and the excess of consumption spending over output,

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W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

generated by the production function f (qR). We presume the country to be small
so that the price of the exportable good is "xed relative to the consumption
good. Without loss of generality, in the analytical part, we assume p"1.9 The
interest payment, h(B), may include a premium on default risk.10

3. Critical debt
In the context of the control problem (1), (2) we may raise the following
question. Given an initial stock R and an initial debt B is it possible, by
0
0
chosing the extraction rate appropriately, to steer the debt B(t) to zero? Our
result is that this can be achieved for B below a critical level of debt which
0
depends on R .11 Above this level B(t) will increase exponentially no matter how
0
the control q(t) is chosen. It is obvious that the critical level BH(R ) is nothing but
0
the present value of R in case h(B)"dB for some d that is
0
BH(R )" Max :=e~dtf (q(t) R(t) dt
0
0
q
s.t.
RQ "g(R)!qR, R(0)"R .
0
For non-constant h(B)B~1, however, one might ask how the present value of
R is de"ned at all. We propose to de"ne it as BH(R ) the formal de"nition of
0
0
which is as follows:
De,nition. Call B subcritical for R if there is a measurable function
0
0
q: [0,#R]P[0, Q] such that if RQ "g(R)!qR, BQ "h(B)!f (qR), R(0)"
R , B(0)"B then B(t)"0 for some "nite t'0 or lim
B(t)"0 and
0
0,
t?=
BH(R)"supMB D B subcritical for RN
R C (R, BH(R))"xH(R) is called the critical curve.
The critical curve is piecewise a solution to a certain initial value problem of
some ODE which is associated with one of two &extremal' vector "elds in the
(R, B) space. The initial value will satisfy in most cases h(B)"f (g(R)). The
construction of xH(R) is easily explained in an informal way by referring to
Figs. 1a and 1b.

9 Note, however, that exchange rate depreciation following a sudden reversal of capital #ows may
worsen the situation of an indebted country by shifting the critical debt curve down.
10 See, for example, Bhandari et al. (1990) for an elaborate study of country speci"c default risks
giving rise to convex interest rate payments.
11 Note that a safety margin for minimal consumption would move the critical debt curve down.
Note that also that public debt for which the Ricardian equivalence theorem holds, i.e. where the
debt is serviced by a non-distortionary tax, would cause the critical debt curve to shift down.

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

1127

Fig. 1. Debt dynamics with G~ the save region.

The dashed line stands for the critical curve which, in Fig. 1a, is tangent to the
curve h(B)"f (g(R)) at the critical point (RH, BH). In Fig. 1b the critical curve
does not approach the h(B)"f (g(R)) curve but approaches B"0 instead.
Above the critical curve trajectories shown in the "gure tend to B"R. Below
the critical curve they tend to B"0 in "nite time.
Let
G~ "
: M(R, B) D 04R,B; h(B)(f (g(R))N,
G` "
: M(R, B) D 04R,B; h(B)'f (g(R))N.
G~ is the safe region where debt may be paid o! with a stationary resource.
If one steers B(t) as steeply downward as possible by chosing the extraction
rate q(t) appropriately * say by decreasing the resource * and still B(t)'0 for
all t 50 then the trajectory tP(R(t),B(t)) will lie above the critical curve,
i.e. B(t)5BH(R(t)) for all t50. If, however, for some t50 B(t)"0 or
h(B(t)"f (g(R(t))) then one can reduce B(t) to zero with stationary R chosing
q"
: g(R). The critical curve therefore is the upper envelope of all trajectories
tP(R(t),B(t)) which run into h(B)"f (g(R)) or B"0 in "nite time, minimizing
the slope all the way downward. The critical curve either is tangent to
h(B)"f (g(R)) or runs into the origin (0,0), having extremal slope everywhere, see
Theorem 1 for a more formal statement. There are two kinds of extremal slopes,
however, one where the resource is increased and one where it is decreased.
To minimize the slope in G` means either to minimize
BQ
h(B)!f (qR)
"
while qR(g(R) and 0(q(Q
RQ
g(R)!qR

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W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

in which case the corresponding trajectory t C (R(t), B(t)) is increasing in R(t), or
to maximize
BQ
h(B)!f (qR)
"
while qR'g(R) and 0(q(Q.
RQ
g(R)!qR
The corresponding trajectory t C (R(t), B(t)) will have decreasing R(t) in this
case.
Parameterized by R the trajectory in the "rst case solves
x5 "v`(x), v`(R, B) "
: (1, t`(R, B)),
h(B)!f (qR)
t`(R, B) "
:
Min
,
g(R)!qR
qR:g(R)
and in the second case
x5 "v~(x), v~(R, B) "
: (!1,!t~(R, B)),
h(B)!f (qR)
t~(R, B) "
: Max
.
g(R)!qR
qR;g(R)
The vector "elds v~, v` are called extremal vector "elds, because solutions to
x5 "v~(x) and x5 "v`(x) respectively have extremal slope among solutions to (1)
and (2). The point xH, which serves as initial value for both ODEs, x5 "v~(x) and
x5 "v`(x), is the one with h(B)"f (g(R)), i.e where v` are tangent to
h(B)"f (g(R)). Hence, xH"(RH, BH) satis"es
g@(RH)"h@(BH),

h(B)"f (g(R)),

where the "rst equation is the tangency condition. xH is also a stationary point
for both of the processes (R(t), B(t)) which run into xH from the left and right
respectively with minimal BQ /RQ . Fig. 2 depicts a more complicated yet interesting
situation where there are two critical points xH"(RH, BH) and xHH"(RHH, BHH)
with a point x8 "(RI , BI ) in between which is a source but not stationary for the
process of extremal debt reduction. Such points are also called Skiba points in
the literature, see Brock and Malliaris (1996, Chapter 6).
De,nition. Let x : [0,#R)PR2 be a solution to
x5 "v~(x), x(0)3G`.
Since x(t)"(R(t), B(t)) is decreasing in R(t) as t increases, x de"nes an &upper
region' upx as follows
upx"M(RH, BH)3G` D & t50 RH"R(t), BH5B(t)N.

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

1129

Fig. 2. Multiple critical points.

In the same way, upx is de"ned for x5 "v`(x), x(0)3G`. The extremal vector
"elds v`,v~ are de"ned as to make the following propositions intuitively
obvious. We use these propositions in Theorem 2 to show that above the critical
curve debt explodes.
Proposition 1. For every solution x of x5 "v~(x) with x(0)3G` the upper region
upx dexned by x is invariant, that is, if y(t) solves (1) and (2), and y(0)3upx, then
y(t)3upx for all t50.
Proof. See the appendix.
Proposition 2. For every solution x of x5 "v`(x) with x(t)3G` for all t50 the
upper region of x is invariant, that is, if y(t) solves (1) and (2) and y(0)3upx, then
y(t)3upx for all t50.

4. Computing critical debt
We want to show that the critical curve has extremal slope v~(x) or v`(x) and
either runs into the curve h(B)"f (g(R)) tangentially or runs into (0, 0). In order
to simplify the analysis we make the following

1130

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

Assumption. There is at most one point (RH, BH) with h(B*)"f (g(R*)) where
v(RH, BH) is tangent to the curve h(B)"f (g(R)).
¹heorem 1. Let v`(RH,BH) be tangent to h(B)"f (g(R)) and RPx~(R)"
(R, B~(R)) be the solution to x5 "v~(x), x(RH)"(RH, BH), RPx`(R)"(R, B`(R))
be the solution to x5 "v`(x), x(RH)"(RH, BH), RPx0(R)"(R, B0(R)) be the
solution to x5 "v~(x), x(0)"(0,0). Then the critical curve x(RH)"(R, BH(R)) is
given by BH(R)"max(B~(R), B`(R), B0(R), 04R4R . If there is no such
.!9
(RH, BH) then x0 is the critical curve.
Remark. Solutions to ODE initial value problems do not necessarly exist for all
times (all R'0 in our case) and need not be unique if the vector "eld is not
Lipschitz. In our case it is easy to verify that at least one of B~(R),B`(R),B0(R)
exists for every R. v` on the other hand is not Lipschitz near (0,0). Therefore, the
solutions to x5 "v`(x) with x(0)"(0,0) is understood as the maximal solution to
this problem, see Piccinini et al. (1984).
Proof. Note "rst that the set of solutions x to x5 "!v~(x) is ordered: if x, x8 are
two such solutions x(R)"(R, B(R)) and x8 (R)"(R, BI (R)) and for some R both
1
are de"ned and B(R )(BI (R ), then B(R)(BI (R) whenever both are de"ned in
1
1
R. So we may consider the in"mum x~ of all solutions for which B(R)'0 and
h(B(R))'f (g(R)), (R50). By Ascoli's theorem x~"!v~(x~). There are several cases to be considered.
Case 1: There is a point x~(RH)"(RH,BH) such that h(BH)"f (g(RH)) and
v~(RH,BH) is tangent to the curve h(B)"f (g(R)) at (RH,BH). In this case (RH,BH) is
called a critical point. According to our assumption it is uniquely determined.
To compute it note that applying the chain rule to h~1"f"g at RH we obtain

K K
d
dR

H

R/R

f @g(RH))g(RH)
h~1"f"g(R)"
h@(BH)

Furthermore, if t(RH,BH)"f @(gRH) for some q3[0, Q], then t(RH,BH)"f @(g(RH))
by concavity and hence
h@(BH)"f @(g(RH))
otherwise
h(BH)!f (QRH) f @(g(RH))g@(RH)
"
.
g(RH)!QRH
h@(BH)
Case 2: x~ is not tangent to h(B)"f (g(R)) but x~(0)"(0, 0).

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

1131

Fig. 3. Initial value problem with more than one solution.

Observation 1. If x~ is not tangent to h(B)"f (g(R)), then x~"xH (the critical curve), see Fig. 1b.
Remark. x~ in case 2 is the maximal solution to the initial value problem
x5 "!v(x), x(0)"(0,0). We cannot exclude the possibility that there are several
solutions. If this initial value problem admits more than one solution, then there
is a region strictly below the critical curve where it is not possible to steer the
debt to zero in "nite time, see Fig. 3.
The shadowed region in Fig. 3 consists of points below the critical curve
where debt may be steered to zero but not in "nite time.
In case 1 one can solve the initial value problem x5 "v`(x), x(RH)"(RH, BH).
Call the solution of it x`.
Observation 2. If x~ is tangent to h(B)"f (g(R)) and x`(R)'0 for 0(R(RH,
then xH"x, where

G

x(R)"

x~(R)

for R5RH,

x`(R)

for R4RH,

see Fig. 1a.
The proof of Observations 1 and 2 is omitted, since it will be clear from the one
of observation 3.

1132

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

Fig. 4. Critical curve consisting of three parts.

There is still a more complicated situation conceivable, namely
Case 3: x~is tangent to h(B)"f (g(R)) but x`(R)"0 for some R3(0, RH). In
this case let x/ be the maximal solution of x5 "!v~(x), x(0)"(0, 0).
Observation 3. Suppose a critical point (RH, BH) exists but x`(R)"0 for some
R3(0,RH), then xH"x, where

G

x(R)"

x~(R)

for R5RH,

max(x/(R),x`(R))

for 04R4RH.

Proof. See the appendix.
Case 3 is represented in Fig. 4.
The critical curve (dashed line) consists of three parts, x/ and x~ are extremal,
resource decreasing trajectories while x` is an extremal, resource increasing
trajectory. RH is stationary for the extremal debt pay-o! process whereas for
RHH there are two possibilities, namely to decrease R along x/ or to increase it
along x`.
Remark. Fig. 4 suggests a general form of the critical curve in case several
critical points exist. The general form is the upper envelope of all curves with
extremal velocity running into (0, 0) or into h(B)"f (g(R)) tangentially. This
form is equivalent to an algorithm which "rst computes a number of initial value

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

1133

problems and then takes its upper envelope. We call this Ferebee's algorithm
since Brooks Ferebee proposed such an algorithm as an alternative to dynamic
programming.12 The same type of "gure can be found in Brock and Malliaris
(1996, Chapter 6).
Remark. So far we have demonstrated (proof of Observation 3) that above the
critical curve lim
B(t)"#R. If h grows linearly this implies exponential
t?=
growth of B(t).
Assume
h(B)
inf
'd'0 for some constant d'0.
B
B;0
Then
lim
B(t)"#R
implies
BQ (t)"h(B(t))!f (q(R))'h(B(t))!
t?=
f (QR )'dB(t) for large t and BQ (t)'d edt for some constant d '0. Similarly,
.!9
1
1
quadratic growth of h would imply that B(t) becomes #R in "nite time.
Summing up, we have
Theorem 2. Above the critical curve debt explodes; below it may be steered to zero.
Formally, let y(t)"(R(t), B(t)) solve (1), (2). Then:
(a) If B(0)'BH(R(0)) then lim
B(t)"#R.
t?=
(b) If B(0)"BH(R(0)) it is possible to steer B(t) bounded (by extremal control ).
(c) If B(0)(BH(0) there is a control q which steers B(t) to zero in xnite or inxnite
time.
For computational purposes we present the following notes:
Note 1: If h(B)'f (g(R)) and
h(B)!f (l)
t(R, B)"Max
g(R)!l
t;g(R)
then t(R, B)"f @(l(R,B)) where l"l(R,B) is uniquely determined by the equation
F(R, B, l) "
: h(B)!f (l)!f @(l) (g(R)!l)"0
if this equation admits a solution l4QR; else
h(B)!f (l)
.
t(R,B)"
g(R)!QR
Proof. See the appendix; similarly:

12 A further elaboration on this algorithm is given in Sieveking and Semmler (1998).

1134

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

Note 2: If h(B)'f (g(R)) and
h(B)!f (l)
tH(R, B)" Min
g(R)!l
0:l:g(R)
then tH(R, B)"f @(l(R, B)) where l"l(R, B) is uniquely determined by the equation
F(R, B, l)"h(B)!f (l)!f @(l)(g(R)!l)"0, l(g(R)
if this equation admits a solution. If it does not admit a solution, then
h(B)
tH(R, B)"
.
g(R)

5. The optimal control problem
We now consider the trajectories R(t), B(t) for the optimal control problem
(P ) allowing for various discount rates. In order to establish existence of
d
solutions to (P ) we "rst consider its convexi"cation which, in our case, is
d
achieved by replacing f (qR) by the interval 04f H4f (qR) where f H is a control
variable:

G

Max
s.t.

(P#)
d

:=e~dt;(R, c) dt
0
RQ "g(R)!qR, 04q4Q,
BQ "h(B)!f H#c, 04c4C,
B4xH(R), 04f H4f (qR),

(R(0),B(0))"(R , B ).
0 0
According to a standard existence theorem (see e.g., Berkovitz, 1974), (P#) has
d
a solution for every initial state R , B 4xH(R ), 04R 4R . If the asso0 0
0
0
.!9
ciated control is (q(t), f H(t), c(t)), then, as it is easy to verify, (q(t), c(t)) is a solution
of (P ). Hence,
d
Proposition 4. For every initial state (R , B )3[0, R ]][0, xH(R )] problem (P )
0 0
.!9
0
d
has a solution.
Unfortunately we do not su$ciently understand how optimally controlled
paths tP(R(t), B(t)) behave as t becomes in"nite. Rauscher (1990) and Feichtinger and Novak (1991) observed the interesting phenomenon of debt cyles for
certain discount rates. Such a limit cycle is simulated in Section 6 using dynamic
programming. Limit cycles may be related to actually observed debt cycles.
Another important issue is whether or not an optimally controlled path

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

1135

tP(R(t), B(t)) becomes critical in "nite time, i.e.
B(tH)"BH(R(tH)) for some tH(R.
The economic implication would be that consumption is impossible for t'tH
and that slight perturbations of (R( ) ), B( ) )), say by a slight perturbation of the
optimal consumption, lead to debt explosion, i.e. insolvency. If this happens in
our model, it may be related to real problems indebted countries face when they
are forced to optimize consumption and get exposed to economic shocks. If,
however, the model is used as a planning instrument we should make sure that
a safety margin for minimal consumption is built into the model.
The mathematical implication of a solution to P being critical in "nite time is
d
that such a solution is incompatible with the maximum principle as used by
Blanchard (1983), Rauscher (1990) and Feichtinger and Novak (1991).
Although, as is well known, the maximum principle applies up to the "rst time
t, when the state control
B4BH(R)
becomes binding, it ceases to apply from then on. This is seen most easily if we
use Q"#R,;(R, c)"dceRo with o'0, d'0, 0(e(1 since the maximum principle (see below) requires
L
;(R, c)#j "0
2
Lc
for the shadow price of debt j which is impossible for c"0, the consumption at
2
a critical state. This means that investigations which unconditionally use the
maximum principle as a necessary condition will miss all those solutions which
start below the critical debt but become critical in "nite time. Numerical
simulations indicate that solutions exist which become critical in "nite time for
small discount rates d as well as for large (but realistic) ones, see Figs. 5 and
6 below. The range of parameters and initial values for which solutions become
critical in "nite time is unknown and constitutes in our opinion an interesting
mathematical problem.
Claim. If a solution tP(R(t), B(t)) to P satisxes inf R(t)'0 and
d
t;0
d(MinMh@(B) D 04B4BH(R )N then it becomes critical in xnite time. Note that
.!9
resource depletion, i.e. lim
R(t)"0, is not optimal if the discount rate d is small.
t?=
Proof. This follows from the maximum principle, see below. In fact, the shadow
price j of the debt satis"es
2
L
;(R, c)#j "0,
2
Lc
jQ "j (d!h@(B)).
2
2

1136

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

This implies lim
j (t)"0, lim
; (R(t), c(t))"0, lim
c(t)"#R and
t?= 2
t?= c
t?=
lim
B(t)"#R in contradictions to our assumption that (R ( ) )B( ) ))
t?=
solves P .
d
It seems to us that also for large d solutions may become critical in "nite time
but we do not have proof for that.
For the convenience of the reader we now state the maximum principle for
P and the so-called canonical equations. These di!er from the ones of Rauscher
d
(1990) since we use extraction rate qR instead of q.
We state the maximum principle for Q"#R, C"#R.
The Hamiltonian H for P reads
d
H"
: j ;(R, c)#j (g(R)!qR)#j (h(B)!f (qR)#c).
0
1
2
According to the maximum principle, if tP(R(t), B(t)) is a solution to P with
d
optimal consumption c(t) and extraction rate q(t), which is uncritical on [0,q)
then there are j 3M0,1N and &shadow prices' j ,j (for R,B) such that
0
1 2
RQ "g(R)!qR,
BQ "h(B)!f (qR)#c,
j ; #j "0,
0 c
2
j #f @(qR)j "0,
1
2
LH
jQ "! #dj "j (d!g@(R))!j ; ,
1
1
1
0 R
LR
LH
jQ "! #dj "j (d!h@(B)).
2
2
2
LB
It is easy to verify that j "1, j '0, j '0 where j ,j are interpreted as
0
1
2
1 2
shadow prices.
If a solution tP(R(t), B(t)) to P converges,
d
e"lim (R(t), B(t)),
t?=
then e is an optimally controlled stationary state. Stationary states have to
satisfy certain equations:
5.1. Stationary states
Obviously, (0,0) and (RH,BH) * the critical point * are optimally controlled
stationary states. A stationary solution to P which is uncritical and has
d
constant shadow prices has to satisfy
B"h@~1(d)

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

1137

and
; (R, f (g(R))!h(h@~1(d)))
f @(g(R))(d!g@(R))" R
.
; (R, f (g(R))!h(h@~1(d)))
c
For a demonstration of the above, see Rauscher (1990). Note that the latter
equation may have none or several solutions R50.
There is, a priori, also the possibility, rarely noted in the literature, that an
optimally controlled stationary state (R, B) admits a non-stationary costate
tP(j (t), j (t)). This possibility, however, does not exist for our model as is seen
1
2
from the maximum principle above. Of course, (RH, BH) is not mentioned in
Rauscher (1990) or Feichtinger Novak (1991) as they consider the canonical
equations a necessary condition for solutions to P .
d
6. Simulations
Feichtinger and Novak (1991) have found, for their chosen parameter constellation, that a discount rate of 11.95 will give rise to limit cycles (a discount rate
probably still considered unrealistic).
For the numerical simulation we employ the following functional forms:
;(R, c)"dceRo;

f (qR)"p[(1#qR)c!1];

g(R)"eR(1!R);

h(B)"aBp

with parameters: a"0.1, c"0.76, e"0.21, o"0.77, p"1.01, e"0.22,
d"0.19, p"24.85. Most of the parameters are directly taken from Feichtinger
and Novak (1991).
The critical curve is numerically computed by employing vector "eld analysis
as proposed in Sections 3 and 4. We pursue here the simple Case 1 of Section 4.
In addition, we employ a dynamic programming algorithm as described in
Sieveking and Semmler (1997a), which iterates on the value function. Hereby the
two controls c, q are obtained in feedback form from the state equations so that
at each grid point of the state space the optimal controls c,q are known. The
optimal solutions of the state variables can be computed from these.
Fig. 5 exhibits the critical curve and the trajectory R(t), B(t), resulting from
optimal actions at each grid point of R, B, for a very large discount rate, d"1.13
As Fig. 5 shows, the trajectory runs into the critical curve in "nite time. The
dashed line is the critical curve.

13 We have also undertaken simulations for a discount parameter d"50 which gave us roughly
the same trajectories as for the case of d"1.

1138

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

Fig. 5. Simulated trajectory: d"1, R(0)"0.7, B(0)"2.0.

Fig. 6. Simulated trajectory: d"0.1, R(0)"0.8, B(0)"9.

Also for d"0.1 the country's debt runs into the critical curve in "nite time,
see Fig. 6.
For a further decrease of the discount rate we could not observe the trajectories approaching R(t)"0. The trajectories instead again ran into the critical
curve even for very small discount rates, for example, for d"0.000005.

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

1139

Fig. 7. Simulated trajectory: d"0.05, R(0)"0.48, B(0)"5.2.

Finally, for an intermediate discount rate of d"0.05 we see a limit cycle
arising, see Fig. 7.14
An economic interpretation of a limit cycle in the context of a model such as
above is given in Feichtinger and Novak (1991).

7. Conclusions
The paper presents an intertemporal version of an open economy with current
account surpluses and de"cits. If de"cits occur they have to be "nanced externally. We study a resource-based economy with a tradable commodity
obtained from an exploitable renewable resource. The intertemporal decisions
to extract the resource and to consume determine the current account de"cit
and thus the dynamics of the resource and foreign debt. The country's welfare
depends on consumption as well as on a renewable resource.
We have shown that the usual non-explosiveness condition of debt * also
called intertemporal budget constraint * is equivalent to a state constraint.
This means if and only if the debt is below a certain critical debt, it is possible to

14 The fact that the trajectories of the limit cycle are not very smooth stems from the chosen grid
size for the state and control variables. We have tried further grid re"nement yet the trajectories did
not change signi"cantly.

1140

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

satisfy the non-explosiveness condition. The advantage of the state constraint
formulation is threefold. First, in contrast to the usual condition which is written
in terms of a limit as t tends to in"nity our new condition tells the agent exactly
what to do (or not to do) at any time t. Second, it reminds us not to use naively
the maximum principle in order to determine the solutions to the optimal
control problem. Third, it shows that these solutions are fragile in a potentially
dangerous sense: any additional consumption may render the borrower
insolvent.
We also show that to compute the critical debt is a non-trivial task which,
however, may be done solving certain initial value problems where debt is made
as small as possible by either increasing or decreasing the resource, see our use of
extremal vector "elds in Sections 3 and 4.15
Finally, we want to remark that the analysis of the solutions to the optimal
control problem (see Feichtinger and Novak, 1991; Rauscher, 1990) has been
incomplete. Yet Feichtinger and Novak (1991) have indicated the existence of
limit cycles for certain parameters. A further contribution of our paper is thus to
show that for large discount rates and most likely also for very small ones debt
becomes critical in "nite time which implies that consumption becomes zero in
"nite time. Consequently, one should be careful to adopt the optimization
model for practical purposes unless su$cient safety margins for consumption
are built into the model (i.e. into the production function). Numerical simulations are necessary to explore the dynamics of the system which analytically is
understood only partially. Our simulations of critical debt and some optimally
controlled paths also stress potential applicability of the model to risk control.16

Appendix. Some proofs
Proof of Proposition 1. It su$ces to construct a function H which increases
along solutions y of (1) and (2) the gradient of which points into upx for every
solution (1) and (2) with y(t)3G` for t50. De"ne
vM(R, B)"(!t~(R, B),1).
y satis"es: y5 (t)"(g(R)!qR, h(B)!f (qR)).
15 It might be worthwhile to explore of whether the above result also holds for intertemporal
models with households', "rms' and public debt; for a survey of such models, see Blanchard and
Fischer (1989, Chapter 2). For a study of critical debt which includes the capital stock as state
variable, see Sieveking and Semmler (1997b).
16 Historically, of course, for example in the 1980s, there have been many policies to reduce the
risk from debt overhang, ranging from debt rescheduling, temporary reduction of debt service,
debt-equity swaps to debt relief and debt forgiveness; see Krugman (1992, Chapter 7}9). Yet,
a sudden reversal of capital #ows carries potential dangers, see Milesi-Ferretti and Razin (1997).

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

1141

If q(t)R(t)'g(R(t)) then by de"nition of t~(R, B) we have
y5 (t)vM(R(t), B(t))50.
If g(R(t))'q(t)R(t) then for all l'g(R)
f (g(R)) h(B) h(B)!f (qR)
f @(l)4
4
4
.
g(R)
g(R)
g(R)!qR
Here we use concavity of f for the "rst and last inequality signs and the
assumption f (g( R))(h(B) for the middle part.
This again implies
y5 (t)vM(R(t), B(t))50.
There is a function H which is continuously di!erentiable and satis"es
vM(R, B)"a(R, B) grad H(R, B)
for (R, B)3G` and some positive function a. It is intuitively clear that such
a function exists. Formally, to obtain H we "rst solve
Lbt Lb
!
# "0
LB
LR
for b'0 (which is possible since vM is parallelizable on G`) and then de"ne
H(R, B) by a path integral from some (R , B ) to (R, B) of the vector "eld bvM,
0 0
which is possible since G` is simply connected.
It follows that
d
H(y(t))"grad H(y(t))y5 (t)
dt
1
"
vM(y (t)) y5 (t)50.
a(y(t))
Hence y(t)3upx (t50) where x is the solution to x5 "v~(x) through y(0).
Proof of Observation 3. It is obvious how to steer B into zero below x: use v~ for
(R, B) with R5RH, v` for RHH4R4RH and again v~ for 04R4RHH, where
RHH is the point where x/ and x` coincide, see Fig. 4. This proves that x is below
xH. Suppose we are above x, y(t)"(R(t), B(t)) solves RQ "g(R)!gR,
BQ "h(B)!f (qR) for some q : [0,#R) C [0, Q] and B(0)'x (0). It follows
2
from Propositions 1 and 2 of Section 3 that y stays above x, i.e. if y(t)"(R(t),B(t))
then B(t)'x (R(t)) when x(R)"(R, x (R)). According to the proof of Proposi2
2
tion 2 there is a di!erentiable function H the gradient of which is perpendicular
to v~, points upwards and increases along y(t).

1142

W. Semmler, M. Sieveking / Journal of Economic Dynamics & Control 24 (2000) 1121}1144

Suppose b"sup B(t)(#R. Then y is bounded. By Ascoli's theorem
t?0
it admits a limit function y8 which again solves (1), (2) for some control q"q8 . H,
however, is a constant along y8 . Hence the range of y8 is contained in the range
of z for some solution z of x5 "v~(x). The same is true for v` due to Proposition 2 and its proof. Hence, the range of y8 also belongs to the range of a
solution u of x5 "v`(x). Hence, the range of y8 belongs to the intersection of
both ranges and therefore is a single point which, however, is impossible
except for the critical point (RH, BH) which is however not above xH. This
contradiction proves lim
B(t)"#R. This in turn proves that x lies above
t?=
xH, therefore x"xH.
Proof of Note 1.

A

B

d h(B)!f (l)
F(R, B, l)
,
"
dl g(R)!l
(g(R)!l)2
d
F( R, B, l)"!f A(l)(g(R)!l)(0 for l'g(R).
dl
It follows that if t(R, B, q)"(h(B)!f (gR))/(g(R)!gR) has a maximal value
for qR'g(R) then this value is f @(l) where l is the unique solution to
F(R, B, l)"0. By the implicit function theorem l"l(R, B) will have continuous
partial derivatives equal to
F
!g@(R) f @(l(R, B))
l "! R"
,
R
F
f A(l(R, B))(g(R)!l(R,B))
l
h@(B)
F
.
l "! B"
B
f A(l(R, B))(g(R)!l(R, B))
F
l
If however t(R, B, q) does not have a maximal value for qR'g(R), then
h(B)!f (l)
f (l)
t~(R, B)"lim
"lim
g(R)!l
l
l?=
l?=
"lim f @(l)"f @(R)
l?=
and this happens for all (R, B)3G`. Evidently then
t~(R, B)"t~(R, B)"0.
B
R

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1143

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