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 stockRa critical levelBH(R) of debt above which debt tends to in"nity but below which it may be steered to zero. We demonstrate how to computeBH(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 debtBH(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& Computa-tional 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.
income between countries.1On 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.2This 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.5In an open economy, the decisions to consume and invest imply a decision to borrow whenever output is less than investment and consumption.6Current account de"cits lead to an increase in the country's debt.7In 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.8We consider a renewable resource and substitute the
1See, for example, Barro et al. (1995).
2Such 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.
3In the latter work there are also variants considered with credit constrained open economies.
4Blanchard (1983), in an extension of his model, also includes the disutility of debt in the utility function.
5See Swensson and Razin (1983) and Sachs (1982).
6We 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).
7This, of course, presumes a free access to capital markets which does not always exists, see Sen (1991)
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 ex-plosion, 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 con-sumption.
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$ciently small. It is more plausible * and also ver"ed in a numerical example (in Section 6)* that debt becomes critical in "nite time if the discount rate of utility is large compared to interest rate on debt. In general, even a rough estimate of the region of initial states which become critical in"nite time, as well as an estimate of the time, remains an unresolved problem. The answer will not be simple, since we know from Rauscher (1990) and Feichtinger and Novak (1991) and from our own simulations in Section 6 that limit cycles exist for certain parameters and initial values.
is one ODE for steepest debt descent with increasing resource and one for decreasing resource.
We believe that the kind of analysis presented in our paper might be practic-ally useful in risk analysis. For this purpose, however, one would have to take stochastic in#uences into account, thus increasing the number of state variables to at least 3. If another state variable for example capital stock is incorporated we count 4 state variables and there will be corresponding di$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 consump-tion 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.
B(0)4sup
which is the present value of the resource stockR(0) for a discount rated. Thus, the present value is the maximal initial debtB
0, which may be held bounded by an appropriate control q* given the initial stock R
0. The latter one is our de"nition of BH(R) no matter what h(B) is, see Section 3 for a more formal de"nition. Note also that we useqRas extraction rate instead ofq. This prevents the resource from becoming negative.
We make the following assumptions:d'0,g,h,f,; continuously di!
erenti-able cP;(R,c) concave, fconcave, ;
c'0,;R'0,Q'max0yRyR.!9g(R)/R,
C'f(QR
.!9). More speci"cally, we posit
1. f'0,f(0)"0,f@'0, fA(0,fAcontinuous,
4. in order to simplify the analysis we shall assume in Section 4 that there is at most one critical point on the curveh(B)"f(g(R)) where the extremal vector "eldvis tangent (see Section 3).
generated by the production functionf(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 assumep"1.9The 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 stockR
0and an initial debt B0 is it possible, by chosing the extraction rate appropriately, to steer the debtB(t) to zero? Our result is that this can be achieved for B
0below a critical level of debt which depends onR
0.11Above this levelB(t) will increase exponentially no matter how the controlq(t) is chosen. It is obvious that the critical levelBH(R0) is nothing but
For non-constanth(B)B~1, however, one might ask how the present value of
R
0is de"ned at all. We propose to de"ne it asBH(R0) the formal de"nition of which is as follows:
De,nition. Call B
0 subcritical for R0 if there is a measurable function
q: [0,#R]P[0, Q] such that if RQ"g(R)!qR, BQ"h(B)!f(qR), R(0)"
R
0, B(0)"B0,thenB(t)"0 for some"nitet'0 or limt?=B(t)"0 and
BH(R)"supMBDB subcritical forRN
RC(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.
9Note, 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.
10See, for example, Bhandari et al. (1990) for an elaborate study of country speci"c default risks giving rise to convex interest rate payments.
Fig. 1. Debt dynamics withG~the save region.
The dashed line stands for the critical curve which, in Fig. 1a, is tangent to the curveh(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 toB"R. Below the critical curve they tend toB"0 in"nite time.
Let
G~": M(R,B)D04R,B; h(B)(f(g(R))N,
G`": M(R,B)D04R,B;h(B)'f(g(R))N.
G~is the safe region where debt may be paid o!with a stationary resource. If one steersB(t) as steeply downward as possible by chosing the extraction rateq(t) appropriately*say by decreasing the resource*and stillB(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 reduceB(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 intoh(B)"f(g(R)) orB"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 inG`means either to minimize
BQ
RQ"
h(B)!f(qR)
in which case the corresponding trajectorytC(R(t),B(t)) is increasing inR(t), or
Parameterized byRthe trajectory in the"rst case solves
x5"v`(x), v`(R,B) :"(1,t`(R,B)),
The vector"eldsv~,v`are called extremal vector"elds, because solutions to
x5"v~(x) andx5"v`(x) respectively have extremal slope among solutions to (1) and (2). The pointxH, 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.xHis also a stationary point for both of the processes (R(t),B(t)) which run intoxHfrom the left and right respectively with minimalBQ/RQ . Fig. 2 depicts a more complicated yet interesting situation where there are two critical pointsxH"(RH,BH) andxHH"(RHH,BHH) with a pointx8"(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. Letx: [0,#R)PR2be a solution to
x5"v~(x), x(0)3G`.
Sincex(t)"(R(t),B(t)) is decreasing inR(t) as t increases,x de"nes an &upper region' upxas follows
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.
Proposition1. For every solution x ofx5"v~(x)withx(0)3G`the upper region upx dexned by x isinvariant,that is,ify(t)solves(1)and(2),andy(0)3upx,then
y(t)3upxfor allt50.
Proof. See the appendix.
Proposition2. For every solutionxofx5"v`(x) withx(t)3G`for all t50the upper region ofxisinvariant,that is,ify(t)solves(1)and(2)andy(0)3upx,then
y(t)3upxfor allt50.
4. Computing critical debt
Assumption. There is at most one point (RH,BH) withh(B*)"f(g(R*) ) where
Remark. Solutions to ODE initial value problems do not necessarly exist for all times (allR'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 ofB~(R),B`(R),B0(R) exists for everyR.v`on the other hand is not Lipschitz near (0,0). Therefore, the solutions tox5"v`(x) withx(0)"(0,0) is understood as the maximal solution to this problem, see Piccinini et al. (1984).
Proof. Note"rst that the set of solutionsxtox5"!v~(x) is ordered: ifx, x8 are
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 criti-cal curve),seeFig. 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 problemx5"v`(x), x(RH)"(RH,BH). Call the solution of itx`.
Observation2. Ifx~is tangent toh(B)"f(g(R))andx`(R)'0for0(R(RH, thenxH"x,where
x(R)"
G
x~(R) for R5RH,x`(R) for R4RH, see Fig. 1a.
Fig. 4. Critical curve consisting of three parts.
There is still a more complicated situation conceivable, namely
Case 3:x~is tangent toh(B)"f(g(R)) butx`(R)"0 for someR3(0,RH). In this case letx/be the maximal solution ofx5"!v~(x), x(0)"(0, 0).
Observation 3. Suppose a critical point (RH,BH) exists but x`(R)"0for some
R3(0,RH),thenxH"x,where
x(R)"
G
x~(R) for R 5RH, 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/andx~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
RHHthere are two possibilities, namely to decreaseRalongx/or to increase it alongx`.
B;0 B
Then lim
t?=B(t)"#R implies BQ(t)"h(B(t))!f(q(R))'h(B(t))!
f(QR
.!9)'dB(t) for largetandBQ(t)'d1edtfor some constantd1'0. Similarly, quadratic growth ofhwould imply thatB(t) becomes#Rin"nite time.
Summing up, we have
Theorem 2. Above the critical curve debt explodes;below it may be steered to zero. Formally,lety(t)"(R(t),B(t))solve(1), (2).Then:
(a) IfB(0)'BH(R(0))thenlim
t?=B(t)"#R.
(b) IfB(0)"BH(R(0))it is possible to steerB(t)bounded(by extremal control). (c) IfB(0)(BH(0)there is a controlqwhich steersB(t)to zero inxnite or inxnite
time.
For computational purposes we present the following notes: Note 1:Ifh(B)'f(g(R))and
t(R,B)"Max
t;g(R)
h(B)!f(l)
g(R)!l
thent(R,B)"f@(l(R,B))wherel"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 solutionl4QR;else
t(R,B)"h(B)!f(l)
g(R)!QR.
Proof. See the appendix; similarly:
Note 2:Ifh(B)'f(g(R))and
if this equation admits a solution. If it does not admit a solution,then
tH(R,B)"h(B)
g(R).
5. The optimal control problem
We now consider the trajectoriesR(t),B(t) for the optimal control problem (P
d) allowing for various discount rates. In order to establish existence of solutions to (P
d) we "rst consider its convexi"cation which, in our case, is achieved by replacingf(qR) by the interval 04fH4f(qR) wherefHis a control
According to a standard existence theorem (see e.g., Berkovitz, 1974), (P# d) has a solution for every initial state R
0, B04xH(R0), 04R04R.!9. If the asso-ciated control is (q(t),fH(t),c(t)), then, as it is easy to verify, (q(t), c(t)) is a solution of (P
d). Hence,
Proposition 4. For every initial state(R
0,B0)3[0,R.!9]][0,xH(R0)]problem(Pd) has a solution.
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 max-imum principle (see below) requires
L
Lc;(R,c)#j2"0
for the shadow price of debtj2which is impossible forc"0, the consumption at 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 ratesdas 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
d satisxes inft;0R(t)'0 and d(MinMh@(B)D04B4BH(R
.!9)Nthen it becomes critical inxnite time. Note that resource depletion,i.e.lim
t?=R(t)"0,is not optimal if the discount ratedis small.
Proof. This follows from the maximum principle, see below. In fact, the shadow pricej2of the debt satis"es
L
Lc;(R,c)#j2"0,
jQ2"j
This implies lim
It seems to us that also for largedsolutions 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
dand the so-called canonical equations. These di!er from the ones of Rauscher (1990) since we use extraction rateqRinstead ofq.
We state themaximum principleforQ"#R, C"#R. The HamiltonianHforP
optimal consumptionc(t) and extraction rate q(t), which is uncritical on [0,q) then there arej03M0,1Nand&shadow prices' j
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
d which is uncritical and has constant shadow prices has to satisfy
equations a necessary condition for solutions toP
d.
6. Simulations
Feichtinger and Novak (1991) have found, for their chosen parameter constel-lation, 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 controlsc,qare obtained in feedback form from the state equations so that at each grid point of the state space the optimal controlsc,qare 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 ofR,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.
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 ford"0.1 the country's debt runs into the critical curve in"nite time, see Fig. 6.
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 ex-ternally. 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
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 asttends to in"nity our new condition tells the agent exactly what to do (or not to do) at any timet. 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$cient safety margins for consumption are built into the model (i.e. into the production function). Numerical simula-tions are necessary to explore the dynamics of the system which analytically is understood only partially. Our simulations of critical debt and some optimally controlled paths also stress potential applicability of the model to risk control.16
Appendix. Some proofs
Proof of Proposition 1. It su$ces to construct a function H which increases along solutionsyof (1) and (2) the gradient of which points intoupxfor every solution (1) and (2) withy(t)3G`fort50. De"ne
vM(R,B)"(!t~(R,B),1).
ysatis"es:y5(t)"(g(R)!qR,h(B)!f(qR)).
15It 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).
y5(t)vM(R(t),B(t))50.
There is a functionHwhich 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 obtainHwe"rst solve
!Lbt LB#
Lb LR"0
for b'0 (which is possible sincevMis parallelizable onG`) and then de"ne
H(R,B) by a path integral from some (R
0,B0) to (R,B) of the vector"eldbvM, which is possible sinceG`is simply connected.
It follows that
d
dtH(y(t))"grad H(y(t))y5(t)
" 1
a(y(t))vM(y(t))y5(t)50.
Hencey(t)3upx(t50) wherexis the solution tox5"v~(x) throughy(0). Proof of Observation 3. It is obvious how to steerBinto zero belowx: usev~for (R,B) withR5RH,v`forRHH4R4RHand againv~for 04R4RHH, where
RHHis the point wherex/andx`coincide, see Fig. 4. This proves thatxis 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
2(0). It follows from Propositions 1 and 2 of Section 3 thatystays abovex, i.e. ify(t)"(R(t),B(t)) thenB(t)'x
2(R(t)) whenx(R)"(R,x2(R)). According to the proof of Proposi-tion 2 there is a di!erentiable functionHthe gradient of which is perpendicular
Suppose b"sup
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