We derive a new transversality condition (which we prove to be necessary) for a class of infinite horizon optimal control problems.Transversality condition, Optimal control theory

Koji Shimomura

Ngo Van Long

English

Research Papers in Economics

 
Discussion Paper Series No.144 
  
 
A Note on Transversality Conditions 
 
Ngo  Van  Long    McGill  University 
Koji  Shimomura  Kobe  University 
 
November   2003 
 
 
Discussion papers are a series of manuscripts in their draft form. 
They are not intended for circulation or distribution except as 
indicated by the author. For that reason Discussion Papers may 
not be quoted, reproduced or distributed without the written 
consent of the author. 
 
 A Note on Transversality Conditions
Ngo Van Longa and Koji Shimomurab
We derive a new transversality condition (which we prove to be necessary) for a
class of in…nite horizon optimal control problems.
a: Department of Economics, McGill University, 855 Sherbrooke Street West,
Montreal, Quebec, Canada H3A 2T7. Fax: (514)398-4938, Tel: (514)398-4844.
Email: ngo.long@mcgill.ca
b: Research Institute for Economics and Business Administration, Kobe
University, 2-1, Rokkodai-cho, Nada-ku, Kobe, Japan, 657-8501. Fax:
81-78-803-7059, Tel: 81-78-803-7002, Email: coe@rieb.kobe-u.ac.jp
Date: 11 September 2003
JEL Classi…cation: C61
Key Words:Transversality condition, optimal control theoryA Note on Transversality Conditions 1
1 Introduction
While transversality conditions for optimal control problems (in continuous
time1) with a …xed, …nite time horizon are parts of the necessary conditions,
and are relatively straightforward to derive2, the same cannot be said for
transversality conditions for in…nite horizon problems. For example, with
a …xed, …nite time horizon, when a state variable is free (not …xed) at the
terminal time T, it is necessary that the associated co-state variable, ¼(t),
be zero at T: However, when the time horizon is in…nite, it is not true that
limT!1¼(T) = 0 is anecessarycondition. (A counterexample, duetoHalkin,
is reported in Arrow and Kurz (1970, p. 46), and discussed in Takayama
(1985, p. 625)).
For in…nite time horizon problems, various transversality conditions have
been stated as parts of the su¢cient conditions (see, for example, Seierstad
and Sydsaeter, 1977, Leonard and Long, 1992) for an optimal solution, or
as necessary conditions for certain problems with a special structure. For
example, for a special investment problem with adjustment costs, Takayama
(1985, p. 699) stated as necessary the condition
lim
T!1
¸(T)e¡rT = 0
while Arrow (1968) claimed that in the standard one-sector optimal growth
model, witha positivediscount rate, thefollowing conditions are appropriate:
lim
T!1
¸(T)e
¡rT ¸ 0
lim
T!1
¸(T)e¡rTk(T) = 0
For a macro-economc model with a stock of bonds b, and a capital stock k,
Turnovsky (1995, p 236) stated the transversality conditions
lim
T!1
¸(T)e¡rTk(T) = 0
lim
T!1
¸(T)e
¡rTb(T) = 0
1We are not dealing with discrete-time problems. For transversality conditions in
discrete-time problems, see Weitzman (1973), Ekeland and Scheinkman (1986), and Michel
(1990).
2For heuristic derivations, see, for example, Leonard and Long (1992). For rigorous
proofs, see Hestenes (1966).A Note on Transversality Conditions 2
However, there were no proofs that those conditions are necessary conditions
for the special problems under considerations.
A number of theorems have been proved on the necessary transversality
conditions for various special cases of the reduced-form model
max
x
Z 1
0
v(x; _ x;t)dt
subject to
x(0) = x0, (x; _ x) 2 W µ (R
N)
2
For example, Benveniste and Scheinkman (1982), underthe assumptions that
v is non-negative and that v(x; _ x;t) is integrable, established the necessity of
the following standard transversality conditions (STC):
lim
t!1
[¡v2(x; _ x;t)]x = 0 (1)
Kamihigashi (2001), by assuming that the boundedness of v1 and v2, demon-
strated, under certain additional hypotheses, the necessity of a variant of
STC:
lim
t!1[¡v2(x; _ x;t)]x ￿ 0
Michel (1982) proves another transversality condition: as time tends to in-
…nity, the Hamiltonian tends to zero.
The purpose of this note is to prove the necessity of a new transversality
condition, for an in…nite time horizon problem with n state variables. As
we shall see, this new necessary condition implies that two commonly used
transversality conditions are in fact equivalent.
2 The theorem
Let x be a vector of n state variables, and _ x denote its derivative with respect
to time. Consider the problem
max
Z 1
0
v(x; _ x;t)dt (2)
subject to
(x(t); _ x(t)) 2 W µ (Rn)2 for all t ¸ 0A Note on Transversality Conditions 3
x(0) = x0
The function v is twice di¤erentiable.
In what follows, x, _ x;vx and v_ x are column vectors in Rn. If y is a column
vector, then yT denotes y tranposed, a row vector. Let the time path x¤(:)
be a solution of the problem (2). We make the following assumptions:
Assumption A1:For all t ¸ 0
(x
¤(t); _ x
¤(t)) 2 intW
Assumption A2: There exists a small ± > 0 such that for all ® 2
(1 ¡ ±;1 + ±) the pair (y; _ y) generated by y(0) = x0 and _ y(t) = ®_ x¤(t) has
the property that
(y(t); _ y(t)) 2 intW for all t ¸ 0 (3)
Remark: Assumption 1 does not ensure that (3) is satis…ed. Consider
the following counter-example.Let
W =
©
(a;b) 2 R
2 : a 2 [¡1;1] and ¡ (1¡ a) ￿ b ￿ 2(1¡ a)
ª
Consider the problem
max
x
Z 1
0
_ x(t)dt
subject to
(x; _ x) 2 W
and x(0) = 0. Thus, it is required that x 2 [¡1;1] and ¡(1 ¡ x) ￿ _ x ￿
2(1 ¡ x). Clearly, given x(0) = 0; x(t) < 1 for all …nite t. A solution for
this problem is x¤(t) = 1 ¡ e¡t which is less than 1 for all …nite t, and
limt!1x¤(t) = 1. Yet, with y(0) = x0 and _ y(t) = ®_ x¤(t), we have
y(t) = ®[x¤(t)¡ x0] + x0 = ®x¤(t) for x0 = 0
and
lim
t!1y(t) > 1 for any ® > 1;
which is not feasible. In other words, we cannot have _ y = ®_ x¤ for ® > 1.A Note on Transversality Conditions 4
Theorem: Under Assumptions A1 and A2, the following transversality
condition is a necessary condition:
lim
t!1
[x
¤(t) ¡ x0]
T v_ x(x
¤(t); _ x
¤(t);t) = 0 (4)
Corollary: If, inadditiontoassumptions A1and A2;the vector v_ x(x¤(t); _ x¤(t);t)
(evaluated along the optimal path) is non-positive, and x0 is strictly positive,
then the following two conditions are equivalent:
lim
t!1
v_ x(x
¤(t); _ x
¤(t);t) = 0 (5)
lim
t!1[x
¤(t)]
T v_ x(x
¤(t); _ x
¤(t);t) = 0 (6)
Proof of the theorem: we construct a time path y(:) generated by
y(0) = x0
_ y(t) = ®_ x¤(t)
Then
y(t)¡ y(0) =
Z t
0
_ y(t)dt = ®
Z t
0
x¤(t)dt = ®x¤(t) ¡ ®x0
i.e.
y(t) = ®[x
¤(t)¡ x0] + x0
By Assumption A2, ( y(t); _ y(t)) 2 intW for ® su¢ciently close to unity.
De…ne
Ã(®) =
Z 1
0
v(y; _ y;t)dt¡
Z 1
0
v(x¤; _ x¤;t)dt
=
Z 1
0
v(®x¤ + (1¡ ®)x0;®_ x¤;t)dt¡
Z 1
0
v(x¤; _ x¤;t)dt
Then, there exists ± > 0 such that for all ® 2 (1¡±;1+±), we have Ã(®) ￿ 0.
In particular, Ã(1) = 0. In addition, since x¤(:) is an interior solution, it must
hold that Ã
0(®) = 0 at ® = 1. Thus
0 =
dÃ(®)
d®
p®=1=
Z 1
0
£
(x
¤ ¡ x0)
Tvx(x
¤; _ x
¤;t) + (_ x
¤)
Tv_ x(x
¤; _ x
¤;t)
¤
dtA Note on Transversality Conditions 5
Now, recall Euler’s equation
vx(x¤; _ x¤;t) =
d
dt
[v_ x(x¤; _ x¤;t)]
Thus
0 =
Z 1
0
f(x
¤)
T d
dt
[v_ x(x
¤; _ x
¤;t)] +
µ
d_ x¤
dt
¶T
v_ x(x
¤; _ x
¤;t)gdt
¡
Z 1
0
xT
0
d
dt
[v_ x(x¤; _ x¤;t)]dt
=
Z 1
0
d
dt
f(x¤)Tv_ x(x¤; _ x¤;t)gdt¡
Z 1
0
xT
0
d
dt
[v_ x(x¤; _ x¤;t)]dt
= lim
t!1
(x¤)Tv_ x(x¤; _ x¤;t)¡ (x¤
0)Tv_ x(x¤
0; _ x¤(0);0)
¡x
T
0 lim
t!1v_ x(x
¤; _ x
¤;t) + (x
¤
0)
Tv_ x(x
¤
0; _ x
¤(0);0)
Thus we obtain equation (4) as a necessary condition.
Proof of the corollary: This follows from (4) and the assumptions
that the vector v_ x(x¤(t); _ x¤(t);t) (evaluated along the optimal path) is non-
positive, and x0 is strictly positive.
3 Discussion
Our theorem implies that the following path cannot be optimal:
x(t) = e
rt with v_ x = ¡e
¡rt , r > 0:
Concerningtheexample by Halkin (mentionedin the introduction), which
violates our transversality condition (4), we note that he imposes the con-
straint
¡(1 ¡ x) ￿ _ x ￿ 1¡ x
which violates Assumption A2.A Note on Transversality Conditions 6
References
Arrow, K. J. andM.Kurz, 1970, PublicInvestment, the Rateof Return,
and Optimal Fiscal Policy, Johns Hopkins Press, Baltimore, Md.
Benveniste, L. M. and J. A. Sheinkman, 1982, Duality Theory for Dy-
namic Optimization Models of Economics: The Continuous Time Case,
Journal of Economic Theory, 27, 1-19.
Ekeland, I., and J. A. Sheinkman, 1986, Transversality Conditions for
some In…nite Horizon Discrete Time Optimization Problems, Mathe-
matics of Operations Research, 11, 216-229.
Hestenes, Magnus R.,1966, Calculus of Variations and Optimal Control
Theory, New York, Wiley.
Kamihigashi, Takashi, 2001, Necessity of Transversality Conditions for
In…nite-Horizon Problems, Econometrica, 69, 995-1012.
Leonard, Daniel, and Ngo Van Long, 1992, Optimal Control Theory
and Static Optimization in Economics, Cambridge University Press,
Cambridge and New York.
Michel, P., 1982, On the Transversality Condition in In…nite-Horizon
Problems, Econometrica, 50, 975-985.
Michel, P.,1990, Some Clari…cations on the Transversality Conditions,
Econometrica, 58, 705-723.
Takayama, A., 1985, Mathematical Economics, Cambridge University
Press, Cambridge, U.K
Turnovsky, S., 1995, Methods of Macroeconomic Dynamics, MITPress,
Cambridge, MA.
Weitzman, M., 1973, Duality Theory for In…nite Horizon Convex Mod-
els, Management Science, 19, 783-789.

Duality Theory for Dynamic Optimization Models of Economics: The Continuous Time Case,

Duality Theory for In…nite Horizon Convex Models,

Mathematical Economics,

On the Transversality Condition in In…nite-Horizon Problems,

Optimal Control Theory and Static Optimization in Economics,

PublicInvestment, the Rateof Return, and Optimal Fiscal Policy, Johns Hopkins Press,

Some Clari…cations on the Transversality Conditions,

Transversality Conditions for some In…nite Horizon Discrete Time Optimization Problems,

A Note on Transversality Conditions

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