CHAPTER \ 3 TRANSVERSALITY CONDITI ONS FOR VARIABLE- ENDP OINT PROBLEMS The Euler equation-the basic necessary condition in the calculus of varia-tions-is normally a second-order differential equation containing two arbi- trary constants. For problems with fixed initial and terminal points, the two given boundary conditions provide sufficient information to definitize the two arbitrary constants. But if the initial or terminal point is variable (subject to discretionary choice), then a boundary condition will be missing. Such will be the case, for instance, if the dynamic monopolist of the preceding chapter faces no externally imposed price at time T , and can treat the PT choice as an integral part of the optimization problem. In
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Chiang Chap 3. Transversality Conditions for Variable-Endpoint Problems
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CHAPTER\
3TRANSVERSALITY CONDITIONS FOR VARIABLE- ENDP
OINT PROBLEMS
The Euler equation-the basic necessary condition in the
calculus of varia-tions-is normally a second-order
differential equation containing two arbi-trary constants.
For problems with fixed initial and terminal points, the
two given boundary conditions provide sufficient
information to definitize the two arbitrary constants.
But if the initial or terminal point is variable (subject
to discretionary choice), then a boundary condition will
be missing. Such will be the case, for instance, if the
dynamic monopolist of the preceding chapter faces noexternally imposed price at time T,and can treat the PT
choice as an integral part of the optimization problem. In
that case, the boundary condition P(T) = PT will nolonger be available, and the void must be filled by atransversality condition. In this chapter we shall develop
the transversality conditions appropriate to various types
of variable termi-nal points.
3.1 THE GENERAL TRANSVERSALITY
CONDITION
For expository convenience, we shall assume that only
the terminal point is variable. Once we learn to deal with
that, the technique is easily extended to the case of avariable initial point.
The Variable-Terminal-Point Problem Our new
objective is to
Maximize or minimize
Yy] = /*~(t,y, yf)dt0
(3.1) subject to
y(0) = A (A given)
and Y (~)=Y T (T,~~fr
This differs from the previous version of the problem in
that the terminal time T and terminal state y, are now"free" in the sense that they have become a part of the
optimal choice process. It is to be understood that
although T is free, only positive values of T are relevant
to the problem.
To develop the necessary conditions for anextremal, we shall, as before, employ a perturbing curvep(t), and use the variable E to generate neighboring
paths for comparison with the extremal. First, supposethat T * is the known optimal terminal time. Then anyvalue of T in the immediate neighborhood of T* can be
expressed as
where E represents a small number, and AT
represents an arbitrarily chosen (and fixed) small change
in T. Note that since T* is known and AT is a prechosen
magnitude, T can be considered as a function of E,T(E)
with derivative
7rn
The same E is used in conjunction with the
perturbing curve p(t) to generate neighboring paths of
the extremal y*(t):
However, although the p(t) curve must still satisfy the
condition p(0) = 0 [see (2.2)1 to force the neighboring
paths to pass through the fixed initial point, the otherIcondition-p(t) = 0-should now be dropped, because y,
is free. By substituting (3.4) into the functional V[y], weget a function V(E) akin to (2.121, but since T is afunction of E by (3.2), the upper limit of integration in
the V function will also vary with E:
(3.5) ~(c=
kT'"~[t, y*(t) + rp(t), y*'(t) + rpf(t)] dt
Y(t> ~'(tOur problem is to optimize this V function with respect
to E.
definite integral (3.5) falls into the general form of (2.10).
The V/d~is therefore, by (2.111,
on the
right closely resembles the one encountered in (2.13)
arlier development of the Euler equation. In fact, much
of the rocess leading to (2.17) still applies here, with oneexception.
[Fyrp(t)];f in (2.16) does not vanish in the present problem,
value [Fy,p(t)],=, = [F,,],=,p(T), since we have assumed that
p(T) z 0. With this amendment to the earlier result in (2.171,
term in (3.6) = p(t) Fy - -F'! dt + [FY*],=,p(T[ [ 1can also write
Second term in (3.6) = [Fit=, AT
these into (3.6), and setting dV/d~= 0, we obtain the
new
p(t)[Fy - :FYr] dt + [Ff]t=T~(T)+ [F]~=TIT= 0
three terms on the left-hand side of (3.71, each contains
its own arbitrary element: p(t) (the entire perturbing
curve) in the (T) (the terminal value on the perturbing
curve) in the second arbitrarily chosen change in T) in
the third. Thus we cannot offsetting or cancellation of
terms. Consequently, in order tocondition (3.7), each of
the three terms must individually be set o
T FIGURE 3.1FIGURE 3.1
Step ii To this end, we first get rid of the arbitrary
quantity p(T) by transforming it into terms of AT and
AyT-the changes in T and y,, the two principal
variables in the variable-terminal-point problem. This
can be done with the help of Fig. 3.1. The AZ' curverepresents a neighboring path obtained by perturbing the
AZ path by ~p(t ),with E set equal to one for
convenience. Note that while the two curves share the
same initial point because p(0) = 0, they have different
heights at t = T because p(T) # 0 by construction of the
perturbing curve.' The magnitude of p(T), shown by
the vertical distance ZZ', measures the direct change in
y, resulting from the perturbation. But inasmuch as T
can also be altered by the amount EAT (= AT since E =11, the AZ' curve should, if AT > 0, be extended out to
2".2 As a result, yT is further pushed up by the vertical
distance between Z ' and 2". For a small AT, this second
change in y, can be approximated by y'(T) AT. Hence,
the total change in y, from point Z to point Z" is
Rearranging this relation allows us to express p(T) in
terms of AyT and
AT:3
'~echnically, the point T on the horizontal axis in Fig. 3.1 should
be labeled T*. We are omitting the * for simplicity. This is
justifiable because the result that this discussion is leading
to-(3.9)-is a transversality condition, which, as a standard
practice, is stated in terms of T (without the *) anyway.'~lthough we are initially interested only in the solid portion of the AZ
path, the equation for that path should enable us also to plot the
broken portion as an extension of AZ. The perturbing curve is then
applied to the extended version of AZ.
3~heresult in (3.8) is valid even if E is not set equal to one as in Fig.
3.1.
Step iii Using (3.8) to eliminate p(T) in (3.7), and
dropping the first term in (3.7), we finally arrive at the
desired general transversality condition
This condition, unlike the Euler equation, is relevant
only to one point of time, T.Its role is to take the place
of the missing terminal condition in the present problem.
Depending on the exact specification of the terminal line
or curve, however, the general condition (3.9) can be
written in various specialized forms.
3.2 SPECIALIZED TRANSVERSALITYCONDITIONS
In this section we consider five types of variable
The vertical-terminal-line case, as illustrated in Fig.
1.5a, involves g fixed T .Thus AT = 0, and the first
term in (3.9)drops out. But since Ay, is arbitrary and
can take either sign, the only way to make the second
term in (3.9) vanish for sure is to have Fy r= 0 at t =T.This gives rise to the transversality condition
which is sometimes referred to as the natural boundary
condition
Horizontal Terminal Line
(Fixed-Endpoint Problem)
For the horizontal-terminal-line case ,as illustrated in
Fig. 1.5b,the situa-tion is exactly reversed; we now have
Ay, = 0 but AT is arbitrary. So the second term in (3.9
automatically drops out, but the first does not. Since AT
is arbitrary, the only way to make the first term vanish
for sure is to have the bracketed expression equal to zeroThus the transversality condition is
It might be useful to give an economic
interpretation to (3.10 )and (3.11) .To f ix ideas, let usinterpret F(t ,y ,y') as a profit function ,where yrepresents capital stock, and y' represents net
entails taking resourceaway from the current profit-making busines
operation, so as to build up capital which will enhance
future profit. Hence there exists a tradeoff between
current profit and future profit. At any time t, with agiven capital stock y, a specific investmen
decision-say, a decision to select the investment rate
yl,--will result in the current profi F(t, y, y',). As to the
effect of the investment decision on future profits, i
enters through the intermediary of capital. The rate of
capital accumulation is y',; if we can convert that into avalue measure, then we can add it to the current profi
F(t, y, y',) to see the overall (current as well as future
profit implication of the investment decision. The
imputed (or shadow) value to the firm of a unit of capita
is measured by the derivative Fy,.This means that if we
decide to leave (not use up) a unit of capital at the
terminal timeit will entail a negative value equal to -Fyg.Thus, at t = T, the value
measure of y', is -y', Fyto.Accordingly, the overall profit implication of the
decision to choose the investment rate y', is F(t, y,y',) - yb Fyg.The general expression for this is F -y'Fy,, as in (3.11).
Now we can interpret the transversality condition (3.11) to mean that
in a problem with a free terminal time, the firm should
select a T such that a decision to invest and accumulate
capital will, at t = T, no longer yield any overal
(current and future) profit. In other words, all the
profit opportunities should have been fully taken
advantage of by the optimally chosen terminal time. In
addition, (3.10)-which can equivalently be writ-ten as[-FY,],=, = O-instructs the firm to avoid any sacrifice
of profit that will be incurred by leaving a positive
terminal capital. In other words, in a free-terminal-state
problem, in order to maximize profit in the interva
[0, T ] but not beyond, the firm should, at time T, us up all
the capital it ever accumulated.
Terminal Curve
With a terminal curve yT = +(T),as illustrated in Fig.
1.5c, neither AyT nor AT is assigned a zero value, soneither term in (3.9) drops out. However, for a small
arbitrary AT, the terminal curve implies that Ay, =q S AT.
So it is possible to eliminate Ay, in (3.9) and combine the
two terms into the form
[F-y1FYp+ F,vqS]t-TAT = 0
Since AT is arbitrary, we can deduce the transversality
condition
Unlike the last two cases, which involve a single
unknown in the terminal point (either y, or TI, the
terminal-curve case requires us to determine both y,and T. Thus two relations are needed for this purpose.The transversality condition (3.12) only provides one; the
other is supplied by the equation y, = +(T 1.
Truncated Vertical Terminal Line
The usual case of vertical terminal line, with AT = 0,
specidixes (3.9) to
When the line is truncated-restricted by the terminal
condition yT 2 ymi where yminis a minimum permissible
level of y-the optimal solution can have two possible