Forthcoming in Journal of Economic Dynamics and Control

Necessity of the Transversality Condition for
Stochastic Models with
Bounded or CRRA Utility
Takashi Kamihigashi∗
RIEB, Kobe University
Rokkodai, Nada, Kobe 657-8501, Japan
October 8, 2004
Abstract
This paper shows that the standard transversality condition (STVC) is necessary for optimality in stochastic models with bounded or constant-relativerisk-aversion (CRRA) utility under fairly general conditions. We consider an
inﬁnite-horizon stochastic maximization problem that takes a general form ofa multi-sector growth model with a single consumption good. We show that
the STVC is necessary if utility is bounded or logarithmic. We also show that
the STVC is necessary in the case of non-logarithmic CRRA utility as longas lifetime utility is ﬁnite at the optimum. These results apply to various
stochastic growth models, including real business cycle models with endogenous labor supply. Since unbounded utility functions that do not belong to
the CRRA class are rather rare in applications, our results provide a fairly
complete set of solutions regarding necessity of the STVC in practice.
Keywords: Transversality condition, stochastic optimization, stochastic
growth, CRRA, real business cycle.
JEL Classification Numbers: C61, D90, G12∗

Tel./fax: +81-78-803-7015, Email address: tkamihig@rieb.kobe-u.ac.jp

I would like to thank Editor Wouter J. den Haan and two anonymous referees for helpful comments and suggestions. Section 2 owes much to Editor Haan’s suggestions. Section
6 owes much to a referee’s comments. The other referee’s comments and suggestions havebeen helpful in clarifying many issues.

1Introduction

Since Shell (1969) and Halkin (1974), necessity of transversality conditions(TVCs) has been an uneasy matter to economic theorists who use inﬁnitehorizon optimization problems. The most standard TVC, which we call thestandard TVC (STVC), is the condition that the value of optimal stocks at
inﬁnity must be zero. Necessity of the STVC becomes an issue particularly
for models with unbounded utility, such as constant-relative-risk-aversion(CRRA) utility.
Models with CRRA utility are widely used in macroeconomics, for example, in the endogenous growth literature (e.g., Barro and Sala-i-Martin,
1995), the indeterminacy literature (e.g., Benhabib and Farmer, 1999), andthe real business cycle (RBC) literature (e.g., King and Rebelo, 1999). Although some results on necessity of the STVC for models with CRRA utility
are available in the literature, they only deal with deterministic cases (Ekeland and Scheinkman, 1986; Alvarez and Stokey, 1998; Kamihigashi, 2000,
2001).
This paper shows that the STVC is necessary in stochastic models withbounded or CRRA utility under fairly general conditions. In particular,
we consider an inﬁnite-horizon stochastic maximization problem that takesa general form of a multi-sector growth model with a single consumption
good. Our model encompasses various stochastic growth models (e.g., Brockand Mirman, 1972), including RBC models with endogenous labor supply(e.g., King and Rebelo, 1999). Our assumptions on the technology side of
the model are quite general. They require only a weak concavity property
and some technical conditions that are rarely violated in applications.
We establish three results in this paper. Our ﬁrst result is that the STVC
is necessary if utility is bounded. This of course means that the STVC may
fail to be necessary only if utility is unbounded. Hence we next turn to thecase of unbounded utility. We restrict attention to the CRRA class sinceunbounded utility functions that do not belong to this class are rather rare
in applications. Our second result is that the STVC is necessary in the caseof logarithmic utility. This is a strong result since it does not require any
additional condition (other than the technical assumptions on the technology
side of the model). Our last result is that the STVC is necessary in the caseof non-logarithmic CRRA utility as long as lifetime utility is ﬁnite at the
optimum. It is important to emphasize that we do not assume the ﬁniteness oflifetime utility for all feasible paths, for such an assumption is often violated1

in models with unbounded utility.
Since these results are established for a general stochastic growth model,
they generalize similar results shown for deterministic models (e.g., Kamihigashi, 2000, 2001). Though our results are based on a general theorem
shown in Kamihigashi (2003), they are surprisingly clean and powerful. Dueto their general applicability, they provide a fairly complete set of solutions
regarding necessity of the STVC in practice since utility is usually assumed
to be bounded or of the CRRA class in applications. It is useful to know that
the STVC is necessary in these cases, not only because it helps characterize
optimal paths, but also becasue it has important implications for the stability
properties of balanced growth paths, as recently shown by Martı́nez-Garcı́a(2003).
The rest of the paper is organized as follows. Section 2 shows how theBellman equation is related to the STVC, examines two examples in which
the STVC is not necessary, and discusses related results in the literature.
Section 3 presents our general model and discusses our assumptions. Our
main results are stated in Section 4. The proofs appear in Appendix B.
Section 5 explains how one can apply our results to models with endogenous
labor supply such as RBC models. Section 6 discusses how one can apply
our results to other models with control variables. Section 7 concludes thepaper. Appendix A contains preliminary lemmas.

2

Informal Discussions

This section clariﬁes some of the issues regarding necessity of the STVC.1
In particular, we show how the Bellman equation is related to the STVC,
and discuss when the STVC possibly fails to be necessary by examining two
simple examples. Related results in the literature are also discussed. Though
this paper is concerned with stochastic models, most of the known issues on
the STVC are deterministic in nature. Hence for the most part we consider
only deterministic models in this section.
1

This section owes much to Editor Woulter J. den Haan’s suggestions.

2

2.1

The Squeezing Argument

Consider the following maximization problem:max{xt }∞t=0∞


β t v(xt , xt+1 ) s.t. x0 = x0 ,∀t ∈ N, xt+1 ≥ 0,

(1)

t=0

where β ∈ (0, 1) is the discount factor, v is assumed to be nonincreasingin the second argument and concave, and x0 is given exogenously. Theremay be other constraints, but we assume that they are not binding at the
optimum.
For the moment we assume that the objective function, or lifetime utility,
is ﬁnite for all feasible paths. Then we may assume without loss of generalitythatv(0, 0) ≥ 0.
(2)
Essentially it has been known since Weitzman (1973) that in this case, any(interior) optimal path {x∗t } satisﬁes
lim β t [−v2 (x∗t , x∗t+1 )]x∗t+1 = 0.
t↑∞

(3)

This condition means that the value of optimal stocks must be zero at inﬁnity.
We call (3) the standard TVC (STVC).2
Perhaps the easiest way to derive (3) is to consider the Bellman equation
V (x∗t ) = v(x∗t , x∗t+1 ) + βV (x∗t+1 ).

(4)

The ﬁniteness of lifetime utility for all feasible paths implies
lim β t V (x∗t ) = 0.
t↑∞

(5)

Assuming interiority, one obtains the Euler equation
v2 (x∗t , x∗t+1 ) + βv1 (x∗t+1 , x∗t+2 ) = 0.

(6)

By the envelope theorem,V  (x∗t ) = v1 (x∗t , x∗t+1 ).
2(7)

An alternative form of the condition, limt↑∞ β t v1 (x∗t , x∗t+1 )x∗t = 0, is often used.
Though this condition is equivalent to (3) under the Euler equation (6), it has no counterpart in the finite-horizon case as well as in the continuous-time case.

3

From (6) and (7),−v2 (x∗t , x∗t+1 )x∗t+1 = βV  (x∗t+1 )x∗t+1 ≤ βV (x∗t+1 ),(8)

where the last inequality holds since V (0) ≥ 0 by (2) and V is concave by the
concavity of v. Since v is nonincreasing in the second argument, the leftmost
side of (8) is nonnegative. Now TVC (3) follows from (8) and (5) by the“squeezing” argument.3

2.2

Counterexamples

The crucial conditions used in the above argument are (2) and (5), which are
obtained from the assumed ﬁniteness of lifetime utility for all feasible paths.
If this assumption is violated, however, the STVC may fail to be necessary for
optimality. Let us study two simple examples in which the STVC is indeed
not necessary.
We ﬁrst consider the discounted Ramsey model:

∞


t

β u(f (xt ) − xt+1 ),
max
{xt }
(9)
t=0


s.t. x0 = x0 , ∀t ∈ Z+ , 0 ≤ xt+1 ≤ f (xt ),where u is the utility function, and f is the production function. We assume
that f satisﬁes the standard neoclassical properties. Since the main focus of
this paper is on the CRRA case, let us assume∃α ∈ (−∞, 1] \ {0},u(c) =

cα
.
α

(10)

It is well-known that in the discounted Ramsey model, the optimal path{x∗t } from any nonzero initial capital stock converges to the modiﬁed golden
rule capital stock xβ , which is given by βf  (xβ ) = 1. Hencelim β t u (f (x∗t ) − x∗t+1 )x∗t+1 = lim β t u (f (xβ ) − xβ )xβ .
t↑∞

t↑∞

(11)

The right-hand side is of course zero since β ∈ (0, 1). Thus the STVC holds.
It is also well-known that optimal paths converge to the golden rule capitalstock x1 when β = 1. In this case, however, the right-hand side of (11) is
3

See Kamihigashi (2002) for an elementary proof that does not use the envelop theorem.

4strictly greater than zero, so that the STVC is violated. Perhaps partly forthis reason, the violation of the STVC is often associated with the absenceof discounting. But the actual factor that determines necessity of the STVC
is not whether discounting is present or not.
To see this point clearly, consider the following exogenous growth model:

∞


t
t
γ

β u(θ (xt ) − xt+1 ),
max
{xt }
(12)
t=0

t
γ 
s.t. x0 = x0 , ∀t ∈ Z+ , 0 ≤ xt+1 ≤ θ (xt ) ,where θ > 1 is the growth factor, and γ ∈ (0, 1). Let g = θ1/(1−γ) . Note thatg satisﬁes(13)
θg γ = g.
For t ∈ Z+ , deﬁnex̃t = xt /g t−1 .

(14)

The maximization problem (12) can now be written as follows:4

∞



β̃ t u(g −γ (x̃t )γ − x̃t+1 ),
max
{xt }

t=0

s.t. x̃0 = gx0 ,∀t ∈ Z+ , 0 ≤ x̃t+1 ≤ g −γ (x̃t )γ ,(18)

where β̃ = βg α . If β̃ = 1, this problem is a special case of the undiscountedRamsey model (i.e., (9) with β = 1). Hence in this case, the STVC is notnecessary for optimality. But the original problem (12) has strictly positive
discounting since β̃ = 1 is equivalent to β = g −α < 1.
What is common in the preceding two “counterexamples” is not that
there is no discounting, but that lifetime utility is not ﬁnite at the optimum.
Note that the ﬁniteness of lifetime utility at the optimum is a much weakerassumption than the ﬁniteness of lifetime utility for all feasible paths. In fact,
4

To see this, note thatθt (xt )γ − xt+1 = θt (g t−1 x̃t )γ − g t x̃t+1
γ t −γ(x̃t ) − g x̃t+1−γ

γ= (θg ) gt

= g [gwhere the last equality uses (13).

5

γ

t

(x̃t ) − x̃t+1 ],

(15)
(16)
(17)

if α < 0, the latter assumption is violated in the above models regardless ofβ, even when lifetime utility is ﬁnite at the optimum.
It can therefore be expected that the sharpest result one can hope to proveis that the STVC is necessary in the CRRA case if lifetime utility is ﬁnite
at the optimum. This is exactly what we show in Section 4 for a general
stochastic growth model. This result generalizes similar results known for
deterministic models with CRRA utility (e.g., Kamihigashi 2000, 2001).
It should be mentioned that the ﬁniteness of lifetime utility at the optimum is a meaningless assumption unless one focuses on a speciﬁc classof utility functions. In fact, it can always be made true by subtracting anappropriate constant from the utility function, as is usually done in the undiscounted Ramsey model. This type of normalization is prohibited by (10), sothat the ﬁniteness of lifetime utility is a meaningful assumption here.

2.3

Related Results in the Literature

The argument in Section 2.1 is essentially due to Benveniste and Scheinkman(1982, p. 15), but Weitzman (1973) used a similar argument based on support
prices instead of derivatives. The diﬃcult part of the actual proof is to show
(7).5 The key assumptions used in the argument are (2) and (5), as mentioned
above.
The ﬁrst attempt to relax (2) was made by Ekeland and Scheinkman(1986), who studied a model with CRRA-type utility. Instead of assuming
the ﬁniteness of lifetime utility for all feasible paths, they assumed that thefeasible sequences of utility are bounded above by a common summable sequence. This assumption was relaxed by Kamihigashi (2000) to the ﬁnitenessof lifetime utility at the optimum.
Michel (1990) extended Weitzman’s (1973) analysis to a discrete-time
optimal control problem with possibly unbounded returns. Instead of theSTVC, however, he focused on TVCs that an optimal path must satisfy
against all feasible paths that do not incur an inﬁnite loss.
Kamihigashi (2001) uniﬁed the approaches of Ekeland and Scheinkman(1986) and Michel (1990), generalizing earlier results and obtaining new results in a continuous-time reduced-form model. The basic idea of Kamihigashi (2001) is that the STVC is necessary if the optimal path can be shifted
proportionally downward without causing an inﬁnite loss. See Kamihigashi
5

See Stokey and Lucas (1989, p. 85) for a proof of (7).

6

(2001) for a more detailed literature review on necessity of TVCs for deterministic models.
Stochastic versions of Weitzman’s (1973) result were obtained by Zilcha(1976) and Takekuma (1992) using inﬁnite-dimensional separation arguments.6
Their results on the STVC were generalized by Kamihigashi (2003) for a
discrete-time stochastic reduced-form model using the approach of Kamihigashi (2001).
As suggested in the Introduction, the contribution of this paper is toprovide a fairly complete set of solutions regarding necessity of the STVC
for stochastic growth models in applications. Though our results are basedon a general theorem shown in Kamihigashi (2003), they are more directly
applicable. Indeed, since unbounded utility functions that do not belong to
the CRRA class are rather rare in applications, our results seem to covermost cases in practice.

3

The ModelThis section presents our general model and discusses our assumptions. Someof the material here is borrowed from Kamihigashi (2003), which oﬀers further discussions on the general structure of the model.
Let (Ω, F, P ) be a probability
space. Let E denote the associated expec
tation operator; i.e., Ez = z(ω)dP (ω) for any random variable z : Ω → R.
When it is important to make explicit the dependence of z on ω, we writeEz(ω) instead of Ez. Consider the following problem:

∞



t

β
Eu(g
(x
(ω),
x
(ω),
ω))”
“ max

t
t
t+1

{xt }∞
t=0
t=0
(19)

s.t. x0 = x0 ,∀t ∈ Z+ , (xt , xt+1 ) ∈ Xt .7
In what follows, we list and discuss all our assumptions.
Let n ∈ N.8 Let F be the set of all functions from Ω to Rn . The following
assumption means that each xt is a random variable in Rn .
6

Foldes (1978) extended Weitzman’s result to a one-sector stochastic growth modelusing directional derivatives and martingale arguments.
7
Z+ ≡ {0, 1, 2, . . .}.
8
N ≡ {1, 2, 3, . . .}.

7

Assumption 3.1. x0 ∈ F and ∀t ∈ Z+ , Xt ⊂ F × F .
Since F consists of all functions from Ω to Rn , and since Xt is only asubset of F × F , Xt can be chosen in such a way that xt and xt+1 must be
measurable with respect to the information available in period t and periodt + 1, respectively. Restricting Xt this way is not necessary, however, since
our results require no such information structure.
The next assumption collects our general restrictions on the preference
side of the model.
Assumption 3.2. (i) β ∈ (0, 1); and (ii) u : R+ → [−∞, ∞) is C 1 on R++ ,concave, and strictly increasing.
The next assumption simply means that the expression
Eu(gt (xt (ω), xt+1 (ω), ω))(20)

makes sense for all (xt , xt+1 ) ∈ Xt .
Assumption 3.3. ∀t ∈ Z+ , ∀(y, z) ∈ Xt , (i) ∀ω ∈ Ω, gt (y(ω), z(ω), ω) ≥0, (ii) the mapping gt (y(·), z(·), ·) : Ω → R+ is measurable, and (iii) the
expectation Eu(gt (y(ω), z(ω), ω)) exists in [−∞, ∞).
To simplify expressions like (20), let gt (xt , xt+1 ) denote the random variable gt (xt (·), xt+1 (·), ·) : Ω → R for (xt , xt+1 ) ∈ Xt . We say that a sequence{xt }∞
t=0 is a feasible path if x0 = x0 and ∀t ∈ Z+ , (xt , xt+1 ) ∈ Xt . Since inapplications the objective function is often not guaranteed to be ﬁnite orwell-deﬁned for all feasible paths, we use weak maximality (Brock, 1970) as
our optimality criterion. We say that a feasible path {x∗t } is optimal if forany feasible path {xt },
lim

T


T ↑∞ t=0

β t [Eu(gt (xt , xt+1 )) − Eu(gt (x∗t , x∗t+1 ))] ≤ 0.9(21)

Our optimality criterion (i) reduces to the standard maximization criterion whenever the latter makes sense, (ii) applies even when the standard
9

To be precise, this inequality requires that the left-hand side be well-defined. Thismeans that the left-hand side may not involve expressions like “∞ − ∞” and “−∞ + ∞.”Thus optimality implies that Eu(gt (x∗t , x∗t+1 )) is finite for all t ∈ Z+ ; for otherwise theleft-hand side of (21) is undefined for {xt } = {x∗t }.

8

criterion fails, and (iii) is weaker than the similar criterion with lim replacing
lim in (21). Using this optimality criterion does not limit the applicabilityof our results in any way since we are only interested in necessary conditions
for optimality.
For the same reason, assuming the existence of an optimal path imposesno restriction on the model.
Assumption 3.4. There exists an optimal path {x∗t }.
For t ∈ Z+ and d : Ω → Rn such that (x∗t , x∗t+1 + d) ∈ Xt for suﬃciently
small  > 0, we deﬁne the random variable gt,2 (x∗t , x∗t+1 ; d) as follows:
gt (x∗t , x∗t+1 + d) − gt (x∗t , x∗t+1 ),↓0

gt,2 (x∗t , x∗t+1 ; d) = lim(22)

where lim ↓0 is applied pointwise (i.e., for each ω ∈ Ω separately). The righthand side of (22) is called a lower Dini directional derivative. Note that if gt
is diﬀerentiable in the second argument, thengt,2 (x∗t , x∗t+1 ; d) = D2 gt (x∗t , x∗t+1 )d,(23)

where D2 gt denotes the derivative of gt with respect to the second argument.
We use the general derivative deﬁned in (22) only because assuming diﬀerentiability does not oﬀer any simpliﬁcation and, at the same time, requires
additional assumptions that are somewhat cumbersome to state.
As mentioned in Section 2.1, the STVC is the condition that the valueof optimal stocks at inﬁnity must be zero. It takes the following form in thecurrent framework:
lim β t Eu (gt (x∗t , x∗t+1 ))gt,2 (x∗t , x∗t+1 ; −x∗t+1 ) = 0.
t↑∞

(24)

Recalling (23), one can see that (24) reduces to the familiar TVC (e.g., (3)) in
the diﬀerentiable case. The following assumption is a minimum requirementfor (24) to make sense.
Assumption 3.5. The mapping gt,2 (x∗t , x∗t+1 ; −x∗t+1 ) : Ω → R is measurable.
It is diﬃcult to think of a situation in which this assumption is violated.
It must be assumed nonetheless since the other assumptions do not imply it.
It is satisﬁed, for example, if gt depends on ω through a technology shock
9and is continuously diﬀerentiable in xt , xt+1 , and the technology shock (andif these variables are measurable).
The assumptions stated thus far are preliminary requirements that are inmany cases assumed implicitly. In what follows, we state three assumptions
that are actually needed to prove the necessity of the STVC. Each of them isfairly general though it has a speciﬁc role. The following is our assumption
on Xt .
Assumption 3.6. ∀t ∈ Z+ , ∃λt ∈ [0, 1), ∀λ ∈ [λt , 1), (x∗t , λx∗t+1 ) ∈ Xt and∀τ ≥ t + 1, (λx∗τ , λx∗τ +1 ) ∈ Xτ .
This assumption means that the optimal path {x∗t } can be shifted proportionally downward starting from any period. The assumption is common
to most results on the STVC, which basically means that no gain should beachieved by shifting the optimal path proportionally downward.
Assumption 3.6 is satisﬁed if Xt is convex, (0, 0) ∈ Xt , and (x∗t , 0) ∈ Xt for
all t ∈ Z+ . We use Assumption 3.6 in order to accommodate cases in which it
is not always feasible to reduce capital stocks to zero in one step, as in models
with irreversible investment (e.g., Olson, 1989). In a deterministic model,
Ekeland and Scheinkman (1986, Proposition 5.1(iii), (iv)) used a special caseof Assumption 3.6 in which λt = 0 for all t ∈ Z+ .
Assumption 3.6 with λt = 0 implies that the convex combinations of
the optimal path and the constant path at zero are feasible (except for the
initial condition). It is natural to consider such comparison paths to derive
the STVC, which is essentially the optimality requirement that no gain beachieved by shifting the optimal path proportionally downward, as mentioned
above. Such comparison paths however need not be feasible in nonconvex
models (e.g., Skiba, 1978), but in some cases other comparison paths can beconstructed to obtain a diﬀerent TVC. For example, in Skiba’s model, undersome conditions, the convex combinations of the optimal path and a certain
constant path are feasible. Ekeland and Scheinkman (1986, Proposition 5.3)
used this idea to show a result that can accommodate Skiba’s model. This
paper, on the other hand, is mainly concerned with convex models.
The next two assumptions impose some structure on gt . In what follows,
any property on a random variable such as gt (λx∗t , λx∗t+1 ) is understood asholding with probability one.
Assumption 3.7. ∀t ∈ Z+ , (i) gt (x∗t , x∗t+1 ) > 0 and (ii) ∀t ∈ N, gt (λx∗t , λx∗t+1 )is concave in λ ∈ [λ0 , 1], where λ0 is given by Assumption 3.6.
10

Part (i) above holds if u(·) satisﬁes the Inada condition at zero, as longas there is a feasible path satisfying (i). This part is not needed in the caseof bounded utility. Part (ii) is used to ﬁnd a lower bound on gt (λx∗t , λx∗t+1 )that depends only on λ and gt (x∗t , x∗t+1 ) (see Lemma A.2). This part canbe weakened considerably though it seems already general enough for most
purposes.
In our notation, Ekeland and Scheinkman (1986, Proposition 5.1(iii))
assumed that∀t ∈ N+ , ∀λ ∈ [0, 1],

λgt (x∗t , x∗t+1 ) ≤ gt (λx∗t , λx∗t+1 ).

(25)

Our results hold even if Assumption 3.7(ii) is replaced by (25). In fact,
Lemma A.2 in Appendix A shows that Assumption 3.7 implies a weakerversion of (25), and that property is all we need from Assumption 3.7(ii) toshow our main results.
The following is our last assumption.
Assumption 3.8. ∀t ∈ Z+ , gt (x∗t , λx∗t+1 ) is nonincreasing and continuous inλ ∈ (λt , 1], where λt is given by Assumption 3.6.
That gt (x∗t , λx∗t+1 ) is nonincreasing in λ basically means that there is atradeoﬀ between consumption and investment. Such a tradeoﬀ is typicallyrequired for the existence of an optimal path. In this paper, this restriction is used to express the STVC as an equality condition as in (24). The
continuity of gt (x∗t , λx∗t+1 ) in λ plays a technical role that allows us to write
gt (x∗t , x∗t+1 ) as the argument of u (·) in (24). Without this continuity requirement, gt (x∗t , x∗t+1 ) must be replaced by limλ↑1 gt (x∗t , λx∗t+1 ) in (24); see theproof of Lemma A.3 for details.

4

Main Results

We are now ready to state our main results, which assume all the assumptions
stated in the preceding section. Let us start from the case of bounded utility.
Proposition 4.1. Let Assumptions 3.1–3.8 hold. Suppose u is bounded.
Then TVC (24) holds.
Hence the STVC is necessary as long as u is bounded. As discussed in
Section 2.1, Proposition 4.1 is a well-known result in the deterministic case. If
11additional assumptions are made, Proposition 4.1 follows from Zilcha (1976,
Theorem 1) or Takekuma (1992, Theorem 5.2). The main advantage of ourresult is that it does not require any technical condition that is diﬃcult toverify in applications. For example, while the results of Zilcha (1976) andTakekuma (1992) require the state variable xt to be essentially bounded, ourresult imposes no such restriction.
Proposition 4.1 makes it clear that the STVC may fail to be necessary only
if u is unbounded. Thus we next turn to the case of unbounded utility. But
since unbounded utility functions that do not belong to the CRRA class arerather rare in applications, we restrict attention to this class. The following
result deals with the case of logarithmic utility.
Proposition 4.2. Let Assumptions 3.1–3.8 hold. Supposeu(·) = ln(·).

(26)

Then TVC (24) holds.
Hence the STVC is necessary in the case of logarithmic utility. Proposition 4.2 generalizes a similar result shown in Kamihigashi (2001) for a deterministic model. It makes it clear that once logarithmic utility is assumed,
there is no issue about necessity of the STVC even for stochastic models.
The following result deals with the case of non-logarithmic CRRA utility.
Proposition 4.3. Let Assumptions 3.1–3.8 hold. Suppose
(·)α
,
α

(27)

β t Eu(gt (x∗t , x∗t+1 )) < ∞.

(28)

∃α ∈ (−∞, 1] \ {0},
−∞ <

∞


u(·) =

t=0

Then TVC (24) holds.
Thus under (27), the STVC is guaranteed to be necessary unless lifetime
utility is allowed to be inﬁnite at the optimum. Such cases are usually ruledout in practice, and (28) is often assumed implicitly or taken for granted inapplied studies. Proposition 4.3 generalizes similar results shown in Kamihigashi (2000, 2001) for deterministic models.
It is worth emphasizing that (28) requires lifetime utility to be ﬁnite only
for the optimal path. It does not require lifetime utility to be ﬁnite for any
12other feasible path. This is important since lifetime utility is often −∞ formany feasible paths when α < 0. Proposition 4.3 makes it clear that once
non-logarithmic CRRA utility is assumed, there is no issue about necessityof the STVC for all practical purposes.

5

Endogenous Labor Supply

One can easily apply the results shown in the preceding section to models
with endogenous labor supply such as RBC models (e.g., King and Rebelo,1999). To do so, one can take an optimal labor (or, equivalently, leisure)path as given and consider the maximization problem over consumption andcapital paths.
To be more speciﬁc, consider the following problem:

∞


t


β
Er(c
,
l
)”
“ max
t
t


{xt }∞

t=0

t=0
(29)
s.t. x0 = x0 ,∀t ∈ Z+ , ct + xt+1 = ft (xt , 1 − lt ),ct , xt+1 ≥ 0, 0 ≤ lt ≤ 1,
where ct is consumption, lt is leisure, r is the utility function, and ft is the
production function (which depends on ω in the same way as gt does).10 In
this section we assume n = 1, i.e., xt ∈ R for all t ∈ Z+ .
First consider the case in which utility is additively separable in consumption and leisure, i.e.,(30)r(ct , lt ) = u(ct ) + n(lt )for some functions u(·) and n(·). In this case one can take an optimal leisurepath {lt∗ } as given and simply set gt (xt , xt+1 ) = ft (xt , 1 − lt∗ ) − xt+1 . The
model then reduces to (19), ignoring utility from leisure (which is taken as
given).
When utility is not additively separable, it is typically the case that utility
can be written as(31)r(ct , lt ) = u(ct e(lt ))10More precisely, ft (xt , 1 − lt ) denotes the random variable ft (xt (ω), 1 − lt (ω), ω) in
the same way as gt (xt , xt+1 ) denotes the random variable gt (xt (ω), xt+1 (ω), ω); recall theparagraph below Assumption 3.3.

13

for some functions u(·) and e(·) (e.g., King and Rebelo, 1999, p. 945). Typically, u(·) is assumed to satisfy (27) (or (26), which leads to (30)), and e(·)
is required to satisfy certain regularity conditions (King and Rebelo, 1999,
footnote 69). In the case of (31), one can take an optimal leisure path {lt∗ }as given and set gt (xt , xt+1 ) = [ft (xt , 1 − lt∗ ) − xt+1 ]e(lt∗ ). The model thenreduces to (19) again.
We have shown that maximization problems of the form (29) reduce to(19) in typical parametric cases if an optimal leisure path is taken as given.
There is no problem in taking an optimal leisure path as given, since necessary
optimality conditions such as Euler equations and TVCs are expressed taking
an optimal path as (implicitly) given anyway. The discussions here along with
our main results indicate that there is practically no issue about necessity of
the STVC for standard parametric stochastic growth models, whether labor
supply is exogenous or endogenous.

6

Other Models with Control Variables

The preceding discussions suggest that our results apply even to models that
do not take the exact form of (19), which contains no explicit “control”
variable. Since many models involve both state and control variables, we
brieﬂy discuss here how a model with control variables can be reduced to(19).11 To be speciﬁc, consider the following problem:

∞



t

β
Eu(h
(x
,
a
))”
“ max

t
t
t

∞
{xt }t=0
t=0(32)
s.t. x0 = x0 ,∀t ∈ Z+ , xt+1 = ft (xt , at ), xt+1 , at ≥ 0,
N
for some N ∈ N, and ht and ft depend on ω
where at ≡ (a1t , . . . , aNt ) ∈ Rin the same way as gt does (recall footnote 10).
If N = n (where n is the dimension of xt ), it is often possible to reduce(32) to (19) by solving the constraint equation for at and substituting into
the objective function. If there are multiple solutions for at (or if N = n),
one way to reduce (32) to (19) is to deﬁne

gt (xt , xt+1 ) = max{ht (xt , at ) | xt+1 = ft (xt , at ), at ≥ 0},
11This section owes much to the comments made by an anonymous referee.

14

(33)

where the maximum is taken over at with xt and xt+1 taken as given.12 Thisis a special case of the standard reduced-form approach, which is to express
the return function as a function of state variables by maximizing it in termsof control variables.
An alternative approach is to eliminate some control variables and toassume that the rest take optimal values. This is the approach used in
Section 5. If applicable, it seems easier in most cases than the standard
reduced-form approach. For example, the standard approach applied to (29)means expressing ct and lt as functions of xt and xt+1 . This procedure does
not reduce (29) to (19) under (30), in which case (29) does not take theexact form of (32) due to the additive term n(lt ). But if we take an optimal
path for {lt } as given, then (29) eﬀectively becomes a special case of (32).
Although the standard approach does reduce (29) to (19) under (31), theresulting STVC is rather complicated (though it may be simpliﬁed using theenvelope theorem). Thus even when the standard approach is applicable, thealternative approach seems easier.

7

Concluding Remarks

In this paper we showed the necessity of the standard transversality condition (STVC) for a multi-sector growth model with a single consumption goodunder fairly general conditions. We considered the case of constant-relativerisk-aversion (CRRA) utility as well as that of bounded utility. In particular, it was shown that the STVC is necessary if utility is either boundedor logarithmic. It was also shown that the STVC is necessary in the caseof non-logarithmic CRRA utility as long as lifetime utility is ﬁnite at the
optimum. These results apply to various stochastic growth models, including RBC models with endogenous labor supply. Because of their generalapplicability, our results provide a fairly complete set of solutions regarding
necessity of the STVC for stochastic growth models in applications. Indeed,since utility functions that do not belong to the CRRA class are rather rare
in applications, our results seem to cover most cases in practice.
12

The existence of the maximum in (33) requires some assumptions, such as uppersemicontinuity of ht (xt , at ) in at and nonemptiness and compactness of the set {at ≥0 | xt+1 = ft (xt , at )}.

15

Appendix A

Preliminary Lemmas

The proofs of our propositions use a result shown in Kamihigashi (2003) fora general reduced-form model. We state it without proof after introducing
some notation. After stating the result, we show two further lemmas. All
the assumptions stated in Section 3 are maintained here though not all ofthem are used.
For t ∈ Z+ , (y, z) ∈ Xt , and ω ∈ Ω, deﬁnevt (y(ω), z(ω), ω) = β t u(gt (y(ω), z(ω), ω)).

(A.1)

Like gt (xt , xt+1 ), vt (xt , xt+1 ) denotes the random variable vt (xt (·), xt+1 (·), ·) :
Ω → R ∪ {−∞} for (xt , xt+1 ) ∈ Xt . For t ∈ N and λ < 1 with (λx∗t , λx∗t+1 ) ∈Xt , deﬁne
Evt (x∗t , x∗t+1 ) − Evt (λx∗t , λx∗t+1 ),
1−λŵt (λ) = sup wt (z),wt (λ) =z∈[λ,1)

(A.2)
(A.3)

where ŵt (λ) is deﬁned for λ < 1 such that ∀z ∈ [λ, 1), (zx∗t , zx∗t+1 ) ∈ Xt .
Lemma A.1. Suppose∃{bt }∞ŵt (λ) ≤ bt ,t=1 ⊂ R, ∃λ ∈ [λ0 , 1), ∀t ∈ N,∞
bt exists in [−∞, ∞),

(A.4)
(A.5)

t=1

where λ0 is given by Assumption 3.6. Thenlim Evt,2 (x∗t , x∗t+1 ; −x∗t+1 ) ≤ 0,
t↑∞

(A.6)

where vt,2 is defined as in (22).
Proof. See Kamihigashi (2003, Theorem 2.2).
Conditions (A.4) and (A.5) basically mean that the optimal path canbe shifted proportionally downward without causing an inﬁnite loss. Using
these conditions, one can show (A.6) by a generalized version of the squeezingargument (recall Section 2.1).
For simplicity, we use the following notations for t ∈ Z+ :gt∗ = gt (x∗t , x∗t+1 ),gt (λ) = gt (λx∗t , λx∗t+1 ).
16

(A.7)

Lemma A.2. We have∀t ∈ N, ∀λ ∈ [λ0 , 1),gt (λ) ≥ m(λ)gt∗ ,(A.8)

where λ0 is given by Assumption 3.6 andm(λ) =λ − λ0.
1 − λ0(A.9)

Proof. It is easy to see thatm(λ) + [1 − m(λ)]λ0 = λ.

(A.10)

Let t ∈ N. Since gt (λ) is concave in λ ∈ [λ0 , 1] by Assumption 3.7, forλ ∈ [λ0 , 1],gt (λ) ≥ m(λ)gt∗ + [1 − m(λ)]gt (λ0 )≥ m(λ)gt∗ ,(A.11)(A.12)

where the last inequality holds since gt (λ0 ) ≥ 0 by Assumptions 3.6 and 3.3.
Now (A.8) follows.
Lemma A.3. TVC (A.6) is equivalent to TVC (24).
Proof. Since Assumption 3.8 implies vt,2 (x∗t , x∗t+1 ; −x∗t+1 ) ≥ 0, TVC (A.6) isequivalent to(A.13)
lim Evt,2 (x∗t , x∗t+1 ; −x∗t+1 ) = 0.
t↑∞

Thus it suﬃces to verify the following for t ∈ Z+ :vt,2 (x∗t , x∗t+1 ; −x∗t+1 ) = β t u (gt∗ )gt,2 (x∗t , x∗t+1 ; −x∗t+1 ).

(A.14)

Let t ∈ Z+ . Let  > 0 be such that (1 − ) ∈ [λ0 , 1). Note thatvt (x∗t , x∗t+1 − x∗t+1 ) − vt (x∗t , x∗t+1 )= β t [u(gt (x∗t , x∗t+1 − x∗t+1 )) − u(gt∗ )]= β t u (g̃)[gt (x∗t , x∗t+1 − x∗t+1 ) − gt∗ ],(A.15)(A.16)(A.17)

where (A.17) holds by the mean value theorem for someg̃ ∈ [gt∗ , gt (x∗t , (1 − )x∗t+1 )].

(A.18)

Since gt (x∗t , (1 − )x∗t+1 ) is continuous in  ∈ [0, 1 − λt ) by Assumption 3.8, g̃
converges to gt∗ as  ↓ 0. Thus dividing (A.15) and (A.17) by  and applying
lim ↓0 yields (A.14).
17

Appendix B

ProofsBy Lemma A.3, to conclude TVC (24) from Lemma A.1, it suﬃces to verify(A.4) and (A.5). These conditions are veriﬁed in each of the following three
proofs.

B.1

Proof of Proposition 4.1

Let λ ∈ (λ0 , 1), λ ∈ [λ, 1), and t ∈ N. By Assumption 3.7(ii) and theconcavity of u, wt (λ) is nonincreasing in λ ∈ [λ, 1). Thusŵt (λ) ≤ wt (λ) = β tu(gt (x∗t , x∗t+1 )) − u(gt (λx∗t , λx∗t+1 ))
.
1−λ

(B.1)

Let bt equal the rightmost side above. Then {bt } clearly satisﬁes (A.4). It
also satisﬁes (A.5) by the boundedness of u.

B.2

Proof of Proposition 4.2

Let λ ∈ (λ0 , 1), λ ∈ [λ, 1), and t ∈ N. We have
(1 − λ)β −t wt (λ) = E[ln gt∗ − ln gt (λ)]≤ E [ln gt∗ − ln (m(λ)gt∗ )]= − ln m(λ),

(B.2)
(B.3)
(B.4)

where (B.3) uses Lemma A.2. It follows thatβ −t wt (λ) ≤where− ln m(λ)≤ A,
1−λ− ln m(λ).
1−λλ∈[λ,1)A = sup

(B.5)

(B.6)

Note that A is ﬁnite since the supremand is continuous on [λ, 1) and converges
to 1/(1 − λ0 ) as λ ↑ 1. Now (A.4) and (A.5) hold with bt = β t A.

18

B.3

Proof of Proposition 4.3

Let λ ∈ (λ0 , 1), λ ∈ [λ, 1), and t ∈ N. We have
(1 − λ)β −t wt (λ) = E[u(gt∗ ) − u(gt (λ))]≤ E[u(gt∗ ) − u(m(λ)gt∗ )]= [1 − (m(λ))α ]Eu(gt∗ ),

(B.7)
(B.8)
(B.9)

where (B.8) holds by Lemma A.2, and (B.9) holds by (27). It follows that
1 − (m(λ))ααEu(gt∗ )(1 − λ)α≤ AαEu(gt∗ ),β −t wt (λ) ≤where

1 − (m(λ))α
.
A = sup(1 − λ)αλ∈[λ,1)(B.10)(B.11)(B.12)

Note that A is ﬁnite since the supremand is continuous in λ ∈ [λ, 1) andconverges to 1/(1 − λ0 ) as λ ↑ 1. Now (B.10), (B.11), and (28) imply (A.4)
and (A.5) with bt = β t AαEu(gt∗ ).

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21