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If a path satisﬁes both Euler equation and transversality condition, then it isovertaking optimal. This can easily be shown using the usual method. However,the necessity of the transversality condition is diﬃcult to verify. In usual model,
we can verify it by using a perturbation technique to the inner solution. (SeeKamihigashi [7].) This logic is no longer used if the existence of the inner solutionof the model is hard to verify directly. However, we can show the necessity of thetransversality condition for an overtaking optimal solution of the AK model under
mild conditions. (Theorem 3.3)
There is another merit in treating the overtaking optimality. That is, we cantreat a model without time-discounting. The time-discount rate was introducedin the literature by Cass [5] and Koopmans [8] independently. The existence of apositive time-discount rate makes the problem much easier to solve. However, sucha discount rate does not appear in the work of Ramsey [9], which is the origin of thecapital accumulation model. To our understanding, Ramsey considered that thecapital accumulation model belongs to not positive theory but normative theory,and from the normative viewpoint, the ideal social planner must not discriminate
against future people. Fortunately, we can treat Ramsey’s original problem using
the overtaking optimality.
We stress that our assumption of the model is weak. For example, we assume
that the instantaneous utility is continuously diﬀerentiable but do not assume that
it is twice diﬀerentiable. Additionally, we do not assume the boundedness of theutility function, and thus we treat all constant relative risk aversion (CRRA) utility
functions. If the instantaneous utility function is a CRRA function, then we canderive a simple formula for the solution. (Theorem 4.1) Although the same formula
was presented by Barro and Sara-i-Martin [2], we additionally give a necessary andsuﬃcient condition for the existence of the solution, and our condition is wider thantheir requirement.
Section 2 presents our model and prepares the necessary notions and deﬁnitions.
The main formula and general results are given in section 3. Section 4 considers
the case in which the instantaneous utility is a CRRA function.

2. Model and Basic Notions
2.1. Model. The classical AK model is written as∫ ∞e−ρt u(c(t))dtmax0

subject to. k(0) = k̄ > 0, k(t) ≥ 0, c(t) ≥ 0,k̇(t) = γk(t) − c(t) a.e.,c(·) ∈ W,
where k is the amount of capital, c is the amount of private consumption, and ρ isthe time discount rate. The set W is some functional space. The equationk̇(t) = γk(t) − c(t)FORMULA OF THE SOLUTION FOR AN AK GROWTH MODEL

941

represents two economic relationships. The ﬁrst relationship is the equality between
production and consumption. That is,κk(t) = c(t) + i(t),where the positive value κ represents the linear production technology and i denotes
the amount of investment. The second relationship is between the capital stock andinvestment. That is,k̇(t) = i(t) − dk(t),where d ≥ 0 is the capital wastage ratio. Combining these two equations, we havek̇(t) = κk(t) − dk(t) − c(t) ≡ γk(t) − c(t),where γ = κ − d, as desired.
In this paper, however,u is not necessarily bounded and ρ is not necessarily∫∞positive, and thus, 0 e−ρt u(c(t))dt may not be deﬁnable. Therefore, we modify
the above problem as:∫ Tmax lim T →∞e−ρt u(c(t))dt0(2.1)

subject to. k(0) = k̄ > 0, k(t) ≥ 0, c(t) ≥ 0,k̇(t) = γk(t) − c(t) a.e.,c(·) ∈ W,

where lim means either lim sup or lim inf.
We make the following three assumptions in this paper.
Assumption 1. The function u is a continuous, strictly concave, and increasing
function deﬁned on R+ and continuously diﬀerentiable on R++ .
Assumption 2. The value γ is positive.
Assumption 3. The set W denotes the set of all locally integrable functions onR+ .
Note that, under Assumption 3, k must be absolutely continuous on every compact set in R+ .
2.2. Overtaking Optimality. We have converted the classical AK problem toproblem (2.1). However, even for this model, some problematic cases exist. If thereexist (k1 (t), c1 (t)) and (k2 (t), c2 (t)) such that∫ Te−ρt u(ci (t))dt = +∞lim T →∞0for each i, then these must be equivalent, even though c1 (t) > c2 (t) for every t.
This is problematic, and thus we need a criterion that can compare these processes
appropriately.
First, we deﬁne the notion of admissibility.
Deﬁnition 2.1. A pair of functions (k(t), c(t)) deﬁned on R+ is called admissible
if the following properties hold.

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1. c(t) is integrable and k(t) is absolutely continuous on any compact interval in
R+ . 1
2. k(t) ≥ 0 and c(t) ≥ 0 for all t.
3. The following diﬀerential equationk̇(t) = γk(t) − c(t)holds for almost all t ∈ R+ .
Let A be the set of all admissible pairs, and Ak̄ be the set of all admissible pairs
such that k(0) = k̄.
Deﬁne a binary relation ≻∗ such that for any (k1 (t), c1 (t)), (k2 (t), c2 (t)) ∈ Ak̄ ,∫ T
(k1 (t), c1 (t)) ≻∗ (k2 (t), c2 (t)) ⇔ lim T →∞e−ρt [u(c1 (t)) − u(c2 (t))]dt > 0,0

and deﬁne (k1 (t), c1 (t)) ≿∗ (k2 (t), c2 (t)) if and only if (k2 (t), c2 (t)) ̸≻∗ (k1 (t), c1 (t)).
If lim is lim sup, the maximal element of ≿∗ is called the overtaking optimalsolution of problem (2.1), and if lim is lim inf, the maximal element of ≿∗ is called
the weak overtaking optimal solution of problem (2.1). See Carlson, Haurie,and Leizarowitz [4] for more detailed arguments.2
We mention that for (k ∗ (t), c∗ (t)) ∈ Ak̄ , if∫ ∞e−ρt u(c∗ (t))dt0

is deﬁned and ﬁnite, then it is a solution of (2.1) if and only if it is a weakly
overtaking optimal solution of (2.1), if and only if it is an overtaking optimal solutionof (2.1). Therefore, the notion of overtaking optimality is an extension of usual
optimality.
2.3. Euler Equation and Transversality Condition. Let (k(t), c(t)) be admissible. This pair is said to satisfy the Euler equation on an interval I if and only
if c(t) is continuous, u ◦ c is continuously diﬀerentiable, andd(u ◦ c)(t) = (ρ − γ)(u ◦ c)(t)(2.2)
dt
holds for all t.
In the case of the traditional model, it is said that (k ∗ (t), c∗ (t)) is a solution only
if it satisﬁes Euler equation on R+ . However, in our model, c∗ (t) is not necessarily
continuous, and thus this statement is a little incorrect. Therefore, we extend theabove deﬁnition. An admissible pair (k(t), c(t)) satisﬁes the Euler equation a.e. on
an interval I if and only if there exists a continuous function c̃(t) such that (k(t), c̃(t))
is also admissible and satisﬁes Euler equation on I. Then, we can modify the above
claim to the following one: “(k ∗ (t), c∗ (t)) is a solution only if it satisﬁes the Euler
equation a.e. on R+ ”.
1In this paper, the term ‘interval’ means a convex set in R that includes at least two diﬀerent
points.
2In economics, our overtaking optimality is sometimes called catching-up optimality, and in
this case, our weak overtaking optimality is called overtaking optimality.

FORMULA OF THE SOLUTION FOR AN AK GROWTH MODEL943∫∞However, in our model, 0 e−ρt u(c∗ (t))dt may be not deﬁned, and thus thisstatement is not obvious. Additionally, Euler equation is only a necessary conditionof an inner solution. This problem was solved by Hosoya [6], although the proof
is very long.
Next, again let (k(t), c(t)) be admissible. This pair is said to satisfy transversality condition if and only if
lim e−ρt u′ (c(t))k(t) = 0.
t→∞
It is also said that usually (k ∗ (t), c∗ (t)) is a solution if and only if it satisﬁes Euler(2.3)

equation and transversality condition. The necessity of transversality condition foroptimality cannot easily be derived. However, in this paper, the necessity of thiscondition can be derived.
2.4. Linear Diﬀerential Equation. Consider the following linear diﬀerential equation:ẋ(t) = a(t)x(t) + b(t),where a(t), b(t) are locally integrable functions. The general solution of the aboveequation is[]∫∫ttx(t) = e 0 a(s)ds x(0) +∫se− 0 a(τ )dτ b(s)ds .

0In the AK model, both the capital accumulation equation and Euler equation arelinear. Thus, this formula is frequently used in the proof of our main result. Thisis the key idea of this paper.
3. Results
3.1. The Known Facts. In this subsection, we present results of Hosoya [6]. Theseresults are heavily used in this paper.
First, we deﬁne a notion of the inner solution.
Deﬁnition 3.1. Suppose that (k(t), c(t)) is admissible. We say that this pair ispositive on [0, T ] if and only if the following two statements hold.
1) k(t) > 0 for all t ∈ [0, T ].
2) There exists c > 0 such that the Lebesgue measure of the set ∆(c) = {t ∈[0, T ]|c(t) < c} is zero.
If (k ∗ (t), c∗ (t)) is a (weak, or strong) overtaking optimal solution, then it is called
an inner solution if and only if it is positive on [0, T ] for all T > 0.
Then, the following results hold.
Fact 1. Let T > 0, and suppose that (k ∗ (t), c∗ (t)) is a weak overtaking optimal
solution of (2.1) and positive on [0, T ]. Then, (k ∗ (t), c∗ (t)) satisﬁes Euler equation
a.e. on [0, T ].
Fact 2. Suppose thatlim u′ (c) = +∞.

c→0If (k ∗ (t), c∗ (t)) is a weak overtaking optimal solution of (2.1), then (k ∗ (t), c∗ (t)) isan inner solution, and thus Euler equation holds a.e. on R+ .

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Fact 3. Suppose that (k ∗ (t), c∗ (t)) is admissible, c∗ (t) is a continuous function, andboth Euler equation on R+ and transversality condition hold. Then, (k ∗ (t), c∗ (t))
is an overtaking optimal solution of (2.1).
3.2. Main Results. Hereafter, we assume the following.
Assumption 4. limc→0 u′ (c) = +∞ and limc→∞ u′ (c) = 0.
Note that, by Assumption 4, we can deﬁne (u′ )−1 : R++ → R++ , and (u′ )−1 issurjective and decreasing.
For C > 0, deﬁne∫ ∞(3.1)
φ(C) =
e−γt (u′ )−1 (Ce(ρ−γ)t )dt0andB1 = {C > 0|φ(C) ≤ k̄}, B2 = {C > 0|φ(C) ≥ k̄}.
Note that φ is a positive and nonincreasing function of C. We can in fact show that
if both B1 and B2 are nonempty, then inf B1 ∈ B1 .
Theorem 3.2. If there exists a weak overtaking optimal solution of (2.1), then both
B1 , B2 are nonempty and C ∗ ≡ inf B1 > 0. Moreover, the pair (k ∗ (t), c∗ (t)) deﬁnedby[]
∫ t
∗′ −1
∗ (ρ−γ)t∗γt−γs ∗(3.2)
c (t) = (u ) (C e), k (t) = e k̄ −e c (s)ds0

is a weak overtaking optimal solution of (2.1). Furthermore, every weakly overtaking
optimal solution of (2.1) is the same as the above solution a.e..
Proof. By Facts 1-2, if (k ∗ (t), c∗ (t)) is a weak overtaking optimal solution of (2.1),
then c∗ (t) satisﬁes Euler equation a.e., and thus we can assume that c∗ (t) is continuous, andd ′ ∗(u ◦ c )(t) = (ρ − γ)(u′ ◦ c∗ )(t).
dtThis implies that u′ ◦ c∗ is a solution of the following linear diﬀerential equationẋ(t) = (ρ − γ)x(t),and thus,c∗ (t) = (u′ )−1 (u′ (c∗ (0))e(ρ−γ)t ).
Moreover,k̇ ∗ (t) = γk ∗ (t) − c∗ (t), k ∗ (0) = k̄,and thus, we have[∗k (t) = eγtk̄ −∫ te−γs ∗]c (s)ds .

0Because k ∗ (t) ≥ 0 for all t ≥ 0, we have
∫ ∞e−γt c∗ (t)dt = φ(u′ (c∗ (0))),k̄ ≥0FORMULA OF THE SOLUTION FOR AN AK GROWTH MODEL

945

which implies that u′ (c∗ (0)) ∈ B1 . Therefore, B1 is nonempty. If C < u′ (c∗ (0)) andC ∈ B1 , deﬁne[]∫ t+′ −1(ρ−γ)t+γt−γs +c (t) = (u ) (Ce), k (t) = e k̄ −e c (s)ds .
0+++∗Then, (k (t), c (t)) is admissible and c (t) > c (t) for all t ≥ 0. This implies that(k ∗ (t), c∗ (t)) is not weakly overtaking optimal, which is a contradiction. Therefore,
we have that C ∈ B2 for every C < u′ (c∗ (0)), and hence, B2 is nonempty andC ∗ = inf B1 = u′ (c∗ (0)). This completes the proof.
□The following result presents the necessary and suﬃcient condition for the existence of the solution.
Theorem 3.3. Suppose that either φ is continuous or there exists C > 0 such thatk̄ ≤ φ(C) < +∞. Then, the following three requirements are equivalent.
1) There exists a weak overtaking optimal solution of (2.1).
2) There exists an overtaking optimal solution of (2.1).
3) There exists C ∗ > 0 such that φ(C ∗ ) = k̄.
Moreover, the overtaking optimal solution is given by equation (3.2) in this case.
Note that, this is a kind of results that proves equivalence between weak overtaking optimality and overtaking optimality. A similar result is found in Zaslavski[11], where the setup and assumptions of model is diﬀerent from ours.3
We will show in the proof of this theorem that 3) implies that (k ∗ (t), c∗ (t)) deﬁnedin (3.2) satisﬁes the transversality condition. Therefore, we can say that, under
Assumptions 1, 2, 3, and 4, and either the continuity of φ or the existence of C > 0such that k̄ ≤ φ(C) < +∞, the Euler equation and the transversality condition are
the necessary and suﬃcient condition for the overtaking optimality.
Proof. The fact that 2) implies 1) is obvious.
Suppose that 1) holds. By Theorem 3.2, we have that both B1 and B2 arenonempty, and C ∗ = inf B1 > 0. If φ is continuous, then we have that φ(C ∗ ) = k̄,and thus 3) holds. Next, suppose that there exists C > 0 such that k̄ ≤ φ(C) <+∞. If φ(C) = k̄, then 3) holds. Otherwise, we have that k̄ < φ(C) < +∞.
n∗′C ∗ and Cn′ = n+1
Choose Cn = n+1n C . Then, Cn ∈ B2 and Cn ∈ B1 . Because ofthe monotone convergence theorem, we have that limn→∞ φ(Cn′ ) = φ(C ∗ ), which
implies that φ(C ∗ ) ≤ k̄ and C < C ∗ . Therefore, for suﬃciently large n, C ≤ Cn andk̄ ≤ φ(Cn ) ≤ φ(C) < +∞. Again by the monotone convergence theorem, we havethat limn→∞ φ(Cn ) = φ(C ∗ ), which implies that φ(C ∗ ) ≥ k̄, and thus φ(C ∗ ) = k̄and 3) holds.
Suppose that 3) holds. Let (k ∗ (t), c∗ (t)) be deﬁned by (3.2). Then,[]∫ t−γs ∗−ρt ′ ∗∗∗e c (s)dse u (c (t))k (t) = C k̄ −0∗→ C (k̄ − φ(C ∗ )) = 0,3See ch.6 of Zaslavski [11]. Note that, because our model includes e−ρt , it is nonautonomous,and thus, in this monograph, not Theorem 5.20 but Theorem 6.29 corresponds to our Theorem 3.3.

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which asserts that (k ∗ (t), c∗ (t)) satisﬁes both Euler equation and transversality condition. By Fact 3, we must have that this pair is an overtaking optimal solution,and 2) holds. This completes the proof.
□4. Example
For θ > 0, the following function uθ is called the CRRA utility function.
{ 1−θc−1if θ ≠ 1,1−θuθ (c) =log cif θ = 1.
Barro and Sara-i-Martin [2] showed that under a CRRA utility function, if γ, θ, ρsatisfy several inequalities, then the admissible pair deﬁned by (3.2) is a (not overtaking optimal, but usual) solution. We can extend their results. That is, there
exists an inequality that holds if and only if there exists a weak overtaking optimal
solution of (2.1), and in this case, this solution is given by (3.2).
Theorem 4.1. Suppose that u = uθ . Then, there exists a weak overtaking optimal
solution of (2.1) if and only ifρ − (1 − θ)γ > 0,(4.1)
and if so,(4.2)

γ−ρ(ρ − (1 − θ)γ)k̄ γ−ρ t ∗e θ , k (t) = k̄e θ tθc∗ (t) =is overtaking optimal.4
Proof. In this case,u′ (c) = c−θ , (u′ )−1 (x) = x− θ .
1Therefore,{φ(C) =θC − θ ρ−(1−θ)γif ρ − (1 − θ)γ > 0,∞otherwise.

1Thus, by Theorems 3.2 and 3.3, there exists a weak overtaking optimal solution of(2.1) if and only if (4.1) holds. In this case,()θθφ(C ∗ ) = k̄ ⇔ C ∗ =,(ρ − (1 − θ)γ)k̄and thus,(ρ − (1 − θ)γ)k̄ γ−ρ te θ ,θwhich implies that (c∗ (t), k ∗ (t)) deﬁned by (4.2) coincides with that deﬁned by(3.2), and thus this is an overtaking optimal solution of (2.1). This completes theproof.
□(u′ )−1 (C ∗ e(ρ−γ)t ) =4Check that this solution is the same as that given by (3.2).

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947

Acknowledgement
The research agenda pursued in this paper was proposed by Hiroyuki Ozaki. We
are grateful to him for pointing out an important issue in economic dynamics. We
also appreciate the helpful comments from an anonymous reviewer.
References[1] D. Acemoglu, Introduction to Modern Economic Growth, Princeton University Press, 2009.
[2] R. J. Barro and X. I. Sara-i-Martin, EconomicGgrowth, MIT Press, 2003.
[3] O. Blanchard and S. Fischer, Lectures on Macroeconomics, MIT Press, 1989.
[4] D. Carlson, A. B. Haurie and A. Leizarowitz, Inﬁnite Horizon Optimal Control: Deterministicand Stochastic Systems, Springer, 2011.
[5] D. Cass, Optimum growth in an aggregative model of capital accumulation, Rev Econ Stud 32(1965), 233–240.
[6] Y. Hosoya, On the euler equation and the transversality condition of overtaking optimal solution
in macroeconomic dynamics, IERCU Discussion Paper No.320, 2019.
[7] T. Kamihigashi, Necessity of transversality conditions for inﬁnite horizon problems, Econometrica 69 (2001), 995–1012.
[8] T. C. Koopmans, On the concept of optimal economic growth, in: The Econometric Approachto Development planning, J. Johansen, (ed.), North Holland, Amsterdam, 1965.
[9] F. P. Ramsey, A mathematical theory of saving, Econ J 38 (1928), 543–559.
[10] D. Romer, Advanced Macroeconomics, McGraw-Hill Education, 2011.
[11] A. J. Zaslavski, Turnpike Conditions in Inﬁnite Dimensional Optimal Control, Springer, 2019.
Manuscript received May 29 2020
revised July 2 2020

Y. Hosoya
1-50 1601, Miyamachi, Fuchu, Tokyo, 183-0023, Japan
E-mail address: hosoya@tamacc.chuo-u.ac.jp
S. Kuwata
E-mail address: kwtssm1114@gmail.com