Dynamic Optimization with a Nonsmooth,Nonconvex Technology:
The Case of a Linear Objective Function

Takashi Kamihigashi*
RIEB
Kobe University
tkamihig@rieb.kobe-u.ac.jp

Santanu Roy
Department of Economics
Southern Methodist University
sroy@smu.edu

July 31, 2004

Abstract
This paper studies a one-sector optimal growth model with linear utility inwhich the production function is generally nonconvex, nondifferentiable, anddiscontinuous. The model also allows for a general form of irreversible investment. We show that every optimal path either converges to zero or reaches
a positive steady state in finite time (and possibly jumps among different
steady states afterwards). We establish conditions for extinction (convergence to zero), survival (boundedness away from zero), and the existenceof a minimum safe standard of conservation. They extend the conditions
known for the case of S-shaped production functions to a much large classof technologies. We also show that as the discount factor approaches one,
optimal paths converge to a small neighborhood of the golden rule capitalstock.
Keywords: Nonconvex, nonsmooth, and discontinuous technology; extinction; survival; turnpike; linear utility.
*Corresponding author: RIEB, Kobe Univeristy, Rokkodai, Nada, Kobe 657-8501, Japan
Phone/Fax: +81-78-803-7015, Email: tkamihig@rieb.kobe-u.ac.jp

1

Introduction

The aggregative model of optimal economic growth is an important theoretical paradigm that is widely used for analyzing economic issues related
to intertemporal allocation of resources and capital accumulation. Muchof the existing literature focuses on “classical” versions of the model where
the underlying technology is smooth and convex. Yet, there exist a large
variety of economic settings where the technology exhibits nonconvexities,
nonsmoothness, and even discontinuities due to various factors such as fixed
costs, threshold effects, increasing returns to scale, economies of scope, anddepensation in biological reproduction processes. This paper presents a comprehensive analysis of a one-sector optimal growth model with linear utility
that admits nonconvex, nonsmooth, and discontinuous technology as well as
irreversible investment.1
The literature on optimal growth with nonconvex technology dates backto Clark (1971), who analyzed the problem of optimal dynamic consumptionof a biological resource where the production function is S-shaped and theobjective function is linear in consumption.2 A full characterization of optimal paths in the context of optimal growth was provided by Majumdar and
Mitra (1983), who showed among other things that if the discount factor is in
an intermediate range, there exists a critical stock (minimum safe standardof conservation) that determines whether an optimal path converges to zeroor to a positive steady state. Mitra and Ray (1984) studied a more generalmodel with concave utility in which the production function is only required
to be strictly increasing and continuous.
While various results are known on nonconvex one-sector optimal growthmodels with strictly concave utility (e.g., Skiba, 1978; Majumdar and Mitra,
1982; Majumdar and Nermuth, 1982; Dechert and Nishimura, 1983; Kamihigashi and Roy, 2003),3 many of the arguments used in the case of strictly
concave utility cannot readily be applied to the linear utility case. For example, arguments based on the Euler equation cannot directly be applied to thelinear utility case, where optimal paths are often not in the interior of thefeasible set. In addition, though optimal paths are known to be monotone in
1

The case of strictly concave utility is studied in Kamihigashi and Roy (2003). This
paper mostly focuses on arguments specific to the linear utility case.
2
Optimal growth models with convex technology and linear utility have been studied,among others, by Srinivasan (1964), Uzawa (1964), Malinvaud (1965), and Shell (1967).
3
See Kamihigashi and Roy (2003) for a discussion of this literature.

1the case of strictly concave utility, this is not true in the linear utility case,as shown in Proposition 3.2 of this paper.
We show however that every optimal path is strictly monotone until itreaches a steady state. Furthermore, every optimal path either converges
to zero or reaches a positive steady state in finite time (and possibly jumps
among different steady states afterwards). This sharpens, in the linear utility
case, the result by Mitra and Ray (1984) that every optimal path approaches
the set of steady states asymptotically, and extends it to cases with discontinuous production functions and irreversible investment. We also establish
conditions for extinction (convergence to zero), survival (boundedness awayfrom zero), and the existence of a minimum safe standard of conservation.
These conditions extend those established by Clark (1971) and Majumdar
and Mitra (1983) in the case of S-shaped production functions to a muchlarger class of technologies.
Moreover we show that despite the nonclassical features of the model,
as the discount factor approaches one, optimal paths converge to a smallneighborhood of the golden rule capital stock. This result allows us to extend
the turnpike theorem of Majundar and Nermuth (1982) to the case of linear
utility. Though turnpike properties are normally associated with concavity(e.g., McKenzie, 1986), our model essentially involves no concavity.
Our analysis has a number of applications outside optimal economic
growth. First, it can be used to study the optimal exploitation of renewable resources (e.g., Clark, 1971). Second, our model can be viewed as the
problem of a firm that maximizes the discounted sum of retained profits(dividends), faces a constant rate of interest, and invests part of its current
flow of net revenue that in turn determines next period’s flow of net revenue.
Third, many of the techniques developed in this paper can be applied to awider class of models where the objective function is additively separable incurrent and future state variables (e.g., Spence and Starrett, 1975; Clark,
1990, Section 7.7).
The rest of the paper is organized as follows. Section 2 describes themodel. Section 3 develops various properties that constitute the essential
tools of our analysis. Section 4 shows results on monotonicity and convergence of optimal paths. Section 5 offers conditions for survival, extinction,and the existence of a minimum safe standard of conservation. Section 6
establishes turnpike properties of optimal paths.

2

2

The Model

Consider the following maximization problem:
∞
X
(2.1)
max∞
δ t ct{ct ,xt }t=0

(2.2)
(2.3)
(2.4)
(2.5)

t=0

s.t. ∀t ∈ Z+ , ct + xt+1 = f (xt ),ct ≥ 0,xt+1 ≥ r(xt ),x0 given,

where ct is consumption in period t, xt is the capital stock at the beginningof period t, δ is the discount factor, f is the production function, and (2.4)
means that capital cannot be decreased below its depreciated level r(xt );we call r the depreciation function for convenience. Our formulation allows
for nonlinear depreciation. The standard case of reversible investment is a
special case in which r(x) = 0 for all x ≥ 0.4
We use the following standard definitions. A path {ct , xt }∞t=0 is feasible if
it satisfies (2.2)–(2.4). A capital path {xt } is feasible if there is a consumption
path {ct } such that {ct , xt } is feasible. A path from x0 is a path {c0t , x0t } suchthat x00 = x0 . A capital path from x0 is defined similarly. A feasible path
{ct , xt } is optimal (from x) if it solves the maximization problem (2.1)–(2.5)(with x0 = x). A feasible capital path {xt } is optimal (from x) if there is aconsumption path {ct } such that {ct , xt } is optimal (from x). A stationary(capital) path is a constant feasible (capital) path. A capital stock x ≥ 0
is a steady state if the stationary path from x is optimal. The assumptions
stated below are maintained throughout the paper.5
Assumption 2.1. δ ∈ (0, 1).
Assumption 2.2. (i) f : R+ → R+ is strictly increasing and upper semicontinuous. (ii) f (0) = 0.
The production function f is not required to be continuous or differentiable. To state our assumption on the depreciation function r, for h : R+ →R, we define
(2.6)
4
5

h− (x) = lim h(y),h+ (x) = lim h(y),

y↑x

y↓x

See Kamihigashi and Roy (2003) for a more detailed discussion of (2.4).
Assumptions 2.2 and 2.3 are discussed in more detail in Kamihigashi and Roy (2003).

3

provided that the right-hand sides are well-defined.
Assumption 2.3. (i) r : R+ → R+ is nondecreasing and lower semicontinuous. (ii) ∀x > 0, r+ (x) < x and r(x) < f (x).
Like the production function f , the depreciation function r is not required
to be differentiable or continuous. Part (ii) means that the irreversibility constraint (2.4) is never binding at a steady state and that positive consumption
is available from any positive initial capital stock. The following assumption
rules out unbounded growth
Assumption 2.4. ∃x̂ > 0, ∀x > x̂, f (x) < x.
Remark 2.1. If ∃x > 0, f (x) ≤ x, then any feasible capital path {xt } fromx0 ∈ [0, x] satisfies ∀t ∈ Z+ , xt ≤ x.
Assumption 2.4 together with Remark 2.1 implies that all feasible paths
are bounded. We rule out unbounded growth since it does not seem to be arobust phenomenon with linear utility.6 The assumptions made above ensure
the existence of an optimal path from any initial capital stock x0 ≥ 0 by astandard argument (e.g., Ekeland and Scheinkman, 1986, Proposition 4.1).

3

Useful Properties

3.1

The Value Function and Some Definitions

This subsection establishes basic properties of the value function in addition
to defining useful functions and correspondences. Let v : R+ → R+ be
the value function for the maximization problem (2.1)–(2.5).7 The Bellman
equation is given by(3.1)

v(x) = max {f (x) − y + δv(y)},y∈F (x)6

For example, suppose f (x) = Ax for some A > 0 and r(x) = 0 for all x ≥ 0. Let
x0 > 0. Then it is easy to see that if A < 1/δ, the unique optimal path is given by xt+1 = 0
for all t ≥ 0; if A > 1/δ, no optimal path exists; and if A = 1/δ, any feasible capital pathsatisfying limT ↑∞ δ T xT +1 = 0 is optimal. Hence unbounded growth is possible, but is
expected only in a knife edge case even for a more general production function.
7
Because of the irreversibility constraint (2.4), v need not be nondecreasing, as discussed
in Kamihigashi and Roy (2003, Section 3.1).

4

where F (x) = {y ≥ 0 | r(x) ≤ y ≤ f (x)}. Let K denote the optimal policy
correspondence. For x ≥ 0, define
(3.2)
(3.3)
(3.4)

g(x) = f (x) − x,γ(x) = δf (x) − x,λ(x) = δv(x) − x.

The function g(x) gives the stationary level of consumption associated withthe capital stock x. We call γ(x) the partial gain function for the following
reason. An investment of x units of capital generates f (x) units of output
in the next period. Hence the “net gain” is δf (x) − x. This gain however ispartial in the sense that it does not take account of future gains obtained byreinvestment. The “total net gain” is captured by λ(x), which we call the
total gain function.
Lemma 3.1. (i) v is upper semicontinuous. (ii) ∀x > 0, v(x) > 0.
Proof. Part (i) can be shown by a standard argument (e.g., Kamihigashi andRoy, 2003, Lemma 3.3). To see (ii), let {ct , xt } be the path from x such that∀t ∈ Z+ , xt+1 = r(xt ). Since this path is feasible and ∀t ∈ Z+ , ct ≥ 0,
(3.5)v(x) ≥ c0 = f (x) − r(x) > 0,

where the last inequality holds by Assumption 2.3(ii). Now (ii) holds.

3.2

Properties Based on γ(x)This subsection shows some basic results using the partial gain function γ(x).
Our first result here reduces the maximization problem (2.1)–(2.5) to one
in which the discounted sum of γ(xt+1 ) is to be maximized. This type of
reduction was used by Spence and Starrett (1975) and Clark (1990) to study
optimality of “most rapid approach” paths. We use it here for more general
purposes.
Lemma 3.2. A feasible capital path {xt } is optimal iff it solves the following
maximization problem:
(3.6)

max

{xt }∞
t=0∞X

δ t γ(xt+1 ) s.t. ∀t ∈ Zt , xt+1 ∈ F (xt ), x0 given.

t=05

Proof. For any feasible capital path {xt },
(3.7)

∞
X

δ t [f (xt ) − xt+1 ]

t=0

(3.8)
(3.9)

= f (x0 ) − x1 + δf (x1 ) − δx2 + δ 2 f (x2 ) − δ 2 x3 + · · ·∞X= f (x0 ) +δ t γ(xt+1 ),t=0where the last sum exists since {γ(xt+1 )} is bounded by boundedness of {xt }and monotonicity of f . The lemma now follows.
Though Lemma 3.2 may seem rather obvious, it does not necessarily holdif Assumption 2.4 is dropped.8 The following result generalizes Mitra andRay (1984, Theorem 4.2) in the case of linear utility.9
Proposition 3.1. (i) Any x̃ ∈ argmaxx≥0 γ(x) is a steady state.10 (ii) If∃y > 0, γ(y) ≥ 0, then there exists a nonzero steady state.
Proof. Immediate from Lemma 3.2 and upper semicontinuity of γ.
Proposition 3.2. Let x1 , x2 > 0 be steady sates such that x1 ∈ F (x2 ) andx2 ∈ F (x1 ). Then any capital path {xt } such that ∀t ∈ Z+ , xt ∈ {x1 , x2 } is
optimal.
Proof. Since x2 ∈ F (x1 ), γ(x2 ) ≤ γ(x1 ) by Lemma 3.2. Likewise γ(x1 ) ≤γ(x2 ). Thus γ(x1 ) = γ(x2 ). The conclusion now follows from Lemma 3.2.
By Proposition 3.1(i), the hypothesis of Proposition 3.2 holds if there are
two maximizers x1 , x2 of γ(x) and if x1 ∈ F (x2 ) and x2 ∈ F (x1 ). In this case,cycles of any period as well as nonperiodic paths are possible by Proposition
3.2. The result can easily be extended to any number of steady states withmore complicated transition patterns (e.g., Kamihigashi, 1999, Figure 2).
The following result is useful in determining the direction in which an
optimal capital path possibly moves.
8

For example, suppose f (x) = x/δ. Then γ(x) = 0 for all x ≥ 0. Hence any feasible
capital path trivially solves (3.6). In particular, the capital path {xt } from x0 > 0 given
by xt+1 = f (xt ) for all t ≥ 0 solves (3.6). But in this case, ct = f (xt ) − xt+1 = 0 for allt ∈ Z+ , so this is in fact the worst feasible path.
9
See Kamihigashi and Roy (2003, Propositions 3.3, 3.4) for the case of strictly concave
utility.
10
The maximum exists since γ is upper semicontinuous and γ(x) < 0 for x > x̂, wherex̂ is given by Assumption 2.4.

6

Lemma 3.3. Let {xt } be an optimal capital path from x0 ≥ 0. Then either(i) ∃t ∈ Z+ , γ(xt+1 ) > γ(x0 ) or (ii) ∀t ∈ Z+ , γ(xt+1 ) = γ(x0 ).
Proof. Suppose both cases (i) and (ii) are false. Then ∀t ∈ Z+ , γ(xt+1 ) ≤γ(x0 ) and ∃t ∈ Z+ , γ(xt+1 ) < γ(x0 ). Thus
(3.10)

0 ≤ f (x0 ) +∞Xδ t γ(xt+1 ) < f (x0 ) +t=0

γ(x0 )g(x0 )=.
1−δ
1−δ

Hence the stationary path from x0 is feasible. But then by the above strict
inequality and Lemma 3.2, {xt } cannot be optimal, a contradiction.
Lemma 3.3 means that an optimal capital path must achieve a higher
partial gain γ(xt+1 ) at some point unless partial gains are constant over time.
In other words, it never moves in a direction in which only lower partial gains
are available.11

3.3

Properties Based on λ(x)This subsection shows some basic results using the total gain function λ(x).
The following result characterizes the policy correspondence by a single static
maximization problem.
Lemma 3.4. Let x, y ≥ 0. Then y ∈ K(x) iff y ∈ argmaxz∈F (x) λ(z).
Proof. Note from (3.1) and (3.4) that
(3.11)

v(x) = max {f (x) + λ(z)} = f (x) + max λ(z).
z∈F (x)z∈F (x)The conclusion now follows.
Corollary 3.1. x ≥ 0 is a steady state iff x ∈ argmaxz∈F (x) λ(z).
Proof. Immediate from Lemma 3.4.
For x ≥ 0, define(3.12)

K(x) = min{y | y ∈ K(x)},K(x) = max{y | y ∈ K(x)}.

The above minimum and maximum exist since K(x) is compact by Lemma
3.4 and upper semicontinuity of λ. The next result follows from Topkis (1978,
Theorem 6.1), but can easily be shown by using the total gain function λ.
11

A slightly different version of Lemma 3.3 holds in the case of strictly concave utility(Kamihigashi and Roy, 2003, Lemma 3.5).

7

Lemma 3.5. (i) Let x, y ≥ 0, y ∈ K(x), and y 0 ∈ K(x0 ). Then min{y, y 0 } ∈K(min{x, x0 }) and max{y, y 0 } ∈ K(max{x, x0 }). (ii) K and K are nondecreasing.

3.4

Implications of Lemma 3.5

This subsection shows two lemmas that require only the properties of K
stated in Lemma 3.5. The first one shows that an optimal capital path canchange direction only at a steady state, and that if it changes direction, there
is a “shortcut.”
Lemma 3.6. Let x ≥ 0, y ∈ K(x), and z ∈ K(y). Suppose (a) x ≤ y ≥ z or(b) x ≥ y ≤ z. Then (i) y is a steady state and (ii) z ∈ K(x).
Proof. In case (a), y = max{y, z} ∈ K(max{x, y}) = K(y) and z = min{y, z} ∈K(min{x, y}) = K(x) by Lemma 3.5(i). Case (b) is similar.
The next result shows that two different capital stocks achieved along an
optimal capital path can always be connected by a monotone path that is
also optimal. It allows us to simplify some of our arguments.
Lemma 3.7. Let n ≥ 2. Let(3.13)

x1 ≥ 0, x2 ∈ K(x1 ), . . . , xn ∈ K(xn−1 ).

Then there exists a monotone (i.e., nondecreasing or nonincreasing) path
1
1 m{y i }m= xn , and y 2 ∈ K(y 1 ), . . . , y m ∈i=1 with m ≥ 2 such that y = x , y
m−1
K(y
).
Proof. If {xi }ni=1 is monotone, the conclusion is obvious. So suppose it is not
monotone, which requires n ≥ 3. Then ∃j ∈ {1, . . . , n − 2}, xj ≤ xj+1 ≥ xj+2
or xj ≥ xj+1 ≤ xj+2 . By Lemma 3.6(ii), xj+2 ∈ K(xj ). For i = 1, . . . , n − 1,
define x̃i = xi if i ≤ j and x̃i = xi+1 if i > j (i.e., remove xj+2 ). Then {x̃i }n−1i=1satisfies (3.13) with appropriate modifications. If {x̃i } is not monotone, apply
the same procedure to remove another element. The procedure can be applied
at most n − 2 times since each application removes one element. Since x1
and xn are never removed, this process yields a desired monotone path.

8

3.5

Euler Inequalities

For h : (a, b) → R with a < b, define(3.14)

h(x) − h(x − ²),²²↓0h0− (x) = limh(x + ²) − h(x).
²↓0²h0+ (x) = lim

These generalized derivatives allow us to obtain “Euler inequalities” insteadof an Euler equation. Lemmas 3.8 and 3.9 below are adapted from Kamihigashi and Roy (2003, Lemmas 3.6, 3.7); the proofs are omitted.
Lemma 3.8. Let {ct , xt } be an optimal path. Let t ∈ Z+ . If ct > 0 and
xt+2 > r+ (xt+1 ), then
(3.15)

1 ≥ δf+0 (xt+1 ).

If xt+1 > r(xt ) and ct+1 > 0, then
(3.16)

1 ≤ δf−0 (xt+1 ).

Lemma 3.9. If x > 0 is a steady state, then g(x) > 0 and(3.17)

δf+0 (x) ≤ 1 ≤ δf−0 (x).

The following result provides a link between the generalized derivativesof f and the monotonicity properties of γ.
Lemma 3.10. Let 0 ≤ a < b. (i) If δf+0 > 1 on [a, b), γ is strictly increasing
on [a, b]. (ii) If δf−0 < 1 on (a, b], γ is strictly decreasing on [a, b].
Proof. Since γ+0 = δf+0 − 1 and γ−0 = δf−0 − 1, the lemma follows from Giorgi
and Kolmósi (1992, Theorem 1.13).

4

Monotonicity and Convergence

In the case of strictly concave utility, optimal capital paths are known to be
monotone (Kamihigashi and Roy, Lemma 3.2). As shown in Proposition 3.2,
this property does not carry over to the linear utility case. The following
result however shows that an optimal capital path is strictly monotone until
it reaches a steady state.
9

Proposition 4.1. Let {xt } be an optimal capital path. Suppose ∃T ∈ N, ∀t ≤+1
T, xt is not a steady state. Then {xt }Tt=0
is strictly monotone.
Proof. Since x0 is not a steady state, x0 6= x1 . Assume x0 < x1 without
loss of generality. If x1 ≥ x2 , then x1 is a steady sate by Lemma 3.6(i), a
contradiction. Thus x1 < x2 . By induction, ∀t ≤ T, xt < xt+1 .
It can be expected from Proposition 4.1 that an optimal capital path
reaches a steady state either in finite time or asymptotically. The next resultshows that an optimal capital path either (i) converges to zero or (ii) reaches
a positive steady state in finite time and possibly jumps among different
steady states afterwards. Let S be the set of steady states.
Proposition 4.2. Let {xt } be an optimal capital path. Then one of thefollowing holds:(4.1)

lim xt = 0;

(4.2)

∃T ∈ Z+ , ∀t ≥ T,t↑∞

xt ∈ S.

The proof appears in Subsection 4.1. Mitra and Ray (1984, Theorem 6.1)
showed that every optimal capital path approaches the set of steady states
asymptotically in the case of concave utility and reversible investment with acontinuous production function. Proposition 4.2 offers a sharper characterization in the case of linear utility and irreversible investment with a generally
discontinuous production function.
Proposition 4.2 shows that there are only two possibilities for the asymptotic behavior of an optimal capital path. One might wonder if the firstcase (4.1) could be combined into the second, but convergence to zero simply
cannot occur in finite time if r(x) is strictly increasing near x = 0. Hencean additional assumption is needed to ensure that convergence to zero also
occurs in finite time.

4.1

Proof of Proposition 4.2

Lemma 4.1. Let x > 0 be such that g− (x) ≤ 0.12 Then for every feasible
capital path {xt } from x0 ∈ [0, x), ∀t ∈ Z+ , xt < x.
12

By upper semicontinuity, g− (x) ≤ g(x) (= g+ (x)).

10

Proof. Since x1 ≤ f (x0 ) < f− (x), x1 − x < g− (x) ≤ 0. By induction,∀t ∈ N, xt < x.
Lemma 4.2. Let x > 0 be such that g− (x) ≤ 0. Then ∃x ∈ (0, x), ∀y ∈[0, x), K(y) < x.
Proof. Suppose g(x) = g− (x) ≤ 0. Then ∀y ∈ F (x), y ≤ x by Remark 2.1,
and x 6∈ K(x) by Lemma 3.9; thus x ≡ K(x) < x. Hence the conclusion
holds by Lemma 3.5(ii).
Now suppose g(x) > 0. Let f˜ be a production function satisfying Assumptions 2.2–2.4 such that ∀y ∈ [0, x), f˜(y) = f (y) and f˜(x) = f− (x). LetK̃ be the corresponding policy correspondence. By Lemma 4.1, every feasible capital path {xt } from y ∈ [0, x) satisfies ∀t ∈ Z+ , xt < x for the original
problem. This implies ∀y ∈ [0, x), K(y) = K̃(y). Hence the argument in theprevious case applies.
For the rest of the proof, we take an optimal capital path {xt } as given.
Lemma 4.3. Suppose there exists a subsequence {xti }∞i=1 of {xt } such thatx ≡ limi↑∞ xti > 0. Then (i) limi↑∞ g(xti ) > 0. Furthermore, (ii) ∀I ∈N, ∃i ≥ I, x ∈ F (xti ).
Proof. Taking a further subsequence, we may assume {xti } is monotone. To
show (i), suppose(4.3)

lim g(xti ) ≤ 0,i↑∞where the limit exists by monotonicity. Since g− (x) ≤ limi↑∞ g(xti ) ≤ 0,
{xti } is nonincreasing by Lemma 4.2. Applying Lemma 3.7 to each pair of
the form (xti , xti+1 ), we see that there is a nonincreasing optimal capital path{yt } from xt1 with limt↑∞ yt = x. If ∃T ∈ Z+ , yt = x, then x is a steady stateand g(x) ≤ 0 by (4.3), contradicting Lemma 3.9. It follows that
(4.4)

∀t ∈ Z+ ,x < yt .

We have(4.5)

lim[f (yt ) − yt+1 ] = lim[f (yt ) − yt ] = lim g(yt ) ≤ 0,
t↑∞

t↑∞

t↑∞

11

where the last inequality holds by (4.3). Let a = − limt↑∞ g(yt ). Suppose
a > 0. Let b ∈ (0, a). Then yt+1 − yt ≤ f (yt ) − yt < −b for large t,13 which
implies limt↑∞ yt = −∞, a contradiction. Thus
(4.6)

0 = lim g(yt ) = g+ (x) = f (x) − x,t↑∞where the last equality holds by upper semicontinuity. By Fatou’s lemma,(4.7)

lim v(yt ) = lim
t↑∞

(4.8)

t↑∞

≤

∞
X
i=0

∞
X

δ i [f (yt+i ) − yt+i+1 ]i=0

δ i lim[f (yt+i ) − yt+i+1 ] ≤ 0,t↑∞

where the last inequality uses (4.5). On the other hand, by (3.5), (4.4), (4.6),
and Assumption 2.3(ii).
(4.9)(4.10)

lim v(yt ) ≥ lim[f (yt ) − r(yt )]
t↑∞

t↑∞

= f (x) − r+ (x) = x − r+ (x) > 0,

contradicting (4.7) and (4.8).
To show (ii), suppose ∃I ∈ N, ∀i ≥ I, x 6∈ F (xti ). Since r+ (x) < x byAssumption 2.3(ii), r(xti ) < x for large i. Thus x > f (xti ) for large i. But
then limi↑∞ g(xti ) = limi↑∞ f (xti ) − x ≤ 0, contradicting part (i).
Lemma 4.4. If g(xt ) ≤ 0 infinitely often, then limt↑∞ xt = 0.
Proof. Let {xti }∞i=1 be a subsequence of {xt } such that ∀i ∈ N, g(xti ) ≤ 0.
Taking a further subsequence, we may assume {xti } is convergent. Sincelimi↑∞ g(xti ) ≤ 0, limt↑∞ xt = 0 by Lemma 4.3(i). By Remark 2.1, ∀i ∈N, ∀t ≥ ti , xt ≤ xti . Hence limt↑∞ xt = 0.
Lemma 4.5. Suppose ∀t ∈ Z+ , g(xt ) ≥ 0. Then λ(xt ) is nondecreasing in t.
Proof. Let t ∈ Z+ . Since g(xt ) ≥ 0, xt ∈ F (xt ). Thus by Lemma 3.4,λ(xt ) ≤ λ(xt+1 ).
13“For large t” means “for t ∈ Z+ sufficiently large.” Similar conventions apply to similar
expressions.

12Lemma 4.6. Suppose limt↑∞ xt > 0. Then ∃T ≥ 0, ∀t ≥ T, (i) λ(xt+1 ) =limt↑∞ λ(xt ) ≡ λ and (ii) xt+1 ∈ S.
Proof. By Lemma 4.4, g(xt ) > 0 for large t. Hence we may assume ∀t ∈Z+ , g(xt ) > 0 without loss of generality. By Lemma 4.5, λ(xt ) is nondecreasing in t. Since limt↑∞ xt > 0, there is a subsequence {xti } of {xt } withx ≡ limt↑∞ xti > 0. By monotonicity and upper semicontinuity,(4.11)

λ = sup λ(xt ) = lim λ(xt ) ≤ λ(x).

Z

t∈ +

t↑∞

By Lemma 4.3(ii), there is i ∈ N with x ∈ F (xti ). Let T = ti . Then by(4.11), Lemma 3.4, and monotonicity of λ(xt ), ∀t ≥ T, λ ≤ λ(x) ≤ λ(xT +1 ) ≤λ(xt+1 ) ≤ λ; thus (i) follows. To see (ii), let t ≥ T . Suppose xt+1 6∈ S. Thenby Corollary 3.1, λ(xt+2 ) > λ(xt+1 ), contradicting part (i).
Proposition 4.2 now follows from Lemma 4.6.

5

Survival, Extinction, and a Minimum Safe
Standard of Conservation

This section gives conditions for survival, extinction, and the existence of a“minimum safe standard of conservation.” Our first result here provides a
condition for survival, i.e., for an optimal capital path to be bounded awayfrom zero. Together with Proposition 4.2, it extends Majumdar and Mitra(1983, Propositions 2, 3) to our general setting.
Proposition 5.1. Suppose(5.1)∃x > 0, ∀x ∈ (0, x),δf+0 (x) > 1.

Then every optimal capital path {xt } from x0 > 0 satisfies xt ≥ x for large t.
Proof. By (5.1) and Lemma 3.9, there is no steady state in (0, x). Let x ∈(0, x). We claim K(x) > x. Suppose K(x) ≤ x. Since x is not a steadystate, K(x) < x. By Lemma 3.5(ii), there is an optimal capital path {xt }from x with ∀t ∈ N, xt ≤ K(x) < x. By Lemma 3.10, γ is strictly increasing
on [0, x],14 so ∀t ∈ Z+ , γ(xt+1 ) < γ(x0 ), contradicting Lemma 3.2. We haveshown that ∀x ∈ (0, x), K(x) > x. Since there is no steady state in (0, x),
the conclusion holds by Proposition 4.2.
14

The origin is included since γ is continuous at 0 by Assumption 2.2.

13

The next result offers a condition for global extinction, i.e., for every
optimal capital path to converge to zero. It extends Majumdar and Mitra(1983, Proposition 1) to our general setting.
Proposition 5.2. Suppose(5.2)∀x > 0,δf−0 (x) < 1.

Then every optimal capital path converges to zero.
Proof. By (5.2) and Lemma 3.10, γ is strictly decreasing on [0, ∞). Thus
every optimal capital path is nonincreasing by Lemma 3.3. It converges tozero by Proposition 4.2 since there is no steady state in (0, ∞) by (5.2) andLemma 3.9.
The next result gives a condition for local extinction, i.e., for any optimal
capital path starting below a certain level to converge to zero. It generalizes
Majumdar and Mitra (1983, Lemma 4) to our general setting.
Proposition 5.3. Suppose(5.3)

lim δf−0 (x) < 1.
x↓0Then ∃x > 0, any optimal capital path from x ∈ (0, x] converges to zero.
Before proving Proposition 5.3, let us show the existence of a “minimum
safe standard of conservation” (e.g., Clark, 1971) under an additional condition. The following result along with Proposition 4.2 extends the analysis of
Majumdar and Mitra (1983, Section 3c) to our general setting.
Proposition 5.4. Assume (5.3). Suppose maxx≥0 γ(x) > 0. Then there
exists x > 0 such that from any x ∈ (0, x) there exists an optimal capital
path converging to zero, and from any x > x there exists no optimal capital
path converging to zero.
Proof. Let x = sup{x > 0 | there is an optimal capital path from x converging
to zero}. By Proposition 5.3, x > 0. Note from Lemma 3.5 that if there is an
optimal capital path from x > 0 converging to zero, then from any x0 ∈ (0, x)
there is an optimal capital path converging to zero. It follows that x has thedesired property except for its finiteness. Since maxx≥0 γ(x) > 0, there exists
a steady state x̃ > 0 by Proposition 3.1. Since no optimal capital path fromx̃ converges to zero by Lemma 3.3, we obtain x ≤ x̃.
14

5.1

Proof of Proposition 5.3

Note that (5.3) implies(5.4)∃z > 0, ∃² > 0, ∀x ∈ (0, z],δf−0 (x) < 1 − ².

Lemma 5.1. Under (5.4), ∃x ∈ (0, z], any optimal capital path from x ∈(0, x] is bounded above by z, where z is given by (5.4).
Proof. If ∃x ∈ (0, z], g(x) ≤ 0, then the conclusion follows from Remark 2.1.
So suppose(5.5)∀x ∈ (0, z],g(x) > 0.

Suppose the conclusion is false. Then ∀x ∈ (0, z], ∃x ∈ (0, x], there is an
optimal capital path {xt } from x with the property that ∃t ∈ N, xt > z. By
Lemma 3.5, this means that ∀x ∈ (0, z], there is an optimal capital pathwith the same property. For i ∈ N, define xi = f −i (z); this is well-definedsince f is strictly increasing, (5.4) implies that f is continuous on [0, z], andf (z) > z by (5.5). By definition and (5.5),(5.6)∀i ∈ N,(a) f i (xi ) = z,(b) z > xi = f (xi+1 ) > xi+1 .

Fix i ∈ N for the moment. Let {xit } be an optimal capital path from xi
such that ∃t ∈ N, xit > z. Let s ∈ N be the first t ∈ N with xit > z. Then
(5.7)(a) s > i,(b) z < xis ≤ f (z).

Since there is no steady state in (0, z] by (5.4) and Lemma 3.9, {xit }st=0 isstrictly increasing by Proposition 4.1. Thus ∀t ∈ {0, 1, . . . , s − 1}, xit+1 >xit > r(xit ), where the last inequality holds by Assumption 2.3(ii). Hence byLemma 3.8 and (5.4),(5.8)∀t ∈ {1, . . . , s − 1},

xit+1 = f (xit ) (i.e., cit = 0).

Since the stationary path from xi is feasible by (5.5), it follows that
(5.9)

g(xi )≤ v(xi ) = f (xi ) − xi1 + δ s v(xs ) < g(xi ) + δ s v,
1−δwhere v = maxx∈[z,f (z)] v(x), the equality follows from (5.8), and the lastinequality holds by (5.7)(b) since xi1 > xi . From (5.5), (5.9), and (5.7)(a),(5.10)0<δg(xi )< δ s v ≤ δ i+1 v.
1−δ15Thus δ −i g(xi ) ≤ (1 − δ)v. Since the right-hand side is finite and does notdepend on i, to obtain a contradiction it suffices to show
(5.11)lim δ −i g(xi ) = ∞.
i↑∞Note that ∀i ∈ N,
(5.12)(5.13)(5.14)

δ −i g(xi )f (xi ) − xi=δ
δ −(i+1) g(xi+1 )f (xi+1 ) − xi+1
f (xi ) − f (xi+1 )=δxi − xi+10≤ δf− (x̃i ) ≤ 1 − ²,

where (5.13) uses (5.6)(b), the first inequality in (5.14) holds for some x̃i ∈[xi+1 , xi ] by the generalized mean value theorem (Giorgi and Kolmósi, 1992,
Theorem 1.8), and the last inequality holds by (5.4). Now (5.11) follows.
To complete the proof of Proposition 5.3, let x be as given by Lemma 5.1.
Since there is no steady state in (0, z] by (5.4) and Lemma 3.9, any optimal
capital path from x ∈ (0, x] converges to zero by Proposition 4.2.

6

Turnpike Properties

This section shows that for any ² > 0, for δ close to one, any optimal capital
path from x0 ≥ ² “converges” to a small neighborhood of the golden rule
capital stock. We start by assuming that maximum sustainable consumption
is strictly positive.
Assumption 6.1. g ∗ ≡ maxx≥0 g(x) > 0.
If g ∗ ≤ 0, extinction occurs from every initial capital stock by Lemma 3.9
and Proposition 4.2. Assumption 6.1 rules out this trivial case.
Assumption 6.2. There exists a unique x∗ ≥ 0 such that g(x∗ ) = g ∗ .15
We call x∗ the golden rule capital stock. The following is our last assumption.
15

Along the lines of Kamihigashi and Roy (2003, Section 8) it is possible to deal withcases in which there are more than one capital stock at which g(x) is maximized.

16

Assumption 6.3. ∀x ∈ (0, x∗ ], g− (x) > 0.
If g− (x) ≤ 0 for some x ∈ (0, x∗ ], no feasible capital path from x0 ∈ (0, x)
can reach x∗ by Lemma 4.1. Assumption 6.3 ensures that it is possible toreach x∗ from any x ∈ (0, x∗ ]. We are now ready to state the main result of
this section.
Proposition 6.1. Let ²∗ ∈ (0, x∗ ) and ² ∈ (0, x∗ − ²∗ ). Then for δ close toone, any optimal capital path {xt } from x0 ∈ [², ∞) satisfies(6.1)∃T ∈ Z+ , ∀t ≥ T,

xt ∈ (x∗ − ²∗ , x∗ + ²∗ ).

Before proving this result, let us show a simple consequence that extends
Majumdar and Nermuth (1982, Theorem 3.4) to the case of linear utility.16
Corollary 6.1. Let ²∗ ∈ (0, x∗ ). Suppose(6.2)

lim f+0 (x) > 1.
x↓0

Suppose there exists a unique steady state x(δ) in (x∗ − ²∗ , x∗ + ²∗ ) for δ
close to one. Then for δ close to one, every optimal capital path {xt } fromx0 ∈ (0, ∞) converges to x(δ).
Proof. By (6.2), ∃µ > 1, ∃x > 0, ∀x ∈ (0, x), f+0 (x) > µ. Hence for δ close toone, ∀x ∈ (0, x), δf+0 (x) > 1. Thus the conclusion follows from Propositions
5.1, 6.1 (with ² = x), and 4.2.

6.1

Proof of Proposition 6.1

Lemma 6.1. ∀² ∈ (0, x∗ − ²∗ ), for δ close to one, there exists no steady statein [², x∗ − ²∗ ].
Proof. It is easy to see that ∃T ∈ N, ∀x0 ∈ [², x∗ − ²∗ ], there is a feasible
capital path from x0 such that ∀t ≥ T, xt = x∗ .17 Let g = maxy∈[²,x∗ −²∗ ] g(y).
By Assumption 6.2, for δ close to one, g/(1 − δ) < δ T g ∗ /(1 − δ), so thestationary path from any x ∈ [², x∗ − ²∗ ] is not optimal.
16

For extensions of their result in the case of strictly concave utility, see Kamihigashi
and Roy (2003, Section 8).
17
Let T ∈ N be such that there is a feasible capital path from ² with ∀t ≥ T, xt = x∗ .
This T works for all x0 ∈ [², x∗ − ²∗ ].

17

Lemma 6.2. For δ close to one, there exists no steady state in [x∗ + ²∗ , ∞).
Proof. By Assumption 2.4, there is no steady state in [x̂, ∞) for any δ ∈(0, 1). The argument of the proof of Lemma 6.1 shows that there is no
steady state in [x∗ + ², x̂] for δ close to one.
Lemma 6.3. Let x ∈ (0, x∗ − ²∗ ) be such thatf (x) < g ∗ .18(6.3)

Then for δ close to one, K(x) > x.19
Proof. Let T ∈ N be such that there is a feasible capital path from x with∀t ≥ T, xt = x∗ . Then
(6.4)

∀δ ∈ (0, 1),v(x) ≥ δ T g ∗ /(1 − δ).

Let δ ∈ (0, 1) be such thatf (x) < δ T g ∗ .

(6.5)

This inequality holds for δ close to one by (6.3). Suppose K(x) ≤ x. Thenby Lemma 3.5(ii), there is an optimal capital path {xt } from x such that∀t ∈ Z+ , xt ≤ x. Hence(6.6)v(x) =∞Xδ t [f (xt ) − xt+1 ] ≤ f (x)/(1 − δ),

t=0which together with (6.5) contradicts (6.4).
Let us now complete the proof of Proposition 6.1. If f (²) ≥ g ∗ , make ²
smaller so that f (²) < g ∗ . From Lemmas 6.1–6.3, for δ close to one, there is
no steady state in [², x∗ − ²∗ ] ∪ [x∗ + ²∗ , ∞), and K(²) > ². Hence any optimal
capital path {xt } from x0 ≥ ² satisfies ∀t ∈ Z+ , xt ≥ K(²) > ² by Lemma
3.5(ii), and thus satisfies (6.1) by Proposition 4.2.
18
19

Such x exists since limx↓0 f (x) = 0 by Assumption 2.2 and g ∗ > 0 by Assumption 6.1.
For simplicity, we do not express the dependence of K and v on δ.

18

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