A Nonsmooth, Nonconvex Model of
Optimal Growth
Takashi Kamihigashi*
RIEB
Kobe University
tkamihig@rieb.kobe-u.ac.jp

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

December 25, 2003

Abstract
This paper analyzes the nature of economic dynamics in a one-sector optimal
growth model in which the technology is generally nonconvex, nondifferentiable, and discontinuous. The model also allows for irreversible investment
and unbounded growth. We provide sufficient conditions for boundedness,
extinction (convergence to zero), survival (boundedness away from zero), andunbounded growth. These conditions reveal that boundedness and survival
are symmetrical phenomena, so are extinction and unbounded growth. Sincemany of the conditions are only local, it is possible that extinction occurs
from small capital stocks, while unbounded growth occurs from large capitalstocks. We also show that in most cases, if the discount factor is close toone, any optimal path from a given initial capital stock converges to a smallneighborhood of what we define as the golden rule capital stock. If this stock
is not finite, i.e., if sustainable consumption is maximized at infinity, thenas the discount factor approaches one, unbounded growth at least almost
occurs.
Keywords: Nonconvex, nonsmooth, and discontinuous technology; optimal
growth; unbounded growth; extinction; neighborhood turnpike.
*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 vast majority of growth models in the economic literature assume smoothtechnologies. In most cases, smoothness, or differentiability, is assumed
purely for analytical convenience. In reality, however, various types of indivisibility and discreteness are present, so that the actual aggregate technology
is most likely nonsmooth and even discontinuous. In addition, nonconvexities enter the aggregate technology through numerous sources, such as fixed
costs, threshold effects, increasing returns to scale, economies of scope, anddepensation in the reproduction of natural resources. Yet, there has been noattempt in the literature to conduct a systematic analysis of an aggregative
optimal growth model that admits a nonconvex, nonsmooth, and discontinuous technology.1
This paper offers such an analysis by studying the nature of economic
dynamics in a one-sector optimal growth model in which the technology is
generally nonconvex, nondifferentiable, and discontinuous. The model also
allows for irreversible investment and unbounded growth. The main resultsof the paper are categorized into two groups. The first group provides sufficient conditions for boundedness, extinction (convergence to zero), survival(boundedness away from zero), and unbounded growth. The second group
establishes the neighborhood turnpike property of optimal paths as the discount factor approaches one.
One of the contributions of this paper is to fill the gap between two separate strands of the optimal growth literature. The first is the literature on
optimal unbounded growth in one-sector models. This literature dates backat least to McFadden (1967, Sec. 6) and Gale and Sutherland (1968), though
they were not primarily concerned with conditions for unbounded growth.
More recently, Jones and Manuelli (1990) obtained a condition for unbounded
growth in a convex one-sector model with multiple capital stocks.2 Essentially, however, the literature on optimal unbounded growth in one-sector
models has dealt only with convex technologies.3
1

Various results are known for nondifferentiable, convex growth models (e.g., McKenzie,1986). The discrete-choice problems studied in Kamihigashi (2000a, b) have discontinuousfeatures. More relevant works are discussed below.
2
The analysis was extended to a multisector model by Dolmas (1996). See De Hek andRoy (2001) for results on stochastic models.
3
A notable exception is Romer (1986, Sec. V.B), who analyzed a one-sector optimal
growth model with a special form of increasing returns. The technology in his model,

1The implications of nonconvex technologies have been examined in theliterature on optimal growth models with a nonconcave production function.
The analysis of optimal paths is fairly extensive for the case of an S-shaped
production function (Skiba, 1978; Majumda and Mitra, 1982, 1983; Dechert
and Nishimura, 1983).4 Models with a more general production function
were analyzed by Majumdar and Nermuth (1982), Mitra and Ray (1984),
and Amir et al. (1991).5 Majumdar and Nermuth also allowed for irreversible
investment. The literature on nonconvex optimal growth models, however,has ruled out unbounded growth by assuming the existence of a maximum
sustainable capital stock.6
The optimal growth literature therefore has lacked a comprehensive analysis of nonconvex one-sector models in which unbounded growth is possible.
This paper provides such an analysis. Aside from technical conditions required to rule out trivial cases or to ensure the existence of optimal paths,
we only assume that the utility function is strictly increasing and striclty concave, that the production function is strictly increasing, and that the lower
bound on next period’s capital is nondecreasing in current capital. The lastassumption, which is trivially satisfied in the case of reversible investment,
allows for a general form of irreversible investment.
Another contribution of this paper is to provide the first systematic analysis of an optimal growth model in which the technology is generally nonconvex, nondifferentiable, and discontinuous. In terms of generality, the stationary framework used by Mitra and Ray (1984) and Amir et al. (1991)
is the closest to ours. In their framework, the production function is only
required to be continuous and strictly increasing, though discontinuity, irreversible investment, and unbounded growth are ruled out. Amir et al. (1991)
showed the monotonicity of the value function and the optimal consumption
policy correspondence with respect to changes in the discount factor. These
comparative statics properties are not considered in this paper. Mitra andRay (1984) established the monotonicity of optimal paths and the existence
however, does not take the standard form.
4
Early contributions to this literature are studies of optimal management of fisheries
and other renewable resources (e.g., Clark, 1971).
5
The discrete-time models mentioned in this paragraph are discussed in detail at theend of Section 2.
6
Stochastic models with a nonconvex technology (e.g., Majumdar et al., 1989; Joshi,
1997) are distinct from deterministic ones in that the effect of exogenous shocks, rather
than that of initial conditions, essentially determines the long run dynamics.

2

of a nonzero steady state. This monotonicity property, which carries over toour model, is our point of departure. We also extend their result concerning
the existence of a steady state.
The generality of our model poses several technical challenges. The absence of differentiability makes the standard Euler equation invalid. The
discontinuity of the technology implies that the value function is generally
discontinuous, and that the optimal policy correspondence is generally not
upper hemi-continuous. The irreversibility of investment implies that thevalue function is not necessarily increasing. These difficulties make various
familiar techniques inapplicable, but, for this very reason, help gain a deeper
insight into the fundamental mechanisms of economic dynamics.
There are four essential tools for overcoming these difficulties. The firstis Dechert and Nishimura’s (1983) monotonicity argument. In fact, we impose only the very minimum set of assumptions under which their argument
works. The second tool is what we call the gain function, which measures
discounted net returns on investment. We show that optimal paths nevermove in a direction in which higher discounted net returns on investment, or
higher “gains,” will never be available. This result helps determine the direction in which an optimal path possibly moves. The third tool is our findingthat a bounded optimal path converges to an optimal steady state despite
the discontinuity of the technology. The forth tool is generalized one-sided
derivatives (called Dini derivatives) that are well-defined even for nondifferentiable or discontinuous functions.7 Using these derivatives, we derive Euler
inequalities instead of an Euler equation, and obtain necessary conditions fora steady state. With these four tools, our general one-sector model can behandled in a rather standard way in many cases.
An interesting feature of the model that our analysis reveals is symmetry
between downward and upward dynamics. Our results show that boundedness and survival are completely symmetrical phenomena, so are extinction
and unbounded growth. For example, boundedness occurs if higher gains will
never be available along any increasing path, while survival occurs if higher
gains will never be available along any decreasing path. Since many of ourconditions are only local, it is possible that unbounded growth occurs fromlarge capital stocks, while extinction occurs from small capital stocks, i.e.,
the rich get richer and the poor get poorer.
7

Dini derivatives were used in Kamihigashi (2003) to obtain transversality conditions
for general stochastic problems.

3Despite such nonclassical features, the model behaves much like a classicalone as the discount factor approaches unity. Specifically, in most cases themodel exhibits the neighborhood turnpike property, which is well-known forconvex models (e.g., McKenzie, 1982, 1986; Yano, 1984; Montrucchio, 1994,
1995; Guerrero-Lechtenberg, 2000). It is the property that as the discountfactor approaches one, any optimal path from a given initial capital stock“converges” to a small neighborhood of the golden rule capital stock, which
is defined as the unique steady state in the undiscounted case. Since thisdefinition does not work in our nonconvex model, we define the gold rule
capital stock as the smallest capital stock that achieves maximum sustainableconsumption. We show that the neighborhood turnpike property holds if theinitial capital stock is less than the golden rule level. If this level is not
finite, i.e., if sustainable consumption is maximized at infinity, then as thediscount factor approaches one, unbounded growth at least almost occurs:
an optimal path either grows unboundedly or converges to an arbitrarily
large capital stock. We also show that the neighborhood turnpike property
holds regardless of the initial capital stock if irreversible investment is ruled
out, or if there is a unique capital stock that achieves maximum sustainableconsumption. Moreover, we argue that the neighborhood turnpike property
holds rather generically.
An immediate consequence of our results is the turnpike theorem shownby Majumdar and Nermuth (1982) for a special case of our model that assumes, among other things, differentiability and the existence of a uniquesteady state and a maximum sustainable capital stock. Their result has been
the most general turnpike theorem on nonconvex optimal growth models.8
Our contribution here is to show that the neighborhood turnpike property
holds even when unbounded growth is possible and the technology is nondifferentiable and discontinuous.
The rest of the paper is organized as follows. Section 2 presents the model
along with the assumptions that are maintained throughout the paper. Section 3 develops the essential tools discussed above and shows some resultsof independent interest. Sections 4, 5, 6, and 7 give conditions for boundedness, extinction, survival, and unbounded growth, respectively. Section 8
establishes several neighborhood turnpike results. Section 9 concludes thepaper. Longer proofs are relegated to the appendices.
8

A turnpike theorem for the S-shaped case was shown by Majumdar and Mitra (1982)
and Dechert and Nishimura (1983).

4

2

The Model

Consider the following maximization problem:
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)

max

∞
X

{ct ,xt }∞t=0

δ t u(ct )t=0s.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 beginning of
period t, δ is the discount factor, u is the utility function, f is the production
function, and r(xt ) is the lower bound on xt+1 . The objective function canbe treated as a Lebesgue integral under our assumptions.
Except for (2.4), the structure of the model is that of a standard onesector growth model. In the standard case, r(x) = 0 for all x ≥ 0, and f (x)can be written as(2.6)

f (x) = f˜(x) + (1 − d)x

for some function f˜ and constant d, where f˜ is the net production function
and d is the depreciation rate (possibly equal to one). In models with irreversible investment, it is typically assumed that r(x) = (1−d)x and f satisfies(2.6). These are special cases of our general formulation. For convenience,
we interpret r(xt ) as the depreciated capital stock at the end of period t, andcall (2.4) the irreversibility constraint and r(x) the depreciation function. In
this interpretation, (2.4) means that capital cannot be decreased below its
depreciated level. Our formulation allows for nonlinear depreciation, which
seems natural but has not received attention in the literature.
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 pair (c, x) is a steady5state if the stationary path {ct , xt } such that ct = c and xt = x for all t ∈ Z+
is optimal. A capital stock x ≥ 0 is a steady state if (c, x) is a steady state
for some c ≥ 0.
Aside from technical conditions required to rule out trivial cases or toensure the existence of optimal paths, we only assume that u is strictlyincreasing and strictly concave, that f is strictly increasing, and that r is
nondecreasing. The precise assumptions are stated and discussed in what
follows. They are maintained throughout this paper.
Assumption 2.1. (i) u : R+ → [−∞, ∞) is strictly increasing and strictly
concave. (ii) limc↓0 u(c) = u(0) ∈ [−∞, ∞). (iii) δ ∈ (0, 1).
The utility function u is not required to be differentiable. Since the caseu(0) = −∞ is permitted, u can be logarithmic or, more generally, of the
CRRA class.
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. Upper semicontinuity, which is essentially a normalization for monotone functions, is assumed here to ensure the existence of optimal paths.
To state our assumption on the depreciation function r, for h : R+ → R,
we define(2.7)

h− (x) = lim h(y),(2.8)

h+ (x) = lim h(y),

y↑x

y↓x

provided that the right-hand sides are well-defined. Any nondecreasing function h clearly satisfies h− (x) ≤ h(x) ≤ h+ (x).
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. Lower semicontinuity, which is essentiallya normalization for monotone functions, is assumed here to ensure the existence of optimal paths. The inequality r+ (x) < x basically means thatthe irreversibility constraint (2.4) is never binding at a steady state. It is6easy to see that if r+ (x) ≥ x for some x > 0, and if r is strictly increasing,then any feasible capital path from x0 > x is bounded below by x. Such atechnological restriction on feasible paths is ruled out here. The inequalityr(x) < f (x) means that strictly positive consumption is feasible at x. Thisis necessary for the maximization problem (2.1)–(2.5) to make sense in thecase u(0) = −∞. Assumption 2.3 is satisfied if r(x) = 0 for all x > 0, or if
f satisfies (2.6) with d ∈ (0, 1], r(x) = (1 − d)x, and f˜(x) > 0 for all x > 0.
Assumption
2.4. ∀x > 0, there exists a feasible path {ct , xt } from x suchPtthat ∞
δu(ct ) > −∞.
t=0
This assumption is trivially satisfied if u is bounded below. The assumption is required for the maximization problem (2.1)–(2.5) to make sense.
Ptt
9Assumption 2.5. ∀x > 0, ∞t=0 δ u(f (x)) < ∞.
The only role of this assumption is to ensure the existence of optimal
paths and the upper semicontinuity of the value function. It is satisfied, forexample, if u is bounded above, or if u(c) = ln(c) and f (x) = Ax for someconstant A > 0. In general Assumption 2.5 is a joint restriction on u, f , andδ, and there are various other cases in which it is satisfied.
Many of the conditions stated above are assumed only to ensure the existence of optimal paths. Their roles can be better understood by considering
the value function for the maximization problem (2.1)–(2.5):(2.9)

v(x0 ) = sup∞X{xt }∞
t=0 t=0

δ t u(f (xt ) − xt+1 ),where the supremum is taken over all feasible capital paths from x0 , i.e.,
capital paths {xt } from x0 such that r(xt ) ≤ xt+1 ≤ f (xt ) for all t ∈ Z+ . The
set of feasible capital paths from any x0 ≥ 0 is nonempty by Assumption 2.3.
Assumptions 2.1 and 2.2 imply that u(f (xt ) − xt+1 ) is upper semicontinuousin (xt , xt+1 ). Assumptions 2.2 and 2.3 imply that the set {(xt , xt+1 ) | r(xt ) ≤xt+1 ≤ f (xt )} is closed in R2 . Assumptions 2.4 and 2.5 ensure −∞ < v(x) <∞ for all x > 0. Hence the existence of an optimal path from any initial
capital stock x0 ≥ 0 is guaranteed by a standard argument (e.g., Ekelandand Scheinkman, 1986, Proposition 4.1).
9 2

f (x) = f (f (x)), f 3 (x) = f (f (f (x))), etc.

7

Various special cases of our model have been analyzed in the literature.
In the framework adopted by Majumdar and Mitra (1982) and Dechert andNishimura (1983), u and f are twice continuously differentiable, u0 (0) = ∞,f is S-shaped, r(x) = 0 for all x ≥ 0, and there is a maximum sustainable
capital stock, i.e.,(2.10)∃x > 0, ∀x > x,

f (x) < x.

This assumption is common to existing studies on nonconvex one-sector optimal growth models. In Majumdar and Nermuth’s (1982, Sec. 3) framework,u and f are also twice continuously differentiable with u0 (0) = ∞, and forsome d ∈ (0, 1], r(x) = (1 − d)x and f 0 (x) > (1 − d) for all x > 0. In Mitra
and Ray’s (1984, Sec. 5) framework, f is continuous and r(x) = 0 for allx ≥ 0. The infinite-horizon case of the model studied by Amir et al. (1991)
is another special case of our model in which f is continuous and r(x) = 0
for all x ≥ 0. Many of the arguments used in the literature apply to ourgeneral model as well. We use some of them in the next section.

3

Fundamental Properties

This section establishes fundamental properties of optimal paths. In particular, we show that an optimal capital path is monotone, that a bounded
optimal path converges to a steady state, and that an optimal capital pathnever moves in a direction in which higher gains, or higher discounted net
returns on investment, will never be available. We also establish the almost
everywhere uniqueness of the policy correspondence, a sufficient condition
for the existence of a nonzero steady state, and necessary conditions for asteady state. Many of the results here become essential tools in our subsequent analysis. Some of them are of independent interest.

3.1

Monotonicity and Convergence

The Bellman equation for the maximization problem (2.1)–(2.5) is given by(3.1)

v(xt ) =maxr(xt )≤xt+1 ≤f (xt ){u(f (xt ) − xt+1 ) + δv(xt+1 )}.

Let K : R+ → 2R+ denote the policy correspondence:(3.2) K(xt ) = {xt+1 ∈ [r(xt ), f (xt )] | v(xt ) = u(f (xt ) − xt+1 ) + δv(xt+1 )}.
8

We begin by showing a monotonicity property of K.
Lemma 3.1. ∀x0 ≥ 0, ∀y0 > x0 , ∀x1 ∈ K(x0 ), ∀y1 ∈ K(y0 ), x1 ≤ y1 .
Proof. Let 0 ≤ x0 < y0 , x1 ∈ K(x0 ), and y1 ∈ K(y0 ). If x0 = 0, we triviallyhave x1 = 0 ≤ y1 . Suppose x0 > 0 and x1 > y1 . Then
(3.3)

r(x0 ) ≤ r(y0 ) ≤ y1 < x1 ≤ f (x0 ) < f (y0 ).

Hence y1 is feasible from x0 , and x1 is feasible from y0 . The rest of the proofis the same as the first paragraph of the proof of Dechert and Nishimura(1983, Theorem 1).10
The next result shows a monotonicity property of optimal capital paths.
Lemma 3.2. Let {xt } be an optimal capital path. Then ∀t ∈ Z+ , xt ≤ xt+1
or ∀t ∈ Z+ , xt ≥ xt+1 .
Proof. See Appendix A.
The same result was shown by Majumdar and Nermuth (1982, Theorem
3.1) and Dechert and Nishimura (1983, Corollary 1) for differentiable cases,
and by Mitra and Ray (1984, Lemma 5.2) for a continuous case. Lemma 3.2
follows from an argument used in the working paper version of Mitra andRay (1984) (Mitra and Ray, 1983).
It is immediate from Lemma 3.2 that every bounded optimal path converges. It is not obvious, however, whether it converges to a steady state(which in our terminology means an optimal steady state). To see why, notethat an optimal path {ct , xt } satisfies v(xt ) = u(ct ) + δv(xt+1 ) for all t ∈ Z+
by the principle of optimality. If xt → x > 0 and ct → c > 0,11 and if(3.4)

lim v(xt ) = v(x),t↑∞then v(x) = u(c) + δv(x) by the continuity of u. This implies that (c, x) is a
steady state. But since f and r are not continuous, neither is v. Thus (3.4)
need not hold if {xt } is an arbitrary convergent sequence. Nevertheless (3.4)
can be shown if {xt } is an optimal capital path. A first step toward this is
to show the upper semicontinuity of v.
10

Given (3.3), their argument goes through as long as u is strictly concave and f isstrictly increasing. Lemma 3.1 can alternatively be shown by applying Topkis (1978,
Theorem 6.3).
11
If x = c = 0, then (c, x) is trivially a steady state.

9

Lemma 3.3. v is upper semicontinuous.
Proof. See Appendix A.
Hence limt↑∞ v(xt ) ≤ v(x) for any convergent capital path {xt } withx = limt↑∞ xt , in particular, for any bounded optimal capital path. The“reverse” inequality, v(x) ≤ limt↑∞ v(xt ), can be verified by arguing thatwhen xt is close to x, the cost of jumping from xt to x is small, so that the“benefit,” v(x) − v(xt ), must be likewise small.
Proposition 3.1. Any optimal path that is bounded converges to a steadystate.
Proof. See Appendix A.
Except for immediate consequences of our assumptions, upper semicontinuity is the only property of v that we use. In fact, v need not even bemonotone. If r(x) = 0 for all x ≥ 0, then a higher capital stock is always
better because it expands the set of feasible capital stocks for next period.
But if r is not constant, then a higher capital stock, which does not always
expand the feasible set, is not necessarily better.12 The non-monotonicity ofv complicates some of our proofs, but only slightly.
The next result is immediate from Lemma 3.1. We state it here for easy
reference.
Lemma 3.4. Let y0 > 0. If every optimal capital path from y0 is nonincreasing, then every optimal capital path from x0 ∈ [0, y0 ] is bounded aboveby y0 . Likewise, if every optimal capital path from y0 is nondecreasing, thenevery optimal capital path from x0 ≥ y0 is bounded below by y0 .

3.2

Almost Everywhere Uniqueness

Although optimal paths are in general not unique in this model, the policy
correspondence for the Bellman equation (3.1) can be shown to be singlevalued almost everywhere.
12

For example, suppose r(x) = 0 for x ∈ [0, z] and r(x) = f (x)−² for x > z, where z > 0
and ² ∈ (0, f (z)). Assume u(0) = 0, and δ = 0 for the moment. Then v(x) = u(f (x)) forx ≤ z, but v(x) = u(²) < v(f (z)) for x > z. It is easy to see that v has a similar structurefor δ > 0 close to zero.

10

Proposition 3.2. The policy correspondence K is single-valued almost everywhere. More specifically, the set {x ≥ 0 | K(x) is multivalued} is countable.
Proof. See Appendix A.
Dechert and Nishimura (1983, p. 352) showed the same result for theirmodel using the envelope theorem and the monotonicity (thus almost everywhere differentiability) of v. Their approach does not work in our model,
where v is not monotone and u is not differentiable. The proof of Proposition
3.2 uses only the monotonicity property of K stated in Lemma 3.1.

3.3

The Gain Function

For x ≥ 0, define(3.5)

Γ(x) = δf (x) − x.

We call this function the gain function for the following reason. If x units ofcapital are invested today, it generates f (x) units of output tomorrow. Thus
the discounted net return, or “gain,” is δf (x) − x.
The gain function plays a central role in our analysis. The same function
was used by Majumdar and Nermuth (1982, p. 358), Dechert and Nishimura(1983, Lemmas 2, 3), and Mitra and Ray (1984, p. 160, 164) to examine theproperties of steady states. We use it for more general purposes.
It is useful to note that for any feasible path {ct , xt } and T ∈ N,
(3.6)

T
X
t=0

t

δ ct =TX

δ t [f (xt ) − xt+1 ]

t=0

(3.7)

= f (x0 ) − x1 + δ[f (x1 ) − x2 ] + · · · + δ T [f (xT ) − xT +1 ](3.8)

= f (x0 ) +T −1Xδ t Γ(xt+1 ) − δ T xT +1 .

t=0Thus Γ(xt+1 ) is the contribution of xt+1 to the present discounted value ofconsumption. The following lemma is shown by generalizing an argumentused by Majumdar and Nermuth (1982, p. 358) and Dechert and Nishimura(1983, Lemma 2).
11

Lemma 3.5. Let {xt } be an optimal capital path that is nonstationary. Then∃t ∈ N, Γ(x0 ) < Γ(xt ).
Proof. See Appendix A.
Therefore a nonstationary optimal capital path must always achieve ahigher gain at some point in the future. In other words, optimal capital
paths move in a direction in which higher gains will eventually be available.
If the highest gain is available at the current capital stock, then it is optimal
to stay there forever. Thus the following result is an immediate consequenceof Lemma 3.5.
Proposition 3.3. Suppose ∃x̂ > 0, Γ(x̂) = supx≥0 Γ(x). Then x̂ is a steadystate.
As a consequence, we obtain a sufficient set of conditions for the existenceof a nonzero steady state.
Proposition 3.4. Suppose (i) ∃x̃ > 0, Γ(x̃) ≥ 0 and (ii) limx↑∞ Γ(x) <supx>0 Γ(x). Then there exists a nonzero steady state.
Proof. Since f is upper semicontinuous, so is Γ. This together with (ii)
implies that ∃x̂ ≥ 0, Γ(x̂) = supx≥0 Γ(x) ≡ s. If s > 0, then x̂ > 0 sinceΓ(0) = 0. If s = 0, then x̂ can be chosen to be strictly positive by (i). Thus
the conclusion follows by Proposition 3.3.
If f satisfies (i), then it is called δ-productive in Mitra and Ray’s (1984,
p. 164) terminology. Condition (ii) holds if there is a maximum sustainable
capital stock; for then Γ(x) = δf (x) − x < f (x) − x < 0 for large x (recall(2.10)).13 Hence Proposition 3.4 extends Mitra and Ray (1984, Theorem 4.2)
to our general model.14 Our argument is similar to theirs, but more direct
since we do not consider support prices.
13“For large x” means “for all x sufficiently large.” Similar remarks apply to similar
expressions.
14
Strictly speaking, since they did not assume the strict concavity of u for their corresponding result, Proposition 3.4 does not generalize their result. But in fact Proposition
3.4 holds even if u is only concave. This is because x̂ as given in the proof of Proposition
3.4 is a steady state by (A.14) even if u is only concave.

12

3.4

Nonsmooth Analysis

Even in the absence of differentiability, generalized versions of derivatives areavailable. For h : (a, b) → R with a < b, define(3.9)(3.10)

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

These generalized derivatives are called the lower left and the upper right Dini
derivative of h evaluated at x. They allow us to obtain “Euler inequalities”
instead of an Euler equation.
Lemma 3.6. Let {ct , xt } be an optimal path. Let t ∈ Z+ . If ct > 0 and
xt+2 > r+ (xt+1 ), thenu0− (ct ) ≥ δu0+ (ct+1 )f+0 (xt+1 ).

(3.11)

If xt+1 > r(xt ) and ct+1 > 0, thenu0+ (ct ) ≤ δu0− (ct+1 )f−0 (xt+1 ).

(3.12)
Proof. See Appendix A.

If u and f are differentiable, then (3.11) and (3.12) reduce to the Euler
equation u0 (ct ) = δu0 (ct+1 )f 0 (xt+1 ). The cost of not assuming differentiability
is that the Euler equation must be replaced by the two Euler inequalities
above. This cost is rather small here since we use them only to obtainnecessary conditions for a steady state. For this purpose, for x ≥ 0, define(3.13)

g(x) = f (x) − x,which is the stationary consumption level associated with capital stock x.
For x > 0 with g(x) > 0, define(3.14)

u0 (g(x)) 0
Φ(x) = δ 0+
f (x),u− (g(x)) +u0 (g(x)) 0
Ψ(x) = δ 0−
f (x).
u+ (g(x)) −If one divides (3.11) through by u0− (ct ) and sets xt+1 = x and ct = ct+1 = g(x),then the resulting right-hand side is Φ(x). One obtains Ψ(x) similarly. Hencethe Euler inequalities (3.11) and (3.12) evaluated at a steady state imply(3.15) below.
13

Lemma 3.7. If x > 0 is a steady state, then g(x) > 0 and(3.15)

Φ(x) ≤ 1 ≤ Ψ(x).

Proof. See Appendix A.
As u is concave, u0− ≥ u0+ ; thus
(3.16)∀x > 0,Φ(x) ≤ δf+0 (x),δf−0 (x) ≤ Ψ(x).

If u is differentiable, Φ(x) = δf+0 (x) and Ψ(x) = δf−0 (x). If u and f aredifferentiable, Φ(x) = Ψ(x) = δf 0 (x). In this case, (3.15) implies δf 0 (x) =1, a well-known necessary condition for a steady state in the differentiablecase. The following result gives useful relationships among the functions
Φ, Ψ, f−0 , f+0 , and Γ.
Lemma 3.8. Let x > 0 and 0 ≤ a < b.
(i) If Φ(x) > 1, then δf+0 (x) > 1.
(ii) δf+0 ≥ 1 on [a, b) iff Γ is nondecreasing on [a, b].
(iii) If Ψ(x) < 1, then δf−0 (x) < 1.
(iv) δf−0 ≤ 1 on (a, b] iff Γ is nonincreasing on [a, b].
Proof. See Appendix A.
Remark 3.1. If ∃x > 0, f−0 (x) ≤ 1 or Ψ(x) < 1, then f is continuous at x.
Indeed, any condition implying f−0 (x) < ∞ implies that f is continuous
at x. To see this, note that if f is discontinuous at some x > 0, thenf−0 (x) = ∞ since f− (x) < f+ (x) and f (x) = f+ (x) by upper semicontinuity.
Some of our conditions for boundedness and extinction imply the local or
global continuity of f . None of our conditions for survival and unbounded
growth implies any continuity of f .

4Boundedness

This section offers conditions under which optimal paths are bounded. We
begin by stating a result that restricts the gain function Γ for large capitalstocks.

14

Proposition 4.1. Suppose(4.1)∃x > 0, ∀x > x,

Γ(x) ≥ Γ(x).

Then every optimal capital path from x0 ∈ [0, x] is bounded above by x.15
Proof. By (4.1) and Lemma 3.5, any optimal capital path from x is nonincreasing. Hence the conclusion follows by Lemma 3.4.
Recall from Lemma 3.5 that optimal paths never move in a directionin which higher gains, or higher discounted net returns on investment, will
never be available. Condition (4.1) means that gains will never be higher
than Γ(x) if capital is increased above x. Under (4.1) therefore it is never
optimal to cross x from below, which is the conclusion of Proposition 4.1.
A sufficient condition for (4.1) is that Γ(x) is nonincreasing for large x
or, equivalently, δf−0 (x) ≤ 1 for large x (recall Lemma 3.8). This condition
is used in the following.
Corollary 4.1. Suppose(4.2)∃x > 0, ∀x > x,

δf−0 (x) ≤ 1.

Then (i) every optimal capital path is bounded. Furthermore, (ii) every optimal capital path from x0 ≥ x is nonincreasing.
Proof. By (4.2) and Lemma 3.8, Γ is nonincreasing on [x, ∞). Thus (ii) holds
by Lemma 3.5. Proposition 4.1 and (ii) imply (i)If u and f are differentiable, conclusion (ii) of Corollary 4.1 can be shownusing the Euler equation u0 (ct ) = δu0 (ct+1 )f 0 (xt+1 ). Indeed, if {xt } is an
optimal capital path from x0 ≥ x that is nondecreasing and nonstationary,then the Euler equation together with (4.2) implies that {ct } is nonincreasing
and nonstationary. This leads to a contradiction since the stationary pathfrom x0 can be shown to be a better alternative.
Though it may be possible to generalize this approach to the nondifferentiable case, the proof of Corollary 4.1 as well as that of Proposition 4.1
makes no use of the Euler inequalities (3.11) and (3.12). In fact these results
rely solely on the shape of the gain function Γ through Lemma 3.5.
15

To simplify the statements of our results, we focus on conclusions about capital paths,
from which relevant conclusions about consumption paths easily follow.

15

Recall that Proposition 4.1 restricts Γ for large capital stocks. It is basedon the idea that growth is not worthwhile if higher gains will never be available along any increasing path. The next result by contrast restricts Γ only
for small capital stocks. It is based on the idea that growth is not worthwhile
if higher gains will not be available until capital is increased substantially.
Proposition 4.2. Suppose ∃z > 0, there exists a sequence {zi }∞i=1 in (0, z)
such that(4.3)(i) lim zi = 0,i↑∞(ii) ∀i ∈ N, ∀x ∈ (zi , z], Γ(zi ) ≥ Γ(x).

Further suppose
u0+ (0) = ∞.

(4.4)

Then ∃i ∈ N, every optimal capital path from x0 ∈ (0, zi ] is bounded above byzi .
Proof. See Appendix B.
The first hypothesis of Proposition 4.2 holds if Γ is nonincreasing on aneighborhood of zero. The second hypothesis, (4.4), is a nonsmooth versionof an Inada condition.
Proposition 4.2 can be understood quite intuitively. Condition (4.3)(ii)
means that if the initial capital stock is zi , capital must be increased above z
to possibly achieve higher gains. If higher gains are not available even there,
then it is never optimal to choose an increasing capital path (by Lemma
3.5). However, when no information is available about Γ(x) for x ≥ z, it
is potentially optimal to increase capital above z. For large i, increasing
capital above z requires decreasing current consumption significantly. But(4.4) means that decreasing current consumption is extremely costly when it
is already very low. Hence it is never optimal to increase capital from a very
low level under (4.3) and (4.4).

5

Extinction

This section gives conditions under which extinction occurs, i.e., an optimal path converges to zero. Extinction occurs if an optimal capital path isbounded and if there is no nonzero steady state to which it can converge.
16Nonzero steady states can be ruled out if a necessary condition given by
Lemma 3.7 is not met. Conditions for boundedness are given by Propositions 4.1 and 4.2. The following result assumes the condition in Proposition
4.1.
Proposition 5.1. Assume (4.1). Suppose(5.1)∀x ∈ (0, x],Ψ(x) < 1, 16

where x is given by (4.1). Then every optimal capital path from x0 ∈ (0, x]
converges to zero.
Proof. By (4.1) and Proposition 4.1, any optimal capital path {xt } fromx0 ∈ (0, x] is bounded above by x. By Proposition 3.1, {xt } converges to asteady state. But by (5.1) and Lemma 3.7, there is no steady state in (0, x].
Hence {xt } converges to zero.
Note from Remark 3.1 that (5.1) implies the continuity of f on (0, x].
If f is discontinuous at some x ∈ (0, x], then a small decrease from x incapital results in a large loss in output. Thus it is possible that an optimal
capital path stays at x instead of going to zero. Condition (5.1) rules out thispossibility by ensuring the continuity of f on (0, x] as well as the nonexistenceof steady states in (0, x]. A similar argument applies to (5.2) below. Ourconditions for survival and unbounded growth, given in the next two sections,do not imply any continuity of f .
If u and f are continuously differentiable, then Ψ(x) = δf 0 (x), so that
(5.1) holds if δf 0 (0) < 1. This is the condition for local extinction shownby Dechert and Nishimura (1983, Lemma 5). While Proposition 5.1, whichassumes (4.1), does not generalize their result, the following does.
Proposition 5.2. Assume (4.4). Suppose(5.2)∃z > 0, ∀x ∈ (0, z],

Ψ(x) < 1.

Then ∃x ∈ (0, z], every optimal capital path from x0 ∈ (0, x] converges tozero.
Proof. Let {zi }∞i=1 be a sequence in (0, z) satisfying (4.3)(i). By (5.2) andLemma 3.8, Γ is nonincreasing on (0, z]. Thus {zi } also satisfies (4.3)(ii).
16

Recall (3.14) for the definition of Ψ.

17

By Proposition 4.2, ∃i ∈ N, every optimal capital path from x0 ∈ (0, zi ] isbounded above by zi . Let x = zi . Then every optimal capital path fromx0 ∈ (0, x] converges to a steady state by Proposition 3.1. By (5.2) andLemma 3.7, there is no steady state in (0, x]. Hence it converges to zero.
As mentioned above, Proposition 5.2 generalizes Dechert and Nishimura(1983, Lemma 5). Since they made extensive use of the Euler equationand their assumptions that f is S-shaped and that there is a maximum
sustainable capital stock (which implies that every optimal path converges
to a steady state), their argument cannot easily be extended to our model.
Olson and Roy (1996, Theorem 6.2) used a more general argument to show
local extinction in a slightly different context, but their argument still relies
heavily on the Euler equation. Our argument by contrast uses the Euler
inequalities (through Lemma 3.7) only to rule out nonzero steady states.
Our last result in this section gives a condition for global extinction.
Proposition 5.3. Suppose(5.3)∀x > 0,Ψ(x) < 1.

Then every optimal capital path converges to zero.
Proof. Let x0 > 0. By (5.3) and Lemma 3.8, Γ is nonincreasing on R++ .
Thus (4.1) holds with x = x0 . Hence by Proposition 5.1, any optimal capital
path from x0 converges to zero.
If u and f are differentiable, (5.3) reduces to the condition that δf 0 (x) < 1
for all x > 0. This is the condition for global extinction shown by Majumdar
and Mitra (1982, p. 122) and Dechert and Nishimura (1983, p. 346) for an
S-shaped case. Proposition 5.3 is a direct generalization of their result.

6

SurvivalThis section offers conditions under which survival occurs, i.e., an optimal
path is bounded away from zero. The results here are symmetrical to thosein Section 4, which gave conditions for boundedness, or “boundedness awayfrom infinity.” We begin by stating a result that restricts the gain function
Γ for small capital stocks.

18

Proposition 6.1. Suppose∃x > 0, ∀x ∈ [0, x),(6.1)

Γ(x) ≤ Γ(x).

Then every optimal capital path from x0 ≥ x is bounded below by x.
Proof. Similar to the proof of Proposition 4.1
Recall once again from Lemma 3.5 that optimal paths never move in a
direction in which higher gains, or higher discounted net returns on investment, will never be available. Condition (6.1) means that gains will never behigher than Γ(x) if capital is decreased below x. Under (6.1) therefore it isnever optimal to cross x from above, which is the conclusion of Proposition
6.1.
A sufficient condition for (6.1) is that Γ(x) is nondecreasing for small xor, equivalently, δf+0 (x) ≥ 1 for small x (recall Lemma 3.8). This condition
is used in the following.
Corollary 6.1. Suppose(6.2)∃x > 0, ∀x ∈ [0, x),δf+0 (x) ≥ 1.

Then (i) every optimal capital path is bounded away from zero. Furthermore,(ii) every optimal capital path from x0 ∈ (0, x] is nondecreasing.
Proof. Similar to the proof of Corollary 4.1.
Recall that Proposition 6.1 restricts Γ for small capital stocks. It is basedon the idea that decline is not desirable if higher gains will never be available
along any decreasing path. The next result by contrast restricts Γ only forlarge capital stocks. It is based on the idea that decline is not desirable if
higher gains will not be available until capital is decreased substantially.
Proposition 6.2. Suppose ∃z > 0, there exists a sequence {zi }∞i=1 in (z, ∞)such that(6.3)(i) lim zi = ∞,i↑∞(ii) ∀i ∈ N, ∀x ∈ [z, zi ), Γ(x) ≤ Γ(zi ).

Further suppose(6.4)

lim u0+ (c)c < ∞.

c↑∞Then ∃i ∈ N, every optimal capital path from x0 ≥ zi is bounded below by zi .
19

Proof. See Appendix C.
The first hypothesis of Proposition 6.2 holds if Γ(x) is nondecreasing forlarge x. There are various cases in which the second hypothesis, (6.4), holds.
For example, it holds if u is bounded above, as implied by the following.
Remark 6.1. If u is bounded above, thenlim u0+ (c)c = 0.

(6.5)

c↑∞

Proof. See Appendix C.
Another important case in which (6.4) holds is when u(c) = ln c, which
implies u0 (c)c = 1 for all c > 0.
Like Proposition 4.2, Proposition 6.2 can be understood intuitively. Condition (6.3)(ii) means that if the initial capital stock is zi , capital must be
reduced below z to possibly achieve higher gains. If higher gains are notavailable even there, then it is never optimal to choose a decreasing capital path (by Lemma 3.5). However, when no information is available about
Γ(x) for x < z, it is potentially optimal to reduce capital below z. For i
sufficiently large, (6.3) implies that the stationary level of consumption atzi is arbitrarily large. Thus if capital is reduced below z, future consumption is decreased substantially while current consumption is increased. But(6.4) means that marginal utility declines fast as consumption grows, i.e., thebenefit of increasing current consumption is small when high consumption isalready feasible. Hence it is never optimal to reduce capital from a very highlevel under (6.3) and (6.4).

7

Unbounded GrowthThis section gives conditions under which unbounded growth occurs, i.e.,
an optimal capital path goes to infinity. The results in this section are
symmetrical to those in Section 5, which gave conditions for extinction, orconvergence to zero. Unbounded growth occurs if an optimal capital pathis bounded away from zero and if there is no steady state to which it canconverge. Steady states can be ruled out if a necessary condition given by
Lemma 3.7 is not met. Conditions for boundedness away from zero are given
by Propositions 6.1 and 6.2. The following result assumes the condition in
Proposition 6.1.
20

Proposition 7.1. Assume (6.1). Suppose(7.1)∀x ≥ x,Φ(x) > 1,

where x is given by (6.1). Then every optimal capital path from x0 ≥ x goesto infinity.
Proof. Similar to the proof of Proposition 5.1.
If u and f are differentiable, then Φ(x) = δf 0 (x), so that the last inequality in (7.1) reduces to δf 0 (x) > 1.
The following result uses the conditions for survival in Proposition 6.2.
Proposition 7.2. Assume (6.4). Suppose(7.2)∃z > 0, ∀x ≥ z,

Φ(x) > 1.

Then ∃x ≥ z, every optimal capital path from x0 ≥ x goes to infinity.
Proof. Similar to the proof of Proposition 5.2.
The following result gives a condition for global unbounded growth.
Proposition 7.3. Suppose(7.3)∀x > 0,Φ(x) > 1.

Then every optimal capital path goes to infinity.
Proof. Similar to the proof of Proposition 5.3.
For a differentiable convex model, Jones and Manuelli (1990, p. 1014)
showed that unbounded growth occurs if δf 0 is bounded below away from 1.17
If u and f are differentiable, (7.3) reduces to the condition that δf 0 (x) > 1
for all x > 0. This condition is necessary for unbounded growth in the
differentiable convex case by a standard argument (e.g., Jones and Manuelli,
1997, p. 78). Proposition 7.3 shows that this necessary condition is alsosufficient for global unbounded growth even without convexity.18
17

The sufficient conditions for unbounded growth used by Dolmas (1996, Assumption(P)) and Kaganovich (1998, Assumption 7) reduce to Jones and Manuelli’s (1990) condition in the one-sector case with a single capital good.
18
One can add the following statement to Propositions 7.1–7.3: the associated consumption path also goes to infinity. This can easily be shown by using (3.11), Lemma C.1, andthe fact that v(x) ≥ u(g(x))/(1 − δ) whenever g(x) ≥ 0.

21

To discuss an implication of convexity, suppose u and f are differentiable
and f is concave. Then since f 0 is nonincreasing, either δf 0 (x) > 1 for allx > 0 or δf 0 (x) ≤ 1 for large x > 0. This means that unbounded growtheither occurs globally (by Proposition 7.3) or never occurs (by Corollary 4.1).
Likewise, since either δf 0 (x) < 1 for all x > 0 or δf 0 (x) ≥ 1 for small x > 0,
extinction either occurs globally (by Proposition 5.3) or never occurs (byCorollary 6.1). Hence unbounded growth and extinction may not coexist in
the convex (and differentiable) case. We close this section by pointing outthat unbounded growth and extinction may coexist in the nonconvex case.
Corollary 7.1. Assume (4.4), (5.2), (6.4), and (7.2). Then ∃x, x > 0
with x ≤ x, every optimal capital path from an initial capital stock in (0, x]
converges to zero, while every optimal capital path from an initial capitalstock in [x, ∞) goes to infinity.
Proof. Immediate from Propositions 5.2 and 7.2.
This result may not be particularly surprising in view of Dechert andNishimura (1983, Theorem 4). Many of their arguments, however, rely heavily on the existence of a maximum sustainable capital stock (recall (2.10)),and thus cannot easily be modified to allow for unbounded growth. Corollary
7.1 suggests the possibility that the rich get richer and the poor get poorer,even without any distortion.19

8

The Neighborhood Turnpike Property

Despite the nonclassical results shown above, our model behaves much likea classical one as δ approaches unity. In particular, we show that in mostcases, for δ close to one, any optimal capital path from a given initial stock“converges” to a small neighborhood of what we define as the gold rule capitalstock.20 Unlike in the classical case, this stock is not required to be finite.
We begin by assuming that maximum sustainable consumption is strictly
positive.
19

The corresponding situation in Dechert and Nishimura’s model is that the poor getpoorer, the modestly rich get richer, and the very rich get less rich.
20
This is in fact slightly more general than the standard statement of the neighborhood
turnpike theorem. Our version is equivalent to the standard statement under typicalhypotheses for the latter.

22

Assumption 8.1. g ∗ ≡ supx≥0 g(x) ∈ (0, ∞].
It is easy to see that if g ∗ ≤ 0, extinction occurs from every capital stock.21
The role of Assumption 8.1 is to rule out this trivial case. Note that we do
not require g ∗ to be finite.
We define the golden rule capital stock x∗ as follows.
(min{x ≥ 0 | g(x) = g ∗ } if ∃x ≥ 0, g(x) = g ∗ ,
(8.1)
x∗ =
∞
otherwise.
By Assumption 8.1, x∗ > 0. The case x∗ = ∞ means that sustainableconsumption is maximized at infinity.
This section maintains all the assumptions stated in Section 2 for eachδ ∈ (0, 1). The following is our last assumption.
Assumption 8.2. ∀x ∈ (0, x∗ ], g− (x) > 0.22
It is easy to see that if g− (x) ≤ 0, i.e., f− (x) ≤ x, for some x ∈ (0, x∗ ],then no feasible capital path from x0 < x can reach x. Hence Assumption 8.2
is a minimum requirement for the neighborhood turnpike property to holdglobally.
For the rest of this section, we take an arbitrary initial capital stock
x0 ∈ R++ as given. For each δ ∈ (0, 1), let {xδt } be an optimal capital pathfrom x0 with the discount factor equal to δ. The neighborhood turnpike
property is now expressed as follows.
(8.2)

lim lim xδt = x∗ .23
δ↑1 t↑∞

This equation means that for δ close to one, {xδt } converges to a small neighborhood of x∗ .
The first step to establishing (8.2) is the following result, which is similar
to Scheinkman’s (1976) “visit lemma.”
21

To see this, suppose ∀x > 0, g(x) ≤ 0. Let {xt } be an optimal capital path fromx0 > 0. Then {xt } is nonincreasing since ∀t ∈ Z+ , xt+1 − xt ≤ f (xt ) − xt = g(xt ) ≤ 0. By
Lemma 3.7, there is no nonzero steady state. Since {xt } converges to a steady state by
Proposition 3.1, it converges to zero.
22
Recall (2.7) for the definition of g− .
23
Of course, we do not assume that the expression on the left-hand side is well-defineda priori. The same remark applies to Lemma 8.1.

23

Lemma 8.1. limδ↑1 supt∈Z+ g(xδt+1 ) = g ∗ .
Proof. See Appendix D.
This lemma implies that if x∗ is the unique maximizer of g(x) (or, more
precisely, under (8.6) and (8.7) below), then for δ close to one, {xδt } “visits”
a small neighborhood of x∗ at least once. In general, {xδt } approaches somemaximizer of g(x) at least once for δ close to one. The following result showsthat if x0 ≤ x∗ , then {xδt } in fact converges to a small neighborhood of x∗for δ close to one.
Proposition 8.1. Suppose x0 ≤ x∗ .24 Then (8.2) holds.
Proof. See Appendix D.
An immediate implication of the above result is that if x∗ = ∞, i.e., ifsustainable consumption is maximized at infinity, then for δ close to one,
unbounded growth at least almost occurs.
Corollary 8.1. Suppose x∗ = ∞. Then
(8.3)

lim lim xδt = ∞.

δ↑∞ t↑∞

Proof. Immediate from Proposition 8.1.
This is a “unbounded growth” version of the neighborhood turnpike theorem. Equation (8.3) means that for δ close to one, unbounded growth atleast almost occurs: {xδt } either goes to infinity or converges to an arbitrarily
large steady state.
The only case that is not covered by Proposition 8.1 is when x∗ < ∞ andx0 > x∗ . In this case, (8.2) need not hold since {xδt } could go to infinity or
converge to a neighborhood of some x > x∗ with g(x) = g ∗ . However, there
are two important cases in which (8.2) is guaranteed to hold even if x0 > x∗ .
The first case is when it is feasible to jump from x0 to x∗ .
Proposition 8.2. Suppose(8.4)

r(x0 ) ≤ x∗ .

Then (8.2) holds.
24

Since x0 ∈ R++ , x0 < x∗ if x∗ = ∞.

24

Proof. See Appendix D.
An obvious sufficient condition for (8.4) is(8.5)∀x ≥ 0,r(x) = 0,i.e., the irreversibility constraint is effectively absent. In this case, (8.4) holds
regardless of x0 , so does (8.2).
Note from Assumption 2.3 that (8.4) holds whenever x0 ≤ x∗ . Suppose
x0 > x∗ . Then under (8.4), it is suboptimal to stay above x∗ since current consumption can be increased by jumping down to x∗ , and since themaximum sustainable amount can be consumed every period afterwards bystaying at x∗ forever. If (8.4) does not hold, however, it is infeasible to jumpfrom x0 to x∗ , so that a transition to x∗ could require many periods of low
consumption. This is the case, for example, if r ≈ f on (x∗ , x0 ).
The following is a useful restatement of Proposition 8.2.
Corollary 8.2. Let x = sup{x ∈ R+ | r(x) ≤ x∗ }. If x0 ∈ (0, x], then (8.2)
holds.25
Assumption 2.3 implies x > x∗ . Under (8.5), x = ∞, so that (8.2) holds
regardless of x0 , as mentioned above. Corollary 8.2 makes it clear that there
is an interval strictly larger than (0, x∗ ] on which the neighborhood turnpike
property holds.
The second case in which (8.2) holds for x0 > x∗ is when x∗ is the “uniquemaximizer” of g.
Proposition 8.3. Suppose x∗ < ∞. Suppose(8.6)∀x ∈ (0, ∞) \ {x∗ },(8.7)g(x) < g ∗ ,lim g(x) < g ∗ .

x↑∞Then (8.2) holds.
Proof. See Appendix D.
Assume (8.6). Suppose (i) there is a maximum sustainable capital stock,(ii) there is a neighborhood of zero on which Φ > 1 for δ close to one, and(iii) there is a neighborhood of x∗ that contains only one steady state x(δ) for25

If x < ∞, then r(x) ≤ x∗ by the lower semicontinuity of r.

25δ close to one. In this case, (8.7) holds by (i) (recall (2.10)), and Proposition
8.3 along with Lemma 3.7 and Corollary 6.1 implies that for δ close to one,all optimal capital capital paths from all initial stocks converge to x(δ). This
result was shown by Majumdar and Nermuth (1982, Theorem 3.4) for adifferentiable case.26
Let us conclude this section by arguing that the neighborhood turnpike
property (8.2) is a rather generic one. The results in this section show that
(8.2) holds unless (i) x∗ < ∞, (ii) x∗ < x0 , (iii) it is infeasible to jump fromx0 to x∗ , and (iv) g has multiple maximizers, possibly including infinity. The
situations in which the neighborhood turnpike property may not hold are
unstable since a small perturbation (e.g., in the sup norm sense) to f easily
eliminates all but one maximizer of g.
Now consider stability of the situations in which the neighborhood turnpike property does hold regardless of the initial capital stock. Corollary 8.1
and Proposition 8.3 show that (8.2) holds if (i) x∗ = ∞ or if (ii) x∗ < ∞ andg has a unique maximizer in the sense of (8.6) and (8.7). If x∗ = g ∗ = ∞,then this remains to be true under small perturbations. On the other hand,the case in which x∗ = ∞ and g ∗ < ∞ is unstable since x∗ can be finite after
a small perturbation to f . But this case is rather exceptional since it implies
that the graph of f (x) approaches from below a straight line parallel to the
45 degree line gradually or infinitely often as x ↑ ∞. If x∗ < ∞ and g has aunique maximizer, a small perturbation to f can create multiple maximizersof g in a small neighborhood of x∗ . But this does not affect the neighborhood
turnpike property since Lemma 8.1 implies that for δ close to one, convergence to this neighborhood must occur even after a small perturbation to f ,and since Corollary 8.2 implies that the new golden rule capital stock is the
only “attractor” in the neighborhood.
To summarize, the discussions in the preceding two paragraphs show thatthe neighborhood turnpike property holds rather generically.

9

Concluding Remarks

This paper has analyzed the nature of economic dynamics in a one-sector
optimal growth model in which the technology is generally nonconvex, non26

The global convergence result shown by Majumdar and Mitra (1982, Theorem 5.5)
and Dechert and Nishimura (1983, Theorem 3) for the “mild discount” case can be viewed
as a special case of Majumdar and Nermuth’s turnpike result in this context.

26

differentiable, and discontinuous. The model also allows for irreversible investment and unbounded growth. One of the contributions of the paper is tofill a gap in the optimal growth literature by analyzing a nonconvex optimal
growth model in which unbounded growth is possible. Another contribution
is to provide the first systematic analysis of an optimal growth model in which
the technology is generally nonconvex, nondifferentiable, and discontinuous.
These are properties that the actual aggregate technology is likely to havefor various reasons. We have shown sufficient conditions for boundedness,
extinction, survival, and unbounded growth. We have also shown that theneighborhood turnpike property holds in most cases.
An interesting feature of the model that our analysis has revealed is symmetry between downward and upward dynamics. In particular, boundedness
and survival are symmetrical phenomena, so are extinction and unbounded
growth. For example, boundedness occurs if higher gains, or higher discounted net returns on investment, will never be available along any increasing path. Likewise, survival occurs if higher gains will never be available
along any decreasing path. All our conditions for boundedness and survival
have this symmetrical feature, so do our conditions for extinction and unbounded growth. Since many of these conditions are only local, it is possiblethat extinction occurs from small capital stocks, while unbounded growthoccurs from large capital stocks.
Despite such nonclassical features, in most cases, if the discount factor isclose to one, any optimal path from a given initial capital stock converges to asmall neighborhood of the golden rule capital stock, which is in our definition
the smallest capital stock that achieves maximum sustainable consumption.
The neighborhood turnpike property holds, for example, if the initial capitalstock is smaller than the golden rule level. If the latter is not finite, i.e., ifsustainable consumption is maximized at infinity, then as the discount factor
approaches one, unbounded growth at least almost occurs: an optimal patheither grows unboundedly or converges to an arbitrarily large capital stock.
The neighborhood turnpike property also holds if investment is reversible asin the standard case, or if sustainable consumption is maximized at a uniquecapital stock. We have also argued that the neighborhood turnpike property
holds rather generically.
Many of our techniques are directly applicable to other one-dimensional
problems with generally nonconvex, nondifferentiable, and discontinuous constraints. The results shown in this paper would provide basic insight into suchproblems. Though our methods may seem rather specific to one-dimensional
27

cases, some of our arguments are of wider interest. For example, it seems possible to extend the neighborhood turnpike property to multi-sector growthmodels and other multi-dimensional problems. The insight gained in thispaper would be useful in analyzing such complex problems.

Appendix A
A.1

Proofs of Section 3 Results

Proof of Lemma 3.2

Lemma A.1. Let {xt } be an optimal capital path. If x0 < x1 , then ∀t ∈Z+ , xt ≤ xt+1 . Likewise, if x0 > x1 , then ∀t ∈ Z+ , xt ≥ xt+1 .
Proof. 27 Suppose x0 < x1 . Suppose ∃t ∈ Z+ , xt > xt+1 . Let s ∈ Z+ be
the first t ∈ Z+ with xt > xt+1 . Note that xs−1 ≤ xs . If xs−1 < xs ,then xs ≤ xs+1 by Lemma 3.1, a contradiction. Thus xs−1 = xs . Let s ≤s be the first t ∈ Z+ with xt = xs . Note that xs−1 < xs and that thecapital path {xs−1 , xs , xs+1 , xs+2 , · · · } is optimal from xs−1 by the principleof optimality. Since xs−1 < xs , we must have xs ≤ xs+1 by Lemma 3.1, again
a contradiction. The case x0 > x1 is similar.
Let us now prove Lemma 3.2. If {xt } is stationary, it trivially has thedesired property. Suppose it is nonstationary. Let s ∈ Z+ be the first t ∈ Z+
with xt 6= xt+1 . If xs < xs+1 , then by Lemma A.1, ∀t ≥ s, xt ≤ xt+1 ; thus∀t ∈ Z+ , xt ≤ xt+1 . The case xs > xs+1 is similar.

A.2

Proof of Lemma 3.3i
Let x ≥ 0. Let {xi }∞i=1 be a sequence in R+ such that limi↑∞ x = x. Fori ∈ N, let {cit , xit } be an optimal path from xi . Let y > x. Without loss ofgenerality, we may assume that ∀i ∈ N, xi < y. Clearly,(A.1)∀i ∈ N,cit ≤ f t (y).

Taking a subsequence, we may assume that ∀t ∈ Z+ , xit converges to somext ≥ 0 as i ↑ ∞. By the upper semicontinuity of f , ∀t ∈ Z+ ,(A.2)

0 ≤ lim cit = lim[f (xit ) − xit+1 ] ≤ f (xt ) − xt+1 ≡ ct .
i↑∞

27

The following argument is due to Mitra and Ray (1983). We include this proof here
since their paper, which is the working paper version of Mitra and Ray (1984), is not easily
available.

28

By the lower semicontinuity of r, ∀t ∈ Z+ , 0 ≤ limi↑∞ (xit+1 − r(xit )) ≤ xt+1 −r(xt ). Thus the path {ct , xt } is feasible from x. By Fatou’s lemma, (A.1),
and Assumption 2.5,
∞
X
(A.3)
δ t u(cit )lim v(xi ) = lim
i↑∞

i↑∞

(A.4)

≤

(A.5)

≤

∞
X
t=0
∞
X

t=0

δ t lim u(cit )i↑∞δ t u(ct ) ≤ v(x),t=0where the first inequality in (A.5) uses (A.2), and the second uses the feasibility of {ct , xt }. It follows that v is upper semicontinuous.

A.3

Proof of Proposition 3.1

Lemma A.2. Let {xt } be a convergent feasible capital path with x ≡ limt↑∞ xt ∈(0, ∞). Then ∃ν ∈ R, for large t,28u(0)< ν < v(xt ).
1−δ
Proof. Let z ∈ (r+ (x), x). This interval is nonempty by Assumption 2.3. Letc ∈ (0, x − z). Since limt↑∞ xt = x, for large t,
(A.6)

(A.7)

r(xt ) < z < z + c < xt+1 ≤ f (xt ).

Let t ∈ Z+ be large enough to satisfy
By Assumption 2.4, there is
P∞(A.7).
sa feasible path {ĉs , x̂s } from z with s=0 δ u(ĉs ) > −∞. Define {c̃s , x̃s } asfollows.
(A.8)

x̃0 = xt , x̃1 = z, c̃0 = f (xt ) − z,∀s ∈ N, x̃s+1 = x̂s , c̃s = ĉs−1 .

Then by (A.7), x̃1 ∈ (r(xt ), f (xt )) and c̃0 > c. Hence by the feasibility of{ĉs , x̂s } from x̃1 (= z), {c̃s , x̃s } is feasible from xt . Thus
(A.9)

v(xt ) ≥∞Xsδ u(c̃s ) > u(c) + δ

s=0

∞
X

δ s u(ĉs ) ≡ ν.

s=0

Since the last sum is finite and c > 0, (A.6) follows.
28“For large t” means “for all t sufficiently large.” Similar remarks apply to similar
expressions.

29Lemma A.3. Let {ct , xt } be an optimal path. If x ≡ limt↑∞ xt ∈ (0, ∞),then limt↑∞ ct > 0.
Proof. Note that limt↑∞ ct exists since {f (xt )} is monotone in t and thus
(A.10)

lim ct = lim[f (xt ) − xt+1 ] = lim f (xt ) − x.
t↑∞

t↑∞

t↑∞

Suppose limt↑∞ ct = 0. Note that
(A.11)

lim v(xt ) = lim
t↑∞

t↑∞

∞
X

δ i u(ct+i ) ≤

i=0

u(0)
,
1−δ

where the inequality holds by Fatou’s lemma since {ct } is bounded. But(A.11) contradicts Lemma A.2.
Lemma A.4. Let {xt } be an optimal capital path that is bounded. Supposex ≡ limt↑∞ xt > 0. Then limt↑∞ v(xt ) = v(x).
Proof. Since limt↑∞ v(xt ) ≤ v(x) by Lemma 3.3, it suffices to show
(A.12)

v(x) ≤ lim v(xt ).
t↑∞

Let {ct } be the associated consumption path. Let c = limt↑∞ ct . By Lemma
A.3, c > 0. Thus by Assumption 2.3 and (A.10), r+ (x) < x < limt↑∞ f (xt ).
Hence for large t, r(xt ) < x < f (xt ). Thus
(A.13)u(f (xt ) − x) + δv(x) ≤ u(ct ) + δv(xt+1 ).

Since limt↑∞ u(ct ) = limt↑∞ u(f (xt ) − x) = u(c) by (A.10) and continuity,
applying limt↑∞ to (A.13) yields (A.12).
Let us now prove Proposition 3.1. Let {ct , xt } be an optimal path that isbounded. Let x = limt↑∞ xt and c = limt↑∞ ct . By the upper semicontinuityof f , c + x ≤ f (x). Note from Assumption 2.3 that x ≥ r(x). If x = 0,then c = 0, so (c, x) is trivially a steady state. Suppose x > 0. For t ∈ Z+ ,we have v(xt ) = u(ct ) + δv(xt+1 ). By Lemma A.4 and the continuity of u,v(x) = u(c) + δv(x). It remains to verify c + x = f (x). But if c + x < f (x),then v(x) ≥ u(f (x) − x) + δv(x) > u(c) + δv(x) = v(x), a contradiction. It
follows that (c, x) is a steady state.
30

A.4

Proof of Proposition 3.2

Let k : R+ → R+ be an arbitrary selection from K; i.e., ∀x ≥ 0, k(x) ∈ K(x).
Lemma A.5. (i) k is nondecreasing. (ii) If K is multivalued at x > 0, thenk is discontinuous at x.
Proof. Lemma 3.1 implies (i). For (ii), suppose K is multivalued at x0 >0, i.e., ∃x1 ≥ 0, ∃y1 > x1 , {x1 , y1 } ⊂ K(x0 ). Then by Lemma 3.1, ∀x ∈[0, x0 ), k(x) ≤ x1 . Thus k− (x0 ) ≤ x1 . On the other hand, ∀x > x0 , k(x) ≥ y1 .
Thus k+ (x0 ) ≥ y1 > x1 ≥ k− (x0 ), i.e., k is discontinuous at x0 .
Let us now prove Proposition 3.2. By Lemma A.5(i), k is nondecreasing;thus it has only countably many discontinuities. Thus by Lemma A.5(ii), K
can be multivalued only at finitely many points.

A.5

Proof of Lemma 3.5

Lemma A.6. For any feasible path {ct , xt } with x0 > 0,
(A.14)∞X

δ t u(ct ) ≤t=0u((1 − δ)f (x0 ) + Γ),
1−δ

where Γ = supt∈Z+ Γ(xt+1 ). The inequality is strict if {ct } is not constant.
Proof. It follows from (3.6)–(3.8) that
(A.15)TXtδ ct ≤ f (x0 ) +

T −1
X

t=0

δ t Γ.

t=0

Hence(A.16)∞X

δ t ct ≤ f (x0 ) +

t=0

Γ
.
1−δ

Multiplying through by 1 − δ and recalling that u is increasing, we get
Ã
!
∞
X
¢
¡
(A.17)u (1 − δ)δ t ct ≤ u (1 − δ)f (x0 ) + Γ .
t=0Applying Jensen’s inequality to the left-hand side yields (A.14). Since u isstrictly concave, (A.14) holds with strict inequality if {ct } is not constant.
31

Lemma A.7. Let {ct , xt } be a nonstationary optimal path with x0 > 0 suchthat
(A.18)∀t ∈ Z+ ,Γ(xt+1 ) ≤ Γ(x0 ).

Then {ct } is not constant.29
Proof. Suppose {ct } is constant. By Lemma A.6 and (A.18),(A.19)u(c0 ) ≤ u((1 − δ)f (x0 ) + Γ(x0 )) = u(g(x0 )),where g is defined by (3.13). It follows that c0 ≤ g(x0 ). If c0 < g(x0 ), thenthis contradicts optimality since the stationary path from x0 is feasible (recall
Assumption 2.3(ii)). Suppose c0 = g(x0 ). Then x1 = f (x0 ) − g(x0 ) = x0 .
Since {ct } is constant, it follows that ∀t ∈ N, xt = x0 , contradicting thenonstationarity of {ct , xt }.
Let us now prove Lemma 3.5. Let {ct } be the associated consumption
path. Assume (A.18). Since {xt } is nonstationary, {ct } is not constant by
Lemma A.7. Thus by Lemma A.6 and (A.18),(A.20)∞
Xt=0

δ t u(ct ) <u((1 − δ)f (x0 ) + Γ(x0 ))u(g(x0 ))
=
.
1−δ
1−δ

This requires g(x0 ) > 0, which together with Assumption 2.3(ii) implies thatthe stationary path from x0 is feasible. But this contradicts the optimalityof {ct , xt } by (A.20) again.

A.6

Proof of Lemma 3.6

Proof. We only prove (3.12). The proof of (3.11) is similar. Suppose(A.21)

xt+1 > r(xt ),ct+1 > 0.

If f is not left continuous at xt+1 , then f−0 (xt+1 ) = ∞, so (3.12) triviallyfollows. Suppose f is left continuous at xt+1 . Consider increasing ct by ²,
decreasing xt+1 by ², and decreasing ct+1 by µ(²) ≡ f (xt+1 ) − f (xt+1 − ²),29

The nonstationarity of {ct , xt } only implies that {xt } is not constant. It is possiblethat {ct } is constant while {xt } is not constant. On the other hand, if {xt } is constant,{ct } is obviously constant.

32

while keeping the rest of the path unchanged. By (A.21), this perturbation
is feasible for small ² > 0. For small ² > 0, by optimality,(A.22)u(ct + ²) + δu(ct+1 − µ(²)) ≤ u(ct ) + δu(ct+1 ).

Rearranging and dividing through by ², we get(A.23)u(ct + ²) − u(ct )u(ct+1 ) − u(ct+1 − µ(²)) µ(²)
≤δ
.
²
µ(²)
²

By concavity the left-hand side is monotone in ², so is [u(ct+1 ) − u(ct+1 −µ(²))]/µ(²). Since lim²↓0 µ(²) = 0 by the left continuity of f at xt+1 , applying
lim²↓0 to both sides of (A.23) yields (3.12).

A.7

Proof of Lemma 3.7

If g(x) < 0, the stationary path from x is not feasible. Thus g(x) ≥ 0. To
verify g(x) > 0, it suffices
to show that there is a feasible path {c0t , x0t } fromPt
0
x such that c00 > 0 and ∞t=0 δ u(ct ) > −∞; for this implies that a feasible
path along which consumption is zero every period cannot be optimal.
Let c00 = (f (x) − r(x))/2 > 0 and x01 = f (x) − c00 > r(x), where the firstinequality holds by Assumption
By Assumption 2.4, there is a feasible
P 2.3.
i0path {ĉi , x̂i } from x01 with ∞δu(ĉi ) > −∞. For t ∈ N, let ct = ĉt−1 andi=0
x0t+1 = x̂t . Then clearly {c0t , x0t } is feasible and has the desired property. It
follows that g(x) > 0.
From Assumption 2.3 and this, r+ (x) < x < f (x). Hence (3.11) and(3.12) hold with ct = ct+1 = g(x) and xt+1 = x. Both inequalities in (3.15)
now follow.

A.8

Proof of Lemma 3.8

Lemma A.8. Let h : [a, b] → R be upper semicontinuous, where −∞ < a <b < ∞.
(i) If h is nondecreasing (nonincreasing), then h0− ≥ (≤) 0 on (a, b] and0h+ ≥ (≤) 0 on [a, b).30(ii) If h0+ ≥ 0 on [a, b), then h is nondecreasing on [a, b].
(iii) If h0− ≤ 0 on (a, b], then h is nonincreasing on [a, b].
30This part does not require upper semicontinuity.

33

Proof. See Giogi and Komlósi (1992, pp. 13–14).
Let us prove Lemma 3.8. Parts (i) and (iii) are immediate from (3.16).
Parts (ii) and (iv) hold by Lemma A.8 because Γ is upper semicontinuous,
Γ0+ = δf+0 − 1, and Γ0− = δf−0 − 1.

Appendix B

Proof of Proposition 4.2

Lemma B.1. Let {ct , xt } be an optimal path satisfying(B.1)(i) g(x0 ) > 0,(ii) ∃T ∈ N, ∀t ≤ T − 1,Γ(xt+1 ) ≤ Γ(x0 ).

Thenδ[v(xT +1 ) − v(x0 )] ≥ u0+ (g(x0 ))(xT +1 − x0 ).

(B.2)

Proof. By (3.6)–(3.8) and (B.1)(ii),
T
X

(B.3)

t

δ ct ≤ f (x0 ) +

T −1
X

t=0

δ t Γ(x0 ) − δ T xT +1

t=0

(B.4)

=

T
X

δ t [f (x0 ) − x0 ] + δ T x0 − δ T xT +1 .

t=0

Since g(x0 ) = f (x0 ) − x0 (recall (3.13)), it follows that
T
X

(B.5)

δ t (ct − g(x0 )) ≤ −δ T (xT +1 − x0 ).

t=0Define {x̃t } as follows: x̃t = x0 for t ≤ T +1 and x̃t = xt−T −1 for t ≥ T +2.
Clearly {x̃t } is feasible and {x̃T +1+i }∞i=0 (= {xt }) is optimal from x0 . Thus
(B.6)

T
X

t

δ u(g(x0 )) + δ

T +1v(x0 ) ≤t=0

TXδ t u(ct ) + δ T +1 v(xT +1 ).

t=0This rearranges to
(B.7)

−δ

T +1

[v(xT +1 ) − v(x0 )] ≤

T
X
t=0
T
X

δ t [u(ct ) − u(g(x0 ))]δ t u0+ (g(x0 ))(ct − g(x0 ))

(B.8)

≤

(B.9)

≤ −u0+ (g(x0 ))δ T (xT +1 − x0 ),

t=0

34

where (B.8) holds by concavity, and (B.9) by (B.5). Now (B.2) follows.
Lemma B.2. Under the hypotheses of Proposition 4.2, ∃i ∈ N, every optimal
capital path from zi is bounded above by z.
Proof. Suppose the conclusion is false. Then ∀i ∈ N, there is an optimal
path {cit , xit } from xi0 = zi such that limt↑∞ xit > z. Let z > 0 be such thatz
(B.10)
f (z) < z,< z.
1−δWithout loss of generality, assume ∀i ∈ N, xi0 < z. Note thatlim g(xi0 ) = 0,

(B.11)(B.12)i↑∞∀i ∈ N,g(xi0 ) = f (xi0 ) − xi0 > f (xi0 ) − xiti ≥ 0,

where ti is the first t with xit > xi0 .
Let i ∈ N. Let Ti be the first t ∈ Z+ with xit+1 > z. Note thatxi0 < z < xiTi ≤ z < xiTi +1 ≤ f (xiTi ) ≤ f (z).

(B.13)
By Lemma B.1,
(B.14)

δv(xiTi +1 ) ≥ u0+ (g(xi0 ))(xiTi +1 − xi0 ) + δv(xi0 ).

Recalling (B.13) and (B.12), we get(B.15)m ≡ δ maxy∈[z,f (z)]δu(g i )0iv(y) ≥ u+ (g )(z − z) +,1−δwhere g i = g(xi0 ). By concavity, u(g i ) ≥ u(z)−u0+ (g i )(z−g i ) ≥ u(z)−u0+ (g i )z.
Thusδ(B.16)∞ > m ≥ u0+ (g i )(z − z) +[u(z) − u0+ (g i )z]
1
−
δ
·
¸
z
δu(z)
0
i
(B.17)= u+ (g ) z −
+
.
1−δ
1−δ
Since the expression in square brackets is strictly positive by (B.10), andsince limi↑∞ u0+ (g i ) = ∞ by (B.11) and (4.4), the right-hand side of (B.17)
goes to ∞ as i ↑ ∞, a contradiction.
Let us now prove Proposition 4.2. By Lemma B.2, ∃i ∈ N, any optimal
capital path from zi is bounded above by z. Thus by (4.3) and Lemma 3.5,
any optimal capital path from zi is nonincreasing. Thus the conclusion holds
by Lemma 3.4.
35

Appendix C
C.1

Proofs of Section 6 Results

Proof of Remark 6.1

By concavity, ∀c, ĉ ≥ 0, u(c) ≤ u(ĉ) + u0+ (ĉ)(c − ĉ). Thus
(C.1)

∀c, ĉ ≥ 0,

u0+ (ĉ)ĉ ≤ u(ĉ) − u(c) + u0+ (ĉ)c.

Since u is bounded above, limĉ↑∞ u0+ (ĉ) = 0. It follows that(C.2)∀c ≥ 0,lim u0+ (ĉ)ĉ ≤ lim u(ĉ) − u(c).

ĉ↑∞

ĉ↑∞

Applying limc↑∞ yields (6.5).

C.2

Proof of Proposition 6.2

Lemma C.1. Suppose ∃z > 0, there exists a sequence {zi }∞i=1 in (z, ∞)satisfying (6.3). Then ∃θ > 0,
(C.3)

∃i ∈ N, ∀i ≥ i,zi ≤ θg(zi ).

Proof. By (6.3), ∀i ∈ N, δf (zi ) − zi ≥ δf (z) − z; equivalently,(C.4)∀i ∈ N,f (zi ) − zi ≥ f (z) −z+ µzi ,
δ

where µ = 1/δ − 1. From (C.4), ∀i ∈ N, g(zi )/zi ≥ [f (z) − z/δ]/zi + µ. Let
θ > 1/µ. Then since limi↑∞ zi = ∞, g(zi )/zi ≥ 1/θ for large i, i.e., (C.3)
holds.
Lemma C.2. Under the hypotheses of Proposition 6.2, ∃i ∈ N, any optimal
capital path from zi is bounded below by z.
Proof. Suppose the conclusion is false. Then ∀i ∈ N, there is an optimal
path {cit , xit } from xi0 = zi such that limt↑∞ xit < z. Let θ > 0 and i ∈ N be
as given by Lemma C.1. Without loss of generality, assume i = 1. Then
(C.5)

∀i ∈ N,xi0 < θg(xi0 ).

Let i ∈ N. Let Ti be the first t ∈ Z+ with xit+1 < z. By Lemmas B.1 and
C.1,
(C.6)

δ[v(xi0 ) − v(xiTi +1 )] ≤ u0+ (g i )(xi0 − xiTi +1 ) ≤ u0+ (g i )θg i ,36where g i = g(xi0 ). Since the stationary path from xi0 is feasible by (C.5),·¸u(g i )
(C.7)
δ− max v(y) ≤ θu0+ (g i )g i .
1 − δ y∈[0,z]Suppose u is bounded above. Then by (C.5), the left-hand side tends to·¸supc≥0 u(c)(C.8)δ− max v(y) > 0y∈[0,z]1−δas i ↑ ∞. On the other hand, by (C.5) and Remark 6.1, the right-hand sideof (C.7) goes to zero as i ↑ ∞, a contradiction.
Now suppose u is unbounded above. Then by (C.5), the left-hand sideof (C.7) goes to ∞ as i ↑ ∞, while by (6.4) the right-hand side is uniformly
bounded above for large i, again a contradiction.
Let us now prove Proposition 6.2. By Lemma C.2, ∃i ∈ N, any optimal
capital path from zi is bounded below by z. Thus by (6.3) and Lemma 3.5,
any optimal capital path from zi is nondecreasing. Thus the conclusion holds
by Lemma 3.4.

Appendix D
D.1

Proofs of Section 8 Results

Preliminary Lemmas

Lemma D.1. (i) ∀x > 0, ∀x ≥ x, inf x∈[x,x] (x − r(x)) > 0. (ii) ∀x ∈(0, x∗ ), ∀x ∈ [x, x∗ ] ∩ [x, ∞),31 inf x∈[x,x] g(x) > 0.
Proof. We prove (ii) first. Suppose inf x∈[x,x] g(x) = 0. Then there exists a
sequence {xi } ⊂ [x, x] such that limi↑∞ g(xi ) = 0. Taking a subsequence,
we may assume that {xi } converges to some y ∈ [x, x]. Now we obtainthe following contradiction: 0 = limi↑∞ g(xi ) ≥ g− (y) > 0, where the lastinequality holds by Assumption 8.2. To prove (i), replace g(x) by (x − r(x))
and note from Assumption 2.3(ii) that y − r+ (y) > 0.
Lemma D.2. Let y ∈ R++ be such that y ≤ x∗ . Then ∀x > 0, there exists
a feasible path {ct , xt } from x such that
(D.1)
31

∃T ∈ Z+ ,∀t ≤ T, ct > 0,∀t ≥ T + 1, ct = g(y), xt = y.

If x∗ < ∞, then x ≤ x∗ ; if x∗ = ∞, then x < x∗ .

37

Proof. Suppose x ≥ y first. We construct {ct , xt } recursively as follows:(D.2)

x0 = x,∀t ∈ Z+ , xt+1 = max{r(xt ), y}, ct = f (xt ) − xt+1 .

Let a = inf z∈[y,x] (z −r(z)). By Lemma D.1, a > 0. If ∃t ∈ Z+ , r(xt ) > y, thenxt+1 − xt = r(xt ) − xt ≤ −a. Thus the inequality r(xt ) > y holds only finitely
many times or never holds. Let T be the first t ∈ Z+ such that r(xt ) ≤ y. If
T ≥ 1, then for t ≤ T −1, ct = f (xt )−r(xt ) > 0 by Assumption 2.3. We havexT +1 = y and cT = f (xT ) − y ≥ f (y) − y = g(y) ≥ g− (y) > 0, where the lastinequality holds by Assumption 8.2 since y ≤ x∗ . For t ≥ T + 1, ct = g(y).
It follows that {ct , xt } is feasible and satisfies (D.1).
Now suppose x < y. Let c ≡ inf z∈[x,y] g(z). Since y ≤ x∗ , c > 0 by Lemma
D.1. Let c ∈ (0, c). We construct {ct , xt } recursively as follows:(D.3)

x0 = x,∀t ∈ Z+ , xt+1 = min{f (xt ) − c, y}, ct = f (xt ) − xt+1 .

Note that(D.4)

ct = max{c, f (xt ) − y} ≥ c.

∀t ∈ Z+ ,If ∃t ∈ Z+ , f (xt )−c < y, then xt+1 −xt = f (xt )−c−xt = g(xt )−c ≥ c−c > 0.
Thus the inequality f (xt ) − c < y holds only finitely many times or neverholds. Since {xt } is nondecreasing, the irreversibility constraint (2.4) always
holds. It follows that {ct , xt } is feasible and satisfies (D.1).

D.2

Proof of Lemma 8.1

Lemma D.3. Let y ∈ R++ be such that y ≤ x∗ . Thenlim sup g(xδt+1 ) ≥ g(y).

(D.5)

Z

δ↑1 t∈ +Proof. Let η > 0. Let {c̃t , x̃t } be any feasible path from x0 with
(D.6)

sup g(x̃t+1 ) ≤ g(y) − η.

Z

t∈ +

We show that {c̃t , x̃t } is suboptimal for δ close to one. Since ∀x ≥ 0, Γ(x) =δf (x) − x ≤ f (x) − x = g(x), it follows by Lemma A.6 that(D.7)

∀δ ∈ (0, 1),(1 − δ)∞X

δ t u(c̃t ) ≤ u((1 − δ)f (x0 ) + g(y) − η).

t=0

38

By Lemma D.2, there is a feasible path {ct , xt } from x0 satisfying (D.1).
Hence
(D.8)

∀δ ∈ (0, 1),(1 − δ)TX

δ t u(ct ) + δ T +1 u(g(y)) ≤ (1 − δ)v(x0 ).

t=0Note that T does not depend on δ and that v implicitly depends on δ. As
δ ↑ 1, the left-hand side of (D.8) goes to u(g(y)), while the right-hand sideof (D.7) goes to u(g(y) − η) < u(g(y)). It follows that for δ close to one, any
feasible path {c̃t , x̃t } satisfying (D.6) is suboptimal.
Since {xδt } is optimal for each δ ∈ (0, 1), we must have supt∈Z+ g(xδt+1 ) >g(y) − η for δ close to one. Since this is true for any η > 0, (D.5) follows.
Now we prove Lemma 8.1. By the definition of g ∗ , limδ↑1 supt∈Z+ g(xδt+1 ) ≤g ∗ . Thus it suffices to show
(D.9)

lim sup g(xδt+1 ) ≥ g ∗ .

Z

δ↑1 t∈ +If x∗ < ∞, (D.9) follows from Lemma D.3 with y = x∗ . If x∗ = ∞, applying
supy∈R++ to (D.5) yields (D.9).

D.3

Proof of Proposition 8.1

Lemma D.4. Suppose ∃x̂ > 0, g(x̂) = g ∗ . Then for any δ ∈ (0, 1), any
optimal capital path from x̂ is nonincreasing.
Proof. Note that ∀x ≥ x̂, Γ(x) = δf (x) − x = δ(f (x) − x) − (1 − δ)x ≤δg(x̂) − (1 − δ)x̂ = Γ(x̂). Thus the conclusion holds by Lemma 3.5.
Lemma D.5. If x0 ≤ x∗ , then limδ↑1 limt↑∞ xδt ≤ x∗ .
Proof. The inequality is trivial if x∗ = ∞. If x∗ < ∞, it holds by Lemmas
3.4 and D.4 with x̂ = x∗ .
Lemma D.6. If x0 < x∗ , then (8.2) holds.
Proof. By Lemma D.5, it suffices to show limδ↑1 limt↑∞ xδt ≥ x∗ . Suppose
this inequality does not hold; i.e., ∃x ∈ (x0 , x∗ ), there is a sequence {δi }∞
i=1
in (0, 1) with limi↑∞ δi = 1 such that ∀i ∈ N, limt↑∞ xδt i ≤ x. By monotonicity,ii∀i ∈ N, ∀t ∈ Z+ , xδt+1≤ x. Thus ∀i ∈ N, supt∈Z+ g(xδt+1) ≤ maxy∈[0,x] g(y) <∗g , contradicting Lemma 8.1.
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Lemma D.7. Let y ∈ (0, x∗ ). Then for δ close to one, every optimal capital
path from y is nondecreasing.
Proof. Note that Lemma D.6 holds for any x0 ∈ (0, x∗ ) and any set of optimal
capital paths {{xδt }}δ∈(0,1) from x0 such that each {xδt } is optimal when the
discount factor equals δ. Thus the conclusion follows from monotonicity andLemma D.6 with x0 = y.
Lemma D.8. If x∗ ≤ x0 , then x∗ ≤ limδ↑1 limt↑∞ xδt .
Proof. By Lemmas D.7 and 3.4, ∀y ∈ (0, x∗ ), y ≤ limδ↑1 limt↑∞ xδt . Lettingy ↑ x∗ gives the desired inequality.
To complete the proof of Proposition 8.1, suppose x0 ≤ x∗ . If x0 < x∗ ,then (8.2) holds by Lemma D.6. Suppose x0 = x∗ . Then by Lemmas 3.4
and D.4 with x̂ = x∗ , limδ↑1 limt↑∞ xδt ≤ x∗ . This together with Lemma D.8
shows (8.2).

D.4

Proof of Proposition 8.2

Throughout this proof, we assume (8.4). Since Proposition 8.1 covers thecase x0 ≤ x∗ , we also assume x∗ < ∞ and x∗ < x0 .
Lemma D.9. Let z ∈ (x∗ , x0 ]. Then for any δ ∈ (0, 1), there exists no
optimal capital path from z that is nondecreasing.
Proof. Let {c0t , x0t } be a feasible path from z such that {x0t } is nondecreasing.
Then(D.10)∀t ∈ Z+ ,c0t = f (x0t ) − x0t+1 ≤ f (x0t ) − x0t = g(x0t ).

Define {x̃t , c̃t } as follows.
(D.11)

x̃0 = z, c̃0 = f (z) − x∗ ,∀t ≥ 1, x̃t = x∗ , c̃t = g(x∗ ).

By (8.4), x∗ is feasible from z. Since x01 > x∗ ,(D.12)c̃0 > f (z) − x01 = c00 ≥ 0.

Thus {c̃t , x̃t } is feasible. Since x∗ < z, c̃0 > g(z). Recalling (D.10) and(D.12), we see that c00 < c̃0 and ∀t ∈ N, c0t ≤ c̃t . Since {c̃t , x̃t } was arbitrary, itfollows that any nondecreasing feasible capital path from z is suboptimal.
Let us now prove Proposition 8.2. By Lemma D.9, ∀δ ∈ (0, 1), limt↑∞ xδt ≤x . Thus limδ↑1 limt↑∞ xδt ≤ x∗ . This together with Lemma D.8 shows (8.2).
∗

40

D.5

Proof of Proposition 8.3

If x0 ≤ x∗ , (8.2) holds by Proposition 8.1. Suppose x0 > x∗ . By Lemma
D.8, it suffices to verify limδ↑1 limt↑∞ xδt ≤ x∗ . Suppose limδ↑1 limt↑∞ xδt > x∗ .
Then ∃x ∈ (x∗ , x0 ), there is a sequence {δi }∞i=1 in (0, 1) with limi↑∞ δi = 1
such that ∀i ∈ N, limt↑∞ xδt i ≥ x. By monotonicity, ∀i ∈ N, ∀t ∈ Z+ , xδt i ≥ x.
i
) ≤ supx≥x g(x) < g ∗ , where the last inequality
Hence ∀i ∈ N, supt∈Z+ g(xδt+1
holds by (8.6) and (8.7). But this contradicts Lemma 8.1.

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