The stochastic turnpike property without uniformity in convex aggregate growth models

Journal of Economic Dynamics & Control 27 (2003) 1289–1315www.elsevier.com/locate/econbase

The stochastic turnpike property withoutuniformity in convex aggregate growth models

Sumit Joshi∗

Department of Economics, 624 Funger Hall, The George Washington University, 2201 G Street N.W.,Washington DC 20052, USA

Abstract

The proof of the “late” turnpike property in optimal growth theory requires constructing abounded value-loss process that records a strictly positive value-loss when paths of capital accu-mulation from di0erent initial stocks diverge. Uniformity assumptions strengthen this sensitivityby ensuring that value-loss is independent of time and state of environment in which the di-vergence occurs, and are acknowledged as strong restrictions on the model. This paper arguesthat uncertainty can obviate the need for uniformity. The multiplicity of states a0orded by astochastic framework permits constructing a value-loss process over an “extended” time-line thatis a martingale; if capital stocks diverge, then the martingale registers an upcrossing across aband of uniform width on its extended time-line, thereby giving uniform value-loss. The Mar-tingale Upcrossing theorem and the 5rst Borel–Cantelli lemma then clinch the turnpike property.? 2002 Elsevier Science B.V. All rights reserved.

JEL classi)cation: C60; D90

Keywords: Turnpike; Martingales; Stochastic; Optimal growth; Uniformity assumptions

1. Introduction

Optimal growth theory, in Ramsey-type normative models with convex preferencesand technology, has identi5ed an important stability property referred to variously as thelate or twisted turnpike. It asserts that two in5nite horizon optimal programs of capitalaccumulation from distinct initial stocks converge (almost surely) in a suitable metric.A critical input in the derivation of this property is a strong uniformity assumption(Brock and Majumdar, 1978; Brock and Scheinkman, 1975; Chang, 1982; F@ollmer and

∗ Tel.: +1-202-994-6154; fax: +1-202-994-6147.E-mail address: [email protected] (S. Joshi).

0165-1889/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S0165 -1889(02)00060 -X

1290 S. Joshi / Journal of Economic Dynamics & Control 27 (2003) 1289–1315

Majumdar, 1978; Joshi, 1997; Majumdar and Zilcha, 1987; McKenzie, 1976). 1 It isgenerally invoked in both discounted and undiscounted frameworks which eschew thetime-stationarity restriction on preferences, technology and the evolution of the stochas-tic environment. It ensures independence from both time and state (of the stochasticenvironment) of the sensitivity of a key process—the value-loss process—which tracksthe divergence in optimal programs from di0erent initial stocks. The primary objectiveof this paper is to demonstrate, within the context of a convex aggregate stochasticgrowth model, that the late turnpike property can be derived without the encumbranceof a uniformity restriction.

A general derivation of the late turnpike property, without any concession to unifor-mity, is warranted by the preeminent position this result occupies in the various strandsof the growth literature. To substantiate, we o0er a brief review.

(i) Optimal Growth Theory: In normative models, the late turnpike property iscentral by virtue of asserting the global asymptotic stability of optimal programs. 2

Under the assumption of time-stationary preferences, technology and the stochasticenvironment, and with the added restriction of no discounting, it has been shown inBrock and Mirman (1973), Dana (1974) and Mirman and Zilcha (1977) that all goodprograms converge in an appropriate topology to the golden rule (or optimal stationary)program. 3 In the discounted context, as exempli5ed in Brock and Mirman (1972)and Mirman and Zilcha (1975), time-stationarity yields convergence in distributionof optimal programs from distinct initial stocks to the modi5ed golden rule program.In non-stationary models, the uniformity restriction has been critical in establishingconvergence in probability (Brock and Majumdar, 1978) or the stronger property ofalmost sure convergence (Chang, 1982; F@ollmer and Majumdar, 1978; Joshi, 1997;Majumdar and Zilcha, 1987).

(ii) Patience and chaos: In optimal growth theory, the late turnpike property rulesout the possibility of optimal programs exhibiting chaotic dynamics. In reduced formmodels with two or more sectors, the existence of complicated dynamics has generallybeen obtained for low values of the discount factor (Boldrin and Montrucchio, 1986;Mitra, 1998; Nishimura et al., 1994; Sorger, 1992). Some additional features of thesemodels include felicity functions that are concave but not strictly concave, and opti-mal programs that are possibly non-interior. In an aggregate growth model, Majumdarand Mitra (1994) have shown the existence of complicated dynamics when the felicityfunction depends on both consumption and the capital stock. This raises the followingissue: in aggregate growth models with strictly concave felicity functions that de-pend on consumption alone, and in which optimal programs are interior, does the late

1 Optimal growth theory has identi5ed two other turnpike properties—the early and the middle—of whichthe latter also relies on a uniformity assumption (for instance, McKenzie, 1976).

2 A comprehensive review of the deterministic literature on turnpike theory, along with a discussion ofthe role of uniformity, is provided in McKenzie (1998). The review, however, does not cover the stochasticcase. For this reason, we have mostly limited the discussion here to optimal growth under uncertainty.

3 Good programs, 5rst identi5ed in Gale (1967), are feasible programs which do not perform in5nitelyworse in utility terms than the golden rule program and which include non-stationary programs that areoptimal in terms of catching-up or overtaking of partial utility sums.

S. Joshi / Journal of Economic Dynamics & Control 27 (2003) 1289–1315 1291

turnpike property obtain for all values of the discount factor? Majumdar and Zilcha(1987) answered this in the aNrmative under a uniform lower bound on the degreeof concavity of the production function and a particular relative distance function tomeasure the divergence in capital stocks. This paper attempts to extend the result tothe case with no uniformity restrictions and no restriction to a particular metric.

(iii) Competitive equilibrium and the turnpike property: In contrast to normativemodels of optimal growth, the deterministic analysis of Bewley (1982), and its stochas-tic extension by Marimon (1989), consider a positive model of equilibrium growthwith 5nitely many (in5nitely-lived) consumers and perfectly competitive 5rms. A com-petitive equilibrium in their model corresponds to the solution of an optimal growthproblem where the social welfare function is a weighted sum of the consumers’ util-ity functions, the weights being the inverses of the marginal utility of expendituresin equilibrium. In particular, the stationary competitive equilibrium with transfer pay-ments corresponds to the (modi5ed) golden rule program with respect to this socialwelfare function. The late turnpike property highlights the global asymptotic stabilityof interior competitive equilibria by showing that they converge to the stationary com-petitive equilibrium with transfer payments for suNciently high values of the discountfactor.

(iv) Imperfectly competitive equilibria of endogenous growth theory: The volumi-nous literature on endogenous growth, following the seminal contributions of Lucas(1988) and Romer (1986), has considered the dynamic general equilibria of imper-fectly competitive markets characterized by sustained growth at endogenously deter-mined levels. The late turnpike property addresses the issue of whether the time pathof an imperfectly competitive equilibrium converges to the path of balanced growth.Another facet of this literature has been to explain the di0erence in growth rates ofdeveloping and developed economies (Barro and Sala-i-Martin, 1995). Identifying con-ditions under which long run convergence to the same growth rate does (or does not)obtain bears formal similarity to the late turnpike property.

The pervasive nature of the late turnpike property in economic dynamics, as attestedby the above review, provides a compelling reason to re-examine this issue under thegreatest generality. In this regard, uniformity assumptions pose a strong restriction onoptimal growth models. This paper demonstrates that the potential to exploit multiplestates of the environment a0orded by a stochastic paradigm can eradicate the need forany strong uniformity restriction. Towards this end, we organize the paper as follows.The convex optimal growth model is presented in Section 2. The nature of the unifor-mity assumption is examined in Section 3. A non-technical description of our methodis provided in Section 4. The mathematical underpinnings of this method, along witha formal statement of results, is available in Section 5. Applications of our method arediscussed in Section 6. All proofs are relegated to Appendix A. Our conclusions arecontained in Section 7.

2. The aggregate growth model

Our description of the aggregate growth model generalizes Brock and Mirman (1972),Majumdar and Zilcha (1987), and Mirman and Zilcha (1977) by allowing


non-stationarities in preferences, technology and the evolution of the stochastic envi-ronment. From now on, we will let I+={0; 1; 2; : : :}, and let 〈ht〉 denote the sequence,h0; h1; h2; : : : ; ht ; : : : ; t ∈I+.

The possible states, !t , of the environment at any date t ∈I+ is given by an un-countably in5nite set, �t , that is a compact metric space in an appropriate topology.Let Et denote the Borel �-5eld of subsets of �t generated by the open sets in thistopology. By assumption, �t satis5es the second countability axiom (Munkres, 1975,Section 4–1), i.e. �t has a countable basis, Ht = 〈Hn;t〉, n∈I+, for its topology. Itfollows that Et = �(Ht).

Let � = ×∞t=0�t denote the set of all sequences, ! = 〈!t〉, !t ∈�t , and F denote

the �-5eld on � generated by open sets in the product topology on �. The stochasticenvironment is represented by the probability space (�;F; ), where is a probabilitymeasure on �. Let 〈Ft〉 denote the 5ltration on �. Ft is the sub-�-5eld on � inducedby partial history till date t, i.e. Ft = �(E0 × E1 × · · · × Et × �t+1 × �t+2 × · · ·). 4Technology is described by a sequence of possibly time-varying production functions

〈ft〉, ft :R+ × �t+1 → R+, where for each t ∈I+:

(A.1) ft is continuous on R+ × �t+1.(A.2) For each !t+1 ∈�t+1, ft(0; !t+1) = 0, ft(k; !t+1) is strictly concave for k¿ 0,

and f′t(k; !t+1) ≡ 9ft(k; !t+1)=9k exists and is strictly positive for k ¿ 0.

Preferences are represented by a sequence of possibly time-varying felicity functions〈ut〉, ut :R+ → R, such that for each t ∈I+:

(A.3) ut(c) is continuous and strictly concave for c¿ 0.(A.4) u′t(c) exists and is strictly positive for c¿ 0 with u′t(c) ↑ +∞ as c ↓ 0.

The initial stock, s, can be random and is drawn from the set L ≡ L∞(�;F0; ;R++)of all essentially bounded F0-measurable functions from � into R++. A real-valued 〈Ft〉-adapted process, 〈(kt ; ct)〉, is a feasible program from s∈L if withprobability 1:

k0 + c06 s; (1)

kt+1 + ct+16ft(kt ; !t+1); t ∈I+; (2)

kt¿ 0; ct¿ 0; t ∈I+: (3)

The set of all feasible programs from a given initial stock s∈L is denoted by �(s).A program, 〈(kst ; cst )〉 ∈�(s), is optimal if for any other program, 〈(kt ; ct)〉 ∈�(s):

lim supN→∞

N∑t=0

E[ut(ct)− ut(cst )]6 0:

Existence of an optimal program follows under a joint boundedness restriction on pref-erences and technology (Majumdar and Zilcha, 1987, Theorem 1; Mitra and Nyarko,

4 Note that E0 ×E1 × · · ·×Et ×�t+1 ×�t+2 × · · · denotes the collection of cylindrical sets of the form,A0 × A1 × · · · × At × �t+1 × �t+2 × · · ·, where Ai ∈Ei for i = 0; 1; : : : ; t.


1991, Condition E). From (A.2) and (A.3), the optimal program in �(s) is unique. Fur-ther, from (A.4), it is interior and (in addition to appropriate transversality conditionsfor discounted and undiscounted models) satis5es the stochastic Euler equations:

u′t(cst ) = E[u′t+1(c

st+1)f

′t(k

st ; !t+1) ‖Ft] -a:s: t ∈I+: (4)

De5ne the competitive price process, 〈pst 〉, associated with 〈(kst ; cst )〉 as ps

t = u′t(cst );

t ∈I+. Further, let

�s0 = 1; �st+1 =t∏

i=0

f′i(k

si ; !i+1); t ∈I+;

where �st is Ft-measurable and strictly positive (almost surely) from the interiority ofoptimal programs. Multiplying both sides of (4) by �st , and using its Ft-measurability,the Euler equations can be rewritten succinctly as

pst �

st = E[ps

t+1�st+1 ‖Ft] -a:s: t ∈I+: (5)

Therefore, the process 〈pst �

st 〉 is a 〈Ft〉-martingale with E[ps

t �st ]=E[ps

0�s0]=E[u′0(c

s0)];

t ∈I+. We can interpret 〈pst �

st 〉 as the sequence of valuations of future increments of

a unit of capital invested optimally in time period 0 from the initial stock s.Consider an initial stock y∈L, y �= s. Let � ≡ E[u′0(c

s0)−u′0(c

y0 )]. An examination of

(5) shows that shifting to the process, 〈pst �

st +�〉, leaves the Euler equations una0ected.

However, given that the two sides of (5) are strictly positive (almost surely), thisoperation conveniently bounds each side from below by �. Letting Mi

t ≡ pit�

it + �,

i= s; y, the Euler equations become Mit =E[Mi

t+1 ‖Ft] almost surely for each t ∈I+.

3. The uniformity assumption

In this section, to illuminate the precise nature of the uniformity assumption, wereview the methodology that underlies the stochastic late turnpike property in non-stationary models.

The 5rst step entails the construction of a value-loss process, 〈Vt〉, by utilizing thecompetitive conditions which characterize an optimal program (the “turnpike”). Theseare the Euler conditions in aggregate models (Joshi, 1997; Majumdar and Zilcha, 1987)and the reduced utility maximization conditions in multisector models (Brock andMajumdar, 1978; Chang, 1982; F@ollmer and Majumdar, 1978). The process 〈Vt〉 hasthe convenient property of being either a martingale (Majumdar and Zilcha, 1987,Eqs. (6.20) and (6.21)), a submartingale (Brock and Majumdar, 1978) or a supermartin-gale (Joshi, 1997; Marimon, 1989) thereby permitting a passage to the rich theory ofmartingales. The second step is to ensure that 〈Vt〉 is uniformly bounded in expectation.This is achieved in aggregate models through interiority of optimal programs, and inmultisector models through the transversality (bounded capital value) condition. Thethird step is to endow 〈Vt〉 with the sensitivity to record a strictly positive di0erence—called value-loss—when optimal programs from di0erent initial stocks diverge by some


Fig. 1. Value-loss process exhibiting non-uniform sensitivity to a divergence in capital stocks.

pre-speci5ed critical amount. This value-loss will in general depend on the time period,t, and the state of environment, !, in which the divergence occurs. It is shown inFig. 1 where the graph on top tracks the time path of ‖kst (!)−kyt (!)‖, where ‖·‖ is anydistance function, and s; y∈L, s¡y with probability 1. It permits us to observe thosetime periods when the divergence in capital stocks exceeds some arbitrary constant�¿ 0. The graph below shows a value-loss process which exhibits sensitivity to an�-divergence in capital stocks.

At this stage, under Assumptions (A.1)–(A.4) only, F@ollmer and Majumdar (1978,Theorem 3.1) note that the following weak version of the late turnpike property canbe obtained: for any arbitrary constant �¿ 0, 〈Vt〉 will almost surely leave a set onwhich value-loss exceeds � in )nite time. This is a consequence of the uniform boundon the expectation of 〈Vt〉 and the Martingale Convergence theorem (Billingsley, 1979,Theorem 35.4). Coupled with the sensitivity property, it implies that capital stocks


cannot diverge for in5nitely many periods by an amount that causes value-loss toexceed �. It is a weak characterization, however, because convergence is not implied:capital stocks can diverge for in)nitely many periods by any amount that causesvalue-loss to be ¡�.

This is precisely the point where the uniformity assumption enters into the analysisto add to Assumptions (A.1)–(A.4) and to force the convergence of optimal programsfrom di0erent initial stocks by strengthening the sensitivity of 〈Vt〉. In particular, forany �¿ 0, if capital stocks diverge by more than � in period t and state of environment!, then uniformity dictates that 〈Vt〉 record a value-loss of at least (�)¿ 0, where (�)is independent of the tuple (t; !). In other words, value-loss should be uniformly sensi-tive to a critical divergence in capital stocks. Since 〈Vt〉 converges almost surely fromthe Martingale Convergence theorem, the (contrapositive of the) uniformity assump-tion ensures that capital stocks generated by optimal programs from di0erent initialstocks converge too. This is how the twin properties of uniformly bounded expecta-tion (which allows an application of the Martingale Convergence theorem to 〈Vt〉) andthe uniformity assumption (which ties the convergence of optimal programs from dis-tinct initial stocks to convergence of 〈Vt〉) act in conjunction to yield the late turnpikeproperty.

While the uniform bound on expectation is relatively easier to impose, ensuringuniform sensitivity poses the diNcult problem of identifying the precise restrictions onthe fundamentals of the model—the felicity functions, the production technology, andthe discount factor—that permit value-loss to record a uniform jump in those statesof the environment where capital stocks diverge while being uniformly bounded onaverage across all states. The literature has addressed the problem in two ways whichmay be classi5ed as direct and indirect. The former method directly imposes uniformsensitivity on the value-loss process by appealing to appropriate curvature restrictionson technology and preferences without making them explicit (Brock and Majumdar,1978; Chang, 1982; F@ollmer and Majumdar, 1978; Joshi, 1997; McKenzie, 1976). Thelatter method proves uniform sensitivity from 5rst principles by explicitly imposingbounds on the degree of concavity of the felicity functions (Brock and Scheinkman,1975; Guerrero-Luchtenberg, 1998; McKenzie, 1976, in the multisector case) or theproduction functions (Majumdar and Zilcha, 1987, in the aggregate case).

In either their direct or indirect guise, uniformity assumptions are acknowledged asconstituting strong restrictions on the growth model. In the direct approach, they havebeen characterized as “strong uniformity” (Brock and Majumdar, 1978, Assumption(A.4)) or as a “strong value-loss assumption” (F@ollmer and Majumdar, 1978, p. 281).Further, as noted in McKenzie (1976), they are diNcult to extend to discounted modelswithout additional restrictions on the discount factor. In the indirect approach, they pre-clude growth models with time-varying preferences and technology that asymptoticallyapproach the linear case. For instance, consider the sequence of functions 〈ht :R+ →R+〉, ht(x) = x1−1=(t+2). This sequence of strictly increasing, strictly concave functionsis precluded from describing felicity or production functions. The degree of concavityof ht , given by −xh′′t (x)=h

′t(x), is equal to 1=(t + 2), and approaches zero as t → ∞.

The uniformity assumption, however, requires that the degree of concavity of each htbe uniformly bounded from below by a strictly positive constant.


4. Methodology: a descriptive view

For an informal description of our method, consider Fig. 2.The graph on top once again tracks the time path of ‖kst (!)−kyt (!)‖ permitting us to

observe those time periods when the divergence in capital stocks exceeds �¿ 0 (in ourexample, t = 1; 4; 5). The graph for 〈Vt〉 shows that for these time periods, value-loss

Fig. 2. Value-loss process exhibiting uniform sensitivity to a divergence in capital stocks through upcrossingsof the interval [0; 2�].


exceeds the strictly positive constant, 2�¿ 0. 5 In the other time periods, when thereis less than �-divergence (in our example, t = 0; 2; 3), value-loss is non-positive. Thedi0erence from Fig. 1 is that we now extend the time-line by including the mid-pointbetween periods t and t + 1 for t = 0; 1; 2; : : : ; and appropriately choose value-loss,Vt+1=2, at this intermediate point so that uniform sensitivity follows. We let "=2t; t=0; 12 ; 1; 1

12 ; 2; 2

12 ; : : : ; denote this extended time-line and let 〈V"〉 denote the value-loss

process over ". Since we will have occasion to switch between our original and newtime-lines, it is useful to note that even-valued " correspond to the original time-linewhile odd-valued " refer to the 5ctitiously introduced “intermediate” points on thetime-line.

We now observe that if we have more than �-divergence in capital stocks, for instancein t = 1, then 〈V"〉 moves upwards across a band of uniform width equal to 2� overthe time interval t = 1

2 to t = 1 (or, in the lexicon of martingale theory, it registersan upcrossing 6 over the range [0; 2�] during this time interval). This upcrossing, byconstruction, is immediately followed by a downcrossing from t=1 to t=11

2 . If there isless than �-divergence in any period, for instance in t=2, then V" remains non-positiveand no upcrossing is registered across [0; 2�].

In implementing the above idea, we have to ensure that 〈V"〉 is a martingale. Themultiplicity of states a0orded by a stochastic paradigm permits us the facility to induceupcrossings across [0; 2�] of 〈V"〉 for those realizations where ‖kst (!) − kyt (!)‖¿�while simultaneously manipulating those states where ‖kst (!) − kyt (!)‖ 6 � in orderto endow 〈V"〉 with a martingale structure. 7 The construction has the Vavor of anoptimal stopping argument. Suppose 〈(kyt ; cyt )〉 is the “turnpike”, and the planner (orrepresentative consumer) is o0 the turnpike on the path 〈(kst ; cst )〉. Staying o0 theturnpike subjects the planner to value-loss and, therefore, the objective is to switchfrom 〈(kst ; cst )〉 to the turnpike. However, such a switch involves an adjustment costwhich depends on the distance of 〈(kst ; cst )〉 from the turnpike. Hence, given a realization!, the planner would consider “stopping” value-loss and e0ecting a transition to theturnpike in those time periods t(!) in which ‖kst(!)(!)− kyt(!)(!)‖6 � and not when‖kst(!)(!)− kyt(!)(!)‖¿�. Given the adjustment cost, in an optimal stopping problemthe objective is to determine the optimal stopping time to e0ect the switch to theturnpike to minimize total cost (value-loss plus the adjustment cost). Our constructionconsiders the reverse problem: given any t ∈I+ and realization !, what is the size ofthe adjustment cost which just makes it optimal for the planner to “stop” value-loss attime t if ‖kst (!)− kyt (!)‖6 � but not on the complement of this set. The value-lossprocess is shown to imbibe a martingale structure as a consequence of such a choice ofadjustment costs. This argument of course presupposes that in each period t ∈I+ there

5 In convex growth models, monotonicity of the optimal consumption program ensures that � ≡ E[u′0(cs0)−

u′0(cy0 )]¿ 0 if y¿ s almost surely.

6 In martingale theory, an upcrossing across an interval can take place over more than one time periodwhile in our analysis it takes place in one time period. We use the terminology of an upcrossing because ofthe formal analogy between our resolution of the turnpike property and the Martingale Upcrossing theorem(Billingsley, 1979, Theorem 35.3).

7 Note that optimal programs are de5ned on the original time-line while the value-loss process is con-structed over the extended time-line.


will exist a set of states of strictly positive measure on which ‖kst (!) − kyt (!)‖ 6 �.This technical requirement is met via a reachability assumption on initial stocks.

We have now achieved our twin objectives: the uniform bound on the expectationof 〈V"〉 is realized because a potentially unbounded increasing process is replaced withan oscillatory martingale process, and martingales have the convenient property thattheir expectation in any period "¿ 0 is equal to their expectation in period " = 0.Interiority of the optimal programs will ensure that expectation in period "=0 is 5nite.Uniform sensitivity of value-loss is attained because, when capital stocks are morethan �-distance apart, 〈V"〉 upcrosses a band of uniform and strictly positive width.We, therefore, obtain the turnpike property under Assumptions (A.1)–(A.4) only andimpose no additional restrictions as in the received literature.

5. Methodology: a formal analysis

Our construction of a value-loss process has to be tempered by four requirements:〈Ft〉-adaptability, the martingale property, uniform bound on expectation, and upcross-ing across a band of uniform width following a critical divergence in capital stocks.For each t ∈I+, we isolate a collection of subsets of � generating Ft and construct avalue-loss process satisfying the requisite properties with respect to this collection. Wethen invoke Dynkin’s �–� theorem (Billingsley, 1979, Theorem 3.2) to extend theseproperties to the �-5eld Ft generated by this collection. As a 5rst step, we exploit thesecond countability axiom to explicitly characterize Ft in terms of the countable basisof �i; i = 0; 1; : : : ; t.

Proposition 1. Let �t; t ∈I+; be a compact metric space and consider the countablecollection Bt =H0 ×H1 × · · · ×Ht ×�t+1 ×�t+2 × · · · ; t ∈I+. Then; Ft = �(Bt).Further; Bt can be assumed to be a partition of �.

Our construction of the value-loss process introduces the “intermediate” points, t+ 12 ,

between points t and t + 1 on the time-line. With regard to these points we have

Corollary 1. There exists a countable collection; Bt+1=2; t ∈I+; such that Ft+1=2 =�(Bt+1=2) is )ner than Ft but coarser than Ft+1.

Now consider the set, S�t = {!: ‖kst (!) − kyt (!)‖¿�}; t ∈I+, and note that it is

Ft-measurable by virtue of the 〈Ft〉-adaptability of optimal programs. The next criticalstep is to ensure that (� \ S�

t )¿ 0 for each t ∈I+ so that we are assured a set ofstates of strictly positive measure on which to manipulate the value-loss process. Thiswill necessitate a reachability restriction on the initial stocks.

De$nition 1. Consider an initial stock y∈L; y¿ s with probability 1. We say that yis reachable from s if; for some N ¡∞; there exists a real-valued 〈Ft〉-adapted 5niteprocess; 〈(kt ; ct)〉Nt=0; which satis5es (1); (2) and (3) along with kN ¿y almost surely.


Clearly, in non-stationary models, initial stocks must satisfy some reachability con-dition if the late turnpike property is to be obtained. 8 Our de5nition of reachability isquite weak and corresponds to the expansibility notion that we can reach one initialstock from another via some feasible program in 5nite time with probability 1.

Proposition 2. Consider initial stocks s; y∈L; s¡y -a.s.; such that y is reachablefrom s. Under Assumptions (A.1)–(A.4); (� \ S�

t )¿ 0 for each t ∈I+ and for all�¿ 0.

Proposition 2 hints at the late turnpike property. The onerous task still remains todemonstrate that optimal programs from reachable initial stocks will be arbitrarily closefor all suNciently large t ∈I+ on a set of full probability measure. Unfortunately, inthis regard, it is not enough to just show that (�\S�

t )¿ 0, t ∈I+. What we require isthat ((�\S�

t )∩Bn; t)¿ 0 for each Bn; t ∈Bt , a property that we refer to as the non-emptyintersection property. 9 This is because a process, 〈Vt〉, is a 〈Ft〉-martingale if and onlyif for each Ft ∈Ft ;

∫FtVt+1 d=

∫FtVt d. Since Ft =�(Bt) from Proposition 1, it suf-

5ces to show this for each Bn; t ∈Bt . In this regard, the non-empty intersection propertywill permit us to let the value-loss process register a jump of 2� for realizations inS�t+1 ∩ Bn; t while simultaneously manipulating realizations in (� \ S�

t )∩ Bn; t (through asuitable choice of adjustment costs) such that the martingale property holds.

There is no reason, however, why the collection Bt will exhibit the non-emptyintersection property. In general, it will be made up of the following two sub-collectionsof mutually disjoint sets:

B′t = {B′

n; t ∈Bt : ((� \ S�t ) ∩ Bn; t)¿ 0};

B′′t = {B′′

n; t ∈Bt : ((� \ S�t ) ∩ Bn; t) = 0}: (6)

Since Bt covers �, the sub-collection B′t is non-empty. Without loss of generality,

it can be assumed that B′t and B′′

t are countably in5nite. 10 We can now de5ne apartition, Ct = 〈Cn; t〉, of � by letting Cn; t =B′

n; t ∪B′′n; t ; n∈I+. By construction, ((� \

S�t )∩Cn; t)¿ 0 for all Cn; t ∈Ct . Since Ct is a coarser partition of � than Bt , �(Ct) ⊆Ft . We, therefore, need to augment this collection while preserving the non-emptyintersection property. For this purpose, re-index sets in B′′

t such that Cn; t ∩B′′n; t = ∅ for

each n∈I+. We can now prove:

Proposition 3. Consider Dt = 〈Dn;t〉; Dn; t = Cn; t ∪ B′′n; t for Cn; t∈Ct ; B′′

n; t ∈B′′t ; Cn; t ∩

B′′n; t=∅; n∈I+. Then Dt displays the non-empty intersection property and �(Dt)=Ft .

8 The precise function of reachability in the proof of the late turnpike property is to bound the value-lossprocess (McKenzie, 1976). Reachability is put towards the same end in our framework, albeit somewhattangentially. By facilitating the construction of a value-loss process that is a martingale, it indirectly boundsit in expectation.

9 This terminology is for brevity only for in fact the property requires not only non-empty intersection of� \ S�t with each Bn;t ∈Bt but also that this intersection have strictly positive measure.

10 As subsets of a compact metric space, S�t and (� \ S�t ) are Lindel@of spaces (Munkres, 1975, Chapter 4)and, therefore, can be covered by a countable collection of open sets, S�t =

⋃∞n=1 O

′′n; t and �\S�t =

⋃∞n=1 O

′n; t .

B′t and B′′

t can be the basis sets generating 〈O′n; t〉 and 〈O′′

n; t〉, respectively.


The collection Dt is not a partition of �. Rather, 5nite intersections of the elementsof Dt , coupled with the operations of 5nite unions and set-theoretic di0erences, yieldsBt . With regard to the collection Bt+1=2; t ∈I+, we let

B′t+ 1

2= {B′

n; t+ 12∈Bt+ 1

2: ((� \ S�

t+1) ∩ Bn; t+ 12)¿ 0};

B′′t+ 1

2= {B′′

n; t+ 12∈Bt+ 1

2: ((� \ S�

t+1) ∩ Bn; t+ 12) = 0}: (7)

We can similarly obtain a countable collection, Ct+1=2, and extend it as in Proposition 3to a collection Dt+1=2 generating Ft+1=2 and satisfying the non-empty intersectionproperty.

At this point, it is convenient to switch to the “extended” time-line " = 2t wheret = 0; 12 ; 1; 1

12 ; 2; : : : . Recall once again that even-valued " refer to points on the origi-

nal time-line and odd-valued " to the arti5cially introduced “intermediate” points. Onthe time-line ", we can de5ne the adjustment cost parameters, 11 )"+1 :D" → R and*"+1 :D" → R, "∈I+, in a manner which ensures that value-loss is a martingale.The details are provided in the proof of Proposition 4. While these parameters havea collection of (possibly non-singleton) subsets of � as their domain, the value-lossprocess has to be constructed pointwise on � so that we can keep track of more than�-divergence in capital stocks for each tuple (t; !). This motivates the next step whichis to use the adjustment cost parameters to de5ne the adjustment cost functions overrealizations in �. Let

)"+1(!) = )"+1(Dn;"); !∈Cn;"; Dn;" = Cn;" ∪ B′′n;"; n∈I+; (8)

*"+1(!) = *"+1(Dn;"); !∈Cn;"; Dn;" = Cn;" ∪ B′′n;"; n∈I+: (9)

Since C" is a partition of �, the adjustment cost functions are well de5ned. By con-struction, they are F"-measurable and assume a constant value over each Dn;" ∈D".

We are now equipped with the requisite preliminaries to construct two comple-mentary processes that together make up the value-loss process. Consider 5rst thereal-valued stochastic process, 〈X"〉, which at even-valued " (i.e. for points on theoriginal time-line) measures total cost (value-loss plus adjustment costs) incurred when〈(ky" ; cy" )〉 is the “turnpike” and the planner is o0 the turnpike on the path 〈(ks" ; cs")〉.In even ", given a realization !, if !∈ S�

" , then the planner stays o0 the turnpikeincurring a value-loss Ms

" . However, if !∈� \ S�" , then the planner switches to the

turnpike eradicating any value-loss but incurring an adjustment cost in the transition.This motivates the following de5nition for the 〈X"〉 process, where ,(C) represents theindicator function of a set C:

X" = ,(S�")M

s" + ,(� \ S�

"))"; " even (10)

with the initial condition, X0 = �. A symmetric argument is used to construct thereal-valued stochastic process 〈Y"〉 which measures value-loss when 〈(ks" ; cs")〉 is the

11 We refer to )"+1 and *"+1 as “parameters” since, being F"-measurable, or predictable, their value is5xed in period " + 1 by the information available till period ".


“turnpike” and the planner is o0 the turnpike on the path 〈(ky" ; cy" )〉:Y" = ,(S�

")My" + ,(� \ S�

")*"; " even (11)

with the initial condition, Y0 =−�. We now extend the de5nition of 〈Xt〉 and 〈Yt〉 overodd values of " by letting:

X" = ,

(⋃n

B′′n;"

)[Ms

"−1 −My"−1] + ,

(⋃n

B′n;"

))"; " odd; (12)

Y" = ,

(⋃n

B′′n;"

)[My

"−1 −Ms"−1] + ,

(⋃n

B′n;"

)*"; " odd: (13)

Note that since capital stocks are de5ned only at even-valued ", we have replaced S�"

and � \ S�" by

⋃n B

′′n;" and

⋃n B

′n;", respectively, when " is odd. We can prove:

Proposition 4. Under Assumptions (A.1)–(A.4); 〈X"〉 and 〈Y"〉 are 〈F"〉-martingaleprocesses on the time-line " which are uniformly bounded in expectation.

Let V"=X"+Y"; "∈I+. Culling together the properties of the above two processes,we show that 〈V"〉 is the desired value-loss process on the time-line ".

Proposition 5. Under Assumptions (A.1)–(A.4); there exists a zero-mean 〈F"〉-martingale process 〈V"〉 such that

V"(!) =

0; "= 0; !∈�;6 0; "= 2n+ 1; n∈I+; !∈�;¡ 0; "= 2n; n∈I+ \ {0}; !∈� \ S�

" ;¿ 2�; "= 2n; n∈I+ \ {0}; !∈ S�

" :

The martingale and upcrossing properties of 〈V"〉, which are fundamental to the resolu-tion of the late turnpike property, are invariant to an aNne transformation of value-loss.Consider any transformed process, 〈V2;3

" 〉; 2; 3∈R++, where V2;3" (!)=3V"(!)+2 for

all !∈� and "∈I+. Any 〈V2;3" 〉 can assume the role of a value-loss process since it is

a martingale (and hence uniformly bounded in expectation) and registers an upcrossingacross the interval [2; 2�3+ 2] when there is more than �-divergence in capital stocks.A change in 2 adjusts the upper and lower limits of the upcrossing band by the sameamount. The positivity of 3 ensures that we upcross a band given some critical diver-gence in capital stocks; if 3 was negative, a symmetric argument can be constructedwhere 〈V2;3

" 〉 downcrosses the band [2�3 + 2; 2] following a critical divergence.For any 2; 3∈R++, and !∈�, de5ne the process 〈K2;3

" 〉, K2;3" :� → {0; 1}; "∈I+,

as

K2;3" (!) =

0; "= 0;

0; "∈I+ \ 0; V 2;3"−1(!)¿ 2�3 + 2;

1; "∈I+ \ 0; V 2;3"−1(!)¡ 2�3 + 2:


This process is predictable (i.e. K2;3" is F"−1-measurable) and it indicates one upcross-

ing across [2; 2�3+2] of 〈V2;3" 〉 when a chain of 1’s is followed by a 0. This is shown

in the third graph in Fig. 2. Note that in whatever manner the process may oscillate,only one upcrossing is registered across the upcrossing interval over any chain of 1’sVanked on either side by 0’s (for instance, consider the chain of 1’s stretching from"= 2 to "= 4 in Fig. 2). 12

To count the number of upcrossings given !∈�, for each "∈I+ let

Z2;3" (!) =

{1; K2;3

" (!) = 1; K2;3"+1(!) = 0;

0; otherwise:

Given an arbitrary planning horizon N ∈I+ and state !, let U�N (!) =

∑N"=1 Z

2;3" (!)

represent the total number of upcrossings over [2; 2�3+2] of 〈V2;3" (!)〉 on the time-line

" following more than �-divergence in capital stocks. Our version of the MartingaleUpcrossing result states:

Proposition 6. For any �¿ 0 and N ∈I+; the expected number of upcrossings acrossthe interval [2; 2�3 + 2] of the value-loss process 〈V2;3

" 〉 over a planning horizon oflength N + 1 is bounded from above independently of N .

With the help of Proposition 6, we have a resolution of the late turnpike propertywithout the invocation of any strong uniformity restriction.

Theorem 1. Consider initial stocks s; y∈L; s6y almost surely; such that y is reach-able from s. Under Assumptions (A.1)–(A.4); ‖kst −kyt ‖ → 0 almost surely as t → ∞.

6. Discussion

Our method of proving the turnpike property is to use the Euler characterization ofthe optimal program to generate a martingale process 〈Mt〉. The convex structure ofthe model is then used to show that (a modi5cation of) this process shows sensitivityto a divergence in capital stocks and therefore satis5es the technical condition that(� \ S�

t )¿ 0; t ∈I+ and �¿ 0 (Proposition 2). The value-loss, however, depends onthe tuple (t; !). At this point in the received literature, we have to invoke a unifor-mity assumption stating that value-loss is independent of the tuple (t; !) in order toobtain the turnpike property. In contrast, we have shown in Sections 4 and 5 how themultiplicity of states in a stochastic environment allow using 〈Mt〉 and reachability as

12 It may be noted that our de5nition of K"(!) does not correspond to those in standard probability texts(for instance, Billingsley, 1979, Theorem 35.3) where a sequence of 1’s also indicates the duration ofan upcrossing. We, on the contrary, are not interested in the duration (since it is 5xed at one period byconstruction) but rather in the number of upcrossings over the given interval (since each is indicative ofmore than �-divergence in capital stocks); for the same reason, as opposed to martingale theory, we do notwish to count downcrossings over any interval.


building blocks towards a new martingale process which displays the required uniformsensitivity.

In this section we provide some instances where our turnpike method could beapplied with pro5t. In optimal growth theory, we show how the important middleturnpike property can be proved without a uniformity assumption using our methods.It is interesting to note that long run invariance properties, akin to the late turnpikeproperty, exist in diverse areas of economics. These results demonstrate the convergencein a suitable metric of the time paths (or the invariance of the steady state) of somevariable of interest to alternative speci5cations of a given parameter. We show theapplicability of our methods to this issue by considering an example from the 5eld ofpublic 5nance.

6.1. The middle turnpike property

The middle turnpike theorem compares the time path of a 5nite horizon optimalprogram from a given initial stock to the time path of an in5nite horizon optimalprogram from a di0erent initial stock. Given a F0-measurable strictly positive initialstock y, and a FT -measurable non-negative terminal stock 8, a 5nite horizon programof length T +1 is feasible if it satis5es (1)–(3) along with kT ¿ 8 with probability 1.A program 〈(ky;8;Tt ; cy;8;Tt )〉 is optimal in the set of feasible programs if it maximizesthe expected 5nite sum of felicities, E

∑Tt=0 ut(ct). The middle turnpike property states

that for any arbitrary constant �¿ 0, if 8 is chosen appropriately, then the expectednumber of time periods 〈ky;8;Tt 〉 spends outside the �-neighbourhood of 〈kst 〉 is boundedfrom above independently of the length of the horizon.

We can mimic the proof of Proposition 5 to construct a value-loss process, 〈V2;3";T 〉,

which compares the capital stocks corresponding to the 5nite and in5nite horizon op-timal programs. In particular, if in any even-valued " there is more than �-divergencein the capital stocks, then V2;3

";T ¿ 2�T3, where �T ≡ E[u′0(cs0)− u′0(c

y;8;T0 )]. Therefore,

value-loss is a function of T , the planning horizon. The proof of the middle turnpiketheorem, however, requires independence from T . This can be achieved by suitablyrestricting the terminal stock via a reachability assumption: 13

De$nition 2. A FT -measurable non-negative terminal stock 8 is reachable from anin5nite horizon optimal program 〈(kyt ; cyt )〉 if there exists a constant T (8; y)¡∞ suchthat kyt ¿ 8 with probability 1 for all t¿T (8; y).

Theorem 2. Consider initial stocks s; y∈L; s �=y; and a terminal stock 8 which isreachable from 〈(kyt ; cyt )〉. Under Assumptions (A.1)–(A.4); for each �¿ 0 there ex-ists a :¡∞ such that the expected number of time periods for which ‖kst −ky;8;Tt ‖¿�is bounded from above by :; where : is independent of the planning horizon; T .

13 A similar restriction has been utilized by Majumdar and Zilcha (1987, p. 124) to obtain the turnpikeproperty.


6.2. Long run invariance results

Our turnpike technique can be pro5tably applied to all areas where the objective isto obtain the asymptotic stability of the time paths of a variable of interest generatedfrom di0erent initial conditions, and the underlying dynamic model is characterized byconvex preferences and technology. For example, in industrial organization, a resultbearing formal resemblance to the late turnpike property identi5es conditions underwhich two 5rms with di0erent initial conditions (technology gap) can close the gapover time through optimal investment in research and development (Budd et al., 1993).This analysis can be recast in a growth setting as shown in Joshi and Vonortas (2001)where the capital stocks represent investment in R&D and the variable of interest isa 5rm’s unit cost of production in the product market. The 5rms start with di0erentinitial stocks, and therefore di0erent unit costs of production. This has implications forboth current and future pro5t possibilities of the 5rms, and hence on how much R&Dinvestment a 5rm can 5nance from its product market pro5ts. In a convex model, theturnpike technique of this paper can be used to show that 5rms gravitate towards asymmetric outcome.

In public 5nance, a classical invariance proposition states that in a neoclassicalgrowth model a capital income tax is completely shifted to the labour input in thelong run (Becker, 1985). Reframing this result in the turnpike vernacular, the timepaths of the after-tax return to capital per unit, corresponding to di0erent rates ofcapital income taxation, converge almost surely. We now discuss brieVy how this re-sult can be derived using our method in the recursive competitive equilibrium (RCE)framework of Becker (1985) and Coleman (1991).

The production sector is represented by a unique competitive 5rm facing a sequenceof production functions 〈ft〉 satisfying (A.1)–(A.2). Given the tuple (t; !) and thegross return to capital per unit Rt , the 5rm chooses the aggregate level of capital, kt ,to maximize ft(kt ; !t+1)− Rtkt . 14 Assuming that the production functions satisfy anInada condition at the origin, the interior solution to the 5rm’s problem is characterizedby Rt = f′

t(kt ; !t+1).The consumption sector comprises of a continuum of identical consumers distributed

on the unit interval [0; 1] according to some distribution m. It is assumed that a con-sumer supplies an inelastic amount of labor. The representative consumer’s preferencesare described by a sequence of felicity functions 〈ut〉 satisfying (A.3)–(A.4). Letting0¡=¡ 1 denote the discount factor, the representative consumer maximizes the dis-counted sum of felicities over the set of all consumption and capital input programssatisfying the restriction that with probability 1, c0+k0=s and ct+1+kt+1=(1−:)Rtkt+Wt + gt ; t¿ 1. Here, 0¡:¡ 1 is the capital income tax rate, kt is the representativeconsumer’s holding of capital, Wt is the wage rate and gt is the lump sum transferfrom the government in period t. The government’s function in this model is to taxcapital at rate : and distribute the tax revenue in a lump sum fashion. Let 〈(k:t ; c:t )〉be the solution to the representative consumer’s problem.

14 In the competitive equilibrium model, we have to distinguish between aggregate capital in period t, kt ,and the representative consumer’s holding of capital, kt .


Aggregate capital holdings and consumption is then equal to k:t =

∫ 10 k:t dm and

c:t =∫ 10 c:t dm. We say that 〈(k:

t ; c:t )〉 is a :-RCE if 〈k:

t 〉 solves the 5rm’s problem,and aggregate demand equals aggregate supply with probability 1, i.e. c:0 + k:

0 = s andc:t+1 + k:

t+1 = ft(k:t ; !t+1). The :-RCE satis5es the stochastic Euler equation:

u′t(c:t ) = =E[u′t+1(c

:t+1)(1− :)f′

t(k:t ; !t+1) ‖Ft] -a:s: t ∈I+: (14)

The after-tax return to capital is given by r:t = (1 − :)f′t(k

:t ; !t+1). Letting p:

t =u′t(c

:t ); �

:0 = 1; �:t+1 =

∏ti=0 r

:i , the Euler equations can be rewritten as

p:t �

:t = E[p:

t+1�:t+1 ‖Ft] -a:s: t ∈I+: (15)

We now have a martingale process as in (5). Let 0¡�¡ 1 be a di0erent capitalincome tax rate. A similar Euler expression is obtained for a �-RCE, � �= :. Next, wehave to show as in Proposition 2 that (! : ‖r:t −r�t ‖6 �)¿ 0 for all t ∈I+ and �¿ 0.Recall from the discussion in Appendix A (Lemma 2) that this requires constructinga process 〈Wt〉 exhibiting value-loss which may depend on the tuple (t; !). To derivethis let

qt = min[r:t ; r�t ]; t ∈I+: (16)

Further, let �0 = 1 and �t+1 =∏t

i=0 qi. Then

pit�t¿E[pi

t+1�t+1 ‖Ft] -a:s:; t ∈I+; i = :; �: (17)

Now letting Wt = p:t �t + p�

t �t , it can be veri5ed that

E[Wt −Wt+1 ‖Ft] = E[p:t �t − p:

t+1�t+1 ‖Ft]¿ 0; r:t ¿ r�t ;

E[Wt −Wt+1 ‖Ft] = E[p�t �t − p�

t+1�t+1 ‖Ft] ¿ 0; r:t ¡ r�t : (18)

Therefore, following the argument of Proposition 2, (! : ‖r:t − r�t ‖ 6 �)¿ 0 for allt ∈I+ and �¿ 0. We now have all the necessary ingredients in place to use the methoddeveloped in Section 5 to derive the following long run invariance property:

Theorem 3. Let :; �∈ (0; 1) be any two distinct rates of capital income taxation and〈rt〉; i= :; �, the after-tax return to capital per unit under these tax rates. Assuming(A.1)–(A.4), ‖r:t − r�t ‖ → 0 almost surely as t → ∞.

7. Conclusion

In this paper, we have exploited the underlying stochastic primitive to obtain thelate turnpike property in convex aggregate growth models without imposing any stronguniformity restriction. In this regard, our paper echoes the argument put forward inAmir (1997) that growth models under uncertainty should not be simply extensionsof the deterministic case with the stochastic element as a mere addendum. Rather,uncertainty should add in an essential way to the results derivable from the certainty


case. A similar consideration had also motivated Chang (1982) to put forth a expectedvalue-loss assumption conditioned to the particular dictates of a stochastic paradigm.

There remains the issue of whether our technique can be extended to multisector op-timal growth models. As noted earlier, such models can display complicated dynamicsfor low values of the discount factor. However, even for discount factors suNcientlyclose to unity, some form of uniformity is generally invoked to obtain the late turn-pike property (Guerrero-Luchtenberg, 1998; McKenzie, 1976; Montrucchio, 1995). Itis an interesting question, therefore, whether critical manipulation of the stochastic en-vironment, along with a suitable restriction of reachability on initial stocks, permitsa derivation of the late turnpike property for suNciently high values of the discountfactor without recourse to strong uniformity restrictions.

Acknowledgements

I would like to thank F.R. Chang, Santanu Roy and seminar participants at ErasmusUniversity, Rotterdam, for their comments. I would also like to thank two anonymousreferees for comments. This paper was completed while I was visiting the Centrefor Development Economics, Delhi School of Economics. I would like to thank themfor providing me the resources to complete this project. I remain responsible for anyerrors.

Appendix A

Proof of Proposition 1. Note 5rst of all that

Bt ⊆ �(H0)× �(H1)× · · · × �(Ht)× �t+1 × �t+2 × · · ·= E0 × E1 × · · · × Et × �t+1 × �t+2 × · · ·⊆ �(E0 × E1 × · · · × Et × �t+1 × �t+2 × · · ·) =Ft :

Therefore, �(Bt) ⊆ Ft . To prove the converse, we use a result from Yeh (1995,Lemma 1.3) that �(H0) × �(H1) × · · · × �(Ht) ⊆ �(H0 ×H1 × · · · ×Ht). Usingthis result:

�(H0)× �(H1)× · · · × �(Ht)× �t+1 × �t+2 × · · ·⊆ �(H0 ×H1 × · · · ×Ht)× �t+1 × �t+2 × · · ·⊆ �(H0 ×H1 × · · · ×Ht × �t+1 × �t+2 × · · ·) = �(Bt):

Hence, Ft = �(�(H0)× �(H1)× · · · × �(Ht)× �t+1 × �t+2 × · · ·) ⊆ �(Bt).For i = 0; 1; : : : ; t, since �i is open and can be expressed as the countable union of

the elements of Hi, it follows that Bt covers �. To show that Bt can be assumedto be a countable collection of mutually disjoint sets, de5ne one possible countablepartition, B′

t = 〈B′n; t〉, as B′

0; t = B0; t ; B′n; t = Bn; t \

⋃n−1j=0 Bj; t for n∈I+ \ {0}. Let F′

t


denote the �-5eld generated by B′t . For each !∈�, and a set Bn; t ∈Bt containing !,

by construction there is a set B′n; t ∈B′

t such that !∈B′n; t ⊆ Bn; t . Hence, it follows that

Ft ⊆ F′t . Conversely, because a �-5eld is closed under set-theoretic di0erence (given

closure under complementation and 5nite intersections) and countable unions, we haveB′

t ⊆ Ft . Hence, F′t = �(B′

t) ⊆ Ft .

Proof of Corollary 1. Consider any t ∈I+. For each !∈Bn; t ∈Bt , there exists aBm;t+1 ∈Bt+1 such that !∈Bm;t+1 ⊂ Bn; t . Recall the partition of Bt given by (6).Construct the collection Bt+1=2 by letting Bn; t+1=2 = B′

n; t+1; further, partition the set� \⋃n B

′n; t+1 by choosing Bn; t+1=2 such that B′′

n; t+1 ⊂ Bn; t+1=2 ⊂ B′′n; t . Then, Bn; t+1=2 is

the required partition.

To prove Proposition 2, in addition to reachability, we will need two additionalresults. The 5rst is a standard monotonicity property of both optimal capital input andconsumption programs with respect to the initial stock in a convex aggregate model(Majumdar and Zilcha, 1987). Since we are operating within a convex one-sectormodel, a direct application of these methods shows that this monotonicity propertyobtains in our model as well. We, therefore, record this property without proof. Thismethod also yields strict monotonicity for interior optimal programs.

Lemma 1. Consider any s; y∈L; s¡y -a:s: Under Assumptions (A.1)–(A.4),kst ¡ kyt and cst ¡ cyt almost surely for each t ∈I+.

The second result exploits the stochastic Euler equations to construct a submartingalevalue-loss process of the “conventional” variety in the sense that value-loss dependson both time and state in which the critical divergence in optimal capital stocks beingcompared occurs.

Lemma 2. Consider initial stocks s; y∈L; s¡y -a:s:, such that y is reachable froms. Under Assumptions (A.1)–(A.4), there exists a 〈Ft〉-submartingale, 〈Wt〉, which isuniformly bounded in expectation and displays the following sensitivity property: forany �¿ 0, and any tuple (t; !), if !∈ S�

t then there exists a constant t(!; �)¿ 0such that E[Wt+1 −Wt ‖Ft](!)¿ t(!; �).

Proof of Lemma 2. Since s¡y -a.s., (A.2) and Lemma 1 imply that f′t(k

yt ; !t+1)¡

f′t(k

st ; !t+1) with probability 1. In particular, given �¿ 0, if !∈ S�

t , then there existsa :′t(!; �)¿ 0 such that f′

t(kst ; !t+1)¿f′

t(kyt ; !t+1)+ :′t(!; �). Now suppose 〈(kyt ; cyt )〉

is considered as the turnpike. From the Euler equations characterizing the turnpike(see (4)) it follows that for the given !∈ S�

t (where reference to the argument ! issuppressed in some terms for notational ease)

pyt = E[py

t+1f′t(k

yt ; !t+1) ‖Ft]

¡ E[pyt+1f

′t(k

st ; !t+1) ‖Ft]− E[py

t+1:′t(!; �) ‖Ft]: (19)


Multiplying (19) by �st , and combining with (5) for the turnpike, yields:

pyt �

st + py

t �yt + ′t(!; �)¡E[py

t+1�st+1 + py

t+1�yt+1 ‖Ft]; (20)

where ′t(!; �) ≡ E[pyt+1�

st :

′t(!; �) ‖Ft] is value-loss if !∈ S�

t ; evaluated at the pricesystem 〈py

t 〉 supporting the turnpike. An identical argument; assuming 〈(kst ; cst )〉 is theturnpike; establishes that if !∈ S�

t

pst �

st − ps

t �yt + ′′t (!; �)¡E[ps

t+1�st+1 − ps

t+1�yt+1 ‖Ft]; (21)

where ′′t (!; �) is value-loss evaluated at the price system 〈pst 〉. Letting Wt ≡ py

t �st +pyt �

yt + ps

t �st − ps

t �yt ; and t(!; �) ≡ ′t(!; �) + ′′t (!; �); t ∈I+; it follows by adding

(20) and (21) that

E[Wt+1 −Wt ‖Ft](!)¿

{0; !∈� \ S�

t ;

t(!; �); !∈ S�t :

(22)

From (22) we can infer that 〈Wt〉 is a 〈Ft〉-submartingale displaying the requisitesensitivity to more than �-divergence in the optimal capital stocks from s and y. Further;given (A.2) and (A.3); py

t ¡pst and �yt ¡�st with probability 1. Using the fact that

〈pst �

st 〉 is a 〈Ft〉-martingale; EWt ¡ 3E[u′0(c

s0)].

Proof of Proposition 2. Let W0 =W0 and

Wt =t∑

i=1

E[Wi −Wi−1 ‖Fi−1] +W0: (23)

By construction; 〈Wt〉 is an increasing process and; therefore; a 〈Ft〉-submartingale.Further

∫Wt d=

∫ t∑i=1

E[Wi −Wi−1 ‖Fi−1] d+∫

W0 d

=t∑

i=1

∫E[Wi −Wi−1 ‖Fi−1] d+

∫W0 d

=t∑

i=1

∫[Wi −Wi−1] d+

∫W0 d

=∫ t∑

i=1

[Wi −Wi−1] d+∫

W0 d=∫

Wt d¡ 3E[u′0(cs0)]: (24)

Now let "0(!) = inf{t ∈I+ : Wt(!)¿ 3E[u′0(cs0)] + �}. Since 〈Wt〉 is a 〈Ft〉-adapted

strictly increasing process, "0(!) is a stopping time such that {!: "0(!)¡∞} = 1.Also consider the stopping times, "n(!) = "0(!) + n, n∈I+ \ {0}. From Billingsley(1979, Section 35), the stopped process, 〈W"n〉 is a 〈F"n〉-submartingale. Using the


Kolmogorov inequality for submartingales (Billingsley, 1979, Theorem 35.2), it followsthat for any "n; n∈I+:

({! : ‖ks"n(!)− ky"n(!)‖¿�})6 ({! : W"n(!)¿ 3E[u′0(cs0)] + �})

= ({

!: max06i6"n

W"n(!)¿ 3E[u′0(cs0)] + �

})

¡EW"n

3E[u′0(cs0)] + �

¡3E[u′0(c

s0)]

3E[u′0(cs0)] + �

≡ �¡ 1: (25)

Therefore, ({! : ‖ks"n(!)−ky"n(!)‖6 �})¿ (1−�)¿ 0 for each "n, n∈I+. The resultnow follows by noting that, without loss of generality, we can re-index the time-lineto start at "0, 15 given our concern with the asymptotic behaviour of optimal programsand the fact that the optimal programs, 〈(kit ; cit)〉, i=s; y, continue to be optimal startingfrom the initial stock f"0 (k

i"0 ; !"1 ) in the initial period "0.

Proof of Proposition 3. Since each Dn;t ∈Dt is a 5nite union of sets in Bt , it followsthat Dt ⊂ Ft and, therefore, �(Dt) ⊆ Ft . Conversely, it can be veri5ed that eachBn; t ∈Bt can be derived from Dt through the operations of 5nite intersection, 5niteunion, and set-theoretic di0erence. 16 Therefore, Bt ⊂ �(Dt), and hence, Ft ⊆ �(Dt).

Proof of Proposition 4.17 We start by de5ning the adjustment cost parameters. For"= 1, let

)1(Dn;0) =1

(⋃

n B′n;1 ∩ Dn;0)

[�(Dn;0)−

∫⋃

n B′′n;1∩Dn;0

(Ms0 −My

0 ) d

]; (26)

*1(Dn;0) =1

(⋃

n B′n;1 ∩ Dn;0)

[−�(Dn;0)−

∫⋃

n B′′n;1∩Dn;0

(My0 −Ms

0) d

]: (27)

15 Our subsequent analysis permits altering the 5xed time-line, t ∈I+, to one indexed by !; "n :� →I+; n∈I+. This is because the di0erent elements of the value-loss process on an extended time-line areconstructed pointwise on �, and the crucial properties of this process are inherited by the process which is“stopped” at the stopping times 〈"n〉. For notational convenience, however, we continue to employ a 5xedtime-line.

16 For instance, suppose D1; t =B′1; t ∪B′′

1; t ∪B′′2; t . By construction, there are distinct sets, Dm;t ; Dn; t ∈Dt , of

the form Dm;t =B′m;t ∪B′′

2; t ∪B′′m;t and Dn;t =B′

n; t ∪B′′1; t ∪B′′

n; t . Therefore, B′′2; t =D1; t ∩Dm;t , B′′

1; t =D1; t ∩Dn;t ,and B′

1; t = D1; t \ (B′′1; t ∪ B′′

2; t).17 This proof has bene5ted from the comments of an anonymous referee.


For even-valued "¿ 2, we de5ne the cost parameters inductively as

)"(Dn;"−1) =1

(� \ S�" ∩ Dn;"−1)

[∫⋃

n B′′n;"−1∩Dn;"−1

(Ms"−2 −My

"−2) d

−∫S�"∩Dn;"−1

Ms" d+ )"(Dn;"−2)

(⋃n

B′n;"−1 ∩ Dn;"−1

)];

*"(Dn;"−1) =1

(� \ S�" ∩ Dn;"−1)

[∫⋃

n B′′n;"−1∩Dn;"−1

(My"−2 −Ms

"−2) d

−∫S�"∩Dn;"−1

My" d+ *"(Dn;"−2)

(⋃n

B′n;"−1 ∩ Dn;"−1

)]; (28)

where Dn;"−2 ∈D"−2 is such that Dn;"−1 ⊂ Dn;"−2 (this is possible since D"−2 is coarserthan D"−1). For odd-valued "¿ 3, we let

)"(Dn;"−1) =1

(⋃

n B′n;" ∩ Dn;"−1)

[−∫⋃

n B′′n;"∩Dn;"−1

(Ms"−1 −My

"−1) d

+∫S�"−1∩Dn;"−1

Ms"−1 d+ )"−1(Dn;"−2)(� \ S�

"−1 ∩ Dn;"−1)

];

*"(Dn;"−1) =1

(⋃

n B′n;" ∩ Dn;"−1)

[−∫⋃

n B′′n;"∩Dn;"−1

(My"−1 −Ms

"−1) d

+∫S�"−1∩Dn;"−1

My"−1 d+ *"−1(Dn;"−2)(� \ S�

"−1 ∩ Dn;"−1)

]; (29)

where Dn;"−2 ∈D"−2 is such that (� \ S�"−1 ∩ Dn;"−1) = (� \ S�

"−1 ∩ Dn;"−2) (this ispossible since by construction B′

"−1 = B′"−2 so that the subsets of Dn;"−2 and Dn;"−1

whose intersection with � \ S�"−1 have strictly positive measure are the same). The

adjustment cost parameters are well de5ned and 5nite by virtue of the non-emptyintersection property; further, they are 〈F"〉-adapted by construction.

The proof is provided for 〈X"〉 and is identical for 〈Y"〉. By construction, 〈F"〉-adaptability follows. From Billingsley (1979, Section 35), to establish the martingaleproperty we need to demonstrate that for any F ∈F"∫

FX"+1 d=

∫FX" d: (30)

We establish this result by means of Dynkin’s �–� theorem. Note that (30) is triv-ially true if (F) = 0. Therefore, without loss of generality, we restrict attention toF"-measurable sets of strictly positive measure.


We start with the collection D". First, let "+1 be even-valued. Taking the expectationof X"+1 over Dn;"∫

Dn;"

X"+1 d=∫S�"+1∩Dn;"

M s"+1 d+

∫(�\S�

"+1)∩Dn;"

)"+1 d: (31)

By de5nition )"+1 assumes the constant value of )"+1(Dn;") on Dn;". Therefore∫Dn;"

X"+1 d=∫S�"+1∩Dn;"

M s"+1 d+ )"+1(Dn;")(� \ S�

"+1 ∩ Dn;"): (32)

Substituting for )"+1(Dn;") from (28), and recalling that from construction Dn" ⊂ Dn;"−1,we observe that (30) is satis5ed:∫

Dn;"

X"+1 d=∫⋃

n B′′n;"∩Dn;"

(Ms"−1 −My

"−1) d+ )(Dn;"−1)

(⋃n

B′n;" ∩ Dn;"

)

=∫⋃

n B′′n;"∩Dn;"

(Ms"−1 −My

"−1) d+∫⋃

n B′n;"∩Dn;"

)" d

=∫Dn;"

X" d:

Next, let "+ 1 be odd-valued. Then∫Dn;"

X"+1 d=∫⋃

n B′′n;"+1∩Dn;"

(Ms" −My

" ) d+∫⋃

n B′n;"+1∩Dn;"

)"+1 d: (33)

Substituting once again for )"+1(Dn;") from (29) in (33), while recalling that fromconstruction Dn" ⊂ Dn;"−1, we observe that (30) is satis5ed. Now let O" denote thecollection of sets which are a countable union of disjoint sets in D". For any Ok;" =⋃∞

j=1 Dj;" ∈O"; k ∈I+:

∫Ok;"

X"+1 d=∫⋃∞

j=1 Dj;"

X"+1 d=∞∑j=1

∫Dj;"

X"+1 d

=∞∑j=1

∫Dj;"

X" d=∫⋃∞

j=1 Dj;"

X" d=∫Ok;"

X" d: (34)

Therefore, (30) holds for all sets in O". Also note that O" constitutes a �-system,i.e. it is closed under 5nite intersections. Given any Ok;"; Om;" ∈O", one possibility isOk;" ∩ Om;" is empty, in which case (30) holds trivially. The other possibility is thatOk;" ∩ Om;" is non-empty, and is, therefore, a countable collection of disjoint sets inD"; in this case a string of equalities as in (34) establishes (30).

Now consider the collection, C, of all F-measurable subsets of � for which (30)holds. We now show that C constitutes a �-system. First of all note that �∈C, giventhat � is open in the product topology which generates F. Second, C is closed under


proper di0erences. To verify, consider F;G ∈C such that G ⊂ F . Since F=(F \G)∪G,∫F\G

X"+1 d=∫FX"+1 d−

∫GX"+1 d

=∫FX" d−

∫GX" d=

∫F\G

X" d: (35)

Third, C is closed under monotone limits. To verify, consider a sequence 〈Fn〉 ofF-measurable sets and F ∈F such that Fn ↑ F . Then 06 ,(Fn) ↑ ,(F). 18 Using theMonotone Convergence theorem it follows that∫

FX"+1 d= lim

n→∞

∫Fn

X"+1 d= limn→∞

∫Fn

X" d=∫FX" d: (36)

This indicates closure under monotone limits. Note that O" ⊂ C since sets in O" satisfy(30) and are F-measurable. From Dynkin’s �−� theorem, �(O") ⊆ C. The martingaleproperty now follows, since an argument similar to that in Lemma 3 establishes that�(O")=�(D")=F". To establish the bound on the expectation of 〈X"〉, note that since�∈F", we have

∫� X"+1 d=

∫� X0 d= �.

Proof of Proposition 5. Recall that the value-loss process 〈V"〉 is de5ned as

V" = X" + Y"; "∈I+ \ {0} (37)

with the initial value; V0 = 0. 19 We will 5rst prove by induction that

)" + *"6 0; "∈I+ \ {0}: (38)

From (26) and (27), this is true for "=1. Now suppose that (38) is true for any "¿ 1.Let "+ 1 be even. Then, from (28) at even-valued points, )"+1 + *"+16 0. Let "+ 1be odd. Then, from (29), at odd-valued points

)"+1(Dn;") + *"+1(Dn;") =1

(⋃

nB′n;"+1 ∩ Dn;")

[∫S�"∩Dn;"

(Ms" +My

" ) d

− (� \ S�" ∩ Dn;")

(� \ S�" ∩ Dn;"−1)

∫S�"∩Dn;"−1

(Ms" +My

" ) d

]: (39)

But, from construction, B′" =B′

"−1 when "+ 1 is odd. This implies (� \ S�" ∩Dn;") =

(� \ S�" ∩ Dn;"−1). Therefore, )"+1 + *"+16 0.

If " is odd, then by construction, V" = 0 if !∈⋃n B′′n;", and V" = )" + *"6 0 if

!∈⋃n B′n;". Now suppose that " is even. If !∈ S�

" , then V" =Ms" +My

" ¿ 2�. On the

18 Fix �¿ 0. If ! �∈ F , then ! �∈ Fn for all n and hence ,(F) − ,(Fn) = 0¡� for all n. If !∈F , thenthere exists m∈I+ such that !∈Fn for all n¿m and hence ,(F) − ,(Fn) = 0¡� for all n¿m.

19 Since our time-line starts at zero, 5xing the initial value V0 = 0 implies that we ignore a potentialdeviation in capital stocks by more than � in the initial period. But ignoring one possible deviation in period0 is inconsequential given our concern with the long run behaviour of optimal programs.


other hand, if !∈ (�\S�"), then V"=)"+*"6 0. Further, as the sum of 〈F"〉-martingales

〈X"〉 and 〈Y"〉 with means � and −�, respectively, 〈V"〉 is a zero-mean 〈F"〉-martingale.

Proof of Lemma 6. Consider any 〈V2;3" 〉; 2; 3∈R++. For arbitrary N ∈I+ and any

!∈�, on the time-line "

V 2;3N ¿ V2;3

N − V2;30

=N∑"=1

(V2;3" − V2;3

"−1)

=N∑"=1

K2;3" (V2;3

" − V2;3"−1) +

N∑"=1

(1− K2;3" )(V2;3

" − V2;3"−1): (40)

The reference to the argument, !, is suppressed for notational ease. Now consider anupcrossing represented by a chain of 1’s Vanked by 0’s: K2;3

m =0, K2;3m+1= · · ·=K2;3

n =1,K2;3n+1 = 0. Note by construction that V2;3

n ¿ 2�3 + 2 and V2;3m 6 2 (where the latter

follows from the fact that, with the exception of "=0, K2;3" =0 only for odd-valued "

which succeed an upcrossing and never at any even-valued "; further, at any odd-valued", value-loss takes a value 6 2). Therefore

n∑"=m+1

K2;3" (V2;3

" − V2;3"−1) = V2;3

n − V2;3m ¿ 2�3: (41)

It now follows that if the number of upcrossings are equal to U�N for the given !; then

N∑"=1

K2;3" (V2;3

" − V2;3"−1)¿ 2�3U�

N : (42)

Also note from the martingale property of 〈V2;3" 〉 and the predictability of K2;3

" that

N∑"=1

∫(1− K2;3

" )(V2;3" − V2;3

"−1) d

=N∑"=1

∫(1− K2;3

" )E[V2;3" − V2;3

"−1 ‖F"−1] d= 0: (43)

Combining (40), (42) and (43) yields

2�3∫

U�N d6

∫V2;3N d: (44)

The result follows by noting that the expectation of V2;3N is bounded from above by 2.


Proof of Theorem 1. Revert to the original time-line (i.e. even-valued ") since thevariable Z2;3

t = 1 only in time periods t = 1; 2; : : : ; and not in periods t = 12 ; 1

12 ; : : : :

Further; note that an upcrossing takes place over [2; 2�3 + 2] if and only if there ismore than �-divergence in capital stocks; i.e. Z2;3

t = 1 if and only if ,(S�t ) = 1. From

(44); and the fact that EU�N = E

∑Nt=1 Z

2;3t =

∑Nt=1 (S

�t ); we can conclude that

2�3N∑t=1

(S�t ) = 2�3

∫ N∑t=1

,(S�t ) d

= 2�3∫ N∑

t=1

Z2;3t d= 2�3

∫U�

N d6∫

V2;3N d: (45)

Since the expectation of V2;3N is bounded from above independently of N , letting N →

∞ in (45) yields∑∞

t=1 (S�t )¡∞. An application of the 5rst Borel–Cantelli lemma

then yields ({! : lim supt→∞S�t }) = 0.

Proof of Theorem 2. First of all we show that value-loss is bounded from below in-dependently of the horizon, T . Note that for any T¿ 1, the truncation of 〈(kyt ; cyt )〉 tothe 5rst T + 1 periods is a 5nite horizon optimal program with terminal stock kyT . LetT¿T (8; y). From a well-known result on terminal stock monotonicity (See Majumdarand Zilcha, 1987), with probability 1

ky;8;Tt 6 kyt t = 0; 1; : : : ; T: (46)

Therefore; from interiority; 0¡cy0 = y − ky0 6y − ky;8;T0 = cy;8;T0 . From the concavityof u0 it now follows that

� ≡ E[u′0(cs0)− u′0(c

y0 )]6E[u′0(c

s0)− u′0(c

y;8;T0 )] ≡ �T : (47)

Recall that an upcrossing takes place if and only if there is more than �-divergence incapital stocks. Therefore; the number of periods in which ‖kst − ky;8;Tt ‖¿�; is simplythe number of upcrossings over the horizon T ; U �

T . It follows from Proposition 6 that∫U�

T 6 2=2�3 ≡ :.

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The stochastic turnpike property without uniformity in convex aggregate growth models

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