Small noise asymptotics for a stochastic growth model

Journal of Economic Theory 119 (2004) 271–298

Small noise asymptotics for a stochasticgrowth model

Noah Williamsa,b,�

aDepartment of Economics, Princeton University, Princeton, NJ 08544-1021, USAbNBER, 1050 Massachusetts Avenue, Cambridge, MA 02138, USA

Received 25 September 2002; final version received 15 December 2003

Abstract

We develop analytic asymptotic methods to characterize time-series properties of nonlinear

dynamic stochastic models. We focus on a stochastic growth model which is representative of

the models underlying much of modern macroeconomics. Taking limits as the stochastic

shocks become small, we derive a functional central limit theorem, a large deviation principle,

and a moderate deviation principle. These allow us to calculate analytically the asymptotic

distribution of the capital stock, and to obtain bounds on the probability that the log of the

capital stock will differ from its deterministic steady-state level by a given amount. This latter

result can be applied to characterize the probability and frequency of large business cycles. We

then illustrate our theoretical results through some simulations. We find that our results do a

good job of characterizing the model economy, both in terms of its average behavior and its

occasional large cyclical fluctuations.

r 2004 Elsevier Inc. All rights reserved.

JEL classification: E32; C32

Keywords: Economic fluctuations; Asymptotic methods; Large deviations

1. Introduction

Modern macroeconomics is built on the foundation of nonlinear dynamicstochastic general equilibrium (DSGE) models. In particular, the stochastic growthmodel is one of the most widely used models in all of economics, and is the standard

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�Department of Economics, Princeton University, Princeton, NJ 08544-1021, USA.

E-mail address: [email protected].

0022-0531/$ - see front matter r 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.jet.2003.12.007

model for business cycle analysis. However, because the model can only be solved inclosed form under very restrictive assumptions (such as log utility and fulldepreciation of capital), analysis of the model must resort to approximations. Forexample, a standard practice is to linearize the Euler equations which characterizethe optimal solution around a deterministic steady state.1 In many cases, there islittle discussion of the quality of such approximations, particularly for the stochasticproperties of the economy.2 In this paper we provide some steps in this direction. Weprovide analytic asymptotic results which characterize the average behavior of thestochastic growth model and its occasional large fluctuations.Our analysis considers limits as the standard deviation (s) of the stochastic

technology shocks converges to zero. We show that the capital accumulationtrajectories converge to the corresponding trajectories from a deterministic model.The limiting deterministic models are typically easier to analyze, particularly in theneighborhood of a steady state. The results provide analytic, theoretically justifiedapproximations for stochastic models with small noise. There is less, if any, need fornumerical methods and simulation. Further, the analytic expressions we obtain areuseful for comparative statics and dynamics, and can potentially be used as a meansfor estimation. Interestingly, we find that many asymptotic properties of theeconomy can be described by a linear approximation. However for largerfluctuations, and to describe potential asymmetries in the time series, nonlinearmethods are needed. While we focus on a relatively simple and standard model, themethods we develop can be applied to more general nonlinear DSGE models whichmay provide a closer match to the data.In our analysis below, we obtain three different characterizations of the rate at

which the stochastic model converges to the deterministic one. We first formulate afunctional central limit theorem which shows that at rate s; the centered capitaltrajectories are asymptotically normal. While this result holds for both for the levelof capital and its logarithm, for our other results we consider solely the log of thecapital stock. We then apply a large deviation principle, which shows that theabsolute differences of the log capital trajectories converge to zero exponentially fast.We also obtain estimates of the average time it takes the log capital stock to differfrom its steady-state level by a given amount. Finally, we present a moderatedeviation principle which provides similar results, but for an intermediate range ofasymptotics between the functional central limit theorem and the large deviationprinciple.Both the functional central limit theorem and the moderate deviation principle are

expressed in terms of a linear approximation to the deterministic model. Theseresults thus suggest that in order to consider the average behavior of the economyand even to consider some ‘‘extreme’’ events, a linear approximation is sufficient.

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1Papers which use linearizations are too numerous to list, but some notable applications in contexts

similar to this paper include Magill [27], Kydland and Prescott [25], King et al. [19], and Campbell [6]. A

summary of a variety of numerical methods can be found in Judd [14].2An exception is the literature on perturbation methods, discussed in more detail below. A related issue,

which we do not address is the approximation of welfare associated with linearized models. Kim and Kim

[18] have shown the potential problems this may cause.

N. Williams / Journal of Economic Theory 119 (2004) 271–298272

However, the linear approximation does not fully capture the large deviationproperties, and it abstracts from certain asymmetries in the model due tononlinearities.3 By applying the large deviation results, we show that the modeleconomy is slightly asymmetric and is slightly more likely to exhibit recessions(appropriately defined) than booms.This appears to be the first paper to apply these type of convergence results for the

stochastic growth model. However, there are several of related papers in theliterature. In a continuous time setting, Prandini [28] used the large deviation resultsof Azencott [3] to analyze a stochastic Solow growth model. He showed that thecapital trajectories in the stochastic model converges uniformly on a finite timehorizon to the corresponding trajectories from the standard deterministic model. Weobtain similar results in our section on large deviations. However, we provide a moredetailed analysis with more explicit calculations, in addition to the wider array ofasymptotic results we establish. Perhaps more importantly, our model incorporatesexplicit optimization, which broadens the potential scope of applications. However italso provides some complications in the analysis, as we must establish that the policyfunctions converge in an appropriate sense and satisfy some additional regularityconditions.As we noted above, there is a closely related and extensive branch of the literature

focusing on the properties of approximate solutions. However, the functionalapproximation results do not characterize the stochastic properties of the modeleconomy, which is our focus. In the mathematics literature, key papers includeFleming [9] and Fleming and Souganidis [11] who prove the uniform convergence ofpolicy functions in continuous time models when the noise goes to zero. In theeconomics literature, an early contribution was Magill [27] who derived anasymptotic linear-quadratic approximation in a continuous time stochastic growthmodel. Judd [14] contains a comprehensive overview of perturbation methods andtheir applications in economics. Judd and Guu [15,16] present numerical methodsbased on Taylor expansions to analyze stochastic and deterministic growth models,respectively. Gaspar and Judd [12] provide high-order expansions for multivariatemodels. Most of these papers focus on the local analytic properties of the solution,with a few providing global numerical results. But again, none of these touch on theissues we analyze.

2. The model

In this section we lay out the benchmark model for the analysis. It is a specializedBrock–Mirman [5] economy with production, capital accumulation, and stochasticproductivity growth. To simplify the presentation and analysis, we assume that

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3There are higher-order asymptotic results for continuous time diffusions in the literature, such as

Fleming and Souganidis [11] and Fleming and James [10], which may capture some of these nonlinearities

in an analytically tractable way. However, to our knowledge there are no such results for our discrete time

setting.

N. Williams / Journal of Economic Theory 119 (2004) 271–298 273

technology shocks are permanent, which implies that the model has a single statevariable. This assumption can be relaxed. Further, in the development of the paper itshould be evident that our results have applications to more general nonlinearstochastic models.

2.1. The stochastic growth model

We assume that output is produced according to a standard constant returns toscale Cobb–Douglas production function with parameter a:

FðK ;LÞ ¼ KaðALÞ1�a;

where K is the capital stock, L is the labor supply and A is the labor-augmentingtechnology parameter. For simplicity, we fix the total labor supply at L ¼ 1: Weassume that A evolves exogenously as a unit root process in logarithms:

logAtþ1 ¼ kþ logAt þ sWtþ1; ð1Þ

where W is a standard normal random variable and kX0 is the mean rate oftechnology growth. The unit root assumption is made for simplicity, and is roughlyconsistent with US time-series data. Let d be the depreciation rate of capital, and Ct

be consumption. Then the evolution equation for capital is given by

Ktþ1 ¼ A1�at Ka

t � Ct þ ð1� dÞKt: ð2Þ

Although the technological process is nonstationary, the ratios of capital totechnology, kt ¼ Kt=At; and that of consumption to technology, ct ¼ Ct=At; arestationary. We therefore represent the problem in terms of the stationary variables.Normalizing by the technology level, (2) becomes

ktþ1 ¼ yZstþ1ðka

t � ct þ ð1� dÞktÞ; ð3Þ

where we define the lognormal random variable Zs and constant y as

Zstþ1 ¼ expð�sWtþ1Þ; y ¼ expð�kÞ:

A representative agent has time-additively separable preferences over consumption,with CRRA period utility:

UðCÞ ¼ C1�g

1� g¼ A1�g c1�g

1� g:

The social planner’s problem is to choose a consumption sequence to maximize theexpected discounted utility of the representative agent. Thus we solve

supfCtg

EXNt¼0

btUðCtÞ ð4Þ

subject to (2) and (1). Note that expressing utility in terms of ct makes the effective

subjective discount factor bðyZstþ1Þ

g�1; and thus introduces a form of preference

shocks. Straightforward calculations, detailed in Appendix A, show that thisMarkov optimization problem has a solution which is a feedback control of the

ARTICLE IN PRESSN. Williams / Journal of Economic Theory 119 (2004) 271–298274

form: ct ¼ csðktÞ: This implies that we can write the optimal capital evolution asktþ1 ¼ yZs

tþ1ðkat � csðktÞ þ ð1� dÞktÞ

%f sðZstþ1; kt; csÞ; ð5Þ

where the notation in the second line emphasizes the dependence on the unknownconsumption policy function cs: As is standard, this policy function satisfies thestochastic Euler equation, derived in Appendix A:

csðkÞ�g

¼ bZ

ðyZsÞgcsð %f sðZs; k; csÞÞ�g½a %f sðZs; k; csÞa�1 þ 1� d� dGsðZsÞ; ð6Þ

where Gs is the relevant lognormal distribution function.In our analysis to follow, it helps to split the capital evolution into its conditional

expectation and its martingale component. Therefore, we define the expectation ofthe right side of (5), conditioned on kt ¼ k as

f sðkÞ ¼ y exps2

2

� �ðka � csðkÞ þ ð1� dÞkÞ; ð7Þ

and the random component is then

nsðkÞ ¼ y exp �sWtþ1 �s2

2

� �ðka � csðkÞ þ ð1� dÞkÞ: ð8Þ

Again, both of these depend on the unknown function cs; but we suppress thisdependence in the notation. Thus, we have an alternate expression for the capitalevolution:

ktþ1 ¼ f sðktÞ þ nsðktÞ: ð9ÞWhile the optimization problem is most naturally stated in terms of the level of thecapital stock, for most of our analysis we will work with the logarithm of the capitalstock. The multiplicative nature of the noise term in (5) makes the logs particularlyeasy to work with. Thus if we let lt ¼ logðktÞ; we can take logs and rewrite:

ltþ1 ¼ gsðltÞ � sWtþ1; ð10Þwhere we define:

gsðlÞ ¼ log½expðalÞ � csðexpðlÞÞ þ ð1� dÞ expðlÞ� � k: ð11ÞThe conditional normality of lt greatly simplifies many of the results that follow.

2.2. A deterministic growth model

Corresponding to the stochastic growth model, we can define a deterministicgrowth model by setting s ¼ 0 in the equations above. This yields a discrete timeversion of a standard Ramsey–Cass–Koopmans model in which the technologygrows at a constant rate. The implied evolution for capital is therefore:

ktþ1 ¼ y½kat � ct þ ð1� dÞkt�: ð12Þ

ARTICLE IN PRESSN. Williams / Journal of Economic Theory 119 (2004) 271–298 275

The optimal growth problem is then the deterministic analogue of (4), with law ofmotion (12). Again, straightforward calculations show that this problem has a

feedback solution of the form ct ¼ c0ðktÞ: This implies that capital and log capitalfollow the deterministic difference equations:

ktþ1 ¼ f 0ðktÞ; ð13Þ

ltþ1 ¼ g0ðltÞ; ð14Þ

where f 0 and g0 are obtained by replacing cs with c0 in (7) and (11), respectively. Theoptimal consumption policy satisfies the Euler equation analogous to (6):

c0ðkÞ�g ¼ bygc0ðf 0ðk; c0ÞÞ�g½af 0ðk; c0Þa�1 þ 1� d�: ð15Þ

2.3. Comparisons of the models

Let fkst g be a realization of the capital stock trajectory from the stochastic growth

model, fk0t g be the capital stock trajectory from the deterministic model, and flst gand fl0t g their respective logarithms. As s-0 we expect that, starting from the same

initial value, the sample paths of the solution of the stochastic growth model wouldapproach that of the deterministic model. In the rest of the paper we show that this isindeed the case, and we obtain explicit characterizations of the asymptotics atdifferent rates of convergence. These results provide us with approximatecharacterizations of the stochastic economy when the noise is small.Our results consider the normalized differences, for rX0:

Xs;rt ¼ 1

s1�r ðlst � l0t Þ: ð16Þ

In Section 3, we consider the case r ¼ 0; and present a functional central limittheorem (FCLT). We show that at rate s; the normalized differences converge to aGaussian linear autoregressive process. In Section 4, we consider the case r ¼ 1; andformulate a large deviation principle (LDP). Thus, we obtain bounds on theprobability that (on a given finite horizon) the stochastic capital trajectory differsfrom the deterministic one by a given amount. We also consider the exit problem,which provides estimates of how long it typically takes for the log capital stock todepart from its steady state level by a given amount. In Section 5, we consider arange of cases where 0oro1; and apply a so-called moderate deviation principle(MDP). These results are similar to the large deviation principle, but for s-dependentneighborhoods. This modification leads to explicit results. In Section 6, we illustrateour results through some explicit calculations and simulations in a calibrated model.Our results show that the theoretical predictions provide a good explanation of thebehavior of the model as observed in simulations. We also illustrate some of thedifferences between the large deviation and moderate deviation results, which haveimplications for linear solution methods. Finally, Section 7 concludes. Throughoutwe make smoothness assumptions on the consumption policies, and we also assumethat the stochastic policies converge to the deterministic one. In Appendix A, we


formally establish these results. Appendices B and C collect proofs of some of theresults in the text.

3. A functional central limit theorem

In this section we begin our analysis of the stochastic growth model. We present afunctional central limit, which follows Klebaner and Nerman [21], and shows that

the normalized differences Xs;0t from (16) converge to a Gaussian linear

autoregression. We also show that a similar result holds for the capital levels. Asnoted above, in order to make clear which features of the model are required for eachresult we make direct assumptions on the policy functions. These assumptions areverified in our application in Appendix A. Because this model has been so widelystudied, some of the smoothness and stability conditions we require follow fromknown results in the literature. We also require that the policy functions convergeuniformly, and this assumption requires the most argument. We establish it bybuilding up from properties of the value functions.To cover both logs and levels, in the assumptions we use hðxÞ as a stand in for

either gðlÞ or f ðkÞ; and we always use a subscript x for a derivative of a function.Klebaner and Nerman [21] impose boundedness conditions on the derivatives ofpolicy functions which cause some slight complications in our analysis. Due to theInada condition on the utility function, the slopes of the policy functions (in levels)increase to infinity at zero capital. However, on any compact set bounded away fromzero, the boundedness conditions are satisfied. Therefore, we extend the results bytruncating the state evolution to a compact set, but relaxing the truncation in thelimit. In what follows, we require xAX with X a compact set. This notation is astand in for lALCR and kAKCRþþ with L and K compact.

Assumption 1. On any compact set X; the function hs is continuous, twicecontinuously differentiable, and has bounded derivatives hs

x and hsxx for all sX0:

Assumption 2. On any compact set X; hs-h0 uniformly as s-0:

Assumption 3. On a compact setX; h0 has a unique fixed point x� which is stable, i.e.

jh0xðx�Þjo1; and whose domain of attraction includes all of X:

Theorem 1. Suppose that Assumptions 1 and 2 hold for h ¼ g: Then as s-0; the

normalized differences fXs;0t g converge weakly to a process fXtg: The limit process

follows the linear Gaussian autoregression, dependent on the deterministic process fl0t g:

Xtþ1 ¼ g0xðl0t ÞXt þ Btþ1; ð17Þ

where fBtþ1g is an i.i.d. sequence of standard normal random variables.

Proof. See Appendix B. &


Although we focus on the case of log-normal technology shocks Zst ; Theorem 1

can be extended to a more general setting, although not as general as a typical centrallimit theorem. The theorem would continue to hold for any distribution whichconverges to a normal with small noise, i.e. for which:

logðZst Þ

s) Nð0; 1Þ as s-0:

Obvious examples include a Student’s t distribution or a multinomial approximationto a normal, wherein each case we increase the precision of the approximation as weshrink the standard deviation. Of more direct importance, for small s a lognormaldistribution also converges weakly to a normal (see Johnson and Kotz [13]). Thisunderlies our results on the limit distribution of the capital stock levels kt below.Since there is a unique steady state of the deterministic model, Theorem 1 implies

that the differences from the steady state are asymptotically normal. As this resulthas implications for approximate solution methods, we state it explicitly.

Corollary 2. Suppose that Assumptions 1–3 hold for h ¼ g: Then as s-0 and t-N

the following holds asymptotically:

lstþ1 � l� ¼ g0xðl�Þðlst � l�Þ þ sBtþ1:

Therefore, as s-0 and t-N; flst g converges to a stationary Gaussian process with

mean l� and variance s2

1�g0xðl�Þ2:

This result shows that the log capital stock asymptotically follows a Gaussianlinear autoregression centered on the deterministic steady state. One simpleapplication of this result concerns the calibration of models. Typically, models arecalibrated to match the first moments of key time series. However, Corollary 2allows the use of analytic asymptotic second moments as well. In particular, itprovides a means of calibrating the risk aversion parameter ðgÞ by using the(asymptotic) autocorrelation of the capital stock, which equals the derivative of thepolicy function at the steady state. Notice that this calibration is completely analytic,and does not require simulation of the model. We use this result to calibrate themodel in Section 6.

For the case of the levels, we substitute kt for lt in our definition of Xs;0t : Then,

making the appropriate substitutions, we have the following results.

Theorem 3. Suppose that Assumptions 1 and 2 hold for h ¼ f : Then as s-0; the

normalized differences converge weakly to the process:

Xtþ1 ¼ f 0x ðk0t ÞXt þ f 0ðk0t ÞBtþ1:

Suppose in addition that Assumption 3 holds for h ¼ f : Then the results of Corollary 2

hold with sf 0ðk�Þ in place of s and the other obvious substitutions.


4. A large deviation principle

In the previous section, we showed that as the technology shock standarddeviation s goes to zero, the log capital stock from the stochastic growth modelconverges to the deterministic one and the normalized differences are asymptoticallynormal. In this section we consider the asymptotics of the differences, withoutnormalizing by s: We provide a large deviation principle (LDP) which shows that,on a given finite horizon, the differences converge to zero exponentially fast. We usethis to analyze episodes where the log capital stock differs from its steady-state valueby a given amount. We show that the time between such events increasesexponentially, and we provide an estimate of the typical length. We also show thatthe curvature of the policy function determines whether large increases or decreasesare more likely. In Section 6, we illustrate the implications of these results forbusiness cycles. The results in this section follow Klebaner and Zeitouni [22], withour proofs in Appendix C.1.We now define some terminology. Let a sequence fZeg be defined on a probability

space ðO;F;PÞ and taking values in a Polish space X: A rate function S :X-½0;N�has the property that for any MoN the level set fxAX : SðxÞpMg iscompact.

Definition 4. A sequence fZeg satisfies a large deviation principle on X with ratefunction S and speed e if the following two conditions hold.

(1) For each closed subset F of X; Ze satisfies the large deviation upper bound:

lim supe-0

e log PfZeAFgp� infxAF

SðxÞ:

(2) For each open subset G of X; Ze satisfies the large deviation lower bound:

lim infe-0

e log PfZeAGgX� infxAG

SðxÞ:

If the limit of the sequence is not in F and G; the probability the sequence entersthese sets converges to zero. A LDP insures that the convergence is exponential, withthe leading exponent determined by the rate function. The definitions also show howin large deviation theory the evaluation of a probabilistic statements is characterizedby an optimization problem. This makes the theory amenable for analysis and leadsto natural solution methods to apply it in practice.We now develop a LDP for finite time paths of the log capital stock, considered as

elements of the product space LTCRT : We equip the space with Euclidean metric,denoted dT ; and let ½u�T ¼ ðu0; u1;y; uTÞ: Then we have the following result.

Theorem 5. Suppose that Assumptions 1 and 2 hold for h ¼ g: Then on a given finite

horizon ½0;T � the sequence flst g satisfies a large deviation principle on ðLT ; dTÞ with


speed s2 and rate function:

SðT ; ½u�TÞ ¼1

2

XT�1

t¼0ðutþ1 � g0ðutÞÞ2: ð18Þ

Thus, Theorem 5 shows that the probability the stochastic and deterministic pathsdiffer by a given amount converges to zero exponentially fast, and determines therate of convergence. The rate function for the time paths is the cumulation of eachone-step transition. Since each step is conditionally normal, the rate function isquadratic. The theorem also implies that once the capital stock reaches itsdeterministic steady-state level, it has a small probability of exiting a neighborhoodof the steady state.We now spell out this implication of the LDP by considering what is known as the

exit or escape problem. To begin with this analysis, we define:

Vðx; yÞ ¼ inff½u�T : u0¼x;uT¼y;ToNg

SðT ; ½u�T Þ: ð19Þ

This gives the minimized ‘‘cost’’ of moving from x to y in some finite horizon, asevaluated by the rate function S: A path ½u�T which achieves the minimum clearly

exists and is called a dominant escape path. We focus mainly on escapes from thedeterministic steady state, so we define:

%VðeÞ ¼ inffy: jy�l�jXeg

Vðl�; yÞ: ð20Þ

Let Y � be the set of minimizers in (20) and define the exit time from the interval as:

tsðeÞ ¼ infft40: jlst � l�jXe; jls0 � l�joeg:

We now show that the exit times increase to infinity exponentially fast at a given rate,and we determine the end of the interval where an exit will most likely take place.

Theorem 6. Suppose that Assumptions 1–3 hold with h ¼ g: Then given e40; for any

Z40 we have:

lims-0

P exp%VðeÞ � Zs2

� �ptsðeÞpexp

%VðeÞ þ Zs2

� �� ¼ 1:

For any exit which occurs at lsts and for any Z40 there exists a y�AY � such that:

lims-0

Pðjlsts � y�joZÞ ¼ 1:

To apply these results, we must solve the minimization problem in (19)–(20), whichdoes not generally have an explicit solution. In Section 6 we solve the problemnumerically, while we now provide a local characterization. We show that for smallescapes, the curvature of the policy function determines whether a positive ornegative escape is more likely. Loosely speaking, with a convex (log) capitalaccumulation function, the proportion of income saved increases with l and hence


there is a slower return to l� following a positive shock. The formal result applies aperturbation argument, as detailed in Appendix C.2.

Theorem 7. Suppose that Assumptions 1–3 hold for h ¼ g: In a neighborhood of the

stable equilibrium l� the rate function in (19) has the expansion:

Vðl�; yÞ ¼ ðl� � yÞ2

2ð1� g0xðl�Þ

2Þ � ðl� � yÞ3

2g0xxðl�Þg0xðl�Þ

2 1� g0xðl�Þ2

1� g0xðl�Þ3

þ Oðjl� � yj4Þ:

Thus if g0 is strictly convex (respectively, strictly concave), an exit from the interval

ðl� � e; l� þ eÞ happens at l� þ e (respectively, l� � eÞ with probability converging to one

as s-0 and e-0: If g0 is linear the exit is equally likely to happen at either endpoint.

5. A moderate deviation principle

In this section we derive some asymptotic results which in effect combine theinsights of the two previous sections. We present a moderate deviation principle

(MDP), which considers the sequence Xs;rt for the intermediate case 0oro1;

whereas we have previously considered r ¼ 0 and 1. In essence, a moderate deviationprinciple is a large deviation principle for the normalized process over a slower rangeof speeds. Our results in this section provide simple, explicit asymptoticcharacterizations.Following Klebaner and Liptser [20], we first provide some of the heuristic

arguments which convey the main ideas. For rAð0; 1Þ we can write Xs;rt ¼ srX

s;0t :

Now by Theorem 1, we know that Xs;0t ) Xt where fXtg is the Gaussian linear

autoregression in (17). Now suppose that the processes fXs;rt g and f eXXs;r

t g ¼ fsrXtgsatisfy the same large deviation principle.4 Recall from (17) that f eXXs;r

t g satisfies:eXX s;rtþ1 ¼ g0xðl0t Þ eXX s;r

t þ srBtþ1:

Then relying on Theorem 5, we could show that on an infinite horizon the sequence

f eXXs;rt g satisfies a large deviation principle with speed s2r and rate function:

Jð½u�Þ ¼ 1

2

XNt¼0

ðutþ1 � g0xðl0t ÞutÞ2; ð21Þ

where and ½u� ¼ ðu0; u1;yÞ: The rate function J is quadratic in u; as it relies on thecentral limit results, and this allows for explicit analytic characterization.We now verify that these heuristics do in fact hold. The results here are the parallel

of Theorems 5 and 6, with proofs also in Appendix C.1. We focus on the infinitehorizon MDP, which implies a corresponding MDP over any finite horizon. We thus

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4As emphasized by Kushner [23], this will not hold in general, as there is only weak convergence.


extend the state space to LN and equip it with the metric:

dð½u�; ½v�Þ ¼XNt¼0

2�t�1 jut � vtj1þ jut � vtj

:

We now define the exit time for the normalized process:

ts;rðeÞ ¼ infft40jX s;rt jXeg:

This is the first time the process flst g exits from an window of width 2es1�r centered

on the path fl0t g: The MDP exit sets thus shrink with s; unlike the LDP. Then wehave the following results, providing the MDP and a characterization of the exitproblem.

Theorem 8. Let 0oro1; and suppose Assumptions 1–2 hold for h ¼ g:

1. Then the sequence fXs;rt g satisfies a moderate deviation principle on the infinite

product space ðLN; dÞ with speed s2r and rate function J given in (21).

2. Suppose that Assumption 3 holds as well, and let l00 ¼ l�: Then we have:

lims-0

s2r log Ets;rðeÞ ¼ e2

2ð1� g0xðl�Þ

2Þ %JðeÞ:

6. An application

In this section we illustrate our theoretical results in a calibrated model. Wechoose the parameters of the model so that it matches certain features of US time-series data. To calibrate the process for At; we used data on the cumulative Solowresidual from the DRI database, following the construction of Stock and Watson[32].5 The mean growth rate k and standard deviation s of the At process are chosento match the implied Solow technology process at an annual frequency.The remaining parameters are chosen in accordance with our theoretical

predictions. From our Corollary 2 we predict that for small s the mean of the logcapital stock should be l� and its autocorrelation g0xðl�Þ: In Appendix A, we provideanalytic expressions for these variables in terms of the model parameters. From ourdata sample, we find that the log capital stock has mean 1.396 and annualautocorrelation of 0.943. Given the values of a and k above, for any choice of therelative risk aversion parameter g the expressions for ðl�; g0xðl�ÞÞ determine pairsðb; dÞ which are consistent with the data. However for arbitrary g; we are notguaranteed that b and d will be between zero and one. By experimenting, we foundthat setting g to the plausible, if slightly high, value of 4 led to reasonable results for

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5The data are quarterly from 1959:Q1 to 1999:Q2 and are constructed from output (GDP less farm,

housing and government), capital (interpolation of annual values of fixed non-residential capital stock

using quarterly investment), and labor (hours of employees on non-agricultural payrolls). We then

construct the technology shock process using a labor’s share value of a ¼ 0:65:


the subjective discount rate and depreciation. Our specific parameter choices aresummarized in Table 1.To provide a basis of comparison for our theoretical results, we numerically solve

for the policy functions. Appendix C.3 describes our solution method. In Fig. 1, weplot the log capital accumulation function from the deterministic model along withits linear approximation. Here we see that the policy function is nearly linear, with aslight convexity which is apparent far from the steady state. This provides a preview

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Table 1

The baseline model parameters

Parameter Description Annual value

k Technology growth 0.0176

s Technology shock standard deviation 0.0492

a Labor’s share 0.65

d Depreciation 0.0517

b Subjective discount factor 0.9847

g Relative risk aversion 4

-4 -2 0 2 4 6 8 10 12 14 16

-4

-2

0

2

4

6

8

10

12

14

16Log Capital Accumulation Functions

Log Capital Stock

NonlinearLinear Approximation

Fig. 1. The log capital accumulation function from the numerical solution of the deterministic growth

model and its linear approximation.


of some of the results to follow: the linear approximation provides a goodcharacterization of the policy function except far from the steady state. Furthermoreby Theorem 7, the slight convexity suggests that positive escapes will be more likely.We will identify large movements in the log capital stock with business cycles,

which is justified in Fig. 2. The figure plots simulated data from the baselineparameterization, showing the value of the log capital stock minus its deterministicsteady-state level and the detrended log output. The figure clearly shows the negativecorrelation between the log capital stock and log output, with large changes inoutput corresponding to the large changes of opposite direction in the capital stock.The dotted lines in the figure plot the escape sets that we consider below. Notice thatthe negative escapes correspond to booms (peaks of output relative to trend), whilepositive escapes correspond to recessions (falls in output relative to trend).We now illustrate our asymptotic characterizations. Table 2 summarizes the

simulation results for different levels of the standard deviation of the technologyshock process, which we index by multiples of our benchmark s from Table 1.Turning first to the functional central limit theorem predictions, the first twocolumns of the table give the autocorrelation and standard deviation of the logcapital stock lt from simulations of 5000 periods. Corollary 2 above providestheoretical predictions for the mean (1.396), autocorrelation (0.943), and thestandard deviation of the series (0.148), which formed the basis of our calibration.For the benchmark s; the predictions are very close to the simulated values in the

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0.3

0.4

0.5

Capital Stock and Detrended Output

Detrended Log Output Log Capital Stock - Steady State

Fig. 2. A simulation of the logarithm of the capital stock and detrended output. The dashed lines show the

large deviation bands.


table. Further, the standard deviation decreases proportionately to s in thesimulations, just as the theory predicts.We next apply the LDP and MDP to characterize rare events in the log capital

stock process. In both cases, the probability of an escape and the mean escape timesare characterized by the rate function. For the MDP, Theorem 8 provides thisexplicitly, while for the LDP we must use numerical methods, as described in

Appendix C.3. The rate functions %V and %J for the LDP and MDP are shown in Fig. 3for different values of the escape set e: We know that the MDP rate function issymmetric, while the nonlinearity of the policy function causes the LDP ratefunction to be slightly asymmetric. For small escape sets, the two are similar, as thepolicy function is well-approximated by a linear function over much of the statespace. Therefore, even for the analysis of some rare events, the linear approximationmay provide acceptable results. However, just as the linear approximation breaksdown far from the steady state, for large escapes the rate functions differsubstantially. Due to the convexity of the policy function, the LDP rate functionis lower for positive escapes, which confirms the local analysis of Theorem 7. Buteven the small differences for small escape sets matter, as they lead to differences inthe exponential increase in the time between escapes. Therefore, for small s bothlarge booms and large recessions become increasingly unlikely, but we expectrecessions to occur more frequently than booms.We now illustrate these results through simulations. The last four columns of

Table 2 provide a summary of 5000 simulated escape paths for different levels of thetechnology shock. For each simulation run, we initialize the log capital stock at thesteady state and simulate until either a positive or negative escape happens.6 We letthe size of the escape set be fixed at e ¼ 0:222; corresponding to a nearly 16% changein the log capital stock, a sizable fluctuation. For the MDP, we let r ¼ 0:5: Recallthat the MDP escape sets shrink with s; and thus escapes are more likely in this case.

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Table 2

Simulation results for different levels of the technology shock standard deviation

Tech. FCLT LDP MDP

Std. Mean level Auto. Std. Mean Positive Mean Positive

dev. corr. dev. time (percent) time (percent)

2 s 1.32 0.946 0.301 10.1 49.6 18.9 49.8

s 1.37 0.947 0.152 42.3 51.6 42.3 51.6

0.75 s 1.38 0.947 0.114 108.3 52.8 65.5 53.5

0.5 s 1.38 0.947 0.076 1008.1 58.2 132.0 53.1

6As s goes to zero, lt converges to l�; the fixed point of g0: But for s40 the fixed point of gs is different

from l�: In our simulations, we thus look at escapes from the fixed point instead of from l�: This hasessentially no effect on the mean escape times, but it does affect the results on the asymmetry of escapes.

Note that in Table 2 the mean of lst converges to l� from below. By analyzing escapes from l� we would

distort the results in favor of negative escapes.


The table clearly shows the rapid increase in the mean escape times in both cases.The asymmetry is also evident, especially in the LDP results. At the baseline s; wefind that recessions account for 52% of the escapes. When s is cut in half, thisincreases to 58% of the LDP escapes (and over 53% for the MDP). For small noisethe asymmetry of the large business cycle fluctuations becomes more apparent, andas predicted we find recessions to be more likely than booms.Our results also give accurate predictions of the rate of increase in escape times.

For the LDP, Theorem 6 shows that for small s the escape times are approximately:

log EtdsElogCL þ %V=s2;

for some constant CL: Similarly, for the MDP with r ¼ 0:5; Theorem 8 shows thatfor small s the mean escape times are approximately:

log Etds;0:5ElogCM þ %J=s;

for some constant CM : Thus, the escape times increase exponentially with the inverseof standard deviation for the MDP and the inverse of the variance for the LDP, withthe rate of increase determined by the rate function. This is shown graphically inFig. 4, where we plot the log of the mean escape times from Table 2 along with ourpredictions. We choose the constants CL and CM in order to get a good fit, as ourresults only determine the slopes of the lines. The figure shows that our predictions

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0.05Rate Functions for Different Escape Sets

Multiple of Sqrt

Large Deviation Moderate Deviation

Fig. 3. Comparisons of the rate functions for the large deviation principle and the moderate deviation

principle, for varying escape sets (indexed by e ¼ nffiffiffis

p; for different n).


accurately characterize the rate of increase in the mean escape times. Further, boththe LDP and MDP predictions are much closer for small s; as our theory predicts.

7. Conclusion

In this paper we have presented and applied a variety of asymptotic methodswhich characterize the properties of the stochastic growth model. We have shownthat as the standard deviation of the technology shock process gets small, the logcapital stock process converges to a Gaussian linear autoregression. Further, wehave characterized the probability and frequency of large fluctuations in the logcapital stock. Our results have shown that for small noise, the capital stock processconverges to the deterministic steady state, and that the autocorrelation and varianceof the log capital stock process can be characterized analytically. Additionally, wehave shown that for small noise large booms and recessions become increasinglyunlikely, although recessions are more likely than booms. Finally, we showed thatthe average time between business cycles increased exponentially, and providedaccurate predictions of the rate of increase.For several reasons, in this paper we have focused on the stochastic growth model.

It is one of the most widely used models, and is a standard test case for new methods.Further, the fact that there was a single state variable provided substantial technicalsimplifications. However it should be clear that our results apply much morebroadly, and could be extended to multidimensional cases. Of particular interest maybe our implications for linearization methods. We show that for small noise, a linearapproximation provides a characterization of the average behavior of the model andalso certain occasional large fluctuations. As linearization remains one of the mostcommon solution methods, these results have many potential applications.

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7Log Mean Large Deviation Escape Times

1/Multiple of σ2

SimulatedPredicted

SimulatedPredicted

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3

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3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5Log Mean Moderate Deviation Escape Times

1/Multiple of σ(a) (b)

Fig. 4. Logarithm of mean escape times for different standard deviations. The solid lines plot the results

from simulations, while the dashed lines plot predicted results. The left panel plots the results for the large

deviation principle, and the right panel plots the moderate deviation principle.


Acknowledgments

I thank William (Buz) Brock, Matthias Doepke, Han Hong, Ken Judd, MakotoNirei, Simon Potter, Chris Sims, and seminar participants at Columbia University,Stanford University, the Federal Reserve Bank of New York, the 2002 Society forEconomic Dynamics meetings, the 2003 Econometric Society Winter Meetings, the2003 Society for Computational Economics Meetings, and the Conference onLearning and Bounded Rationality at the University of Illinois for helpfulcomments. Comments by an associate editor and an anonymous referee helpedimprove the paper substantially. Financial support from the National ScienceFoundation is gratefully acknowledged.

Appendix A. Properties of the policy functions

As described previously it is convenient to scale consumption and capital by thetechnology level. This allows us to derive Bellman equations that do not depend onA; but only on k; thus simplifying our analysis. As is well-known, the Bellmanequation for the original problem is

vðK ;AÞ ¼ maxC

C1�g

1� gþ b

ZvðK 0;A0Þ dGðZÞ

� �s:t: ð1Þ; ð2Þ;

where G is the log-normal distribution function. We use the normalizations k ¼ K=A

and c ¼ C=A; then verify that the value functions satisfy vðK ;AÞ ¼ A1�gvðkÞ: Thisleads to the reduced Bellman equation:

vðkÞ ¼ maxc

c1�g

1� gþ b

ZðyZÞg�1vðk0Þ dGðZÞ

� �s:t: ð3Þ: ðA:1Þ

The utility function satisfies the Inada conditions, so that we know solutions must beinterior. The first-order condition and an application of the envelope theorem thenyield the Euler equation (6) in the text. Analogous results in the deterministic caselead to the Euler equation (15).Standard results, as in Stokey et al. [33] show that in both the deterministic and

stochastic cases the policy functions k0 ¼ f 0ðkÞ and f sðkÞ are continuous andbounded for finite k: Simple arguments also show that the policy functions arestrictly increasing, and further that the consumption policies are also strictlyincreasing. Results from Araujo [2] and Santos [29] show that the policy functionsare continuously differentiable in the deterministic case, and Santos [29] provides abound on the derivative. By adapting results from Amir [1], we also have that thepolicy functions are twice continuously differentiable in the stochastic case (see alsoBlume et al. [4]), and are bounded on the interior of the state space. Finally, theresults in Santos [30] suggest that in our model the policy function in thedeterministic case is also twice continuously differentiable. As an optimal path isalways interior and, as we show below, there is a unique interior stable steady state,


we can apply the implicit function theorem to establish the higher-order smoothnessof the policy function.We next turn to deriving analytic expressions for the deterministic steady state l�

and the derivative of the policy function at the steady state g0xðl�Þ: These expressionsform the basis of our calibrations. From the Euler equation (15), the unique interiordeterministic steady state is easily seen to be

k� ¼ 1� byg þ dbyg

abyg

� �1=a�1:

Taking logs gives l�: Then from (3), we find the steady-state consumption c� ¼ðk�Þa þ y�1�dy

y k�: This in turn allows us to evaluate the derivatives of the utility

function at the steady state: u0 ¼ ðc�Þ�g; and u00 ¼ �gðc�Þ�g�1: Following Juddand Guu [16], we apply the implicit function theorem to the Euler equation (15) tofind an expression for the derivative of the consumption function at the steady

state c0 ¼ c0xðk�Þ:ðc0Þ2½bygþ1u00ðaðk�Þa�1 þ 1� dÞ� þ c0½u00 � bygþ1u00ðaðk�Þa�1 þ 1� dÞ2�

� bygu0aða� 1Þðk�Þa�2 ¼ 0:

Using the expression for k�; this can be reduced to

ðc0Þ2 þ D1c0 þ D2 ¼ 0; with:

D1 ¼1

y� 1

byg; D2 ¼ �bygþ1

c�

gað1� aÞðk�Þa�2:

Since D2o0; this insures that c0 is real, and since we know c0 is positive, we have that:

c0 ¼ �D1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD21 � 4D2

q4� D1:

With this expression, we can then evaluate g0xðl�Þ:

g0xðl�Þ ¼aðk�Þa � k�c0 þ ð1� dÞk�

ðk�Þa � c� þ ð1� dÞk� :

As f 0 is strictly increasing, g0x ¼ kf 0x =f 0 is strictly positive. Using the expressions

for c0; c�; and k� we have:

c04� D1 ¼1

y� 1

byg¼ aðk�Þa�1 þ 1� d� 1

y

¼ c�

k� � ð1� aÞðk�Þa�1

and therefore g0xðl�Þo1; so that the unique interior steady state is stable.All that remains to be shown is the convergence of the stochastic policy function

to the deterministic one. As above, we confine our attention to a compact setKCð0;NÞ: Our proof adapts results in Santos and Vigo [31]. First note thatthe sequence fZsg converges weakly to 1 as s-0: Next consider the deterministic

value function v0ðkÞ; and note that ðZsÞg�1v0ðZsxÞ is continuous and uniformly


integrable (since v0 is bounded above). Therefore,

lims-0

ZðZsÞg�1v0ðZsxÞ dGsðZsÞ ¼ v0ðxÞ: ðA:2Þ

Furthermore, since vðK ;AÞ is a concave function for all sX0;

ðZsÞg�1v0ðZsxÞpv0ðxÞ;

and therefore the convergence in (A.2) is monotone, and therefore by Dini’sTheorem (see [8]) uniform on compact sets. Next, we define the Bellman operator on

the right side of (A.1) as Ts; with the corresponding operator T0 in the deterministic

case. By our uniform convergence result, we then have that Tsv0ðkÞ-T0v0ðkÞ:Additionally, since Tsv0ðkÞpT0v0ðkÞ ¼ v0ðkÞ; this convergence is again uniform oncompact sets. Finally, letting jj � jj be the sup norm on the compact set K; we have:

jjv0 � vsjj ¼ jjT0v0 � Tsvsjj

p jjT0v0 � Tsv0jj þ jjTsv0 � Tsvsjj

p jjT0v0 � Tsv0jj þ byg�1jjv0 � vsjj

by the triangle inequality and the contraction property of Ts: Thus,

jjv0 � vsjjp 1

1� byg�1jjT0v0 � Tsv0jj

and so vs converges uniformly to v0; which implies that cs converges pointwise to c0:

This in turn implies the pointwise convergence of gs to g0:To show that the convergence of the policy functions is uniform, for a given k

define the function:

GðcÞ ¼ c1�g

1� gþ byg�1v0ðyðka � c þ ð1� dÞkÞÞ:

Let c0 ¼ c0ðkÞ and cs ¼ csðkÞ and note that Gðc0Þ ¼ v0ðkÞ and GðcsÞXTsv0ðkÞ sothat Gðc0Þ � GðcsÞpT0v0 � Tsv0: Then note that G0ðc0Þ ¼ 0 and GðcÞ is concavewith a bounded second derivative on K (since the utility and value functions havefinite second derivatives for k40). Call the bound Z: Then we have:

Z2ðc0 � csÞ2pGðc0Þ � GðcsÞ

and therefore:

jc0 � csjpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ2jT0v0 � Tsv0j

r:

Since this holds for all k; we can take the sup over k; and the uniform convergencethus follows.

Moreover, the convergence of the value functions onK is exponential at rate s2;implying that the convergence of the policy functions is exponential at rate s: This is


evident from the following:

jjTsv0 � T0v0jj ¼ maxc

c1�g

1� gþ b

ZðyZÞg�1v0ðk0ÞdGðZÞ

� ��max

c

c1�g

1� gþ byg�1v0ðk0Þ

� ��p byg�1 max

k0

ZðyZÞg�1v0ðk0Þ dGðZÞ � v0ðk0Þ

� �� p byg�1 max

x

ZZg�1v0ðZxÞ dGðZÞ � v0ðxÞ

� �� p byg�1 max

xv0xðxÞxE½expððg� 1ÞsWÞ � expðgsWÞ�

�� ¼ byg�1 max

xv0xðxÞx exp

1

2ð1� gÞ2s2

� �� exp 1

2g2s2

� �� :Here the third line uses the definition of k0 in the stochastic model, the fourth uses

the concavity of v0; the boundedness of the derivatives of v0 onK; and the definitionof Z; and the fifth uses the properties of lognormal random variables. On thecompact set K the capital stock and the derivative of the value function are bothbounded, yielding the result.

Appendix B. Functional central limit theorem

In this appendix, we prove Theorem 1 by extending the results of Klebaner andNerman [21]. The basis of our result is their Theorem 3, which requires thatAssumption 1 hold globally, not just on restrictions to compact sets. We use atruncation argument following Kushner and Yin [24] in order to extend the resultand show that the truncation is not needed in the limit. As in most of the paper, wefocus on the results in logarithms, i.e. for l ¼ log k: The results for the levels aresimilar, and follow from the following preliminary result, which follows simply fromthe definitions of the functions and L’Hopital’s rule.

Lemma 9. Let Assumptions 1 and 2 hold for h ¼ f : Then for each fixed k; we have:

lims-0

1

s2EnsðkÞ2 ¼ f 0ðkÞ2:

In this section, since we focus on the case r ¼ 0 we suppress the second superscript

and write Xst ¼ X

s;0t : First we truncate the process. That is, for every integer M let

qM be a continuous function on the line satisfying: �MpqMðxÞpM; qMðxÞ ¼ x for

jxjpM; qMðxÞ ¼ M for xXM þ 1; and qMðxÞ ¼ �M for xp� ðM þ 1Þ: Note thatqM

x ðxÞ ¼ 1 for jxjpM and 0 for jxjXM þ 1: Then let ls;M0 ¼ ls0 and for tX0 define


the truncated process:

ls;Mtþ1 ¼ gsðqMðls;Mt ÞÞ � sWtþ1: ðB:1Þ

Then we define the normalized differences: Xs;Mt ¼ l

s;Mt �l0t

s : Note that we do not need

to truncate l0t as long as jl00 joM; which we assume. Under our assumptions, the

truncated process (B.1) satisfies the conditions of Klebaner and Nerman [21],Theorem 3. Since the truncation is not applied to the deterministic process, we have

that truncated normalized differences fXs;Mt g converge weakly to the autoregression

fXtg in (17).We now show that the truncation is unnecessary in the limit by establishing the

tightness of the untruncated normalized process. The results of Klebaner andNerman [21] establish weak convergence in the product space, so it is enough toverify tightness at each date. Thus, we need to show that for any t we have:

limK-N

sups

PðjXst jXKÞ ¼ 0:

By Prohorov’s Theorem (see Kushner and Yin [24], Theorem 7.3.1), tightness impliesthe existence of a weakly converging subsequence. As the truncation is not applied inthe limit, the weak limit of the untruncated process must be the same as the truncatedprocess.Notice that from any initial condition xAL; we have:

X s1 ¼ ls1 � l01

s¼ gsðxÞ � g0ðxÞ

sþ W1:

The tightness of this sequence thus follows by the tightness of the normal randomvariable W1: Then we proceed by induction. Thus for any t we have:

Xs;0tþ1 ¼

lstþ1 � l0tþ1s

¼ gsðlst Þ � sWtþ1 � g0ðl0t Þs

¼ 1s½gsðlst Þ � g0ðlst Þ þ g0ðlst Þ � g0ðl0t Þ� � Wtþ1:

Therefore, we have

PðjX st jXKÞpPðjgsðlst Þ � g0ðlst ÞjXKsÞ þ Pðjg0ðlst Þ � g0ðl0t ÞjXKsÞ

þ PðjWtþ1jXKsÞ: ðB:2Þ

By the tightness of the normal distribution, the third term goes to zero with K : Thennotice that we have:

jg0ðlst Þ � g0ðl0t Þj ¼ jg0xðl0t Þjjlst � l0t j þ Oðjlst � l0t j2Þ:

Since x ¼ l00pl0t pl� the derivative is bounded and by the induction hypothesis

jlst � l0t j converges to zero with s; and the second term is of order s2: This impliesthat the second term in (B.2) converges to zero at rate s; which in turn implies that itgoes to zero with K : Finally, we split the first term in (B.2) depending on whether lst


is in L: Notice that:

Pðjgsðlst Þ � g0ðlst ÞjXKÞ ¼Pðjgsðlst Þ � g0ðlst ÞjXK jjlst ALÞPðlst ALÞ

þ Pðjgsðlst Þ � g0ðlst ÞjXK jjlst ALcÞPðlst ALcÞ: ðB:3Þ

Notice that by the LDP results in Section 4, the probability of being inL at any datet is of order 1 as s-0 and the probability of being inLc goes to zero (exponentially

fast) with s2: Further, we have previously established that onL the policy functionsconverge uniformly at rate s: Thus the fist term on the right of (B.3) converges to

zero at rate s: To bound the term on Lc notice that limk-0 f sðkÞ=f 0ðkÞ ¼ 1 and

limk-N f sðkÞ ¼ Hk for some H40 for all sX0: Thus jgsðkÞ � g0ðkÞj is bounded onLc: Putting this together implies that both terms on the right of (B.3) converge tozero at rate s; and thus the first term in (B.2) converges to zero with K : Thiscompletes the proof of tightness, and so the result follows.

Appendix C. Additional proofs and calculations

C.1. Proofs of LDP and MDP Results

We begin with an intermediate result on the one-step transitions of the log capitalstock as in (10). We define lsx as the random variable with distribution identical to

ltþ1 conditioned on lt ¼ x:

lsxBNðgsðxÞ; s2Þ:

Lemma 10. Suppose that Assumptions 1 and 2 hold for h ¼ g: Then for any given finite

x; flsxg satisfies a large deviation principle on LCR with speed s2 and rate function:

Iðx; yÞ ¼ 12ðy � g0ðxÞÞ2: ðC:1Þ

This one-step LDP is a building block for the results to follow.

Proof. The result follows from the Gartner–Ellis Theorem (see [7], Theorem 2.3.6).To apply the theorem, we need to calculate the limiting logarithmic momentgenerating function:

Hðx; lÞ ¼ lims-0

s2 log E expllsxs2

� �¼ lg0ðxÞ þ l2

2; ðC:2Þ

where the second equality follows from Assumption 2. Then the rate function isgiven by the Legendre transform of H:

Iðx; yÞ ¼ suplAR

½ly � Hðx; lÞ�;

which gives the result.


Proof of Theorem 5. Follows from Klebaner and Zeitouni [22, Lemma 2.1]. Thenecessary conditions of their theorem are easily verified given our assumptions andthe form of the rate function I established in Lemma 10. In particular, their Lemma2.5 holds under Assumption 2 and implies the uniformity of the large deviationprinciple. The continuity of I and the other technical conditions follow as in theproof of their Theorem 3.1 under our Assumptions 1 and 2.

Proof of Theorem 6. Follows from Klebaner and Zeitouni [22, Theorem 2.1] and theremark which follows it. The necessary conditions are shown to hold using the samearguments as in our Theorem 5.

Proof of Theorem 8. Part 1 follows from Klebaner and Liptser [20, Theorem 2].Their necessary conditions are easily verified given our assumptions and the form ofthe H function defined in (C.2). Under our assumptions, Part 2 follows from Part 1and Klebaner and Liptser [20], Theorem 4.

C.2. Local expansion of the LDP rate function

In this appendix we derive the local expansion in Theorem 7. The idea adapts acontinuous time result of Kasa [17], but the derivation here is rather different andmore involved. We expand the function VðyÞ Vðl�; yÞ around y ¼ l�: At this pointthe optimal path is ut ¼ l� for all t: For a given y reached at date T ; each step ut ofthe optimal path will be an implicit function of y: However, an application of theenvelope theorem to problem (C.15) gives:

dV

dy¼ y � g0ðuT�1Þ; ðC:3Þ

d2V

dy2¼ 1� g0xðuT�1Þ

duT�1dy

; ðC:4Þ

d3V

dy3¼ �g0xðuT�1Þ

d2uT�1dy2

� g0xxðuT�1ÞduT�1

dy

� �2: ðC:5Þ

Note that the first derivative (C.3) is zero when evaluated at y ¼ uT�1 ¼ l�:To evaluate the higher-order derivatives, we use the first-order conditions from

problem (C.15):

ut � g0ðut�1Þ � ðutþ1 � g0ðutÞÞg0xðutÞ ¼ 0; ðC:6Þ

for t ¼ 1;y;T � 1: Differentiating (C.6) gives implicit expressions for dut

dy: When

evaluated at l� this gives:

ð1þ g0xðl�Þ2Þdut

dy� g0xðl�Þ

dut�1dy

� g0xðl�Þdutþ1

dy¼ 0; ðC:7Þ


with boundary conditions du0dy

¼ 0 and duT

dy¼ 1: For use below, note that (C.7) is a

homogenous second order difference equation, whose characteristic equation has

roots g0xðl�Þ and 1=g0xðl�Þ: Therefore, the general solution can be written:dut

dy¼ c1g

0xðl�Þ

T�t þ c2g0xðl�Þ

t�T

for some c1; c2: To satisfy the boundary conditions (for T-N) we must have c1 ¼ 1and c2 ¼ 0; so that

dut

dy¼ g0xðl�Þ

T�t: ðC:8Þ

Substituting this expression into (C.4) gives

d2V

dy2

��l�¼ 1� g0xðl�Þ

2: ðC:9Þ

Note that this agrees with the expressions in Lewis [26], who on pp. 32–35 explicitlysolves this problem with a linear state evolution. This follows since a problem with

linear evolution whose slope is g0xðl�Þ yields the identical difference equation (C.7).Using similar logic, we evaluate the second derivative with respect to the end point

y by implicitly differentiating (C.6) again. After simplifying and evaluating at l� weobtain:

ð1þ g0xðl�Þ2Þ d2ut

dy2� g0xðl�Þ

d2ut�1dy2

� g0xðl�Þd2utþ1

dy2

þ 3g0xðl�Þg0xxðl�Þdut

dy

� �2�g0xxðl�Þ 2

dut

dy

dutþ1dy

þ dut�1dy

� �2 !¼ 0:

Using (C.8), this simplifies to

ð1þ g0xðl�Þ2Þ d2ut

dy2� g0xðl�Þ

d2ut�1dy2

� g0xðl�Þd2utþ1

dy2

þ g0xxðl�Þg0xðl�Þ2ðT�tÞ�1ð3g0xðl�Þ

2 � 2� g0xðl�Þ3Þ ¼ 0; ðC:10Þ

with boundary conditions d2u0dy2

¼ d2uT

dy2¼ 0: Thus (C.10) is a nonhomogeneous

equation of the same form as (C.7). We transform it to a homogeneous equation

by defining the variable zt ¼ d2ut

dy2� mt; where

mt ¼g0xxðl�Þð3g0xðl�Þ

2 � 2� g0xðl�Þ3Þ

g0xðl�Þ4 � g0xðl�Þ

3 � g0xðl�Þ þ 1g0xðl�Þ

2ðT�tÞ: ðC:11Þ

Then zt follows the same difference equation as (C.7), but now with boundaryconditions z0 ¼ �m0 ¼ 0; zT ¼ �mT : This means that we have the solution:

zt ¼ �mT g0xðl�ÞT�t


and therefore:

d2ut

dy2¼ zt þ mt ¼ mTðg0xðl�Þ

2ðT�tÞ � g0xðl�ÞT�tÞ: ðC:12Þ

Evaluating (C.12) at T � 1 and using (C.11) gives:

d2uT�1dy2

¼ g0xxðl�Þg0xðl�Þð3g0xðl�Þ2 � 2� g0xðl�Þ

3Þg0xðl�Þ

4 � g0xðl�Þ3 � g0xðl�Þ þ 1

ðg0xðl�Þ � 1Þ: ðC:13Þ

Then substituting (C.8) and (C.13) into (C.5) and simplifying gives:

d3V

dy3

��l�¼ �3g0xxðl�Þg0xðl�Þ

2 1� g0xðl�Þ2

1� g0xðl�Þ3: ðC:14Þ

Collecting (C.9) and (C.14) along with Taylor’s theorem gives the expansion statedin Theorem 7. The conclusions follow from Theorem 6, where we note that by

Assumption 3 the sign of the third derivative term is determined by g0xx:

C.3. Numerical methods

To find the policy functions numerically, we solve the Euler equations (6) and (15)on a grid of 15 001 points centered on the steady-state level, using log-linearinterpolation. For the stochastic model, we approximate the conditional expectationusing Gauss–Hermite quadrature with 51 nodes. We also apply numerical methodsto solve for the dominant escape path for the LDP. Using (C.1) we can re-state theminimization problem as

inffToN;½u�Tg

1

2

XT�1

t¼0ðutþ1 � g0ðutÞÞ2 ðC:15Þ

subject to u0 ¼ l�; uT ¼ l�7e: Our solution strategy works with the first-orderconditions. For any given horizon T ; problem (C.15) is a two-point boundary valueproblem for a path ½u�T which starts at l� and ends at l�7e at date T :We convert theboundary value problem to an initial-value problem. The first-order conditions in(C.6) can be rewritten so that they imply a recursion for the optimal path. Inparticular, we have for 1ptpT � 1:

utþ1 ¼ g0ðutÞ þut � g0ðut�1Þ

g0xðutÞ: ðC:16Þ

With an arbitrary u1; we can then iterate on (C.16) until we hit a terminal uT which isat least e away from l�: Evaluating the rate function along this path, we obtain avalue SðT ; ðl�; u1;y; uTÞÞ: Then minimizing over the initial step u1; we obtain the

optimized rate function %V: This algorithm determines T endogenously, and provedto be very fast in practice. Note that the entire algorithm must be carried outnumerically, as (C.16) at each step uses the numerical policy function and itsderivative.


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Small noise asymptotics for a stochastic growth model

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