Waelde2009wp_ProdTechnologiesInStochContTimeModels

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Production Technologies in StochasticContinuous Time Models

Klaus Wlde

CESIFO WORKING PAPERNO.2831CATEGORY 6:FISCAL POLICY,MACROECONOMICS AND GROWTH

OCTOBER2009

An electronic version of the paper may be downloaded

from the SSRN website: www.SSRN.com

from the RePEc website: www.RePEc.org

from the CESifo website: Twww.CESifo-group.org/wpT


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CESifo Working Paper No. 2831

Production Technologies in StochasticContinuous Time Models

Abstract

Properties of dynamic stochastic general equilibrium models can be revealed by either usingnumerical solutions or qualitative analysis. Very precise and intuition-building results areobtained by working with models which provide closed-form solutions. Closed-formsolutions are known for a large class of models some of which, however, have someundesirable features. This paper offers closed-form solutions for models which are just astractable but do not suffer from these shortcomings.

JEL Code: E20, O40.

Keywords: dynamic stochastic general equilibrium models, closed-form solution, continuoustime, jump diffusion.

Klaus Wlde

Mainz School of Management and Economics

University of Mainz

Jakob-Welder-Weg 4

55131 Mainz

Germany

[email protected]

October 2009I am grateful to Florian Nolte for very valuable research assistance.


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constraint with differential formulations of the technology such that the formulation of thetechnology is standard. In this reading, our contribution is this reinterpretation and thegeneralization of results to a setup with a Wiener process and many Poisson processes.

2 The model

2.1 The planner

We study a classic saving-investment problem. A social planner chooses the amount ofconsumption () so as to maximize a social welfare function representing expected utilityof infinitely-lived households,

=

Z

[] (()) , (1)

where denotes the time preference rate. The instantaneous utility function is of the CRRAclass,

(()) =

(()11

1for 6= 1

log () for = 1 (2)

We see this central planner view as a short-cut to a decentralized economy without anydistortions. All of the results which will follow below will therefore also hold in an economywith many households.

2.2 Differential representation of the technology

This section presents the approach which specifies technologies by a differential. In orderto be comparable as much as possible to the approach appearing in section 2.3.2 below, wemodify the differential setup, as it can be found in the literature, in two ways. First, we makethe depreciation rate explicit. The depreciation rate is is usually implicitly contained inthe constant production technology factor We replace by which is of course just anotational matter and does not alter any of the implications of the differential model. Second,instead of adding a stochastic component in the equation for the production technology wesubtract this term. Again, this is just a notational matter and only changes the signs ofthe corresponding parameters of the stochastic processes considered. It will, however, beadvantageous later when comparing the approach we suggest with this differential approach.

The production process of an economy can now be specified - following a large part ofthe literature as cited in the introduction - by

() = ( ) () () () , (3)

where () is aggregate output in , ( ) is a constant measure of the TFP less thedeterministic depreciation rate , () is the capital stock, and the increment () will bespecified below. The flow of output is used for capital accumulation and for consumption,modeled as

() = () + () . (4)

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Combining equations (3) and (4) gives the expression for the capital accumulation which isequal to investment2

() = (( ) ()()) () () (5)

This and similar processes are used for example by Evans and Kenc (2003), Gertler andGrinols (1982), Gokan (2002), Gong and Zou (2002, 2003), Grinols and Turnovsky (1998a,1998b), Steger (2005), or Turnovsky (1993). In most of these papers the stochastic incrementin eq. (3) is defined to be () = () where () is a Brownian motion and the issome variance measure of output.

While some authors acknowledge (see e.g. footnote 4 in Grinols and Turnovsky (1998a))the fact that this implies the possibility of a negative , we briefly recapitulate this propertyhere by simply solving the differential equation (3) starting in = 0 with (0) 0 and(0) 0,

() = 0 + ( )Z

0

() Z

0

() ()

To see that this is in fact a solution of the above flow-representation just apply Itos Lemmaand recover (3).3 This says that with a constant factor input , i.e. () = 0+( ) (), output in is determined by a deterministic and a stochastic part. The latter one isoften interpreted to reflect various random elements affecting output production (e.g. Gongand Zou, 2003). Ignoring 0 the deterministic part ( ) implies linear growth (i.e. notexponential growth as is usually the case and empirically more relevant) and the stochasticpart () implies deviations from the trend. As () is Brownian motion, the sum of thedeterministic and the stochastic part can become negative.

The unusual formulation in (3) and the property that () can become negative might be

two reasons why this approach is not as popular as it could be. We will now present standardformulations of technologies which make models just as tractable as with this formulation.

2.3 Standard representation of the technology

We now present technologies in the standard way. We use a typical structure and intro-duce uncertainty either through uncertainty in TFP - standard I - or through uncertaintyin the capital accumulation process - standard II.

2.3.1 Standard I: stochastic TFP

This section will now preserve the spirit of the technology used above - the AK-structure -but expresses the technology in a way such that it has standard properties. The economy

2 This equation (5) can be obtained under the additional assumption that that () = () where...consumption over the instant occurs at the non-stochastic rate () . (cf. Turnovsky and Smith(2006) on page 251). The expression () = () can be found expicitly e.g. in Grinols and Turnovsky(1998a) on page 503 or in Turnovsky (2000) on page 400 and is at odds with the prediction of this model (seefurther below) that optimal consumption is a constant fraction of the capital stock, i.e. () = () making consumption stochastic and not just a function () times the deterministic time differential .

3 For an introduction to the methods behind this and for more background, see Wlde (2009, part IV).

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produces the final good according to

() = () () . (6)

Capital is the only factor of production and TFP is given by (). It follows the stochasticprocess

() = () + () () , (7)

where the increment () in its most general formulation will be specified below as acombination of one Wiener and many Poisson processes.

Capital accumulation is equal to investments () = ()() net of depreciation,

() = ( () ()()) . (8)

2.3.2 Standard II: stochastic depreciation

There is an alternative where output is left in the standard formulation (6), TFP remainsconstant at , i.e. () = 0, and hence (7) is not valid. In contrast, following Rebeloand Xie (1999), we assume that capital accumulation is stochastic. Let there be a standardgoods market clearing condition () = () + () but let capital accumulation be risky,e.g. by a stochastic depreciation process:

() = () () ( + ()) .

Again, () is a combination of a Wiener and Poisson processes. This capital accumula-tion equation gives, together with the goods market clearing condition and the productiontechnology, an expression of

() = (() ()()) () () . (9)

Notice that this equation is identical to the capital accumulation process of the differentialsetup in (5). Hence, if () is a simple Brownian motion we have a similar formulation ofthe resource constraint as Eaton (1981) and others, only that we have obtained it with amore standard setup with standard output and a stochastic capital accumulation process.

The interpretation that the stochastic part of the capital accumulation process is theresult of stochastic depreciation was already suggested by Eaton (1981, footnote 2). Inthe differential approach, stochastic capital accumulation is the result of combining theassumptions about the stochastic evolution of output in (3) and the goods market clearingcondition in (4). By contrast, in the standard II approach proposed here, the fact that capitalaccumulation is risky is an assumption which results in a stochastic evolution of output.

This has the advantage of explicitly specifying where uncertainty comes from and avoidsspecifications of output which can easily be criticized. With more conventional technologiesas proposed here, the continuous time approach to uncertainty could become more popular.

3 Closed form solutions

It is well-known that closed form solutions exist for a variety of models. Table 1 summarizesthe structure of the models presented above and provides a preview of the optimal consump-tion levels. This section generalizes existing closed-form solutions and compares them to the

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new ones - which are based on models with standard production functions.

Table 1: Model overview and preview of results

differential standard I standard II

(stoch. output) (stoch. TFP) (stoch. depreciation)output = ( ) = =

TFP constant

= + constantcapital = = =

investment = = = stoch. process = +=1 identical identical

log-utility = = = CES-utility = n.a. =

The differential approach widely used in the literature and its features are presented in thesecond column. Approaches with standard production functions appear in columns threeand four. The notation used in this table is as in the text. The only new parameter is

1

( ) + (1 )

1

22

1

=1

(1 )

1 1

(10)

Following the idea of Merton (1976), we allow the stochastic component () in thetechnology (column 2), in TFP (column 3) or in depreciation (column 4) to be a mixture ofboth a continuous and countably many jump processes, () = ()+=1 (), implyingthat

() = () +

=1 () . (11)The differential () stands for the increment of standard Brownian motion, i.e. () (0 ), whereas the () for = 1 describe increments of independent Poisson pro-cesses with arrival rates . By setting = 0 the formulation used by Eaton (1981) and inthe bulk of the literature thereafter is obtained. With = 0 a pure jump setup results.

3.1 Differential approach

The definition for uncertainty in (11) applied to the capital accumulation process in (5)yields

() = (( ) ()()) () ()=1() () (12)

In order to obtain a closed-form solution, one needs to solve the following nonlinear ordinarydifferential equation called the maximized Bellman equation (for the derivation, see app.6.1.1),

() = log

1

+

( )

1

+

1

222+ (13)

=1 ( ( ) ())

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Solving requires an educated guess about the value function whose structure can be shownto be identical (cf. app. 6.1.2) to that one of the well-known case where = 0 (cf. e.g.Merton, 1969), i.e.

() log()

+ (14)

with

= , = 1

22 +=1 log (1 )

2.

This explicit form of the value function leads to the conclusion that optimal consumption isjust the constant fraction of the capital stock,

() = () (15)

The result that is a constant fraction of is robust to allowing a more general in-stantaneous utility function as in (2). This implies optimal consumption to be given by (cf.

app. 6.1.3) () = () (16)

where the constant is the one from (10).

3.2 Standard approaches

3.2.1 Standard I

In contrast to the differential approach, we now let TFP () to be a function of time andstochastic disturbances. Allowing 6= 0 in the formulation for the TFP process (cf. eq.(7) with (11)), we specify the parameters , , , and the arrival rates of the Poisson

processes such that () is a martingale, i.e. the expectation about the TFP level atsome future point in time is given by the current level, [ ()] = () for implyingthat () has no trend. Given the technology in (6), this assumption is made in orderto preserve a stationary expected growth rate of the economy which is governed by the

marginal product of capital () (cf. app. 6.2.1). As [ ()] = () (+=1)() (cf.

app. 6.2.2) the restriction of no trend in the TFP-process requires that

+=1 = 0. (17)

Economically speaking, given that the arrival rates are positive by construction, technolo-gical progress is either technological regress ( 0) or there are occasional negative shocks,

i.e. 0 We find the second interpretation more plausible where oil price shocks, naturalor other disasters disrupt the smooth evolution of an economy. But it should be kept inmind that without an specification, one could allow for both positive and .

With this setup and a logarithmic utility, the maximized Bellman equation is a secondorder nonlinear partial differential equation (cf. app. 6.2.3),

( ) = log

1

+

1

+ +

1

2

22+ (18)

=1 ( ( + ) ( ))

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It is standard in the literature to provide an educated guess and prove a verificationtheorem. Finding such a guess can sometimes be very time-consuming. For our case here in(18), we can be more systematic and derive a solution.

Use the definition of the value function as the expected utility out of current and allfuture optimal consumption, ( ) max{()} subject to constraints where is theobjective function from (1). Looking at the results from the literature as well as from theprevious section, a plausible guess concerning optimal consumption is

() () , (19)

where the is an arbitrary constant. Inserting this conjecture into the objective functionwe can analyze whether the resulting integral can be solved. In fact, doing so by derivingexpressions for () from (8) and for () from (7) with (11), respectively, together withthe guess (19) gives (cf. app. 6.2.4)

( ) =

Z

() log() (())

, (20)where

= ( )1

22 ( ) + () + log

=1

1 +

=1

()

Simplifying and solving this integral and dropping the time subscript (as this expressionholds for each point in time) yields the value function,

=log

+

2

+

2. (21)

It is easy to verify that, for = , this is indeed a solution to the maximized Bellman

equation. Therefore it is confirmed that optimal consumption is a constant fraction of thecapital stock,

() = () . (22)

Unfortunately unlike in the case with the differential technology this result holds only forthe logarithmic utility (2) and not for the CRRA case of 6= 1

3.2.2 Standard II with Brownian motion and many Poisson processes

Fortunately, however, closed from solution can be found for CRRA utility functions with 6=1 and standard technologies. The alternative to stochastic TFP we propose is a stochastic

capital accumulation technology via e.g. stochastic depreciation. Let the () in (9) againstand for a combination of a Wiener process and many Poisson processes as in (11):

() = (()()) () ( + () + =1 ()) (23)

Of course this equation is identical to the corresponding expression in the differential setup in(12). Therefore, computing the solution for the maximization problem of the social plannerboth in the case with log-utility (2) and in the case with the more general instantaneousutility function (2) leads to the same rule concerning optimal consumption, i.e. consumptionis just the constant fraction (cf. eq. (15)) and (cf. eq. (16)), respectively, of the capitalstock.

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4 Conclusion

The differential approach to specifying technologies is widely used. Specifications of tech-nologies of this type in continuous time models with uncertainty have undesirable features.These features might be one reason why modeling uncertainty in continuous time is not as

popular in macroeconomics as it could be. One major advantage of these specifications istheir analytical tractability.

This paper generalizes existing closed-form solutions for differential technologies. Moreimportantly, it proposes two alternative setups where technologies are modeled by standardproduction functions. In both cases, the undesirable features of the differential approach arenot present and the analytical tractability is preserved. Closed form solutions are providedfor both alternatives.

One alternative implies solutions which are identical to solutions of models followingthe differential approach. This shows that all results obtained so far in the literature can bepreserved even without the undesirable features. For all future papers, analytical tractability

can be obtained just as easily as before.

5 Appendix

The appendix is available upon request

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