An optimizing IS-LM framework with endogenous investment

Journal of Macroeconomics 28 (2006) 621–644

www.elsevier.com/locate/jmacro

An optimizing IS-LM frameworkwith endogenous investment

Miguel Casares a, Bennett T. McCallum b,*

a Departamento de Economıa, Universidad Publica de Navarra, 31006 Pamplona, Spainb Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Received 17 May 2004; accepted 9 November 2004Available online 17 August 2006

Abstract

Dynamic optimizing models with an IS-LM-type structure and slow price/wage adjustments havebeen used for much recent monetary-policy analysis, but usually with capital and investment treatedas exogenous – a significant restriction. This paper demonstrates that investment decisions can beendogenized without undue complexity in such models and that these can be calibrated to providereasonably realistic dynamic behavior. It is necessary, however, to include capital adjustment costs;models with no adjustment costs match cyclical data very poorly. Indeed, their match is considerablypoorer than models with constant capital. The paper also finds that the preferred adjustment-costspecification is not close to quadratic.� 2006 Elsevier Inc. All rights reserved.

JEL classification: E10; E22; E30

Keywords: Optimizing; Endogenous investment; Adjustment costs; IS-LM

1. Introduction

Numerous recent papers have effectively utilized dynamic macroeconomic models withoptimizing behavior that can be expressed in terms of relationships that are similar inmany respects to traditional models of the ‘‘IS-LM’’ variety. Some noteworthy examples

0164-0704/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmacro.2004.11.005

* Corresponding author.E-mail addresses: [email protected] (M. Casares), [email protected] (B.T. McCallum).

mailto:[email protected]

mailto:[email protected]

622 M. Casares, B.T. McCallum / Journal of Macroeconomics 28 (2006) 621–644

of such papers include Clarida et al. (1999), Kerr and King (1996), Jeanne (1998), Galı andGertler (1999), McCallum and Nelson (1999a,b), Rotemberg and Woodford (1997, 1999),and Woodford (1995),1 plus discussions in the textbooks of Walsh (1998) and Woodford(2003). The main difference between the models in these papers and semi-traditionalIS-LM formulations – i.e., ones that distinguish between real and nominal interestrates2 – is that the counterpart of the IS function includes an additional term (with a unitcoefficient) involving expected future expenditure.

The inclusion of this term gives a forward-looking aspect to expenditure decisions, afeature that will typically result in significantly altered dynamic behavior in these models,relative to the semi-traditional type. Of course, these recent optimizing models still possesssignificant limitations.3 One of the more notable of these is that capital investment is typ-ically taken to be either non-existent (e.g., Woodford, 1995) or exogenous (e.g., McCallumand Nelson, 1999b).4 Clearly, this limitation is quite significant.5 Not only does it elimi-nate the possibility of studying issues relating to capital formation and growth, but inaddition it removes the possibility of explaining endogenously the contrasting variabilityof consumption and investment spending, a contrast featured in much of the real-busi-ness-cycle literature.

The first purpose of the present paper is to extend the basic optimizing IS-LM frame-work so as to incorporate endogenous investment. This extension results in a model withan IS ‘‘sector’’ rather than an IS function, but retains much of the attractive simplicity ofthe exogenous-investment models and continues to facilitate a similar conceptual organi-zation of behavioral forces. A priori, it would seem likely that for capital and investmentmovements to be even moderately realistic, it might be necessary to posit some form ofadjustment costs. Here we will consider a familiar adjustment-cost specification.6 For pur-poses of realism, the model presented includes sluggish adjustment of prices and wages; inthat regard we utilize a price–wage adjustment assumption related to Calvo (1983).7 Torepresent the medium-of-exchange role of money, we utilize a specification, very commonin the recent literature, in which the period utility function is separable in terms of

1 This list includes only papers published prior to 2000 that use or mention the IS-LM terminology. There arenumerous other papers with related models that make no such mention, including Goodfriend and King (1997),Kimball (1995), King and Wolman (1996, 1999), and Yun (1996).

2 The IS-LM setups that we will be concerned with are ones that are intended to represent only the aggregatedemand portion of a macroeconomic system, not ones that treat the price level as a constant. In such models, realand nominal interest rates are clearly distinguished. For more discussion, see McCallum and Nelson (1999a).

3 Most of the models being referred to are ones in which product prices are presumed to adjust slowly. There ismuch disagreement concerning various models of price adjustment, but in this paper we wish to focus uponlimitations to the ‘‘IS-LM’’ portion of the model, not the ‘‘AS’’ portion (AS representing aggregate supply).

4 There are some notable exceptions, including King and Wolman (1996), Yun (1996), and Kollmann (1997). Inall of these cases, however, investment considerations are discussed very briefly. Also, in each the existence of aneconomy-wide rental market for capital services is assumed whereas our analysis includes producer-specificcapital accumulation that realistically prevents immediate reallocations of capital goods across producers.Another early contribution by Koenig (1993) is insightful, but does not develop an explicit dynamic model that issuitable for quantitative application.

5 For some purposes this weakness may not be serious, as is argued by McCallum and Nelson (1999a), but forothers it can be crucial.

6 It is worth noting that with adjustment costs recognized, capital becomes a distinct asset differing fromgovernment and private bonds, even abstracting from uncertainty.

7 Recently this model has been the subject of considerable interest and controversy; see Walsh (1998, pp. 224–226), Galı and Gertler (1999), Estrella and Fuhrer (1998), and Erceg et al. (2000).

M. Casares, B.T. McCallum / Journal of Macroeconomics 28 (2006) 621–644 623

consumption and (the services of) real money balances. This specification is less plausiblethan ones involving a non-separable transaction-cost function, but the differences areinsignificant for present purposes, as we explain in Section 2.

Besides developing this extension, the paper explores various related issues. One of con-siderable importance is whether enrichment of the optimizing IS-LM framework, so as toincorporate endogenous investment, has a major impact on properties of the model thatare crucial for the study of alternative monetary-policy rules. Does output or inflation,for example, respond differently in terms of its magnitude and/or dynamic pattern to a mon-etary-policy shock? Or a technology shock? A second principal objective of the paper,accordingly, is to conduct an investigation of this issue. Surprisingly, we find that appropri-ate calibration of a constant-capital model permits it to closely match the endogenousinvestment model in terms of these properties. Essentially the same point has been devel-oped, since the first version of the present paper was circulated (Casares and McCallum,2000), by Woodford (2003, pp. 252–278). Our analysis differs from Woodford’s, however,in several respects. One is that our model includes sticky wages, as well as prices. In addition,our treatment of adjustment costs is different (and implies steady-state effects absent fromWoodford’s specification) and our calibration is substantially different. Perhaps mostimportantly, we obtain striking results for specifications with endogenous investment butno adjustment costs, which are not considered by Woodford. In addition, we explore theidea that investment adjustment costs are necessary to avoid unrealistic movements of cap-ital and investment magnitudes. In the process of doing so, we obtain results suggesting thatsuch costs are necessary but that the relatively familiar quadratic specification is inadequate.

None of the paper’s analysis depends, of course, on use of the IS-LM terminology,which some economists find attractive and others find objectionable. Our own use of theseterms is partly motivated by a desire to illustrate that actual analytical differences betweenproponents and opponents of ‘‘IS-LM analysis’’ can easily be overstated.

Organizationally, we begin in Section 2 with the specification of a model of the type typ-ically used in the recent papers featuring dynamic optimizing models of the IS-LM type.Then in Section 3 we extend the setup so as to include endogenous investment with – and,partly for comparative purposes, without – adjustment costs. Section 4 is devoted to aquantitative calibration of the three model variants, and their dynamic properties arecompared in Section 5. A short summary appears as Section 6.

2. Reference model with exogenous investment

We begin by specifying a typical model of the optimizing IS-LM form in which invest-ment is not determined endogenously. Thus, we consider an economy populated by a largenumber of households making decisions in the presence of uncertainty about the future. Atypical household seeks at time t to maximize the intertemporal utility function

Et

X1j¼0

bjUðctþj; ltþj;mtþj; ttþj; ftþjÞ;

where b = (1 + q)�1, q > 0, is the household’s discount factor, ct is consumption during t,lt is leisure time during t, and mt is the stock of real money balances at the end of t.8 Also,

8 We could alternatively let mt denote real balances at the beginning of t. This would result in a slightly differentmoney-demand function analogous to (17) below; on this point see McCallum and Nelson (1999a).


Et is the expectation operator conditional on information (assumed complete) available inperiod t. Preference shocks tt and ft enter the utility function in ways that will be specifiedbelow. Different households produce different goods; for the typical household ct repre-sents a Dixit and Stiglitz (1977) index denoting the number of bundles consumed. The rea-son that mt appears in U(Æ) is that holdings of the economy’s medium of exchange yieldtransaction services that reduce the amount of time or material resources needed for‘‘shopping’’ – i.e., conducting transactions – for the many goods that comprise consump-tion bundles. Such effects are often represented indirectly by inclusion of mt. It wouldnot alter our results substantially to use a theoretically preferable formulation that spec-ified explicitly the time or material resources that are needed for conducting transactions,with the amount needed negatively dependent upon the quantity of real money balancesheld. (For details, see Casares, 2000; McCallum, 2000). If the relevant cost function isnot separable in ct and mt, there would then be implications as discussed below in footnote11.

The typical household’s production of its specialized output is governed by the produc-tion function yt ¼ f ðAtnd

t ; ktÞ, where At is a labor-augmenting technology shock, ndt is the

demand for labor, and kt is the stock of capital held at the start of t. This production func-tion f(Æ) is homogeneous of degree 1 in its two inputs. Sales of the household’s specializedoutput are restricted by the demand function Y A

t ðP t=P At Þ�hP , where Y A

t is aggregate demand,Pt is the money price of the household’s product, P A

t is the aggregate price level, and hP isthe elasticity of substitution across the differentiated consumption goods. In addition, thehousehold supplies differentiated labor services in a monopolistically competitive labormarket in which the quantity demanded is nA

t ðW t=W At Þ�hW , where nA

t is aggregate labor,Wt is the household’s nominal wage, W A

t is the aggregate nominal wage, and hW is theelasticity of substitution for the differentiated labor services. Simultaneously, householdspurchase time units of a Dixit–Stiglitz composite labor input at the real wage ratewt ¼ W A

tP A

t. Finally, bt+1 is the number of one-period government bonds purchased in t.

Their real price is (1 + rt)�1 and each is redeemed in t + 1 for enough money to purchase

one unit of consumption. Letting pt ¼ ðP At =P A

t�1Þ � 1 denote the inflation rate in period t,the household’s budget constraint for period t can be written in consumption-bundle unitsas

Y At ðP t=P A

t Þ1�hP � txt � wtðnd

t � nAt ðW t=W A

t Þ1�hW Þ

¼ ct þ ktþ1 � ð1� dÞkt þ mt � ð1þ ptÞ�1mt�1 þ ð1þ rtÞ�1btþ1 � bt ð1Þ

in which txt denotes lump-sum taxes (net of transfers) levied on the household while d isthe capital depreciation rate. As discussed above, each household’s production is equal tothe quantity demanded of its good:

f ðAtndt ; ktÞ ¼ Y A

t ðP t=P At Þ�hP : ð2Þ

In a similar way, labor supply is determined by the monopolistic competition demandequation and enters the time constraint as

nAt ðW t=W A

t Þ�hW þ lt ¼ 1; ð3Þ

where lt is leisure in period t and total time has been normalized to 1.


In this setting, the household’s optimality conditions include (1)–(3), and9

U 1ðct;mt; lt; tt; ftÞ � kt ¼ 0; ð4Þ

U 2ðct;mt; lt; tt; ftÞ � kt þ bEt½ktþ1ð1þ ptþ1Þ�1� ¼ 0; ð5ÞU 3ðct;mt; lt; tt; ftÞ � nt ¼ 0; ð6Þ� ktwt þ utAtf1ðAtnd

t ; ktÞ ¼ 0; ð7Þ� kt þ bEtktþ1ð1� dÞ þ bEt½utþ1f2ðAtþ1nd

tþ1; ktþ1Þ� ¼ 0; ð8Þ

� ktð1þ rtÞ�1 þ bEtktþ1 ¼ 0; ð9Þ

ktY At ð1� hP ÞP�hP

t =ðP At Þ

1�hP þ utYAt hP P�ðhPþ1Þ

t =ðP At Þ�hP ¼ 0; ð10Þ

ktwtnAt ð1� hW ÞW �hW

t =ðW At Þ

1�hW þ ntnAt hW W �ðhW þ1Þ

t =ðW At Þ�hW ¼ 0: ð11Þ

Notice that kt, ut and nt are the Lagrange multipliers attached to (1)–(3) respectively.Assuming that the relevant transversality conditions are satisfied, these eleven relationsdetermine the household’s choices of ct, mt, lt, nd

t , kt+1, bt+1, Pt, Wt, kt, ut and nt giventhe values of pt, wt, rt, P A

t , W At , and txt that it faces.

For general equilibrium, the government’s budget constraint must also be satisfied.In per-household terms, it is

gt � txt ¼ mt � ð1þ ptÞ�1mt�1 þ ð1þ rtÞ�1btþ1 � bt; ð12Þwhere gt denotes government consumption of goods and services. In addition, there are themarket-clearing conditions

nAt ¼

Xnd

t ð13Þ

and

mt ¼Mt

P At

; ð14Þ

where Mt is the per-household nominal money supply, and also the identity

pt ¼P A

t

P At�1

� 1: ð15Þ

In the absence of nominal rigidities, symmetry across households would hold in equilib-rium with P t ¼ P A

t and W t ¼ W At . Then we have 15 conditions (apart from the relevant

transversality conditions) to determine the time paths of c, m, l, nA, nd, k, b, P, W, k, u,n, w, r, and p in response to exogenous, government-chosen paths of Mt, gt, and txt. Alter-natively, the government could be thought of as determining bt instead of gt or txt, withthe other variable becoming endogenous. Similarly, the government could implementmonetary policy by means of the nominal interest rate Rt, defined by 1 + Rt = (1 + rt)(1 +Etpt+1), instead of Mt.

To express this system in an IS-LM fashion, we combine (4) and (9) to yield

U 1ðct;mt; lt; tt; ftÞ ¼ bEt½U 1ðctþ1;mtþ1; ltþ1; ttþ1; ftþ1Þ�ð1þ rtÞ ð16Þ

9 Here and below, fi(Æ) represents the partial derivative of the function f(Æ) with respect to its ith argument.


and also combine (4), (5), (9), and the definition of Rt to yield

U 2ðct;mt; lt; tt; ftÞ ¼ U 1ðct;mt; lt; tt; ftÞRt

1þ Rt: ð17Þ

Next we assume that the period utility function can be expressed or approximated bythe form

Uðct; lt;mt; tt; ftÞ ¼ � expðttÞc1�r

t

1� rþ ð1� � Þ expðftÞ

m1�ct

1� cþ U

l1�#t

1� # ð18Þ

with 0 < � < 1 and r, #, c, U > 0. Accordingly, we have U 1ðct;mt; lt; tt; ftÞ ¼ � expðttÞc�rt

and U 2ðct;mt; lt; tt; ftÞ ¼ ð1� � Þ expðftÞm�ct . Then (16) reduces to

expðttÞc�rt ¼ Et½expðttþ1Þc�r

tþ1�bð1þ rtÞ: ð19Þ

Here and below, we employ the techniques described extensively in Uhlig (1999) to obtainlog-linear approximations to non-linear relations such as (19). In doing so, ‘‘hat’’ variablesare defined as (logarithmic) fractional deviations from steady-state values, e.g., bct ¼log ct

css

� �with the ss superscript denoting the steady-state value. Thus, Eq. (19) can be trans-

formed into10

bct ¼ Etbctþ1 � r�1ðrt � rssÞ � r�1ðEtttþ1 � ttÞ: ð20ÞWe assume that the consumption preference shock follows the AR(1) process

tt ¼ qttt�1 þ ett; ð21Þwith jqtj < 1 and ett drawn from a normal distribution, i.e., ett � Nð0; rettÞ. Substituting(21) into (20) and using rt = Rt � Etpt+1 yields

bct ¼ Etbctþ1 � r�1ðRt � Etptþ1 � rssÞ þ r�1ð1� qtÞtt: ð22ÞRegarding the money-demand formulation, for the utility function specification (18) the

general expression (17) reduces to

mt ¼ expðc�1ðft � ttÞÞcrct

�Rt

ð1� � Þð1þ RtÞ

� ��1c

; ð23Þ

which can be approximated taking Rt ’ Rt1þRt

by the semi-log-linearized relation

bmt ¼rcbct �

1

cRss ðRt � RssÞ þ 1

cðft � ttÞ: ð24Þ

Defining the composite disturbance vt ¼ 1c ðft � ttÞ, Eq. (24) becomes the following money-

demand expression:

bmt ¼rcbct �

1

cRss ðRt � RssÞ þ vt: ð25Þ

Clearly, (25) is of the familiar ‘‘LM’’ form, in which real balances demanded depend pos-itively upon a transaction variable and negatively upon an opportunity-cost variable.

10 Here we also use log(1 + rt) ’ rt and logb = �log (1 + q) ’ �q = �rss. Regarding the latter, note that in aSidrauski-type optimizing model, the steady-state real interest rate rss is equal to the rate of intertemporalpreference (q = rss).


Next, to put (22) into an ‘‘IS’’ form, let us first suppose – as in Woodford (1995) andelsewhere – that kt+1 is not a variable but is fixed at the steady-state value kss. Then we canwrite the overall resource constraint

f ðAtndt ; ktÞ ¼ ct þ ktþ1 � ð1� dÞkt þ gt ð26Þ

as

yt ¼ ct þ dkss þ gt: ð27ÞFor a log-linear approximation to (27) we take

by t ¼ x1bct þ x2bgt; ð28Þwhere x1 and x2 are the steady-state shares in output of consumption and governmentexpenditure, i.e., x1 = css/yss and x2 = gss/yss. Then substitution of (22) into (28) results in

by t ¼ Etby tþ1 �x1

rðRt � Etptþ1 � rssÞ þ x2ðbgt � Etbgtþ1Þ þ

x1ð1� qtÞr

tt: ð29Þ

The latter is an ‘‘expectational IS function’’, in which demand for current output isexpressed not only as a negative function of rt, a positive function of ðbgt � Etbgtþ1Þ, anda positive function of the consumption preference shock tt, but is also related positivelyto Etby tþ1 (hence the designation expectational). The presence of the last term representsthe only significant difference from a semi-traditional IS specification, but this differencecan obviously be important for various dynamic issues.11

In the fully flexible price-and-wage system just described, the symmetry conditionP t ¼ P A

t combined with Eqs. (7) and (10) imply that the real marginal cost wt is constant

wt ¼wt

Atf1ðAtndt ; ktÞ

¼ hP � 1

hPð30aÞ

and the nominal wage symmetry condition W t ¼ W At inserted in (11) leads to a constant

ratio of the leisure/consumption marginal rate of substitution over the real wage

U 3ðct;mt; lt; tt; ftÞwtU 1ðct;mt; lt; tt; ftÞ

¼ hW � 1

hW: ð30bÞ

Now, let us specify the production technology by means of the Cobb–Douglas function

f ðAtndt ; ktÞ ¼ ðexpðAtÞnd

t Þ1�aka

t ; 0 < a < 1: ð31ÞUsing the production function (31) under the constant-capital assumption in (30a), theutility function (18) in (30b), and combining the resulting expressions with the log-linearoverall resource constraint (28) and the log-linearized time constraint (3), yields the fol-lowing log-linear expression for fluctuations of output:

a1� a

þ #nss

ð1� aÞlss þrx1

� �by t ¼ 1þ #nss

lss

� �At þ tt þ

rx2

x1

bgt: ð32Þ

11 In the case that a transactions-cost formulation is used, as we suggested above, the IS-type function (29) willalso include a term involving bmt � Et bmtþ1 unless the transaction-cost function is separable. McCallum (2000)suggests a plausible specification that is not separable. Quantitative calibration of the model’s parameterssuggests, however, that the coefficient attached to this term will be quite small – e.g., less than 0.02. Accordingly,it is satisfactory to neglect such effects for the purposes of the present paper.


In this flexible price-and-wage system, output by t responds in a positive direction to threeexogenous processes: the technology shock At, the consumption preference shock tt, and(exogenous) government expenditures bgt. Notice that monetary conditions do not affectoutput even in the short-run dynamics.

Furthermore, analysis of the IS-LM type can be carried out as follows. With exogenousprocesses specified for Mt, bgt, At, tt, and vt, Eqs. (29), (25) and (32) determine the behaviorof yt, Rt and pt, given the identities mt ¼ Mt=P A

t and pt ¼ P At =P A

t�1 � 1.12 Alternatively, themonetary authority could specify the behavior of Rt, as central banks do in reality, inwhich case the model would determine the behavior of Mt.

The model may also include the possibility that either Pt or Wt (or both) do not adjustfully within each period – i.e., by assuming either price stickiness or nominal wage stick-iness. Suppose, for example, that households cannot adjust the price with fixed probabilitygP as in Calvo (1983). Then, the first-order condition on Pt becomes

Et

X1i¼0

bigiP ½ktþiY A

tþið1� hP ÞP�hPt =ðP A

tþiÞ1�hP þ utþiY

AtþihP P�ðhPþ1Þ

t =ðP AtþiÞ

�hP � ¼ 0; ð100Þ

which, after log-linearizing, and using definitions of pt and the Dixit–Stiglitz aggregateprice level P A

t with staggered prices a la Calvo, leads to the (New Keynesian) inflationequation

pt � pss ¼ bðEtptþ1 � pssÞ þ ð1� bgP Þð1� gP Þð1� aÞgP ð1� aþ ahP Þ

bwt; ð33Þ

where bwt represents (log) deviations from steady state of the average real marginal cost,as defined in (30a).13

Now, the IS-LM set of equations would comprise (29), (25) and (33) which would deter-mine dynamic paths for yt, Rt and pt, given the exogenous processes mentioned above andone extra equation defining bwt. This variable is computed as the difference between the realwage and the marginal product of labor in fractional (log-linear) deviations from steadystate. The latter is directly obtained from the Cobb–Douglas production function. Asfor the real wage, its dynamic behavior depends upon the wage setting assumption. Thus,if we assume fully flexible nominal wages and W t ¼ W A

t , one obtains14

bwt ¼#nss

lssð1� aÞ þrx1

� �by t �rx2

x1

bgt �#nss

lss At � tt: ð34Þ

12 This argument does not imply that money-demand and saving behavior, represented by (25) and (29)respectively, are entirely independent, for the structural parameter r appears in both of these relations. Also, theirdisturbances are stochastically dependent.13 See Sbordone (2002) for the explicit derivation of the New Keynesian Phillips curve with constant capital.

Sveen and Weinke (2004) and Woodford (2005) show that such an inflation equation with constant capital isbarely affected by the introduction of endogenous capital subject to a one-period time-to-build requirement. Thisresult will be used later in the paper for the model with endogenous capital.14 This result is obtained by first log-linearizing (30b) computed for the specific utility function (18). Then, we

used the log-linear overall resources constraint (28), and log-linearized approximations to the time constraint (3)and Cobb–Douglas production technology (31) to get to the real wage Eq. (34).


Alternatively, a real wage equation with slowly adjusting (sticky) wages will be introducedin Section 4.

3. Endogenous investment

Now we are prepared to endogenize investment. Conceptually, this could be donestraightforwardly by merely recognizing kt+1 as a variable and including relation (8) inthe system of first-order conditions. The resulting model would be unrealistic, however,in that it would imply much more volatility than exists in reality for investment, capital,and the marginal product of capital. This important weakness will be documented below.

It is possible to develop a well-behaved and tractable investment specification, neverthe-less, by adopting the highly plausible assumption that the process of capital investment –i.e., of ‘‘installing’’ capital goods – requires the usage of some resources. Many previousstudies have utilized such an assumption, including Eisner and Strotz (1963), Lucas andPrescott (1971), and Hayashi (1982). An alternative approach would be to include exoge-nous ‘‘time to build’’ constraints, as in Kydland and Prescott (1982) and many subsequentpapers. We consider it methodologically preferable, however, to endogenize the postulatedsluggishness of capital stock adjustments. The particular adjustment-cost specification thatwe use is taken from Hayashi (1982). Specifically, the assumption is that gross investmentxt, defined as

xt ¼ ktþ1 � ð1� dÞkt ð35Þentails adjustment costs C(xt,kt) according to the formula

Cðxt; ktÞ ¼ xthðxt=ktÞ;

where the unit cost of installing capital depends positively on the ratio xt/kt, i.e.,Cðxt ;ktÞ

xt¼ hðxt=ktÞ, with the properties h 0(Æ) > 0 and 2h 0(Æ) + d h00(Æ) > 0. Thus, the total adjust-

ment cost of investment in period t varies with both xt and kt. With this functional form,the production function including adjustment costs is homogeneous of degree 1. As a con-sequence, the ratio Cðxss;kssÞ

yss is independent of the size of the production plant. Moreover,Hayashi (1982) demonstrates that the average value of Tobin’s q is equal to its marginalvalue under this specification. The theoretically attractive features of this representationhave led it to be used with some frequency in the literature (e.g., Abel and Blanchard,1983; Blanchard and Fischer, 1989, p. 59; Obstfeld and Rogoff, 1996; Jermann, 1998).Assuming the functional form hðxt=ktÞ ¼ H1ðxt=ktÞH2 , one obtains

Cðxt; ktÞ ¼ H1

xH2þ1t

kH2t

ð36Þ

in which H2 = 1 represents the case with quadratic adjustment costs.Recognition of adjustment costs implies that �C(xt,kt) appears as an additional term

on the left-hand side (lhs) of the household’s budget constraint (1), and also on the lhsof the overall resource constraint (26). With this change, the first-order optimality condi-tion with respect to kt+1 (8) becomes,

�ktð1þ C1tÞ þ bEtktþ1½1� d� C2tþ1� þ bEt½utþ1f2tþ1� ¼ 0: ð80ÞHere, for simplicity, we let C1t = C1(xt,kt) and C2t+1 = C2(xt+1,kt+1) denote the marginaladjustment costs related to the choice of kt+1, and use f2t+1 to denote next period marginal


product of capital f2ðAtþ1ndtþ1; ktþ1Þ. Now recalling (7) one-period ahead and the definition

of next period’s real marginal cost wtþ1 ¼ wtþ1

Atþ1f1ðAtþ1ndtþ1;ktþ1Þ

, we get

ktð1þ C1tÞ ¼ bEtktþ1½1� d� C2tþ1� þ bEtktþ1½wtþ1f2tþ1�: ð37ÞUsing bEtkt+1 = kt(1 + rt)

�1 from (9), and eliminating kt on both sides of (37), the house-hold first-order condition regarding the choice of capital becomes

1þ rt ¼1� d� EtC2tþ1 þ Et½wtþ1f2tþ1�

1þ C1t: ð38Þ

The left-hand side of (38) is the return on the financial asset – i.e., the opportunity cost forone unit of kt+1 – and the right-hand side is the return on the last unit of capital invested,which depends on the depreciation rate d, the expected marginal product adjusted by themarginal cost Et[wt+1f2t+1], and on both the marginal adjustment costs paid in this periodC1t and those expected for the next period EtC2t+1. Substituting the marginal adjustmentcosts implied by (36), one obtains

1þ rt ¼1� dþ ð1� dÞH1ðH2 þ 1Þ Etxtþ1

ktþ1

� �H2

þH1H2

Etxtþ1

ktþ1

� �H2þ1

þ Et½wtþ1f2tþ1�

1þH1ðH2 þ 1Þ xt

kt

� �H2:

ð39ÞUsing Uhlig’s (1999) linearization rules, and assuming that the real interest rate, the mar-ginal product of capital, and the marginal adjustment cost are small relative to 1, we canapproximate the latter with the following semi-log-linear investment equation

bxt ¼1

1þ dEtbxtþ1 þ

wssf ss2

ð1þ dÞH2Css1

Etbwtþ1 þ

1

ð1þ dÞH2Css1

ðEtf2tþ1 � d� rtÞ þd

1þ dbkt;

ð40Þwith Css

1 ¼ H1ðH2 þ 1ÞdH2 as the marginal adjustment cost in steady state, and both wss

and f ss2 also denoting steady-state figures for the real marginal cost and the marginal prod-

uct of capital. This is an ‘‘expectational investment equation’’ in which current period’sinvestment depends on next period’s expected investment Etbxtþ1, on next period’s real mar-ginal cost Et

bwtþ1, on the gap between the expected return on physical capital and the re-turn on the financial asset Etf2t+1 � d � rt, and on current capital bkt.

Thus representation (40) of our investment equation indicates that current investmentdepends on current and all expected future premiums on investing in real assets (by meansof the forward-looking term Etbxtþ1). Let us define X ¼ 1

ð1þdÞH2Css1

from (40) as the semi-elas-ticity of investment with respect to (wrt) the real asset’s premium. This number directlyaffects the variability of investment over the economic cycle. Clearly, both parametersof the adjustment-cost function at hand influence the size of this number. To illustrate thisinfluence, Table 1a reports figures of the semi-elasticity of the physical asset premium in(40) for various calibrations of the parameters H1 and H2 of the adjustment-cost function.

Thus, a higher value of the scale parameter H1 reduces the semi-elasticity X. Variabilityof investment goes down as responses to changes in the real asset premium are weaker.The adjustment cost elasticity parameter H2 has the opposite effect on X, so a higher valueof H2 leads to a higher X. Investment becomes more volatile because it responds moreaggressively to changes in the real asset premium.

Table 1bUnit adjustment costs in steady state, hðxss=kssÞ ¼ H1d

H2

H1 = 1000 H1 = 3000 H1 = 5000 H1 = 7000

H2 = 1.0 25.0 75.0 125.0 175.0H2 = 2.0 0.62 1.87 3.12 4.37H2 = 3.0 0.0156 0.0469 0.0781 0.1094H2 = 4.0 0.0004 0.0012 0.0020 0.0027

Table 1aSemi-elasticity of investment wrt the real asset premium in Eq. (40), X

H1 = 1000 H1 = 3000 H1 = 5000 H1 = 7000

H2 = 1.0 0.0195 0.0065 0.0039 0.0028H2 = 2.0 0.260 0.087 0.052 0.037H2 = 3.0 5.20 1.73 1.04 0.74H2 = 4.0 124.9 41.6 25.0 17.8


Another relevant characteristic influenced by the adjustment cost parameters is the unitadjustment cost hðxt=ktÞ ¼ H1ðxt=ktÞH2 . Table 1b displays how H1 and H2 determine theunit adjustment cost in steady state. As expected, a larger scale parameter H1 leads tohigher unit adjustment costs in steady state. On the contrary, large values of H2 implylow unit adjustment costs in steady state. More details on the steady-state solution ofthe model are described in Appendix A of this paper.

Finally, with the recognition of variable capital and investment adjustment costs, thesystem includes an ‘‘IS sector’’ consisting of

bct ¼ Etbctþ1 � r�1ðrt � rssÞ þ r�1ð1� qtÞtt; ð41aÞ

bxt ¼1

1þ dEtbxtþ1 þ Xðwssf ss

2 Etbwtþ1 þ Etf2tþ1 � d� rtÞ þ

d1þ d

bkt; ð41bÞ

bwt ¼ bwt �a

1� aðbkt � by tÞ � At; ð41cÞ

f2t ¼ f ss2 ðby t � bktÞ; ð41dÞ

bktþ1 ¼ ð1� dÞbkt þ dbxt; ð41eÞ

by t ¼ x1bct þ x2bgt þ x3bxt ð41fÞ

in which (41a) and (41b) are the structural equations for consumption and investment.15

The next two equations, (41c) and (41d), respectively are a log-linear approximation to thereal marginal cost bwt and the marginal product of capital f2t for a Cobb–Douglas func-tional form (31). The overall resource constraint is (41f), which now includes investment.Consumption decisions (41a), investment decisions (41b), and the amount of (exogenous)government purchases give rise to current real demand by t as in (41f). The value of x3 in(41f) reflects the weight of investment in total output from a steady-state perspective(x3 = (xss(1 + h(xss, kss)))/yss).

15 Note that if prices are fully flexible the value of Etbwtþ1 is constant as implied by (30a).


To determine the system’s endogenous variables, this six-equation IS sector can be usedtogether with the LM equation (25), the Fisher equation rt = Rt � Etpt+1, two equationsregarding both price and wage adjustment specifications – for example, (33) and (34)incorporating variable capital – and policy rules for bgt, and either Mt or Rt. The resulting12 relations will determine time paths of by t;bct; bxt, bgt, bmt, bktþ1, bwt, bwt, rt, f2t, pt, and eitherRt or Mt.

4. Calibration

In this section we complete the specification of our model in numerical terms, so as topermit quantitative analysis of its properties (considering several variants). One step is toadopt a monetary-policy rule. For the sake of realism we will treat Rt as the monetaryauthority’s instrument variable, using a version of the well-known policy rule proposedby Taylor (1993). Studies by Clarida et al. (1998) and others have indicated that estimatedversions of this rule generally benefit from inclusion of the variable Rt�1, to reflect interestrate smoothing, with a positive coefficient in the vicinity of 0.75–0.85. Accordingly, we rep-resent monetary-policy behavior as follows:

Rt � Rss ¼ ð1� l3Þ ð1þ l1Þðpt � pssÞ þ l2Et�1~yt½ � þ l3ðRt�1 � RssÞ þ eRt; ð42Þ

where eRt is a policy shock. Here, and in what follows, ~yt denotes by t � by t, the fractionaldifference between output and its potential value. We will refer to this as the outputgap. Now, let potential (natural-rate) output yt be defined as the quantity of output thatwould prevail in an economy without nominal rigidities (i.e. with fully flexible prices andwages). Thus, the Cobb–Douglas technology leads to quarter-to-quarter fluctuations ofpotential output by t evolving as follows:

by t ¼ ð1� aÞAt þ ð1� aÞbnt þ abkt; ð43Þ

where bnt and bkt denote natural-rate labor and capital respectively and can be expressed asa function of the exogenous processes entering the model.16 The presence of the outputgap ~yt in (42) represents policy feedback intended to stabilize ~yt as in Taylor (1993) andClarida et al. (1998). The response is applied in (42) to Et�1~yt rather than ~yt since actualcentral banks do not have observations on by t or by t when setting Rt.

For the same reason it might be desirable to utilize Et�1pt or Et�1pt+j rather than pt

itself in (42). In the case with full price flexibility, however, this would lead to a form ofprice level indeterminacy, because the relevant portion of the model would then includeno reference to the current inflation rate. Thus we have assumed that pt appears in (42)so as to be able to use the same policy rule for flexible and sticky-price variants. Thisdeparture from operationality in the specification is not serious since percentage forecasterrors are much smaller for pt than for ~yt in actual economies. For parameter values weadopt l1 = 0.5, l2 = 0.1, and l3 = 0.8 , which are quite standard in the literature.

Next we discuss the price (wage) adjustment scheme to be used in the variant of themodel with nominal rigidities. We have already mentioned the Calvo formula (33) above,which will be used for our sticky-price specification. Similarly, let us assume that nominal

16 Notice that Eq. (43) leads to Eq. (32) when the constant-capital assumption is considered.


wages cannot be adjusted with some fixed probability gW. In turn, the nominal wage first-order condition would be

Et

X1i¼0

bigiW ½ktþiwtþinA

tþið1� hW ÞW �hWt =ðW A

tþiÞ1�hW þ ntþin

AtþihW W �ðhW þ1Þ

t =ðW AtþiÞ

�hW � ¼ 0:

ð110ÞNow, log-linearizing (11 0) and combining it with other relations of the model, brings aboutthe following real wage equation:

bwt ¼1

1þ bbwt�1 þ

b1þ b

Et bwtþ1 �1

1þ bðpt � pssÞ þ b

1þ bðEtptþ1 � pssÞ þ j

1þ bbst;

ð340Þwhere j ¼ ð1�bgW Þð1�gW Þ

gW ð1þhW #ðnss=lssÞÞ and bst are (log) deviations from steady state of the ratio of the

average leisure-consumption marginal rate of substitution over the real wage, i.e., theleft-hand side of (30b).17 The parameters gP and gW represent the degree of price/wagestickiness as the probability that the household is not able to set a new price/wage. Inour sticky-price and sticky-wage calibrated model we set gP = gW = 0.75 as in Erceget al. (2000) which implies that on average both prices and nominal wages are adjustedonce per year. The economy with fully flexible prices and wages (that corresponds tothe parameterization gP = gW = 0.0) will also be examined in the impulse–response ana-lysis carried out below.

It remains to assign values to the other parameters, i.e., to complete calibration of themodel. We do so with regard to quarterly US data. For the IS sector we adopt the follow-ing values: r = 5, d = 0.025, a = 0.36, q = rss = 0.005, H1 = 5363, H2 = 3.14, hP = 6.0,hW = 4.0, x1 = 0.78, x2 = 0, and x3 = 0.22. In the cases without adjustment costs, wehave x1 = 0.7, x2 = 0, and x3 = 0.3. We set r so as to imply an intertemporal elasticityof substitution in consumption of 1/5, a magnitude consistent with empirical studies ofthe US data such as Hall (1988), Ogaki and Reinhart (1998), and Fuhrer (2000). Thedepreciation rate and capital-share parameters, d and a, are quite conventional. The valueof q = rss implies an average real interest rate of 2% per year.

Calibration of the two parameters of the adjustment-cost function C(xt,kt) is done tosatisfy some two-part criteria. First, the unit adjustment cost is in steady state equal to5% of investment, i.e., h(xss,kss) = 0.05, which implies that total adjustment costs representa figure slightly higher than 1% of output in the steady state. Secondly, after analyzingimpulse–response functions and some simulation experiments, we found that the semi-elas-ticity X in the investment function should be around 1.5 to yield business-cycle fluctuationsclose to the empirical evidence in the sense that investment variability is several times thatof consumption. The pair of values that satisfies both criteria is H1 = 5363 and H2 = 3.14(check Tables 1a and 1b). Note that the figure selected for H2 is far from the quadraticadjustment cost case (H2 = 1.0). If the quadratic calibration were to be chosen, either thesemi-elasticity X or the steady-state size of adjustment costs would be much higher.18

17 See Sbordone (2001) for a step-by-step derivation of this real wage equation under sticky wages a la Calvo.18 Concretely, if we calibrate H1 under the quadratic adjustment cost case H2 = 1.0 and a steady-state unit

adjustment cost equal to 5% of investment h(xss,kss) = 0.05, the semi-elasticity in the investment equationbecomes X = 9.75. Alternatively, if we calibrate H1 under the quadratic adjustment cost case H2 = 1.0 and a semi-elasticity X = 1.5, the steady-state unit adjustment cost is h(xss,kss) = 0.325.


The elasticity of substitution between differentiated consumption goods is the quitestandard business-cycle figure hP = 6.0, implying a 16.67% mark-up of price over marginalcost in steady state. The elasticity of substitution across labor services is hW = 4.0, as sug-gested by Griffin (1992, 1996) based on his micro-level empirical study. Another parameterthat determines labor and real wage fluctuations is the leisure relative risk aversion coef-ficient # which appears in the utility function. It is set at # = 2.0 so as to imply a unit realwage elasticity of the labor supply when leisure takes twice the time of labor hours insteady state; thus nss/lss = 1/2.

The money-demand or LM relation (25) shows that the interest rate semi-elasticityequals 1/(cRss). Thus if we were to adopt c = 7 and Rss = rss + pss = 0.01, we would obtaina value of 14.3. But with Rt used as the policy instrument, and with a separable periodutility function, there is no need to calibrate the money-demand equation as it plays norole in the evolution of most of the model’s variables.

Finally, we need to assign magnitudes to the parameters of the stochastic disturbanceprocesses. The technology shock follows the process At = qAAt�1 + eAt, where eAt isGaussian white noise, and we take qA = 0.95 and reA ¼ 0:01. These calibrated valueslead to realistic business-cycle fluctuations of output as documented below. The othernon-zero serial correlation parameter pertains to the process (21) generating the consump-tion preference shock tt; we use 0.3 for qt and a value of 0.01 for ret .

Regarding the variability of the monetary-policy rule white-noise disturbance eRt, wehave set reR ¼ 0:0017 as estimated for the United States by McCallum and Nelson (1999b).

5. Model properties and comparisons

We are now prepared to develop the paper’s main quantitative results, which indicatehow the model’s properties depend upon the exogeneity vs. endogeneity of investment andthe presence or absence of adjustment costs. The comparisons are developed for threemodel variants with both fully flexible and nominal rigidities a la Calvo in prices/wagesusing (33) and (34 0). In describing these properties we focus on impulse response functionsfor two shocks: the technology shock, eAt, and the random component of monetary policy,eRt.

Fig. 1 reports paths for the variables by t, by t, pt � pss, bct, bxt, bkt, f2t � f ss2 , rt � rss, and

Rt � Rss in response to a unit realization of the technology innovation eAt when bothprices and wages adjust fully within the period of the shock. There are three model vari-ants represented: the constant-capital case (dashed lines), the endogenous capital case withno adjustment costs (dotted lines), and the endogenous capital case with investmentadjustment costs (solid lines). As mentioned above, the ‘‘hat’’ figures represent deviationsof these variables from their steady-state values. Thus a magnitude of 0.5 indicates that thevariable’s response is, in percentage terms, half as large as the shock.

Our first variant treats capital as a constant, therefore, there is no response of bxt or bkt toa unit realization of eAt. Consequently, the response of bct exceeds that of by t, which jumpsup and then gradually tails off. The marginal product of capital rises, as indicated by Eq.(41d), but the real rate of interest on securities, rt, falls. This somewhat surprising responseoccurs in order to induce an increase in consumption so as to equate the current use ofgoods ðby t ¼ ðcss=yssÞbctÞ to the enhanced capacity to produce goods ðby tÞ. With rt thus given,the policy rule (42) and the Fisher relation rt = Rt � Etpt+1 together determine inflationand the nominal interest rate. In this determination the presence of 0.8Rt�1 in the policy

Fig. 1. Responses to a technology unit shock in economies with fully flexible prices and wages (gP = gW = 0.0).


rule leads to a prolonged departure of Rt from its steady-state value, and thereby influ-ences the paths plotted.

In the second variant (dotted lines), investment and capital are generated endogenouslyand there are no adjustment costs. Both investment and capital respond positively to thetechnology shock. The response of bxt is about three times as large as for by t, and about tentimes as large as for bct. As for capital, its peak in response to the technology impact isreached many quarters after the shock and has a size of 1/2 of the shock, approximately.In this case the real interest rate equals the expected value of next period’s marginal prod-

uct of capital, adjusted by the constant real marginal cost hP�1hP

and net of depreciation

rt ¼ hP�1hP

Etf2tþ1 � d�

, so it must rise with the positive technology shock. Again, the policy

rule and the Fisher relation determine paths of Rt and pt; in this case there is a (small)upward blip in inflation.

The last result is reversed, however, in the case with adjustment costs (solid lines). Herethe presence of these costs of investing reduces the demand for investment by enough thatits increase is only about four times as large as for bct. Accordingly, the value of rt does notneed to rise to equate overall supply and demand; instead a slight fall is recorded. Thisleads to a temporary fall in the inflation rate. The response of the capital stock to theshock is somewhat smaller relative to that of the model without adjustment costs.

We need also to consider the effects of a shock to the policy rule, an upward blip in eRt.But for all the cases discussed thus far, the responses to eRt are the same for each of thevariables plotted. This is so because with flexible prices and wages the model possessesmonetary neutrality, as shown by Eq. (32). Thus, the only effect is a one-time drop in


the price level (a one-period negative blip in pt). Interestingly, this drop is just sufficient sothat the net response of Rt is zero, even though the Rt policy rule (42) is the equationshocked. This is so because with neutrality rt is not affected, and neither is expected futureinflation. No figures are presented for these cases, therefore, because all three are alike,and in each there is no response for eight of the nine variables!

Now we turn to the variant of the model in which goods prices and wages respondslowly, rather than instantaneously, to the shocks. Here we take the Calvo fixed probabil-ities setup to determine prices/wages optimally as mentioned above. In our opinion, this isthe more realistic specification, although we readily admit that the particular price/wageadjustment scheme adopted is open to objections.19 Figs. 2 and 3 report impulse responsefunctions respectively for the technology and policy rule shocks. Again, three cases areexamined: constant capital (dashed lines), endogenous investment without adjustmentcosts (dotted lines), and endogenous investment with adjustment costs (solid lines). Webegin with the responses to a technology shock, shown in Fig. 2. In the constant-capitalcase, we see that by t responds much as in Fig. 1 and that bxt and bkt do not move at all(by assumption). For the remaining variables, the patterns are distinctly different fromthose in Fig. 1. The responses are in the same directions but the movements are more grad-ual and in some cases are smaller as well.

In the next case, with endogenous investment and no adjustment costs, there is a largespike (almost four times the size of the shock) in bxt observed right after the shock, followedby a smooth return to the steady-state level thereafter. Similar spikes appear in by t andf2t � f ss

2 . It is interesting to consider why the response of xt is so much greater and less per-sistent than in Fig. 1. The fact is that without nominal rigidities inflation responds differ-ently and, via the expected inflation rate, affects the response of the real interest rate, rt.Therefore with no adjustment costs it is the case that Etf2t+1 must satisfy the arbitrage con-dition rt ¼ hP�1

hPEtf2tþ1 � d. Consequently, bktþ1 must jump upward to keep Etf2t+1 from ris-

ing in response to the technology shock. In other words, investment must be very large andshow less inertia to maintain rt ¼ hP�1

hPEtf2tþ1 � d when prices are not fully flexible. With

the recognition of adjustment costs, by contrast, the responses of bxt, by t, and f2t � f ss2 are

moderate and gradual. The maximum increase in bkt is considerably smaller than withno adjustment costs (compare solid lines to dotted lines in Fig. 2).

Responses to monetary-policy rule shocks are even more interesting in these cases withnominal rigidities (see Fig. 3). In the model with constant capital, we observe that anupward blip in eRt brings about a similar-sized jump in Rt, which decays slowly. Inflationfalls and then gradually returns to its steady-state value. The magnitude of the fall is smallenough, however, that the real interest rate rt behaves much like Rt. The temporary policyrestriction drives by t;bct, and f2t � f ss

2 down, after which all of these gradually return to theirsteady-state values as time passes.

With endogenous investment but no adjustment costs, the responses of bxt, by t, bktþ1, andf2t � f ss

2 to an interest-rate policy shock are extremely large. These huge reactions led us toseparate this case from the other cases in Fig. 3 (if shown together, that would make theothers too small for the plot). The sharpest fall is observed in the response of investmentbxt, which is truly enormous, falling proportionally by nearly 400 times (!) as much as the

19 As are all others. McCallum (1999) argues that price adjustment is the most poorly understood aspect ofmacroeconomics; that there is great disagreement among scholars as to the formulation that most nearly satisfiesthe dual requirement of theoretical plausibility and empirical veracity.

Fig. 2. Responses to a technology unit shock in economies with sticky prices and wages (gP = gW = 0.75).


increase in eRt. Output also falls drastically by 100 times as much (in relative terms) as theshock. It seems clear that this case is highly unrealistic.

With adjustment costs, however, the responses are quite moderate (see solid lines ofFig. 3). There bct falls by less than the shock magnitude, and investment by six times asmuch. The maximum fall in bkt is (in percentage terms) almost one half as much as theshock magnitude. Since output is a weighted average of consumption and investment,its response reaches an intermediate value – about two times the size of the shock. Theeffect on inflation of the brief policy tightening is not large, about one-tenth the size ofthe shock. However, part of the effect is still noticeable even after 4–6 quarters.

What is the verdict on the issue mentioned in the introduction, namely, whether theendogeneity vs. exogeneity of investment strongly affects the model’s properties in otherrespects? If we take the variant with nominal rigidities to be more realistic than that withfully flexible prices and wages, and take the presence of adjustment costs to be more real-istic than their absence, this question concerns the similarity of dashed and solid lines ineconomies with sticky prices and wages (Figs. 2 and 3). In fact, as can be seen, there isconsiderable similarity between these two, in responses to both shocks. Of course the mag-nitude of the output response is noticeably greater in the variable-capital case, since itincorporates investment responses, but the shapes of the seven plots excluding bxt andbktþ1 are remarkably alike. In fact, calibration of r can be modified in the constant-capitalcase so as to yield strikingly similar responses to the variable-capital case. Specifically, theconstant-capital economy calibrated as above except with r = 1.6 brings about responsesof by t, pt, rt, and Rt to monetary and technology shocks that are almost identical to those of

Fig. 3. Responses to a monetary policy unit shock in economies with sticky prices and wages (gP = gW = 0.75).


Fig. 4. Model comparison under sticky prices/wages. Responses to a technology unit shock (first panel), and to amonetary policy unit shock (second panel).


Table 2Standard deviations (in percentage units)

Model 1 Model 2 Model 3 US dataa Euro area data b

ry 0.61 16.7 1.48 1.49 1.03rc 0.82 1.59 0.90 1.34 0.84rx 0 65.4 3.41 5.35 2.70rx=rc 0 41.1 3.79 4.00 3.21

a Quarterly observations were obtained for the US economy for the period 1970.1–2002.4. Output is Real GDP,consumption is Real Personal Consumption Expenditures on non-durable goods, and investment is Real FixedPrivate Domestic Investment. All the original series are expressed in chained 1996 US dollars and seasonallyadjusted. To calculate the statistics, each series was logged and detrended using the Hodrick–Prescott filter(k = 1600) as is common in the literature. Source: US Department of Commerce, Bureau of Economic Analysis.

b Output is measured by the Real Gross Domestic Product. Consumption definition is Real Private FinalConsumption Expenditures whereas investment is Real Gross Fixed Capital Formation. The sample period andthe way of calculating the statistics are the same as for US data. Source: updated area-wide model data base byFagan et al. (2001).

Table 3Coefficients of correlation

Model 1 Model 2 Model 3 US data Euro area data

qyt ;ct1 0.30 0.97 0.88 0.80

qyt ;xt0 0.99 0.98 0.92 0.89

qct ;xt0 0.23 0.94 0.85 0.83

Table 4Coefficients of autocorrelation

Model 1 Model 2 Model 3 US data Euro area data

qyt ;yt�10.91 0.04 0.94 0.87 0.83

qct ;ct�10.91 0.96 0.92 0.88 0.76

qxt ;xt�10 0.02 0.93 0.91 0.85


the variable-capital economy with the calibrated figure r = 5.0. For these results, seeFig. 4.20 Thus it would appear that, judiciously utilized, analysis with constant-capital –i.e., exogenous investment – can provide a useful guide to the behavior of by t, pt, rt, andRt. Obviously it cannot do so for bct � by t, nor for bxt or bktþ1.21

Another message provided by the exercise is that the presence or absence of nominalrigidities in the model’s formulation has important effects on its response properties. Thesedifferences are sharper for responses to monetary shocks, but are also noticeable forresponses to technology shocks – compare Figs. 1 and 2.

20 If we were to comment on some small difference, it could be said that after a monetary shock inflation dropsslightly more in the constant-capital model with r = 1.6. Responses to a technology shock are surprisingly similarin both models.21 Much the same can be said for the flexible-price variant of the model, as is suggested by Figs. 1 and 3.


In addition, the responses shown in dotted lines of Figs. 2 and 3, for the sticky-price–sticky-wage case with endogenous investment but no adjustment costs, are extremely largefor the variables bxt, by t, bktþ1, and f2t � f ss

2 – especially in response to monetary shocks. Reli-ance on any single sticky-price specification is risky, without a robustness study. Neverthe-less, the results provided in dotted lines of Fig. 3 are so implausible that we are inclined tosuggest that the inclusion of investment adjustment costs (or some other sluggishnessmechanism)22 is virtually mandatory for models with endogenous investment and stickyprices.

We have also obtained some stochastic simulation results regarding second moments toillustrate the models’ properties in another fashion and to provide a stringent check ontheir realism. The models with constant capital (Model 1), with endogenous investmentand no adjustment costs (Model 2), and with endogenous investment and adjustment costs(Model 3) were simulated using the sticky-price–sticky-wage specifications, (33) and (34 0),and the calibration described in Section 4. Standard deviations, coefficients of correla-tion, and coefficients of autocorrelation for the main variables of the models – and forthe US and euro area economies – are shown in Tables 2–4.

The simulation results lead to basically the same conclusion as obtained from theimpulse response functions. When capital is endogenously introduced, some sort of slug-gishness mechanism such as adjustment costs must be included to avoid excessive variabil-ity of investment. Indeed, the introduction of adjustment costs diminishes the standarddeviation of investment by nearly 20 times! (see Table 2).

As for the coefficients of correlation and autocorrelation, our model with adjustmentcosts again matches the actual US and euro area values moderately well. By contrast,the model without adjustment costs generates far too little contemporaneous co-move-ment of consumption and output and extremely low (close to zero) serial correlation inoutput and investment (see Tables 3 and 4).

6. Conclusions

In the foregoing sections we have explored the inclusion of endogenous investmentdecisions in dynamic optimizing models that are expressed in an optimizing IS-LM frame-work. Such models, with nominal rigidities, have been used in much recent monetary-pol-icy research but usually with fixed or exogenous capital. We have found that capital/investment decisions can be endogenized readily without undue complexity or loss ofthe IS-LM-style conceptual framework that is attractive heuristically. Specifically, we havefound that endogenous investment in the presence of capital adjustment (‘‘installation’’)costs can be calibrated to yield reasonably realistic dynamic behavior using a simpleadjustment-cost function, that implies constant returns to scale in production. Our calibra-tion suggests, however, that a quadratic specification for the adjustment-cost function,which has been used by various researchers, is apparently inappropriate. Specifically, itimplies an excessively large responsiveness of investment to monetary-policy shocks, atleast under the realistic assumption that these are implemented with an interest rateinstrument.

22 Such as the Kydland and Prescott (1982) time-to-build mechanism, as mentioned above.


The paper’s analysis also yields two additional conclusions that are quite noteworthy.First, models with endogenous capital/investment choices but no adjustment costs implyhighly unrealistic responses to monetary-policy shocks under the assumption that price–wage stickiness obtains. Consequently, such models appear to be less appropriate thanones with exogenous investment. Second, although explicit inclusion of endogenousinvestment is both attractive and readily accomplished, analyses with exogenous capitalare not fundamentally misleading with respect to the cyclical behavior of output and realinterest rates. With suitable calibration, that is, such models can match those with endog-enous capital very closely. Thus such models can, if judiciously utilized, be useful for someissues in business-cycle and monetary-policy analysis.

Acknowledgements

The authors thank Andrew Abel, Michael Dotsey, Alexander Wolman, Michael Wood-ford, and two referees for helpful comments.

Appendix A. Steady-state analysis in an optimizing IS-LM model with endogenous

investment and adjustment costs

Let us take the capital first-order condition with adjustment costs of investment (38) insteady state. Recalling that in a Sidrauski-type optimizing model rss = q, we obtain

1þ q ¼ 1� dþ ð1� dÞH1ðH2 þ 1ÞdH2 þH1H2dH2þ1 þ wssf ss

2

1þH1ðH2 þ 1ÞdH2;

which can be simplified to yield

1þ q ¼ 1� dþ ð1� dþH2Þhss þ wssf ss2

1þ Css1

;

where hss ¼ H1dH2 is the steady-state unit adjustment cost and Css

1 ¼ H1ðH2 þ 1ÞdH2 is thesteady-state marginal adjustment cost. The previous expression implies, after some alge-bra, that the steady-state marginal product of capital f ss

2 becomes

f ss2 ¼

qð1þ Css1 Þ þ dð1þ hssÞ

wss :

Marginal product of capital in steady state depends positively on the rate of intertemporalpreference q, the rate of depreciation d, the steady-state marginal adjustment cost Css

1 andthe steady-state unit adjustment cost hss, and negatively on the steady-state real marginalcost wss. As for the stock of capital in a steady state, k, we can illustrate how it is deter-mined by using a Cobb–Douglas technology with steady-state labor normalized to 1:

aka�1 ¼ qð1þ Css1 Þ þ dð1þ hssÞ

wss :

After rearrangement, the steady-state capital is

k ¼ wssaqð1þ Css

1 Þ þ dð1þ hssÞ

� � 11�a

:


Thus the stock of capital depends positively on the steady-state real marginal cost wss, andon the capital share in the production function a. On the contrary, it is affected negativelyby the rate of depreciation d, the rate of intertemporal preference q, and both the steady-state marginal and unit adjustment cost, Css

1 and hss.

References

Abel, A.B., Blanchard, O.J., 1983. An intertemporal equilibrium model of saving and investment. Econometrica51, 575–692.

Blanchard, O.J., Fischer, S., 1989. Lectures on Macroeconomics. MIT Press, Cambridge, MA.Calvo, G.A., 1983. Staggered pricing in a utility-maximizing framework. Journal of Monetary Economics 12,

383–396.Casares, M., 2000. Dynamic analysis in an optimizing monetary model with transaction costs and endogenous

investment. Doctoral Thesis, pp. 60–101.Casares, M., McCallum, B.T., 2000. An optimizing IS-LM framework with endogenous investment. NBER

Working Paper Series No. 7908.Clarida, R., Galı, J., Gertler, M., 1998. Monetary policy rules in practice: Some international evidence. European

Economic Review 42, 1033–1067.Clarida, R., Galı, J., Gertler, M., 1999. The science of monetary policy: A new Keynesian perspective. Journal of

Economic Literature 37, 1661–1707.Dixit, A.K., Stiglitz, J.E., 1977. Monopolistic competition and optimum product diversity. American Economic

Review 67, 297–308.Eisner, R., Strotz, R., 1963. Determinants of business investment. In: Impacts of Monetary Policy. Prentice-Hall

Pub. Co., Englewood Cliffs, NJ.Erceg, C.J., Henderson, D.W., Levin, A.T., 2000. Optimal monetary policy with staggered wage and price

contracts. Journal of Monetary Economics 46, 281–313.Estrella, A., Fuhrer, J.C., 1998. Dynamic inconsistencies: Counterfactual implications of a class of rational

expectations models. Federal Reserve Bank of Boston Working Paper No. 98-5.Fagan, G., Henry, J., Mestre, R., 2001. An area-wide model for the euro area. Econometric Modelling Division,

European Central Bank, ECB Working Paper Series No. 42.Fuhrer, J.C., 2000. Habit formation in consumption and its implications for monetary-policy models. American

Economic Review 90, 367–390.Galı, J., Gertler, M., 1999. Inflation dynamics: A structural econometric analysis. Journal of Monetary

Economics 44, 195–222.Goodfriend, M., King, R.G., 1997. The new neoclassical synthesis and the role of monetary policy. In: Bernanke,

B.M., Rotemberg, J.J. (Eds.), NBER Macroeconomics Annual 1997. MIT Press.Griffin, P., 1992. The impact of affirmative action on labor demand: A test of some implications of the Le

Chatelier principle. Review of Economics and Statistics 74, 251–260.Griffin, P., 1996. Input demand elasticities for heterogeneous labor: Firm-level estimates and an investigation into

the effects of aggregation. Southern Economic Journal 62, 889–901.Hall, R.E., 1988. Intertemporal substitution in consumption. Journal of Political Economy 96, 339–357.Hayashi, F., 1982. Tobin’s marginal and average q: A neoclassical interpretation. Econometrica 50, 213–224.Jeanne, O., 1998. Generating real persistent effects of monetary shocks: How much nominal rigidity do we need?

European Economic Review 42, 1009–1032.Jermann, U.J., 1998. Asset pricing in production economies. Journal of Monetary Economics 41, 257–275.Kerr, W., King, R.G., 1996. Limits on interest rate rules in the IS model. Federal Reserve Bank of Richmond

Economic Quarterly 82, 47–75.Kimball, M.S., 1995. The quantitative analytics of the basic neomonetarist model. Journal of Money, Credit, and

Banking 27, 1241–1277.King, R.G., Wolman, A.L., 1996. Inflation targeting in a St. Louis model of the 21st century. Federal Reserve

Bank of St. Louis Review 78, 83–107.King, R.G., Wolman, A.L., 1999. What should the monetary authority do when prices are sticky? In: Taylor, J.B.

(Ed.), Monetary Policy Rules. University of Chicago Press.Kollmann, R., 1997. The exchange rate in a dynamic optimizing current account model with nominal rigidities:

A quantitative investigation. Working Paper 97-7, International Monetary Fund.


Koenig, E., 1993. Rethinking the IS in IS-LM: Adapting Keynesian tools to non-Keynesian economies, part 2.Federal Reserve Bank of Dallas Economic Review (Fourth Quarter).

Kydland, F.E., Prescott, E.C., 1982. Time to build and aggregate fluctuations. Econometrica 50, 1345–1370.Lucas Jr., R.E., Prescott, E.C., 1971. Investment under uncertainty. Econometrica 39, 659–681.McCallum, B.T., 1999. Issues in the design of monetary policy rules. In: Taylor, J.B., Woodford, M. (Eds.),

Handbook of Macroeconomics. North-Holland Pub. Co., Amsterdam.McCallum, B.T., 2000. Theoretical analysis regarding a zero lower bound on nominal interest rates. Journal of

Money, Credit and Banking 32, 870–904.McCallum, B.T., Nelson, E., 1999a. An optimizing IS-LM specification for monetary policy and business cycle

analysis. Journal of Money, Credit, and Banking 31, 296–316.McCallum, B.T., Nelson, E., 1999b. Performance of operational policy rules in an estimated semiclassical

structural model. In: Taylor, J.B. (Ed.), Monetary Policy Rules. University of Chicago Press, Chicago, IL.Obstfeld, M., Rogoff, K., 1996. Foundations of International Macroeconomics. MIT Press.Ogaki, M., Reinhart, C.M., 1998. Measuring intertemporal substitution: The role of durable goods. Journal of

Political Economy 106, 1078–1098.Rotemberg, J.J., Woodford, M., 1997. An optimizing-based econometric framework for the evaluation of

monetary policy. In: Bernanke, B.M., Rotemberg, J.J. (Eds.), NBER Macroeconomics Annual 1997. MITPress.

Rotemberg, J.J., Woodford, M., 1999. Interest rate rules in an estimated sticky price model. In: Taylor, J.B. (Ed.),Monetary Policy Rules. University of Chicago Press.

Sbordone, A. M., 2001, An optimizing model of US wage and price dynamics, manuscript, Rutgers University.Sbordone, A.M., 2002. Prices and unit labor costs: A new test of price stickiness. Journal of Monetary Economics

49, 265–292.Sveen, T., Weinke, L., 2004. Pitfalls in the modeling of forward-looking price setting and investment decisions.

Research Department, Norges Bank, Working Paper 2004/1.Taylor, J.B., 1993. Discretion versus policy rules in practice. Carnegie Rochester Conference Series on Public

Policy 39, 195–214.Uhlig, H., 1999. A toolkit for analyzing nonlinear dynamic stochastic models easily. In: Marimon, Ramon, Scott,

Andrew (Eds.), Computational Methods for the Study of Dynamic Economics. Oxford University Press.Walsh, C.E., 1998. Monetary Theory and Policy. MIT Press, Cambridge, MA.Woodford, M., 1995. Price-level determinacy without control of a monetary aggregate. Carnegie-Rochester

Conference Series on Public Policy 43, 1–46.Woodford, M., 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University

Press.Woodford, M., 2005. Firm-specific capital and the New Keynesian Phillips curve. International Journal of

Central Banking 1, 1–46.Yun, T., 1996. Nominal price rigidity, money supply endogenity, and business cycles. Journal of Monetary

Economics 37, 345–370.

An optimizing IS-LM framework with endogenous investment

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