On the stability of the two-sector neoclassical growth model with …bherrend/Published Papers/JEDC 2006.pdf · 2008. 9. 19. · Journal of Economic Dynamics & Control 30 (2006) 1339–1361

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Journal of Economic Dynamics & Control 30 (2006) 1339–1361

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On the stability of the two-sector neoclassicalgrowth model with externalities

Berthold Herrendorfa, Akos Valentinyib,c,d,�

aArizona State University, Tempe, USAbUniversity of Southampton, Southampton, UK

cInstitute of Economics, Hungarian Academy of Sciences, Budapest, HungarydCEPR, London, UK

Received 11 February 2005; accepted 12 May 2005

Available online 30 August 2005

Abstract

We study a class of two-sector growth models with sector-specific externalities, in which one

sector produces consumption and the other sector produces investment. The novelty is that

investment allocated to the consumption sector is an imperfect substitute for investment

allocated to the investment sector. We show analytically that in this case local indeterminacy

near the steady state is impossible for every empirically plausible specification of the model

parameters. More specifically, we show that a necessary condition for local indeterminacy is

an upward-sloping aggregate labor demand curve in the investment sector, which requires a

counterfactual strength of the externality. We show numerically that an elasticity of

substitution of plausible size implies determinacy near the steady state. These findings differ

sharply from the standard result for two-sector models that if the investments allocated to the

two sectors are perfect substitutes, then local indeterminacy occurs for a wide range of

plausible parameter values.

r 2005 Elsevier B.V. All rights reserved.

JEL classification: E0; E3

Keywords: Imperfect substitutability; Sector-specific externality; Determinacy; Local indeterminacy

see front matter r 2005 Elsevier B.V. All rights reserved.

.jedc.2005.05.006

nding author. Economics Division, School of Social Sciences, University of Southampton,

uthampton SO17 1BJ, UK. Tel.: +44 23 80592536; fax: +44 23 80593858.

dress: [email protected] (A. Valentinyi).

www.elsevier.com/locate/jedc

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B. Herrendorf, A. Valentinyi / Journal of Economic Dynamics & Control 30 (2006) 1339–13611340

1. Introduction

We study the local stability properties of the neoclassical growth model at thesteady state. Local stability analysis provides important information about the localuniqueness of equilibrium at the steady state and about the way in which businesscycles can occur in the model economy. If the steady state is saddle-path stable, thenequilibrium is locally unique at the steady state. This is called determinacy and itimplies that business cycles must come from shocks to the fundamentals of the model(typically technology shocks). In contrast, if the steady state is locally stable, then acontinuum of equilibrium paths converge to the steady state, implying a severe formof local non-uniqueness of equilibrium. This is called local indeterminacy and itimplies that business cycles can come from self-fulfilling shocks to individual beliefs.Since both determinacy and local indeterminacy are theoretically possible when thereis some form of non-convexity, we ask which one prevails for empirically plausiblechoices of parameter values.

We restrict our attention to a class of two-sector neoclassical growth models withsector-specific, positive externalities. In these models, one sector producesconsumption and the other sector produces investment. This class of models hasbeen the focus of the recent research on local indeterminacy, which started withBoldrin and Rustichini (1994) and Benhabib and Farmer (1996).1 The main findingof this research is that local indeterminacy can occur for mild, empirically plausibleexternalities in the investment sector, which are consistent with a downward-slopingaggregate labor demand curve. In contrast, in the class of standard one-sectorneoclassical growth models, local indeterminacy requires strong externalities thatmake the aggregate labor demand curve upward sloping (Benhabib and Farmer,1994; Farmer and Guo, 1994). Such strong externalities are empirically implausibleand an upward-sloping aggregate labor demand curve has awkward economicimplications (Aiyagari, 1995).

This literature assumes that investment allocated to the consumption sector is aperfect substitute for investment allocated to the investment sector. Perfectsubstitutability means that the two investments and the two installed capital stocksare perfect substitutes. Perfect substitutability implies that the production possibilityfrontier of the investment sector is linear and installed capital is not sector specific(‘putty-putty’). In contrast to this literature, we assume here that investmentallocated to the consumption sector is an imperfect substitute for investmentallocated to the investment sector. Imperfect substitutability means that the twoinvestments and the two installed capital stocks are imperfect substitutes. This is theempirically plausible case (Huffman and Wynne, 1999). Imperfect substitutability

1Subsequent contributions fall into two groups. The first group has private constant returns and

aggregate increasing returns. Examples include Perli (1998), Weder (1998, 2000), Schmitt-Grohe (2000),

Harrison (2001), and Harrison and Weder (2001). The second group has private decreasing returns and

aggregate constant returns. Examples include Benhabib and Nishimura (1998), Benhabib et al. (2000,

2002), and Nishimura and Venditti (2002). Benhabib and Farmer (1999) provide a review.

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B. Herrendorf, A. Valentinyi / Journal of Economic Dynamics & Control 30 (2006) 1339–1361 1341

implies that the production possibility frontier of the investment sector is strictlyconcave and installed capital is sector specific (‘putty-clay’).

The main result of this paper is that local indeterminacy depends criticallyon whether or not the investments are perfect substitutes. In fact, the sta-bility properties of the two-sector model with imperfect substitutability and sector-specific externalities are similar to those of the one sector model with increasingreturns. We show this in two steps. First, we show analytically that local in-determinacy does not occur if the two investments are imperfect substitutes and theaggregate labor demand curve slopes downward. Surprisingly, this holds truewhenever substitutability is imperfect no matter what the elasticity of substitution is.This result differs sharply from that of the literature, which finds that with perfectsubstitutability local indeterminacy can easily occur if the aggregate labor demandcurve slopes downward. Second, we show numerically that equilibrium isdeterminate (instead of unstable) for empirically plausible values of the elasticityof substitution and the externality. This result is robust to reasonable changes in theparameter values.

Imperfect substitutability of the two investments can be due to capital adjust-ment costs. In Herrendorf and Valentinyi (2003), we explore the local stabilityproperties of the two-sector model with the standard capital adjustment costsused by Lucas and Prescott (1971). We show that it matters whether the adjust-ment costs apply to the total capital stock of the economy or to each sector’scapital stock: local indeterminacy is easier to obtain in the first case than in thesecond case. All results of Herrendorf and Valentinyi (2003) are numerical.The present paper goes beyond it in three aspects. First, the key result of thepresent paper is analytical. Second, the key result of the present paper is stronger:given a downward-sloping aggregate labor demand curve (which is the empiricallyplausible case), we show here that local indeterminacy does not occur for any

positive elasticity of substitution between the two investments irrespective of where itcomes from. In contrast, previously we have shown only that it does not occur forsome typical parameter choices. Third, the present paper provides the economicintuition for the effects of imperfect substitutability, which was missing from ourprevious paper.

The intuition follows from the relationship between the relative price of the twoinvestments and the composition of the investment production. We start with thecase in which the two investments are perfect substitutes (which happens, forexample, when it is costly to change total capital but not each sector’s capitalstock). Then the relative price between the two investments is one and the firmsin the investment sector are indifferent between different compositions of theirproductions, so that composition is entirely determined by the demand for theinvestments. Since there are sector-specific externalities, changes in beliefs aboutfuture returns can then lead to self-fulfilling changes in the composition of theinvestment production. This mechanism does not work when the two investmentsare imperfect substitutes (which happens, for example, when it is costly to changeeach sector’s capital stock). The relative price between them then varies withthe composition of the investment production and the firms in the investment

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sector are not indifferent between different compositions of their productions. Therelative price of one investment in terms of the other then determines thecomposition of the investment production. Since that relative price is determinedat each point in time by past consumption–savings decisions, changes in beliefsabout future returns can no longer lead to self-fulfilling changes in the compositionof the investment production.

Wen (1998), Guo and Lansing (2002), and Kim (2003) also studied theimplications of capital adjustment costs for local indeterminacy, but they employedthe one-sector neoclassical growth model with an externality. They all found athreshold result: given a strength of increasing returns that implies localindeterminacy, there is a positive, minimum size of the capital adjustment coststhat makes local indeterminacy impossible.2 The threshold behavior of the one-sector model is similar to the behavior of the two-sector model when the adjust-ment costs apply to the total capital stock (Herrendorf and Valentinyi, 2003).The threshold behavior of the one-sector model is very different from the behaviorof the two-sector model when the adjustment costs apply to each sector’s capitalstock separately and indeterminacy cannot occur for any strength of increas-ing returns that leaves the aggregate labor demand curve downward sloping.The reason is that adjustment costs of each sector’s capital stock lead to im-perfect substitutability of the two investments, so the results of the present paperapply.

The paper is organized as follows. In Section 2 we present the model.Sections 3 and 4 contain our analytical and numerical results about the stabilityproperties of the two-sector growth model with imperfect substitutability andsector-specific externalities. Section 5 concludes the paper. All proofs are in anAppendix.

2. Model economy

Consider the following environment. Time is continuous and runs forever. Thereis a representative households and two representative firms. One representativefirm produces a perishable consumption good and the other one produces theinvestments for both sectors. The representative household is endowed with theinitial capital stocks, with the property rights for the representative firms, andwith one unit of time at each instant. We assume that installed capital is sectorspecific, which is consistent with the evidence collected by Ramey and Shapiro(2001) that it is very costly to reallocate installed capital to other sectors. At eachpoint in time five commodities are traded in sequential markets: consumption,the investment for the consumption sector and the investment for the invest-ment sector, working time in the consumption sector, and working time in theinvestment sector.

2Lahiri (2001) shows that opening up capital mobility has the opposite effect as capital adjustment costs:

it decreases the strength of increasing returns required for indeterminacy.

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The representative household solves:

maxfCt;X ct ;X xt ;Lct;Lxt ;Kct;Kxtg

1t¼0

Z 10

e�rt½logCt þ ðL� Lct � LxtÞ�dt ð1aÞ

s.t. Ct þ pctX ct þ pxtX xt ¼ pct þ pxt þ wctLct þ wxtLxt

þ rctKct þ rxtKxt, ð1bÞ

_Kct ¼ X ct � dcKct; _Kxt ¼ X xt � dxKxt, ð1cÞ

Kc0 ¼ Kc0 given; Kx0 ¼ Kx0 given, ð1dÞ

0pCt;Lct;Lxt;X ct;X xt;Kct;Kxt; Lct þ LxtpL. ð1eÞ

The notation is as follows: r40 is the discount rate; Ct denotes the consumptiongood at time t (which is the numeraire); L 2 ð0;1Þ is the time endowment; thesubscripts c and x indicate variables from the consumption and the investmentsector; Lct and Lxt are the working times, wct and wxt are the wages, X ct and X xt arethe investments, pct and pxt are the relative prices of the investments, Kct and Kxt arethe installed capital stocks, rct and rxt are the real interest rates, dc and dx are thedepreciation rates, and pct and pxt are the profits (which will be zero in equilibrium).Two features of the representative household’s problem deserve further comment.First, we restrict X ct and X xt to be non-negative, meaning that installed capital issector specific. Nevertheless the capital stock of a sector can be reduced by notreplacing depreciated capital, so close to the steady state (the existence of which wewill prove below) the non-negativity constraints will not be binding. Second, wechoose the functional form for utility that is most commonly used in the literature.We focus on an infinite equilibrium labor supply elasticity because the existingstudies identify this to be the best case for local indeterminacy. An economicjustification for an infinite labor supply elasticity is the lottery argument of Hansen(1985) and Rogerson (1988).

Denoting by mct and mxt the current value multipliers attached to the accumulationequations (1c), the necessary and sufficient conditions for a solution to thehousehold’s problem are (1b)–(1e) and

pct

Ct

¼ mct;pxt

Ct

¼ mxt, (2a)

Ct ¼ wct ¼ wxt, (2b)

_mctpmctðdc þ rÞ �rct

Ct

ðwith equality if X ct40Þ, (2c)

_mxtpmxtðdx þ rÞ �rxt

Ct

ðwith equality if X xt40Þ, (2d)

limt!1

pctKct

Ct

e�rt ¼ limt!1

pxtKxt

Ct

e�rt ¼ 0. (2e)

Note that, as usual, the dynamic first-order conditions (2c) and (2d) hold only fort40. Note too that the wage rates will be equalized across sectors but the real

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interest rates will only be equalized across sectors if the two investments are perfectsubstitutes, in which case their shadow prices are equal.

We now turn to the production side of the model economy. The problem of therepresentative firm of the consumption sector is

maxct;kct ;lct

pct � ct � rctkct � wctlct ð3aÞ

s.t. ct ¼ Atkactl

1�act ; ct; lct; kctX0, ð3bÞ

where AtX0 denotes total factor productivity in the sector and a 2 ð0; 1Þ. Thenecessary and sufficient conditions for a solution are (3b) and

rct ¼ aAtka�1ct l1�a

ct , (4a)

wct ¼ ð1� aÞAtkactl�act . (4b)

The problem of the representative firm of the investment sector is

maxxxt ;xct ;lxt;kxt

pxt � pxtxxt þ pctxct � rxtkxt � wxtlxt ð5aÞ

s.t. f ðxct;xxtÞ ¼ Btkbxtl

1�bxt ; xxt;xct; kxt; lxtX0, ð5bÞ

where BtX0 denotes total factor productivity in the sector, b 2 ð0; 1Þ, and f is a twicecontinuously differentiable function that is non-negative, increasing in botharguments, linear homogeneous, and quasi-convex.3 A functional form that satisfiesthese requirements is

f ðxct;xxtÞ ¼ ðf cx1þect þ f xx1þe

xt Þ1=ð1þeÞ, (6)

where f c, f x, and e are positive constants.Denoting the multiplier attached to the equation of (5b) by lt, the necessary and

sufficient conditions for the solution to problem (5) are (5b) and

rxt ¼ ltbBtkb�1xt l1�b

xt , (7a)

wxt ¼ ltð1� bÞBtkbxtl�bxt , (7b)

pctpltf cðxct;xxtÞ ðwith equality if xct40Þ, (7c)

pxtpltf xðxct;xxtÞ ðwith equality if xxt40Þ, (7d)

where f c and f x denote the partial derivatives of f with respect to xct and xxt.The assumption of quasi-convexity implies that for a given f 2 Rþ the lower setsfðxxt;xctÞ 2 R

2þjf ðxxt;xctÞpf g are convex, so the production possibility frontier

between the two investments, xct and xxt, is concave. In other words, the twoinvestments are imperfect substitutes. This is relevant only if the two installedinvestments are also imperfect substitutes, otherwise any reallocation of total capitalbetween the two sectors can be achieved by reallocating installed capital. It is for thisreason that we have assumed that installed capital is sector specific. The standard

3Homogeneity is required for the existence of a balanced growth path.

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assumption in the literature is that f is linear:

f ðxct;xxtÞ ¼ f cxct þ f xxxt, (8)

where f c and f x are positive constants, which are often set to one.4 If f is linear, thenthe production possibility frontier between the two investments is linear too. In otherwords, the two investments are perfect substitutes. If this is the case, then it isirrelevant for the local stability properties whether the two installed investments areperfect or imperfect substitutes. The reason is that installed capital depreciates, soclose to the steady state any change in the capital stocks of the two sectors can beachieved by a corresponding change in the composition of the investmentproduction. In any case, we find it convenient to maintain sector-specificity alsowhen we study perfect substitutability of investments.5

The total factor productivities are specified so that there can be positiveexternalities at the level of each sector:

At ¼ kycact lycð1�aÞ

ct ; Bt ¼ kyxbxt lyxð1�bÞ

xt , (9)

where yc; yxX0. Substituting (9) back into the production functions, the sectors’aggregate outputs become:

ct ¼ ka1ct la2xt ; a1 � ð1þ ycÞa; a2 � ð1þ ycÞð1� aÞ, (10a)

xt ¼ kb1xt l

b2xt ; b1 � ð1þ yxÞb; b2 � ð1þ yxÞð1� bÞ. (10b)

Note that (9) implies that the externalities on capital and labor are the same.6 Notetoo that the externalities are not taken into account by the firms, so a competitiveequilibrium exists. In equilibrium, profits are zero and the capital and labor sharesare the usual ones: rctkxt=ct ¼ a, wctlxc=ct ¼ 1� a, rxtkxt=kt ¼ b, wxtlxt=kt ¼ 1� b.Moreover, in equilibrium, the total factor productivities on which the firms basetheir decisions must be equal to those that results from these decisions:

Definition 1 (Competitive equilibrium). A competitive equilibrium is (shadow) pricesfwct;wxt; rct; rxt; pct; pxt;mct;mxt; ltg

1t¼0, an allocation fCt; ct;X ct;xct;X xt;xxt;Lct; lct;

Lxt; lxt;Kct; kct;Kxt; kxtg1t¼0, and total factor productivities fAt; Btg

1t¼0 such that:

(i)

4Th

choic5N

How6Th

impo

witho

fCt;X ct;X xt;Lct;Lxt;Kct;Kxtg1t¼0 solve the problem of the representative house-

hold, (1), that is, (1c) and (2a)–(2e) hold;
(ii) fct; lct, kctg
1t¼0 solve the problem of the representative firm of the consumption

sector, (3), that is, (3b) and (4a)–(4b) hold;
(iii) fxxt;xct; lxt; kxtg
1t¼0 solve the problem of the representative firm of the investment

sector, (5), that is, (5b) and (7a)–(7d) hold;

e choice of f c and f x amounts to a choice of the units in which xct and xxt are denominated. This

e does not matter for the local stability properties of the steady state.

ote that at t ¼ 0 both kc0 and kx0 are given because installed capital is assumed to be sector specific.

ever, this does not invalidate the previous argument.

e results of Harrison and Weder (2001) suggest that imposing this constraint does not affect in an

rtant way the stability properties of the steady state of the two-sector neoclassical growth model

ut capital adjustment costs.

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(iv)

7If

capit

econo

indet

conti8Th

secto

stead

markets clear, that is, Ct ¼ ct, X ct ¼ xct, X xt ¼ xxt, Lct ¼ lct, Lxt ¼ lxt, Kct ¼ kct,Kxt ¼ kxt;

(v)
At and Bt are determined consistently, that is, the two equations in (9) hold.
3. Analytical results

3.1. Local stability properties

We start with the case of linear f (perfect substitutability). Since the results withlinear f are known, we just summarize them here and refer the reader to Benhabiband Farmer (1996), Harrison and Weder (2001), or the working-paper version of thispaper, Herrendorf and Valentinyi (2004), for the technical derivations. There, thefollowing is shown. First, there is a unique steady state and a neighborhood of thesteady state such that the equilibrium reduced-form dynamics can be described bythe dynamics of the state variable kt � f ckct þ f xkxt and the dynamics of the controlvariable mct. Second, there are constants yx 2 ð0; b=ð1� bÞÞ and yx 2 ð�rb=ðrbþ

dxÞ; ð1� bÞ=bÞ such that either 0oyxoyx or yxo0oyx. In the first case, the steadystate is saddle-path stable for yx 2 ½0; yxÞ, locally stable for yx 2 ðyx; yxÞ, and unstablefor yx 2 ðyx; ð1� bÞ=bÞ. In the second case the steady state is saddle-path stable foryx 2 ½0; yxÞ and locally stable for yx 2 ðyx; ð1� bÞ=bÞ.

Typical calibrations of our two-sector model are consistent with the assumptionsbo1� b and yxoð1� bÞ=b. The inequality bo1� b ensures that the capital share inthe investment sector’s income is smaller than one half and that b=ð1� bÞoð1� bÞ=b. The inequality yxoð1� bÞ=b ensures that the aggregate returns to capitalare not too large so there cannot be endogenous growth in steady state. There aretwo relevant subcases of yxoð1� bÞ=b: for yx 2 ½0; b=ð1� bÞÞ the aggregate labordemand curve of the investment sector slopes downward and for yx 2 ðb=ð1� bÞ;ð1� bÞ=bÞ it slopes upward. The labor demand curve is obtained by equating themarginal product of labor in the investment sector with the wage which giveslx ¼ ½wx=ð1� bÞkbð1þyxÞ

x �1=½ð1�bÞð1þyxÞ�1�. So the results summarized above say that ifthe aggregate labor demand curve in the investment sector slopes downward, thenthe steady state can be saddle-path stable, unstable, and locally stable.7 The fact thatthe steady state of the two-sector model with sector-specific externalities can belocally stable when the aggregate labor demand curve in the investment sector slopesdownward implies that local indeterminacy does not require unreasonably strongexternalities.8

the steady state is saddle-path stable then the equilibrium is determinate, that is, given the initial

al stocks close to the steady-state values there are unique initial shadow prices such that the model

my converges to the steady state. If the steady state is locally stable, then the equilibrium is locally

erminate, that is, given the initial capital stocks close to the steady-state values there exists a

nuum of shadow prices such that the model economy converges to the steady state.

e literature typically derives this when capital is fully mobile across sectors whereas here capital is

r specific. As Christiano (1995) showed, though, this does not matter for the stability properties of the

y state.

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We now turn to the case of strictly quasi-convex f (imperfect substitutability).

Proposition 1 (Reduced-form dynamics for strictly quasi convex f ). There is a unique

steady state. There is a neighborhood of the steady state such that the equilibrium

reduced-form dynamics can be described by the dynamics of the two state variables kct

and kxt and the two control variables mct and mxt.

Proof. See Appendix A. &

In words, the equilibrium reduced-form dynamics close to the steady state are twodimensional when f is linear and four dimensional when f is strictly quasi-convex.The reason is that with linear f the two investments are perfect substitutes, so onlythe total capital stock and its shadow price are needed to describe the dynamics.With a strictly quasi-convex f the two investments are imperfect substitutes, so bothof them and two shadow prices are needed to describe the dynamics.

We now explore analytically the stability properties of the steady state. The steadystate is saddle-path stable if there are as many stable roots (i.e. roots with negativereal part) as states and as many unstable roots (i.e. roots with positive real part) ascontrols. It is locally stable if there are more stable roots than states and it is unstableif there more unstable roots than controls. Since it is not feasible to computeanalytically the four eigenvalues, we will only compute the determinant and the traceof the linearization of the reduced-form equilibrium dynamics at the steady state.Although this does not allow for a full characterization of the local stabilityproperties, it provides important information because the determinant equals theproduct of the eigenvalues and the trace equals the sum of the real parts of theeigenvalues (complex eigenvalues occur in conjugates, implying that the imaginaryparts cancel in the summation). This leads to the next proposition, which constitutesthe main result of our paper.

Proposition 2 (Local stability properties of the steady state for strictly quasi-convex

f ). Let bo1� b and yx 2 ½0; ð1� bÞ=bÞ.

(a)
yx 2 ½0; b=ð1� bÞÞ is a necessary condition for the steady state to be saddle-path
stable;
(b) yx 2 ðb=ð1� bÞ; ð1� bÞ=bÞ is a necessary condition for the steady state to be locally
stable.

Proof. See Appendix B. &

In words, this proposition says that if the aggregate labor demand curve in theinvestment sector slopes upward, then the steady state can be locally stable orunstable but not saddle-path stable and if the aggregate labor demand curve in theinvestment sector slopes downward, then the steady state can be saddle-path stable orunstable but not locally stable. Thus, a strictly quasi-convex f rules out localindeterminacy at the steady state if the aggregate labor demand curve slopesdownward. This is our key analytical result, which holds for any strictly quasi-convex f. In other words, the local stability properties of the two-sector neoclassical

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growth model with strictly quasi-convex f differ strikingly from those with a linear f.In fact, the local stability properties of the two-sector model with strictly quasi-convex f are much more like those of the one-sector neoclassical growth modelwithout capital adjustment costs, in which local indeterminacy requires an upward-sloping aggregate labor demand curve (Benhabib and Farmer, 1994).

3.2. Intuition

Here, we seek to understand why imperfect substitutability precludes the possi-bility of local indeterminacy when the aggregate labor demand of the investmentsector slopes downward.

The first possibility that comes to mind is that there is a discontinuity at the steadystate, that is, as the strictly quasi-convex f converges to the linear f, the steady statesof the model economies with strictly quasi-convex f do not converge to the steadystate of the economy with linear f. In the working-paper version of this paper,Herrendorf and Valentinyi (2004), we establish that this is not the case, so adiscontinuity at the steady state is not the explanation for our results.

To explain our results, it is useful to recall how with perfect substitutability localindeterminacy can occur for mild strengths of the externality.9 To this end, supposethe model economy is on an equilibrium path to the steady state and ask whetherthere can be another equilibrium path with the same initial capital stock buttemporarily higher capital stocks subsequently. Such a path requires a differentcomposition of investment output: initially more investments for the investmentsector and then fewer investments for the investment sector. This results in initiallylower and then higher capital stocks in the consumption sector. Consumptiongrowth is therefore first lower and then higher, which is optimal for the householdonly if the returns on installed capital are initially higher and then lower. Since withperfect substitutability the relative price of the two investments is constantirrespective of the composition of investment output, only the shape of theaggregate production possibility frontier (ppf henceforth) between consumption andthe composite capital good matters. The aggregate ppf is strictly convex, so a lowerratio of consumption to composite investment good is associated with a lowerrelative price of the composite investment good in terms of consumption and viceversa. Along the alternative path, capital gains can therefore generate the requiredmovements of the returns to capital.

Two crucial ingredients bring about the capital gains when the two investments areperfect substitutes. First, the aggregate ppf between consumption and compositeinvestment is strictly convex at the steady state. The working-paper version of thispaper, Herrendorf and Valentinyi (2004), shows that this ingredient is still presentwhen the two investments are imperfect, but arbitrarily close, substitutes. The secondcrucial ingredient that brings about the capital gains with perfect substitutability isthe indeterminate composition of the investments production. Formally this followsfrom the fact that the relative price of the two investments is constant and so the firm

9The arguments of this section follow Christiano (1995).

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is indifferent between different compositions of its investment production. Hence,the household’s demand determines the composition of the investment production,implying that changes in beliefs about future returns can be accommodated bychanges in the composition of the investments production. This second ingredient isnot present when the two investments are imperfect substitutes. The reason is thatthe relative price of one investment good in terms of the other then determines thecomposition of the investment production. Formally, this follows by combining (2a),(7c), and (7d) (the last two with equality):

pct

pxt

¼mct

mxt

¼f cðxct=xxt; 1Þ

f xðxct=xxt; 1Þ. (11)

To see how this rules out local indeterminacy, assume that dc ¼ dx, for simplicity,and use the arbitrage conditions (2c) and (2d) together with (2a) to derive

rct

pct

þ_pct

pct

¼rxt

pxt

þ_pxt

pxt

.

Consequently,

d

dt

pct

pxt

� �¼

pct

pxt

rxt

pxt

�rct

pct

� �. (12)

Consider now an equilibrium path that converges to the steady state and ask whetherthere can be another equilibrium path that too converges to the steady state and thatinitially has less capital allocated to the consumption sector, so initially xct=xxt issmaller. Eq. (11) shows that initially pct=pxt must then be smaller too. Since with lesscapital in the consumption sector the marginal product of capital in the consumptionsector is larger than in the investment sector, we also have that initially

rxt

pxt

�rct

pct

o0.

The arbitrage equation (12) therefore implies that pct=pxt must decrease further and(11) implies that xct=xxt must decrease further, and so on and so forth.Consequently, along the alternative path, the capital stock of the consumptionsector is ever decreasing, implying that the alternative path cannot converge to thesteady state and thus violates the transversality condition. Therefore, it cannot be anequilibrium path.

It is important to emphasize that capital gains are possible irrespective of whetherwe have perfect or imperfect substitutability. The difference is that with perfectsubstitutability the capital gains have no direct effect on the composition of theinvestments production. In contrast, with imperfect substitutability the capital gainsaffect the composition of the investments production. This is the key feature of ourmodel that rules out local indeterminacy for downward-sloping aggregate labordemand curve.

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Table 1

Benchmark calibration of parameters that affect local stability

a ¼ 0:41 b ¼ 0:34 dc ¼ 0:018 dx ¼ 0:020 r ¼ 0:02 e 2 ð0; 0:5Þ yx 2 ð0; 0:9Þ


4. Numerical results

The analytical results derived so far for imperfect substitutability show that anecessary condition for local indeterminacy is an upward-sloping aggregate labordemand curve and a necessary condition for determinacy is a downward-slopingaggregate labor demand curve. Since our dynamical system is four dimensional, it isimpossible to fully characterize the local stability properties at the steady stateanalytically. We therefore calibrate our model to quarterly data and numericallycompute the four eigenvalues for the calibrated model. We adopt the functionalforms from Huffman and Wynne (1999).

We start with the calibration of L, f c, and f x. The values of these parameters donot affect the local stability properties of the steady state. In fact, we do not need tospecify a value for L at all. Note that we could choose L to match the share ofworking time in total time. In contrast, we do need to specify values for f c and f x.We impose f x ¼ 1� f c and choose f c such that mc ¼ mx in steady state, which isconvenient.

For most technology parameter, we use the estimates of Huffman and Wynne(1999). Before going into details, we should mention that Huffman and Wynne had atwo sector with constant returns whereas we have one with increasing returns. Thisimplies that their parameter values may be inappropriate for our purposes.Fortunately, Huffman and Wynne estimate a, b, dc, dx, and e in such a way thatthe results do not depend on whether or not there are increasing returns. Specifically,they directly estimate the capital shares and the depreciation rates from postwar U.S.data. This gives the values reported in Table 1. Huffman and Wynne’s estimation ofthe elasticity parameter e is a little more involved. It uses postwar U.S. data on thenominal and real investments of the two sectors and the equilibrium relationship

pcxc

pxxx

¼f c

f x

xc

xx

� �1þe

,

which implies that the percentage deviations from trend satisfy

dpcxc � dpxxx ¼ ð1þ eÞð bxc � bxxÞ.

Since (7c)–(7d) imply that these relationships hold in our model too, we may useHuffman and Wynne’s estimate for e, which depending on the procedure is 0:1 or0:3. Since e is a key parameter determining the local stability properties of the steadystate and since it is hard to draw a sharp line between empirically plausible andimplausible values for it, we will vary it and explore the local stability properties ofthe steady state for all e 2 ð0:000; 0:500Þ.10

10e ¼ 0:000000001 is the closest value to zero that we use in these computations.

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This leaves the preference parameter r and the two increasing returns parametersyc and yx. We set r ¼ 0:02. Independent of the values of yc and yx, this implies thatin steady state the investment–output ratio is 0:2 and the capital–output ratio is 2:6.We do need to choose a value for yc, as the reduced-form equilibrium dynamics areindependent of it; see (B.3)–(B.5e).11 We restrict the value of yx from the availableevidence on increasing returns. This evidence is somewhat mixed. However, it is clearthat Hall’s (1988) initial estimates of large aggregate increasing returns around 0:5were upward biased. More recent empirical studies find estimates between constantreturns and milder increasing returns up to 0:3; see e.g. Bartelsman et al. (1994),Burnside et al. (1995), Burnside (1996), Basu and Fernald (1997), and Harrison(2003). According to Basu and Fernald (1997) and Harrison (2003) these aggregateincreasing returns are mainly due to increasing returns in the investment sector. Forexample, Harrison finds decreasing to constant returns in the consumption sectorand mildly increasing returns in the investment sector (the point estimate isyx ¼ 0:108). Since yx is a key parameter determining the local stability properties ofthe steady state and since it is hard to draw a sharp line between empirically plausibleand implausible values for it, we will vary it explore the local stability properties ofthe steady state for all yx 2 ð0:000; 0:900Þ.

Our numerical results are reported in Fig. 1. The figure displays the stabilityproperties of the economy for e40 only, which is indicated by the small distancebetween the shaded area and the vertical axis. The numerical results confirm theanalytical one that an upward-sloping (downward-sloping) aggregate labor demandcurve in the investment sector is a necessary condition for local indeterminacy(determinacy).12 The numerical results go beyond the analytical ones in threerespects. First, they show that an upward-sloping (downward-sloping) aggregatelabor demand curve in the investment sector becomes a sufficient condition for localindeterminacy (determinacy) when the substitutability between the investments aresufficiently low (eX0:411). Second, they show that for capital adjustment costswithin the range calibrated by Huffman and Wynne, e 2 ½0:1; 0:3�, the steady state isdeterminate if the increasing returns do not exceed 0:313. The range yx 2 ½0; 0:313�includes most values of increasing returns that are usually considered reasonable. So,given e 2 ½0:1; 0:3�, the local stability properties with a strictly quasi-convex f aresummarized by determinacy for every empirically plausible specification of yx. Third,our numerical results show that e ¼ 0:000000001 makes the equilibrium determinatefor yx 2 ð0; 0:078Þ. In contrast, for e ¼ 0 the equilibrium is locally indeterminate foryx 2 ð0:053; 0:078Þ. Thus, the steady state with small degree of imperfect substitut-ability is saddle-path stable in the region of increasing returns in which the steadystate with perfect substitutability is locally stable.13

11Weder (2000) and Harrison (2001) show that is also the case with perfect substitutability.12For the calibration used here the equilibrium labor demand curve slopes upward if and only if

yx40:51.13Note the possibility for global indeterminacy, which we do not pursue any further in this paper. This

follows from the additional piece of information that at the bifurcation to ‘instability’ two of the

eigenvalues are complex and their real parts change sign, that is, a Hopf bifurcation occurs. The Hopf

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Fig. 1. Local stability if investments are imperfect substitutes and r ¼ 0:02, dc ¼ 0:018, dx ¼ 0:020,a ¼ 0:41, b ¼ 0:34.


We complete this section with a brief discussion of the robustness of our numericalfindings. First, the working-paper version of this paper, Herrendorf and Valentinyi(2004), shows that our numerical determinacy result survives for reasonablevariations of the parameter values used above. Second, Herrendorf and Valentinyi(2003) show that our numerical determinacy result survives when the imperfectsubstitutability comes from capital adjustment costs of the form suggested by Lucasand Prescott (1971).

5. Conclusion

We have explored the conditions under which indeterminacy of equilibrium occursnear the steady state in a class of two-sector neoclassical growth models with sector-

(footnote continued)

bifurcation theorem implies the existence of limit cycles, which may or may not be stable. If they are stable,

then a form of global indeterminacy occurs.

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specific externalities. Our main finding has been that imperfect substitutability of thetwo investments precludes local indeterminacy for every empirically plausiblespecification of the model parameters. This analytical result contrasts sharply withthe standard result that with perfect substitutability local indeterminacy can occur inthe two-sector model for a wide range of plausible parameter values. It can beinterpreted to mean that local indeterminacy is not a robust property of the class oftwo-sector neoclassical growth models with sector-specific externalities. Weconjecture that this result is likely to carry over to models with more than twosectors and more than two investments.

Our findings are relevant for several reasons. To begin with, if local indeterminacyis impossible for plausible specifications of the parameter values, then self-fulfillingbusiness cycles are impossible for plausible specifications of the parameter values.This has important implications for the debate about whether or not governmentpolicy should aim to stabilize business cycles; see Christiano and Harrison (1999).Second, models from the class of two-sector neoclassical growth models that we havestudied here are widely used; see for example Fisher (1997), Huffman and Wynne(1999), and Boldrin et al. (2001). Our results provide a better understanding ofthe local stability properties of this important class of models. Finally, the results ofthis paper contribute to a recent debate about the robustness of multiple andindeterminate equilibria. Even though Morris and Shin (1998) and Herrendorf et al.(2000) studied rather different environments with externalities, they share a commontheme with the present paper: the introduction of frictions can substantially reducethe scope for the multiplicity or local indeterminacy of equilibrium.

Acknowledgements

We are particularly indebted to Manuel Santos for his help. Moreover, we haveprofited from the comments of the editor Peter Ireland, an anonymous referee, JessBenhabib, Michele Boldrin, Robin Mason, Salvador Ortigueira, Juuso Valimaki, MarkWeder, the audiences at Carlos III, Frankfurt, the Helsinki School of Economics, andSouthampton. Herrendorf acknowledges research funding from the Spanish DireccionGeneral de Investigacion (Grant BEC2000-0170), from the European Union (Project721 ‘New Approaches in the Study of Economic Fluctuations’), and from the InstitutoFlores de Lemus (Universidad Carlos III de Madrid).

Appendix A. Proof of Proposition 1

A.1. Reduced-form dynamics

Suppose that all first-order conditions hold with equality. Eqs. (1c) and (2b)–(2d)then imply

_kct ¼ xct � dckct; _kxt ¼ xxt � dxkxt, (A.1a)

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_mct ¼ mctðdc þ rÞ �rct

wct

; _mxt ¼ mxtðdx þ rÞ �rxt

wxt

. (A.1b)

To represent the model economy as a dynamical system in kct, kxt, mct, and mxt, weneed to express all endogenous variables, i.e. xct, xxt, lct, lxt, rct, rxt, pct, pxt, wct, andwxt, as functions of these four variables. Establishing this is the first step of the proof.

To begin with, note that (2a) implies that pct=pxt ¼ mct=mxt, so (7c) and (7d) (withequality) together with the strict quasi-convexity of t imply that there is a function g

such that

gmct

mxt

� ��

f c

f x

� ��1 mct

mxt

� �¼

xct

xxt

. (A.2a)

Next, observe that dividing (4a) by (4b) and (7a) by (7b) and using (A.3a), we canexpress the factor price ratios as functions of the corresponding factors:

rct

wct

¼a

1� a

lct

kct

;rxt

wxt

¼b

1� b

lxt

kxt

. (A.2b)

Now, we derive labor in the consumption sector. Combining (2b), the firstequation of (3b), and (4b) gives

lct ¼ 1� a. (A.3a)

Turning to labor in the investment sector, observe that (2a) and (2b) imply1 ¼ mxtðwxt=pxtÞ. Substituting (7b) and (7b) into this leads to

1 ¼ ð1� bÞmxtkb1xt l

b2�1xt f x g

mct

mxt

� �; 1

� �� 1,

where we used the fact that f ð � ; � Þ is homogeneous of degree one, and (A.2a).Rearranging leads to the reduced form for labor in the investment sector:

lxt ¼ lxðkxt;mct;mxtÞ � ½ð1� bÞmxt�1=ð1�b2Þf x g

mct

mxt

� �; 1

� �1=ðb2�1Þ

kb1=ð1�b2Þxt .

(A.3b)

Substituting (A.3a) and (A.3b) into (A.2b) for lc and lx, rearranging and pluggingthe result into (A.1b) gives

_mct ¼ Fmcðkct; kxt;mct;mxtÞ � ðrþ dcÞmct �a

kct

, (A.4a)

_mxt ¼ Fmxðkct; kxt;mct; mxtÞ � ðrþ dxÞmxt �b

1� b½ð1� bÞmxt�

1=ð1�b2Þ

� f x gmct

mxt

� �; 1

� �1=ðb2�1Þ

kðb1þb2�1Þ=1�b2xt . ðA:4bÞ

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Next, we derive the expressions for each type of investment. Substituting (9) and(A.2a) into the equation of (5b) gives

kb1xt l

b2xt ¼ xct

f ðgðmct=mxtÞ; 1Þ

gðmct=mxtÞ¼ xxtf g

mct

mxt

� �; 1

� �.

To eliminate lxt from these expressions, we use (A.3b). Solving afterwards for xct andxxt gives

xct ¼ xcðkxt;mct;mxtÞ � ½ð1� bÞmxt�b2=ð1�b2Þ

�gðmct=mxtÞf xðgðmct=mxtÞ; 1Þ

b2=ðb2�1Þ

f ðgðmct=mxtÞ; 1Þkb1=ð1�b2Þxt ,

xxt ¼ xxðkxt;mct;mxtÞ � ½ð1� bÞmxt�b2=ð1�b2Þ

f xðgðmct=mxtÞ; 1Þb2=ðb2�1Þ

f ðgðmct=mxtÞ; 1Þkb1=ð1�b2Þxt .

Substituting the above reduced forms for xct, xxt, into (A.1a) and rearranging, wefind the reduced-form equilibrium dynamics:

_kct ¼ Fkcðkct; kxt;mct; mxtÞ � ½ð1� bÞmxt�b2=ð1�b2Þ

�gðmct=mxtÞf xðgðmct=mxtÞ; 1Þ

b2=ðb2�1Þ

f ðgðmct=mxtÞ; 1Þkb1=ð1�b2Þxt � dckct, ðA:4cÞ

_kxt ¼ Fkxðkct; kxt; mct;mxtÞ � ½ð1� bÞmxt�b2=ð1�b2Þ

�f xðgðmct=mxtÞ; 1Þ

b2=ðb2�1Þ

f ðgðmct=mxtÞ; 1Þkb1=ð1�b2Þxt � dxkxt. ðA:4dÞ

A.2. Existence and uniqueness of steady state

Representing variables in steady state by dropping the time index t and assumingthat all first-order conditions hold with equality, the steady-state versions of (A.4b)and (A.4d) are found to be

dxkð1�b1�b2Þ=ð1�b2Þx ¼ ½ð1� bÞmx�b2=ð1�b2Þ

f xðgðmc=mxÞ; 1Þb2=ðb2�1Þ

f ðgðmc=mxÞ; 1Þ, (A.5a)

ðrþ dxÞkð1�b1�b2Þ=ð1�b2Þx ¼ b½ð1� bÞmx�

b2=ð1�b2Þf xðgðmc=mxÞ; 1Þ1=ðb2�1Þ. (A.5b)

Dividing the second equation by the first one leads to

rþ dx

bdx

¼f ðgðmc=mxÞ; 1Þ

f xðgðmc=mxÞ; 1Þ. (A.6)

Given the assumed properties of f, this expression can be solved uniquely for mc=mx,so the steady-state shadow price ratio is uniquely determined by the parameters of

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the model. From now on we will therefore write f, f x, and g for the unique steady-state values of these functions. We can then write (A.4a), (A.4c), and (A.4d)evaluated at the steady state as follows:

mct ¼a

rþ dc

k�1ct , (A.7a)

dckc ¼½ð1� bÞmx�

b2=ð1�b2Þgf b2=ðb2�1Þx

fkb1=ð1�b2Þ

x , (A.7b)

dxkx ¼½ð1� bÞmx�

b2=ð1�b2Þf b2=ðb2�1Þx

fkb1=ð1�b2Þ

x . (A.7c)

To show uniqueness, we will show that kc, mx, and mc are functions of kx. We willthen show that kx is uniquely determined by the parameters of the model. Dividing(A.7b) by (A.7c) gives kc as a function of kx:

kc ¼dx

dcgkx. (A.8)

Since from (A.7a) mc is a function of kc, (A.8) implies that mc is a function of kx.Since from (A.6) mx is a function of mc, (A.8) implies that mx is a function of kx.Finally, substituting mxðkxÞ into (A.7c), we find that kx is uniquely determined by theparameters of the model.

We complete this part of the proof by noting that the non-negativity constraints onthe investments are not binding in either steady state, because xi ¼ diki is strictlypositive for di 2 ð0; 1Þ. This justifies the above assumption that all first-order conditionshold with equality at the steady state. This also implies that there will be neighborhoodof the steady state in which all first-order conditions hold with equality.

Appendix B. Proof of Proposition 2

B.1. Computation of the determinant and the trace

We represent the steady values of f, g, and their derivatives by dropping theirarguments, so f � f ðxc=xx; 1Þ, g � gðxc=xxÞ, etc. We start the proof by listing somehelpful identities that have to hold in our model. First, the definition of g as theinverse of f c=f x implies that

g0 ¼f 2

x

f ccf x � f cf xc

. (B.1a)

Second, the linear homogeneity of f implies:

f ¼ gf c þ f x; 0 ¼ gf cc þ f cx; 0 ¼ f xx þ gf cx. (B.1b)

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Third, (A.6) and (B.1b) give

rþ dxð1� bÞ

bdx

¼gf c

f x

;rþ dxð1� bÞ

rþ dx

¼gf c

f. (B.2a)

Finally, using this and (B.1a), we find

f xc

f x

g0mc

mx

¼f xcf c

f ccf x � f cf xc

¼ �gf c

f x þ gf c

¼ �rþ dxð1� bÞ

rþ dx

. (B.2b)

The first step of the derivation of the determinant and the trace is to linearize thereduced-form dynamics at the steady state. Indicating steady-state variables bydropping the time subscript, the result is

_kct

_kxt

_mct

_mxt

266664

377775 ¼

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

26664

37775

kct � kc

kxt � kx

mct � mc

mxt � mx

266664

377775, (B.3)

where

a11 ¼ �dc; a12 ¼b1

1� b2

dckc

kx

,

a13 ¼g0

g

mc

mx

�b2

1� b2

f xc

f x

g0mc

mx

�gf c

f

g0

g

mc

mx

� �dckc

mc

,

a14 ¼b2

1� b2�

g0

g

mc

mx

þb2

1� b2

f xc

f x

g0mc

mx

þf c

fg0

mc

mx

� �dckc

mx

; a21 ¼ 0,

a22 ¼b1

1� b2

dxkx

kx

� dx; a23 ¼b2

1� b2

f xc

f x

g0mc

mx

�f c

fg0

mc

mx

� �dxkx

mc

,

a24 ¼b2

1� b2�

b21� b2

f xc

f x

g0mc

mx

þgf c

f

g0

g

mc

mx

� �dxkx

mx

,

a31 ¼ðrþ dcÞmc

kc

; a32 ¼ 0; a33 ¼ rþ dc; a34 ¼ 0,

a41 ¼ 0; a42 ¼b1 þ b2 � 1

1� b2

ðrþ dxÞmx

kx

; a43 ¼1

1� b2

f xc

f x

g0mc

mx

ðrþ dxÞmx

mc

,

a44 ¼ ðrþ dxÞ �1

1� b2ðrþ dxÞ �

1

1� b2ðrþ dxÞ

f xc

f x

g0mc

mx

.

To find these expressions we have repeatedly used the fact that if a function is of theform hðx1;x2;x3Þ ¼ xa

1xb2 � ax3, then its partial derivative can be written as

qh=qx1 ¼ aðf ðx1;x2;x3Þ þ ax3Þ=x1.

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To simplify these expressions, it is useful to define the elasticity of the investmentratio with respect to the relative price evaluated at the steady state. Denoting theinverse of that elasticity by eX0,14 we have

e �gðmc=mx; 1Þ

g0ðmc=mx; 1Þ

1

mc=mx

. (B.4)

Now, using (B.2a) and (B.2b), the previous terms can be rewritten:

a11 ¼ �dc; a12 ¼b1

1� b2

dckc

kx

,

a13 ¼b2

1� b2þ

1

e1� ð1þ eÞb2

1� b2

dxb

rþ dx

� �dckc

mc

, ðB:5aÞ

a14 ¼ �1

e1� ð1þ eÞb2

1� b2

dckc

mx

dxb

rþ dx

; a21 ¼ 0; a22 ¼ dx

b1 þ b2 � 1

1� b2,

(B.5b)

a23 ¼ �1

e1� ð1þ eÞb2

1� b2

dxkx

mc

rþ dxð1� bÞ

rþ dx

,

a24 ¼b2

1� b2þ

1

e1� ð1þ eÞb2

1� b2

rþ dxð1� bÞ

rþ dx

� �dxkx

mx

, ðB:5cÞ

a31 ¼ðrþ dcÞmc

kc

; a32 ¼ 0; a33 ¼ rþ dc; a34 ¼ a41 ¼ 0, (B.5d)

a42 ¼ �b1 þ b2 � 1

1� b2

mx

kx

ðrþ dxÞ; a43 ¼ �1

1� b2½rþ dxð1� bÞ�

mx

mc

,

a44 ¼ ðrþ dxÞ �1

1� b2dxb. ðB:5eÞ

The second step is to combine the terms just derived and actually compute thedeterminant and the trace. Using the fact that a32 ¼ a34 ¼ a41 ¼ 0, the determinantcan be written as

Det ¼ a31a42ða13a24 � a14a23Þ þ a22a31ða14a43 � a13a44Þ

þ a11a33ða22a44 � a24a42Þ þ a12a31ða23a44 � a24a43Þ.

Using the previous expressions, the four terms in that determinant are found toequal:

a31a42ða13a24 � a14a23Þ ¼ �1

eb2

1� b2

b1 þ b2 � 1

1� b2dcdxðrþ dcÞðrþ dxÞ,

14Note that if f is parameterized by e according to (6), then the inverse elasticity in the case is also given

by e.

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a22a31ða14a43 � a13a44Þ ¼b1 þ b2 � 1

1� b2

b21� b2

ð1þ eÞdxb� eðrþ dxÞ

edcdxðrþ dcÞ,

a11a33ða22a44 � a24a42Þ ¼ �b1 þ b2 � 1

1� b2

1þ ee

dxdcðrþ dcÞ½rþ dxð1� bÞ�,

a12a31ða23a44 � a24a43Þ ¼b1

1� b2

b21� b2

1þ ee

dcdxðrþ dcÞ½rþ dxð1� bÞ�.

Using these expressions and simplifying, we find the determinant:

Det ¼1þ ee

dcdxðrþ dcÞ½rþ dxð1� bÞ�ð1� b1Þ1� b2

. (B.6)

In general form the trace is given by

Tr ¼ a11 þ a22 þ a33 þ a44.

Substituting in the previous expressions for aii, we find the trace:

Tr ¼ 2rþ dx

b1 � b

1� b2. (B.7)

B.2. Characterization of the stability properties

We start with the case yx 2 ½0; b=ð1� bÞÞ, implying that b2o1. Then Det40 andTr40.15 Now suppose that the steady state were locally stable. Then (B.3) wouldhave three or four eigenvalues with negative real parts. If (B.3) had four eigenvalueswith negative real parts, then the trace would have to be negative, which is acontraction. If (B.3) had three eigenvalues with negative real part, then thedeterminant would have to be negative, which is a contradiction.

We continue with the case yx 2 ½b=ð1� bÞ; ð1� bÞ=bÞ, implying that b241. ThenDeto0. Suppose that the steady state were saddle-path stable. Then (B.3) wouldhave two eigenvalues with negative real part and two eigenvalues with positive realpart. Irrespective of whether they are real or complex conjugates, this would implythat the determinant must become positive, which is a contraction.

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On the stability of the two-sector neoclassical growth model with …bherrend/Published Papers/JEDC 2006.pdf · 2008. 9. 19. · Journal of Economic Dynamics & Control 30 (2006) 1339–1361