INTRODUCTION
The aim of this paper is to discuss the role of the elasticity of substitution or factor substitutability in the local determinacy properties of a steady state or a stationary balanced growth path in a general multisector economy with CES technologies. We prove two new results. First, local indeterminacy occurs without a capital intensity reversal, provided the elasticity of substitution between factors is “weak” (less than 1). Second, allowing the elasticity of substitution between factors to be different from unity provides sufficient flexibility in these models to recognize that conditions for local indeterminacy in exogenous growth models are qualitatively different from those for endogenous growth models, unlike what the literature might suggest.
Under perfect competition, it is now well known since the contributions of Kehoe and Levine (1985), Muller and Woodford (1988), and Kehoe et al. (1990), that equilibria are generically locally determinate in models with a finite number of infinitely lived agents whereas local indeterminacy, that is, the existence of a continuum of equilibrium paths, is a rather standard feature of models with an infinite number of finitely lived agents, such as OLG models.1
See also Diamond (1965) and Galor (1992).
On the contrary, under imperfect competition, locally indeterminacy also arises in economies “à la Ramsey” with infinitely lived agents.
There is a large literature on indeterminacy in macroeconomics. Increasing returns, imperfect competition, and money are some of the main transmission mechanisms that provide room for the existence of sunspot fluctuations based on shocks on expectations.2
See the recent survey by Benhabib and Farmer (1999).
Following Woodford (
1986) who proves that local indeterminacy is a sufficient condition for the existence of sunspot equilibria, many contributions have focused on coordination problems raised by multiple equilibria
3 See Evans and Guesnerie (2003).
and on macroeconomic instability based on the volatility of expectations.
4 See Benhabib and Rustichini (1994), Benhabib (1998).
Indeterminacy and sunspots indeed provide a possible explanation of business-cycle fluctuations. The most recent contributions, such as those from Benhabib and Nishimura (
1998,
1999), show that exogenous-growth multisector models are compatible with realistic market imperfections and can provide a good match to macroeconomic time series.
5 See also Benhabib and Farmer (1996), Benhabib et al. (2002), Nishimura and Venditti (2002).
Most of the papers in the literature deal with infinitely lived agents in order to match the frequency of fluctuations observed in the data.
6 From this point of view, two-period OLG models are not appropriate to deal with short-run business-cycle fluctuations.
Since Romer (1986) and Lucas (1988), much attention has also been given to stability of equilibria in endogenous growth models.7
The common feature of most of these contributions is the consideration of Cobb-Douglas technologies and sector-specific and/or intersectoral external effects.
8 All these papers are representative-agent two-sector models. However, these models possess features of agents' interactions through external effects. See Becker and Tsyganov (2002) for a model that explicitly introduces a heterogeneity of agents.
Focusing on production functions with unitary elasticity of substitution prevented any discussion of the role of factor substitutability.
Although Cobb-Douglas technologies are widely used in growth theory, recent papers have questioned the empirical relevance of this specification. Duffy and Papageorgiou (2000), for instance, consider a panel of 82 countries over a 28-year period to estimate a CES production function specification. They find that for the entire sample of countries the assumption of unitary elasticity of substitution may be rejected. Moreover, dividing the sample of countries into several subsamples, they find that capital and labor have an elasticity of substitution greater than unity in the richest group of countries whereas the elasticity is less than unity in the poorest group of countries. It therefore seems necessary to question the robustness of indeterminacy with respect to the elasticity of substitution.9
On the basis of interest-rate elasticity of U.S. demand for money, Benhabib and Farmer (2000) use a CES production function, with labor and real money balance as inputs, to investigate the monetary transmission mechanism.
Boldrin and Rustichini (1994) and Drugeon et al. (2003) consider two-sector growth models with general neoclassical technologies and intersectoral externalities. Bond et al. (1996) also examine the effects of distortionary factor taxation, which is formally equivalent to sector-specific external effects. These authors provide conditions for local indeterminacy of equilibrium paths. However, no clear condition on the size of the elasticity of substitution is given.
We consider in this paper an (n+1)-sector continuous-time economy with CES technologies and sector-specific externalities. Our goal is to characterize the local stability of a steady state or a stationary balanced growth path and to evaluate the influence of the input elasticity of substitution. The model with exogenous growth consists of a pure consumption good and n capital goods that are produced using capital and labor. We assume that the instantaneous utility function is linear. The model with endogenous growth consists of one consumable capital good and n pure capital goods. We assume that each good is produced without fixed factors. As in the standard literature, we assume that the instantaneous utility function is homogeneous with a constant intertemporal elasticity of substitution.
Building on some recent empirical studies of disaggregated U.S. data by Basu and Fernald (1997), we assume that the aggregate technology of each sector has constant social returns, which implies that individual firms exhibit small decreasing returns. This divergence between private and social returns is explained by the existence of mild external effects. It follows that the standard duality between Rybczynski and Stolper-Samuelson effects, which holds in optimal growth models, is broken in our framework.
Based on the destruction of this duality, it has been shown by Benhabib and Nishimura (1998) in a two-sector exogenous growth model with sector-specific externalities that the steady state is locally indeterminate if the consumption good is capital intensive at the private level but labor intensive at the social level.10
Similar results have been derived in endogenous growth models by Bond et al. (1996), Mino (2001) and Benhabib et al. (2000).
Local indeterminacy then requires a capital intensity reversal.
However, all these results have been established using Cobb-Douglas technologies. We use CES production functions and study the role of the elasticity of substitution in the stability properties of the steady state or the stationary balanced growth path in general multisector models. We show that local indeterminacy takes place even without a capital intensity reversal if factor substitutability is weak, that is, the elasticity of substitution is less than 1. Moreover, we prove that factor substitutability must be sufficiently weak in exogenous growth models for local indeterminacy to take place whereas this is not necessarily true in endogenous growth models.
This paper is organized as follows. The next section sets up the basic model of production. Section 3 examines the conditions for indeterminacy in the exogenous growth model while Section 4 deals with the endogenous growth model. Section 5 contains some concluding comments. Most of the proofs are gathered in the Appendix.
BASIC MODEL OF PRODUCTION
Output yj is produced by inputs x0j, …, xnj. There are n+1 outputs y0, y1, …, yn. The production functions are CES, and the representative firm in each industry faces the following function:
with ρj>−1 and σj=1/(1+ρj)[ges ]0 the elasticity of substitution. The positive externalities, ej(Xj), will be equal to
, with bij[ges ]0 and Xj=(X0j, X1j, …, Xnj) where Xij denotes the average use of input i in sector j. We assume that these economywide averages are taken as given by the individual firm. At the equilibrium, since all firms of sector j are identical, we have Xij=xij and we may define the social production functions as follows:
We assume that in each sector [sum ]ni=0(βij+bij)=1 so that the production functions collapse to Cobb-Douglas in the particular case ρj=0. Notice also that the returns to scale are constant at the social level, and decreasing at the private level.
11 Denoting by fj(x0j, …, xnj, ej(Xj)) the technology of sector j, we have for any λ>1
Our formulation is, however, compatible with constant returns at the private level if we assume that there exists a factor in fixed supply such as land.
12 The technology of sector j may be formulated as follows:
with Tj the amount of land used in the production of good j, which is normalized to 1 in our formulation.
In this case, the income of the representative consumer will be increased by the rental of land.
A firm in each industry maximizes its profit, given output price pj and input prices w0, …, wn. Its profit is
Assumption 1. For any i, j=0, …, n, βij>0.
It is well known that with CES technologies, depending on the value of the elasticity of substitution, the Inada conditions may not be satisfied. It follows that Assumption 1 is necessary but not sufficient to guarantee that a positive amount of every good is produced. Therefore, in the rest of the paper, we restrict our analysis to the case of interior solutions for which every good is produced by a positive amount and every input is used by a positive amount in the production of every good.
To focus the analysis on the elasticity of substitution, we assume that it is identical accross sectors, that is, ρj=ρ, j=0, …, n. The first-order conditions subject to the private technologies (1) are the following:
From (3), we have
We call aij the input coefficients from the private viewpoint. If the agents take account of externalities as endogenous variables in profit maximization, the first-order conditions subject to the social technologies (2) are
and the input coefficients become
with
. We call
the input coefficients from the social viewpoint. However, as we show below, the factor-price frontier, which gives a relationship between input prices and output prices, is not exactly expressed with the input coefficients from the social viewpoint. We define
which we will call the quasi-input coefficients from the social viewpoint, and it is easy to derive that
Notice that
if bij=0, i.e., there is no externality coming from input i in sector j, or ρ=0, that is, the production function is Cobb-Douglas.
On the basis of these input coefficients, we now establish various lemmas. We first show that the factor-price frontier is determined by the quasi-input coefficients from the social viewpoint.
LEMMA 1. Denote p=(p0, …, pn)′, w=(w0, …, wn)′ and
. Then,
.
Proof. Substituting (4) into the real production function (2) gives
It follows that
Multiplying both sides of this equality by p1/(1+ρ)j gives
The total stock of factors is a vector x=(x0, …, xi)′ with xi=[sum ]nj=0xij. From the full employment conditions, we derive the factor market-clearing equation, which depends on the input coefficients from the private perspective.
LEMMA 2. Denote x=(x0, …, xn)′, y=(y0, …, yn)′, and A(w, p)=[aij(wi, pj)]. Then A(w, p)y=x.
Proof. By definition, xij=aijyj, and thus,
We now examine some comparative statics. Since the function
is homogeneous of degree zero in w and p, the envelope theorem implies that the factor-price frontier satisfies Lemma 3.
LEMMA 3.
.
Proof. Differentiating equation (5) gives
and therefore,
The factor market-clearing equation finally satisfies
LEMMA 4.
Proof. Starting from [sum ]nj=0aijyj=xi, we have
The total derivative of the system can be summarized by the following equation:
Under Assumption 1, we now define (n+1)×(n+1) positive matrices
Assumption 2. B and
are nonsingular matrices.
Let us also define the two following diagonal matrices:
From (4), we get A=W−1BP,
, and thus under Assumption 2, A−1=P−1B−1W,
. Note also from Lemmas1–2 and the above diagonal matrices that
and
Factor rentals are functions of output prices only,
, whereas outputs are functions of factor stocks and output prices, yi=yi(x, p), i=0, …, n. Finally, we obtain
Without external effects, that is, bij=0, the matrix [∂y/∂x] reflects the Rybczynski theorem whereas thematrix [∂w/∂p] reflects the Stolper–Samuelson theorem. From the duality betweenthese two effects well known in trade theory, we get [∂y/∂x]=[∂w/∂p]′. However, in the presence of externalities, the Rybczynski effects depend on theinput coefficients from the private perspective whereas the Stolper–Samuelson effects depend onthe quasi-input coefficients from the social perspective. The duality between these two effects isthus destroyed. This follows from the fact that with market distortions, true costs are not beingminimized. Local indeterminacy of equilibria will be a consequence of this property.
EXOGENOUS GROWTH MODEL WITH LINEAR UTILITY
Model
A representative agent optimizes a linear additively separable utility function with discount rate δ [ges ] 0. This problem can be described as
where x0(t), interpreted as labor, is always equal to 1; y0(t) is the output of the pure consumption good; and g [ges ] 0 is the depreciation rate of the capital stocks.
13 We assume that the instantaneous utility is linear. This implies that the objective functional coincides with the social production function of the consumption-good sector which is nonlinear and concave.
We can write the modified Hamiltonian incurrent value as
Here, pj(t) and wi(t) are, respectively, co-state variable and Lagrange multipliers, representing utility prices of the capital goods, their rental rates, and the wage rate, with p0(t)=1. The static first-order conditions are given by
for j=1, …, n, s=0, 1, …, n, and they are equivalent to (3). Let x1=(x1, …, xn)′,
, p1=(p1, …, pn)′ and w1=(w1, …, wn)′.It follows from (6) and (7) that thenecessary conditions that describe the solution to problem (9) are given by theequations of motion:
Assumption 3. There exists a stationary point (x1*, p1*) of the dynamical system (10) that solves
, i=1, …, n.14
From a general point of view, existence of steady state is proved under conditions (continuity of functions, some boundary properties of the functions on the domain) that are fairly independent from the local stability properties at the steady state. In our model, following the procedure introduced by Benhabib and Nishimura (1979) and Benhabib and Rustichini (1990) for optimal growth models and Benhabib and Nishimura (1998) for growth models with externalities, the existence of a steady state may be proved by constructing CES production functions from some values of the prices pj, the rental rates wj and the coefficients βij, bij that satisfy the local indeterminacy conditions. However, to avoid unnecessary complications, we simply assume the existence of the steady state to begin with.
Linearizing around (x1*, p1*) gives the 2n×2n Jacobian matrix
Since in this model we have one pure consumption good, we need toeliminate from equality (8) the columns and rows that are associated with x0, y0, p0, and w0. To do so, we introduce the following n×n matrices:
for i, j=1, …, n. The Jacobian matrix is thus as follows:
In the current economy, there are n capital goods whose initial values are given. Any solution from (10) that converges to the steady state (x1*, p1*) satisfies thetransversality condition and constitutes an equilibrium. Therefore, given x(0), if thereis more than one set of initial prices p(0) in the stable manifold of, the equilibrium path from x(0) will not be unique. In particular, if J has more than n roots with negative realparts, there will be a continuum of converging paths and thus a continuum of equilibria.
DEFINITION 1. If the locally stable manifold of the steady state (x1*, p1*) has dimension greater than n, then (x1*, p1*) is said to be locally indeterminate.
The roots of J are determined by the roots of [A−11−gI] and
. A1 is the matrix of factor intensity differences from the privateviewpoint and
is the matrix of quasi factor intensity differences from the socialviewpoint. Using the definitions of input coefficients given in Section 2,we may indeed interpret the elements of A1 and
as follows:
DEFINITION 2. The consumption good is said to be
- more intensive in capital good i than the capital good j at the private level if aija00−ai0a0j<0
- more quasi intensive in capital good i than the capital good j at the social level if
- more intensive in capital good i than the capital good j at the social level if .
As in a Cobb-Douglas framework, it is usual to formulate the factor intensity differences in terms of the βij and
coefficients. A similar convenient formulation canbe achieved with CES technologies. To do so at the private level, we need to define an n×n matrixB1 as follows:
for i, j=1, …, n. Considering also
we easily obtain from (4) A1=W−11B1P1. Similarly, using the quasi inputcoefficients at the social level and defining an n×n matrix
as
for i, j=1, …, n, it follows that
. Under Assumption 2, the matricesB1 and
are invertible. By the steady state conditions, δ+g=wi/pi, the Jacobian matrix J becomes
We may thus relate the input coefficients to the CES parameters.
PROPOSITION 1. Let Assumptions 1–3 hold. At the steady state
- the consumption good is more intensive in capital good i than the capital good j from the private perspective if and only if βijβ00−βi0β0j<0
- the consumption good is more quasi labor intensive than the capital good j from the social perspective if and only if
- the consumption good is more intensive in capital good i (labor) than the capital good j from the social perspective if and only if .
Two-Sector Model
In the two-sector model with n=1, the matrices A1 and
arescalars. From Definition 2, if a11a00−a10a01<0, the consumption good is capital intensive from theprivate viewpoint. Moreover, if
, the consumption good isquasi capital intensive from the social viewpoint. The following proposition establishesthat local indeterminacy requires a capital intensity reversal from the private input coefficientsto the quasi input coefficients.
PROPOSITION 2. Let n=1 and Assumptions 1–3 hold. The steady state is locally indeterminate if and only if the consumption good is capital intensive from the private perspective, but quasi labor intensive from the social perspective.
To get indeterminacy in a framework with constant returns to scale at the social level, we need amechanism that nullifies the duality between the Rybczynski and Stolper–Samuelson effects. Asshown in Section 2, the Rybczynski effect is given by the input coefficients from theprivate perspective whereas the Stolper–Samuelson effect is given by the quasi input coefficientsfrom the social perspective. In the presence of external effects, the duality between these coefficients is broken and local indeterminacy may appear. Starting from an arbitrary equilibrium, consider an increase in the rate of investment induced by an instantaneous increasein the relative price of the investment good. When the investment good is labor intensive at theprivate level, an increase in the capital stock decreases its output at constant prices through the Rybczynski effect. If, on the contrary, the investment good is quasi capital intensive at thesocial level, the initial rise in its price causes, through the Stolper–Samuelson effect, an increase in one of the components of its return and requires a price decline to maintain theover all return to capital equal to the discount rate. This offsets the initial rise in the relative price of the investment good so that the transversality condition still holds.
This mechanism is very similar to the one exhibited in the contribution of Benhabib and Nishimura (1998). There is, however, a major difference with the current paper, which is based on the fact that as soon as the factor elasticity of substitution is non unitary, the quasi input coefficient sat the social level depend on the elasticity of substitution whereas the social input coefficients do not. It follows that, depending on the valueof the elasticity of substitution, the capital intensity reversal from the private input coefficients to the quasi input coefficients does not necessarily requires a capital intensity reversal from the private to the social level.
Let us now precisely study the role of the factor elasticity of substitution. Consider (ii) of Proposition 1. When ρ[ges ]0, if
and β11β00−β10β01[les ]0, then the consumption-good sector is always quasi labor intensive from the social perspective. When −1<ρ<0, even if β11β00−β10β01<0, the consumption-good sector may be quasi labor intensive from the social perspective when
and dominates β11β00−β10β01. We may therefore derive the following result, which shows that when the true social input coefficients are considered, local indeterminacy also takes place without a capital intensity reversal.
PROPOSITION 3. Let n=1 Assumptions 1–3 hold and the consumption good be capital intensive from the private perspective. Then,
- if the consumption good is labor intensive from the social perspective, there exists ρ*1 ∈ (−1, 0) such that the steady state is locally indeterminate for any ρ>ρ*1 and saddle-point stable for any ρ∈(−1, ρ*1);
- if the consumption good is also capital intensive from the social perspective and
there exists ρ*2>0 such that the steady state is locally indeterminate for any ρ>ρ*2 and saddle-point stable for any ρ∈(−1, ρ*2).
This proposition shows that when the productive factors are sufficiently substitutable, local indeterminacy occurs if the consumption good is capital intensive at the private level. Note that local indeterminacy is still possible even if the consumption good is capital intensive at the social level. Condition (i) only coincides with the result obtained by Benhabib and Nishimura (1988) in the particular case of Cobb-Douglas technologies. Therefore, when CES production functions are considered, a capital intensity reversal is not always necessary for local indeterminacy.
COROLLARY 1. Let n=1, Assumptions 1–3 hold, the consumption good be capital intensive from the private perspective, and
hold. Then, there exists ρ*>−1 such that for any ρ>ρ*, the steady state is locally indeterminate.
General Multisector Model
In a general multisector model with n capital goods, the local conditions for stability or instability require strong properties on matrices of factor intensity differences B1 or
. In this section, we attempt to provide a generalization of the results obtained for the two-sector model and propose new conditions for local indeterminacy. As in the two-sector case, we have to consider the coefficients βij and
of the CES technologies. We impose restrictions only on the sign of the diagonal terms of B1 and
but not on the sign of the off-diagonal coefficients. We introduce the following sets of indices which characterize the sign of the diagonal terms in B1 and
when ρ=0:
When
, the consumption good is more intensive in capital good j than the capital good j itself at the private level. Similarly, when
, the consumption good is more intensive in capital good k than the capital good k itself at the social level. Let us denote the number of elements in
and
, respectively, by
and
. If
, then βiiβ00−βi0β0i<0 for any i=1, …, n, and if
then
for any i=1, …, n. These restrictions are similar to the Metzler and Minkowsky conditions of Benhabib and Nishimura (1999) which imply the existence of 2n eigenvalues with negative real parts.
15 A Metzler matrix has negative diagonal elements and positive off-diagonal elements, whereas a Minkowski matrix has positive diagonal elements and negative off-diagonal elements.
Our main objective is to give conditions for the existence of at least
n+1 eigenvalues with negative real parts from the sign patterns of the diagonal elements of matrices
B1 and
.
To relate the sign of diagonal elements to the sign of the real parts of eigenvalues, we introduce the following dominant diagonal properties of matrices.
DEFINITION 3. An n×n matrix C=[cij] has a dominant diagonal if |cii|>[sum ]j≠i|cij| for each i=1, …, n or |cii|>[sum ]i≠j|cij| for each j=1, …, n.
This definition is stronger than the quasi-dominant diagonal introduced by McKenzie (1960) since the weighting parameters are here equal to one. We also introduce a strong dominant diagonal that requires both row dominance and column dominance.
DEFINITION 4. An n×n matrix C=[cij] has a strong dominant diagonal if |cii|>[sum ]j≠i|cij| for each i=1, …, n and |cii|>[sum ]i≠j|cij| for each j=1, …, n.
From now on we explicitly parameterize the matrices B1 and
by ρ, namely B1(ρ) and
, in order to simplify the exposition.
Assumption 4. There exists
such that for any
, B1(ρ) has a strong dominant diagonal, and
has only real eigenvalues with dominant diagonal.
Remark 1. In a two-sector model, when we consider constructing an alternative equilibrium with a higher investment rate, we have to decrease the initial level of consumption. If the elasticity of intertemporal substitution in consumption is finite, the desire to smooth consumption over time may overwhelm the technological effects coming from the Rybczynski and Stolper–Samuelson theorems. This is why we assume a linear specification for the utility function. As shown by Benhabib and Nishimura (1998), such an assumption is no longer necessary as soon as a third nonconsumption good is introduced. In this case, indeterminacy may arise from compositional changes in outputs without too much affecting the output of consumption. However, we still assume in the following that the elasticity of intertemporal substitution in consumption is infinite. Such an assumption is necessary to get precise results without resorting to numerical computations. It is easy to notice indeed that if the utility function is nonlinear, the Jacobian matrix J is no longer triangular and the characteristic roots cannot be analyzed. Our strategy is therefore to give conditions on the technological fundamentals to get local indeterminacy when the utility function is linear. As numerically illustrated by Benhabib and Nishimura (1998) in a Cobb-Douglas framework, the argument mentioned above then guarantees that local indeterminacy will persist for finite values of the elasticity of intertemporal substitution in consumption.
THEOREM 1. Let Assumptions 1–4 hold,
, and
for all i=1, …, n. Then, there exists
such that the steady state is locally indeterminate for any
.
Theorem 1 suggests that local indeterminacy cannot arise with high substitutability. We may thus provide conditions for saddle-point stability. We first introduce an alternative restriction to Assumption 4.
Assumption 5. There exists
such that for any
, B1(ρ) has a strong dominant diagonal, and
has only real eigenvalues with dominant diagonal.
PROPOSITION 4. Let Assumptions 1–3 and 5 hold with
. Then, there exists
such that the steady state is saddle-point stable for any
.
Condition
with dominant diagonal guarantee that the Jacobian matrix J has at least n negative eigenvalues. However, strong factor substitutability leads to the existence of a unique equilibrium path.
Examples
In Theorem 1, the lower bound ρ*, above which local indeterminacy is obtained, may be positive or negative depending on the values of
and
. If ρ*<0, then the results cover the Cobb-Douglas case. Under
and suitable dominant diagonal assumptions, it may be shown that ρ*<0 if
whereas ρ*>0 if
. Simple examples may illustrate the possibility of indeterminacy in three different interesting configurations. Following Benhabib and Nishimura (1998), the production parameters are calibrated along the lines of a standard RBC model.16
We use indeed very similar values for the input coefficients. See also Benhabib et al. (1997).
We also consider extremely small external effects.
Example 1. We first illustrate the case
and
. Consider the following matrices of private CES coefficients and external-effects coefficients
It follows that
and, when ρ=0,
We have
for row 1 of matrix B1(ρ), which has a strong dominant diagonal for any ρ>−1. Moreover
and the matrix
has a dominant diagonal for any ρ∈(−1,−0.55)∪(−0.29,+∞). The eigenvalues are positive when ρ>−0.409 and have an opposite sign when ρ∈(−1,−0.409). Then, there exists ρ*=−0.409 such that the steady state is locally indeterminate for any ρ>ρ* and saddle-point stable when ρ∈(−1, ρ*). The values of
and
, respectively, give the number of negative diagonal terms of B1(0) and
. Under dominant diagonal assumptions, these give the number of negative roots of B1(0) and
. Therefore, if
and
, the Jacobian matrix when ρ=0 has n+1 roots with negative real parts.
Example 2. We now illustrate the case
and
. Consider a slight modification of the previous matrices B and b:
We easily obtain
and, when ρ=0,
We have
for row 1 and B1(ρ) has a strong dominant diagonal for any ρ>−1. Condition (15) is satisfied,
and
has a dominant diagonal for any ρ∈(−1, 0.150)∪(0.231,+∞). The eigenvalues are positive when ρ>0.196 and have opposite sign when ρ∈(−1, 0.196). Moreover,
and there exists ρ*=0.196 such that the steady state is locally indeterminate for any ρ>ρ* and saddle-point stable when ρ∈(−1, ρ*). Note that since we use a dominant diagonal property for B1(ρ) and
, local indeterminacy requires at least n+1 negative diagonal terms in B1(ρ) and
. When ρ=0 and under
, this requires that
. Therefore, local indeterminacy cannot occur in the Cobb-Douglas case when
.
Example 3. We finally illustrate the case
and
. Consider a slight modification of matrices (16) as follows:
We easily obtain
and, when ρ=0,
We have
and B1(ρ) has a strong dominant diagonal for any ρ>−1. Condition (15) is satisfied and
.
has a dominant diagonal for any ρ∈(−1,−0.856)∪(−0.596, 0.150)∪(0.231,+∞) and its eigenvalues are positive when ρ>0.195, have the opposite sign when ρ∈(−0.714, 0.195), and are negative when ρ∈(−1,−0.714). Moreover, we have
and there exist ρ*=0.195 and
such that the steady state is locally indeterminate for any
and saddle-point stable when
. The Cobb-Douglas case falls into this configuration. Although the sufficient conditions of Theorem 1 imply that the lower bound ρ* is positive, local indeterminacy of the steady state may still hold for some ρ that is strictly less than ρ*. When
,
, and
, there are at least n+1 negative diagonal terms in B1(0) and
. If, in addition, dominant diagonal properties are satisfied at ρ=0, the steady state is locally indeterminate at ρ=0 even though the lower bound given by Theorem 1 is strictly positive.
ENDOGENOUS GROWTH MODEL WITH NONLINEAR UTILITY
Model
We now consider an economy without fixed factors that exhibits unbounded growth. A representative agent optimizes a nonlinear additively separable utility function with discount rate δ[ges ]0. This problem can be described as
There is no pure consumption good: the good 0 is both a factor of production and a consumption good. The modified Hamiltonian in current value is
Here pj(t) and wi(t) are, respectively, co-state variables and Lagrange multipliers, representing utility prices of the capital goods and their rental rates. The static first-order conditions for this problem are given by
for j=1, …, n, s=0, 1, …, n, and equations (19) are equivalent to (3). It follows from (6) and (7) that the necessary conditions that describe the solution to problem (17) are given by the equations of motion:
The production functions being homogeneous of degree one, let the growth rate of c and xi along the balanced growth path be μ. From equation (18), prices must then decline at the rate σμ. We define discounted variables as
Note that
. Since there are no fixed factors, outputs y are homogeneous of degree one in stocks x, and homogeneous of degree zero in prices p, and the factor prices w are homogeneous of degree one in prices. Then equations (20) can be written as:
The stationary balanced growth rate μ corresponds to the stationary point
of the above system.
Assumption 6. There exists a stationary point
of the dynamical system (21) that solves
, i=0, 1, …, n.17
When endogenous growth is considered, the same comment as in note 14 applies.
From the price equations in the dynamical system (21) evaluated at
and Lemma 1, we have
, which can be reformulated as
Thus, μ is obtained from the Frobenius root of
, that is, (δ+g+σμ)−1, which has w as eigenvector. Linearizing around
gives the 2(n+1)×2(n+1) Jacobian matrix,
where Z is a matrix of zeros except for the element of the first row and the first column, which is (1/σ)p−1−1/σ0. From equation (8), J becomes
In the current economy, there are n+1 capital goods whose initial values are given. Any solution from (21) that converges to the steady state
satisfies the transversality condition and constitutes an equilibrium. Therefore, given
, if there is more than one set of initial prices
in the stable manifold of
, the equilibrium path from
will not be unique. Notice that the Frobenius root of
implies the existence of one zero root for J. Therefore, if J has more than n roots with negative real parts, there will be a continuum of converging paths and thus a continuum of equilibria. Definition 1 of local indeterminacy still applies.
The roots of J are given by the roots of [A−1−(g+μ)I] and
. As in the exogenous growth formulation, we may formulate the factor intensity differences in terms of the βij and
coefficients. From Lemmas 3 and 4 we have
and
. Consider the following two diagonal matrices:
The steady-state conditions give
. Then, we can rewrite J as follows:
Applying the same proof as the one of Proposition 1, we finally obtain Proposition 5.
PROPOSITION 5. Let Assumptions 1, 2, and 6 hold. At the steady state,
- the consumable capital good is more intensive in the pure capital good i than the pure capital good j at the private level iff βijβ00−βi0β0j<0;
- the consumable capital good is more quasi intensive in itself (the pure capital good i) than the pure capital j at the social level iff
- the consumable capital good is more intensive in itself (the pure capital good i) than the pure capital j at the social level iff .
Two-Sector Model
Considering a two-sector model, we first show that, as in the exogenous growth framework, local indeterminacy requires a factor intensity reversal from the private input coefficients to the social quasi input coefficients.
PROPOSITION 6. Let n=1 and Assumptions 1, 2, and 6 hold. If the consumable capital good is intensive in the pure capital good from the private perspective, but it is quasi intensive in itself from the social perspective, then the balanced growth path is locally indeterminate.
This result is based on the quasi input coefficients from the social viewpoint which do not have real economic meaning. Therefore, we need to use the true social input coefficients. Now, however, we have to consider explicitely the factor elasticity of substitution. Applying the same proof as in the exogenous growth case, we show that local indeterminacy occurs without such a reversal when the elasticity of substitution is less than one.
PROPOSITION 7. Let n=1, Assumptions 1, 2, and 6 hold, and the consumable capital good be intensive in the pure capital good from the private perspective. Then,
- if the consumable capital good is intensive in itself from the social perspective, there exists ρ*∈(−1, 0) such that the balanced growth path is locally indeterminate for any ρ>ρ*;
- if the consumable capital good is also intensive in the pure capital good from the social perspective and
there exists ρ*>0 such that the steady state is locally indeterminate for any ρ>ρ*.
When the technologies are close enough to a Leontief function, this Corollary provides a new condition for local indeterminacy based on a consumable capital good intensive in the pure capital good at the private and social levels. Moreover, it extends the conclusions of Bond et al. (1996), Benhabib et al. (2000), and Mino (2001) to any economies with technologies between Cobb-Douglas and Leontief functions.
General Multisector Model
Consider now a general multisector model with n+1 goods. As we can see in the Jacobian matrix (23), n roots are determined by the matrix [A−1−(g+μ)I] without externalities and the other n roots by the matrix
with externalities. If A−1 has a root with negative real part, it makes one root with negative real part in the Jacobian matrix. If
has a root with positive real part, it makes one root with negative real part in the Jacobian matrix unless the root corresponds to the Frobenius root of
. Therefore, to provide sufficient conditions in the multisector case, we start with an assumption that implies that B and thus A have at least one root with negative real part. From now on, we explicitely parameterize the matrices B and
by ρ, namely B(ρ) and
, in order to simplify the exposition.
Assumption 7. For any ρ>−1, B(ρ) has a negative determinant.
From this Assumption, we derive Lemma 5.18
The result of Lemma 5 refers to some diagonal and off-diagonal elements in each column. However, similar results hold true for diagonal and off-diagonal elements in each row, too. In the case in which the dominant diagonal property does not hold, similar inequalities follow for each column and each row. To avoid unnecessary complications, we only use inequalities between diagonal elements and off-diagonal elements for each column throughout the rest of this paper.
LEMMA 5. Under Assumptions 1, 2, 6, and 7, there exist at least one row i∈{0, …, n} such that β1/(1+ρ)ii[les ][sum ]j≠iβ1/(1+ρ)ij.
Now, define the set of rows i that satisfies the inequality given in Lemma 5:
Next, we look at the matrix with externalities and we introduce a dominant diagonal assumption given by Definition 4 in the preceding section for
in order to obtain n+1 roots with positive real parts.
Assumption 8.
has a dominant diagonal.
Moreover, we introduce slight restrictions on some components of matrix B.
Assumption 9. Any row k=0, …, n of B satisfies βkk≠βkj, j≠k.
This implies that the amount of capital k used in its own industry is different from the amount of capital k used in any other industries.
LEMMA 6. Let Assumptions 1–2 and 6–9 hold. Then, there exist
and
such that for any
,
has a dominant diagonal. Moreover.19
The critical value
may be +∞ when n=1 under condition (i) in Proposition 7. However, when n[ges ]2,
if the externalities are mild.
- if there is some and at least one j≠i such that βii<βij, then ;
- let n[ges ]2. If for all and all j≠i, βii>βij, then .
We now introduce Assumption 10, based on the open interval
given in Lemma 6.
Assumption 10. The matrix
has only real eigenvalues for any
.
This finally allows us to state the result in Theorem 2.
THEOREM 2. Let Assumptions 1–2 and –10 hold. Then, the following cases occur:
- If there is some and at least one j≠i such that βii<βij, then there exist and such that for any ρ∈(ρ*1, ρ*2), the stationary balanced growth path is locally indeterminate.
- Let n[ges ]2. If for all and all j≠i, βii>βij, then there exists such that the stationary balanced growth path is locally indeterminate for any ρ∈(−1, ρ*).
This theorem provides a generalization of Proposition 7 (i). In a two-sector model with n=1, if the consumable capital good is intensive in the pure capital good at the private level then B has a negative determinant and Lemma 5 implies that case (i) of Lemma 6 necessarily holds. Moreover, if
has a dominant diagonal, then
and
, which implies that the consumable capital good is intensive in itself from the social perspective.
When compared with the exogenous growth case, Theorem 2 shows that the role of the factor elasticity of substitution in the occurrence of local indeterminacy is quite different in the endogenous growth model. In fact,
is defined by input coefficients and not by capital intensity coefficients in the current formulation. This implies that local indeterminacy does not occur with low factor substitutability when external effects are mild. Moreover, Theorem 2 shows with case (ii) that local indeterminacy occurs with an arbitrarily large elasticity of substitution. This is not the case in exogenous growth models.
In a recent contribution that corresponds to case ρ=0 in our formulation, Benhabib et al. (2000) show that if B has n roots with negative real parts and
has at least two roots with positive real parts, the Jacobian matrix has one zero root and at least n+1 roots with negative real parts, and then the stationary balanced growth path is locally indeterminate. Notice that their conditions are fundamentally based on the private input coefficients while our results rely on the social input coefficients. Indeed, it clearly appears that B being a nonnegative (n+1)×(n+1) matrix, the ocurrence of n roots with negative real parts is difficult to obtain. On the contrary,
being also a nonnegative matrix, a dominant diagonal property ensures the existence of n+1 roots with positive real parts, and local indeterminacy is obtained under a mild additional condition on B.
APPENDIX
PROOF OF PROPOSITION 1
At the steady state,
, and thus
PROOF OF PROPOSITION 2
In the two-sector model, A1 and
are scalars. From Lemma 2, x1=a10y0+a11y1. Moreover, at the steady state, y1=gx1, and it follows that
Therefore,
if and only if a11a00−a10a01<0. From Lemma 1,
. Moreover, at the steady state, (δ+g)p1=w1, and it follows that
Therefore,
if and only if
.
PROOF OF PROPOSITION 3
Since the consumption good is capital intensive at the private level, we have β11β00/β10β01 < 1. When ρ[ges ]0, this implies that (β11β00/β10β01)ρ/(1+ρ)<1 with
. On the contrary, when ρ∈(−1, 0), this implies that (β11β00/β10β01)ρ/(1+ρ)1 with
. Note that (β11β00/β10β01)ρ/(1+ρ) is decreasing in ρ.
- If the consumption good is labor intensive at the social level, then . Therefore, there exists ρ*1∈(−1, 0) such that for all ρ>ρ*1 and for all ρ∈(−1, ρ*1). Local indeterminacy when ρ>ρ*1 and thus saddle-point stability when ρ∈(−1, ρ*1) follow from Propositions 1 and 2.
- If the consumption good is labor intensive at the social level, then . Since we have
there exists ρ*2>0 such that for any ρ>ρ2* and for any ρ∈(−1, ρ*2). Local indeterminacy when ρ>ρ*2 and thus saddle-point stability when ρ∈(−1, ρ*2) follow from Propositions 1 and 2.
PROOF OF THEOREM 1
We need to establish various important lemmas. To prove the first one, we use the following Wielandt (1973) theorem:
THEOREM A.1. Let B=CD, where D is symmetric and C+C′ is positive definite. Let b+, b0, and b− denote the number of positive, vanishing, and negative real parts of eigenvalues of B and let d+, d0, and d− denote the number of positive, vanishing, and negative eigenvalues of D. Then b+=d+, b0=d0, and b−=d−.
Consider the following set of indices
We can now prove Lemma A.1.
LEMMA A.1. Let B=[bij] be an n×n matrix with strong dominant diagonal. Assume that
. Then, B has p eigenvalues with negative real parts and n−p eigenvalues with positive real parts.
Proof. We can write B as the following product: B=CD with C a strong dominant diagonal matrix with positive diagonal and D a diagonal matrix with p diagonal elements equal to −1 and n−p diagonal elements equal to 1. Since C has a strong dominant diagonal, C+C′ is symmetric with positive strong dominant diagonal. Therefore, all the eigenvalues of C+C′ are positive and C+C′ is positive definite. Moreover, D satisfies d−=p, d0=0, and d+=n−p. The result follows from the Wielandt theorem.
LEMMA A.2. Let Assumptions 1–3 hold. If the n×n matrix
has n real positive eigenvalues, then
has n negative eigenvalues.
Proof. From Lemma 1, we have
, that is,
Since at the steady state, (δ+g)p1=w1, we have
It follows that
has a positive quasi-dominant diagonal and thus n roots with positive real parts [see McKenzie (1960) and Takayama (1997)]. Now, denote
an eigenvalue of
and assume that each
is real and positive. It follows that
, which is equivalent to
. Therefore,
has n negative real roots.
As proved by Benhabib et al. (2000), at the steady state, the sign pattern of roots of A1 is the same as that of B1, and the sign pattern of roots of
is the same as that of
. Consider then the set of indices
defined by equation (13).
LEMMA A.3. Let Assumptions 1–3 hold. If B1 has a strong dominant diagonal with
, then [A−11−gI] has at least p roots with negative real parts.
Proof. B1 has a strong dominant diagonal with p strictly negative diagonal coefficients and n−p strictly positive diagonal coefficients. It follows from Lemma A.2 that B1 has p[ges ]1 eigenvalues with negative real parts and n−p<n eigenvalues with positive real parts. Therefore A−11=P−11B−11W1 has p eigenvalues with negative real parts and n−p eigenvalues with positive real parts, and thus [A−11−gI] has at least p roots with negative real parts.
We characterize now the roots of the matrix
.
LEMMA A.4. Let Assumptions 1–4 hold and
for all i=1, …, n. Then, there exists
such that for any
,
has n negative roots.
Proof. The following limits are obtained when ρ goes to −1 or +∞:
Under Assumptions 1–4, if the inequalities (A.2) are satisfied, every diagonal element of
is positive when ρ is sufficiently large. If one diagonal coefficient becomes zero at some value
, the dominant diagonal property of
is lost at
. Therefore,
has a positive dominant diagonal for any
, and under Assumption 4, there exists
such that
has n positive real roots for any
. It follows that
also has n positive real roots for any
. Lemma A.3 then implies that
has n negative real roots.
Consider the set of indices
defined by (14). Theorem 1 is a direct consequence of Lemmas A.2 to A.5.
PROOF OF PROPOSITION 4
Lemma A.4 with
implies that the matrix [A−11−gI] has n roots with negative real parts. Moreover, we have for each i=1, …, n
It follows that if
has a strong dominant diagonal for
with
, then its diagonal is necessarily negative. Assuming now that
has real roots for any
, it follows that the roots of
are real and negative. Then, the same \hbox{argument} as in the proof of Lemma A.3 implies that
has n positive eigen-values.
PROOF OF PROPOSITION 6
In the two-sector endogenous growth model, A and
are 2×2 matrices. The determinant of A−1 is the inverse of the determinant of A. If the consumable capital good is intensive in the pure capital good from the private perspective, the determinant of A is negative, so that one root of A−1 is negative. Therefore, at least one root of A−1−(g+μ)I is negative. Since (δ+g+σμ)−1 is a Frobenius root of
, if the consumable capital good is quasi intensive in itself from the private perspective, the matrix
has one zero root and one negative root [see Benhabib et al. (2000)]. Therefore, J has one zero root and at least two negative roots.
PROOF OF LEMMA 5
We prove this result by contradiction. Assume that
for all i=1, …, n. This is a dominant diagonal property for B. Since βii>0 by Assumption 2, all the roots of B have positive real parts and the determinant of B is positive. This is in contradiction to Assumption 7. Therefore, there is some i such that β1/(1+ρ)ii[les ][sum ]j≠iβ1/(1+ρ)ij.
PROOF OF LEMMA 6
The matrix
has a dominant diagonal if for any i=0, …, n
If
has a dominant diagonal, then there exist
such that this property still holds for any
.
If B(ρ) has a negative determinant, then from Lemma 5 there exists at least one row i∈{0, 1, …, n} such that β1/(1+ρ)ii[les ][sum ]j≠iβ1/(1+ρ)ij. Consider therefore the set of rows
defined by (25). Two cases need to be considered:
- If there is some and at least one j≠i such that βii<βij, then for ρ<0, (βij/βii)−ρ/(1+ρ)>1 and . It follows that
Therefore equation (A.3) cannot hold when ρ is sufficiently close to −1 and .
- If for all and all j≠i, βii>βij, then when ρ<0, we have (βij/βii)−ρ/(1+ρ) < 1. Moreover, any row of B(ρ) will be such that . In this case and under Assumption 9, we necessarily have βkk>βkj for any j≠k. Since has a dominant diagonal, equation (A.3) holds for ρ=0, but the left-hand side of (A.3) increases with ρ. It follows that (A.3) holds for all ρ∈(−1, 0], and has a dominant diagonal within this interval. It follows that .
PROOF OF THEOREM 2
We need first to establish the following lemma.
LEMMA A.5. Let Assumptions 1, 2, and 6 hold. If the (n+1)×(n+1) matrix
has n+1 real positive eigenvalues, then
has n negative eigenvalues and one zero eigenvalue.
Proof. The Frobenius root of
is
. Denote λ{X} an eigenvalue λ of a matrix X. Under Assumption 1,
for all i, j=0, …, n, and the matrix
is indecomposable. It follows that
is a simple eigenvalue. Moreover, if
is real and positive, we have
Since
, we have
. When we consider the Frobenius root of
,
, it follows that
.
Theorem 2 is a consequence of Lemma 5, Lemma 6, and Lemma A.5. As proved by Benhabib et al. (2000), along the balanced growth path, the sign pattern of roots of A is the same as that of B, and the sign pattern of roots of
is the same as that of
. If B has a negative determinant, Lemma 5 implies that [A−1−(g+μ)I] has at least one real negative eigenvalue. Under Assumption 10, the roots of
are real. Case (i) is obtained by continuity from Lemma A.5 and Lemma 6. There exists
such that the stationary balanced growth path is locally indeterminate for any ρ∈(−1, ρ*). Similarly, case (ii) is obtained by continuity from Lemma A.5 and case (ii) in Lemma 6. There exist
and
such that for any ρ∈(ρ*1, ρ*2), the stationary balanced growth path is locally indeterminate.
