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.

Here, p_j(t) and w_i(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, …, x_n)′,

, p¹=(p₁, …, p_n)′ and w¹=(w₁, …, w_n)′.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

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

1. more intensive in capital good i than the capital good j at the private level if a_ija₀₀−a_i0a_0j<0
2. more quasi intensive in capital good i than the capital good j at the social level if
3. 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 β_ij 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=w_i/p_i, 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

1. the consumption good is more intensive in capital good i than the capital good j from the private perspective if and only if β_ijβ₀₀−β_i0β_0j<0
2. the consumption good is more quasi labor intensive than the capital good j from the social perspective if and only if
   
3. 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,

1. 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);
2. 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 β_ij 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 β_iiβ₀₀−β_i0β_0i<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.

.

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=[c_ij] has a dominant diagonal if |c_ii|>[sum ]_j≠i|c_ij| for each i=1, …, n or |c_ii|>[sum ]_i≠j|c_ij| 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=[c_ij] has a strong dominant diagonal if |c_ii|>[sum ]_j≠i|c_ij| for each i=1, …, n and |c_ii|>[sum ]_i≠j|c_ij| 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

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.

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 p_j(t) and w_i(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 x_i 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

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 β_ij 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,

1. the consumable capital good is more intensive in the pure capital good i than the pure capital good j at the private level iff β_ijβ₀₀−β_i0β_0j<0;
2. 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
   
3. 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,

1. 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*;
2. 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

LEMMA 5. Under Assumptions 1, 2, 6, and 7, there exist at least one row i∈{0, …, n} such that β^1/(1+ρ)_ii[les ][sum ]_j≠iβ^1/(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 β_kk≠β_kj, 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

1. if there is some
   and at least one j≠i such that β_ii<β_ij, then
   ;
2. let n[ges ]2. If for all
   and all j≠i, β_ii>β_ij, 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:

1. If there is some
   and at least one j≠i such that β_ii<β_ij, then there exist
   and
   such that for any ρ∈(ρ*₁, ρ*₂), the stationary balanced growth path is locally indeterminate.
2. Let n[ges ]2. If for all
   and all j≠i, β_ii>β_ij, 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.

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.

1. 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.
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=[b_ij] 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 β_ii>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 β^1/(1+ρ)_ii[les ][sum ]_j≠iβ^1/(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 β^1/(1+ρ)_ii[les ][sum ]_j≠iβ^1/(1+ρ)_ij. Consider therefore the set of rows

defined by (25). Two cases need to be considered:

1. If there is some
   and at least one j≠i such that β_ii<β_ij, then for ρ<0, (β_ij/β_ii)^−ρ/(1+ρ)>1 and
   . It follows that
   
   Therefore equation (A.3) cannot hold when ρ is sufficiently close to −1 and
   .
2. If for all
   and all j≠i, β_ii>β_ij, then when ρ<0, we have (β_ij/β_ii)^−ρ/(1+ρ) < 1. Moreover, any row
   of B(ρ) will be such that
   . In this case and under Assumption 9, we necessarily have β_kk>β_kj 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.