3.1 Maximum principle

In the present study that treats a multi-sector model, we have to make explicit whether a given vector is either a row vector or a column vector, in order to avoid possible confusion. In the present study, ℝⁿ refers to a set of all real 1×nrow vectors. Hence if a∈Rn, then a refers to a 1×nrow vector, and a^T refers to an n × 1 column vector.

The problem (1) is solved by defining the current value Hamiltonian ℋ and the current value maximized Hamiltonian ℋ^∗, and by applying the maximum principle to ℋ^∗. Let K=(K1,K2,K3)∈R++3, P=(P1,P2,P3)∈R++3, and W=(W1,W2,W3)∈R+3. The current value Hamiltonian H=H(K,P,C,{Kij}i,j=1,2,3,W) is given by

where C > 0, and Kij≥0,i,j=1,2,3, and where P_i is an imputed price of K_i, and W_i is a rental price of K_i. Note that H=H(K,P,C,{Kij}i,j=1,2,3,W) is concave in (K,C,{Kij}i,j=1,2,3).

Let c_ij denote the (i, j)-element of C for each i,j=1,2,3. Let Wi(P),i=1,2,3 be functions of P∈R++3 defined as

Let W(P):=(W1(P),W2(P),W3(P)). Let H(P) be a 3 × 3 matrix-valued function of P∈R++3 defined as

By construction we have

Since 1C=1, we also have

Let 03:=(0,0,0)T. Let N be a set in R++6 defined as

At the end of the present subsection we shall show that N is a non-empty open subset of R++6. Let Y(K,P)=(Y1(K,P),Y2(K,P),Y3(K,P))∈R++3 be a 1 × 3 vector-valued functions of (K,P)∈N defined as

Let C(P1) be defined as C(P1):=P1−1σ for P1>0. For (K,P)∈N, let Kij(K,P),i,j=1,2,3, be defined as

Then we have

Let H∗:N→R be defined as

Then by the equalities (4), (5), and (6), ℋ^∗ is the maximized Hamiltonian of ℋ for (K,P)∈N. Since ℋ is concave in (K,C,{Kij}i,j=1,2,3), ℋ^∗ is concave in K by the lemma in Kamien and Schwartz (1991, p. 222). Hence we can apply the maximum principle to H∗=H∗(K,P) for (K,P)∈N. We obtain the following system of ordinary differential equations and of boundary conditions.

where K_i is a predetermined variable, P_i is a non-predetermined variable, the condition (9) is an initial condition, and the condition (10) is the transversality condition. The interiority condition (11) guarantees that the maximized Hamiltonian is well defined. If a solution of this system satisfies the summability condition (2), the solution is an optimal solution of the problem (1).

Suppose that an interior BGP exists, and let μ_K and μ_P be balanced growth rates of capital and of its imputed price, respectively. Then we have μP=−σμK from the equation (7), and μP=−σμ from the equation (8). Thus by Assumption 1.1 μK=μ>0, and by Assumption 1.1.b the transversality and summability conditions are satisfied on the BGP.

Before leaving the present subsection, we construct a candidate of an interior BGP of the optimal endogenous growth model (1). Let X¯∈R3 be defined as

By Lemma 1X¯T>03. Let Λ¯ be a one-dimensional manifold in R++6 defined as

Then Λ¯⊂N, because for λ > 0,

Since H(P)KT is continuous in (K,P)∈R++6, N is a non-empty open subset of R++6. Therefore N includes some open-neighborhood of Λ¯. In the next subsection we shall show that Λ¯ constitutes an interior BGP of the growth model (1).

3.2 Decomposition

The present section decomposes the six-dimensional system of equations (7) and (8) into five-dimensional stationary autonomous component and one-dimensional endogenously growing component. Let f1=f1(x,y) and f2=f2(x,y) be functions of (x,y)∈R2 defined as

Let l=l(x) be a function of x∈R given by the definition (33) in Appendix 2. Let L=L(x,y) be a 3 × 3 matrix-valued function of (x,y)∈R2 given by the definition (34) in Appendix 2. Let h=h(x,y) be a function of (x,y)∈R2 given by the definition (35) in Appendix 2. By construction each of these functions is sufficiently smooth, l(0)=0, L(0,0)=O3, and h(0,0)=0, where O₃ denotes the 3 × 3 zero-matrix. Let e3:=(0,0,1)T. Let X_i, i = 1, 2, .., 5 be defined as

Let ki:=log⁡Ki, and pi:=log⁡Pi, i = 1, 2, 3. Let xi:=log⁡Xi, i = 1, 2, .., 5. Then by the relation (36) in Appendix 2 the six-dimensional system of equations (7) and (8) are decomposed into the following three components.

The system (14) is a two-dimensional stationary autonomous component. The system composed of the differential equations (14) and (15) is a five-dimensional stationary autonomous component. The equation (16) is one-dimensional endogenously growing component.

Let (x1∗,x2∗,X3∗,X4∗,X5∗) be defined as

where X¯ is given by the definition (12). Then (x1∗,x2∗,X3∗,X4∗X5∗) is a steady state of the five-dimensional autonomous stationary system composed of the differential equations (14) and (15). Since X¯T>03 by Lemma 1, xi∗=log⁡Xi∗, i = 3, 4, 5 are well defined. Let T(σ) be the 5 × 6 matrix given by the definition (37) in Appendix 3. The rank of T(σ) is five. Consider the following one-dimensional manifold Λ∗ in R++6.

By the relation (38) in Appendix 3, the manifold Λ∗ constitutes a BGP of the optimal growth model (1). We obtain from the construction of T(σ), (K,P)∈Λ∗, if and only if (K,P)=(λX¯,λ−σ1) for λ=P3−1σ. Therefore we have

where Λ¯ and N are the sets given by the definitions (13) and (3). Since Λ∗⊂N, and since H(P)KT>03 for (K,P)∈N, the interiority condition (11) is satisfied on Λ∗ and on some open neighborhood of it. As mentioned in the previous subsection, by Assumption 1.1.b the transversality and the summability conditions are satisfied on Λ∗. Thus we have the following proposition.

Before leaving the present subsection, consider the following set N₁ in R2×R++3 for the later use.

By construction, H(P)KT>03 if and only if (ωC+L(x1))X2T>03∧P3>0 with (x1,X2)=(x1, x₂, X₃, X₄, X5). Therefore, (K,P)∈N, if and only if (x1,x2,X3,X4,X5)∈N1∧P3>0. The set N₁ includes the steady state (0,0,X3∗,X4∗,X5∗) and some open neighborhood of it. Consider also the following set N₂ in ℝ⁵ for the later use.

Then by construction, (K,P)∈N, if and only if (x1,x2,x3,x4,x5)∈N2∧P3>0. The set N₂ includes the point (0,0,x3∗,x4∗,x5∗) and some open neighborhood of it.

3.3 Transitional dynamics

The present section considers the transitional dynamics and the local determinacy of equilibrium around the BGP Λ∗ in terms of predetermined and non-predetermined variables. Let z_i, i = 1, 2, and q_i, i = 1, 2, 3 be variables given by the definition (39) in Appendix 3. And let M(σ) be the 5 × 5 matrix given by the definition (40) in Appendix 3. Then detM(σ)≠0, and we have (z1,z2,q1,q2,q3)=(x1,x2,x3,x4,x5)M(σ)T and (x1,x2,x3,x4,x5)=(z1,z2,q1,q2,q3)(M(σ)−1)T. See the relation (41) in Appendix 3. By construction z_i, i = 1, 2 are predetermined variables and q_i, i = 1, 2, 3 are non-predetermined variables. Let (z1∗,z2∗,q1∗,q2∗,q3∗) be defined as

Then (z1∗,z2∗,q1∗,q2∗,q3∗)(M(σ)−1)T∈N2, and let N₃ be a set in ℝ⁵ defined as

Then (z1∗,z2∗,q1∗,q2∗,q3∗)∈N3, and the set N₃ includes some open neighborhood of it. By construction, (K,P)∈N, if and only if (z1,z2,q1,q2,q3)∈N3∧P3>0. Therefore in some open neighborhood of (z1∗,z2∗,q1∗,q2∗,q3∗), the maximized Hamiltonian is well defined, and local dynamics near the steady state (z1∗,z2∗,q1∗,q2∗,q3∗) is also well defined.

Let J be a 2 × 2 matrix defined as

Then the characteristic roots of the five-dimensional autonomous stationary system composed of the differential equations (14) and (15) evaluated at the steady state (0,0,X¯) are given by two characteristic roots of J and three characteristic roots of ωC−(g+μ)I3. Consider Example 1, Example 2, Example 3 in Section 2. Each of these examples has two stable roots and three unstable roots. (x1,x2,x3,x4,x5) one to one corresponds to (z1,z2,q1,q2,q3) that includes two predetermined and three non-predetermined variables. Thus in each of these examples, the interior BGP is saddle point stable.

Since (z1,z2,q1,q2,q3) includes two predetermined variables zi,i=1,2, the stationary autonomous dynamics (14) and (15) might have a closed orbit that is locally determinate and locally stable around an unstable BGP. In other words, the BGP might not always be saddle point stable. For η > 0, let B¯(η) be a 3 × 3 matrix defined as

Each element of B¯(η) is positive and less than 1, and 1B¯(η)=1. We have

Let c¯ij=c¯ij(η) be the (i, j)-element of the inverse matrix of B¯(η) for each i,j=1,2,3. Then we have

And the corresponding autonomous dynamics (14) has a pair of pure imaginary complex conjugate roots at a steady state.^[5] This suggests that the system (14) undergoes Hopf bifurcation under perturbations of the technology matrix B¯(η). We conduct an exact bifurcation analysis in the next section.

4.1 Hopf bifurcation

The present subsection applies the Hopf bifurcation theorem to the two-dimensional autonomous system (14). For η > 0, let B¯(η) be the 3 × 3 matrix given by the definition (17). For η > 0 and for ν in some neighborhood of 0, let B(ν,η) be a 3 × 3 matrix defined as

As νvaries around 0, the matrix B(ν,η) generates perturbations of the matrix B¯(η). We have 1B(ν,η)=1 and

Suppose η > 0 and 1−3(1+η)ν≠0. Let C(ν,η) denote the inverse matrix of B(ν,η).

Let cij(ν,η) denote the (i, j)-element of C(ν,η) for each i,j=1,2,3. Let J(ν,η,ω) be a 2×2 matrix defined as

Then we obtain the following relation.

And we have

By construction the following four relations hold for η > 0.

1. If −2η+3η25+12η+9η2<ν<1+2η+3η25+12η+9η2, then 0<1−(2+3η)(2+2η)5+12η+9η2−ν<1, and 0<(2+3η)η5+12η+9η2+ν<1.

2. If ν<13(1+η), then detB(ν,η)>0.

3. If −2(1+2)(2+3η)5+12η+9η2<ν<2(2−1)(2+3η)5+12η+9η2, then the characteristic roots of J(ν,η,ω) are a pair of conjugate complex roots.

4. If η≠2−13, then ddν[trJ(ν,η,ω)]|ν=0≠0.