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A two-sector economic growth model with optimal labor and capital allocation
Applied Mathematics and Computation 183 (2006) 1359–1377 www.elsevier.com/locate/amc
A two-sector economic growth model with optimal labor and capital allocation q Donghan Cai College of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China
Abstract In this paper, a two-sector growth model with optimal labor force and capital allocation is given. By treating the quantities of the labor force and capital in the consumption production sector as the control variables in the model, we obtain a two-dimension dynamical system from solving the household utility maximum problem. It is proved that the system has a unique nonzero equilibrium which is a saddle, so there exists an optimal labor force and capital allocation process in the economic growth. The capital accumulation and consumption production strictly increase along the growth path. At the end of this paper, a numerical computation is given to present the allocation process of capital and labor. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Two-sector model; Dynamical optimization; Dynamical system; Labor force and capital allocation; Optimal growth path
1. Introduction One of the most important problems in economic growth theory is how the behavior of consumption affects the long-term economic growth. In the Solow model, the aggregate consumption is the fraction of the output, and the saving rate is fixed. This model implies that reducing the proportion of consumption can accelerate the accumulation of the per capita capital, but it may be not good for the per capita consumption in the long-term view, since there exists so called dynamic inefficiency [1]. In the Ramsey model, the consumption is endogenously decided by an one sector optimal model. One of the notable characters in the Solow model and Ramsey model is that the capital good and consumption good are treated as the same and there does not exist a separate consumption production sector. By the setup of a two-sector economic growth model with separated production good sector and consumption good sector, this paper analyzes the optimal labor force and capital reallocation in the long-term economic growth. We inquire how an economy allocates optimally its resource between the capital production and consumption production in the economic growth process. Harl E. Ryder had presented a model to discuss the optimal accumulation in a two-sector neoclassical economy with nonshiftable capital. The model implies there exists a optimal labor and capital reallocating process. q
Supported by National Natural Science Foundation of China (79970104). E-mail address: [email protected]
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.148
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But the existence and uniqueness of the optimal growth path and the equilibrium is not given. So, the dynamics of the model is unclear [2]. The use of a two-sector framework (or its variants) is broadly used in the theoretical analysis. K. Marx, J.A. Hobson, A. Spiethoff, G. Cassel, and F.A. Hayek, among others, emphasized sectoral imbalance between the consumption and investment good industries as a cause of business cycles or cries [3]. A two-sector growth model with capital and labor shiftable that is presented in [4] gave a solow type model. In our paper, the economy sustains the optimal growth by continuously adjusting the allocation of labor force and capital between the two sectors. In the model, the production of consumption good and capital good increases strictly along the optimal growth path, that is to say, it is not needed to sacrifice the present consumption to get in return the aggregate capital growth. The remainder of the paper is organized as follows. In Section 2, the basic assumptions and the model are presented. The existence and uniqueness of nonzero equilibrium of the model are proved in Section 3. In Section 4, we analysis the dynamics of the model. The dynamics of the capital and consumption production is given in Section 5. In Section 6, an example of numerical computation is presented. A summary is given in Section 7.
2. Setup of the model Consider an economy with a capital production sector and a consumption production sector. They are called Sector I and Sector II, respectively. In each sector all the firms have an identical neoclassical technology and produce output using labor and capital. The capital goods are used in both sectors. The production functions, Fi(Ki, Li), i = 1, 2, are neoclassical and satisfy the Inada conditions, where Ki, Li are the ith sector’s capital and labor force. So, the new investment and consumption production are K_ ¼ F 1 ðK 1 ; L1 Þ dK;
C ¼ F 2 ðK 2 ; L2 Þ;
ð1Þ
where K = K1 + K2. The intensive form of the production function of ith sector is Ki ; 1 ; i ¼ 1; 2 fi ðk i Þ ¼ F i Li and satisfies fi ð0Þ ¼ 0; f i ð1Þ ¼ 1; fi0 ð0Þ ¼ 1; fi0 ð1Þ ¼ 0; fi0 ðk i Þ > 0; fi00 ðk i Þ < 0; k i > 0; i ¼ 1; 2:
ð2Þ
The utility function U(C) is increasing in C and concave, U 0 (C) > 0, U 0 (C) < 0, satisfies that lim U 0 ðCÞ ¼ þ1;
C!0
lim U 0 ðCÞ ¼ 0:
C!1
We further assume that the total labor supply is constant, that is L1 þ L2 ¼ L ¼ const: The economy chooses optimal capital and labor allocation such that the total utility reaches to maximum, that is, it searches for Z 1 max U ðCðtÞÞ expðrtÞ dt; ð3Þ 0
subject to (1); where r > 0 is the rate of time preference. The current value Hamiltonian to solve the optimization problem (3) is H ðK; K 2 ; L2 ; qÞ ¼ U ðCÞ þ q½F 1 ðK K 2 ; L L2 Þ dK:
ð4Þ
D. Cai / Applied Mathematics and Computation 183 (2006) 1359–1377
The first order conditions and transversity are oH ¼ F 1 ðK K 2 ; L L2 Þ dK; K_ ¼ oq oH oF 1 ðK 1 ; L1 Þ ¼ q ðd þ rÞ ; q_ ¼ rq oK oK 1 oH oF 2 ðK 2 ; L2 Þ F 1 ðK 1 ; L1 Þ ¼ U 0 ðCÞ q ; 0¼ oK 2 oK 2 oK 1 oH oF 2 ðK 2 ; L2 Þ oF 1 ðK 1 ; L1 Þ ¼ U 0 ðCÞ q ; 0¼ oL2 oL2 oL1 lim expðrtÞqðtÞKðtÞ ¼ 0: t!1
1361
ð5Þ ð6Þ ð7Þ ð8Þ ð9Þ
By (7) and (8), oF 2 oF 1 oK 2 oK 1 ¼ : oF 2 oF 1 oL2 oL1
ð10Þ
Since oF i ¼ fi0 ðk i Þ; oK i
oF i ¼ fi ðk i Þ k i fi0 ðk i Þ; oLi
i ¼ 1; 2;
Eq. (10) turns into f2 ðk 2 Þ k 2 f 0 ðk 2 Þ f1 ðk 1 Þ k 1 f10 ðk 1 Þ ¼ : f20 ðk 2 Þ f10 ðk 1 Þ
ð11Þ
Let Gi ðk i Þ ¼
fi ðk i Þ k i fi0 ðk i Þ ; fi0 ðk i Þ
i ¼ 1; 2;
ð12Þ
then we have Lemma 1. The functions Gi(ki), i = 1, 2 are strictly increasing and satisfy Gi ðk i Þ > 0;
lim Gi ðk i Þ ¼ 0;
lim Gi ðk i Þ ¼ 1:
k i !1
k i !0
Proof. Since k i ðfi ðk i Þ=k i fi0 ðk i ÞÞ fi ðk i Þfi00 ðk i Þ 0 > 0; G ðk Þ ¼ > 0; i ¼ 1; 2; i i 2 fi0 ðk i Þ ½fi0 ðk i Þ the functions Gi(ki), i = 1, 2 are strictly increasing. It is not difficult to check that limki !0 Gi ðk i Þ ¼ 0. If limki !1 Gi ðk i Þ 6¼ 1; i ¼ 1; 2, then there exists finite Gi > 0, i = 1, 2 such that limki !1 Gi ðk i Þ ¼ Gi . By (12), Gi ðk i Þ ¼
fi0 ðk i Þ 1 ¼ ; fi ðk i Þ Gi ðk i Þ þ k i
i ¼ 1; 2:
For k i > k 0i > 0; i ¼ 1; 2, we have "Z # ki dui 0 fi ðk i Þ ¼ fi ðk i Þ exp : k 0i Gi ðui Þ þ ui Differentiation gives fi0 ðk i Þ
fi ðk 0i Þ ¼ exp Gi ðk i Þ þ k i
"Z
ki
k 0i
# dui ; Gi ðui Þ þ ui
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which implies that "Z ki fi0 ðk i Þ exp k 0i
# Gi ðui Þ Gi fi ðk 0i ÞðGi þ k i Þ dui ¼ : ðGi ðui Þ þ ui ÞðGi þ ui Þ ðGi þ k 0i ÞðGi ðk i Þ þ k i Þ
ð13Þ
Since limki !1 Gi ðk i Þ ¼ Gi , for a fixed > 0, there exist k 0i such that Gi ðk i Þ Gi < ; k i > k 0i . Therefore, the integration Z 1 Gi ðui Þ Gi dui ðGi ðui Þ þ ui ÞðGi þ ui Þ k 0i converges and left hand side of (13) approaches to zero as ki tends to infinite. But the right hand side of (13) approaches
fi ðk 0i Þ . Gi þk 0i
This is a contradiction. h
Lemma 2. Eq. (11) decides a function k2(k1), which satisfies dk 2 ðk 1 Þ > 0; dk 1
lim k 2 ðk 1 Þ ¼ 0;
k 1 !0
lim k 2 ðk 1 Þ ¼ 1:
k 1 !1
Proof. By Lemma 1, Gi(ki), i = 1, 2 are strictly increasing functions and are invertible, so we have k 2 ðk 1 Þ ¼ G1 2 ðG1 ðk 1 ÞÞ and dk 2 ðk 1 Þ G01 ðk 1 Þ > 0; ¼ 0 dk 1 G2 ðk 2 Þ
lim k 2 ðk 1 Þ ¼ 0;
k 1 !0
lim k 2 ðk 1 Þ ¼ 1:
k 1 !1
This completes the proof of Lemma 2. h From Lemma 2, we have K2 K1 k2 ¼ ¼G ¼ Gðk 1 Þ; L2 L1 where GðÞ ¼ G1 2 ðG1 ðÞÞ. Put C = F2(K2, L2) = L2f2(k2) and k2 = G(k1) into (7), then U 0 ðL2 f2 ðGðk 1 ÞÞÞ ¼
qf 01 ðk 1 Þ : f20 ðGðk 1 ÞÞ
ð14Þ
ð15Þ
Since U 0 (Æ) < 0, the function U 0 (Æ) is invertible. Denote that q1 ðk 1 Þ ¼ U 0 Lf2 ðGðk 1 ÞÞ and
f20 ðGðk 1 ÞÞ ; f10 ðk 1 Þ
1 qf 01 ðk 1 Þ 0 1 Iðk 1 ; qÞ ¼ U ; f2 ðGðk 1 ÞÞ f20 ðGðk 1 ÞÞ
ð16Þ
then, for any given k1 > 0, 0 < I(k1, q) < L if and only if q1(k1) < q < 1. Set X ¼ fðk 1 ; qÞj0 < k 1 < 1; q1 ðk 1 Þ < q < 1g; we have the following lemma from (16). Lemma 3. Eqs. (7) and (8) decide a function L2 ¼ Iðk 1 ; qÞ;
ðk 1 ; qÞ 2 X Rþ 2;
which satisfies lim Iðk 1 ; qÞ ¼ L;
q!q1 ðk 1 Þ
lim Iðk 1 ; qÞ ¼ 0:
q!1
ð17Þ
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Therefore, in the domain X, we have L1 ¼ L L2 ¼ L Iðk 1 ; qÞ; K 1 ¼ L1 k 1 ¼ ðL Iðk 1 ; qÞÞk 1 ; K 2 ¼ L2 k 2 ¼ Iðk 1 ; qÞGðk 1 Þ and K ¼ K 1 þ K 2 ¼ ðL Iðk 1 ; qÞÞk 1 þ Iðk 1 ; qÞGðk 1 Þ:
ð18Þ
The derivation of K with respect to t is oIðk 1 ; qÞ oIðk 1 ; qÞ _ K_ ¼ ½L þ ðGðk 1 Þ k 1 Þ þ ðG0 ðk 1 Þ 1ÞIðk 1 ; qÞk_ 1 þ ðGðk 1 Þ k 1 Þ q: ok 1 oq Suppose that I 1 ðk 1 ; qÞ ¼ L þ ðGðk 1 Þ k 1 Þ
oIðk 1 ; qÞ þ ðG0 ðk 1 Þ 1ÞIðk 1 ; qÞ; ok 1
ðk 1 ; qÞ 2 X;
then, from limq!1I(k1, q) = 0 for any given k1, we have the following lemma. Lemma 4. If the function I(k1, q) satisfies lim
q!1
oIðk 1 ; qÞ ¼ 0; ok 1
k 1 > 0;
then, for any given k1 > 0, I1(k1, q) > 0 when q is large enough. Example. When U(C) is constant-relative-risk-aversion utility function I(k1, q) is
C 1h , 1h
0 < h < 1 or logarithm utility h ln C, the
0 1 1 f2 ðGðk 1 ÞÞ h Iðk 1 ; qÞ ¼ f2 ðGðk 1 ÞÞ qf 01 ðk 1 Þ or Iðk 1 ; qÞ ¼
hf20 ðGðk 1 ÞÞ ; 0 qf 1 ðk 1 Þf2 ðGðk 1 ÞÞ
respectively. Both satisfy the condition of Lemma 4. If (k1, q) is a point in X such that I1(k1, q) 5 0, then there exists a neighborhood X1 X such that I1(k1, q) 5 0, (k1, q) 2 X1. Let X ¼ fðk 1 ; qÞjI 1 ðk 1 ; qÞ 6¼ 0; ðk 1 ; qÞ 2 Xg; then X is an open subset in X and not empty under condition of Lemma 4. Therefore, with the condition of Lemma 4, we obtain the following dynamical system in X from the first order conditions (5)–(8) k_ 1 ¼
1 ½ðL Iðk 1 ; qÞÞf1 ðk 1 Þ I 2 ðk 1 ; qÞ dI 3 ðk 1 ; qÞ; I 1 ðk 1 ; qÞ
q_ ¼ ½ðd þ rÞ f10 ðk 1 Þq; where oIðk 1 ; qÞ ; oq I 3 ðk 1 ; qÞ ¼ ðL Iðk 1 ; qÞÞk 1 þ Iðk 1 ; qÞGðk 1 Þ:
I 2 ðk 1 ; qÞ ¼ ðGðk 1 Þ k 1 Þ½ðd þ rÞ f10 ðk 1 Þq
ð19Þ ð20Þ
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3. The existence and uniqueness of equilibrium Let J 1 ðk 1 ; qÞ ¼ ½L Iðk; qÞf1 ðk 1 Þ dI 3 ðk 1 ; qÞ ¼ ½L Iðk 1 ; qÞf1 ðk 1 Þ d½ðL Iðk 1 ; qÞÞk 1 þ Iðk 1 ; qÞGðk 1 Þ ¼ L½f1 ðk 1 Þ dk 1 ½f1 ðk 1 Þ dk 1 þ dGðk 1 ÞIðk 1 ; qÞ; J 2 ðk 1 Þ ¼ ðd þ rÞ f10 ðk 1 Þ; then the dynamical system (19), (20) turns into 1 k_ 1 ¼ J 1 ðk 1 ; qÞ I 2 ðk 1 ; qÞ ; I 1 ðk 1 ; qÞ q_ ¼ J 2 ðk 1 Þq:
ð21Þ ð22Þ
Þ 2 X is an equilibrium of the dynamical system (19), (20) if and only if A point ðk 1 ; q J 1 ðk 1 ; qÞ ¼ J 2 ðk 1 Þ ¼ 0;
ð23Þ
q1 Þ ¼ 0 if J 2 ðk 1 Þ ¼ 0. for I 2 ðk 1 ; Since J2(k1) is a strictly increasing function from negative infinite to d + r > 0, there exists a unique point k 1 > 0 such that J 2 ðk 1 Þ ¼ 0. Let N ðqÞ ¼ J 1 ðk 1 ; qÞ, then oN ðqÞ oIðk 1 ; qÞ ¼ ½f1 ðk 1 Þ dk 1 þ dGðk 1 Þ > 0; oq oq h i k1 ;qÞ if oIðoq < 0; q > q1 ðk 1 Þ for f1 ðk 1 Þ dk 1 ¼ k 1 f1kðk1 1 Þ d > rk 1 > 0 . So, we have the following lemma. Lemma 5. If 0; J 2 ðk 1 Þ ¼ 0.
oIðk 1 ;qÞ oq
< 0; ðk 1 ; qÞ 2 X, then there exists a unique point ðk 1 ; qÞ 2 X such that J 1 ðk 1 ; qÞ ¼
Proof. Since N(q) is strictly increasing in X and satisfies lim N ðqÞ ¼ dIðk 1 ; q1 ðk 1 ÞÞGðk 1 Þ < 0; q!q1 ðk 1 Þ
lim N ðqÞ ¼ L½f1 ðk 1 Þ dk 1 > 0:
q!1
Hence, there exists a unique point q such that N ðqÞ ¼ 0, i.e., J 1 ðk 1 ; qÞ ¼ 0.
h
Theorem 1. If the set X is not empty and ðk 1 ; qÞ belongs to it, the dynamic system (19), (20) have a nonzero equilibrium in a neighborhood of ðk 1 ; qÞ. i When F i ðK i ; Li Þ ¼ Ai K ai i L1a ; 0 < ai < 1; i ¼ 1; 2 are both Cobb–Douglas production functions, then i ai 1a2 1 fi ðk i Þ ¼ Ai k i . By (11), a2 k 2 ¼ 1a k 1 . So, we have, a1
k 2 ¼ Gðk 1 Þ ¼ gk 1 ; 1 where g ¼ 1a 1a2
ð24Þ
a2 . a1
Lemma 6. If U(c) = h ln C, then in the set X, a2 h a1 k ; a1 gA1 L 1 a2 h 1 a1 q k1 ; Iðk 1 ; qÞ ¼ a1 gA1 I 1 ðk 1 ; qÞ ¼ L ð1 a1 Þð1 gÞIðk 1 ; qÞ; q1 ðk 1 Þ ¼
under the Cobb–Douglas production technology.
ð25Þ ð26Þ ð27Þ
D. Cai / Applied Mathematics and Computation 183 (2006) 1359–1377
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i Theorem 2. If U(C) = h ln C, F i ðK i ; Li Þ ¼ Ai K ai i L1a ; 0 < ai < 1; i ¼ 1; 2, then the dynamical system (19), (20) i has a unique nonzero equilibrium in the domain X.
1 a2 Proof. If a1 6 a2, then 1 g ¼ a1að1a 6 0 and 2Þ
I 1 ðk 1 ; qÞ ¼ L ð1 a1 Þð1 gÞIðk 1 ; qÞ P L > 0: If a1 > a2, then 0 < g < 1 for
1a1 a1
2 < 1a . By 0 < I(k1, q) < L and (27), for (k1, q) 2 X, a2
I 1 ðk 1 ; qÞ ¼ L ð1 a1 Þð1 gÞIðk 1 ; qÞ > L½1 ð1 a1 Þð1 gÞ > 0: So, X ¼ X. By Lemma 5, the dynamical system (19), (20) has a unique nonzero equilibrium point. This completes the proof of the theorem. h 4. The dynamics of the model In this section, we inquire the dynamics of the model under the conditions of Theorem 2. f ðk Þ Let k 1 be the unique positive zero point of the equation f1(k1) dk1 = 0, then k 1 < k 1 for 1k 1 ¼ d < d þ r 1 f ðk Þ and k > n > k 1 , where n is the unique point that satisfies f 0 ðnÞ ¼ 1 1 . 1
1
Let
k1
M ¼ fðk 1 ; qÞj0 < k 1 < k 1 ; q1 ðk 1 Þ < q < 1g and J ðk 1 ; qÞ ¼ J 1 ðk 1 ; qÞ I 2 ðk 1 ; qÞ ¼ L½f1 ðk 1 Þ dk 1 ½f1 ðk 1 Þ dk 1 þ dgk 1 þ ðg 1Þðf10 ðk 1 Þ ðd þ rÞÞk 1 Iðk 1 ; qÞ; then J ðk 1 ; qÞ ¼ L½f1 ðk 1 Þ dk 1 ½f1 ðk 1 Þ dk 1 þ dgk 1 þ ðg 1Þðf10 ðk 1 Þ ðd þ rÞÞk 1 Lq1 ðk 1 Þq1 for Iðk 1 ; qÞ ¼
a2 h 1 a a2 h a1 k ¼ Lq1 ðk 1 Þq1 : q k 1 ¼ Lq1 a1 gA1 a1 gA1 L 1
So, we have the following lemma. Lemma 7. The equation J(k1, q) = 0 determines a curve q2(k1) in the domain M X which satisfies that (1) if g = 1, then q2 ðk 1 Þ > q1 ðk 1 Þ; k 1 2 ð0; k 1 Þ; limk1 !k1 q2 ðk 1 Þ ¼ 1; (2) there exists ~k 1 2 ð0; k 1 Þ such that q1 ð~k 1 Þ ¼ q2 ð~k 1 Þ; limk1 !k1 q2 ðk 1 Þ ¼ 1; q1 ðk 1 Þ > q2 ðk 1 Þ; k 1 2 ð0; ~k 1 Þ and q1 ðk 1 Þ < q2 ðk 1 Þ; k 1 2 ð~k 1 ; k 1 Þ if 0 < g < 1; and q2 ðk 1 Þ > q1 ðk 1 Þ; (3) if g > 1, there exists ^k 1 2 ðk 1 ; k 1 Þ; q2 ðk 1 Þ > q1 ðk 1 Þ; k 1 2 ð0; ^k 1 Þ for d < a1rðg1Þ ðg1Þþ1 rðg1Þ q ðk Þ ¼ 1 if d > and lim . k 1 2 ð0; k 1 Þ for d P a1rðg1Þ k 1 !k 1 2 1 ðg1Þþ1 a1 ðg1Þþ1 Proof. Since f1 ðk 1 ; qÞ dk 1 > 0; k 1 2 ð0; k 1 Þ, the equation J(k1, q) = 0 decides a function dg þ ðg 1Þðf10 ðk 1 Þ ðd þ rÞÞ q2 ðk 1 Þ ¼ q1 ðk 1 Þ 1 þ k1 : f1 ðk 1 Þ dk 1 Let gðk 1 Þ ¼ dg þ ðg 1Þðf10 ðk 1 Þ ðd þ rÞÞ;
ð28Þ
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D. Cai / Applied Mathematics and Computation 183 (2006) 1359–1377
then gðk 1 Þ ¼ dg > 0 and
8 > < > 0; if 0 < g < 1; dgðk 1 Þ ¼ ðg 1Þf100 ðk 1 Þ ¼ 0; if g ¼ 1; > dk 1 : < 0; if g > 1:
If g = 1, then g(k1) = dg > 0. So, by (28), dgk 1 q2 ðk 1 Þ ¼ 1 þ q ðk 1 Þ > q1 ðk 1 Þ; f1 ðk 1 Þ dk 1 1
ð29Þ
k 1 2 ð0; k 1 Þ
and limk1 !k1 q2 ðk 1 Þ ¼ 1 for f1 ðk 1 Þ dk 1 ¼ 0. If 0 < g < 1, then from g1 ðk 1 Þ ¼ dg > 0; limk1 !0 gðk 1 Þ ¼ 1 and (29), there exists a unique point ~k 1 2 ð0; k 1 Þ such that gð~k 1 Þ ¼ 0 and gðk 1 Þ > 0; k 1 2 ð~k 1 ; k 1 Þ. So, by (28), (2) holds. If g > 1, then by gðk 1 Þ ¼ dg þ ðg 1Þðf10 ðk 1 Þ ðd þ rÞÞ ¼ dg þ ðg 1Þða1 d ðd þ rÞÞ; we have 8 rðg 1Þ > > ; < 0; if d < > > a1 ðg 1Þ þ 1 > > > < rðg 1Þ ; gðk 1 Þ ¼ 0; if d ¼ > a1 ðg 1Þ þ 1 > > > > rðg 1Þ > > : > 0; if d > : a1 ðg 1Þ þ 1 From (29) and gðk 1 Þ ¼ dg, if d < a1rðg1Þ , then there exists a point ^k 1 2 ðk 1 ; k 1 Þ such that ðg1Þþ1 rðg1Þ q1 ðk 1 Þ < q2 ðk 1 Þ; k 1 2 ð0; ^k 1 Þ. If d P a1 ðg1Þþ1, then q1 ðk 1 Þ < q2 ðk 1 Þ; k 1 2 ð0; k 1 Þ and limk1 !k1 q2 ðk 1 Þ ¼ 1, if . This completes the proof of Lemma 7. h d > a1rðg1Þ ðg1Þþ1 By Lemma 7, we know that the domain M R2þ is divided into four parts by the line k 1 ¼ k 1 and the curve q2(k1). The function f10 ðk 1 Þ is strictly decreasing from infinite to zero. On the left of the line k 1 ¼ k 1 , q_ < 0 and q_ > 0 and on the right the line k 1 ¼ k 1 in the domain M. Above the curve q2(k1), k_ > 0, and k_ < 0 below the curve q2(k1) in the domain M ¼ fðk 1 ; qÞj~k 1 < k 1 < ^k 1 ; q1 ðk 1 Þ < q < 1g M: qÞ is a saddle point by phase portrait analysis. One of the optiTherefore, the unique nonzero equilibrium ðk 1 ; mal path is on the upper-left part and the other is on the low-right part. Therefore, we obtain the following theorem.
Fig. 1. The dynamics of the model.
D. Cai / Applied Mathematics and Computation 183 (2006) 1359–1377
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Theorem 3. The unique nonzero equilibrium of the dynamical system (19), (20) is a saddle and the model has optimal growth path. The phase portrait of the dynamical system (19), (20) is described in Fig. 1. 5. The dynamics of the capital and consumption production Now we inquire the dynamics of the capital K and the consumption production C along the optimal path. _ we have By the representative of K, Iðk 1 ; qÞ _ K_ ¼ ½L ð1 a1 Þð1 gÞIðk 1 ; qÞk_ 1 ðg 1Þk 1 q; ð30Þ q for oI ¼ a1 k 1 1 Iðk 1 ; qÞ; ok 1
oI ¼ q1 Iðk 1 ; qÞ oq
from (26). Obviously, if g P 1, we have K_ > 0 on P1 and K_ < 0 on path P2 for k_ 1 > 0; q_ < 0 on the path P1 and _k 1 < 0; q_ 1 > 0 on the path P2. So, we have Lemma 8. If g P 1, then K_ > 0 on the saddle path P1 and K_ < 0 on the saddle path P2. Now, we assume that 0 < g < 1. Substituting (21), (22) into (30), we obtain K_ ¼ J ðk 1 ; qÞ þ ðg 1Þ½f10 ðk 1 Þ ðd þ rÞk 1 Iðk 1 ; qÞ ¼ L½f1 ðk 1 Þ dk 1 ½f1 ðk 1 Þ dk 1 þ dgk 1 Iðk 1 ; qÞ ¼ L½f1 ðk 1 Þ dk 1 L½f1 ðk 1 Þ dk 1 þ dgk 1 Hence, there exists a curve dgk 1 q3 ðk 1 Þ ¼ 1 þ q ðk 1 Þ; f1 ðk 1 Þ dk 1 1
q1 ðk 1 Þ : q
0 < k1 < k;
ð31Þ
on which K_ ¼ 0. For ln q3 ðk 1 Þ ¼ ln½f1 ðk 1 Þ dk 1 þ dgk 1 ln½f1 ðk 1 Þ dk 1 þ ln q1 ðk 1 Þ and
d ln q1 ðk 1 Þ dk 1
q0 ðk Þ
¼ q1 ðk11 Þ ¼ ak11 . It is not difficult to check that 1 f10 ðk 1 Þ d þ dg f 0 ðk 1 Þ d a1 0 q3 ðk 1 Þ ¼ 1 q ðk 1 Þ f1 ðk 1 Þ dk 1 þ dgk 1 f1 ðk 1 Þ dk 1 k 1 3 dgð1 a1 Þf1 ðk 1 Þ a1 ¼ q ðk 1 Þ ðf1 ðk 1 Þ dð1 gÞk 1 Þðf1 ðk 1 Þ dk 1 Þ k 1 3 a1 dgð1 a1 Þf1 ðk 1 Þk 1 1 q3 ðk 1 Þ: ¼ k 1 a1 ðf1 ðk 1 Þ dð1 gÞk 1 Þðf1 ðk 1 Þ dk 1 Þ
Let pðk 1 Þ ¼ then
½f1 ðk 1 Þ dð1 gÞk 1 ½f1 ðk 1 Þ dk 1 1a1 ¼ A1 k a11 1 þ d2 ð1 gÞA1 dð2 gÞ; 1 k1 f ðk 1 Þk 1 "
0
p ðk 1 Þ ¼ ð1 a1 Þ 1
d2 ð1 gÞ ðA1 k 1a1 1 Þ2
# A1 k 1a1 2 < 0;
0 < k1 < k
for A1 k 1a1 1 > A1 k 1 a1 1 ¼ d. So, the function p(k1) decreases strictly on the interval ð0; k 1 Þ.
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Since, for 0 < g 6 1, k 1 a1 d a1 pðk 1 Þ ¼ ½f10 ðk 1 Þ a1 dð1 gÞ 1 d ¼ ½d þ r a1 dð1 gÞ 1 dþr f1 ðk 1 Þ > ½d þ r a1 dð1 gÞ½1 a1 > dgð1 a1 Þ and
k a1 pðk 1 Þ ¼ ½f10 ðk 1 Þ a1 dð1 gÞ 1 d 1 ¼ 0; f1 ðk 1 Þ
1Þ . From there exists a point k 1 2 ðk 1 ; k 1 Þ such that pðk 1 Þ ¼ dgð1a a1 dgð1 a1 Þ a1 q03 ðk 1 Þ ¼ 1 q ðk 1 Þ; a1 pðk 1 Þ k1 3
we have q03 ðk 1 Þ < 0; 0 < k 1 < k 1 ; q03 ðk 1 Þ ¼ 0 and q03 ðk 1 Þ > 0; k 1 < k 1 < k 1 . By (30), on the curve q3(k1), ½L ð1 a1 Þð1 gÞIðk 1 ; q3 ðk 1 ÞÞq3 ðk 1 Þ q_ ¼ < 0; _k 1 ð1 gÞIðk 1 ; q3 ðk 1 ÞÞk 1
0 < k 1 < k 1 ; k 1 6¼ k 1 :
ð32Þ
Denote the right hand side of the above equation by A(k1), then from L ð1 a1 Þð1 gÞIðk 1 ; q3 ðk 1 ÞÞ L ð1 a1 Þð1 gÞL 1 ¼ ð1 a1 Þ > a1 > ð1 gÞIðk 1 ; q3 ðk 1 ÞÞ ð1 gÞL 1g and 0 < a1
dgð1 a1 Þf1 ðk 1 Þk 1 < a1 ; ½f1 ðk 1 Þ dð1 gÞk 1 ½f1 ðk 1 Þ dk 1
0 < k 1 < k 1 ;
we have 0<
q03 ðk 1 Þ < 1; Aðk 1 Þ
0 < k 1 < k 1 ; k 1 6¼ k 1 ;
that is, q_ q03 ðk 1 Þ > Aðk 1 Þ ¼ ; k_
0 < k 1 < k 1 :
ð33Þ
Since A(k1) is the tangent value of the angle between the tangent vector starting from a point on the curve q3(k1) and k1 axis, (33) implies that the trajectory of the dynamical system (19), (20) cross the curve q3(k1) from northwest to southeast on the interval ð0; k 1 Þ and from southeast to northwest on the interval ðk 1 ; k 1 Þ. Obviously, any trajectory starting at a point on the curve q3(k) cannot come back to it. Lemma 9. The curve q3(k1) under the saddle path P1 on the interval ð0; k 1 Þ and above the saddle path P2 on the interval ðk 1 ; k 1 Þ. Proof. Obviously, the curve q3(k1) cannot cross the saddle path except at the nonzero equilibrium point. On the interval ð0; k 1 Þ, if the curve q3(k1) is above the path P1, then solution of (19), (20) with the initial value at the curve q3(k1) must converge to the nonzero equilibrium ðk 1 ; qÞ, since it cannot cross the path P1. Otherwise, the solution of (19), (20) is not unique. A contradiction. But the convergent path of the system (19), (20) is unique, the curve q3(k1) must be under the saddle path P1. Similarly, we can prove that the curve q3(k1) must be above the path P2 on the interval ðk 1 ; k 1 Þ. h
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Let M 1 ¼ fðk 1 ; qÞj0 < k 1 < k 1 ; q3 ðk 1 Þ < q < 1g; M 2 ¼ fðk 1 ; qÞjk 1 < k 1 < k 1 ; q1 ðk 1 Þ < q < q3 ðk 1 Þg; then K_ > 0 on the region M1 and K_ < 0 on the region M2. By Lemma 9, the saddle path P1 is in the region M1 and P2 is in the region M2 for k 1 < k 1 < k 1 . Therefore, we have the following theorem Theorem 4. The total capital production increases strictly on the optimal path P1 and decreases strictly on the optimal path P2 when g P 1; For 0 < g < 1, it increases on the optimal path P1 and decreases strictly on the optimal path P2 in the region M2. Now, we inquire the dynamics of the consumption production. For C ¼ F 2 ðK 2 ; L2 Þ ¼ L2 f2 ðk 2 Þ ¼ Iðk 1 ; qÞf2 ðGðk 1 ÞÞ; and ln C ¼ ln Iðk 1 ; qÞ þ ln f2 ðGðk 1 ÞÞ ¼ ln
a1 h ln q a1 ln k 1 þ ln A2 ga2 þ a2 ln k 1 ; a1 gA1
the derivative of C with respect to t is _C ¼ 1 q_ þ a2 a1 k_ 1 C: q k1
ð34Þ
If g P 1, then a2 P a1. From k_ 1 > 0; q_ < 0 on the path P1, k_ 1 < 0; q_ > 0 on the path P2 and (34), we have the following lemma. Lemma 10. If g P 1, then C_ > 0 on the saddle path P1 and C_ < 0 on the saddle path P2. Þ q_ Below, we assume that 0 < g < 1, that is, a1 > a2. Substituting k_ ¼ I 1Jðkðk11;qÞ ; q ¼ ðd þ rÞ f10 ðk 1 Þ into (34), we obtain _C ¼ f 0 ðk 1 Þ ðd þ rÞ þ a2 a1 J ðk 1 ; qÞ C: 1 k 1 I 1 ðk 1 ; qÞ
Let Qðk 1 ; qÞ ¼ f10 ðk 1 Þ ðd þ rÞ þ
a2 a1 J ðk 1 ; qÞ ; k 1 I 1 ðk 1 ; qÞ
then from
q ðk 1 Þ J ðk 1 ; qÞ ¼ L½f1 ðk 1 ; qÞ dk 1 1 2 ; q q ðk 1 Þ ; I 1 ðk 1 ; qÞ ¼ L 1 ð1 a1 Þð1 gÞ 1 q
and Lemma 7, we have Qðk 1 ; qÞ ¼ f10 ðk 1 Þ ðd þ rÞ þ
a2 a1 ½f1 ðk 1 Þ dk 1 ½q q2 ðk 1 Þ k 1 q ð1 a1 Þð1 gÞq1 ðk 1 Þ
ð35Þ
and oQðk 1 ; qÞ ða2 a1 Þðf1 ðk 1 Þ dk 1 Þ q2 ðk 1 Þ ð1 a1 Þð1 gÞq1 ðk 1 Þ ¼ < 0; 2 oq k1 ½q ð1 a1 Þð1 gÞq1 ðk 1 Þ for ~k 1 < k 1 < k 1 and (1 a)(1 g) < 1.
ð36Þ
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For any given k 1 2 ð~k 1 ; k 1 Þ, lim Qðk 1 ; qÞ ¼ f10 ðk 1 Þ ðd þ rÞ þ
q!q1 ðk 1 Þ
and lim Qðk 1 ; qÞ ¼
q!1
¼
f10 ðk 1 Þ
a2 a1 ½f1 ðk 1 Þ dk 1 ½q1 ðk 1 Þ q2 ðk 1 Þ >0 k 1 q1 ðk 1 Þ ð1 a1 Þð1 gÞq1 ðk 1 Þ
ð37Þ
a2 a1 1 0 0 ðd þ rÞ þ ½f1 ðk 1 Þ dk 1 ¼ f1 ðk 1 Þ ðd þ rÞ þ ða2 a1 Þ f ðk 1 Þ d a1 1 k1
a2 0 f ðk 1 Þ þ ða1 a2 Þd ðd þ rÞ: a1 1
For f10 ðk 1 Þ strictly decreasing on the interval ð0; k 1 Þ and a2 0 dþr f ðk 1 Þ þ ða1 a2 Þd ðd þ rÞ ¼ ða1 a2 Þ þ d < 0; a1 a1 1 ¼
there exists unique 0 < k 1 < k 1 , such that a2 0 ¼ f ðk 1 Þ þ ða1 a2 Þd ðd þ rÞ ¼ 0: a1 1 ¼ Let k 1 ¼ maxf~k 1 ; k 1 g, then for any given k 1 2 ðk 1 ; k 1 Þ, lim Qðk 1 ; qÞ > 0;
q!q1 ðk 1 Þ
lim Qðk 1 ; qÞ < 0 and
q!1
oQ < 0: ok 1
ð38Þ
Therefore, Q(k1, q) = 0 decides a curve q4(k1) on the interval ðk 1 ; k 1 Þ. From (35), Qðk 1 ; qÞ ¼ 0 for q ¼ q2 ðk 1 Þ and f10 ðk 1 Þ ¼ d þ r. So, q4 ðk 1 Þ ¼ q and the curve q4(k1) passes the equilibrium point ðk 1 ; qÞ. In fact, we can solve q4(k1) from the equation Q(k1, q) = 0 on the interval ðk 1 ; k 1 Þ.
Lemma 11. On the interval ðk 1 ; k Þ, we have ½ð1 a1 Þd þ ð1 a2 Þrða1 a2 Þq1 ðk 1 Þ ; ð1 a2 Þfa1 ½ð1 a1 þ a2 Þd þ r a2 f10 ðk 1 Þg a1 a2 f100 ðk 1 Þ q04 ðk 1 Þ ¼ þ q ðk 1 Þ: k 1 a1 ðð1 þ a2 a1 Þ þ rÞ a2 f10 ðk 1 Þ 4
q4 ðk 1 Þ ¼
ð39Þ ð40Þ
Proof. Since 1g¼
a1 a2 ; a1 ð1 a2 Þ
f1 ðk 1 Þ 1 ¼ f10 ðk 1 Þ k1 a1
and Iðk 1 ; qÞ ¼
Lq1 ðk 1 Þ ; q
q ðk 1 Þ ða1 a2 Þð1 a1 Þ q1 ðk 1 Þ I 1 ðk 1 ; qÞ ¼ L 1 ð1 a1 Þð1 gÞ 1 ¼L 1 ; q a1 ð1 a2 Þ q
we have J ðk 1 ; qÞ 1 1 0 q ðk 1 Þ ¼ f10 ðk 1 Þ d f1 ðk 1 Þ ð1 gÞðf10 ðk 1 Þ rÞ 1 Lk 1 a1 a1 q 1 1 0 a1 a2 q ðk 1 Þ ðf 0 ðk 1 Þ rÞ 1 ¼ f10 ðk 1 Þ d f ðk 1 Þ a1 a1 1 q a1 ð1 a2 Þ 1 1 1 1 a1 0 a1 a2 q1 ðk 1 Þ ¼ ½f10 ðk 1 Þ a1 d f ðk 1 Þ þ r a1 a1 1 a2 1 q 1 a2
ð41Þ
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1371
and ½f10 ðk 1 Þ ðd þ rÞI 1 ðk 1 ; qÞ 1 0 a1 1 a1 q1 ðk 1 Þ ¼ ½f1 ðk 1 Þ ðd þ rÞ : Lða1 a2 Þ a1 a1 a2 1 a2 q
ð42Þ
The equation Q(k1, q) = 0 implies that J ðk 1 ; qÞ ½f10 ðk 1 Þ ðd þ rÞI 1 ðk 1 ; qÞ : ¼ Lk 1 Lða1 a2 Þ Comparing the right hand sides of Eqs. (40) and (41), we have dð1 a1 Þ q ðk 1 Þ 1 ¼ þr 1 ½a2 f10 ðk 1 Þ þ a1 ðdð1 a1 þ a2 Þ þ rÞ: 1 a2 q a1 a2
ð43Þ
So, (39) holds. Taking logarithm on both sides of (39), we have ln q4 ðk 1 Þ ¼ ln q1 ðk 1 Þ ln½a1 ðð1 a1 þ a2 Þd þ rÞ a2 f10 ðk 1 Þ þ ln B;
ð44Þ
2 Þrða1 a2 Þ . where B ¼ ½ð1a1 Þdþð1a ð1a2 Þ Taking the derivative with respect to k1 on both sides of (44), we obtain (40). This completes the proof of Lemma 11. h
Lemma 12. The curve q4(k1) is above the curve q2(k1) on the interval ðk 1 ; k 1 Þ and under the curve q2(k1) on the interval ðk 1 ; k 1 Þ.
Proof. By (38), for a given k 1 2 ðk 1 ; k 1 Þ, the function Qðk 01 ; qÞ decreases strictly with respect to q. From
Qðk 1 ; q2 ðk 1 ÞÞ ¼ f10 ðk 1 Þ ðd þ rÞ > 0;
k 1 2 ðk 1 ; k Þ
Qðk 1 ; q2 ðk 1 ÞÞ ¼ f10 ðk 1 Þ ðd þ rÞ < 0;
k 1 2 ðk 1 ; k Þ;
and
we know that q4 ðk 1 Þ > q2 ðk 1 Þ; k 1 2 ðk 1 ; k 1 Þ and q4 ðk 1 Þ < q2 ðk 1 Þ; k 1 2 ðk 1 ; k 1 Þ. ¼
¼
If k 1 > ~k 1 , then k 1 ¼ k 1 . Denote that ¼
e 11 ¼ fðk 1 ; qÞj~k 1 < k 1 6 k 1 ; q1 ðk 1 Þ < q < 1g; M e 11 for then Qðk 1 ; qÞ P 0; ðk 1 ; qÞ 2 M ¼ a2 lim Qðk 1 ; qÞ ¼ f10 ðk 1 Þ þ ða1 a2 Þd ðd þ rÞ P 0; k 1 2 ð~k 1 ; k 1 : q!1 a1 _ e 11 since C_ ¼ Qðk 1 ; qÞC. Hence, C > 0 on the domain M ¼
For k 1 P ~k 1 , let e1 ¼ M e 11 [ M e 12 ; M e 2 ¼ fðk 1 ; qÞjk 1 < k 1 < k ; q4 ðk 1 Þ < q < q2 ðk 1 Þg; M ¼ e 12 ¼ fðk 1 ; qÞjk 1 < k 1 < k 1 ; q2 ðk 1 Þ < q < q4 ðk 1 Þg. where M Otherwise, let e 1 ¼ fðk 1 ; qÞj~k 1 < k 1 6 k 1 ; q2 ðk 1 Þ < q < q4 ðk 1 Þg; M e 2 ¼ fðk 1 ; qÞjk 1 < k 1 < k ; q4 ðk 1 Þ < q < q2 ðk 1 Þg: M
h
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e 1 and the optimal path P2 lies in the region M e 2. Lemma 13. The optimal path P1 lies in the region M ¼
e 11 in the case of k 1 P ~k 1 . We only need to prove Proof. Obviously, the optimal path lies in the region M e 12 , and P2 above the curve q4(k1) on the that the optimal path P1 is under the curve q4(k1) on the region M e e region M 2 for P1 is above the curve q2(k1) on the region M 12 and P2 is under the curve q2(k1) on the e 2. region M From (34), we have q_ ða2 a1 Þq4 ðk 1 Þ ; ¼ k1 k_
k 1 6¼ k 1
ð45Þ
on the curve q4(k1). ¼ Since a1 ð1 a1 þ a2 Þd a2 f10 ðk 1 Þ > 0 and f100 ðk 1 Þ < 0 on the interval ðk 1 ; k 1 Þ, from (40), we have a1 a2 q_ ¼ þ q ðk 1 Þ > q04 ðk 1 Þ k1 k1 4 k_
ð46Þ
on the curve q4(k1). The formula (46) implies that the trajectory of the dynamical system (19), (20) cross the curve q4(k1) from e 12 and cross the curve q4(k1) from southeast to northwest. Any northwest to southeast on the region M trajectory starting from a point on the curve q4(k1) except the point k 1 ¼ k 1 cannot come back to the curve q4(k1). e 12 , then the trajectory starting from If there is a point on the curve q4(k1) under the path P1 on the region M this point must converge to the equilibrium ðk 1 ; qÞ since it cannot cross the path P1 and come back to the curve q4(k1). Similarly, we can prove that the curve q4(k1) is above ¼ the curve under the path P2. In the same way, we can prove the result in the case of k 1 < ~k 1 . h e 1 and Q(k1, q) < 0 on the region M e 2 , we have the From Lemmas 10 and 13 and Q(k1, q) > 0 on the region M following theorem. Theorem 5. The total consumption production increases on the optimal path P1 when g P 1 and decreases on the e 1 and decreases on the optimal optimal path P2; For 0 < g < 1, it increases on the optimal path P1 in the region M path P2. 6. An example of numerical computation Here we present the results of the numerical computation under the conditions of Theorem 2. The construction parameters of the model are specified below A1 ¼ 2;
A2 ¼ 1:8;
h ¼ 0:7; r ¼ 0:05;
L ¼ 2:
The capital share ai, i = 1, 2 of the two sectors is given there group different values to require the affection of the technical rate of substitution, i.e., they are specified as a1 ¼ 0:33; 0:30; 0:30;
a2 ¼ 0:30; 0:30; 0:33:
Denote that rij, i = 1, 2, j = 1, 2, 3 are the three different technical substitutes of ith sector, then r11 ¼ 2:03;
r12 ¼ 2:33;
r13 ¼ 2:33;
r21 ¼ 2:33;
r22 ¼ 2:33;
r23 ¼ 2:03:
Correspondingly, the values of the parameters of g are g1 = 0.8701, g2 = 1, g3 = 1.1493. Figs. 2 and 3 describe the capital growths of Sector I and II. On the optimal growth path, capital allocated in Sectors I increases sharply at first, then decreases slowly to the equilibrium value. But the capital in Sector II increases strictly to the equilibrium. Although the capital in Section 1 appears to decrease after a short period, the total capital of the economy grows steadily. Fig. 4 shows the growth process of the total capital under the three different parametric values.
D. Cai / Applied Mathematics and Computation 183 (2006) 1359–1377
Fig. 2. The capital of Sector I.
Fig. 3. The capital of Sector II.
Fig. 4. The total capital.
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Fig. 5. The labor of Sector I.
Fig. 6. The labor of Sector II.
Fig. 7. The per capita capital of Sector I.
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Fig. 8. The per capita capital of Sector II.
Fig. 9. The per capita capital.
Figs. 5 and 6 describe the labor allocation between Sector I and Sector II. With the economic growth, the labor in Sector I decreases while the labor in Sector II increases steadily. This means the economy uses more and more labor and capital in the consumption product Sector. (Also see Figs. 2 and 3.) Fig. 7 depicts three growth shapes of per capita capital in Sector I. The per capita capital grows strictly although the total capital in Sector I appears to decrease after a period. This implies that the labor allocated in Sector I decreases more quickly than the total capital. Fig. 8 depicts three growth shapes of per capita capital in Sector II. Figs. 3 and 6 indicate that the capital and labor of Sector II increase simultaneously. Hence, the increasing per capita capital implies that the growth rate of the capital exceeds that of the labor force. The varieties of per capita capital of the economy are given in Fig. 9. The capital production and consumption production are presented in Figs. 10 and 11, respectively. 7. Summary From the analysis above, we can see that the ratio of the technical rate of the two sectors is a key factor that affects the allocation of labor force and capital between the capital production sector and consumption
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Fig. 10. The capital production.
Fig. 11. The consumption production.
production sector. In fact, the economy reaches the instantaneous equilibrium by sustaining two production sectors have same rates of the technical of substitute. The optimal economic growth derives from this continuously adjusting process of the capital and labor. The dynamics of the capital and labor indicate that the economy always chooses the optimal reallocation of the capital and labor between the capital production sector and consumption production sector such that the total capital and total consumption steadily grows along the optimal path, so does the per capita capital and consumption. From the example of the numerical computation, we see that the labor used in Sector I decreases strictly with the economy growth. But the growth of capital used in Sector I appears to have three different situations with three different values of the parameter g. When the technical of substitute in Sector I is low(r11 = 2.03) and the technical of substitute in Sector II is high(r21 = 2.33), the capital used in Sector I grows rapidly then decreases to a steady value and the capital used in Sector II increases strictly. There growths of capital used in Sector I and Sector II are similar for the case where the two sectors have same technical of substitute. When the technical of substitute in Sector I is high and in Sector II is low, the growth of the capital used in Sector I increases rapidly then sustains to steady value. Comparing Fig. 2 with Fig. 3 and Fig. 5 with Fig. 6, we view that more and more capital and labor are allocated into Sector II to produce consumption production with the economy growth. This can be used to explain the ratio of labor in manufacture decreases with economy growth.
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Figs. 10 and 11 show that the economy has the highest capital production when the technical of substitute of Sector I is low and the technical of substitute of Sector II is high but does not reach the highest consumption production. Conversely, the economy has the highest consumption production when the technical of substitute of Sector I is low and the technical of substitute of Sector II is high but does not reach the highest capital production. References [1] J.R. Barro, I.X. Martin, Economic Growth, McGraw-Hill Companies, Inc., New York, 1995. [2] E. Ryder Jr, Optimal accumulation in a two-sector neoclassical economy with non-shiftable capital, Journal of Political Economy 77 (1969) 665–683. [3] A. Ladron-De-Guevara, S. Ortigueir, M.S. Santos, A two-Sector model of endogenous growth with leisure, Review of Economic Studies 66 (1999) 609–631. [4] E. Burmeister, A.R. Dobell, Mathematical Theories of Economic Growth, Gregg Revivals, Hampshire, 1993.
A two-sector economic growth model with optimal labor and capital allocation q Donghan Cai College of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China
Abstract In this paper, a two-sector growth model with optimal labor force and capital allocation is given. By treating the quantities of the labor force and capital in the consumption production sector as the control variables in the model, we obtain a two-dimension dynamical system from solving the household utility maximum problem. It is proved that the system has a unique nonzero equilibrium which is a saddle, so there exists an optimal labor force and capital allocation process in the economic growth. The capital accumulation and consumption production strictly increase along the growth path. At the end of this paper, a numerical computation is given to present the allocation process of capital and labor. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Two-sector model; Dynamical optimization; Dynamical system; Labor force and capital allocation; Optimal growth path
1. Introduction One of the most important problems in economic growth theory is how the behavior of consumption affects the long-term economic growth. In the Solow model, the aggregate consumption is the fraction of the output, and the saving rate is fixed. This model implies that reducing the proportion of consumption can accelerate the accumulation of the per capita capital, but it may be not good for the per capita consumption in the long-term view, since there exists so called dynamic inefficiency [1]. In the Ramsey model, the consumption is endogenously decided by an one sector optimal model. One of the notable characters in the Solow model and Ramsey model is that the capital good and consumption good are treated as the same and there does not exist a separate consumption production sector. By the setup of a two-sector economic growth model with separated production good sector and consumption good sector, this paper analyzes the optimal labor force and capital reallocation in the long-term economic growth. We inquire how an economy allocates optimally its resource between the capital production and consumption production in the economic growth process. Harl E. Ryder had presented a model to discuss the optimal accumulation in a two-sector neoclassical economy with nonshiftable capital. The model implies there exists a optimal labor and capital reallocating process. q
Supported by National Natural Science Foundation of China (79970104). E-mail address: [email protected]
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.148
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But the existence and uniqueness of the optimal growth path and the equilibrium is not given. So, the dynamics of the model is unclear [2]. The use of a two-sector framework (or its variants) is broadly used in the theoretical analysis. K. Marx, J.A. Hobson, A. Spiethoff, G. Cassel, and F.A. Hayek, among others, emphasized sectoral imbalance between the consumption and investment good industries as a cause of business cycles or cries [3]. A two-sector growth model with capital and labor shiftable that is presented in [4] gave a solow type model. In our paper, the economy sustains the optimal growth by continuously adjusting the allocation of labor force and capital between the two sectors. In the model, the production of consumption good and capital good increases strictly along the optimal growth path, that is to say, it is not needed to sacrifice the present consumption to get in return the aggregate capital growth. The remainder of the paper is organized as follows. In Section 2, the basic assumptions and the model are presented. The existence and uniqueness of nonzero equilibrium of the model are proved in Section 3. In Section 4, we analysis the dynamics of the model. The dynamics of the capital and consumption production is given in Section 5. In Section 6, an example of numerical computation is presented. A summary is given in Section 7.
2. Setup of the model Consider an economy with a capital production sector and a consumption production sector. They are called Sector I and Sector II, respectively. In each sector all the firms have an identical neoclassical technology and produce output using labor and capital. The capital goods are used in both sectors. The production functions, Fi(Ki, Li), i = 1, 2, are neoclassical and satisfy the Inada conditions, where Ki, Li are the ith sector’s capital and labor force. So, the new investment and consumption production are K_ ¼ F 1 ðK 1 ; L1 Þ dK;
C ¼ F 2 ðK 2 ; L2 Þ;
ð1Þ
where K = K1 + K2. The intensive form of the production function of ith sector is Ki ; 1 ; i ¼ 1; 2 fi ðk i Þ ¼ F i Li and satisfies fi ð0Þ ¼ 0; f i ð1Þ ¼ 1; fi0 ð0Þ ¼ 1; fi0 ð1Þ ¼ 0; fi0 ðk i Þ > 0; fi00 ðk i Þ < 0; k i > 0; i ¼ 1; 2:
ð2Þ
The utility function U(C) is increasing in C and concave, U 0 (C) > 0, U 0 (C) < 0, satisfies that lim U 0 ðCÞ ¼ þ1;
C!0
lim U 0 ðCÞ ¼ 0:
C!1
We further assume that the total labor supply is constant, that is L1 þ L2 ¼ L ¼ const: The economy chooses optimal capital and labor allocation such that the total utility reaches to maximum, that is, it searches for Z 1 max U ðCðtÞÞ expðrtÞ dt; ð3Þ 0
subject to (1); where r > 0 is the rate of time preference. The current value Hamiltonian to solve the optimization problem (3) is H ðK; K 2 ; L2 ; qÞ ¼ U ðCÞ þ q½F 1 ðK K 2 ; L L2 Þ dK:
ð4Þ
D. Cai / Applied Mathematics and Computation 183 (2006) 1359–1377
The first order conditions and transversity are oH ¼ F 1 ðK K 2 ; L L2 Þ dK; K_ ¼ oq oH oF 1 ðK 1 ; L1 Þ ¼ q ðd þ rÞ ; q_ ¼ rq oK oK 1 oH oF 2 ðK 2 ; L2 Þ F 1 ðK 1 ; L1 Þ ¼ U 0 ðCÞ q ; 0¼ oK 2 oK 2 oK 1 oH oF 2 ðK 2 ; L2 Þ oF 1 ðK 1 ; L1 Þ ¼ U 0 ðCÞ q ; 0¼ oL2 oL2 oL1 lim expðrtÞqðtÞKðtÞ ¼ 0: t!1
1361
ð5Þ ð6Þ ð7Þ ð8Þ ð9Þ
By (7) and (8), oF 2 oF 1 oK 2 oK 1 ¼ : oF 2 oF 1 oL2 oL1
ð10Þ
Since oF i ¼ fi0 ðk i Þ; oK i
oF i ¼ fi ðk i Þ k i fi0 ðk i Þ; oLi
i ¼ 1; 2;
Eq. (10) turns into f2 ðk 2 Þ k 2 f 0 ðk 2 Þ f1 ðk 1 Þ k 1 f10 ðk 1 Þ ¼ : f20 ðk 2 Þ f10 ðk 1 Þ
ð11Þ
Let Gi ðk i Þ ¼
fi ðk i Þ k i fi0 ðk i Þ ; fi0 ðk i Þ
i ¼ 1; 2;
ð12Þ
then we have Lemma 1. The functions Gi(ki), i = 1, 2 are strictly increasing and satisfy Gi ðk i Þ > 0;
lim Gi ðk i Þ ¼ 0;
lim Gi ðk i Þ ¼ 1:
k i !1
k i !0
Proof. Since k i ðfi ðk i Þ=k i fi0 ðk i ÞÞ fi ðk i Þfi00 ðk i Þ 0 > 0; G ðk Þ ¼ > 0; i ¼ 1; 2; i i 2 fi0 ðk i Þ ½fi0 ðk i Þ the functions Gi(ki), i = 1, 2 are strictly increasing. It is not difficult to check that limki !0 Gi ðk i Þ ¼ 0. If limki !1 Gi ðk i Þ 6¼ 1; i ¼ 1; 2, then there exists finite Gi > 0, i = 1, 2 such that limki !1 Gi ðk i Þ ¼ Gi . By (12), Gi ðk i Þ ¼
fi0 ðk i Þ 1 ¼ ; fi ðk i Þ Gi ðk i Þ þ k i
i ¼ 1; 2:
For k i > k 0i > 0; i ¼ 1; 2, we have "Z # ki dui 0 fi ðk i Þ ¼ fi ðk i Þ exp : k 0i Gi ðui Þ þ ui Differentiation gives fi0 ðk i Þ
fi ðk 0i Þ ¼ exp Gi ðk i Þ þ k i
"Z
ki
k 0i
# dui ; Gi ðui Þ þ ui
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which implies that "Z ki fi0 ðk i Þ exp k 0i
# Gi ðui Þ Gi fi ðk 0i ÞðGi þ k i Þ dui ¼ : ðGi ðui Þ þ ui ÞðGi þ ui Þ ðGi þ k 0i ÞðGi ðk i Þ þ k i Þ
ð13Þ
Since limki !1 Gi ðk i Þ ¼ Gi , for a fixed > 0, there exist k 0i such that Gi ðk i Þ Gi < ; k i > k 0i . Therefore, the integration Z 1 Gi ðui Þ Gi dui ðGi ðui Þ þ ui ÞðGi þ ui Þ k 0i converges and left hand side of (13) approaches to zero as ki tends to infinite. But the right hand side of (13) approaches
fi ðk 0i Þ . Gi þk 0i
This is a contradiction. h
Lemma 2. Eq. (11) decides a function k2(k1), which satisfies dk 2 ðk 1 Þ > 0; dk 1
lim k 2 ðk 1 Þ ¼ 0;
k 1 !0
lim k 2 ðk 1 Þ ¼ 1:
k 1 !1
Proof. By Lemma 1, Gi(ki), i = 1, 2 are strictly increasing functions and are invertible, so we have k 2 ðk 1 Þ ¼ G1 2 ðG1 ðk 1 ÞÞ and dk 2 ðk 1 Þ G01 ðk 1 Þ > 0; ¼ 0 dk 1 G2 ðk 2 Þ
lim k 2 ðk 1 Þ ¼ 0;
k 1 !0
lim k 2 ðk 1 Þ ¼ 1:
k 1 !1
This completes the proof of Lemma 2. h From Lemma 2, we have K2 K1 k2 ¼ ¼G ¼ Gðk 1 Þ; L2 L1 where GðÞ ¼ G1 2 ðG1 ðÞÞ. Put C = F2(K2, L2) = L2f2(k2) and k2 = G(k1) into (7), then U 0 ðL2 f2 ðGðk 1 ÞÞÞ ¼
qf 01 ðk 1 Þ : f20 ðGðk 1 ÞÞ
ð14Þ
ð15Þ
Since U 0 (Æ) < 0, the function U 0 (Æ) is invertible. Denote that q1 ðk 1 Þ ¼ U 0 Lf2 ðGðk 1 ÞÞ and
f20 ðGðk 1 ÞÞ ; f10 ðk 1 Þ
1 qf 01 ðk 1 Þ 0 1 Iðk 1 ; qÞ ¼ U ; f2 ðGðk 1 ÞÞ f20 ðGðk 1 ÞÞ
ð16Þ
then, for any given k1 > 0, 0 < I(k1, q) < L if and only if q1(k1) < q < 1. Set X ¼ fðk 1 ; qÞj0 < k 1 < 1; q1 ðk 1 Þ < q < 1g; we have the following lemma from (16). Lemma 3. Eqs. (7) and (8) decide a function L2 ¼ Iðk 1 ; qÞ;
ðk 1 ; qÞ 2 X Rþ 2;
which satisfies lim Iðk 1 ; qÞ ¼ L;
q!q1 ðk 1 Þ
lim Iðk 1 ; qÞ ¼ 0:
q!1
ð17Þ
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1363
Therefore, in the domain X, we have L1 ¼ L L2 ¼ L Iðk 1 ; qÞ; K 1 ¼ L1 k 1 ¼ ðL Iðk 1 ; qÞÞk 1 ; K 2 ¼ L2 k 2 ¼ Iðk 1 ; qÞGðk 1 Þ and K ¼ K 1 þ K 2 ¼ ðL Iðk 1 ; qÞÞk 1 þ Iðk 1 ; qÞGðk 1 Þ:
ð18Þ
The derivation of K with respect to t is oIðk 1 ; qÞ oIðk 1 ; qÞ _ K_ ¼ ½L þ ðGðk 1 Þ k 1 Þ þ ðG0 ðk 1 Þ 1ÞIðk 1 ; qÞk_ 1 þ ðGðk 1 Þ k 1 Þ q: ok 1 oq Suppose that I 1 ðk 1 ; qÞ ¼ L þ ðGðk 1 Þ k 1 Þ
oIðk 1 ; qÞ þ ðG0 ðk 1 Þ 1ÞIðk 1 ; qÞ; ok 1
ðk 1 ; qÞ 2 X;
then, from limq!1I(k1, q) = 0 for any given k1, we have the following lemma. Lemma 4. If the function I(k1, q) satisfies lim
q!1
oIðk 1 ; qÞ ¼ 0; ok 1
k 1 > 0;
then, for any given k1 > 0, I1(k1, q) > 0 when q is large enough. Example. When U(C) is constant-relative-risk-aversion utility function I(k1, q) is
C 1h , 1h
0 < h < 1 or logarithm utility h ln C, the
0 1 1 f2 ðGðk 1 ÞÞ h Iðk 1 ; qÞ ¼ f2 ðGðk 1 ÞÞ qf 01 ðk 1 Þ or Iðk 1 ; qÞ ¼
hf20 ðGðk 1 ÞÞ ; 0 qf 1 ðk 1 Þf2 ðGðk 1 ÞÞ
respectively. Both satisfy the condition of Lemma 4. If (k1, q) is a point in X such that I1(k1, q) 5 0, then there exists a neighborhood X1 X such that I1(k1, q) 5 0, (k1, q) 2 X1. Let X ¼ fðk 1 ; qÞjI 1 ðk 1 ; qÞ 6¼ 0; ðk 1 ; qÞ 2 Xg; then X is an open subset in X and not empty under condition of Lemma 4. Therefore, with the condition of Lemma 4, we obtain the following dynamical system in X from the first order conditions (5)–(8) k_ 1 ¼
1 ½ðL Iðk 1 ; qÞÞf1 ðk 1 Þ I 2 ðk 1 ; qÞ dI 3 ðk 1 ; qÞ; I 1 ðk 1 ; qÞ
q_ ¼ ½ðd þ rÞ f10 ðk 1 Þq; where oIðk 1 ; qÞ ; oq I 3 ðk 1 ; qÞ ¼ ðL Iðk 1 ; qÞÞk 1 þ Iðk 1 ; qÞGðk 1 Þ:
I 2 ðk 1 ; qÞ ¼ ðGðk 1 Þ k 1 Þ½ðd þ rÞ f10 ðk 1 Þq
ð19Þ ð20Þ
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3. The existence and uniqueness of equilibrium Let J 1 ðk 1 ; qÞ ¼ ½L Iðk; qÞf1 ðk 1 Þ dI 3 ðk 1 ; qÞ ¼ ½L Iðk 1 ; qÞf1 ðk 1 Þ d½ðL Iðk 1 ; qÞÞk 1 þ Iðk 1 ; qÞGðk 1 Þ ¼ L½f1 ðk 1 Þ dk 1 ½f1 ðk 1 Þ dk 1 þ dGðk 1 ÞIðk 1 ; qÞ; J 2 ðk 1 Þ ¼ ðd þ rÞ f10 ðk 1 Þ; then the dynamical system (19), (20) turns into 1 k_ 1 ¼ J 1 ðk 1 ; qÞ I 2 ðk 1 ; qÞ ; I 1 ðk 1 ; qÞ q_ ¼ J 2 ðk 1 Þq:
ð21Þ ð22Þ
Þ 2 X is an equilibrium of the dynamical system (19), (20) if and only if A point ðk 1 ; q J 1 ðk 1 ; qÞ ¼ J 2 ðk 1 Þ ¼ 0;
ð23Þ
q1 Þ ¼ 0 if J 2 ðk 1 Þ ¼ 0. for I 2 ðk 1 ; Since J2(k1) is a strictly increasing function from negative infinite to d + r > 0, there exists a unique point k 1 > 0 such that J 2 ðk 1 Þ ¼ 0. Let N ðqÞ ¼ J 1 ðk 1 ; qÞ, then oN ðqÞ oIðk 1 ; qÞ ¼ ½f1 ðk 1 Þ dk 1 þ dGðk 1 Þ > 0; oq oq h i k1 ;qÞ if oIðoq < 0; q > q1 ðk 1 Þ for f1 ðk 1 Þ dk 1 ¼ k 1 f1kðk1 1 Þ d > rk 1 > 0 . So, we have the following lemma. Lemma 5. If 0; J 2 ðk 1 Þ ¼ 0.
oIðk 1 ;qÞ oq
< 0; ðk 1 ; qÞ 2 X, then there exists a unique point ðk 1 ; qÞ 2 X such that J 1 ðk 1 ; qÞ ¼
Proof. Since N(q) is strictly increasing in X and satisfies lim N ðqÞ ¼ dIðk 1 ; q1 ðk 1 ÞÞGðk 1 Þ < 0; q!q1 ðk 1 Þ
lim N ðqÞ ¼ L½f1 ðk 1 Þ dk 1 > 0:
q!1
Hence, there exists a unique point q such that N ðqÞ ¼ 0, i.e., J 1 ðk 1 ; qÞ ¼ 0.
h
Theorem 1. If the set X is not empty and ðk 1 ; qÞ belongs to it, the dynamic system (19), (20) have a nonzero equilibrium in a neighborhood of ðk 1 ; qÞ. i When F i ðK i ; Li Þ ¼ Ai K ai i L1a ; 0 < ai < 1; i ¼ 1; 2 are both Cobb–Douglas production functions, then i ai 1a2 1 fi ðk i Þ ¼ Ai k i . By (11), a2 k 2 ¼ 1a k 1 . So, we have, a1
k 2 ¼ Gðk 1 Þ ¼ gk 1 ; 1 where g ¼ 1a 1a2
ð24Þ
a2 . a1
Lemma 6. If U(c) = h ln C, then in the set X, a2 h a1 k ; a1 gA1 L 1 a2 h 1 a1 q k1 ; Iðk 1 ; qÞ ¼ a1 gA1 I 1 ðk 1 ; qÞ ¼ L ð1 a1 Þð1 gÞIðk 1 ; qÞ; q1 ðk 1 Þ ¼
under the Cobb–Douglas production technology.
ð25Þ ð26Þ ð27Þ
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i Theorem 2. If U(C) = h ln C, F i ðK i ; Li Þ ¼ Ai K ai i L1a ; 0 < ai < 1; i ¼ 1; 2, then the dynamical system (19), (20) i has a unique nonzero equilibrium in the domain X.
1 a2 Proof. If a1 6 a2, then 1 g ¼ a1að1a 6 0 and 2Þ
I 1 ðk 1 ; qÞ ¼ L ð1 a1 Þð1 gÞIðk 1 ; qÞ P L > 0: If a1 > a2, then 0 < g < 1 for
1a1 a1
2 < 1a . By 0 < I(k1, q) < L and (27), for (k1, q) 2 X, a2
I 1 ðk 1 ; qÞ ¼ L ð1 a1 Þð1 gÞIðk 1 ; qÞ > L½1 ð1 a1 Þð1 gÞ > 0: So, X ¼ X. By Lemma 5, the dynamical system (19), (20) has a unique nonzero equilibrium point. This completes the proof of the theorem. h 4. The dynamics of the model In this section, we inquire the dynamics of the model under the conditions of Theorem 2. f ðk Þ Let k 1 be the unique positive zero point of the equation f1(k1) dk1 = 0, then k 1 < k 1 for 1k 1 ¼ d < d þ r 1 f ðk Þ and k > n > k 1 , where n is the unique point that satisfies f 0 ðnÞ ¼ 1 1 . 1
1
Let
k1
M ¼ fðk 1 ; qÞj0 < k 1 < k 1 ; q1 ðk 1 Þ < q < 1g and J ðk 1 ; qÞ ¼ J 1 ðk 1 ; qÞ I 2 ðk 1 ; qÞ ¼ L½f1 ðk 1 Þ dk 1 ½f1 ðk 1 Þ dk 1 þ dgk 1 þ ðg 1Þðf10 ðk 1 Þ ðd þ rÞÞk 1 Iðk 1 ; qÞ; then J ðk 1 ; qÞ ¼ L½f1 ðk 1 Þ dk 1 ½f1 ðk 1 Þ dk 1 þ dgk 1 þ ðg 1Þðf10 ðk 1 Þ ðd þ rÞÞk 1 Lq1 ðk 1 Þq1 for Iðk 1 ; qÞ ¼
a2 h 1 a a2 h a1 k ¼ Lq1 ðk 1 Þq1 : q k 1 ¼ Lq1 a1 gA1 a1 gA1 L 1
So, we have the following lemma. Lemma 7. The equation J(k1, q) = 0 determines a curve q2(k1) in the domain M X which satisfies that (1) if g = 1, then q2 ðk 1 Þ > q1 ðk 1 Þ; k 1 2 ð0; k 1 Þ; limk1 !k1 q2 ðk 1 Þ ¼ 1; (2) there exists ~k 1 2 ð0; k 1 Þ such that q1 ð~k 1 Þ ¼ q2 ð~k 1 Þ; limk1 !k1 q2 ðk 1 Þ ¼ 1; q1 ðk 1 Þ > q2 ðk 1 Þ; k 1 2 ð0; ~k 1 Þ and q1 ðk 1 Þ < q2 ðk 1 Þ; k 1 2 ð~k 1 ; k 1 Þ if 0 < g < 1; and q2 ðk 1 Þ > q1 ðk 1 Þ; (3) if g > 1, there exists ^k 1 2 ðk 1 ; k 1 Þ; q2 ðk 1 Þ > q1 ðk 1 Þ; k 1 2 ð0; ^k 1 Þ for d < a1rðg1Þ ðg1Þþ1 rðg1Þ q ðk Þ ¼ 1 if d > and lim . k 1 2 ð0; k 1 Þ for d P a1rðg1Þ k 1 !k 1 2 1 ðg1Þþ1 a1 ðg1Þþ1 Proof. Since f1 ðk 1 ; qÞ dk 1 > 0; k 1 2 ð0; k 1 Þ, the equation J(k1, q) = 0 decides a function dg þ ðg 1Þðf10 ðk 1 Þ ðd þ rÞÞ q2 ðk 1 Þ ¼ q1 ðk 1 Þ 1 þ k1 : f1 ðk 1 Þ dk 1 Let gðk 1 Þ ¼ dg þ ðg 1Þðf10 ðk 1 Þ ðd þ rÞÞ;
ð28Þ
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then gðk 1 Þ ¼ dg > 0 and
8 > < > 0; if 0 < g < 1; dgðk 1 Þ ¼ ðg 1Þf100 ðk 1 Þ ¼ 0; if g ¼ 1; > dk 1 : < 0; if g > 1:
If g = 1, then g(k1) = dg > 0. So, by (28), dgk 1 q2 ðk 1 Þ ¼ 1 þ q ðk 1 Þ > q1 ðk 1 Þ; f1 ðk 1 Þ dk 1 1
ð29Þ
k 1 2 ð0; k 1 Þ
and limk1 !k1 q2 ðk 1 Þ ¼ 1 for f1 ðk 1 Þ dk 1 ¼ 0. If 0 < g < 1, then from g1 ðk 1 Þ ¼ dg > 0; limk1 !0 gðk 1 Þ ¼ 1 and (29), there exists a unique point ~k 1 2 ð0; k 1 Þ such that gð~k 1 Þ ¼ 0 and gðk 1 Þ > 0; k 1 2 ð~k 1 ; k 1 Þ. So, by (28), (2) holds. If g > 1, then by gðk 1 Þ ¼ dg þ ðg 1Þðf10 ðk 1 Þ ðd þ rÞÞ ¼ dg þ ðg 1Þða1 d ðd þ rÞÞ; we have 8 rðg 1Þ > > ; < 0; if d < > > a1 ðg 1Þ þ 1 > > > < rðg 1Þ ; gðk 1 Þ ¼ 0; if d ¼ > a1 ðg 1Þ þ 1 > > > > rðg 1Þ > > : > 0; if d > : a1 ðg 1Þ þ 1 From (29) and gðk 1 Þ ¼ dg, if d < a1rðg1Þ , then there exists a point ^k 1 2 ðk 1 ; k 1 Þ such that ðg1Þþ1 rðg1Þ q1 ðk 1 Þ < q2 ðk 1 Þ; k 1 2 ð0; ^k 1 Þ. If d P a1 ðg1Þþ1, then q1 ðk 1 Þ < q2 ðk 1 Þ; k 1 2 ð0; k 1 Þ and limk1 !k1 q2 ðk 1 Þ ¼ 1, if . This completes the proof of Lemma 7. h d > a1rðg1Þ ðg1Þþ1 By Lemma 7, we know that the domain M R2þ is divided into four parts by the line k 1 ¼ k 1 and the curve q2(k1). The function f10 ðk 1 Þ is strictly decreasing from infinite to zero. On the left of the line k 1 ¼ k 1 , q_ < 0 and q_ > 0 and on the right the line k 1 ¼ k 1 in the domain M. Above the curve q2(k1), k_ > 0, and k_ < 0 below the curve q2(k1) in the domain M ¼ fðk 1 ; qÞj~k 1 < k 1 < ^k 1 ; q1 ðk 1 Þ < q < 1g M: qÞ is a saddle point by phase portrait analysis. One of the optiTherefore, the unique nonzero equilibrium ðk 1 ; mal path is on the upper-left part and the other is on the low-right part. Therefore, we obtain the following theorem.
Fig. 1. The dynamics of the model.
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Theorem 3. The unique nonzero equilibrium of the dynamical system (19), (20) is a saddle and the model has optimal growth path. The phase portrait of the dynamical system (19), (20) is described in Fig. 1. 5. The dynamics of the capital and consumption production Now we inquire the dynamics of the capital K and the consumption production C along the optimal path. _ we have By the representative of K, Iðk 1 ; qÞ _ K_ ¼ ½L ð1 a1 Þð1 gÞIðk 1 ; qÞk_ 1 ðg 1Þk 1 q; ð30Þ q for oI ¼ a1 k 1 1 Iðk 1 ; qÞ; ok 1
oI ¼ q1 Iðk 1 ; qÞ oq
from (26). Obviously, if g P 1, we have K_ > 0 on P1 and K_ < 0 on path P2 for k_ 1 > 0; q_ < 0 on the path P1 and _k 1 < 0; q_ 1 > 0 on the path P2. So, we have Lemma 8. If g P 1, then K_ > 0 on the saddle path P1 and K_ < 0 on the saddle path P2. Now, we assume that 0 < g < 1. Substituting (21), (22) into (30), we obtain K_ ¼ J ðk 1 ; qÞ þ ðg 1Þ½f10 ðk 1 Þ ðd þ rÞk 1 Iðk 1 ; qÞ ¼ L½f1 ðk 1 Þ dk 1 ½f1 ðk 1 Þ dk 1 þ dgk 1 Iðk 1 ; qÞ ¼ L½f1 ðk 1 Þ dk 1 L½f1 ðk 1 Þ dk 1 þ dgk 1 Hence, there exists a curve dgk 1 q3 ðk 1 Þ ¼ 1 þ q ðk 1 Þ; f1 ðk 1 Þ dk 1 1
q1 ðk 1 Þ : q
0 < k1 < k;
ð31Þ
on which K_ ¼ 0. For ln q3 ðk 1 Þ ¼ ln½f1 ðk 1 Þ dk 1 þ dgk 1 ln½f1 ðk 1 Þ dk 1 þ ln q1 ðk 1 Þ and
d ln q1 ðk 1 Þ dk 1
q0 ðk Þ
¼ q1 ðk11 Þ ¼ ak11 . It is not difficult to check that 1 f10 ðk 1 Þ d þ dg f 0 ðk 1 Þ d a1 0 q3 ðk 1 Þ ¼ 1 q ðk 1 Þ f1 ðk 1 Þ dk 1 þ dgk 1 f1 ðk 1 Þ dk 1 k 1 3 dgð1 a1 Þf1 ðk 1 Þ a1 ¼ q ðk 1 Þ ðf1 ðk 1 Þ dð1 gÞk 1 Þðf1 ðk 1 Þ dk 1 Þ k 1 3 a1 dgð1 a1 Þf1 ðk 1 Þk 1 1 q3 ðk 1 Þ: ¼ k 1 a1 ðf1 ðk 1 Þ dð1 gÞk 1 Þðf1 ðk 1 Þ dk 1 Þ
Let pðk 1 Þ ¼ then
½f1 ðk 1 Þ dð1 gÞk 1 ½f1 ðk 1 Þ dk 1 1a1 ¼ A1 k a11 1 þ d2 ð1 gÞA1 dð2 gÞ; 1 k1 f ðk 1 Þk 1 "
0
p ðk 1 Þ ¼ ð1 a1 Þ 1
d2 ð1 gÞ ðA1 k 1a1 1 Þ2
# A1 k 1a1 2 < 0;
0 < k1 < k
for A1 k 1a1 1 > A1 k 1 a1 1 ¼ d. So, the function p(k1) decreases strictly on the interval ð0; k 1 Þ.
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Since, for 0 < g 6 1, k 1 a1 d a1 pðk 1 Þ ¼ ½f10 ðk 1 Þ a1 dð1 gÞ 1 d ¼ ½d þ r a1 dð1 gÞ 1 dþr f1 ðk 1 Þ > ½d þ r a1 dð1 gÞ½1 a1 > dgð1 a1 Þ and
k a1 pðk 1 Þ ¼ ½f10 ðk 1 Þ a1 dð1 gÞ 1 d 1 ¼ 0; f1 ðk 1 Þ
1Þ . From there exists a point k 1 2 ðk 1 ; k 1 Þ such that pðk 1 Þ ¼ dgð1a a1 dgð1 a1 Þ a1 q03 ðk 1 Þ ¼ 1 q ðk 1 Þ; a1 pðk 1 Þ k1 3
we have q03 ðk 1 Þ < 0; 0 < k 1 < k 1 ; q03 ðk 1 Þ ¼ 0 and q03 ðk 1 Þ > 0; k 1 < k 1 < k 1 . By (30), on the curve q3(k1), ½L ð1 a1 Þð1 gÞIðk 1 ; q3 ðk 1 ÞÞq3 ðk 1 Þ q_ ¼ < 0; _k 1 ð1 gÞIðk 1 ; q3 ðk 1 ÞÞk 1
0 < k 1 < k 1 ; k 1 6¼ k 1 :
ð32Þ
Denote the right hand side of the above equation by A(k1), then from L ð1 a1 Þð1 gÞIðk 1 ; q3 ðk 1 ÞÞ L ð1 a1 Þð1 gÞL 1 ¼ ð1 a1 Þ > a1 > ð1 gÞIðk 1 ; q3 ðk 1 ÞÞ ð1 gÞL 1g and 0 < a1
dgð1 a1 Þf1 ðk 1 Þk 1 < a1 ; ½f1 ðk 1 Þ dð1 gÞk 1 ½f1 ðk 1 Þ dk 1
0 < k 1 < k 1 ;
we have 0<
q03 ðk 1 Þ < 1; Aðk 1 Þ
0 < k 1 < k 1 ; k 1 6¼ k 1 ;
that is, q_ q03 ðk 1 Þ > Aðk 1 Þ ¼ ; k_
0 < k 1 < k 1 :
ð33Þ
Since A(k1) is the tangent value of the angle between the tangent vector starting from a point on the curve q3(k1) and k1 axis, (33) implies that the trajectory of the dynamical system (19), (20) cross the curve q3(k1) from northwest to southeast on the interval ð0; k 1 Þ and from southeast to northwest on the interval ðk 1 ; k 1 Þ. Obviously, any trajectory starting at a point on the curve q3(k) cannot come back to it. Lemma 9. The curve q3(k1) under the saddle path P1 on the interval ð0; k 1 Þ and above the saddle path P2 on the interval ðk 1 ; k 1 Þ. Proof. Obviously, the curve q3(k1) cannot cross the saddle path except at the nonzero equilibrium point. On the interval ð0; k 1 Þ, if the curve q3(k1) is above the path P1, then solution of (19), (20) with the initial value at the curve q3(k1) must converge to the nonzero equilibrium ðk 1 ; qÞ, since it cannot cross the path P1. Otherwise, the solution of (19), (20) is not unique. A contradiction. But the convergent path of the system (19), (20) is unique, the curve q3(k1) must be under the saddle path P1. Similarly, we can prove that the curve q3(k1) must be above the path P2 on the interval ðk 1 ; k 1 Þ. h
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Let M 1 ¼ fðk 1 ; qÞj0 < k 1 < k 1 ; q3 ðk 1 Þ < q < 1g; M 2 ¼ fðk 1 ; qÞjk 1 < k 1 < k 1 ; q1 ðk 1 Þ < q < q3 ðk 1 Þg; then K_ > 0 on the region M1 and K_ < 0 on the region M2. By Lemma 9, the saddle path P1 is in the region M1 and P2 is in the region M2 for k 1 < k 1 < k 1 . Therefore, we have the following theorem Theorem 4. The total capital production increases strictly on the optimal path P1 and decreases strictly on the optimal path P2 when g P 1; For 0 < g < 1, it increases on the optimal path P1 and decreases strictly on the optimal path P2 in the region M2. Now, we inquire the dynamics of the consumption production. For C ¼ F 2 ðK 2 ; L2 Þ ¼ L2 f2 ðk 2 Þ ¼ Iðk 1 ; qÞf2 ðGðk 1 ÞÞ; and ln C ¼ ln Iðk 1 ; qÞ þ ln f2 ðGðk 1 ÞÞ ¼ ln
a1 h ln q a1 ln k 1 þ ln A2 ga2 þ a2 ln k 1 ; a1 gA1
the derivative of C with respect to t is _C ¼ 1 q_ þ a2 a1 k_ 1 C: q k1
ð34Þ
If g P 1, then a2 P a1. From k_ 1 > 0; q_ < 0 on the path P1, k_ 1 < 0; q_ > 0 on the path P2 and (34), we have the following lemma. Lemma 10. If g P 1, then C_ > 0 on the saddle path P1 and C_ < 0 on the saddle path P2. Þ q_ Below, we assume that 0 < g < 1, that is, a1 > a2. Substituting k_ ¼ I 1Jðkðk11;qÞ ; q ¼ ðd þ rÞ f10 ðk 1 Þ into (34), we obtain _C ¼ f 0 ðk 1 Þ ðd þ rÞ þ a2 a1 J ðk 1 ; qÞ C: 1 k 1 I 1 ðk 1 ; qÞ
Let Qðk 1 ; qÞ ¼ f10 ðk 1 Þ ðd þ rÞ þ
a2 a1 J ðk 1 ; qÞ ; k 1 I 1 ðk 1 ; qÞ
then from
q ðk 1 Þ J ðk 1 ; qÞ ¼ L½f1 ðk 1 ; qÞ dk 1 1 2 ; q q ðk 1 Þ ; I 1 ðk 1 ; qÞ ¼ L 1 ð1 a1 Þð1 gÞ 1 q
and Lemma 7, we have Qðk 1 ; qÞ ¼ f10 ðk 1 Þ ðd þ rÞ þ
a2 a1 ½f1 ðk 1 Þ dk 1 ½q q2 ðk 1 Þ k 1 q ð1 a1 Þð1 gÞq1 ðk 1 Þ
ð35Þ
and oQðk 1 ; qÞ ða2 a1 Þðf1 ðk 1 Þ dk 1 Þ q2 ðk 1 Þ ð1 a1 Þð1 gÞq1 ðk 1 Þ ¼ < 0; 2 oq k1 ½q ð1 a1 Þð1 gÞq1 ðk 1 Þ for ~k 1 < k 1 < k 1 and (1 a)(1 g) < 1.
ð36Þ
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For any given k 1 2 ð~k 1 ; k 1 Þ, lim Qðk 1 ; qÞ ¼ f10 ðk 1 Þ ðd þ rÞ þ
q!q1 ðk 1 Þ
and lim Qðk 1 ; qÞ ¼
q!1
¼
f10 ðk 1 Þ
a2 a1 ½f1 ðk 1 Þ dk 1 ½q1 ðk 1 Þ q2 ðk 1 Þ >0 k 1 q1 ðk 1 Þ ð1 a1 Þð1 gÞq1 ðk 1 Þ
ð37Þ
a2 a1 1 0 0 ðd þ rÞ þ ½f1 ðk 1 Þ dk 1 ¼ f1 ðk 1 Þ ðd þ rÞ þ ða2 a1 Þ f ðk 1 Þ d a1 1 k1
a2 0 f ðk 1 Þ þ ða1 a2 Þd ðd þ rÞ: a1 1
For f10 ðk 1 Þ strictly decreasing on the interval ð0; k 1 Þ and a2 0 dþr f ðk 1 Þ þ ða1 a2 Þd ðd þ rÞ ¼ ða1 a2 Þ þ d < 0; a1 a1 1 ¼
there exists unique 0 < k 1 < k 1 , such that a2 0 ¼ f ðk 1 Þ þ ða1 a2 Þd ðd þ rÞ ¼ 0: a1 1 ¼ Let k 1 ¼ maxf~k 1 ; k 1 g, then for any given k 1 2 ðk 1 ; k 1 Þ, lim Qðk 1 ; qÞ > 0;
q!q1 ðk 1 Þ
lim Qðk 1 ; qÞ < 0 and
q!1
oQ < 0: ok 1
ð38Þ
Therefore, Q(k1, q) = 0 decides a curve q4(k1) on the interval ðk 1 ; k 1 Þ. From (35), Qðk 1 ; qÞ ¼ 0 for q ¼ q2 ðk 1 Þ and f10 ðk 1 Þ ¼ d þ r. So, q4 ðk 1 Þ ¼ q and the curve q4(k1) passes the equilibrium point ðk 1 ; qÞ. In fact, we can solve q4(k1) from the equation Q(k1, q) = 0 on the interval ðk 1 ; k 1 Þ.
Lemma 11. On the interval ðk 1 ; k Þ, we have ½ð1 a1 Þd þ ð1 a2 Þrða1 a2 Þq1 ðk 1 Þ ; ð1 a2 Þfa1 ½ð1 a1 þ a2 Þd þ r a2 f10 ðk 1 Þg a1 a2 f100 ðk 1 Þ q04 ðk 1 Þ ¼ þ q ðk 1 Þ: k 1 a1 ðð1 þ a2 a1 Þ þ rÞ a2 f10 ðk 1 Þ 4
q4 ðk 1 Þ ¼
ð39Þ ð40Þ
Proof. Since 1g¼
a1 a2 ; a1 ð1 a2 Þ
f1 ðk 1 Þ 1 ¼ f10 ðk 1 Þ k1 a1
and Iðk 1 ; qÞ ¼
Lq1 ðk 1 Þ ; q
q ðk 1 Þ ða1 a2 Þð1 a1 Þ q1 ðk 1 Þ I 1 ðk 1 ; qÞ ¼ L 1 ð1 a1 Þð1 gÞ 1 ¼L 1 ; q a1 ð1 a2 Þ q
we have J ðk 1 ; qÞ 1 1 0 q ðk 1 Þ ¼ f10 ðk 1 Þ d f1 ðk 1 Þ ð1 gÞðf10 ðk 1 Þ rÞ 1 Lk 1 a1 a1 q 1 1 0 a1 a2 q ðk 1 Þ ðf 0 ðk 1 Þ rÞ 1 ¼ f10 ðk 1 Þ d f ðk 1 Þ a1 a1 1 q a1 ð1 a2 Þ 1 1 1 1 a1 0 a1 a2 q1 ðk 1 Þ ¼ ½f10 ðk 1 Þ a1 d f ðk 1 Þ þ r a1 a1 1 a2 1 q 1 a2
ð41Þ
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and ½f10 ðk 1 Þ ðd þ rÞI 1 ðk 1 ; qÞ 1 0 a1 1 a1 q1 ðk 1 Þ ¼ ½f1 ðk 1 Þ ðd þ rÞ : Lða1 a2 Þ a1 a1 a2 1 a2 q
ð42Þ
The equation Q(k1, q) = 0 implies that J ðk 1 ; qÞ ½f10 ðk 1 Þ ðd þ rÞI 1 ðk 1 ; qÞ : ¼ Lk 1 Lða1 a2 Þ Comparing the right hand sides of Eqs. (40) and (41), we have dð1 a1 Þ q ðk 1 Þ 1 ¼ þr 1 ½a2 f10 ðk 1 Þ þ a1 ðdð1 a1 þ a2 Þ þ rÞ: 1 a2 q a1 a2
ð43Þ
So, (39) holds. Taking logarithm on both sides of (39), we have ln q4 ðk 1 Þ ¼ ln q1 ðk 1 Þ ln½a1 ðð1 a1 þ a2 Þd þ rÞ a2 f10 ðk 1 Þ þ ln B;
ð44Þ
2 Þrða1 a2 Þ . where B ¼ ½ð1a1 Þdþð1a ð1a2 Þ Taking the derivative with respect to k1 on both sides of (44), we obtain (40). This completes the proof of Lemma 11. h
Lemma 12. The curve q4(k1) is above the curve q2(k1) on the interval ðk 1 ; k 1 Þ and under the curve q2(k1) on the interval ðk 1 ; k 1 Þ.
Proof. By (38), for a given k 1 2 ðk 1 ; k 1 Þ, the function Qðk 01 ; qÞ decreases strictly with respect to q. From
Qðk 1 ; q2 ðk 1 ÞÞ ¼ f10 ðk 1 Þ ðd þ rÞ > 0;
k 1 2 ðk 1 ; k Þ
Qðk 1 ; q2 ðk 1 ÞÞ ¼ f10 ðk 1 Þ ðd þ rÞ < 0;
k 1 2 ðk 1 ; k Þ;
and
we know that q4 ðk 1 Þ > q2 ðk 1 Þ; k 1 2 ðk 1 ; k 1 Þ and q4 ðk 1 Þ < q2 ðk 1 Þ; k 1 2 ðk 1 ; k 1 Þ. ¼
¼
If k 1 > ~k 1 , then k 1 ¼ k 1 . Denote that ¼
e 11 ¼ fðk 1 ; qÞj~k 1 < k 1 6 k 1 ; q1 ðk 1 Þ < q < 1g; M e 11 for then Qðk 1 ; qÞ P 0; ðk 1 ; qÞ 2 M ¼ a2 lim Qðk 1 ; qÞ ¼ f10 ðk 1 Þ þ ða1 a2 Þd ðd þ rÞ P 0; k 1 2 ð~k 1 ; k 1 : q!1 a1 _ e 11 since C_ ¼ Qðk 1 ; qÞC. Hence, C > 0 on the domain M ¼
For k 1 P ~k 1 , let e1 ¼ M e 11 [ M e 12 ; M e 2 ¼ fðk 1 ; qÞjk 1 < k 1 < k ; q4 ðk 1 Þ < q < q2 ðk 1 Þg; M ¼ e 12 ¼ fðk 1 ; qÞjk 1 < k 1 < k 1 ; q2 ðk 1 Þ < q < q4 ðk 1 Þg. where M Otherwise, let e 1 ¼ fðk 1 ; qÞj~k 1 < k 1 6 k 1 ; q2 ðk 1 Þ < q < q4 ðk 1 Þg; M e 2 ¼ fðk 1 ; qÞjk 1 < k 1 < k ; q4 ðk 1 Þ < q < q2 ðk 1 Þg: M
h
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e 1 and the optimal path P2 lies in the region M e 2. Lemma 13. The optimal path P1 lies in the region M ¼
e 11 in the case of k 1 P ~k 1 . We only need to prove Proof. Obviously, the optimal path lies in the region M e 12 , and P2 above the curve q4(k1) on the that the optimal path P1 is under the curve q4(k1) on the region M e e region M 2 for P1 is above the curve q2(k1) on the region M 12 and P2 is under the curve q2(k1) on the e 2. region M From (34), we have q_ ða2 a1 Þq4 ðk 1 Þ ; ¼ k1 k_
k 1 6¼ k 1
ð45Þ
on the curve q4(k1). ¼ Since a1 ð1 a1 þ a2 Þd a2 f10 ðk 1 Þ > 0 and f100 ðk 1 Þ < 0 on the interval ðk 1 ; k 1 Þ, from (40), we have a1 a2 q_ ¼ þ q ðk 1 Þ > q04 ðk 1 Þ k1 k1 4 k_
ð46Þ
on the curve q4(k1). The formula (46) implies that the trajectory of the dynamical system (19), (20) cross the curve q4(k1) from e 12 and cross the curve q4(k1) from southeast to northwest. Any northwest to southeast on the region M trajectory starting from a point on the curve q4(k1) except the point k 1 ¼ k 1 cannot come back to the curve q4(k1). e 12 , then the trajectory starting from If there is a point on the curve q4(k1) under the path P1 on the region M this point must converge to the equilibrium ðk 1 ; qÞ since it cannot cross the path P1 and come back to the curve q4(k1). Similarly, we can prove that the curve q4(k1) is above ¼ the curve under the path P2. In the same way, we can prove the result in the case of k 1 < ~k 1 . h e 1 and Q(k1, q) < 0 on the region M e 2 , we have the From Lemmas 10 and 13 and Q(k1, q) > 0 on the region M following theorem. Theorem 5. The total consumption production increases on the optimal path P1 when g P 1 and decreases on the e 1 and decreases on the optimal optimal path P2; For 0 < g < 1, it increases on the optimal path P1 in the region M path P2. 6. An example of numerical computation Here we present the results of the numerical computation under the conditions of Theorem 2. The construction parameters of the model are specified below A1 ¼ 2;
A2 ¼ 1:8;
h ¼ 0:7; r ¼ 0:05;
L ¼ 2:
The capital share ai, i = 1, 2 of the two sectors is given there group different values to require the affection of the technical rate of substitution, i.e., they are specified as a1 ¼ 0:33; 0:30; 0:30;
a2 ¼ 0:30; 0:30; 0:33:
Denote that rij, i = 1, 2, j = 1, 2, 3 are the three different technical substitutes of ith sector, then r11 ¼ 2:03;
r12 ¼ 2:33;
r13 ¼ 2:33;
r21 ¼ 2:33;
r22 ¼ 2:33;
r23 ¼ 2:03:
Correspondingly, the values of the parameters of g are g1 = 0.8701, g2 = 1, g3 = 1.1493. Figs. 2 and 3 describe the capital growths of Sector I and II. On the optimal growth path, capital allocated in Sectors I increases sharply at first, then decreases slowly to the equilibrium value. But the capital in Sector II increases strictly to the equilibrium. Although the capital in Section 1 appears to decrease after a short period, the total capital of the economy grows steadily. Fig. 4 shows the growth process of the total capital under the three different parametric values.
D. Cai / Applied Mathematics and Computation 183 (2006) 1359–1377
Fig. 2. The capital of Sector I.
Fig. 3. The capital of Sector II.
Fig. 4. The total capital.
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Fig. 5. The labor of Sector I.
Fig. 6. The labor of Sector II.
Fig. 7. The per capita capital of Sector I.
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Fig. 8. The per capita capital of Sector II.
Fig. 9. The per capita capital.
Figs. 5 and 6 describe the labor allocation between Sector I and Sector II. With the economic growth, the labor in Sector I decreases while the labor in Sector II increases steadily. This means the economy uses more and more labor and capital in the consumption product Sector. (Also see Figs. 2 and 3.) Fig. 7 depicts three growth shapes of per capita capital in Sector I. The per capita capital grows strictly although the total capital in Sector I appears to decrease after a period. This implies that the labor allocated in Sector I decreases more quickly than the total capital. Fig. 8 depicts three growth shapes of per capita capital in Sector II. Figs. 3 and 6 indicate that the capital and labor of Sector II increase simultaneously. Hence, the increasing per capita capital implies that the growth rate of the capital exceeds that of the labor force. The varieties of per capita capital of the economy are given in Fig. 9. The capital production and consumption production are presented in Figs. 10 and 11, respectively. 7. Summary From the analysis above, we can see that the ratio of the technical rate of the two sectors is a key factor that affects the allocation of labor force and capital between the capital production sector and consumption
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Fig. 10. The capital production.
Fig. 11. The consumption production.
production sector. In fact, the economy reaches the instantaneous equilibrium by sustaining two production sectors have same rates of the technical of substitute. The optimal economic growth derives from this continuously adjusting process of the capital and labor. The dynamics of the capital and labor indicate that the economy always chooses the optimal reallocation of the capital and labor between the capital production sector and consumption production sector such that the total capital and total consumption steadily grows along the optimal path, so does the per capita capital and consumption. From the example of the numerical computation, we see that the labor used in Sector I decreases strictly with the economy growth. But the growth of capital used in Sector I appears to have three different situations with three different values of the parameter g. When the technical of substitute in Sector I is low(r11 = 2.03) and the technical of substitute in Sector II is high(r21 = 2.33), the capital used in Sector I grows rapidly then decreases to a steady value and the capital used in Sector II increases strictly. There growths of capital used in Sector I and Sector II are similar for the case where the two sectors have same technical of substitute. When the technical of substitute in Sector I is high and in Sector II is low, the growth of the capital used in Sector I increases rapidly then sustains to steady value. Comparing Fig. 2 with Fig. 3 and Fig. 5 with Fig. 6, we view that more and more capital and labor are allocated into Sector II to produce consumption production with the economy growth. This can be used to explain the ratio of labor in manufacture decreases with economy growth.
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Figs. 10 and 11 show that the economy has the highest capital production when the technical of substitute of Sector I is low and the technical of substitute of Sector II is high but does not reach the highest consumption production. Conversely, the economy has the highest consumption production when the technical of substitute of Sector I is low and the technical of substitute of Sector II is high but does not reach the highest capital production. References [1] J.R. Barro, I.X. Martin, Economic Growth, McGraw-Hill Companies, Inc., New York, 1995. [2] E. Ryder Jr, Optimal accumulation in a two-sector neoclassical economy with non-shiftable capital, Journal of Political Economy 77 (1969) 665–683. [3] A. Ladron-De-Guevara, S. Ortigueir, M.S. Santos, A two-Sector model of endogenous growth with leisure, Review of Economic Studies 66 (1999) 609–631. [4] E. Burmeister, A.R. Dobell, Mathematical Theories of Economic Growth, Gregg Revivals, Hampshire, 1993.