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An application of traveling wave analysis in economic growth model
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Applied Mathematics and Computation 200 (2008) 261–266 www.elsevier.com/locate/amc
An application of traveling wave analysis in economic growth model Ming-Chun Zhou a,*, Yan-Fang Yang b a
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, PR China b Computer School, Wuhan University, Wuhan 430072, Hubei, PR China
Abstract A temporal–spatial economic growth model is established in this paper. As a useful tool, traveling wave analysis is used to analyze technological growth and diffusion. Numerical simulation shows that this model has perfect performance. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Reaction–diffusion equation; Technology growth and diffusion; The Solow–Swan model; Economic growth; Traveling wave analysis
1. Introduction First, we give a brief introduction of the economic growth model. 1.1. The Solow–Swan model We carry out our economic analysis in the neoclassical economic framework. The production function is Y ¼ F ðK; ALÞ. K is capital, A is technology which is exogenously given, L is labor with exogenous growth rate n, and Y is the output. F ð0; Þ ¼ 0 and F ð; 0Þ ¼ 0 are assumed. The production function is neoclassical, namely, the following three assumptions are satisfied (see [2,3]). Assumption 1. K > 0; L > 0. F ð; Þ exhibits positive and diminishing marginal products with respect to each oF o2 F oF o2 F input: oK > 0; oK < 0: 2 < 0; oL > 0; oL2 Assumption 2. F ð:; :Þ exhibits constant returns to scale, that is, the function has the first-order homogenuity: F ðkK; kLÞ ¼ kF ðK; LÞ, for all k > 0.
*
Corresponding author. E-mail address: [email protected] (M.-C. Zhou).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.11.006
262
M.-C. Zhou, Y.-F. Yang / Applied Mathematics and Computation 200 (2008) 261–266
Assumption 3 (Inada conditions). lim F K ¼ lim F L ¼ 1;
K!0
L!0
lim F K ¼ lim F L ¼ 0:
K!1
L!1
The labor-augmenting model is K_ ¼ sF ðK; ALÞ dK, where s is the constant saving rate and d is the constant discount rate. Letting k be the efficient per capita capital, K=ðALÞ; and f ðkÞ ¼ F ðk; 1Þ, we have K_ ¼ sf ðkÞ dk: AL
ð1Þ
Consider the equality k ¼ K=ðALÞ. Taking logs and differentiations on both sides of this equation, we have _ _ _ _ _ _ K ¼ KK AA LL. Thus k_ ¼ ðKK AA LLÞ AL . By Eq. (1) we have the intensive form A_ k_ ¼ sf ðkÞ d þ n þ k: ð2Þ A
k_ k
_
To obtain the final form of the equation, the expression for AA should be obtained. This will be done in Section 2. Second, we introduce the traveling wave solution obtained from a reaction–diffusion equation. 1.2. Traveling wave solution to the reaction–diffusion equation We consider the temporal–spatial growth and diffusion admits the reaction–diffusion equation ou o2 u ¼ d 2 þ auð1 ux Þ; ot ox where x is a positive constant power law exponent, and d is the technological diffusion coefficient, and a is the growth coefficient (see [1]). Namely, at any point and any point of time technology grows pffiffiffiffiffiffiffi ffi as is described by the 2 auð1 ux Þ term and diffuses as is described by the d ooxu2 term. Let T ¼ at; n ¼ x a=d and we obtain ou ou ¼a ; ot oT
d
o2 u o2 u ¼ a : ox2 on2
Then its dimensionless form is ou o2 u ¼ þ uð1 ux Þ: oT on2 Let the initial and boundary conditions be 1; n < 0 uðn; 0Þ ¼ uð1; T Þ ¼ 1; 0; n > 0;
uðþ1; T Þ ¼ 0:
Let the traveling wave is of the form f ¼ n cT where c is a positive, dimensionless constant wave velocity. Then uðfÞ ¼ uðn; T Þ and we obtain ou du ¼ c ; oT df
o2 u d 2 u ¼ : on2 df2
This equation is transformed into the second-order ordinary differential equation form d2 u du þ c þ uð1 ux Þ ¼ 0: 2 df df In the original independent variable form f ¼ n cT and the initial and boundary conditions become uð1Þ ¼ 1;
uðþ1Þ ¼ 0:
Following Murray’s approach, we will obtain a traveling wave solution to the equation of the form
ð3Þ
M.-C. Zhou, Y.-F. Yang / Applied Mathematics and Computation 200 (2008) 261–266
uðfÞ ¼
1 c; ð1 þ aebf Þ
263
ð4Þ
where a; b and c are positive constants and c¼
2 ; x
xþ4 c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2ðx þ 2Þ
x b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2ðx þ 2Þ
uð0Þ ¼ u ;
a¼
1 x=2 u
1:
ð5Þ
The second derivative provides the inflection point fi ¼
1 x ln ; b 2a
ui ¼
1 ð1 þ x=2Þ
2=x
:
Next, we will establish our economic model whose technology grows and diffuses in a finite circular region. 2. The model Compared with the usually used logistic S-shaped curves to describe the growth and diffusion only in temporal context, we will establish a more realistic model in temporal–spatial context. As this is a model to describe the growth and diffusion of technology, we need more words to explain and give reasonable explanation for the technology growth and diffusion. Assume that there is only one technological center in the economy we considered and all the technology in it other than the center is radially diffused from the center. The technological level at the technological center grows with power law logistic style. We assume the economy lies geographically in a finite circular region (see Fig. 1) whose radius x 2 ½0; xmax and time t 2 ½0; þ1Þ. The technology in the economy radially diffuses from the center where x ¼ 0 and the diffusion is assumed homogenous in all direction. uðx; tÞ is the technological level at point of radius x and point of time t. The aggregate amount of technology at a certain point of time t is the sum of technology from the center to the border of the region. Let the initial technological level at x be uðx; 0Þ ¼ u0 ðxÞ ¼ l mxðl; m > 0Þ. This is reasonable because this pattern just likes what we have always seen in reality: the farther the place away from central cities the lower its technological level is. Then the maximal radius is xmax ¼ l=m. By Eq. (4), the total technology of the economy AðtÞ is the integration of uðfÞ in x from 0 to xmax , Z xmax Z xmax h ic pffiffiffiffiffi AðtÞ ¼ 2pxuðfÞ dx ¼ 2px 1 þ aebðx a=d catÞ dx: ð6Þ 0
0
Differentiating AðtÞ in Eq. (6) we have
Fig. 1. The circular region of the economy.
264
M.-C. Zhou, Y.-F. Yang / Applied Mathematics and Computation 200 (2008) 261–266
9 8 pffiffiffiffiffi , > > = Z xmax h ic pffiffiffiffiffi _A A > 0 ; : 0 1 þ aebðx a=d catÞ
ð7Þ
Let the production function be Cobb–Douglas. That is, f ðkÞ ¼ k s where 0 < s < 1. Then, by Eq. (2), we have the final form of our model _ k_ ¼ sk s ðd þ n þ A=AÞk:
ð8Þ
3. Numerical simulation 3.1. Computation for xmax Set l ¼ 0:1; m ¼ 0:01. The technological level linearly decreases from the technological center, that is, u0 ðxÞ ¼ 0:1 0:01x. Thus we set xmax ¼ 10 where u0 ¼ 0. 3.2. Computation for t The formula for calculating t is given by solving the equation u_ ¼ auð1 ux Þ: The solution is uðtÞ ¼ n
1þ
h x u u0
u
i o1=x ; 1 eaxt
where u is the limit value of u and u0 is the initial value of u at t ¼ 0. The formula for t is x
t ¼
1 ð1=~ u0 Þ 1 ln ; x ax ð1=~uÞ 1
ð9Þ
where ~ u0 ¼ u0 = u and ~ u ¼ u= u which are normalized by u. And we assume pffiffiffiffiffiffi u ¼ 1. Therefore, ~u0 ¼ u0 and ~u ¼ u. When f ¼ 0 and x ¼ xmax =2, wave velocity is c ¼ xmax =2t ¼ c ad , diffusion coefficient is d ¼ x2max =4c2 at2 . 3.3. Results of numerical simulation Let x vary: x ¼ 1; 1:5; 2; 2:5. Set s ¼ 0:33; d ¼ 0:05; n ¼ 0:01; s ¼ 0:2; a ¼ 0:1; u ¼ 0:1. By Eq. (5) the corresponding parameters a; b; c; c then have their values through computation. We give the computational value of various parameters in Table 1. When x ¼ 1, the original reaction–diffusion equation is Fisher equation. When x ¼ 2, the traveling wave solution itself reduces to the ordinary logistic equation uðfÞ ¼
1 : 1 þ aebf
Table 1 List of the values of the parameters x
a
b
c
c
t
d
c
1.0 1.5 2.0 2.5
2.1623 4.6234 9.0000 16.7828
0.4082 0.5669 0.7071 0.8333
0.5000 0.7500 1.0000 1.2500
2.0412 2.0788 2.1213 2.1667
7.4721 7.0707 6.9692 6.9419
1.0746 1.1571 1.1438 1.1051
0.6692 0.7071 0.7174 0.7203
M.-C. Zhou, Y.-F. Yang / Applied Mathematics and Computation 200 (2008) 261–266 1 omega = 1.0 omega = 1.5 omega = 2.0 omega = 2.5
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −25
−20
−15
−10
−5
0
5
10
15
20
25
Fig. 2. The uðfÞ curves for various x.
350
300
250
200
150 omega = 1.0 omega = 1.5 omega = 2.0 omega = 2.5
100
50
0
0
10
20
30
40
50
60
70
Fig. 3. The A(t) curves for various x.
6
5
4
3
2 omega = 1.0 omega = 1.5 omega = 2.0 omega = 2.5
1
0
0
20
40
60
80
100
Fig. 4. The k(t) curves for various x.
120
140
265
266
M.-C. Zhou, Y.-F. Yang / Applied Mathematics and Computation 200 (2008) 261–266
~ are depicted in Fig. 3. The plots of kðtÞ are The plots of uðfÞ curves are depicted in Fig. 2. The plots of AðtÞ depicted in Fig. 4. In Figs. 3 and 4, both AðtÞ and kðtÞ show perfect convergent properties and that larger x gains higher speed for both AðtÞ and kðtÞ. 4. Conclusion As a realistic temporal–spatial model for technological growth and diffusion, we apply traveling wave analysis on this model. Given an initial technological level pattern to determine the border (maximal radius) of the economy, we give numerical simulation on this model and get perfect convergent properties from it, which can be obtained from temporal models either. References [1] R.B. Banks, Growth and Diffusion Phenomena, Springer-Verlag, New York, 1994. [2] R.J. Barro, X. Sala-i-Martin, Economic Growth, McGraw-Hill, New York, 1995. [3] D. Romer, Advanced Macroeconomics, second ed., McGraw-Hill, New York, 2001.
Applied Mathematics and Computation 200 (2008) 261–266 www.elsevier.com/locate/amc
An application of traveling wave analysis in economic growth model Ming-Chun Zhou a,*, Yan-Fang Yang b a
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, PR China b Computer School, Wuhan University, Wuhan 430072, Hubei, PR China
Abstract A temporal–spatial economic growth model is established in this paper. As a useful tool, traveling wave analysis is used to analyze technological growth and diffusion. Numerical simulation shows that this model has perfect performance. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Reaction–diffusion equation; Technology growth and diffusion; The Solow–Swan model; Economic growth; Traveling wave analysis
1. Introduction First, we give a brief introduction of the economic growth model. 1.1. The Solow–Swan model We carry out our economic analysis in the neoclassical economic framework. The production function is Y ¼ F ðK; ALÞ. K is capital, A is technology which is exogenously given, L is labor with exogenous growth rate n, and Y is the output. F ð0; Þ ¼ 0 and F ð; 0Þ ¼ 0 are assumed. The production function is neoclassical, namely, the following three assumptions are satisfied (see [2,3]). Assumption 1. K > 0; L > 0. F ð; Þ exhibits positive and diminishing marginal products with respect to each oF o2 F oF o2 F input: oK > 0; oK < 0: 2 < 0; oL > 0; oL2 Assumption 2. F ð:; :Þ exhibits constant returns to scale, that is, the function has the first-order homogenuity: F ðkK; kLÞ ¼ kF ðK; LÞ, for all k > 0.
*
Corresponding author. E-mail address: [email protected] (M.-C. Zhou).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.11.006
262
M.-C. Zhou, Y.-F. Yang / Applied Mathematics and Computation 200 (2008) 261–266
Assumption 3 (Inada conditions). lim F K ¼ lim F L ¼ 1;
K!0
L!0
lim F K ¼ lim F L ¼ 0:
K!1
L!1
The labor-augmenting model is K_ ¼ sF ðK; ALÞ dK, where s is the constant saving rate and d is the constant discount rate. Letting k be the efficient per capita capital, K=ðALÞ; and f ðkÞ ¼ F ðk; 1Þ, we have K_ ¼ sf ðkÞ dk: AL
ð1Þ
Consider the equality k ¼ K=ðALÞ. Taking logs and differentiations on both sides of this equation, we have _ _ _ _ _ _ K ¼ KK AA LL. Thus k_ ¼ ðKK AA LLÞ AL . By Eq. (1) we have the intensive form A_ k_ ¼ sf ðkÞ d þ n þ k: ð2Þ A
k_ k
_
To obtain the final form of the equation, the expression for AA should be obtained. This will be done in Section 2. Second, we introduce the traveling wave solution obtained from a reaction–diffusion equation. 1.2. Traveling wave solution to the reaction–diffusion equation We consider the temporal–spatial growth and diffusion admits the reaction–diffusion equation ou o2 u ¼ d 2 þ auð1 ux Þ; ot ox where x is a positive constant power law exponent, and d is the technological diffusion coefficient, and a is the growth coefficient (see [1]). Namely, at any point and any point of time technology grows pffiffiffiffiffiffiffi ffi as is described by the 2 auð1 ux Þ term and diffuses as is described by the d ooxu2 term. Let T ¼ at; n ¼ x a=d and we obtain ou ou ¼a ; ot oT
d
o2 u o2 u ¼ a : ox2 on2
Then its dimensionless form is ou o2 u ¼ þ uð1 ux Þ: oT on2 Let the initial and boundary conditions be 1; n < 0 uðn; 0Þ ¼ uð1; T Þ ¼ 1; 0; n > 0;
uðþ1; T Þ ¼ 0:
Let the traveling wave is of the form f ¼ n cT where c is a positive, dimensionless constant wave velocity. Then uðfÞ ¼ uðn; T Þ and we obtain ou du ¼ c ; oT df
o2 u d 2 u ¼ : on2 df2
This equation is transformed into the second-order ordinary differential equation form d2 u du þ c þ uð1 ux Þ ¼ 0: 2 df df In the original independent variable form f ¼ n cT and the initial and boundary conditions become uð1Þ ¼ 1;
uðþ1Þ ¼ 0:
Following Murray’s approach, we will obtain a traveling wave solution to the equation of the form
ð3Þ
M.-C. Zhou, Y.-F. Yang / Applied Mathematics and Computation 200 (2008) 261–266
uðfÞ ¼
1 c; ð1 þ aebf Þ
263
ð4Þ
where a; b and c are positive constants and c¼
2 ; x
xþ4 c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2ðx þ 2Þ
x b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2ðx þ 2Þ
uð0Þ ¼ u ;
a¼
1 x=2 u
1:
ð5Þ
The second derivative provides the inflection point fi ¼
1 x ln ; b 2a
ui ¼
1 ð1 þ x=2Þ
2=x
:
Next, we will establish our economic model whose technology grows and diffuses in a finite circular region. 2. The model Compared with the usually used logistic S-shaped curves to describe the growth and diffusion only in temporal context, we will establish a more realistic model in temporal–spatial context. As this is a model to describe the growth and diffusion of technology, we need more words to explain and give reasonable explanation for the technology growth and diffusion. Assume that there is only one technological center in the economy we considered and all the technology in it other than the center is radially diffused from the center. The technological level at the technological center grows with power law logistic style. We assume the economy lies geographically in a finite circular region (see Fig. 1) whose radius x 2 ½0; xmax and time t 2 ½0; þ1Þ. The technology in the economy radially diffuses from the center where x ¼ 0 and the diffusion is assumed homogenous in all direction. uðx; tÞ is the technological level at point of radius x and point of time t. The aggregate amount of technology at a certain point of time t is the sum of technology from the center to the border of the region. Let the initial technological level at x be uðx; 0Þ ¼ u0 ðxÞ ¼ l mxðl; m > 0Þ. This is reasonable because this pattern just likes what we have always seen in reality: the farther the place away from central cities the lower its technological level is. Then the maximal radius is xmax ¼ l=m. By Eq. (4), the total technology of the economy AðtÞ is the integration of uðfÞ in x from 0 to xmax , Z xmax Z xmax h ic pffiffiffiffiffi AðtÞ ¼ 2pxuðfÞ dx ¼ 2px 1 þ aebðx a=d catÞ dx: ð6Þ 0
0
Differentiating AðtÞ in Eq. (6) we have
Fig. 1. The circular region of the economy.
264
M.-C. Zhou, Y.-F. Yang / Applied Mathematics and Computation 200 (2008) 261–266
9 8 pffiffiffiffiffi , > > = Z xmax h ic pffiffiffiffiffi _A
ð7Þ
Let the production function be Cobb–Douglas. That is, f ðkÞ ¼ k s where 0 < s < 1. Then, by Eq. (2), we have the final form of our model _ k_ ¼ sk s ðd þ n þ A=AÞk:
ð8Þ
3. Numerical simulation 3.1. Computation for xmax Set l ¼ 0:1; m ¼ 0:01. The technological level linearly decreases from the technological center, that is, u0 ðxÞ ¼ 0:1 0:01x. Thus we set xmax ¼ 10 where u0 ¼ 0. 3.2. Computation for t The formula for calculating t is given by solving the equation u_ ¼ auð1 ux Þ: The solution is uðtÞ ¼ n
1þ
h x u u0
u
i o1=x ; 1 eaxt
where u is the limit value of u and u0 is the initial value of u at t ¼ 0. The formula for t is x
t ¼
1 ð1=~ u0 Þ 1 ln ; x ax ð1=~uÞ 1
ð9Þ
where ~ u0 ¼ u0 = u and ~ u ¼ u= u which are normalized by u. And we assume pffiffiffiffiffiffi u ¼ 1. Therefore, ~u0 ¼ u0 and ~u ¼ u. When f ¼ 0 and x ¼ xmax =2, wave velocity is c ¼ xmax =2t ¼ c ad , diffusion coefficient is d ¼ x2max =4c2 at2 . 3.3. Results of numerical simulation Let x vary: x ¼ 1; 1:5; 2; 2:5. Set s ¼ 0:33; d ¼ 0:05; n ¼ 0:01; s ¼ 0:2; a ¼ 0:1; u ¼ 0:1. By Eq. (5) the corresponding parameters a; b; c; c then have their values through computation. We give the computational value of various parameters in Table 1. When x ¼ 1, the original reaction–diffusion equation is Fisher equation. When x ¼ 2, the traveling wave solution itself reduces to the ordinary logistic equation uðfÞ ¼
1 : 1 þ aebf
Table 1 List of the values of the parameters x
a
b
c
c
t
d
c
1.0 1.5 2.0 2.5
2.1623 4.6234 9.0000 16.7828
0.4082 0.5669 0.7071 0.8333
0.5000 0.7500 1.0000 1.2500
2.0412 2.0788 2.1213 2.1667
7.4721 7.0707 6.9692 6.9419
1.0746 1.1571 1.1438 1.1051
0.6692 0.7071 0.7174 0.7203
M.-C. Zhou, Y.-F. Yang / Applied Mathematics and Computation 200 (2008) 261–266 1 omega = 1.0 omega = 1.5 omega = 2.0 omega = 2.5
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −25
−20
−15
−10
−5
0
5
10
15
20
25
Fig. 2. The uðfÞ curves for various x.
350
300
250
200
150 omega = 1.0 omega = 1.5 omega = 2.0 omega = 2.5
100
50
0
0
10
20
30
40
50
60
70
Fig. 3. The A(t) curves for various x.
6
5
4
3
2 omega = 1.0 omega = 1.5 omega = 2.0 omega = 2.5
1
0
0
20
40
60
80
100
Fig. 4. The k(t) curves for various x.
120
140
265
266
M.-C. Zhou, Y.-F. Yang / Applied Mathematics and Computation 200 (2008) 261–266
~ are depicted in Fig. 3. The plots of kðtÞ are The plots of uðfÞ curves are depicted in Fig. 2. The plots of AðtÞ depicted in Fig. 4. In Figs. 3 and 4, both AðtÞ and kðtÞ show perfect convergent properties and that larger x gains higher speed for both AðtÞ and kðtÞ. 4. Conclusion As a realistic temporal–spatial model for technological growth and diffusion, we apply traveling wave analysis on this model. Given an initial technological level pattern to determine the border (maximal radius) of the economy, we give numerical simulation on this model and get perfect convergent properties from it, which can be obtained from temporal models either. References [1] R.B. Banks, Growth and Diffusion Phenomena, Springer-Verlag, New York, 1994. [2] R.J. Barro, X. Sala-i-Martin, Economic Growth, McGraw-Hill, New York, 1995. [3] D. Romer, Advanced Macroeconomics, second ed., McGraw-Hill, New York, 2001.