The life cycle model with optimal individual years of schooling and simulation

Mathematical Computation June 2014, Volume 3, Issue 2, PP.44-48

The Life-Cycle Model with Optimal Individual Years of Schooling and Simulation Donghan Cai #, 1, Huan Yang1, Zhongbin Chen2 1. College of Math. & Stat., Wuhan Univ., Wuhan, 430072, P.R. China 2. School of Econ. & Management, Wuhan Univ., Wuhan, 430072, P.R. China #Email: dhcai@whu.edu.cn

Abstract In this paper, we use a discrete lift-cycle model to inquire the relationship between the individual saving and years of schooling. It is proved that there exist optimal years of schooling for individual to maximize his lifetime utility and the optimal years of schooling increases with respect to the wage growth rate and the quantity of human capital growth rate. The individual saving behavior and his asset change under different years of schooling is presented by the numerical simulation.

Keywords: Lift-Cycle Model; Individual Years of Schooling; Individual Saving; Quantity of Human Capital

1 INTRODUCTION The behavior of individual saving and consumption is an important topic in economic theory [1-5]. The life-cycle theory presented by Modigliani [1-2] argued that individual saves part of earning while working for his or her retirement and his or her net worth is never negative. In Tobin’s model, the negative net worth appears in individual early age [3]. However, the individual years of schooling is not considered in these models. In the paper [6], the years of schooling and the quantity of human capital is introduced in a continuous time overlapping generations’ model to inquire the role of increased life expectancy in raising human capital investment.

In this paper, the years of schooling and the quantity is integrated to a discrete life-cycle model to study the relationship between the individual years of schooling and saving behavior. It is proved that there exist optimal years of schooling for the individual reaching maximal lifetime utility and the optimal individual years of schooling increases with respect to the wage growth rate and the quantity of human capital. At the end of paper, the individual saving behavior and his asset change under different years of schooling is presented by the numerical simulation.

2 SETUP THE MODEL Denote the consumption, asset and wage of individuals who born at time s by cts , ats wts at time t . The finite longevity and retiring age are  and T respectively. We assume that individuals start their lives without assets and end up with no bequest and no debt, i.e.

ass  ass  0

(1)

s  t  T1   0  w   g (T1 ) wt  s  T1  t  s  T   0 s  T  t  s   

(2)

and the wage of individual satisfies s t

Where T1 is the year of schooling and g (T1 ) is the quantity of human capital as a function of taking the years of schooling satisfying g (0)  1 g ()  0 .

The individual budget constraint is



This work is supported by national natural science foundation of China (No.71271158) - 44 www.ivypub.org/mc

ats1  (1  r )ats  wts  cts  t  s s  1

 s    1

(3)

where r is the interest rate.

Hence, the lifetime utility of the individual born at time s is s 

t s s   u (ct )

(4)

t s

where   11 ,   0 is the rate of time preference and the utility function is specified to be u(c)  ln c and the individual optimal problem is to maximize (4) subject to (1) and (3).

3

THE DYNAMICS OF CONSUMPTION AND ASSET

The Lagrangian to solve the individual optimal problem (6) is s 

s 1

t s

t s

L    t  s u (cts )   ts [(1  r )ats  wts  cts  at 1 ]

The first order condition to solve the optimal problem is (3) and

L  ts1  ts (1  r )  0 t  s  1  s   ats L   t  s u(cts )  ts  0 t  s  s    1 cts

(5) (6)

From (5), (6) and u(c)  1c ,

ts1  u(cts1 )  cts 1    s  t  s s  1  s    1 ts 1  r u (cts ) ct 1 So, cts1  11r cts  t  s

 s    1 Let b  11r , then

cts  bt  s css  t  s s  1

 s  

(7)

From (3),

ats1  (1  r )ats  wts  cts  (1  r )[(1  r )ats1  wts1  cts1 ]  wts  cts  (1  r ) 2 ats1  wts  (1  r ) wts1  [cts  (1  r )cts1 ] t

t

is

is

(8)

 (1  r )t 1 s ass   (1  r )t  i wis   (1  r )t i cis  By (1), s 1

s 1

is

is

s 1 i wis   (1  r )s 1i cis  0  (1  r )

So, from (7), s T (1  r )  b s s 1 cs   (1  r )s 1i wis  g (T1 )  (1  r ) s 1i wi i  s i  s T 1 r  b 1

and

css 

s T 1 r  b g (T1 )  (1  r ) s 1i wi    i  s T (1  r )  b 1

Theorem 1. The individual optimal consumption path is given by t s

cts 

s T  1 r  1 r  b g (T1 )  (1  r ) s 1i wi    t  s s  1   i  s T (1  r )  b 1  

 s  

(9)

1

Theorem 2. The individual optimal asset path is given by t

ats1  (1  r )t  (1  r )i (wis  cis ) t  s s  1 is

where wis and cis are given by (2) and (7) respectively.

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 s    1

(10)

4 THE OPTIMAL YEARS OF SCHOOLING In this section, we assume that the wage grows at the rate   r , i.e., wt 1  (1   )wt and the quantity of human capital grows at the rate  , i.e., g (T1 )  (1   )T , then wis  (1   )T (1   )i  s ws  i  s  T1  1

1

s T

s 1 i wis   (1  r )

i  s  T1

 s  T and

(1   )T (1  r ) s 1 s T  1       ws i  s T  1  r  (1   ) s i

1

1

 (1   ) (1  r )  r T1



 1   T 1  1   T       ws   1  r    1  r  1

Therefore,

css 

1  r  b (1   )T (1  r ) (1  r )  b  r 1

 1   T 1  1   T       ws   1  r    1  r  1

(11)

From (7), the individual consumption as a function of quantity of human capital reaches maximum when the initial consumption css reaches maximum. Therefore, from (11), we have follow theorem Theorem 3. If (1   )(1  )  1  r , then the optimal years of schooling is given by T1opt  T  1 

b1  1  ln 1   b1  ln(1   ) 

(12)

where b1  ln 11r .

Proof. Let h( x)  (1   ) x  11r  

T 1

x   11r   , then 

 1     1    (1   )(1   )  h( x)  (1   ) x    ln(1   )    ln 1 r  1 r   1  r   b opt which has a unique zero point T1  T  1  b1 ln 1  ln(1 )  and the function h( x) reaches its minimum at it.

T 1

x

1

1

Therefore, css has maximum value at the time s  T1opt . This completes the proof of the theorem.

Theorem 4. If (1   )(1   )  (1  r) , then the optimal years of schooling increases when the growth rate of the quantity of human capital g (T1 ) increases.

Proof. From (12),

dT1opt d

 (1 ) ln(11) ln

(1 )(1 ) 1r

 0 . The theorem holds.

Lemma 1. Let  b1  p(b1 )  b1  b1  ln(1   ) ln 1    ln(1   )  then p(0) =0 and p(b1 )  0 for  ln(1   )  b1  0 .

Proof. Since p(0)  0 and p(b1 )  ln 1  ln(1b )   0 , the lemma holds.

1

Theorem 5. If (1   )(1   )  (1  r ) and   r , then the optimal years of schooling increases when the wage growth rate  increases.

Proof. From (12) and Lemma 1, dT1opt dT1 db1 p(b1 )    0 d db1 d (1   )[b1  ln(1   )]b12

The theorem holds.

5 NUMERICAL SIMULATION In this section, the structural parameters of the model are given by - 46 www.ivypub.org/mc

s  0 ws  12 r  0065   003 T  60  80  064

Under these parameters, the optimal individual years of schooling is T1opt  22 . Figure 1(a) shows the maximal initial consumption with respect to individual years of schooling and Figure 1(b) presents the optimal individual years of schooling changes with respect to the parameters  and  . css

opt

T1

a

1.16

b

opt

T1

40

opt

T1

30

1.15

20 1.14

10

15

20

25

30

T1

0.03

0.04

0.05

0.06

0.07

FIG. 1 THE OPTIMAL INDIVIDUAL YEARS OF SCHOOLING

Figure 2 presents the individual saving behavior and his asset changes under different years of schooling. wts cts

ats

a

b

100

30

T1 28

50

20

T1 12

T1 22 10

T1 12 20

40

60

20 80

t

50

40

T1 22

60

80

t

T1 28

10

100 20

FIG. 2 THE INDIVIDUAL SAVING AND ASSET CHANGE

6 CONCLUSIONS From Theorem 1 and Theorem 2, we see that the individual optimal consumption and saving behavior is determined by the individual wage, interest rate and years of schooling. When the wage grows at rate  and the quantity of human capital grows at rate  and satisfies (1   )(1   )  1  r , the optimal individual years of schooling is given by T1opt  T  1 

b1  1  ln 1   b1  ln(1   ) 

where b1  ln 11r .

The optimal individual years of schooling increases when the wage growth rate or the quantity of human capital growth rate increases (Theorem 4 and Theorem 5), which is presented by numerical simulation in Figure 1. The effect of schooling years on individual saving and consumption behaviors are shown in Figure 2. Comparing Figure 2 with Figure 1 (a), we see that although the individual has higher wage and saving in his working period (the blue saving curve in Figure 2 (a)), it is not his optimal choice since he must borrow more before he works (the blue asset curve in Figure 2 (b)). Also long working period is not the individual optimal choice and less years of schooling implies fewer wages and saving in the working period (The green saving curve in Figure 2 (a)).

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REFERENCES [1]

Modigliani F, and Brumberg R. “Utility analysis and the consumption function: an interpretation of cross-section data.” In the collected papers of Fronco Modiglianni, Vol.6, 3-45, The MIT Press, 2005

[2]

Modigliani F. “Life Cycle, Individual thrift, and the wealth of nations.” Science, New Series, Vol. 234, 4777(1986): 704-712

[3]

Tobin J. “Life Cycle Saving and Balanced Growth.” In W. Fellner et al., eds., Ten Economic Studies in the Tradition of Irving Fisher, New York, 1967

[4]

Kwack S. Y., and Lee Y. S. “What determines saving rates in Korea? The role of demography.” Journal of Asian Economics, 16(2005):861-873

[5]

Lau S. “Demographic structure and capital accumulation: A quantitative assessment.” Journal of Economic Dynamics & Control, 33(2009): 554-567

[6]

Kalemli-Ozcan S., Ryder H. E., and Weil D. N. “Mortality decline, human capital investment and economic growth.” Journal of development economics, 62(2000):1-23

AUTHORS Donghan Cai, Male, Han Nationality, PhD, Professor, Current

Zhongbin Chen, Male, Han Nationality, PhD, Associate

Research Interests: mathematical economy.

Professor, Current Research Interests: Development Economics.

Huan Yang, Femal, Han Nationality, Graduate student.

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