=Paper=
{{Paper
|id=Vol-2093/paper3
|storemode=property
|title=Optimal control of human capital allocation for piecewise smooth dynamic model
|pdfUrl=https://ceur-ws.org/Vol-2093/paper3.pdf
|volume=Vol-2093
|authors=Bolodurina Irina,Golovatskaia Elena,Khanzhina Natalya,Anciferova Larisa
}}
==Optimal control of human capital allocation for piecewise smooth dynamic model==
https://ceur-ws.org/Vol-2093/paper3.pdf
24
Optimal Control of Human Capital Allocation for
Piecewise Smooth Dynamic Model
Irina Bolodurina1[0000-0003-0096-2587], Elena Golovatskaia, Natalya Khanzhina and Larisa
Anciferova1[0000-0003-6151-6276]
Department of Applied Mathematics, Orenburg State University, Orenburg, Russian Federation
ipbolodurina@yandex.ru, natalyoren@mail.ru
Abstract. The article presents optimal control of human capital allocation under
the condition of smoothness abnormality in the right-hand member of the dy-
namic system. It describes an individual’s behavior who strives to leave to
his/her heirs the maximum of cash and tries to get the maximum utility through
controlling expenditures and time consumption for work and studies. The objec-
tive function of the model maximizes the human life utility. The latter is formed
by the human capital development and financial savings for heirs. The authors
developed special methods how to find optimal control and take into account
smoothness abnormality in the right-hand member of the dynamic system. They
defined the optimality conditions and designed algorithmic support and soft-
ware to find optimal control and interpret its content.
Keywords. Optimal control · human capital · maximum principle · piecewise
smooth dynamic system.
1 Introduction
Nowadays the design and development of new technologies, computing and informat-
ics are becoming of particular importance to all spheres of social life. Because of that
workforce requires excellent qualifications and specific knowledge. In this regard we
can state that human capital is the key factor for society development. Besides the
modeling in human capital allocation as well as its management to achieve the maxi-
mum results are the topical modern issues to study.
Human capital is the combination of physical and mental capacities of a person,
his/her knowledge, abilities and skills, life and professional experience used in public
reproduction [1]. The acquisition of human capital is made up from the investments
into health, safety, life quality, science, education and culture. One of the major fac-
tors leading to human capital growth is the investment in education and development
of employees’ professional qualifications. And the investment to maintain healthy
lifestyle provides the increase in the use duration of human capital and the improve-
ment of its quality.
As time goes on, investments into human capital are to provide incomes to its
holder. Though spending part of his/her employment time on education, a person
25
cannot count on the full volume of potential salary. Thus there is a challenge to allo-
cate a reasonable portion of time for education.
To respond to this challenge let’s apply K. Pohmer model designed for the optimal
management of personal income allocation [2].
2 Mathematical Model of Human Capital Allocation and Its
Numerical Solution
Let’s assume that at the life beginning any individual has an original human capital
H 0 , i.e. innate abilities, health, etc., and an original financial capital K 0 , i.e. finan-
cial funds of parents, a state, etc.
H (0) H 0 , K (0) K 0 . (1)
Human capital changes as the result of education attainment, life and professional
experience acquisition. Let’s consider the function of human capital raising:
F H (t ), s(t ), l (t ) , (2)
where
H (t ) is human capital;
l (t ) is the fraction of total time spent for working;
s (t ) is the amount of employment time spent for education and professional exper-
tise development.
Note that as the time goes, knowledge becomes less relevant and thus loses its val-
ue . The changing of human capital will look like
H (t ) F H (t ), s(t ), l (t ) H (t ) , (3)
where is the loss index of acquired skills, it equals to amortization norm for fixed
capital.
Human capital affects financial capital and we can write the rate of financial capi-
tal change as
K (t ) iK (t ) rH (t ) g (s(t ))l (t ) c(t ) , (4)
where
K (t ) is the financial capital;
i is the capital cost factor;
r is the price per human capital unit;
c(t ) is the consumption;
g (s) is the potential salary in relation to education level which is subject to the
time spent.
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g (s) is the production capability curve that defines the correlation between educa-
tion and earnings: the lower (higher) is s , the more (less) is potential salary though
the level of human capital acquisitions is lower (higher). Thus g (s) satisfies the con-
ditions:
g (0) 1 , g (1) 0 , g s 0 , g ss 0 . (5)
Derived criteria (5) satisfy to the function
g (s) as 2 bs c , a 0 . (6)
Applying to conditions (6) we will define the required function
g ( s) 1 (1 a) s as 2 , a 0 . (7)
So the model for human capital distribution has two processes
H (t ) F H (t ), s(t ), l (t ) H (t ) , (8)
K (t ) iK (t ) rH (t ) g ( s(t ))l (t ) c(t ) , (9)
with initial conditions
H (0) H0 , K (0) K0 , (10)
and restrictions imposed on controllable variables
c(t ) 0 , 0 l (t ) 1 , 0 s(t ) 1 . (11)
Under restrictions we mean a constant consumption of goods and the poverty of
employment time, for example, retirement period. The employment time is to be less
than 1, as an individual needs to have a rest. The fraction of total time spent for a rest,
sleeping included is ( 1 l (t ) ).
One of the priorities for any individual is to give birth to healthy off-springs and
sustain their growth. To achieve these goals a person is to increase his/her human
capital and raise financial capital for his/her heirs. So the objective function of the
model maximizes human life utility made up of human capital development and fi-
nancial savings for heirs.
T
I (c, s, l ) U (t , c, l , H )e t dt Z K (T ) max,
(12)
0
where
T is the duration of an individual’s life;
is the time preference rate of an individual;
27
U (t , c, l , H ) is the utility function dependent on consumption, employment time
and human capital.
Z K (T ) is the utility function for heirs.
The functions U (t , c, l , H ) and Z K (T ) have a typical for economics definitions
c1 (1 l )1 H 1
U t , (13)
1 1 1
K (T )1 k
Z , (14)
1 k
where ratios , , and k point to utility elasticity and , and are the
measures for rest, human capital and heirs capital and t shows that with aging edu-
cation plays a greater role.
Production function for human capital looks like
F H sl , (15)
where is the parameter in the Cobb-Douglas function , is the partial elasticity of
human capital.
So the optimal management problem looks like
T
c(t )1 (1 l (t ))1 H (t )1 t K (T )1k
I (c, s, l )
1
1
t
1
e dt
1 k
max, (16)
0
H (t ) H (t ) s(t )l (t ) H (t ) , (17)
K (t ) iK (t ) rH (t )l (t ) 1 (1 a)s(t ) as(t ) 2 c(t ) , (18)
H (0) H0 , K (0) K0 , (19)
c(t ) 0 , 0 s(t ) 1 , 0 l (t ) 1 . (20)
The problem to solve (16) – (20) is an optimal control problem with a free right
end and control restrictions. Let’s use the Pontryagin maximum principle [3].
1 H sl H 2 iK rHg( s)l c
c1 (1 l )1 H 1 t (21)
t e ,
1 1 1
where conjugate variables 1 and 2 satisfy to
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1 1 H 1sl 2 rg ( s )l tH e t ,
1 (T ) 0,
(22)
2 i 2 ,
k
2 (T ) K (T ) .
Optimal control is defined when the Pontryagin function is at maximum
с 1 t
с1 t
2с e max 2с e , (23)
1 c 0
1
1H ls (a 1) 2rHls a 2 rHls 2
max 1H ls (a 1) 2 rHls a 2rHls2 ,
0 s 1
(24)
H sl rHg ( s)l
1 l
1
e t
1 2
1
(25)
max 1H sl 2 rHg ( s )l
1 l 1 e t .
0 l 1
1
Maximum of expressions (23) – (25) is possible when
,
1
c 2e t (26)
0, s* 0,
a 1 1H 1
s s* , s* [0, 1], where s* , (27)
*
2a 2a 2 r
1, s 1,
1
0, l * 0, H s 2 rHg ( s) t
l where l * 1 1 e .
(28)
l * , l * 0,
Considering the types of formulae received, let’s use numerical solution to find the
optimal control.
To search an optimal path and optimal control we use software algorithm based on
iterative method [3]. The value parameters of the problem are equal to the values in
the table 1.
29
Table 1. The value parameters of the problem
2 1,5 0,8
0,4 0,0015 0,8
0,01 k 0,8 0,35
1 r 1 0,01
a 0,3 i 0,04 T 75
H0 1 K0 40
Figures 1 and 2 demonstrate the results we got after applying the software.
Fig. 1. Temporal dynamics of human capital and financial capital
Fig. 2. Temporal variation of optimal control
30
The resulting graph of the control parameter “employment time” variation shows
its maximum at the age of 14. It certifies the possibility of time saving as a young
body needs less rest. And the graph for s (t ) parameter dynamics shows that all em-
ployment time is spent for education, both at school or university, and self-education
as well. As the time goes, the amount of time for education is decreasing as an indi-
vidual has accumulated some knowledge and tries to use it in his/her activities. The
employment time decreases because of changes in personal life like marriages, birth
of children. According to the statistical data an average person finishes his/her labor
activity at the age of 70.
Under control parameters an individual’s human capital grows by the age of 40. By
that age an individual gets a certain professional expertise. Besides investments in
education characterizing the period of negative capital at the age period between 18
and 37, and acquired experience provide stable income. The financial capital
curve K (t ) starts to grow. The growth of consumption c(t ) follows the growth of an
individual’s income.
3 Optimal Control Problem for Discontinuous Dynamic System
Let’s consider the right-hand smoothness abnormality in dynamic restrictions when i
parameter characterizes capital cost factor and has two values depending on negative
or positive sign of K (t ) :
i1, K (t ) 0,
i (29)
i2 i1, K (t ) 0,
i.e. the problem is:
T
c1 (1 l )1 H 1 t K (T )1 k
I (c, s, l )
1
1
t
1
e dt
1 k
max, (30)
0
H (t ) H sl H , (31)
i1K rHlg ( s) c, K (t ) 0,
K (t ) i2 i1, (32)
i2 K rHlg ( s) c, K (t ) 0,
H (0) H0 , K (0) K0 , (33)
c(t ) 0 , 0 s(t ) 1 , 0 l (t ) 1 , (34)
g ( s) 1 (1 a) s as 2 .
In the optimization problem to solve (30) – (34) the system of differential equa-
tions is the system with the discontinuous right-hand part
31
f (t , x(t ),u (t )), S (t , x) 0,
x (t ) 1
f 2 (t , x(t ),u (t )), S (t , x) 0,
where x ( H , K ) is the absolutely-continuous on the interval [0, T ] vector-function
of phase, u (c, s, l ) is the piecewise-continuous on the interval [0, T ] vector-
function of phase. On the strength of all evidences the switching surface S (t , x) is a
continuously differentiable function that looks like S (t , x) K (t ) .
Let’s consider the case of a single penetration of the switching surface by the path
function at , where is the switching point, i.e. the point where K ( ) 0 .
To formulate the theorem on the necessary optimal conditions for optimal control
we use the Pontryagin function:
(t , x, u, 1 (t )), S ( x) 0,
(t , x, u, (t )) 1
2 (t , x, u, 2 (t )), S ( x) 0,
where
(t ), S ( x) 0,
(t ) 1
2 (t ), S ( x) 0,
1 (t ) : [0, ) R 2 , 2 (t ) : [ , T ] R 2 ,
c1 (1 l )1 H 1 t
j (t , x, u, (t )) t
e j (t ), f j t , x(t ), u (t ) ,
1 1 1
j 1, 2 .
Theorem 1. The process ( x (t ), u (t ), ) , where is the switching point and
which is optimal for the problem to solve (30) – (34). This condition is satisfied with
the necessity of 1 , 2 functions that are not simultaneously equal to zero and
0 0 multiplier. Then the following conditions are fulfilled
1. optimal control u (t ) (c (t ), s (t ), l (t )) , t 0, T in all continuity points affords a
maximum of the Pontryagin function j (t , x (t ), u, j (t )) , j 1, 2 , in all allowa-
ble u (t ) :
j (t , x (t ), u (t ), j (t )) max j (t , x (t ), u, j (t )) , j 1, 2 ;
u (t )
2. the conjugate vector-function j (t ) ( 1j , 2j ) , j 1, 2 , satisfies to the system of
differential equation:
1j (t ) tH e t 1j H 1sl 2j rl g ( s) , j 1, 2 ,
32
2j (t ) i j 2j , j 1, 2 ;
3. transversability conditions:
12 (T ) 0 ,
22 (T ) K (T ) k ;
4. at which is the point of the switching surface penetration by the path function
the conjugate vector-function and the Pontryagin function meet the jump condition
S ( , x ( ))
1 ( 0) 2 ( 0) , S ( , x ( )) 0 ,
x
1( 0) 2 ( 0) ,
( f 2 ( , x( ), u ( 0)) f1 ( , x( ), u ( 0)), 2 ( ))
,
S ( , x( ))
, f1 ( , x( ), u ( 0))
x
where is the magnitude of jump in the point .
Thus we have a boundary value problem of the maximum principle of Pontryagin.
To solve it we use the Lagrange method of multipliers based on the simplifying the
original continuous problem of optimal control (30) – (34) to the discrete one.
To find optimal control and trajectories we used a numerical software algorithm
based on the gradient projection. Figure 3 and Figure 4 present the results of solving
the discontinuous problem of optimal control. On the diagrams the curves H1(t) ,
K1(t) , l1(t) , c1(t) , s1(t) demonstrate the solutions of the discontinuous problem
with the parameters i1 0,04 and i2 0,06 , and the curves H0(t) , K0(t) , l0(t) ,
c0(t) , s0(t) are the solutions for the smooth right-hand problem with
i1 i2 i 0,04 .
Fig. 3. Temporal changes of human and financial capital in smooth and discontinuous cases
33
Fig. 4. Dependency of optimal control on time in smooth and discontinuous cases
Table 2 presents some numerical results like the final time for education ( t1 ), the
final time for professional training ( t2 ), retirement age ( t3 ), left boundary of time
with negative capital ( t4 ), the right boundary of time with negative capital ( t5 ), the
age with maximum employment time ( tl max ) for smooth ( i1 0,04 ) and discontinu-
ous ( i2 0,06 ) cases.
Table 2. Values of some numerical characteristics
t1 t2 t3 t4 t5 tl max
i1 0,04 23,42 45,5 69,5 17,79 37,38 14,05
i2 0,06 10,5 45,5 68,05 24,11 24,15 26,06
The results received after the solution of the discontinuous problem for optimal
control prove the increase of “employment time” parameter in comparison with the
results of the smooth problem and the maximum of this parameter is at the age of 26.
Human capital demonstrates a slight increase. Professional education enters the edu-
cational experience on earlier stages. Because of the capital cost factor increase, the
period with negative capital is reduced considerably. The increase of the capital cost
reduces considerably the period with negative capital (Figure 5) and increases the
growth rate of financial capital curve. The consumption is constant.
34
Fig. 5. Dynamics of capital сhanges close by zero
In conclusion we can state that changes in capital cost factor happen through the
whole individual’s life. These changes when caused by the discontinuity of dynamic
restrictions in optimal control problem influence deeply in the allocation of human
capital and the time period for professional training in education acquisition.
References
1. Maximova, V.F. The investment into human capital: manual / V.F. Maximova. – M.: Eur-
asian Open University, 2010. – 54 p.
2. Pohmer, K. Microeconomic theory of individual and group incomes. – Studies in Contem-
porary Economics, Vol.16. – Berlin; Heidelberg; New York; Tokyo: Springer-Verlag,
1985. – 214 p.
3. Andreeva, E.A. Mathematical modelling: manual / E.A. Andreeva, V.M. Tziruleva. –
Tver: Tver State University, 2004. – 502 p.