A Two-Sector Two-Group Growth Model with
Endogenous Time and Consumer Durables
Wei-Bin Zhang
Keywords: economic structure; heterogeneous households; endogenous time; economic growth
Abstract:
This paper proposes a two-sector two-group growth model with endogenous time distribution and
durable good consumption to examine dynamics of wealth and income distribution in a competitive
economy with capital accumulation as the main engine of economic growth. The model is
influenced by the neoclassical growth theory and the post-Keynesian theory of growth and
distribution. We show how to determine the economic equilibrium and examine long-term effects of
changes in the population, technology and preference on the economic structure. In Appendix we
also show how to find the dynamic equations to follow the motion of the economic system.
Prof. Wei-Bin Zhang
Ritsumeikan Asia Pacific University
1-1 Jumonjibaru, Beppu-Shi, Oita-ken、874-8577 Japan
(Home Tel: 0977-73-9787; Office Tel: 0977-78-1020; Fax: 0977-78-1123;
E-mail: [email protected])
1
1 Introduction
This paper proposes a two-sector two-group growth model to examine dynamics of wealth and
income distribution in a competitive economy with capital accumulation as the main engine of
economic growth. Our approach is influenced by the neoclassical growth theory and the postKeynesian theory of growth and distribution. Kaldor (1955) first formulated this theory in his
seminal article in 1955. Pasinetti (1962) reformulated the Kaldor model and introduced explicitly
the assumption of steady growth. Since then, many models related to distribution and growth have
been published (see, for instance, Samuelson and Modigliani, 1966, Pasinetti, 1974, and Salvadori,
1991). Nevertheless, the most of these models with heterogeneous classes assume that the economic
system has a single production sector. A growth model of two sectors and two classes is proposed
by Stiglitz (1967). The Stigliz model extends the Solow model by a breakdown of the productive
system into two sectors using capital and labor, one of which produces capital goods, the other
consumer good. The model is a synthesis of the post-Keynesian theory and Uzawa’s two-sector model.
But only a few studies have been made to extend and generalize Stiglitz’s model. The purpose of this
study is to examine the economic issues addressed by the Stigliz model but in an alternative approach
to household behavior. Moreover, we make labor supply as an endogenous variable in the model, while
the Stiglitz model fixes the labor supply. The remainder of this study is organized as follows. Section
2 defines the two-sector two-group model. Section 3 identifies equilibrium of the model by
simulation when the production functions take on the Cobb-Douglas form. Section 4 examines
effects of changes in some parameters on the equilibrium of the system. Section 5 concludes the
study. The appendix provides a procedure to find out the explicit expression of two-dimensional
differential equations system how to determine values of all the variables at any point of time.
2 The two-sector two-group model
The production side of the economic system is similar to the traditional two-sector growth models
(Burmesiter and Dobell, 1970, Barro and Sala-i-Martin, 1995). Consumption and capital goods are
different commodities, which are produced in two distinct sectors. Capital goods can be used as an
input in both sectors in the economy. Capital depreciates at a constant exponential rate  k , which is
independent of the manner of use. The two sectors use labor and capital good as inputs. The two inputs
are smoothly substitutable for each other in each sector and are freely transferable from one sector to
the other. Both sectors use neoclassical technology with the standard Inada conditions. The production
functions are given by
F j K j t , N j t , j  i , s ,
where F j are the output of sector j , K j and N j are respectively the capital and labor used in sector
j , F j the production functions, the subscripts i and s denote respectively the capital good sector and
the consumer goods sector. As F j are neoclassical, we have
f j t   f j k j t ,
f j t  
Fj t 
N j t 
, k j t  
K j t 
N j t 
The functions f j have the following properties
(i) f j 0   0 ;
, j  i , s.
2
(ii) f j are increasing, strictly concave on R  , and C 2 on R   ; f j' k j   0 and f j" k j   0; and
(iii) lim k j 0 f j' k j    and lim k j  f j' k j   0 .
We class the labor force into two groups. Let T j t  stand for the work time of a representative
household of group j and N t  for the flow of labor services used at time t for production. We
assume that labor is always fully employed. We measure N t  as follows
2
N t    h jT j t N j ,
(1)
j 1
where h j are the levels of human capital of group j , j  1, 2 .
Markets are perfectly competitive; thus labor and capital earn their marginal products. We assume that
the capital good serves as a medium of exchange and is taken as numeraire. The price of consumer
good is denoted by pt . The rate of interest r t  and wage rate wt  of per unit qualified labor force
are determined by markets. Hence, for any individual firm r t  and wt  are given at each point of
time. The production sector chooses the two variables K j t  and N j t  to maximize its profit. The
marginal conditions are given by


r   k  f i ' ki   pf s' k s , wt   f i ki   ki f i ' ki   pt  f s k s   k s f s' k s  .
(2)
The wage rate of group j is given by w j t   h j wt , j  1, 2 . Total capital stock K t  is allocated to
the two sectors and the households of the two groups. As full employment of labor and capital is
assumed, we have
2
~
Ki t   K s t    K j  K t , Ni t   N s t   N t ,
(3)
j 1
~
where K j t  is the level of consumer durables of the representative consumer of group j , j  1, 2 .
We rewrite the above equations as
2
~
ni t ki t   ns t ks t    n~j t k j t   k t , ni t   ns t   1,
(4)
j
where
~
N q t 
K j t  ~
Nj
~
K t 
k t  
, nq t  
, q  i , s , k j t  
, n j t  
, j  1, 2 .
N t 
N t 
Nj
N t 
Let k j t  denote per capita wealth of group j at t . According to the definitions, we have
2
K t    k j t N j .
j 1
(5)
3
Consumers make decisions on choice of levels of consumer durables, services, commodities, and
saving. This study uses the approach to consumers’ behavior proposed by Zhang (e.g., 2005). A
detailed explanation of the approach, its relations to the other (deterministic) approaches to consumer
behaviors with savings in economics, and many applications of the approach to economic dynamics
are examined in Zhang (2005). Group j ’s per capita current income y j t  from the interest payment
r t k j t  and the wage payment T j t w j t  is defined by
y j t   r t k j t   Tj t wj t .
The sum of money that consumers are using for consuming, saving, or transferring are not necessarily
equal to the current income because consumers can sell wealth to pay, for instance, current
consumption if the temporary income is not sufficient for purchasing goods and services. Retired
people may live not only on the interest payment but also have to spend some of their wealth. The total
value of wealth that consumer j can sell to purchase goods and to save is equal to k j t . Here, we do
not allow borrowing for current consumption. We assume that selling and buying wealth can be
conducted instantaneously without any transaction cost. This is evidently a strict assumption as it may
take time to draw savings from bank or to sell one’s properties. The per capita disposable income of
consumer j is defined as the sum of the current income and the wealth available for purchasing
consumer good and saving
yˆ j t   y j t   k j t   1  r t k j t   Tj t wj t , j  1, 2.
(6)
The disposable income is used for saving and consumption. At each point of time, a consumer would
~
distribute the total available budget among savings s j t , consumer durables k j t , and consumer
good c j t . The budget constraints are given by
~
pt c j (t )  r t    k k j t   s j (t )  yˆ j (t )  1  r t k j t   T j t w j t .
(7)
The total disposable income ŷ j is distributed among consuming consumer good, utilizing consumer
durables, and saving.
Denote Thj t  the leisure time at time t and the (fixed) available time for work and leisure by T0 . The
time constraints are expressed by Tj t   Thj t   T0 . Substituting this relation into (7) yields
~
Thj t w j t   pt c j (t )  r t    k k j t   s j (t )  y j (t )  1  r t k j t   T0 w j t .
(8)
We assume that the utility level U j (t ) that the consumer of group j obtains is dependent on c j t ,
~
k j t  and s j t . The utility level U j t  of group j is specified as follows
~



U j t   Thj 0 j (t )c j 0 j t k j 0 j t s j 0 j t ,  0 j , 0 j , 0 j , 0 j  0 , j  1, 2 ,
(9)
4
where  0 j ,  0 j ,  0 j and 0 j are respectively group j ’s propensities to consume consumer good, to
utilize consumer durables, and to own wealth. Here, for simplicity, we specify the utility function with
the Cobb-Douglas from. Maximizing U j subject to the budget constraints (8) yields
~
w j t Thj t    j y j t , pt c j t    j y j t , r t    k k j t    j y j t , s j t    j y j t ,
(10)
where
 j   j  0 j ,  j   j 0 j ,  j   j 0 j ,  j   j 0 j ,  j 
1
.
 j   j   j  j
According to the definitions of s j t , the wealth accumulation of the consumer of group j is given by
k j t   s j t   k j t .
(11)
The output of the consumer goods sector is consumed by the households. That is
2
 c t N
j 1
j
j
 Fs t .
(12)
As output of the capital goods sector is equal to the depreciation of capital stock and the net savings,
we have
2
 s t N
j 1
j
j
 K t    k K t   Fi t .
(13)
It can be shown that the equation (13) is redundant. We have thus built the dynamic model. We now
examine equilibrium of the model. In the appendix, we show that the dynamics can be described by 2 dimensional differential equations. As the expressions for a genuine dynamic analysis are very tedious,
we are concerned with equilibrium behavior of the model.
3 Equilibrium with the Cobb-Douglas production functions
We specify production functions in the Cobb-Douglas form
f j k j   Aj k j j , Aj  0 , 0   j  1, j  i , s .

(14)
From the equations (14) and (2), we directly obtain
ks  ki ,
where

i s
,  j  1   j , j  i, s .
 s i
(15)
5
The capital intensity of the consumer goods sector is proportional to that of the capital good sector.
By k s  ki and i fi   s pf s , we solve
w  i fi , p 
 i Ai
ki

 s As
i
s
 s
.
(16)
The price of consumer good is positively related to the technological level of the capital good sector
but negatively related to that of the consumer good sector. The price is positively or negatively related
to the capital intensity of the capital good sector, depending on the sign of  i   s . If  i   s , then
the price is constant, p  Ai / As . In the remainder of this section, we require  i   s . If the equality
holds, then labor distribution is invariant in time. The analysis of  i   s is similar. The condition
 i   s (which implies   1 ) guarantees ks t   ki t  for any t .
By (11) we have s j  k j at equilibrium. From s j  k j and s j   j y j , we solve k j   j y j . From
k j   j y j and the definition of y j , we have
 
k j   j ki 
h j ki
i
 j ki  i Ai
(17)
,
where  j  1/  j  1   k and h j  T0 i Ai h j . From the definition of k and (5), we have
 2
1
k    k j N j  .
 j 1
N
By this equation, we can rewrite (12) and (13) as
ns 
s ki
N
,
ni 
i ki
N
,
(18)
where we use c j   j k j /  j p , s j  k j and

 s ki    
j 1
2

 2

N j j k j  1

, i ki    N j k j  k .

 j  p fs
 j 1
 fi
 
~
Substituting k j   j y j and r   k  f i ' ki  into r   k k j   j y j , we solve
~
kj 
 jk j
 j fi ' ki 
(19)
.
From (18) and ni  ns  1, we have
 
 
 
N   ki   s ki   i ki .
(20)
6
By w jThj   j y j , k j   j y j and w j  h j  i f i , we solve
Thj 
 jk j
fi
(21)
,
where  j   j /  j h j i . From (21) and the definition of N , we solve
2 
 k 
N    T0  j j  h j N j ,
fi 
j 1 
(22)
where we also use Tj t   Thj t   T0 . Equalizing the right-hand sides of (20) and (22) yields
2 
 k 
 ki   ki    T0  j j  h j N j  0 .
fi 
j 1 
 
 
(23)
Summarizing the above analysis, we get the following lemma.
Lemma 1
We determine equilibrium of the dynamics of the economic system by the following procedure: ki by
(23) → k j , j  1, 2 by (17) → ks  ki → f j by (14) → y j  k j /  j → p by (16) → N by (20)
~
→ n j by (18) → N j  n j N → r and w by (2) → w j  h j w → Thj by (21) → k j , c j and s j by (10)
→ K  kN → K j  k j N j → F j  f j N j .
It is difficult to explicitly interpret the equations because the interactions among the variables are
complicated. As the lemma provides the procedure to determine all the variables, we can identify
equilibrium of the system by simulation. We specify parameter values as follows:
N1  6 , N 2  18 , Ai  2 , As  1, i  0.38 ,  s  0.34 , h1  2 , h2  0.5 ,  k  0.05 ,
01  0.7 ,  01  0.1, 01  0.1, 01  0.1, 02  0.65 ,  02  0.1, 02  0.07 , 02  0.08. (24)
Group 2' s population is 3 times as many as group 1' s . Group 1' s level of human capital is 4 times
as high as group 2' s . Group 1 has lower (relative) propensity to save (which equals 0.7 ) than group
2' s (which equals 0.65 / 0.9  0.72 ) and group 1 has lower propensity to enjoy leisure time than
group 2 . Figure 1 demonstrates that equation (23) has a unique meaningful solution. The equation has
actually two positive solutions, ki  2.29 , ki  9.30 , as demonstrated in Figure 1. Nevertheless, at
ki  2.29 the work times of the two groups do not belong to 0 , 1.

7
20
100
50
50
100
ki 
1
2
3
15
ki 10
5
4
5
ki 
5
10
15
ki
20
Figure 1. A Unique Meaningful Solution of Equation (23)
We list the equilibrium values of the variables as in (25). As p  2.18 and Fs  13.9 , we see that the
consumer sector makes more contribution to the national economy than the capital good sector
( Fi  11.37 ). Most of the labor force is employed by the consumer good sector. A household of group
1 consumes more consumer good, holds more wealth, uses more consumer durables than a household
of group 2 . A household of group 2 works less hours than group 1.
F  41.74 , Fi  11.37 , Fs  13.94 , K  227.40 , Ki  22.65 , K s  54.14 ,
N  9.37 , Ni  2.44, N s  6.93, fi  4.67 , f s  2.01, ki  9.30 , ks  7.81,
ni  0.26 , ns  0.74 , p  2.18 , r  0.141, w1  5.79 , w2  1.45 , c1  1.32 , c2  0.34,
~
~
(25)
Th1  0.50 , Th 2  0.63, k1  20.10 , k2  5.93, k1  15.06 , k2  3.35,
in which F  Fi  p Fs .
4 Effects of Changes in the Population, the Preference and Technology
This section studies effects of changes in some parameters on the equilibrium. First, we respectively
increase group 1' s population from N1  6 to 7 and group 2' s population from N2  18 to 19 . We
list the results in Table 1. In the table, a variable x j stand for the change rate of the variable x j in
percentage due to changes in the parameter value from N10 (  6 in this case) to N11 (  7 ). That is
x j 
x j N11   x j N10 
x j N10 
 100 .
We will use the symbol  with the same meaning when we analyze other parameters. The numbers
not in (in) parenthesis are the changes in the equilibrium values caused by change in group 1' s
( 2' s ) population. We see that as any group’s population is increased, the output levels of the national
economy, the capital good sector and consumer good sector, the total capital stocks, the total labor
force, and the capital stocks and the labor force employed by the two sectors are increased.
Nevertheless, the effects of the outputs per labor unit and capital intensities of the two sectors, the
wage rates, consumption levels of consumer good, leisure times, and wealth and amount of durable
goods per household are opposite with regard to population increases of the two groups. It is easy to
see why the aggregated variables are increased as the population is increased. To see why the
population changes of the two groups have the opposite effects on the variables in terms of per capita,
we notice that group 1 has lower propensity to save and lower propensity to enjoy leisure time, but
higher level of human capital than group 2 . The differences in human capital and preference between
8
the two groups result in the opposite effects on the variables in terms per household in the population
changes.
Table 1 The Effects of Change in Group 1' s (Group 2' s ) Population
F
9.38(2.43)
Fi
8.94(2.58)
N i
9.49(2.39)
N s
10.11(2.19)
ns
p
0.15 (-0.05) -0.05(0.02)
Th1
0.55(-0.19)
Th 2
0.65(-0.23)
K
Fs
K i
9.61(2.35) 8.94(2.58) 8.04(2.88)
f i
f s
 ki
-0.51(2.19) -0.45(0.16) -1.33(0.48)
 w2
w1
r
K s
11.21(3.47)
ks
-1.33(0.48)
 c1
1.13(-0.04) -0.51(0.18) -0.51(0.18) 0.10(-0.03)
~
~
k2
k2
k1
k1
0.05(-0.01) 0.15(-0.05) -0.78(0.28) -0.68(0.25)
N
9.95(2.44)
 ni
-0.41(0.15)
c2
0.20(-0.07)
We now respectively increase group 1' s human capital as h1 : 2  2.2 and group 2' s human capital
as h2 : 0.5  0.6. We list the results in Table 2. The numbers not in (in) parenthesis are the changes
in the equilibrium values caused by change in group 1' s ( 2' s ) human capital. As any group’s human
capital is improved, the output levels of the national economy, the capital good sector and consumer
good sector, the total capital stocks, the total labor force, and the capital stocks and the labor force
employed by the two sectors are increased. As the effects of the population changes, the effects of the
outputs per labor unit and capital intensities of the two sectors, the wage rates, consumption levels of
consumer good, leisure times, and wealth and amount of durable goods per household are not in the
same directions with regard to human capital improvement of the two groups.
Table 2 A Rise in Group 1' s (Group 2' s ) Human Capital
F
5.63(8.74)
N i
5.70(8.61)
ns
0.09(-0.18)
Th1
0.34(-0.65)
Fi
Fs
5.36(9.28) 5.76(8.47)
f i
N s
K
5.37(9.28)
f s
6.06(7.88) -0.32(0.61)
-0.28(0.55)
r
p
-0.03(0.06) 0.70(-1.35)
w1
9.65(0.61)
Th 2
k2
k1
0.41(-0.77) 10.03(-0.04) 0.09(19.81)
K i
4.82(10.37)
 ki
-0.83(1.62)
 w2
K s
6.73(12.49)
ks
-0.83(1.62)
 c1
N
5.97(8.07)
 ni
-0.26(0.50)
c2
-0.31(20.74) 10.07(-0.11) 0.12(19.73)
~
~
k2
k1
9.47(0.96)
-0.43(21.01)
We now respectively increase group 1' s propensity to save as 01 : 0.7  0.74 and group 2' s
propensity to save as 02 : 0.65  0.7 . The numbers not in (in) parenthesis are the changes in the
equilibrium values caused by change in group 1' s ( 2' s ) propensity to save.
Table 3 Effects of Rises in the Propensities to Save: 01  0.04 ( 02  0.05 )
F
2.14(2.63)
Fi
4.33(5.47)
Fs
1.16(1.35)
f i
K
4.33(5.47)
f s
N
K i
K s
7.03(9.18) 5.13(6.68) 0.52(0.43)
 ni
ks
 ki
N i
N s
2.70(3.25) -0.25(-0.56) 1.58(2.14)
1.41(1.92)
4.21(7.74) 4.21(5.74) 2.17(2.81)
 w2
 c1
r
ns
p
w1
c2
-0.76(-0.99) 0.17(0.22)
-3.42(-4.61) 1.58(2.14)
1.58(2.14) 2.46(-0.33) -0.55(3.55)
~
~
Th1
Th 2
k2
k2
k1
k1
1.01(-2.21) -1.94(1.60) 8.50(-0.11) -0.39(11.76) 5.29(3.41) 2.19(7.43)
9
Table 4 lists the effects of the following change: Ai : 2  2.2 .
Table 4 A Rise in the Capital Sector’s Productivity
F
16.62
Fi
16.62
N i
0
ns
0
Th1
0
N s
0
p
10.68
Th 2
0
Fs
16.62
f i
K
16.62
f s
0
5.37
r
0
w1
0
k2
16.62
k1
16.62
K i
16.62
K s
21.55
 ki
16.62
 w2
ks
16.62
 c1
0
5.37
~
k2
16.62
~
k1
16.62
N
0
 ni
0
c2
5.37
Appendix: The Motion of the System Described by Two Differential Equations
We now analyze dynamics when the production functions take on general forms. From
r t    k  f i ' ki   pf s' k s  in (2), we get
p
f i ' ki 
.
f s' k s 
(A1)
Substituting this equation into f i ki   ki f i ' ki   p f s k s   k s f s' k s  in (2) yields
i ki  
f i ki 
f k 
 ki  s' s  k s  s k s .
'
f i ki 
f s k s 
(A2)
From the properties of f i ki  , we can show that the function i ki  has the following properties:
i 0  0 , i ki   0 for ki  0 , i' ki   
fi ki  fi" ki 
 0.
fi '2 ki 
The function s ks  has the same properties in k s . We see that for any given ks  0 , the equation
(A2) determines ks  0 as a unique function of ki , denoted by ks  ki . We have 0  0 and
'  0 .
The capital intensity of the consumer goods sector is proportional to that of the capital good sector.
By ks  ki  and f i ki   ki f i ' ki   p f s ks   ks f s' ks , we solve fi   s pf s
p
f i ki   ki f i ' ki 
.
f s ki   ki  f s' ki 
The price of consumer goods is a function of ki t .
According to the definitions, we can rewrite (12) and (13) as
10
2
 n~ t c t   n t  f t ,
j 1
j
j
s
s
2
 n~ t s t   k t   n t  f t .
j 1
j
j
i
i
(A3)
Substitute pc j   j y j and s j   j y j into (A3)
1n~1  2 n~2   y1   pns f s 
 n~  n~   y   n f  k  .
2 2  2 
 1 1
 i i

(A4)
Solving (A4) with y j with variables
2 ns pf s  ni f i  k  2
,
n~1
 n f  k   1ns pf s
y2  1 i i
,
n~2
y1 
(A5)
~
where   12  21 (  0 assumed). Substituting (A5) and r   k  f i ' ki  into r   k k j   j y j ,
we solve
~   n pf  n f  k  2 
1 ,
k1   2 s s ~ 'i i



n
f
k
1 i
i


~   n f  k   1ns pf s 
2 .
k2   1 i i '
fi ki n~2


(A6)
Substitute (A6) into (4)
 i ki ni   s ki ns  k ,
(A7)
where
i ki  
 f i ' ki ks  12 pf s  21 pf s 
fi ' ki ki  21 fi  1 2 fi

ns .


,

k

s
i
fi ' ki   2  1
fi ' ki   2  1


Solve (A7) and ni  ns  1
ni   i ki , k  
 s ki   k
k   i ki 
, ns   s ki , k  
.
 s ki    i ki 
 s ki    i ki 
We can now consider ni t  and ns t  as functions of k t  and ki t .
Substitute (A5) into w jThj   j y j
(A8)
11
 12 ns pf s  ni f i  k  1 2
,
n~1
  n f  k    21ns pf s
w2Th 2  2 1 i i
.
n~2
w1Th1 
(A9)


2
Inserting T j  Thj  T0 , wj  h j fi ki   ki fi ' ki  and n~j  N j /  j 1 h j N jT j into (A9), we obtain
T0  1h1 N1  1T1  1h2 N 2T2 ,
T0  h1 N12T1  h2 N 22  1T2 ,
(A10)
where we use T j  Thj  T0 and
1 ki , k  
 12 ns pf s  ni f i  k  1 2
  n f  k    21ns pf s
, 2 ki , k   2 1 i i
'
 fi ki   ki fi ki h1N1
 fi ki   ki fi ' ki h2 N 2 .
Solve (A10) with T j as variables


h2 N 22  1h2 N 2  1
T0 ,
T1  1 ki , k   
 h2 N 22  1h2 N 2  1h1 N12  h2 N 22  11h1 N1  h1 N12  1 


1h1 N1  h1 N12  1
T0 .
T2  2 ki , k   





h
N



h
N

1
h
N


h
N


1

h
N

h
N


1
1 2 2
1 1 2
2 2 2
1 1 1
1 1 2
 2 2 2

(A11)
Insert s j   j y j into (13)
2
 n~  y
j 1
j
j
j
 k  ni fi ki .
(A12)
Substituting (6) into (A12) yields
2
2
j 1
j 1
 n~j  j 1  r k j   n~j  j wj  k  ni fi ki .


Insert r  f i ' ki    k and wj  h j fi ki   ki fi ' ki  into (A10)
2




2
 n~j  j fi ' ki    k j  fi ki   ki fi ' ki   n~j  j hj  k  fi ki ni .
j 1
(A13)
j 1
Insert ni in (A8) into (A13)
2




2
 n~j  j fi ' ki    k j  fi ki   ki fi ' ki   n~j  j h j 
j 1
j 1


fi ki  s ki 
fi ki 
k ,
   
 s ki    i ki  
 s ki    i ki  
(A14)
12
According to the definitions, we can rewrite (5) as
2
kNˆ ki , k    N j k j ,
(A15)
j 1
where
Nˆ ki , k   h1N11 ki , k   h2 N2 2 ki , k .
Substitute n~j  N j / Nˆ ki , k  into (A14)
1 ki , k k1   2 ki , k k2  ki , k 
(A16)
where we use n~j  N j / Nˆ and
 f ' k    
 j N j , j  1, 2 ,
 j ki , k    i i

ˆ
 N ki , k  
2  h N


f k  k 
fi ki 
k  fi ki   ki fi ' ki   j j j .
ki , k   i i s i    
ˆ
 s ki    i ki  
 s ki    i ki  
j 1 N ki , k 


We solve (A15) and (A16) with k1 and k2 as variables
N 2ki , k   k2 ki N ki , k 
,
N 21 ki   N12 ki 
kNki , k 1 ki   N1ki , k 
k2  2 ki , k  
.
N 21 ki   N12 ki 
k1  1 ki , k  
(A17)
Hence, we can express k1 t  and k2 t  as functions of ki t  and k t  at any point of time. Taking
derivatives of (A17) with respect to t yields
k1  1ki ki , k ki  1k ki , k k ,
k2  2ki ki , k ki  2k ki , k k .
(A18)
Insert (A8) and n~j  N j / Nˆ ki , k  into (A5)
  pf  k , k    fi i ki , k   k 2  ˆ
 N ki , k ,
y1   2 s s i
N1


   f  k , k   k   1 pf s s ki , k   ˆ
 N ki , k .
y2   1 i i i
N 2


(A19)
13
We can express y j t  as functions of ki t  and k t .
Insert s j   j y j and (A19) into (11)
  pf  k , k    fi i ki , k   k 2  ˆ
1 N ki , k   1 ki , k ,
k1  ~
s1 ki , k    2 s s i
N1


   f  k , k   k   1 pf s s ki , k   ˆ
2 N ki , k   2 ki , k .
k2  ~
s2 ki , k    1 i i i

N
2


(A20)
Delete k1 t  and k2 t  from (A18) and (A20)
1ki ki , k ki  1k ki , k k  ~
s1 ki , k ,
2 ki ki , k ki  2 k ki , k k  ~
s2 ki , k .
(A21)
Solve (A21) with ki t  and kt  as variables
~
s1 ki , k 2 k ki , k   ~
s2 ki , k 1k ki , k 
,
1ki ki , k 2 k ki , k   1k ki , k 2 ki ki , k 
~
s2 ki , k 1k i ki , k   ~
s1 ki , k 2 k i ki , k 
k 
.
1k i ki , k 2 k ki , k   1k ki , k 2 k i ki , k 
ki 
(A22)
The two differential equations contain two variables. With proper initial conditions ki t  and k t , the
above equations determine values of ki t  and k t  over time. It is straightforward to get the procedure
for determining all the other variables at any time as in Lemma A1.
Lemma A1
The dynamics of the economic system is governed by the 2 -dimensional differential equations (A22)
with ki t  and k t  as the variables. All the other variables can be determined as functions of ki t  and
k t  at any point of time by the following procedure: k j t , j  1, 2 by (A17) → ks t  by (A2) →
f q t   f q kq t  → y j t  by (A19) → pt  by (A1) → ni t  and ns t  by (A8) → r t  and wt  by
~
(2) → w j t   h j wt  , j  1, 2 → T j t  by (A11) → k j t  , c j t  and s j t  by (10) → N t  by (A17)
→ n~ t   N / N t  → N t   n t N t  , q  i, s → K t   k t N t  → K t   k t N t 
j
j
q
q
→ Fq t   f q t N q t  → U j t  by (9).
q
q
q
q