# A Discrete Heterogeneous-Group Economic Growth Model with Endogenous Leisure Time

Authors: Wei-Bin Zhang
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

This paper proposes a one-sector multigroup growth model with endogenous labor supply in discrete time. Proposing an alternative approach to behavior of households, we examine the dynamics of wealth and income distribution in a competitive economy with capital accumulation as the main engine of economic growth. We show how human capital levels, preferences, and labor force of heterogeneous households determine the national economic growth, wealth, and income distribution and time allocation of the groups. By simulation we demonstrate, for instance, that in the three-group economy when the rich group's human capital is improved, all the groups will economically benefit, and the leisure times of all the groups are reduced but when any other group's human capital is improved, the group will economically benefit, the other two groups economically lose, and the leisure times of all the groups are increased.


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## Body

## 1. Introduction

The purpose of this study is to study an economic
growth model with heterogeneous households for providing insights into
relations between economic growth and income and wealth distribution. In the
economic growth literature, the Solow model is the starting point for almost
all analyses of economic growth [1].
Nevertheless, the Solow model does not provide a mechanism of endogenous
savings. Ramsey’s 1928 paper on optimal growth theory has influenced modeling
of consumers’ behavior since the late 1960s [2]. This approach tends
to be associated with higher dimensional dynamic systems. The approach often
makes the analysis intractable even for a simple economic growth problem. In
his original contribution to growth theory with capital accumulation, Diamond
[3] used the overlapping generations structure as proposed by Samuelson
[4] to examine the long-term dynamical efficiency of competitive production
economies. The model has become a standard tool in macroeconomics to study
economic dynamics in discrete time. These seminal papers were technically
refined and generalized in different ways over years [5–7], and many other factors, such as human capital,
population growth, and innovation, have been introduced into these analytical
frameworks (e.g., [8–14]). The purpose of
this study is to examine growth issues with endogenous time and heterogeneous
groups. It should
be remarked that multigroup growth models with endogenous savings can be found
in literature of economic growth within the traditional approaches (see
also, e.g., [15–18]). Our main deviation from these approaches is that we deal with
the problem within a discrete framework with an alternative approach to
behavior of consumers, as illustrated later. The paper is also an extension of
Zhang’s one-sector model with a homogenous population [19]. The paper
is organized as follows. Section 2 defines the one-sector growth model with
leisure time and heterogeneous groups. Section 3 analyzes the dynamic behavior
of the two-group model. Section 4 examines impact of changes in some parameters
on the wealth and income distribution. Section 5 simulates the 3-group model.
Section 6 concludes the study.


## 2. The Multigroup Growth Model in Discrete Time

First, we develop a multigroup
model in discrete time [20]. The economy has an infinite future.
We represent the passage of time in a sequence of periods, numbered from zero
and indexed by t=0,1,2,….
Time 0, being referred to the beginning of period 0,
represents the initial situation from which economy starts to grow. The end of
period t−1 coincides with the beginning of period t; it can also be called period t.
We assume that transactions are made in each period. The population is
classified into groups, indexed by j=1,…,n. Each type of consumers has a fixed number,
denoted by Nj.
As our model exhibits constant returns to
scale, the dynamics (in terms of per capita) will not be affected if we allow
the population to change at a constant growth rate over time. Let K(t) denote the capital existing in period t and N(t) the flow of labor services used at time t for production. Then N(t) is given by (2.1)N(t)=∑j=1nhjNjTj(t), where hj is the level of human capital of group j, j=1,…,n,
and Tj(t) is the work time of a representative household
of group j.The production process is described by a neoclassical production functionF(t)=F(K(t),N(t)) [5, 8]. We assume that F(K(t),N(t)) is
neoclassical. Introduce f(k(t))≡F(k(t),1), where k(t)≡K(t)/N(t). The function f has the following properties:(i)
f(0)=0;(ii)
f is increasing, strictly concave on R+
and C2 on R++;  f′(k)>0 and f′′(k)<0;(iii)
limk→0f′(k)=∞ and limk→+∞f'(k)=0. Let δk denote the fixed rate of capital depreciation.
Markets are competitive; thus labor and capital earn their marginal products,
and firms earn zero profits. We assume that the output good serves as a medium
of exchange and is taken as numeraire. The
rate of interest, r(t), and wage rate, w(t), 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(t) and N(t) to maximize its profit. The marginal conditions are given by (2.2)r(t)+δk=f′(k(t)),w(t)≡f(k(t))−k(t)f′(k(t)). Let kj(t) denote per capita wealth of group j in t. According to the definitions, we have K(t)=∑j=1nkj(t)Nj.Divide the two sides of the
above equation byN(t): (2.3)k(t)=∑j=1nkj(t)nj(t), where nj(t)≡Nj/N(t).
From (2.1), we see that r(t) and wj(t) are functions of kj(t) and nj(t).Consumers
make decisions on choice of consumption levels of services and commodities as
well as on how much to save. In order to provide proper description of
endogenous savings, we should know how individuals perceive the future.
Different from the optimal
growth theory in which utility defined over future consumption streams is used,
we assume that we can find preference structure of
consumers over leisure time, consumption,
and saving at the current state. The preference over current and future
consumption is reflected in the
consumer’s preference structure over leisure,
consumption and saving. This
study uses the approach to consumers’ behavior proposed by Zhang. Theoretical
and empirical implications and applications of the approach are examined in Zhang
[21]. We now describe behavior of consumers. Group j’s per
capita current income yj(t) from
the interest payment r(t)kj(t) and the wage payment wj(t)Tj(t) is defined by (2.4)yj(t)=r(t)kj(t)+wj(t)Tj(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 not only live 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 kj(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 consumption 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 consumption goods and saving: (2.5)y^j(t)=yj(t)+kj(t)=(1+r(t))kj(t)+wj(t)Tj(t),j=1,…,n. The disposable income is used for saving
and consumption. At each point of time, a consumer
would distribute the total available budget among savings, sj(t), and consumption of goods, cj(t). The
budget constraint is given by (2.6)cj(t)+sj(t)=y^j(t). Denote Thj(t) the leisure time at time t and the (fixed) available time for work and
leisure by T0. The time constraint is expressed by Tj(t)+Thj(t)=T0. Substituting this function into the budget constraint
yields (2.7)wj(t)Thj(t)+cj(t)+sj(t)=y¯j(t)≡(1+r(t))kj(t)+wj(t)T0,j=1,…,n. At each point of time, consumers decide the three
variables subject to the disposable income. We assume that
utility level Uj(t) is dependent on the leisure time, Thj(t),
the consumption level of commodity, cj(t), and the savings, sj(t), as follows: (2.8)Uj(t)=Thjσj(t)cjξj(t)sjλj(t),σj,ξj,λj>0,σj+ξj+λj=1,j=1,…,n, where σj,  ξj, and λj are, respectively, group j’s propensities to consume and to hold wealth. Here, for simplicity, we specify the utility function
with the Cobb-Douglas. It is important to examine dynamics with general
utility functions. Maximizing Uj subject to the budget constraints (2.7) yields (2.9)wj(t)Thj(t)=σjy¯j(t),cj(t)=ξjy¯j(t),sj(t)=λjy¯j(t). Per capita wealth of group j in period t+1 is equal to the savings made in period t,
that is, (2.10)kj(t+1)=sj(y¯j(t)),j=1,…,n. We will show that the above
mappings control the motion of the system.As output is either consumed or saved, the sum of net
savings and consumption equals output, that is,(2.11)C(t)+S(t)−K(t)+δkK(t)=F(t), where C(t) is the sum of consumption, S(t)−K(t)+δkK(t) is the sum of net savings of the groups, and C(t)=∑jcj(t)Nj. It can be shown that (2.11) is
redundant in the sense that it can be derived from the other equations in the
system.The
dynamics consist ofn-dimensional
maps. In order to analyze
properties of the dynamic system, it is necessary to express the dynamics in
terms of n variables.
The following lemma, which is proved in
Appendix A, shows that the dynamics is controlled by an n-dimensional
maps system.Lemma 2.1.
The dynamics of  the economic system is governed by the
followingn-dimensional
difference equations:(2.12)kj(t+1)=ϕj(k1(t),…,kn(t)),j=1,…,n, where ϕj are differentiable functions of k1(t),…,kn(t). Moreover,
all the other variables can be determined as functions of k1(t),…,kn(t) in any period by the following procedure: k(t) by (A.9) →Tj(t) by (A.7) →Thj(t)=T0−Tj(t)→N(t) by (2.1) →r(t) and wj(t) by (2.2) →y¯j(t) by (2.7) →cj(t) and sj(t) by (2.9) →K(t)=k(t)N(t)→F(K(t),N(t))→f(t)=F(t)/N(t)→Uj(t) by (2.8).As it is difficult to find explicit conclusions about
dynamic behavior of the system, in the remainder of this study we are concerned
with a few special cases of the general model.


## 3. The Two-Group Model with the Cobb-Douglas Production Function

This section is concerned with the case that there are
two groups of labor force, and the production function takes on the Cobb-Douglas
form byf(k)=Akα, where 0<α<1.
As shown in (A.9) in Appendix A, k is determined as a function of k1(t) and k2(t) by the following equation: (3.1)Λ(k)≡k−ϕ1(k1,k2)kβ−ϕ2(k1,k2)=0, in which (3.2)ϕ1(k1,k2)≡∑jkjσ¯j>0,ϕ2(k1,k2)≡A¯K−αA¯∑jn¯jkj>0,σ¯j≡σjNjδA¯A>0,n¯j=Nj(ξj+λj),A¯≡1(∑jhjn¯j)βT0>0.       Equation (3.1) has a unique solution for given k1(t) and k2(t). The existence of at least one positive
solution is guaranteed by Λ(0)<0 and Λ(k)→+∞ as k→+∞. Let the minimum positive solution of Λ(k)=0 be K. As (3.3)Λ′(K)=1−βϕ1K−α=1−ϕ1K−α+αϕ1K−α=ϕ2K+αϕ1K−α>0,Λ′′(k)=βαϕ1k−1−α>0,∀k>0, we conclude that Λ′(k)>0 for any k≥K. As Λ(K)=0, we see that it is impossible for any k>K to satisfy Λ(k)=0. As K is the minimum positive solution, the equation
thus has a unique positive solution. We denote this solution by k=ϕ(k1,k2). According to (A.1) and with f(k)=Akα, we have (3.4)y¯j(t)=(αAk−β+δ)kj+hjβT0Akα,j=1,2. Insert sj=λjy¯j and (3.4) into the difference
equations (2.10):(3.5)kj(t+1)=λj[(αAϕ−β(t)+δ)kj(t)+hjβT0Aϕα(t)],j=1,2. The two difference equations contain only two
variables, k1(t) and k2(t). With proper initial conditions k1(0) and k2(0), the two difference equations determine values
of k1(t) and k2(t) in any period. According to Lemma 2.1, we can
determine all the other variables in the system. Hence, it is sufficient to
examine the dynamic properties of the two difference equations (3.5).An equilibrium point of the system is given by(3.6)hjβT0Aϕα=(1λj−δ−αAϕ−β)kj. At
equilibrium we also have λjy¯j=kj. From λjy¯j=kj and (3.4), we have (3.7)kj=hjβT0Akα(λ¯j−αAk−β), where λ¯j≡(ξj+σj)/λj+δk>0. To guarantee that kj is positive, we should require λ¯j>αAk−β. From (2.11), we have (3.8)∑j(cj+sj)Nj=(f+δk)N. Insert
(2.9) and λjy¯j=kj into the above equation: (3.9)(λ˜1−Ak−β)N1k1+(λ˜2−Ak−β)N2k2=0, where we
use kN=k1N1+k2N2 and λ˜j≡ξj/λj+δk>0. Substitute (3.7) into the above
equation: (3.10)H(k)≡(kβλ˜1−A)h1N1(kβλ¯1−αA)+(kβλ˜2−A)h2N2(kβλ¯2−αA)=0. As
denominators are positive, for the equation to have solutions, kβλ˜1−A and kβλ˜2−A should have the opposite signs. For
convenience of analysis, in the reminder of this we require λ˜1>λ˜2, that is, ξ1/λ1>ξ2/λ2. This requirement implies that group 1’s “relative”
propensity to save is higher than group 2’s. Under this requirement, we have kβλ˜1−A>0 and kβλ˜2−A<0. In Appendix B, we show that
this equation has a unique (economically meaningful) solution. The following lemma
is proved in Appendix B.Lemma 3.1.
The two-group economy has a unique equilibrium.
It should be noted that as discussed in AppendixB,
the equilibrium may be either stable or unstable, depending on the parameter
values. Now the impact of changes in some parameters is examined.


## 4. Impact of Changes of Some Parameters in the Two-Group Model

This section examines effects of changes in some
parameters on the economic system. First, we study impact of change in group1’s human capital. Taking derivatives of (3.10)
with respect to h1,  (4.1)−dHdkdkdh1=(kβλ˜1−A)N1(kβλ¯1−αA)>0, in which dH/dk<0 is given by (B.5). As h1 is increased, the
capital intensity, k, increases. As the productivity of group 1’s labor force is increased, the average human
capital tends to be increased. In the long term, the capital intensity in terms
of the qualified labor input tends to be increased. From r=αAk−β−δk and wj=hjβAkα, we see that as
group 1’s level of human capital increases, the rate
of interest falls, and the wage rates rise.From (3.7), we have (4.2)dk1dh1=k1h1+(λ¯1−Ak−βλ¯1−αAk−β)αk1kdkdh1,dk2dh1=(λ¯2−Ak−βλ¯2−αAk−β)αk2kdkdh1. We see that if λ¯2<(>)Ak−β, then an increase in group 1’s human capital reduces (increases) group 2’s per capita wealth. As λ¯2=(ξ2+σ2)/λ2+δk, we see that λ¯2>Ak−β tends to be satisfied if group 2’s propensity to
save is relatively low. In the case that group 2’s propensity to
save is relatively low, group 2’s wealth per
capita is increased when group 1’s human capital
rises. If λ¯1>Ak−β, then an increase in group 1’s human capital increases group 1’s per
capita wealth. In the case of λ¯1<Ak−β, the impact on k1 is ambiguous if no further requirement on the
parameter values is added. We see that as group 1’s human capital
is improved, the impact on the capital intensity is certain but the effects on
the levels of wealth per capita are ambiguous. From (B.6), we have (4.3)dThjdh1=−αAβk−β−1Thj(λ¯j−αAk−β)dkdh1<0. As group 1’s level of human capital is increased, the
leisure time of each group falls. As the human capital is improved, the capital
intensity is increased. Consequently, the wage rate is increased. The value of
work hour becomes higher for each group. Hence, the leisure time is reduced. By y¯j=kj/λj and cj=ξjkj/λj, we have (4.4)dy¯jdh1=1λjdkjdh1,dcjdh1=ξjλjdkjdh1. The effects on the disposable incomes and consumption
levels have the same direction as that of the effect on the wealth per capita.To study impact of preference change, we have to
specify change pattern asσj+ξj+λj=1. We are concerned with group 1’s propensity to use leisure time, σ1. We specify the preference change pattern by dσ1=−dξ1 and dλ1=0. That is, as the propensity to use leisure
increases, the propensity to consume goods declines, and the propensity to save
remains invariant. Taking derivatives of (3.10) with respect to σ1,  (4.5)−dHdkdkdσ1=−h1N1(kβλ¯1−αA)λ1<0.  As the propensity to consume leisure increases, k decreases. As
the household of group 1 values more leisure, the capital intensity is
reduced. From (B.6), we have (4.6)dTh1dσ1=Th1σ1−αAβk−β−1Th1(λ¯1−αAk−β)dkdσ1>0,dTh2dσ1=−αAβk−β−1Th2(λ¯2−αAk−β)dkdσ1>0. As group 1’s propensity to use leisure increases, the
leisure time of each group rises. From r=αAk−β−δk and wj=hjβAkα, we see that as
group 1’s propensity to use leisure time increases,
the rate of interest rises, and the wage rates fall. From (3.6), we
have (4.7)dkjdσ1=(λ¯j−Ak−βλ¯j−αAk−β)αkjkdkdσ1. We see that if λ¯j<(>)Ak−β, then an increase in group 1’s propensity to enjoy leisure reduces
(increases) group j’s per capita wealth. By y¯j=kj/λj and cj=ξjkj/λj, we have (4.8)dy¯jdσ1=1λjdkjdσ1,dc1dσ1=ξ1λ1dk1dσ1−k1λ1,dc2dσ1=ξ21λ2dk2dσ1.


## 5. Simulating the 3-Group Model

This section simulates the
model when the economy consists of three different groups. For illustration, we
specifyα=1/3. That is, the
production function takes on the Cobb-Douglas form, f(k)=Ak1/3. The choice does not seem to be unrealistic.
For instance, some empirical studies on the US
economy demonstrate that the
value of the parameter, α, in the Cobb-Douglas production is
approximately equal to 0.3 (e.g., [22]). As shown in (A.9) in Appendix A, k is determined as a function of kj(t),j=1,2,3 by the following equation: (5.1)Λ(k)≡k−ϕ1k2/3−ϕ2=0, in which (5.2)ϕ1≡∑j=13kjσ¯j>0,ϕ2≡A¯K−αA¯∑j=13n¯jkj>0,σ¯j≡σjNjδA¯A>0,n¯j=Nj(ξj+λj),A¯≡1(∑j=15hjn¯j)βT0>0.     Equation (5.1) has a unique solution for given kj(t). The solution is explicitly given by (5.3)k(t)=ϕ({kj(t)})=[ϕ13+21/3ϕ123(2ϕ13+27ϕ2+3ϕ03ϕ2)1/3+(2ϕ13+27ϕ2+3ϕ03ϕ2)1/332]3, where (5.4)ϕ0({kj(t)})≡[4ϕ12{kj(t)}+27ϕ2{kj(t)}]1/2. According to (A.1) and with f(k)=Akα, we have (5.5)y¯j(t)=(Ak−2/33+δ)kj+2hjT0Ak1/33,j=1,2,3. Insert sj=λjy¯j and (5.5) into the difference
equations: (2.10) (5.6)kj(t+1)=λj[(Aϕ−2/3({kj})3+δ)kj+2hjT0Aϕ1/3({kj})3],j=1,2,3. The three difference equations contain three
variables, {kj(t)}. With proper initial conditions {kj(0)}, these difference equations determine values of {kj(t)} in any period. According to Lemma 2.1, we can
determine all the other variables in the system. Hence, it is sufficient to
examine the dynamic properties of the difference equations (5.3) as with the
two-group case in the previous sections.At
equilibrium we haveλjy¯j=kj. From λjy¯j=kj and (5.5), we solve {kj} as in (3.7). Similar to (3.10), the
equilibrium value of k is given by (5.7)H(k)≡∑j=13(k2/3λ˜j−A)hjNj(k2/3λ¯j−A/3)=0. Similar to the requirements
(B.1)–(B.3), we should require k to satisfy certain conditions for the
equilibrium solution to be meaningful.Rather than further examining
these conditions, we simulate the model. To
simulate the model, we specify the groups’ human capital and preferences as
follows:(5.8)(h1h2h3)=(1341),(λ1λ2λ3)=(0.650.550.25),(σ1σ2σ3)=(0.290.30.45),(N1N2N3)=(38215). Group 1 is called the rich class—with the highest level of human capital and the
highest propensity to own wealth. The population share of the rich in the total
population is only 3 percent. Group 2 is termed as the working
class. Its population share is 82 percent. Group 3 is the poor class. The human
capital level of this group is low, and its propensity to save is low. The share
of the population is 15 percent. We specify the rest three parameters
as follows (5.9)A=3.3,δk=0.06,T0=24. The simulation demonstrates a unique equilibrium value
of k=34.08. The equilibrium values of the other variables
are as follows:   (5.10)K=86975.5,F=27305.6,r=0.045,(k1k2k3)=(4506.55885.2157.92),(Th1Th2Th3)=(21.6816.9214.62),(c1c2c3)=(415.99241.4269.51),(w1w2w3)=(92.7328.537.13). The wealth per capita of the rich group is 5 times as
that of the working class, and the wealth per capita of the working class is 15
times as that of the poor class. The rich class’s population is only 3 percent but its shares of income and wealth
are, respectively, 5.65 and 15.54 percent; the middle class’s share of
population is 82 percent, and its shares of income and wealth are,
respectively, 89.6 and 83.5 percent; the poor class’s share of population
is 15 percent, and its shares of income and wealth are,
respectively, 4.72 and 0.01. The rich class enjoys the highest leisure
time. The poor class has the least leisure time. The rich class consumes much
more than the poor class. Due to the human capital differences, the three
groups have different wage rates. The three eigenvalues are given by (5.11)ρ1,2=22998.6±277585i,ρ3=0.69. The steady state is unstable. Further simulation
results demonstrate that the system may be either stable or unstable, depending
on the parameter values. Since the stability conditions are difficult to
interpret, we do not further examine them.As the dynamic system has a
unique equilibrium, we can examine impact of changes in the parameters. First,
we examine impact of change in human capital. We fix the parameter values as in
(5.8) and (5.9) except one parameter h1.
We increase the rich class’s level of human capital from 13 to 14.
We calculate the new equilibrium values as   (5.12)Δk=31,ΔK=1228.27,ΔF=218.72,Δr=−0.001,Δh1=1:(Δk1Δk2Δk3)=(355.141.960.16),(ΔTh1ΔTh2ΔTh3)=(−0.03−0.01−0.01),(Δc1Δc2Δc3)=(32.781.530.20),(Δw1Δw2Δw3)=(7.440.090.02). In (5.12), we denote the difference between equilibrium
values of the variables at the new equilibrium point and old one by Δ. From (5.12), we see that as the rich class’s
human capital is increased, the total output, the total wealth and per capita
wealth of all the groups, the wage rates, and consumption are all increased
but the leisure times of all the groups are slightly reduced. Hence, every
group and the society as a whole benefit from human capital improvement of the
rich class. It should be remarked that the poor class benefits only slightly
from the change. This implies that as the rich class improves its human
capital, the poor class’s relative “social status” might become lower. The wealth
and income gaps are enlarged among the classes.We increase levels of human capital of the other two
classes as follows:(5.13)Δk=−0.52,ΔK=17767.1,ΔF=5914.91,Δr=0.001,Δh2=1:(Δk1Δk2Δk3)=(−13.15217.20−0.27),(ΔTh1ΔTh2ΔTh3)=(0.050.020.01),(Δc1Δc2Δc3)=(−1.2259.24−0.33),(Δw1Δw2Δw3)=(−0.476.95−0.04),Δk=−2.77,ΔK=−4.96,ΔF=1585.57,Δr=0.006,h3=2:(Δk1Δk2Δk3)=(−70.89−17.84111.36),(ΔTh1ΔTh2ΔTh3)=(0.270.130.03),(Δc1Δc2Δc3)=(−6.54−4.87133.63),(Δw1Δw2Δw3)=(−2.58−0.8013.67). We see that as the working class’s human capital is
improved, the levels of per capita wealth and consumption and wage rates of the
working class are increased and the levels of per capita wealth and
consumption and wage rates of the other two classes fall. As the poor class
improves its human capital, the class’s living conditions and wealth are
improved but the other two classes do not benefit, except that they have more
leisure time. When the rich class increases its level of human capital, the
rate of interest decreases but when any of the other two classes increases its
level of human capital, the rate of interest increases. Here, we see that changes
in human capital of different groups have different implications for different
groups.We now examine the impact of technological parameter
on the equilibrium values of the dynamic system. We list up the effects on the
variables as follows:(5.14)Δk=11.40,ΔK=29093.4,ΔF=9133.75,Δr≈0,ΔA=0.7:(Δk1Δk2Δk3)=(1507.45296.1019.38),(ΔTh1ΔTh2ΔTh3)≈(000),(Δc1Δc2Δc3)=(139.1580.7623.25),(Δw1Δw2Δw3)=(31.029.542.39). As the technology is improved, the levels of per
capita wealth, consumption, and wage rate of all the classes are increased. The
rate of interest is slightly affected by the technological change. We see that
the rich class benefits most from the general improvement. Although the
technological change benefits all the classes, the income and wealth gaps among
the classes are enlarged.We also examine effects of change in the preferences. We
increase the propensity to save byΔλj and reduce the propensities to consume and use
leisure time by −Δλj/2, respectively. The simulation results show that
as any class’s propensity to save is increased, the rate of interest declines;
the capital intensity of production increases; the total capital stocks, the total output, and the wage rates of all the classes are increased. When any
class increases its propensity to save, its leisure time is increased but the
two other classes’ leisure times are reduced. We also simulated the effects of
change in each group’s population. As any group’s
labor force increases, the total output level is increased, and the rate of
interest is slightly affected. When the rich class increases its population,
the levels of per capita wealth, the wage rates, and the consumption levels
of all the classes are increased, and the leisure times of all the classes are
reduced. When the working class or poor class increases its population, the
wage rates, and the levels of per capita wealth and of consumption of all the
classes are reduced and the leisure times are increased. As the simulations are
straightforward, we will not provide the results here.


## 6. Conclusions

We
proposed a one-sector growth multigroup model with endogenous labor supply to
provide some insights into dynamics of wealth and income distribution in a
competitive economy with capital accumulation as the main engine of economic
growth. This study treats capital
accumulation as the main engine of economic growth. It is known that almost all
the contemporary growth models with microeconomic foundation are based on Ramsey’s 1928 paper. As the Ramsey [2] model provides a rational mechanism
of household behavior, it is reasonable to expect that the homogenous
population Ramsey model has been extended to economies with heterogeneous
households over years. It has become clear that the Ramsey growth model with
heterogeneous households tends to result in dynamically intractable problems. A
typical model of the Ramsey approach is reflected in a model of heterogeneous
households by Becker [15]. The model forges a link between income
distribution, wealth distribution, and economic growth. The Becker model
demonstrates that if an agent’s lifetime utility function over an infinite
horizon is represented by a stationary, additive, discounted function with a
constant pure rate of time preference, then the income distribution is shown in
the long-run steady state to be determined by the lowest discount rate. The
household (e.g., a single household) with the lowest rate of discount
owns all the capital and earns a wage income; all other households (e.g., other twenty millions households) receive a wage income and have no
wealth. Different from the standard Ramsey model, the model in this paper shows
nondegenerate long-run distribution among
the heterogeneous households. This paper also demonstrates the importance of
introducing heterogeneous households into the growth theory. By simulation we
demonstrate, for instance, that in the three-group economy when the rich
group’s human capital is improved, all the groups will economically benefit, and
the leisure times of all the groups are reduced, but when any other group’s
human capital is improved, the group will economically benefit, the other two
groups economically lose, and the leisure times of all the groups are
increased. We show that the same change in different groups may have different
implications for the national economy. An immediate and important extension of
the model is to make technological change, human capital, preferences as well
as the labor force (which all have been fixed in this study) as endogenous
variables. There is huge amount of literature about endogenous
technological change, human capital accumulation, preference change, and
population in economics. It would be fruitful to examine different issues of
economic growth and development within the framework proposed in this study.



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