#### 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.

#### 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 . Time being referred to the beginning of period , represents the initial situation from which economy starts to grow. The end of period coincides with the beginning of period ; it can also be called period . We assume that transactions are made in each period. The population is classified into groups, indexed by . Each type of consumers has a fixed number, denoted by . 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 denote the capital existing in period and the flow of labor services used at time for production. Then is given by where is the level of human capital of group , , and is the work time of a representative household of group .

The production process is described by a neoclassical production function [5, 8]. We assume that is neoclassical. Introduce where The function has the following properties:(i);(ii) is increasing, strictly concave on and on ; and (iii) and Let 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, , and wage rate, , are determined by markets. Hence, for any individual firm and are given at each point of time. The production sector chooses the two variables and to maximize its profit. The marginal conditions are given by Let denote per capita wealth of group in . According to the definitions, we have

Divide the two sides of the above equation by : where . From (2.1), we see that and are functions of and .

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 ’s per
capita current income from
the interest payment and the wage payment is defined by 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 can
sell to purchase goods and to save is equal
to 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 is defined as the sum of the current income and the
wealth available for purchasing consumption goods and saving: The disposable income is used for saving
and consumption. At each point of time, a consumer
would distribute the total available budget among savings, and consumption of goods, The
budget constraint is given by Denote the leisure time at time and the (fixed) available time for work and
leisure by The time constraint is expressed by Substituting this function into the budget constraint
yields At each point of time, consumers decide the three
variables subject to the disposable income. We assume that
utility level is dependent on the leisure time, ,
the consumption level of commodity, , and the savings, , as follows: where , , and are, respectively, group ’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 subject to the budget constraints (2.7) yields Per capita wealth of group in period is equal to the savings made in period ,
that is, 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, where is the sum of consumption, is the sum of net savings of the groups, and 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 of -dimensional maps. In order to analyze properties of the dynamic system, it is necessary to express the dynamics in terms of variables. The following lemma, which is proved in Appendix A, shows that the dynamics is controlled by an -dimensional maps system.

Lemma 2.1. *The dynamics of the economic system is governed by the
following -dimensional
difference equations:** where are differentiable functions of . Moreover,
all the other variables can be determined as functions of in any period by the following procedure: by (A.9) by (A.7) by (2.1) and by (2.2) by (2.7) and by (2.9) 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 by where . As shown in (A.9) in Appendix A, is determined as a function of and by the following equation: in which Equation (3.1) has a unique solution for given and . The existence of at least one positive solution is guaranteed by and as Let the minimum positive solution of be . As we conclude that for any . As we see that it is impossible for any to satisfy As is the minimum positive solution, the equation thus has a unique positive solution. We denote this solution by According to (A.1) and with we have Insert and (3.4) into the difference equations (2.10): The two difference equations contain only two variables, and With proper initial conditions and the two difference equations determine values of and 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 At equilibrium we also have From and (3.4), we have where To guarantee that is positive, we should require From (2.11), we have Insert (2.9) and into the above equation: where we use and Substitute (3.7) into the above equation: As denominators are positive, for the equation to have solutions, and should have the opposite signs. For convenience of analysis, in the reminder of this we require that is, This requirement implies that group 1’s “relative” propensity to save is higher than group 2’s. Under this requirement, we have and 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 Appendix B,
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 group ’s human capital. Taking derivatives of (3.10) with respect to , in which is given by (B.5). As is increased, the capital intensity, 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 and we see that as group ’s level of human capital increases, the rate of interest falls, and the wage rates rise.

From (3.7), we have We see that if then an increase in group ’s human capital reduces (increases) group ’s per capita wealth. As we see that 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 then an increase in group ’s human capital increases group 1’s per capita wealth. In the case of the impact on 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 As group ’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 and we have 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 as We are concerned with group ’s propensity to use leisure time, We specify the preference change pattern by and 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 , As the propensity to consume leisure increases, decreases. As the household of group values more leisure, the capital intensity is reduced. From (B.6), we have As group ’s propensity to use leisure increases, the leisure time of each group rises. From and we see that as group ’s propensity to use leisure time increases, the rate of interest rises, and the wage rates fall. From (3.6), we have We see that if then an increase in group ’s propensity to enjoy leisure reduces (increases) group ’s per capita wealth. By and we have

#### 5. Simulating the 3-Group Model

This section simulates the model when the economy consists of three different groups. For illustration, we specify That is, the production function takes on the Cobb-Douglas form, The choice does not seem to be unrealistic. For instance, some empirical studies on the US economy demonstrate that the value of the parameter, in the Cobb-Douglas production is approximately equal to (e.g., [22]). As shown in (A.9) in Appendix A, is determined as a function of by the following equation: in which Equation (5.1) has a unique solution for given The solution is explicitly given by where According to (A.1) and with we have Insert and (5.5) into the difference equations: (2.10) The three difference equations contain three variables, With proper initial conditions these difference equations determine values of 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 have From and (5.5), we solve as in (3.7). Similar to (3.10), the equilibrium value of is given by Similar to the requirements (B.1)–(B.3), we should require 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: 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 percent. Group 2 is termed as the working class. Its population share is 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 percent. We specify the rest three parameters as follows The simulation demonstrates a unique equilibrium value of The equilibrium values of the other variables are as follows: 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 percent but its shares of income and wealth are, respectively, and percent; the middle class’s share of population is percent, and its shares of income and wealth are, respectively, and percent; the poor class’s share of population is percent, and its shares of income and wealth are, respectively, and 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 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 . We increase the rich class’s level of human capital from to . We calculate the new equilibrium values as In (5.12), we denote the difference between equilibrium values of the variables at the new equilibrium point and old one by 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: 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: 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 by and reduce the propensities to consume and use leisure time by , 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.

#### Appendices

#### A. Proving Lemma 2.1

We now prove Lemma 2.1 in Section 2. From the definition of in (2.7) and (2.1), we have in which From we have where we use and From , (A.1), and (A.3), we solve where From (2.1) and (2.3), we have Solve (A.5) and (A.6) with as variables As , and are all functions of and , the above equations show that the work times of all the groups are uniquely determined as functions of and .

From (2.11), we have Insert (A.1), (2.1), and and in (2.11) into the above equation: where we use We now show that for any given for all has at least one positive solution. Note that (when at least one ) is a function of According to the definitions of and and the properties of it is straightforward to check the following properties of where we use We see that has at least one positive solution. The solution can be expressed as a function of . Take derivative of with respect to : As the sign of is ambiguous, we are not sure about the uniqueness of solution. As is a function of at any point of time, from (A.1) we see that are functions of Hence, from (2.10), (A.1), and we get

#### B. Proving Lemma 3.1

We now show that (3.10) has a unique solution. From (3.7), we see that it is necessary for to satisfy For (3.10) to positive solution, it is necessary to ask for and to have the opposite sign if If then which also satisfies (B.1). For convenience of analysis, let that is, From (3.10) and (B.1), we should require As we always have Hence for the requirements (B.1) and (B.2) to satisfy, we should require It is straightforward to check the following properties of : Accordingly, has at least positive solution which satisfies (B.3). Take derivatives of with respect to : From (B.4) and (B.5), we conclude that has a unique solution.

From , and we have We see that under (B.3), we have According to Lemma 2.1, we can determine all the other variables. Hence, we proved that the system has a unique equilibrium.

We now determine stability of the unique equilibrium. The Jacobian matrix at equilibrium is given by where in which we calculate from (3.1) where is given by (3.3). We have , The two eigenvalues, and are determined by It is difficult to explicitly judge the stability conditions. Simulation demonstrates that the unique equilibrium can be either stable or unstable, depending on the parameter values.

#### Acknowledgments

The author is grateful to important comments of Editor Huang Weihong and two anonymous referees.