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

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 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 𝑁_𝑗. 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 𝑁(𝑡)=^𝑛_𝑗=1ℎ_𝑗𝑁_𝑗𝑇_𝑗(𝑡), (2.1) where ℎ_𝑗 is the level of human capital of group 𝑗, 𝑗=1,…,𝑛, and 𝑇_𝑗(𝑡) is the work time of a representative household of group 𝑗.

The production process is described by a neoclassical production function 𝐹(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)lim_𝑘→0𝑓^′(𝑘)=∞ and lim_𝑘→+∞𝑓^′(𝑘)=0. Let 𝛿_𝑘 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 𝑟(𝑡)+𝛿_𝑘=𝑓^′(𝑘(𝑡)),𝑤(𝑡)≡𝑓(𝑘(𝑡))−𝑘(𝑡)𝑓^′(𝑘(𝑡)). (2.2) Let 𝑘_𝑗(𝑡) denote per capita wealth of group 𝑗 in 𝑡. According to the definitions, we have 𝐾(𝑡)=∑^𝑛_𝑗=1𝑘_𝑗(𝑡)𝑁_𝑗.

Divide the two sides of the above equation by 𝑁(𝑡): 𝑘(𝑡)=^𝑛_𝑗=1𝑘_𝑗(𝑡)𝑛_𝑗(𝑡), (2.3) where 𝑛_𝑗(𝑡)≡𝑁_𝑗/𝑁(𝑡). From (2.1), we see that 𝑟(𝑡) and 𝑤_𝑗(𝑡) are functions of 𝑘_𝑗(𝑡) and 𝑛_𝑗(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 𝑦_𝑗(𝑡) from the interest payment 𝑟(𝑡)𝑘_𝑗(𝑡) and the wage payment 𝑤_𝑗(𝑡)𝑇_𝑗(𝑡) is defined by 𝑦_𝑗(𝑡)=𝑟(𝑡)𝑘_𝑗(𝑡)+𝑤_𝑗(𝑡)𝑇_𝑗(𝑡). (2.4) 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 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 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: 𝑦_𝑗(𝑡)=𝑦_𝑗(𝑡)+𝑘_𝑗(𝑡)=(1+𝑟(𝑡))𝑘_𝑗(𝑡)+𝑤_𝑗(𝑡)𝑇_𝑗(𝑡),𝑗=1,…,𝑛. (2.5) 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), and consumption of goods, 𝑐_𝑗(𝑡). The budget constraint is given by 𝑐_𝑗(𝑡)+𝑠_𝑗(𝑡)=𝑦_𝑗(𝑡). (2.6) Denote 𝑇_ℎ𝑗(𝑡) the leisure time at time 𝑡 and the (fixed) available time for work and leisure by 𝑇₀. The time constraint is expressed by 𝑇_𝑗(𝑡)+𝑇_ℎ𝑗(𝑡)=𝑇₀. Substituting this function into the budget constraint yields 𝑤_𝑗(𝑡)𝑇_ℎ𝑗(𝑡)+𝑐_𝑗(𝑡)+𝑠_𝑗(𝑡)=𝑦_𝑗(𝑡)≡(1+𝑟(𝑡))𝑘_𝑗(𝑡)+𝑤_𝑗(𝑡)𝑇₀,𝑗=1,…,𝑛. (2.7) At each point of time, consumers decide the three variables subject to the disposable income. We assume that utility level 𝑈_𝑗(𝑡) is dependent on the leisure time, 𝑇_ℎ𝑗(𝑡), the consumption level of commodity, 𝑐_𝑗(𝑡), and the savings, 𝑠_𝑗(𝑡), as follows: 𝑈_𝑗(𝑡)=𝑇^𝜎𝑗_ℎ𝑗(𝑡)𝑐^𝜉𝑗_𝑗(𝑡)𝑠^𝜆𝑗_𝑗(𝑡),𝜎_𝑗,𝜉_𝑗,𝜆_𝑗>0,𝜎_𝑗+𝜉_𝑗+𝜆_𝑗=1,𝑗=1,…,𝑛, (2.8) where 𝜎_𝑗, 𝜉_𝑗, and 𝜆_𝑗 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 𝑈_𝑗 subject to the budget constraints (2.7) yields 𝑤_𝑗(𝑡)𝑇_ℎ𝑗(𝑡)=𝜎_𝑗𝑦_𝑗(𝑡),𝑐_𝑗(𝑡)=𝜉_𝑗𝑦_𝑗(𝑡),𝑠_𝑗(𝑡)=𝜆_𝑗𝑦_𝑗(𝑡). (2.9) Per capita wealth of group 𝑗 in period 𝑡+1 is equal to the savings made in period 𝑡, that is, 𝑘_𝑗(𝑡+1)=𝑠_𝑗(𝑦_𝑗(𝑡)),𝑗=1,…,𝑛. (2.10) 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, 𝐶(𝑡)+𝑆(𝑡)−𝐾(𝑡)+𝛿_𝑘𝐾(𝑡)=𝐹(𝑡), (2.11) where 𝐶(𝑡) is the sum of consumption, 𝑆(𝑡)−𝐾(𝑡)+𝛿_𝑘𝐾(𝑡) is the sum of net savings of the groups, and 𝐶(𝑡)=∑_𝑗𝑐_𝑗(𝑡)𝑁_𝑗. 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 𝑛-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 following 𝑛-dimensional difference equations:𝑘_𝑗(𝑡+1)=𝜙_𝑗(𝑘₁(𝑡),…,𝑘_𝑛(𝑡)),𝑗=1,…,𝑛, (2.12) where 𝜙_𝑗 are differentiable functions of 𝑘₁(𝑡),…,𝑘_𝑛(𝑡). Moreover, all the other variables can be determined as functions of 𝑘₁(𝑡),…,𝑘_𝑛(𝑡) in any period by the following procedure: 𝑘(𝑡) by (A.9) →𝑇_𝑗(𝑡) by (A.7) →𝑇_ℎ𝑗(𝑡)=𝑇₀−𝑇_𝑗(𝑡)→𝑁(𝑡) by (2.1) →𝑟(𝑡) and 𝑤_𝑗(𝑡) by (2.2) →𝑦_𝑗(𝑡) by (2.7) →𝑐_𝑗(𝑡) and 𝑠_𝑗(𝑡) by (2.9) →𝐾(𝑡)=𝑘(𝑡)𝑁(𝑡)→𝐹(𝐾(𝑡),𝑁(𝑡))→𝑓(𝑡)=𝐹(𝑡)/𝑁(𝑡)→𝑈_𝑗(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 by 𝑓(𝑘)=𝐴𝑘^𝛼, 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: Λ(𝑘)≡𝑘−𝜙₁(𝑘₁,𝑘₂)𝑘^𝛽−𝜙₂(𝑘₁,𝑘₂)=0, (3.1) in which 𝜙₁(𝑘₁,𝑘₂)≡∑_𝑗𝑘_𝑗𝜎_𝑗>0,𝜙₂(𝑘₁,𝑘₂)≡𝐴𝐾−𝛼𝐴∑_𝑗𝑛_𝑗𝑘_𝑗>0,𝜎_𝑗≡𝜎_𝑗𝑁_𝑗𝛿𝐴𝐴>0,𝑛_𝑗=𝑁_𝑗(𝜉_𝑗+𝜆_𝑗),1𝐴≡(∑_𝑗ℎ_𝑗𝑛_𝑗)𝛽𝑇₀>0. (3.2) 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 Λ^′(𝐾)=1−𝛽𝜙₁𝐾^−𝛼=1−𝜙₁𝐾^−𝛼+𝛼𝜙₁𝐾^−𝛼=𝜙₂𝐾+𝛼𝜙₁𝐾^−𝛼Λ>0,_′′(𝑘)=𝛽𝛼𝜙₁𝑘^−1−𝛼>0,∀𝑘>0, (3.3) 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)=𝐴𝑘^𝛼, we have 𝑦_𝑗(𝑡)=(𝛼𝐴𝑘^−𝛽+𝛿)𝑘_𝑗+ℎ_𝑗𝛽𝑇₀𝐴𝑘^𝛼,𝑗=1,2. (3.4) Insert 𝑠_𝑗=𝜆_𝑗𝑦_𝑗 and (3.4) into the difference equations (2.10):𝑘_𝑗(𝑡+1)=𝜆_𝑗[(𝛼𝐴𝜙^−𝛽(𝑡)+𝛿)𝑘_𝑗(𝑡)+ℎ_𝑗𝛽𝑇₀𝐴𝜙^𝛼(𝑡)],𝑗=1,2. (3.5) 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 ℎ_𝑗𝛽𝑇₀𝐴𝜙^𝛼1=(𝜆_𝑗−𝛿−𝛼𝐴𝜙^−𝛽)𝑘_𝑗. (3.6) At equilibrium we also have 𝜆_𝑗𝑦_𝑗=𝑘_𝑗. From 𝜆_𝑗𝑦_𝑗=𝑘_𝑗 and (3.4), we have 𝑘_𝑗=ℎ_𝑗𝛽𝑇₀𝐴𝑘^𝛼(𝜆_𝑗−𝛼𝐴𝑘^−𝛽), (3.7) where 𝜆_𝑗≡(𝜉_𝑗+𝜎_𝑗)/𝜆_𝑗+𝛿_𝑘>0. To guarantee that 𝑘_𝑗 is positive, we should require 𝜆_𝑗>𝛼𝐴𝑘^−𝛽. From (2.11), we have _𝑗(𝑐_𝑗+𝑠_𝑗)𝑁_𝑗=(𝑓+𝛿𝑘)𝑁. (3.8) Insert (2.9) and 𝜆_𝑗𝑦_𝑗=𝑘_𝑗 into the above equation: (𝜆^{0𝑥002𝑑𝑐}₁−𝐴𝑘^−𝛽)𝑁₁𝑘₁+(𝜆^{0𝑥002𝑑𝑐}₂−𝐴𝑘^−𝛽)𝑁₂𝑘₂=0, (3.9) where we use 𝑘𝑁=𝑘₁𝑁₁+𝑘₂𝑁₂ and ^0𝑥0𝜆^02𝑑𝑐_𝑗≡𝜉_𝑗/𝜆_𝑗+𝛿_𝑘>0. Substitute (3.7) into the above equation: 𝐻(𝑘)≡(𝑘𝜆^{𝛽0𝑥002𝑑𝑐}₁−𝐴)ℎ₁𝑁₁(𝑘^𝛽𝜆₁+−𝛼𝐴)(𝑘𝜆^{𝛽0𝑥002𝑑𝑐}₂−𝐴)ℎ₂𝑁₂(𝑘^𝛽𝜆₂−𝛼𝐴)=0. (3.10) As denominators are positive, for the equation to have solutions, 𝑘𝜆^{𝛽0𝑥002𝑑𝑐}₁−𝐴 and 𝑘𝜆^{𝛽0𝑥002𝑑𝑐}₂−𝐴 should have the opposite signs. For convenience of analysis, in the reminder of this we require ^0𝑥0𝜆^02𝑑𝑐₁>𝜆^{0𝑥002𝑑𝑐}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 𝑘𝜆^{𝛽0𝑥002𝑑𝑐}₁−𝐴>0 and 𝑘𝜆^{𝛽0𝑥002𝑑𝑐}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 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 1’s human capital. Taking derivatives of (3.10) with respect to ℎ₁, −𝑑𝐻𝑑𝑘𝑑𝑘𝑑ℎ₁=(𝑘𝜆^{𝛽0𝑥002𝑑𝑐}₁−𝐴)𝑁₁(𝑘^𝛽𝜆₁−𝛼𝐴)>0, (4.1) 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−𝛿_𝑘 and 𝑤_𝑗=ℎ_𝑗𝛽𝐴𝑘^𝛼, 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 𝑑𝑘₁𝑑ℎ₁=𝑘₁ℎ₁+(𝜆₁−𝐴𝑘^−𝛽𝜆₁−𝛼𝐴𝑘^−𝛽)𝛼𝑘₁𝑘𝑑𝑘𝑑ℎ₁,𝑑𝑘₂𝑑ℎ₁=(𝜆₂−𝐴𝑘^−𝛽𝜆₂−𝛼𝐴𝑘^−𝛽)𝛼𝑘₂𝑘𝑑𝑘𝑑ℎ₁. (4.2) We see that if 𝜆₂<(>)𝐴𝑘^−𝛽, then an increase in group 1’s human capital reduces (increases) group 2’s per capita wealth. As 𝜆₂=(𝜉₂+𝜎₂)/𝜆₂+𝛿_𝑘, we see that 𝜆₂>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 𝜆₁>𝐴𝑘^−𝛽, then an increase in group 1’s human capital increases group 1’s per capita wealth. In the case of 𝜆₁<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 𝑑𝑇_ℎ𝑗𝑑ℎ₁=−𝛼𝐴𝛽𝑘^−𝛽−1𝑇_ℎ𝑗(𝜆_𝑗−𝛼𝐴𝑘^−𝛽)𝑑𝑘𝑑ℎ₁<0. (4.3) 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=𝑘_𝑗/𝜆_𝑗 and 𝑐_𝑗=𝜉_𝑗𝑘_𝑗/𝜆_𝑗, we have 𝑑𝑦_𝑗𝑑ℎ₁=1𝜆_𝑗𝑑𝑘_𝑗𝑑ℎ₁,𝑑𝑐_𝑗𝑑ℎ₁=𝜉_𝑗𝜆_𝑗𝑑𝑘_𝑗𝑑ℎ₁. (4.4) 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 𝜎_𝑗+𝜉_𝑗+𝜆_𝑗=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, −𝑑𝐻𝑑𝑘𝑑𝑘𝑑𝜎₁ℎ=−¹𝑁₁(𝑘^𝛽𝜆₁−𝛼𝐴)𝜆₁<0. (4.5) 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 𝑑𝑇_ℎ1𝑑𝜎₁=𝑇_ℎ1𝜎₁−𝛼𝐴𝛽𝑘^−𝛽−1𝑇_ℎ1(𝜆₁−𝛼𝐴𝑘^−𝛽)𝑑𝑘𝑑𝜎₁>0,𝑑𝑇_ℎ2𝑑𝜎₁=−𝛼𝐴𝛽𝑘^−𝛽−1𝑇_ℎ2(𝜆₂−𝛼𝐴𝑘^−𝛽)𝑑𝑘𝑑𝜎₁>0. (4.6) As group 1’s propensity to use leisure increases, the leisure time of each group rises. From 𝑟=𝛼𝐴𝑘^−𝛽−𝛿_𝑘 and 𝑤_𝑗=ℎ_𝑗𝛽𝐴𝑘^𝛼, 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 𝑑𝑘_𝑗𝑑𝜎₁=(𝜆_𝑗−𝐴𝑘^−𝛽𝜆_𝑗−𝛼𝐴𝑘^−𝛽)𝛼𝑘_𝑗𝑘𝑑𝑘𝑑𝜎₁. (4.7) 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=𝑘_𝑗/𝜆_𝑗 and 𝑐_𝑗=𝜉_𝑗𝑘_𝑗/𝜆_𝑗, we have 𝑑𝑦_𝑗𝑑𝜎₁=1𝜆_𝑗𝑑𝑘_𝑗𝑑𝜎₁,𝑑𝑐₁𝑑𝜎₁=𝜉₁𝜆₁𝑑𝑘₁𝑑𝜎₁−𝑘₁𝜆₁,𝑑𝑐₂𝑑𝜎₁=𝜉₂₁𝜆₂𝑑𝑘₂𝑑𝜎₁. (4.8)

5. Simulating the 3-Group Model

This section simulates the model when the economy consists of three different groups. For illustration, we specify 𝛼=1/3. That is, the production function takes on the Cobb-Douglas form, 𝑓(𝑘)=𝐴𝑘^1/3. 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 𝑘_𝑗(𝑡),𝑗=1,2,3 by the following equation: Λ(𝑘)≡𝑘−𝜙₁𝑘^2/3−𝜙₂=0, (5.1) in which 𝜙₁≡∑³_𝑗=1𝑘_𝑗𝜎_𝑗>0,𝜙₂≡𝐴𝐾−𝛼𝐴∑³_𝑗=1𝑛_𝑗𝑘_𝑗>0,𝜎_𝑗≡𝜎_𝑗𝑁_𝑗𝛿𝐴𝐴>0,𝑛_𝑗=𝑁_𝑗(𝜉_𝑗+𝜆_𝑗),1𝐴≡(∑⁵_𝑗=1ℎ_𝑗𝑛_𝑗)𝛽𝑇₀>0. (5.2) Equation (5.1) has a unique solution for given 𝑘_𝑗(𝑡). The solution is explicitly given by 𝑘(𝑡)=𝜙({𝑘_𝑗𝜙(𝑡)})=[¹3+2^1/3𝜙²₁3(2𝜙³₁+27𝜙₂+3𝜙₀3𝜙₂)^1/3+(2𝜙³₁+27𝜙₂+3𝜙₀3𝜙₂)^1/33√2]³, (5.3) where 𝜙₀({𝑘_𝑗(𝑡)})≡[4𝜙²₁{𝑘_𝑗(𝑡)}+27𝜙₂{𝑘_𝑗(𝑡)}]^1/2. (5.4) According to (A.1) and with 𝑓(𝑘)=𝐴𝑘^𝛼, we have 𝑦_𝑗(𝑡)=(𝐴𝑘^−2/33+𝛿)𝑘_𝑗+2ℎ_𝑗𝑇₀𝐴𝑘^1/33,𝑗=1,2,3. (5.5) Insert 𝑠_𝑗=𝜆_𝑗𝑦_𝑗 and (5.5) into the difference equations: (2.10) 𝑘_𝑗(𝑡+1)=𝜆_𝑗[(𝐴𝜙^−2/3({𝑘_𝑗})3+𝛿)𝑘_𝑗+2ℎ_𝑗𝑇₀𝐴𝜙^1/3({𝑘_𝑗})3],𝑗=1,2,3. (5.6) The three difference equations contain three variables, {𝑘_𝑗(𝑡)}. With proper initial conditions {𝑘_𝑗(0)}, these difference equations determine values of {𝑘_𝑗(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 have 𝜆_𝑗𝑦_𝑗=𝑘_𝑗. From 𝜆_𝑗𝑦_𝑗=𝑘_𝑗 and (5.5), we solve {𝑘_𝑗} as in (3.7). Similar to (3.10), the equilibrium value of 𝑘 is given by 𝐻(𝑘)≡³_𝑗=1(𝑘𝜆^{2/30𝑥002𝑑𝑐}_𝑗−𝐴)ℎ_𝑗𝑁_𝑗(𝑘^2/3𝜆_𝑗−𝐴/3)=0. (5.7) 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: ℎ₁ℎ₂ℎ₃=41,13𝜆₁𝜆₂𝜆₃=,𝜎0.650.550.25₁𝜎₂𝜎₃=,0.29𝑁0.30.45¹𝑁₂𝑁₃=382.15 (5.8) 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 𝐴=3.3,𝛿_𝑘=0.06,𝑇₀=24. (5.9) The simulation demonstrates a unique equilibrium value of 𝑘=34.08. The equilibrium values of the other variables are as follows: 𝑘𝐾=86975.5,𝐹=27305.6,𝑟=0.045,₁𝑘₂𝑘₃=,4506.55𝑇885.2157.92^ℎ1𝑇_ℎ2𝑇_ℎ3=,𝑐21.6816.9214.62₁𝑐₂𝑐₃=,415.99𝑤241.4269.51¹𝑤₂𝑤₃=.92.7328.537.13 (5.10) 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 𝜌_1,2=22998.6±277585𝑖,𝜌₃=0.69. (5.11) 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 Δ𝑘=31,Δ𝐾=1228.27,Δ𝐹=218.72,Δ𝑟=−0.001,Δℎ₁=1∶Δ𝑘₁Δ𝑘₂Δ𝑘₃=,355.141.960.16Δ𝑇_ℎ1Δ𝑇_ℎ2Δ𝑇_ℎ3=,−0.03−0.01−0.01Δ𝑐₁Δ𝑐₂Δ𝑐₃=,32.781.530.20Δ𝑤₁Δ𝑤₂Δ𝑤₃=.7.440.090.02 (5.12) 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:Δ𝑘=−0.52,Δ𝐾=17767.1,Δ𝐹=5914.91,Δ𝑟=0.001,Δℎ₂=1∶Δ𝑘₁Δ𝑘₂Δ𝑘₃=,−13.15217.20−0.27Δ𝑇_ℎ1Δ𝑇_ℎ2Δ𝑇_ℎ3=,0.050.020.01Δ𝑐₁Δ𝑐₂Δ𝑐₃=,−1.2259.24−0.33Δ𝑤₁Δ𝑤₂Δ𝑤₃=,ℎ−0.476.95−0.04Δ𝑘=−2.77,Δ𝐾=−4.96,Δ𝐹=1585.57,Δ𝑟=0.006,₃=2∶Δ𝑘₁Δ𝑘₂Δ𝑘₃=,−70.89−17.84111.36Δ𝑇_ℎ1Δ𝑇_ℎ2Δ𝑇_ℎ3=,0.270.130.03Δ𝑐₁Δ𝑐₂Δ𝑐₃=,−6.54−4.87133.63Δ𝑤₁Δ𝑤₂Δ𝑤₃=.−2.58−0.8013.67 (5.13) 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: Δ𝑘=11.40,Δ𝐾=29093.4,Δ𝐹=9133.75,Δ𝑟≈0,Δ𝐴=0.7∶Δ𝑘₁Δ𝑘₂Δ𝑘₃=,1507.45296.1019.38Δ𝑇_ℎ1Δ𝑇_ℎ2Δ𝑇_ℎ3≈000,Δ𝑐₁Δ𝑐₂Δ𝑐₃=,139.1580.7623.25Δ𝑤₁Δ𝑤₂Δ𝑤₃=.31.029.542.39 (5.14) 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 Δ𝜆_𝑗 and reduce the propensities to consume and use leisure time by −Δ𝜆_𝑗/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.

Appendices

A. Proving Lemma 2.1

We now prove Lemma 2.1 in Section 2. From the definition of 𝑦_𝑗(𝑡) in (2.7) and (2.1), we have 𝑦_𝑗(𝑡)=𝑓_𝛿(𝑘)𝑘_𝑗+ℎ_𝑗𝑓₀(𝑘), (A.1) in which 𝑓_𝛿(𝑘)≡𝑓^′(𝑘)+𝛿>0,𝑓₀(𝑘)≡[𝑓(𝑘)−𝑘𝑓^′(𝑘)]𝑇₀>0,∀𝑘>0. (A.2) From 𝑤_𝑗𝑇_ℎ𝑗=𝜎_𝑗𝑦_𝑗, we have 𝑇_ℎ1𝑇_ℎ𝑗=𝜎_𝑗𝑦₁𝑦_𝑗,𝑗=2,…,𝑛, (A.3) where we use 𝑤₁/𝑤_𝑗=ℎ₁/ℎ_𝑗 and 𝜎_𝑗≡ℎ_𝑗𝜎₁ℎ₁𝜎_𝑗,𝑗=1,…,𝑛. (A.4) From 𝑇_ℎ𝑗=𝑇₀−𝑇_𝑗, (A.1), and (A.3), we solve 𝑇_𝑗=[(𝜎_𝑗𝑘₁−𝑘_𝑗)𝑓_𝛿+ℎ_𝑗𝑓₀]𝑇₀+(𝑘_𝑗𝑓_𝛿+ℎ_𝑗𝑓₀)𝑇₁(𝑘₁𝑓_𝛿+ℎ₁𝑓₀)𝜎_𝑗,𝑗=2,…,𝑛, (A.5) where ℎ_𝑗≡𝜎_𝑗ℎ₁−ℎ_𝑗. From (2.1) and (2.3), we have ^𝑛_𝑗=1ℎ_𝑗𝑁_𝑗𝑇_𝑗(𝑡)=𝐾(𝑡)𝑘(𝑡). (A.6) Solve (A.5) and (A.6) with 𝑇_𝑗(𝑡) as variables 𝑇₁(𝑡)=Ω₁𝐾(𝑘)≡[𝑘−^𝑛_𝑗=2[(𝜎_𝑗𝑘₁−𝑘_𝑗)𝑓_𝛿+ℎ_𝑗𝑓₀]ℎ_𝑗𝑁_𝑗𝑇₀(𝑘₁𝑓_𝛿+ℎ₁𝑓₀)𝜎_𝑗](ℎ₁𝑁₁+^𝑛_𝑗=2(𝑘_𝑗𝑓_𝛿+ℎ_𝑗𝑓₀)ℎ_𝑗𝑁_𝑗(𝑘₁𝑓_𝛿+ℎ₁𝑓₀)𝜎_𝑗)⁻¹,𝑇_𝑗(𝑡)=Ω_𝑗(𝑘,𝑘₁,…,𝑘_𝑛)≡[(𝜎_𝑗𝑘₁−𝑘_𝑗)𝑓_𝛿+ℎ_𝑗𝑓₀]𝑇₀+(𝑘_𝑗𝑓_𝛿+ℎ_𝑗𝑓₀)Ω₁(𝑘₁𝑓_𝛿+ℎ₁𝑓₀)𝜎_𝑗,𝑗=2,…,𝑛. (A.7) As 𝐾, 𝑓₀,𝑓_𝛿, and 𝑓 are all functions of 𝑘 and {𝑘_𝑗}, the above equations show that the work times of all the groups are uniquely determined as functions of 𝑘 and {𝑘_𝑗}.

From (2.11), we have _𝑗(𝑐_𝑗+𝑠_𝑗)𝑁_𝑗=𝐹+𝛿𝐾=(𝑓+𝛿𝑘)𝑁. (A.8) Insert (A.1), (2.1), and 𝑐_𝑗 and 𝑠_𝑗 in (2.11) into the above equation: Λ(𝑘;𝑘₁,𝑘₂)≡(𝑓(𝑘)𝑘+𝛿)𝐾−_𝑗𝑁_𝑗(𝜉_𝑗+𝜆_𝑗)(𝑘_𝑗𝑓_𝛿(𝑘)+ℎ_𝑗𝑓₀(𝑘))=0, (A.9) where we use 𝑁=𝐾/𝑘. We now show that for any given 𝑘_𝑗≥0 for all 𝑗,Λ(𝑘)=0 has at least one positive solution. Note that 𝐾>0 (when at least one 𝑘_𝑗>0) is a function of {𝑘_𝑗}. According to the definitions of 𝑓₀ and 𝑓_𝛿, and the properties of 𝑓, it is straightforward to check the following properties of Λ(𝑘)∶Λ(𝑘)→+∞as𝑘→0,Λ(𝑘)→−∞as𝑘→+∞, (A.10) where we use 𝑓𝑘𝑓→+∞as𝑘→0,𝑘𝑓→0as𝑘→+∞,_′−𝑓𝑘<0,𝑓^′>0,𝑓^′′<0,∀𝑘>0. (A.11) We see that Λ(𝑘)=0 has at least one positive solution. The solution can be expressed as a function of {𝑘_𝑗}. Take derivative of Λ(𝑘) with respect to 𝑘: 𝑑Λ𝑑𝑘=(𝑓^′−𝑓𝑘)𝐾𝑘−_𝑗𝑁_𝑗(𝜉_𝑗+𝜆_𝑗)(𝑘_𝑗−ℎ_𝑗𝑇₀𝑘)𝑓^′′. (A.12) As the sign of 𝑑Λ/𝑑𝑘 is ambiguous, we are not sure about the uniqueness of solution. As 𝑘(𝑡) is a function of {𝑘_𝑗} at any point of time, from (A.1) we see that 𝑦_𝑗(𝑡) are functions of {𝑘_𝑗}. Hence, from (2.10), (A.1), and 𝑠_𝑗=𝜆_𝑗𝑦_𝑗, we get 𝑘_𝑗(𝑡+1)=𝜙_𝑗(𝑘₁(𝑡),…,𝑘_𝑛(𝑡))≡𝜆_𝑗⌊𝑓_𝛿(𝑘)𝑘_𝑗+ℎ_𝑗𝑓₀(𝑘)⌋,𝑗=1,…,𝑛. (A.13)

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 𝑘 to satisfy 𝑘>max_𝑗{(𝛼𝐴𝜆_𝑗)^1/𝛽}. (B.1) For (3.10) to positive solution, it is necessary to ask for ^0𝑥0𝜆^02𝑑𝑐₁−𝐴𝑘^−𝛽 and ^0𝑥0𝜆^02𝑑𝑐₂−𝐴𝑘^−𝛽 to have the opposite sign if ^0𝑥0𝜆^02𝑑𝑐₁≠𝜆^{0𝑥002𝑑𝑐}₂. If ^0𝑥0𝜆^02𝑑𝑐₁=𝜆^{0𝑥002𝑑𝑐}₂, then 𝑘=(𝐴/𝜆^{0𝑥002𝑑𝑐}₂)^1/𝛽, which also satisfies (B.1). For convenience of analysis, let ^0𝑥0𝜆^02𝑑𝑐₁>𝜆^{0𝑥002𝑑𝑐}₂, that is, 𝜉₁/𝜆₁>𝜉₂/𝜆₂. From (3.10) and (B.1), we should require (_0𝑥0𝐴_02𝑑𝑐𝜆₂)^1/𝛽𝐴>𝑘>(_{0𝑥002𝑑𝑐}𝜆₁)^1/𝛽. (B.2) As ^0𝑥0𝜆^02𝑑𝑐₁>𝜆^{0𝑥002𝑑𝑐}₂, we always have 𝐴/𝜆^{0𝑥002𝑑𝑐}_𝑗>𝛼𝐴/𝜆_𝑗. Hence for the requirements (B.1) and (B.2) to satisfy, we should require (_0𝑥0𝐴_02𝑑𝑐𝜆₂)^1/𝛽>𝑘>𝜆^∗𝐴≡max{(_{0𝑥002𝑑𝑐}𝜆₁)^1/𝛽,(𝛼𝐴𝜆₂)^1/𝛽}. (B.3) It is straightforward to check the following properties of 𝐻(𝑘): 𝐴𝐻((_{0𝑥002𝑑𝑐}𝜆₂)^1/𝛽()=^{0𝑥002𝑑𝑐}𝜆₁/𝜆^{0𝑥002𝑑𝑐}₂−1)ℎ₁𝑁₁(𝜆₁/𝜆^{0𝑥002𝑑𝑐}₂−𝛼)>0,𝐻(𝜆^∗()=^{0𝑥002𝑑𝑐}𝜆₂/𝜆^{0𝑥002𝑑𝑐}₁−1)ℎ₂𝑁₂(𝜆₂/𝜆^{0𝑥002𝑑𝑐}₁−𝛼)<0,if𝜆^∗𝐴=(_{0𝑥002𝑑𝑐}𝜆₁)^1/𝛽,𝐻(𝜆^∗)<0,if𝜆^∗=(𝛼𝐴𝜆₂)^1/𝛽. (B.4) Accordingly, 𝐻(𝑘)=0 has at least positive solution which satisfies (B.3). Take derivatives of 𝐻(𝑘) with respect to 𝑘: 𝑑𝐻(𝑑𝑘=[𝜆₁−𝛼𝜆^{0𝑥002𝑑𝑐}₁)(𝑘^𝛽𝜆₁−𝛼𝐴)²ℎ₁𝑁₁+(𝜆₂−𝛼𝜆^{0𝑥002𝑑𝑐}₂)(𝑘^𝛽𝜆₂−𝛼𝐴)²ℎ₂𝑁₂]𝐴𝛽𝑘^−𝛼<0. (B.5) From (B.4) and (B.5), we conclude that 𝐻(𝑘)=0 has a unique solution.

From 𝑇_ℎ𝑗=𝜎_𝑗𝑦_𝑗/𝑤_𝑗, 𝑦_𝑗=𝑘_𝑗/𝜆_𝑗, and 𝑤_𝑗=𝛽𝐴𝑘^𝛼, we have 𝑇_ℎ𝑗=𝜎_𝑗𝑘_𝑗𝛽𝐴ℎ_𝑗𝑘^𝛼𝜆_𝑗=𝜎_𝑗𝑇₀𝜆_𝑗1(𝜆_𝑗−𝛼𝐴𝑘^−𝛽). (B.6) We see that under (B.3), we have 0<𝑇_ℎ𝑗<𝑇₀. 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 𝑎𝐽=¹¹𝑎₁₂𝑎₂₁𝑎₂₂, (B.7) where 𝑎₁₁=(𝛼𝐴𝑘^𝛽+𝛿)𝜆₁+(ℎ₁𝑇₀−𝑘₁𝑘)𝜆₁𝛼𝛽𝐴𝑘^𝛽𝜕𝜙𝜕𝑘₁,𝑎₁₂=(ℎ₁𝑇₀−𝑘₁𝑘)𝜆₁𝛼𝛽𝐴𝑘^𝛽𝜕𝜙𝜕𝑘₂,𝑎₂₁=(ℎ₂𝑇₀−𝑘₂𝑘)𝜆₂𝛼𝛽𝐴𝑘^𝛽𝜕𝜙𝜕𝑘₁,𝑎₂₂=(𝛼𝐴𝑘^𝛽+𝛿)𝜆₂+(ℎ₂𝑇₀−𝑘₂𝑘)𝜆₂𝛼𝛽𝐴𝑘^𝛽𝜕𝜙𝜕𝑘₂, (B.8) in which we calculate from (3.1) Λ^′𝜕𝜙𝜕𝑘_𝑗=𝜎_𝑗𝑘^𝛽+𝐴(𝑁_𝑗−𝛼𝑛_𝑗)>0,𝑗=1,2, (B.9) where Λ^′>0 is given by (3.3). We have 𝜕𝜙/𝜕𝑘_𝑗>0, 𝑗=1,2. The two eigenvalues, 𝜌₁ and 𝜌₂, are determined by 𝜌_1,2=𝑎₁₁+𝑎₂₂±(𝑎₁₁−𝑎₂₂)²+4𝑎₁₂𝑎₂₁2. (B.10) 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.