Capital and Knowledge: Integrating Arrow’s Learning-by-Doing,the Walrasian Equilibrium Theory and Neoclassical Growth Theory

Abstract

This article proposes a neoclassical growth model with endogenous capital accumulation and knowledge creation. The model integrates Arrow’s learning by doing, Uzawa’s two-sector growth model and Walrasian general equilibrium theory. We use a utility function, which determines leisure time, saving and consumption with utility optimization without leading to a higher dimensional dynamic system like the Ramsey approach. The dynamics of J-household economy is described by a ( J + 1)-dimensional differential equations system. We simulate the motion of the model and demonstrate transitional and long-term effects of changes in the propensity to save, the propensity to use leisure time, the population, the learning-by-doing efficiency and knowledge utilization efficiency. The comparative dynamic analysis provides some insights into issues related to inequality and growth.

JEL Classification: D51, D31, O41, E13

Keywords

Walrasian general equilibrium theory neoclassical growth theory learning by doing elastic labour supply knowledge creation capital accumulation

Introduction

The purpose of this study is to study inequality and growth by building a neoclassical growth model of heterogeneous households with endogenous capital accumulation and knowledge creation. Our model is influenced by three main theories in economics—Arrow’s learning by doing, Walrasian general equilibrium theory and neoclassical growth theory. The Walrasian general equilibrium theory of pure exchange and production economies is the core theory of formal economic theories with microeconomic foundation. The theory is proposed by Walras and further formalized by Arrow, Debreu and others in 1950s (e.g., Arrow, 1974; Arrow and Debreu, 1954; Arrow and Hahn, 1971; Debreu, 1959; Gale, 1955; Mas-Colell et al., 1995; McKenzie, 1959; Nikaido, 1956, 1968; Walras, 1874). The general equilibrium theory describes market equilibrium with economic mechanisms of production, consumption and exchanges with heterogeneous industries and households. Nevertheless, the theory has not been successfully generalized and extended to growth theory of heterogeneous households with endogenous wealth. Although Walras introduced saving and capital accumulation in his general equilibrium theory, he failed to properly integrate his ideas about capital accumulation into the general equilibrium theory (e.g., Impicciatore et al., 2012). Over years, there are different attempts to further develop Walras’ capital accumulation within Walras’ framework of heterogeneous households (e.g., Dana et al., 1989; Diewert, 1977; Eatwell, 1987; Montesano, 2008; Morishima, 1964, 1977). The common problem for these approaches is the lack of proper microeconomic foundation for wealth accumulation. To overcome this problem, Impicciatore et al. (2012) propose a model in which it is assumed that consumers store capital goods in order to supply their services to the production sector in the next period under the condition that capital goods exiting in one period totally depreciate at the end of the period. Nevertheless, the model still relies on a proper treatment of household saving behaviour. This study introduces an alternative approach for modelling wealth accumulation with the traditional Walrasian general equilibrium framework of heterogeneous households.

Although the Walrasian general equilibrium theory shows how economic equilibrium is achieved, the theory is not proper for addressing issues related to growth and structural change with wealth and income distribution. The neoclassical growth theory mainly developed since the 1950s deals with capital accumulation, even though the theory is not effectively for explaining wealth and income distribution as it is mainly for economies with homogeneous population (e.g., Barro and Sala-i-Martin, 1995; Burmeister and Dobell, 1970; Solow, 1956). The neoclassical growth theory directly models endogenous wealth accumulation with microeconomic foundation (e.g., Ramsey model). This study is based on the neoclassical growth theory. This study follows Uzawa’s two sector growth model in describing economic structure (Uzawa, 1961). Uzawa’s two-sector model has been generalized and extended in different ways over years (see Diamond, 1965; Drugeon and Venditti, 2001; Mino, 1996; Stiglitz, 1967). We will integrate the neoclassical growth theory with the Walrasian general equilibrium theory for studying dynamic interactions among growth, wealth and income distribution and economic structures. It should be noted that some attempts have been made to introduce neoclassical growth theory into the general equilibrium analysis (e.g., Jensen and Larsen, 2005). As reviewed by Shoven and Whalley (1992, p. 1), ‘Most contemporary applied general models are numerical analogs of traditional two-sector general equilibrium models popularized by James Meade, Harry Johnson, Arnold Harberger, and others in the 1950s and 1960s.’ From the history of analytical economics, we know it is difficult to properly model economic growth with wealth and income distribution.

Except integrating the two theories in a single analytical framework, we also take account of endogenous knowledge and elastic labour supply in modelling wealth and income distribution. The literature on endogenous knowledge and economic growth has increasingly expanded since Romer (1986) re-examined issues of endogenous technological change and economic growth in his 1986’s paper (see also Aghion and Howitt, 1998; Brecher et al., 2002; Chari and Hopenhayn, 1991; Grossman and Helpman, 1991; Lucas, 1988; Martin and Ottaviano, 2001; Nocco, 2005). The pioneering work on formally modelling endogenous knowledge is by Arrow (1962) who introduced learning by doing into the formal growth theory. In this study, we model the knowledge creation with Arrow’s learning by doing. There are many applications of Arrow’s idea to different issues in economic growth theory (e.g., Albelo and Manresa, 2005; Kitaura and Yakita, 2010; Martin and Rogers, 1997; Shea, 2013; Solow, 1997; Young, 1991, 1993). This study follows an application of learning by doing by Zhang (1993). This article also extends a recent article on economic growth with heterogeneous households by Zhang (2012) by introducing endogenous knowledge into the neoclassical growth model. The rest of this article is organized as follows. The second section defines the heterogeneous-households neoclassical growth model with capital accumulation and knowledge creation. The third section shows that the dynamics of the economy with J types of households can be described by ( J + 1)-dimensional differential equations. As mathematical analysis of the system is too complicated, we demonstrate some of the dynamic properties by simulation when the economy consists of three types of households. The fourth section carries out comparative dynamic analysis with regard to the population, knowledge utilization efficiency, propensity to save and learning-by-doing efficiency. The fifth section concludes the study.

The Model of Heterogeneous Households with Capital and Knowledge

The economic structure is based on the traditional two-sector model initially proposed by Uzawa (1961). The two sectors use capital and labour inputs. The capital goods sector produces capital goods, while the consumer goods sector produces consumer goods. Consumer and capital goods are different commodities. Capital depreciates at a constant exponential rate $_{k}$ . It is assumed that the depreciation rate is independent of the manner of use. We follow the neoclassical growth theory in modelling technology and behaviour of the two sectors (see Barro and Sala-i-Martin, 1995; Burmeister and Dobell, 1970; Solow, 1956). It is assumed that the households own capital of the economy and distribute their incomes to consume and to save. Exchanges take place in perfectly competitive markets. The labour and capital inputs are smoothly substitutable for each other in each sector and are freely transferable from one sector to the other. We select the capital goods to serve as numeraire (whose price is normalized to 1), with all the other prices being measured relative to its price. The population is classified into multiple groups, indexed by j = 1, ..., J. The extreme case is that J equals the population. We use T_j(t) to represent the work time of a typical worker in group j. Each group has a fixed number of population, $\overset{}{N_{j}}$ , ( j = 1, ..., J). Each group’s total labour supply $N_{j} (t)$ , is

N_{j} (t) T_{j} (t) Z^{m_{j}} (t) {\overset{}{N}}_{j},

where Z(t) (> 0) is the knowledge stock at time t and Z(t) (> 0). Here, we interpret $Z^{m_{j}} (t)$ as group j’s level of human capital. We call m_j group j’s knowledge utilization efficiency parameter. The total labour supply of the economy is

N (t)_{j 1}^{J} N_{j} (t)_{j 1}^{J} T_{j} (t) Z^{m_{j}} (t) {\overset{}{N}}_{j} .

(1)

The Production Sectors

The capital and consumer goods sectors use neoclassical technology. We use subscripts i and s to denote the capital goods sector and the consumer goods sector. Let $F_{j} (t)$ stand for the output of sector j, and $K_{j} (t)$ and $N_{j} (t)$ respectively for the capital and labour inputs used in sector j. The production functions are

F_{j} (t) A_{j} K_{j}^{_{j}} (t) N_{j}^{_{j}} (t),_{j} > 0,_{j}_{j} 1, j i, s,

(2)

where j is sector j’s total productivity, $_{j}$ and $_{j}$ are the output elasticities of capital and labour, respectively. We can also write the production functions in terms of the capital intensities as follows:

f_{j} (k_{j} (t)) A_{j} k_{j}^{_{j}} (t), A_{j} > 0, 0_{j} 1, j i, s,

(3)

where

f_{j} (t) f_{j} (k_{j} (t))

f_{j} (t) \frac{F_{j} (t)}{N_{j} (t)}, k_{j} (t) \frac{K_{j} (t)}{N_{j} (t)}, j i, s .

In perfectly competitive markets, labour and capital inputs are paid according to their marginal products. Let $p (t)$ stand for the price of consumer goods. The rate of interest, $r (t)$ and wage rate, $w (t)$ are determined by markets. For individual firms, $r (t)$ and $w (t)$ are exogenous. The production sectors choose the two inputs, $K_{j} (t)$ and $N_{j} (t)$ , to maximize their profits. The marginal conditions are

\begin{matrix} r (t)_{k}_{i} A_{i} k_{i}^{_{i}} (t)_{s} A_{s} p (t) k_{s}^{_{s}} (t), \\ w (t)_{i} A_{i} k_{i}^{_{i}} (t)_{s} A_{s} p (t) k_{s}^{_{s}} (t). \end{matrix}

(4)

Labour and capital are fully employed, that is,

K_{i} (t) K_{s} (t) K (t),

N_{i} (t) N_{s} (t) N (t) .

(5)

Behaviour of Consumers

Each worker may get income from wealth ownership and wages. Consumers make decisions on consumption levels of goods as well as on how much to save. This study uses the approach to consumers’ behaviour proposed by Zhang (1993). Let ${\overset{}{k}}_{j} (t)$ stand for per capita wealth in group j. We note that the wage rate of group j is

w_{j} (t) w (t) Z^{m_{j}} (t), j 1,, J .

Each consumer of group j obtains current income

y_{j} (t) r (t) {\overset{}{k}}_{j} (t) w_{j} (t) T_{j} (t), j 1,, J,

(6)

from the interest payment $r {\overset{}{k}}_{j}$ and the wage payment $w T_{j} Z^{m_{j}} .$ We call $y_{j}$ the current income in the sense that it comes from consumers’ payment and consumers’ current earnings from ownership of wealth. The sum of income that consumers are using for consuming, saving or transferring are not necessarily equal to the temporary income, because consumers can sell wealth to pay, for instance, the current consumption if the current income is not sufficient for buying food. 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 a consumer of group j can sell to purchase goods and to save is equal to ${\overset{}{k}}_{j} (t) .$ Here, we assume that selling and buying wealth can be conducted instantaneously without any transaction cost. The disposable income is

{\overset{}{y}}_{j} (t) y_{j} (t) {\overset{}{k}}_{j} (t)

(7)

The disposable income is used for saving and consumption. It should be noted that the value, ${\overset{}{k}}_{j} (t),$ (i.e., $p_{i} (t) {\overset{}{k}}_{j} (t)$ with $p_{i} (t) 1$ ) in the above equation is a flow variable. Under the assumption that selling wealth can be conducted instantaneously without any transaction cost, we may consider ${\overset{}{k}}_{j} (t)$ as the amount of the income that the consumer obtains at time t by selling all of his wealth. Hence, at time t, the consumer has the total amount of income equalling ${\overset{}{y}}_{j} (t)$ to distribute between consuming and saving.

The disposable income is used for saving and consumption. At each point of time, the representative 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) {\overset{}{y}}_{j} (t) .

(8)

Let $T_{0}$ denote the total available time. The time constraint requires that the amounts of time allocated to each specific use add up to the time available.

T_{j} (t) {\overset{}{T}}_{j}^{} (t) T_{0}, j 1,, J .

(9)

Substituting (9) into (8) yields

\begin{matrix} w_{j}^{} (t) {\overset{}{T}}_{j}^{} (t) c_{j} (t) s_{j} (t) \\ {\overset{}{y}}_{j} (t) (1 r (t)) {\overset{}{k}}_{j} (t) T_{0} w_{j}^{} (t) . \end{matrix}

(10)

At each point of time, households decide the three variables subject to the disposable income. We assume that utility level $U_{j} (t)$ is dependent on ${\overset{}{T}}_{j}^{} (t),$ $c_{j} (t)$ and $s_{j} (t)$ as follows:

U_{j} (t) {\overset{}{T}}_{j}^{_{j 0}} (t) c_{j}^{_{j 0}} (t) s_{j}^{_{j 0}} (t),_{j 0},_{j 0},_{j 0} > 0,

where $_{j},$ $_{j}$ and $_{j}$ are respectively country j’s propensities to use the leisure time, to consume goods and to hold wealth. Maximizing $U_{j}$ subject to (10) yields

w_{j}^{} (t) Z^{m_{j}} (t)_{j} {\overset{}{y}}_{j} (t), c_{j} (t)_{j} {\overset{}{y}}_{j} (t), s_{j} (t)_{j} {\overset{}{y}}_{j} (t),

(11)

where

_{j}_{j}_{j 0},_{j}_{j}_{j 0},_{j}_{j}_{j 0},_{j} \frac{1}{_{j 0}_{j 0}_{j 0}},

where $_{j 0},$ $_{0 j}$ and $_{0 j}$ are respectively household j’s propensities to use the leisure time, to consume and to hold wealth. Here, for simplicity, we specify the utility function with the Cobb–Douglas form. It would provide more insights if we take some other forms of utility functions. In this study, we fix the preference structure. It is quite reasonable to assume that one’s attitude towards the future is dependent on factors such as capital gains, the stock of durables, income distribution and demographic factors.

According to the definitions of $s_{j} (t),$ the wealth accumulation of the representative household in group j is given by

{\overset{}{\overset{}{k}}}_{j} (t) s_{j} (t) {\overset{}{k}}_{j} (t) .

(12)

This equation simply states that the change in wealth is equal to the savings minus dissavings.

Knowledge Accumulation through Arrow’s Learning by Doing

Arrow (1962) first introduced learning by doing into growth theory. Like capital, a refined classification of knowledge and technologies tend to lead new conceptions and modelling strategies. This study uses knowledge in a highly aggregated sense. We assume that knowledge growth is through the so-called learning by doing. We propose the following equation for knowledge growth:

\overset{}{Z} (t) \frac{_{i} F_{i} (t)}{Z^{_{i}} (t)} \frac{_{s} F_{s} (t)}{Z^{_{s}} (t)}_{z} Z (t),

(13)

in which $_{z} (0)$ is the depreciation rate of knowledge, and $_{j}$ and $_{j}$ are parameters. Equation (13) implies that knowledge accumulation is through learning by doing. The parameters $_{j}$ and $_{z}$ are non-negative. We interpret $_{j} F / Z^{_{j}}$ as the contribution to knowledge accumulation through learning by doing by the capital goods sector. To see how learning by doing occurs, assume that knowledge is a function of the total output during some period

Z (t) a_{1} {\{_{0}^{t} F_{j} () d\}}^{a_{2}} a_{3}

in which $a_{1}, a_{2}$ and $a_{3}$ are positive parameters. The above equation implies that the knowledge accumulation through learning by doing exhibits decreasing (increasing) returns to scale in the case of $a_{2} (>) 1$ . We interpret $a_{1}$ and $a_{3}$ as the measurements of the efficiency of learning by doing by the production sector. Taking the derivatives of the equation yields $\overset{}{Z}_{j} F_{j} / Z^{_{j}},$ in which $_{j} a_{1} a_{2}$ and $_{j} 1 a_{2} .$

Balance of Demand and Supply

The total capital stock is equal to the wealth owned by all the households. That is,

K (t) K_{i} (t) K_{s} (t)_{j 1}^{J} {\overset{}{k}}_{j} (t) {\overset{}{N}}_{j} .

(14)

Demand and Supply of the Two Sectors

The demand for consumer goods is equal to the supply of consumer goods at any point of time. That is,

_{j 1}^{J} c_{j} (t) {\overset{}{N}}_{j} F_{s} (t) .

(15)

The national saving is the sum of the households’ saving. As output of the capital goods sector is equal to the net savings and the depreciation of capital stock, we have

S (t) K (t)_{k} K (t) F_{i} (t),

(16)

where $S (t) K (t)_{k} K (t)$ is the sum of the net saving and depreciation and

S (t)_{j 1}^{J} s_{j} (t) {\overset{}{N}}_{j} .

We have thus built the model with trade, economic growth, capital accumulation, knowledge creation and utilization. Irrespective of the obvious strict assumptions in our model, from a structural point of view, the model is quite general in the sense that some well-known models in economics can be considered as its special cases. For instance, if the population is homogeneous, our model is structurally similar to the neoclassical growth model by Solow (1956) and Uzawa (1961). It is structurally similar to the Ricardian models by Pasinetti and Samuelson (e.g., Caravale and Tosato, 1980; Casarosa, 1985; Pasinetti, 1960, 1974; Samuelson, 1959).

The Economic Dynamics

It is reasonable to expect that the dynamics should be described by highly dimensional differential equations. In the Appendix, we show that the dynamics is described by a $(J 1)$ dimensional differential equations system. As it is too difficult to get analytical properties of the dynamic system, we simulate the model for illustrating the behaviour of the system. First, we introduce a variable

z \frac{r_{k}}{w} .

Lemma

The dynamics of the economy is governed by the following $(J 1)$ dimensional differential equations system with $z (t),$ $Z (t)$ and ${\overset{}{k}}_{j} (t),$ $j$ $2,, J,$ as the variables

\overset{}{z} (t)_{1} (z (t), Z (t), \{{\overset{}{k}}_{j} (t)\}),

{\overset{}{\overset{}{k}}}_{j} (t)_{j} (z (t), Z (t), \{{\overset{}{k}}_{j} (t)\}), j 2, ..., J,

\overset{}{Z} (t) (z (t), Z (t), \{{\overset{}{k}}_{j} (t)\}),

(17)

in which $_{j} (t)$ and $(t)$ are unique functions of $z (t),$ $Z (t)$ and $\{{\overset{}{k}}_{j} (t)\} ({\overset{}{k}}_{2} (t), ..., {\overset{}{k}}_{J} (t))$ at any point of time, defined in Appendix. For any given positive values of $z (t),$ $Z (t)$ and $\{{\overset{}{k}}_{j} (t)\}$ at any point of time, the other variables are uniquely determined by the following procedure: $r (t),$ $w (t)$ and $p (t)$ by (A3) → ${\overset{}{k}}_{1} (t)$ by (A17) → $w_{j} (t) w (t) Z^{m_{j}} (t)$ → $T_{j} (t)$ by (A10) → ${\overset{}{T}}_{j} (t) T_{0} T_{j} (t)$ → $N (t)$ by (A11) → $K_{s} (t)$ by (A7) → $K_{i} (t)$ by (A3) → $K (t) K_{i} (t) K_{s} (t)$ → $N_{i} (t) {\overset{}{}}_{i} z (t) K_{i} (t)$ → $N_{s} (t) {\overset{}{}}_{s} z (t) K_{s} (t)$ by (A14) → ${\overset{}{y}}_{j} (t)$ by (A4) → $F_{i} (t)$ and $F_{s} (t)$ by the definitions → $c_{j} (t)$ and $s_{j} (t)$ by (11).

The computational procedure is important for us to plot motion of the economic system with any number of types of households. With regard to the Arrow–Debreu concept of general equilibrium the final stage of analysis is to find a price vector at which excess demand is zero (Judd, 1988). There are numerical approaches for calculating equilibria (e.g., Scarf, 1967; Scarf and Hansen, 1973). We can apply these traditional methods to find how the prices and other variables are related to the variables in the differential equations. As it is difficult to provide analytical properties of the dynamic system, for illustration we simulate the model with the following parameters:

\begin{matrix} T_{0} 1, A_{i} 1, A_{s} 1,_{i} 0.28,_{s} 0.34,_{i} 0.3, \\ _{s} 0.3,_{i} 0.3,_{s} 0.4,_{k} 0.05,_{Z} 0.04, \end{matrix}

\begin{matrix} (\begin{matrix} N_{1} \\ N_{2} \\ N_{3} \end{matrix}) (\begin{matrix} 3 \\ 4 \\ 8 \end{matrix}), (\begin{matrix} m_{1} \\ m_{2} \\ m_{3} \end{matrix}) (\begin{matrix} 0.5 \\ 0.4 \\ 0.3 \end{matrix}), (\begin{matrix} _{10} \\ _{20} \\ _{30} \end{matrix}) (\begin{matrix} 0.8 \\ 0.75 \\ 0.6 \end{matrix}), (\begin{matrix} _{10} \\ _{20} \\ _{30} \end{matrix}) \\ (\begin{matrix} 0.1 \\ 0.15 \\ 0.1 \end{matrix}), (\begin{matrix} _{10} \\ _{20} \\ _{30} \end{matrix}) (\begin{matrix} 0.2 \\ 0.2 \\ 0.2 \end{matrix}) . \end{matrix}

(18)

The population of Group 3 is the largest, while the population of Group 2 is the next. Group 1 applies knowledge mostly effective, while Group 3 applies least effectively. The capital goods and consumer goods sectors’ total productivities are 1. We specify the values of the parameters, $_{j},$ in the Cobb–Douglas productions for the capital goods and consumer goods sectors approximately equal to 0.3 (e.g., Abel et al., 2007; Miles and Scott, 2005). The depreciation rates of physical capital and knowledge are specified at 0.05 and 0.04. Group 1’s propensity to save is 0.8 and Group 3’s propensity to save is 0.6. The value of Group 2’s propensity to save is between the other two groups. We specify the initial conditions as follows

z (0) 0.05, Z (0) 77, {\overset{}{k}}_{2} (0) 16, {\overset{}{k}}_{3} (0) 10 .

The motion of the variables is plotted in Figure 1. In Figure 1, the national income is

Y (t) F_{i} (t) p_{s} (t) F_{s} (t) .

The knowledge stock rises over time, while the level of the national capital stocks falls. The national output is augmented over time. The labour force is increased, while Group $1' s$ and Group $3' s$ wage rates are reduced over time. Groups 1 and 3 raise their work hours, while Group 3 slightly reduces its work hours. The output level of the capital goods sector is increased, while the output level of the consumer goods sector is reduced. The labour and capital inputs of the capital goods sector are increased. The capital input of the consumer goods sector rises, while the labour input of the consumer goods sector falls initially and then rises.

Figure 1.

The Motion of the Economic System

It is straightforward to confirm that the variables become stationary. The simulation confirms that the system has a unique equilibrium. We list the equilibrium values in (19):

\begin{matrix} K 233.7, N 26.42, Y 48.89, F_{i} 11.69, F_{s} 42.21, \\ K_{i} 48.03, K_{s} 185.68, N_{i} 6.74, N_{s} 19.68, r 0.018, \\ p 0.88, w 1.25, w_{1} 11.08, w_{2} 7.16, w_{3} 4.63, \end{matrix}

\begin{matrix} c_{1} 4.4, c_{2} 3.62, c_{3} 1.82, {\overset{}{k}}_{1} 31.04, {\overset{}{k}}_{2} 15.96, \\ {\overset{}{k}}_{3} 9.6, {\overset{}{T}}_{1} 0.7, {\overset{}{T}}_{2} 0.6, {\overset{}{T}}_{3} 0.69 . \end{matrix}

(19)

It is straightforward to calculate the four eigenvalues as follows:

\{0.324, 0.285, 0.207, 0.04\} .

The eigenvalues are real and negative. The unique equilibrium is locally stable. This is important as the stability guarantees that we can effectively conduct comparative dynamic analysis for transitional dynamics as well as long-term state.

Comparative Dynamic Analysis

As we can plot the motion of the national economy, it is straightforward for us to show how the economic system reactions to exogenous changes. As the lemma gives the computational procedure to calibrate the motion of all the variables, we can study effects of change in any parameter on transitional processes as well as stationary states of all the variables. We introduce a variable $\overset{}{} x_{j} (t)$ which stands for the change rate of the variable, $x_{j} (t),$ in percentage due to changes in the parameter value.

Group 1 Augmenting Its Propensity to Save

From the Walrasian general equilibrium theory, it is well established that changes in preferences of different households influence distributions of production factors and incomes and prices. Nevertheless, as the Walrasian general equilibrium theory has not been properly extended to general dynamics with wealth accumulation, important issues such as effects of changes in saving propensities on economic structure cannot be properly addressed by the traditional analytical framework. As our analytical framework integrates the economic mechanisms of the Walrasian general equilibrium theory and neoclassical growth theory, in principle, we can analyze effects of a change in the preference of any people on the dynamic path of the economic growth. For illustration, we now study the case that Group 1 increases its propensity to save in the following way: $_{01} : 0.8 0.81 .$ The simulation results are plotted in Figure 2. An immediate impact of the preference change is that the typical household in Group 1 increases the wealth. As the propensity to enjoy leisure and the propensity to consume consumer goods are reduced, the household consumes less consumer goods and work longer hours in the transitional process. But in the long term, the household’s leisure time is only slightly affected as the households accumulate more wealth and have higher wage rate. It should be noted that in the long term, the household’s consumption is increased as well. This positive impact on consumption is due to the positive effects on knowledge and capital stocks. As the propensity to save is enhanced, the national wealth tends to be increased. The increase in the national capital stocks enables the two sectors to employ more capital stocks, resulting in the augment of the output levels of the two sectors. As the two sectors increase their output levels, knowledge stock is increased. The improved productivity due to the rise in the knowledge stock enhances the wage rates of the three groups. It should be noted that the rise in the capital stocks tends to reduce the wage rates. The net result raises the wage rates. The rate of interest and the price of consumer goods are reduced. As the output levels of the two sectors are increased and the price is only slightly affected, the net impact on the national output is positive. The capital goods sector attracts more labour force, while the labour force to the consumer goods sector is slightly affected. Groups 2 and 3 work more hours and increase their consumption and wealth. In the long terms, the inequalities in the wage rates, wealth levels and consumption levels among Group 1 and the other groups are enlarged.

Figure 2.

A Rise in Group 1’s Propensity to Save

Group 1 Enhancing its Knowledge Utilization Efficiency

The impact of human capital is currently a main topic in economic theory and empirical research (e.g., Bandyopadhyay and Tang, 2011; Barro, 2001; Castelló-Climent and Hidalgo-Cabrillana, 2012; Easterlin, 1981; Hanushek and Kimko, 2000; Krueger and Lindahl, 2001). There are different empirical conclusions about inequalities and human capital (e.g., Could et al., 2001; Fleisher et al., 2011; Tilak, 1989; Tselios, 2008). This study addresses issues related to dynamic interactions among growth, inequality and distribution by assuming heterogeneity in preferences and knowledge utilization efficiencies among different types of people. To see how we discuss the issues, we now allow Group 1 to enhance its knowledge utilization efficiency as follows: $m_{1} : 0.5 0.52 .$ The simulation result is plotted in Figure 3. An immediate result of the change is that the knowledge stock is increased. The wage rates of the three groups are increased. The wealth and consumption levels of the three groups are all increased. We see that national economic growth is sped up and the inequalities in the wage rates, consumption and wealth among Group 1 and the other groups are enlarged. From Figure 2, we also see that the total capital and labour force are augmented in association with the increase in the knowledge stock. As the supplies of two goods are increased, the rate of interest and price are lowered. The change in the price has negative impact on the national economic growth and the changes in the output levels have positive effects. The net result leads to increases in the national output level over time.

Figure 3.

Group 1 Enhancing Its Knowledge Utilization Efficiency

As Forbes (2000) comments, a ‘careful reassessment of the relationship between these two variables (growth rate and income inequality) needs further theoretical and empirical work evaluating the channels through which inequality, growth, and any other variables are related.’ Our simulation demonstrates that as Group 1 improves its knowledge utilization efficiency, national economic growth is sped up and the inequalities in the wage rates, consumption and wealth among Group 1 and the other groups are enlarged. It is straightforward to demonstrate that if Group 3 increases its knowledge utilization efficiency, then the inequalities in the wage rates, consumption and wealth among Group 1 and the other groups are diminished. This implies that in order to reduce inequalities, it is effective for the government to encourage the groups of less education and low income to more effectively accumulate and apply knowledge.

Group 2’s Population Being Increased

There are debates about relationships between population change and economic growth. Theoretical models with human capital predict situation-dependent interactions between population and economic growth (see Boucekkine et al., 2002; Bretschger, 2013; Ehrlich and Lui, 1997; Galor and Weil, 1999). There are also mixed conclusions in empirical studies on the issue (e.g., Furuoka, 2009; Yao et al., 2013). Although this study keeps the population and its structure fixed, we may provide some insights into the relationship by examining effects of changes in the population sizes. As different groups have different levels of knowledge utilization efficiency and creativity, increases in the population sizes may have different effects upon the economy. We now change Group 1’s population to be increased as follows: $N_{1} : 3 3.2 .$ The results are plotted in Figure 4. We see that as far as the aggregate variables are concerned, during transitional processes as well as in the long term, the national output, knowledge stock, total capital stock, total labour force, capital inputs and labour inputs of the two sectors and the output levels of the two sectors are all increased. The rate of interest and price of consumer goods fall in association with the rises in the total capital and supply of consumer goods. The wage rates of the three groups are enhanced in association with the augment in the knowledge stock. The work hours of the three groups are increased. As the group’s population is increased, Group 1’s levels of consumption and wealth fall initially and then rise. The other two groups’ levels of consumption and wealth are increased. We conclude that in the economy with endogenous knowledge, population increase benefits the national economic growth as well as long-term consumption and wealth accumulation. It should be noted that the inequalities are not reduced when the group with high knowledge utilization efficiency increases its population.

Figure 4.

Group 1 Increasing Its Population

Group 3 Increasing its Propensity to Use Leisure Time

We now change Group 3’s propensity to use leisure time as follows: $_{03} : 0.2 0.22 .$ The results are plotted in Figure 5. We see that as far as the aggregate variables are concerned, during transitional processes as well as in the long term, the national output, knowledge stock, total capital stock, total labour force, capital inputs and labour inputs of the two sectors and the output levels of the two sectors are all reduced. The rate of interest and price of consumer goods rise initially and then fall. The wage rates of the three groups are lowered in association with the reduction in the knowledge stock. Group 3 reduces its work hours, while the other two groups’ work hours are slightly affected in the long term. The three groups’ levels of consumption and wealth are reduced.

Figure 5.

Group 3 Increasing Its Propensity to Use Leisure Time

The Capital Goods Sector Learning More Effectively through Doing

We now study the case that the capital goods sector learning more effectively through doing as follows: $_{1} : 0.3 0.32 .$ The results are illustrated in Figure 6. During transitional processes as well as in the long term, the national output, knowledge stock, total capital stock, total labour force, capital inputs and labour inputs of the two sectors and the output levels of the two sectors are all increased. The rate of interest and price of consumer goods rise initially and then fall. In the long term, the rate of interest and price of consumer goods are affected slightly. The wage rates of the three groups are enhanced in association with the augment in the knowledge stock. The work hours of the three groups are increased. The three groups’ levels of consumption and wealth are increased.

Figure 6.

The Capital Goods Sector Learning More Effectively through Doing

Conclusions

This article proposed a neoclassical growth model with endogenous capital accumulation and knowledge creation. The model is influenced by Arrow’s learning by doing, Uzawa’s two-sector growth model and Walrasian general equilibrium theory. Different from the growth models with the Ramsey approach in the literature, we used a utility function, which determines leisure time, saving and consumption with utility optimization without leading to a higher dimensional dynamic system like the traditional approach. The dynamics of $J$ -household economy is controlled by a $(J 1)$ -dimensional differential equations system. We also simulated the motion of the model and demonstrated transitional and long-term effects of changes in the propensity to save, the propensity to use leisure time, the population, the learning-by-doing efficiency and knowledge utilization efficiency. The comparative dynamic analysis provides some insights into issues related to inequality and growth. For instance, we demonstrated that when the group with the highest knowledge utilization efficiency increases its propensity to save, the typical household in the group increases its wealth and consumes less consumer goods and work longer hours in the transitional process. But in the long term, the household’s leisure time is only slightly affected and has higher wage rate. The national wealth is increased. The increase in the national capital stocks enables the two sectors to employ more capital stocks, resulting in the augment of the output levels of the two sectors. As the two sectors increase their output levels, knowledge stock is increased. The wage rates of the three groups are increased. The net impact on the national output is positive. The capital goods sector attracts more labour force, while the labour force to the consumer goods sector is slightly affected. The other two groups work more hours and increase their consumption and wealth. In the long terms, the inequalities in the wage rates, wealth levels and consumption levels among the group and the other groups are enlarged. It is well known that one-sector growth model has been generalized and extended in many directions. It is not difficult to generalize our model along these lines. It is straightforward to develop the model in discrete time. We may analyze behaviour of the model with other forms of production or utility functions.

Footnotes

Proving Lemma 1

By (4) we obtain (A1)

z \frac{r_{k}}{w} \frac{1}{{\overset{}{}}_{i} k_{i}} \frac{1}{{\overset{}{}}_{s} k_{s}},

where ${\overset{}{}}_{j}_{m} /_{m} .$ From (A1) and (4), we also have (A2)

r (z)_{i} A_{i} {(z {\overset{}{}}_{i})}^{_{i}}_{k}, w (z) \frac{r_{k}}{z}, p (z) \frac{w}{_{s} A_{s} k_{s}^{_{s}}} .

Hence, we determine the rate of interest, the wage rate and the price as functions of $z .$ From (A1) and (5), we obtain (A3)

{\overset{}{}}_{i} K_{i} {\overset{}{}}_{s} K_{s} \frac{N}{z} .

From the definitions of ${\overset{}{y}}_{j},$ we have (A4)

{\overset{}{y}}_{j} (1 r) {\overset{}{k}}_{j} T_{0} w Z^{m_{j}} .

Insert $p c_{j}_{j} {\overset{}{y}}_{j}$ in (15) (A5)

_{j 1}^{J}_{j} {\overset{}{N}}_{j} {\overset{}{y}}_{j} p F_{s} .

Substituting (A4) in (A5) yields (A6)

(1 r)_{j 1}^{J}_{j} {\overset{}{N}}_{j} {\overset{}{k}}_{j} T_{0} w_{j 1}^{J}_{j} {\overset{}{N}}_{j} Z^{m_{j}} p F_{s} .

Insert $p F_{s} w N_{s} /_{s}$ in (A6) (A7)

{\overset{}{g}}_{0}_{j 1}^{J}_{j} {\overset{}{N}}_{j} {\overset{}{k}}_{j} \overset{}{g} K_{s},

where we use $N_{s} z {\overset{}{}}_{s} K_{s}$ and

\overset{}{g} (z, Z) \frac{_{s} T_{0}}{z {\overset{}{}}_{s}}_{j 1}^{J}_{j} {\overset{}{N}}_{j} Z^{m_{j}}, {\overset{}{g}}_{0} (z) \frac{R_{s}}{z {\overset{}{}}_{s}}, R (z) \frac{1 r}{w} .

From (A3) and (14), we solve (A8)

K_{s} \frac{{\overset{}{}}_{i}}{({\overset{}{}}_{i} {\overset{}{}}_{s})}_{j 1}^{J} {\overset{}{k}}_{j} {\overset{}{N}}_{j} \frac{N}{({\overset{}{}}_{i} {\overset{}{}}_{s}) z} .

From (A7) and (A8), we have (A9)

({\overset{}{}}_{i} {\overset{}{}}_{s}) \overset{}{g} \frac{N}{z}_{j 1}^{J} {\overset{}{}}_{j} {\overset{}{k}}_{j},

where

{\overset{}{}}_{j} (z) ({\overset{}{}}_{i} ({\overset{}{}}_{i} {\overset{}{}}_{s}) {\overset{}{g}}_{0}_{j}) {\overset{}{N}}_{j} .

Insert (A4) in $w Z^{m_{j}} {\overset{}{T}}_{j}^{}_{j} {\overset{}{y}}_{j}$ in (11), we have

w Z^{m_{j}} {\overset{}{T}}_{j}^{} (1 r)_{j} {\overset{}{k}}_{j} T_{0}_{j} w Z^{m_{j}} .

From this equation and $T_{j} {\overset{}{T}}_{j}^{} T_{0}$ (A10)

T_{j}^{} (1_{j}) T_{0}^{} \frac{_{j} R {\overset{}{k}}_{j}}{Z^{m_{j}}} .

From (A10) and (1), we get (A11)

N R_{j 1}^{J}_{j} {\overset{}{k}}_{j} {\overset{}{N}}_{j},

where

(Z) T_{0}^{}_{j 1}^{J} (1_{j}) Z^{m_{j}} {\overset{}{N}}_{j} .

Insert (A11) in (A9) (A12)

({\overset{}{}}_{i} {\overset{}{}}_{s}) z \overset{}{g}_{j 1}^{J} R_{j} {\overset{}{k}}_{j},

where

R_{j} (z) z {\overset{}{}}_{j} R_{j} {\overset{}{N}}_{j} .

Solve (A12) with regard to ${\overset{}{k}}_{1}$ (A13)

{\overset{}{k}}_{1} (z, Z, \{{\overset{}{k}}_{j}\}) \frac{({\overset{}{}}_{i} {\overset{}{}}_{s}) z \overset{}{g}}{R_{1}} \frac{1}{R_{1}}_{j 2}^{J} R_{j} {\overset{}{k}}_{j},

where $\{{\overset{}{k}}_{j}\} ({\overset{}{k}}_{2}, ..., {\overset{}{k}}_{J}) .$ It is straightforward to confirm that all the variables can be expressed as functions of $z,$ $Z$ and $\{{\overset{}{k}}_{j}\}$ by the following procedure: $r,$ $w$ and $p$ by (A3) → ${\overset{}{k}}_{1}$ by (A17) → $w_{j} w Z^{m_{j}}$ → $T_{j}$ by (A10) → ${\overset{}{T}}_{j} T_{0} T_{j}$ → $N$ by (A11) → $K_{s}$ by (A7) → $K_{i}$ by (A3) → $K K_{i} K_{s}$ → $N_{i} z {\overset{}{}}_{i} K_{i}$ → $N_{s} z {\overset{}{}}_{s} K_{s}$ by (A14) → ${\overset{}{y}}_{j}$ by (A4) → $F_{i},$ $F_{s}$ by the definitions → $c_{s},$ $s_{j}$ by (11). From this procedure, (A13), (A12) and (A13), we have (A14)

{\overset{}{\overset{}{k}}}_{1} (z, Z, \{{\overset{}{k}}_{j}\})_{1} {\overset{}{y}}_{1},

{\overset{}{\overset{}{k}}}_{j}_{j} (z, Z, \{{\overset{}{k}}_{j}\})_{j} {\overset{}{y}}_{j} {\overset{}{k}}_{j}, j 2, ..., J,

(A15)

\overset{}{Z} (z, Z, \{{\overset{}{k}}_{j}\}) \frac{_{i} F_{i}}{Z^{_{i}}} \frac{_{s} F_{s}}{Z^{_{s}}}_{z} Z .

Taking derivatives of equation (A13) with respect to t and then combining with (A15) implies (A16)

{\overset{}{\overset{}{k}}}_{1} \frac{}{z} \overset{}{z} \frac{}{Z}_{j 2}^{J}_{j} \frac{}{{\overset{}{k}}_{j}} .

Equalling the right-hand sizes of equations (A14) and (A16), we get (A17)

\overset{}{z}_{1} (z, Z, \{{\overset{}{k}}_{j}\}) [\frac{}{Z}_{j 2}^{J}_{j} \frac{}{{\overset{}{k}}_{j}}] {(\frac{}{z})}^{1} .

In summary, we proved the lemma.

Acknowledgements

The author is grateful to the constructive comments of the anonymous referee. The author is also grateful for the financial support from the Grants-in-Aid for Scientific Research (C), Project No. 25380246, Japan Society for the Promotion of Science.

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