Silviase/jgov_v1.4_full_ann huggingface dataset:
subject to comparable privacy regulation?
プライバシー規制に服さない第三者へ、個人データをシステマティックに開示したり、かかる第三者が個人データにアクセスできるようにしていないか
Data handlingデータ取扱い	(9) Does the project involve new or changed data collection policies or practices that may be unclear or intrusive?
不明瞭もしくは侵害的なデータ収集ポリシーを新しく盛り込むか、不明瞭もしくは侵害的なデータ収集を新しく行うか、またはデータ収集ポリシーもしくはデータ収集を変更して不明瞭もしくは侵害的なものとなるか
(10) Does the project involve new or changed data quality assurance processes and standards that may be unclear or unsatisfactory?
不明瞭または不十分なデータ保証プロセス及びデータ保証標準を新たに盛り込むか、またはデータ品質保証プロセス及びデータ保証標準を変更して不明瞭または不十分なものとなるか
(11) Does the project involve new or changed data security arrangements that may be unclear or unsatisfactory?
不明瞭または不十分なデータセキュリティを新たに組み込むか、または変更して不明瞭又は不十分なものとなるか
(12) Does the project involve new or changed data access or disclosure arrangements that may be unclear or permissive?
不明瞭または安易になりうるデータアクセスまたは開示を新たに組み込むか、または変更して不明瞭または安易になりうるか
(13) Does the project involve new or changed data retention arrangements that may be unclear or extensive?
不明瞭または広汎になりうるデータ保管を新たに組み込むか、または変更して不明瞭または広汎になりうるか
(14) Does the project involve changing the medium of disclosure for publicly available information in such a way that the data becomes more readily accessible than before?
公開情報の開示方法について、以前よりも容易にデータにアクセスできる方法に変更しているか（??、但し回答がYesだとリスクありとされるので、そうすると訳が変??）
## 第4 カナダ（連邦）

PIA ガイドラインにて、まず、システムにおいて個人情報の収集、使用または開示が行われるかを確認する。個人情報の収集等が行われる場合は、以下のチェックリストにより PIA の実施時期等を検討する。

①以下の行為を行うか

・新しいプログラム又はサービスの設計

・既存プログラム又はサービスの重大な変更

・旧型伝達方法から電子伝達方法へ変更し、顕著なプライバシー問題を有し PIA を実施していない場合

②プログラム上で個人情報の収集、使用又は開示を行うか

③プログラム上で以前よりも機微性の高い個人情報を収集、使用又は開示するか。個人情報の収集方法を、同意に基づく方法から間接的取得方法に変更するか。

④個人のプライバシー権に関する通知又は個人情報の収集、使用及び開示に関する同意取得の機能を設ける必要があるか

⑤プログラム上で、組織内の他のプログラム、他の組織、他の政府又は民間事業者から個人情報を収集するか

⑥プログラムによって生成された個人情報が、当該個人に直接影響を与える決定過程（たとえば、プログラム又はサービスの適格性や執行など）において用いられるか
⑦プログラムによって生成された個人情報が、その他の目的（調査、統計目的なども含む）で用いられるか
⑧当初収集された目的以外のために、個人情報が他の組織との間で共有されるか
⑨新しい共通識別子、又は法令上の根拠なくSIN（社会保障番号）を用いるか
⑩公衆が、プログラム又はサービスに関してプライバシーに対する懸念を有することが予測されるか
⑪個人情報の物理的又は論理的分離に影響を与えたり、個人情報の管理やアクセスコントロールのためのセキュリティ機能に影響を与える変更を、業務システムやインフラに対して行うか
# The Role of Productivity Growth Rates for Rising Inequality in an Economy with Heterogeneous Agents

Atsumasa Kondo

February 2016

Economic and Social Research Institute

Cabinet Office

Tokyo, Japan

The views expressed in “ESRI Discussion Papers” are those of the authors and not those of the Economic and Social Research Institute, the Cabinet Office, or the Government of Japan. (Contact us: https://form.cao.go.jp/esri/en_opinion-0002.html)
# The Role of Productivity Growth Rates for Rising Inequality in an Economy with Heterogeneous Agents $ ^{1} $

Atsumasa Kondo

Economic and Social Research Institute, Cabinet Office $ ^{2} $

$ ^{2} $ 1-6-1, Nagata-cho, Chiyoda-ku, Tokyo 100-8914 Japan
ESRI Discussion Paper Series No.326

The Role of Productivity Growth Rates for Rising Inequality

in an Economy with Heterogeneous Agents

## Abstract

This paper examines the dynamic properties of equilibrium paths in a simple general equilibrium model with heterogeneous agents. Heterogeneity is characterized by productivity growth rates in addition to the discount factors of agents for their future utilities. The consumption levels of impatient agents converge to 0 as time passes if their discount factors are sufficiently small and their productivity growth rates are not too high. The result can be interpreted as the emergence of rising inequality. The sufficient conditions can be reworded as follows: the discount factors of impatient agents are smaller than the “market’s long-term discount factor”, i.e., $ 1/(1+r) $ .

Keywords: inequality, heterogeneous agents, productivity growth rates, discount factors

JEL Classification Numbers: D24, D31
## 1 Introduction

The problem of rising inequality among countries, regions, and social strata has become more serious in recent times. In Japan, this topic has been actively discussed during a past few decades. Using Gini coefficient, Tachibanaki (1998) argued that the disparity between rich people and poor people was spreading, and the middle class was diminishing since 1980's. Ohtake (2005) offered a counterargument in which the rise of Gini coefficient can be primarily attributed to population ageing $ ^{1} $ , and that the data set presented by Tachibanaki does not necessarily imply the disappearance of the middle class. Ohtake also pointed out that inequality between non-regular workers and regular workers and inequality in consumption in young people is growing since late 90's. However, both Tachibanaki (1998) and Ohtake (2005) did not theoretically examine properties of time paths of economic variables from a viewpoint of a rigorous dynamic general equilibrium analysis.

The main purpose of this paper is to explain growing disparity of consumption among various types of agents. For that aim, it introduces heterogeneous agents into a standard dynamic general equilibrium model. One line to examine heterogeneous agents within a dynamic general equilibrium framework can be found in Becker (1980). Becker exploited a one-sector dynamic general equilibrium framework with many heterogeneous agents who discount future utilities at various rates, and proved that impatient agents will own almost no capital assets and consume very little in future periods. $ ^{2} $ These results can be interpreted as the emergence of inequality as a dynamic general equilibrium path. This paper examines the Ramsey–Becker result within a simple dynamic general equilibrium model with heterogeneous agents. Heterogeneity is characterized by productivity growth rates in addition to discount factors for future utilities and the initial conditions of real assets and production levels. Given the various types of inequalities observed in the real world, this is an important research topic to examine how the productivity growth rates of a class of people affect an economy’s dynamic allocation.

As with Becker (1980), in this study rising inequality is explained as a market outcome. More accurately, the consumption levels of impatient agents converge to 0 as time passes if their discount factors are sufficiently small and
their productivity growth rates are not high in comparison with the discount factors of agents. That is, if impatient agents are not too myopic or if their productivity growth rates are sufficiently high, the set of assumptions that creates economic inequality cannot be sustained.

That crucial assumption, which generates the main result of this paper, can be interpreted as follows. The discount factor of impatient agents is smaller than the “market’s long-term discount factor”, an inverse value of a gross real interest rate in a steady state, that is, $ 1/(1+r) $ . Thus, impatient agents are more myopic than the “market” from a long-term perspective. Because the “market’s long-term discount factor” $ 1/(1+r) $ correlates with the discount factor of patient agents and with the productivity growth rate of hard-working agents, the inter-relationships between those parameters are important to prove the main result of this paper.

Since their seminal works, the results of Ramsey (1928) and Becker (1980) have been extended in various directions. Becker and Foias (1987) illustrated the existence of a two-period equilibrium cycle, and proved the turnpike property of that model; that is, total capital will approach a steady state that is independent of the initial conditions. Becker and Tsyganov (2002) extended similar results to a two-sector setting. Sorger (2002) and Bosi and Seegmuller (2010a) introduced progressive taxation by governments into the Ramsey model with heterogeneous agents. They also extended the model in a following paper (2010b) by considering the endogenous labor supply of consumers. Becker (2012) demonstrated the “twisted turnpike property” of the model, which means that optimal paths starting from different initial capital stocks will run closely together in future periods. Nishimura et al. (2014, 2015) introduced public spending with externalities and investigated its (de)stabilizing effects on the equilibrium path. However, the above authors did not investigate the effects of numerous agents’ heterogeneous productivity growth rates on equilibrium allocations.

In addition to the research introduced above, many studies concerning the dynamic allocations of economies have considered the discount factors of economic agents for their future utilities as an important parameter. Scheinkman (1976) and McKenzie (1983) proved that an optimal path converges to a steady state if agents' future utilities are discounted sufficiently weakly. Nishimura and Yano (1994) and Nishimura, Sorger and Yano (1994) demonstrated that if agents' discount factors are sufficiently near 0, the optimal paths in dynamic models may behave in a complicated manner—even chaotically. Nishimura and Yano (1995) showed that even if agents' future
utilities are discounted arbitrarily weakly, parameter constellations exist that produce chaotic solutions as optimal paths. These authors, however, concentrated on cases with a representative agent and did not study the effects of heterogeneous economic agents.

Bewley (1982), Yano (1984, 1991, 1998) and Coles (1985) allowed for a finite and large number of heterogeneous agents, and proved that an equilibrium path will lie in a small neighborhood of a steady state if future utilities are sufficiently important to the agents. More recently, Ghiglino (2005) explored the effects of the heterogeneity of agents' preferences on the dynamic stability of a steady state. Furthermore, the effect of heterogeneity on the indeterminacy of equilibria was investigated by Ghiglino and Olszak-Duquenne (2005). Kondo (2008) extended the neighborhood turnpike theorem to a monetary model with heterogeneous agents. However, these papers did not study cases in which productivity growth rates are different across economic agents.

The remainder of this paper is organized as follows. The next section constructs a model economy on which the following analyses are based. The main result is presented in Section 3. Section 4 offers a numerical example, which focuses on parameter values that is compatible with assumptions made in this paper. Section 5 briefly concludes the paper.

## 2 Model

Think of an economy with discretely indexed time points $ t = 0, 1, 2, \cdots $ . The period between time points t - 1 and t is called period t. There are H types of agents, type $ 1, \cdots, H $ . They engage in the transaction of a consumption good and a real asset in each period. A real asset bears a real interest rate $ r_{t} $ from period t to $ t + 1 $ .

An h-type agent, where $ h = 1, \cdots, H $ , holds the real asset $ a_{ht-1} $ at the beginning of period t. ( $ a_{ht-1} < 0 $ represents real liabilities. $ ^{3} $ ) The agent receives a consumption good endowment $ y_{ht} $ and consumes $ c_{ht} $ in period t. Flow budget constraints for the agent are given by

$$ c_{h t}+a_{h t}\leq y_{h t}+(1+r_{t-1})a_{h t-1}, $$
for any $ t = 1, 2, \cdots $ . The no ponzi-game condition is

$$ \liminf_{T\to\infty}\left(\prod_{t=1}^{T-1}\frac{1}{1+r_{t}}\right)a_{hT}\geq0. $$

The agent obtains utility from the consumption. An h-type agent discounts his or her future utilities relative to the utility from the current consumption with a discount factor $ \rho_{h} \in (0,1) $ . The behavior of the h-type agent is summarized as a maximizing problem:

$$ \max_{\{c_{h t},a_{h t}\}_{t=1}^{\infty}}\sum_{t=1}^{\infty}\rho_{h}^{t-1}\frac{c_{h t}^{1-\sigma_{h}}-1}{1-\sigma_{h}} $$

s.t. Equations (1) and (2)

$$ given\left\{r_{t}\right\}_{t=0}^{\infty},a_{h0} $$

where $ 1/\sigma_{h} $ is the inter-temporal elasticity of the substitution of consumption with $ \sigma_{h} > 0 $ and $ \sigma_{h} \neq 1 $ . It is well-known that the instantaneous utility function in (3) potentially includes the logarithmic function as a special case with $ \sigma_{h} = 1 $ .

The H types of agents interact in the markets. The market clearing conditions are given by

$$ \sum_{h=1}^{H}c_{ht}=Y_{t}\ \mathrm{and}\ \sum_{h=1}^{H}a_{ht}=0 $$

for any $ t = 1, 2, \cdots $ , where $ Y_{t} \equiv \sum_{h=1}^{H} y_{ht} $ is the total products in period t, which can be interpreted as the GDP of that period. A time stream of price and allocation of resources in the economy is determined so that conditions (3) and (4) are simultaneously satisfied.

## 3 The Ramsey–Becker Result

This section presents the main results of this paper. After enumerating a set of assumptions and providing some preliminary considerations, the Ramsey–Becker result is established (Theorem 1): the consumption paths (and the consumption paths relative to GDP) of impatient agents converge to $ 0^{4} $ .
Next, it is demonstrated that the path of the real interest rate converges (Corollary 1). The limit value depends negatively on the discount factors of patient agents and positively on the productivity growth rates of hard-working agents whose productivity growth rates are higher than those of other agents.

### 3.1 Assumptions

To establish the main results of this paper, the following assumptions are made:

Assumption 1 $ 0 < \rho_{h} < \rho_{\alpha} < 1 $ for any $ h = 1, \cdots, H $ with $ h \neq \alpha $ .

Assumption 2 $ y_{ht+1}/y_{ht}=1+\theta_{h} $ for any $ t=1,2,\cdots $ and $ h=1,\cdots,H $ .

Assumption 3 $ \theta_{h} < \theta_{\beta} $ for any $ h = 1, \cdots, H $ with $ h \neq \beta $ , and $ 0 < \theta_{\beta} $ .

Assumption 1 means that $ \alpha $ -type agents are the most patient. Assumption 2 states that the productivity growth rate of each agent is constant over time. Assumption 3 means that the productivity growth rate of $ \beta $ -type agents is greater than that of other types of agents. In this sense, $ \beta $ -type agents are mostly hard-working. Furthermore, $ \theta_{\beta} $ is supposed to be positive. This study does not necessarily exclude the possibility $ \alpha = \beta $ ; that is, some types of agents $ h(=\alpha=\beta) $ may be mostly patient and hard-working.

Note that heterogeneity of the present model is characterized by the following five factors: discount factors for future utilities $ \rho_{h} $ , the inverses of the inter-temporal elasticities of the substitution of consumption $ \sigma_{h} $ , productivity growth rates $ \theta_{h} $ , the distribution of the initial real assets $ a_{h0} $ and the initial level of production $ y_{h1} $ . Given these parameters, other variables, for example, the real interest rate $ r_{t} $ , are determined in the equilibrium. Among these factors that characterize the heterogeneity of the present model, the focus is on discount factors and productivity growth rates.

The following condition are also postulated.

Assumption 4 $ \rho_{\alpha} < (1 + \theta_{\beta})^{\sigma_{\alpha}-1} $

Assumption 5 $ \rho_{h}(1+\theta_{\beta})^{\sigma_{\alpha}}<\rho_{\alpha}(<1) $ for any $ h=1,\cdots,H $ with $ h\neq\alpha $ .
Assumption 4 guarantees that the life-time income is well-defined for any $ h = 1, \cdots, H $ . For this point, see the Appendix.

What does Assumption 5 mean? As will be shown in Corollary 1, which is presented after Theorem 1, the real interest rate in a steady state r satisfies the relationship $ 1 + r = (1 + \theta_{\beta})^{\sigma_{\alpha}} / \rho_{\alpha} $ . Thus, Assumption 5 implies that $ \rho_{h} < 1 / (1 + r) $ . In other words, the discount factor of h-type agents ( $ h \neq \alpha $ ) is smaller than the “market’s long-term discount factor” $ 1 / (1 + r) $ . Thus, impatient agents are more myopic than the “market” from a long-term perspective.

Additionally, note that the condition $ 0 < \theta_{\beta} $ in Assumption 3 guarantees that the real interest rate in the steady state is positive because $ 1 + r = (1 + \theta_{\beta})^{\sigma_{\alpha}} / \rho_{\alpha} $ . Furthermore, Assumption 1 is automatically satisfied under Assumption 5. That is, Assumption 1 is redundant. However, because the heterogeneity of the discount factors is one of the most important factors in this paper, I prefer that it is explicitly stated.

### 3.2 Preliminaries

This subsection provides some preliminary considerations for the analyses conducted below.

First, as a first-order condition of the maximizing problem (3), the following relationship is obtained:

$$ \left(\frac{c_{h t+1}}{c_{h t}}\right)^{\sigma_{h}}=\rho_{h}\left(1+r_{t}\right) $$

for any $ t = 1, 2, \cdots $ .

Next, it can be verified that an h-type agent's production level $ y_{ht} $ to the total production $ Y_{t} $ ratio gradually decreases to 0 for any $ h = 1, \cdots, H $ with $ h \neq \beta $ while that of the $ \beta $ -type agent increases to 1. That is,

$$ \frac{y_{h t}}{Y_{t}}\downarrow0(for any h\neq\beta)\ and\ \frac{y_{\beta t}}{Y_{t}}\uparrow1\ as\ t\rightarrow\infty. $$

The expression $ y_{ht}/Y_{t} \downarrow 0 $ in the equation (6) means that $ y_{ht}/Y_{t} $ converges to 0 by satisfying $ y_{ht}/Y_{t} \geq 0 $ for any $ t = 1, 2, \cdots $ , and $ y_{\beta t}/Y_{t} \uparrow 1 $ can be understood in the same manner. Equation (6) can be proved. Take $ t = 1, 2, \cdots $ , and fix it momentarily. From Assumption 2,

$$ \frac{y_{ht}}{Y_{t}}=\frac{y_{ht}}{\sum_{j=1}^{H}y_{jt}}=\frac{\left(1+\theta_{h}\right)^{t-1}y_{h1}}{\sum_{j=1}^{H}\left(1+\theta_{j}\right)^{t-1}y_{j1}} $$
for any $ h = 1, \cdots, H $ . Dividing both the numerator and the denominator by $ y_{\beta t} $ yields

$$ \frac{y_{ht}}{Y_{t}}=\frac{\left(\frac{1+\theta_{h}}{1+\theta_\beta}\right)^{t-1}\frac{y_{h1}}{y_{\beta1}}}{1+\sum_{j\neq\beta}\left(\frac{1+\theta_{j}}{1+\theta_\beta}\right)^{t-1}\frac{y_{j1}}{y_{\beta1}}} $$

for any $ h = 1, \cdots, H $ . From Assumption 3, it holds that

$$ \frac{y_{h t}}{Y_{t}}\to0(for any h\neq\beta)\text{and}\frac{y_{\beta t}}{Y_{t}}\to1\text{as}t\to\infty. $$

Furthermore, the sequence $ \{y_{ht}/Y_{t}\} $ is monotone decreasing for any $ h=1,\cdots,H $ with $ h\neq\beta $ while $ \{y_{\beta t}/Y_{t}\} $ is monotone increasing. Thus, the desired results (6) hold.

Assumptions 2 and 3 jointly imply that the growth rate of the total production level $ \Theta_{t} \equiv Y_{t+1}/Y_{t} - 1 $ satisfies the following relationship:

$$ 1+\Theta_{t}=\frac{1}{Y_{t}}\left[\sum_{h=1}^{H}\left(1+\theta_{h}\right)y_{h t}\right]=\sum_{h=1}^{H}\left(1+\theta_{h}\right)\frac{y_{h t}}{Y_{t}}<1+\theta_{\beta} $$

for any $ t = 1, 2, \cdots $ . In other words, the gross growth rate of GDP, $ 1 + \Theta_{t} $ , can be expressed as a convex combination of the economic agents' gross productivity growth rates, $ 1 + \theta_{h} $ . The weight of the convex combination is the relative share of the production level of each agent to GDP, $ y_{ht}/Y_{t} $ . The right-hand inequality in (8) implies that the growth rate of GDP, $ \Theta_{t} $ , is less than the productivity growth rate of hard-working agents, $ \theta_{\beta} $ . However, according to the relationships (6), it holds that $ \Theta_{t} $ converges to $ \theta_{\beta} $ from below:

$$ \Theta_{t}\uparrow\theta_{\beta}\quad as\ t\rightarrow\infty. $$

As a result of (9), GDP $ Y_{t} $ diverges to $ \infty $ as time passes:

$$ Y_{t}\to\infty\ \mathrm{as}t\to\infty. $$

Indeed, from (9), for any $\varepsilon\in(0,\theta_{\beta})$, there exists a positive integer $t^{*}$ such that

$$ 1+\varepsilon<1+\Theta_{t}\left(=\frac{Y_{t+1}}{Y_{t}}\right)<1+\theta_{\beta} $$

for any $ t = t^{*}, t^{*} + 1, t^{*} + 2, \cdots $ . Thus, $ Y_{t} $ diverges to $ \infty $ ,
### 3.3 Main Results

The Ramsey–Becker result is demonstrated below.

Theorem 1 Under Assumptions 1-5, the following holds. In the equilibrium path, it holds that $ c_{ht} \to 0 $ as $ t \to \infty $ for any $ h = 1, \cdots, H $ with $ h \neq \alpha $ . Furthermore, it holds in the limit as $ t \to \infty $ that $ c_{ht}/Y_t \to 0 $ for any $ h = 1, \cdots, H $ with $ h \neq \alpha $ , and $ c_{\alpha t}/Y_t \to 1 $ .

Proof. At the outset, I will show the first part of the theorem. Take $ t = 1, 2, \cdots $ , and fix it momentarily. From equation (5), it follows that

$$ c_{h t}^{\sigma_{h}}=\rho_{h}^{t-1}\left(1+r_{1}\right)\cdot\cdot\cdot\left(1+r_{t-1}\right)c_{h1}^{\sigma_{h}} $$

for each $ h = 1, \cdots, H $ . Take $ h = 1, \cdots, H $ such that $ h \neq \alpha $ . From (11), it holds that

$$ \frac{c_{h t}^{\sigma_{h}}}{c_{\alpha t}^{\sigma_{\alpha}}}=\left(\frac{\rho_{h}}{\rho_{\alpha}}\right)^{t-1}\frac{c_{h1}^{\sigma_{h}}}{c_{\alpha1}^{\sigma_{\alpha}}}. $$

Using (12), (4) and (8),

$$ \begin{align*}0\quad&<\quad c_{ht}^{\sigma_{h}}=\left(\frac{\rho_{h}}{\rho_\alpha}\right)^{t-1}\frac{c_{h1}^{\sigma_{h}}}{c_{\alpha1}^{\sigma_\alpha}}c_{\alpha t}^{\sigma_\alpha}\\&<\quad\left(\frac{\rho_{h}}{\rho_\alpha}\right)^{t-1}\frac{c_{h1}^{\sigma_{h}}}{c_{\alpha1}^{\sigma_\alpha}}Y_{t}^{\sigma_\alpha}<\left(\frac{\rho_{h}}{\rho_\alpha}\right)^{t-1}\frac{c_{h1}^{\sigma_{h}}}{c_{\alpha1}^{\sigma_\alpha}}\left[\left(1+\theta_\beta\right)^{t-1}Y_{1}\right]^{\sigma_\alpha}\\&=\quad\left(\frac{\rho_{h}}{\rho_\alpha}\left(1+\theta_\beta\right)^{\sigma_\alpha}\right)^{t-1}\frac{c_{h1}^{\sigma_{h}}}{c_{\alpha1}^{\sigma_\alpha}}Y_{1}^{\sigma_\alpha}\end{align*} $$

From Assumption 5, the desired result $ c_{ht} \rightarrow 0 $ follows.

This result together with (10) implies that $ c_{ht}/Y_{t} \rightarrow 0 $ for any $ h = 1, \cdots, H $ with $ h \neq \alpha $ . Lastly, I will demonstrate that $ c_{\alpha t}/Y_{t} \rightarrow 1 $ . From (4), I obtain

$$ 1=\frac{Y_{t}}{Y_{t}}=\frac{c_{1t}}{Y_{t}}+\cdots\frac{c_{\alpha t}}{Y_{t}}+\cdots\frac{c_{Ht}}{Y_{t}}. $$

Since $ c_{ht}/Y_{t} \rightarrow 0 $ for any $ h = 1, \cdots, H $ with $ h \neq \alpha $ , the desired result follows.

Theorem 1 may be interpreted as the emergence of rising inequality in a market economy because it implies that $ \alpha $ -type agents will consume almost all
of the consumption goods in the future (i.e., $ |c_{\alpha t} - Y_t| \to 0 $ as $ t \to \infty $ ) while the other types of agents will consume almost nothing. To derive that result, Assumption 5 is crucial. The productivity growth rate of $ \beta $ -type agents is not necessarily high in relation to the discount factors of the agents. Thus, if $ \beta $ -type agents increase their productivity growth rates, these continuous efforts may break the inequality, given the discount factors of the economic agents. Assumption 5 may imply that the discount factors of the impatient agents $ \rho_h $ ( $ h \neq \alpha $ ) is not so high in relation to the discount factor of the patient agents $ \rho_{\alpha} $ and the productivity growth rate of the hard-working agents $ \theta_{\beta} $ . Thus, if the impatient agents place a higher value on their future utility, the inequality result does not necessarily hold. (To illustrate these points, the next section will present a numerical example.)

What can the public sector do to suppress the increasing inequality? Theorem 1 suggests the following points. The government should facilitate educational investment for impatient agents to increase their productivity growth rate. It may also be effective to hold a public seminar to educate impatient agents on their futures to cure their short-sightedness.

Using Theorem 1, the following corollary regarding the path of the real interest rate can be established.

Corollary 1 Under Assumptions 1-5, it holds in the equilibrium path that $ 1 + r_{t} \to (1 + \theta_{\beta})^{\sigma_{\alpha}} / \rho_{\alpha} $ as $ t \to \infty $ .

Proof. Take $ t = 1, 2, \cdots $ , and fix it momentarily. On the one hand, it holds according to (5) that

$$ \frac{\sum_{h=1}^{H}c_{ht+1}}{\sum_{h=1}^{H}c_{ht}}=\frac{\sum_{h=1}^{H}\rho_{h}^{1/\sigma_{h}}\left(1+r_{t}\right)^{1/\sigma_{h}}c_{ht}}{\sum_{h=1}^{H}c_{ht}}. $$

On the other hand, as a result of equation (4),

$$ \frac{\sum_{h=1}^{H}c_{ht+1}}{\sum_{h=1}^{H}c_{ht}}=\frac{Y_{t+1}}{Y_{t}}=1+\Theta_{t}. $$

From (13) and (14), it holds that

$$ 1+\Theta_{t}=\frac{\sum_{h=1}^{H}\rho_{h}^{1/\sigma_{h}}\left(1+r_{t}\right)^{1/\sigma_{h}}\left(c_{h t}/Y_{t}\right)}{\sum_{h=1}^{H}\left(c_{h t}/Y_{t}\right)}. $$