# Effects of wariness on economic growth in overlapping generations models

Hai Ha Pham, Ngoc-Sang Pham

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## To cite this version:

Hai Ha Pham, Ngoc-Sang Pham. Effects of wariness on economic growth in overlapping generations models. Journal of Mathematical Economics, 2024, 115, pp.103060. 10.1016/j.jmateco.2024.103060. hal-04918980

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# Effects of wariness on economic growth in overlapping generations models☆

<figure><img src="image_4.png" /></figure>

Hai Ha Pham<sup>a,b,1</sup>, Ngoc-Sang Pham<sup>c,*</sup>

## ARTICLE INFO

Manuscript handled by Editor A Toda

JEL classification:

D14

D5

E71

O41

Keywords:

Wariness

Overlapping generations

Dynamic efficiency

Economic growth

Endogenous growth

## ABSTRACT

We introduce the notion of wariness, defined as a concern for the lowest lifetime utility, in overlapping generations models and explore its effects on economic growth. In an exogenous growth model, under standard assumptions, we prove that the capital stock converges to a steady state. We then explore conditions under which this steady state is increasing (or decreasing) in the wariness level. We also provide a necessary and sufficient condition for the dynamic efficiency of the intertemporal equilibrium. In endogenous growth models (à la Romer (1986) or à la Barro (1990)), we show that the growth rate of capital stock per capita in the economy with wariness is lower (higher, respectively) than that in the economy without wariness if and only if the capital return is high (low, respectively).

## 1. Introduction

We live in a world where numerous factors, such as pandemics, disasters, and policy uncertainty, can cause people to worry. Facing these situations, some people prefer to live better now, and enjoy the present by consuming more.<fnref n="2" /> However, others may save more in order to face potential problems in the future. When making decisions, some people are concerned with their worst consumption over time – a situation we mean that there is a wariness. Motivated by these observations, we aim to investigate the role of this behavior in economic growth. To model the wariness, we consider that individuals take care not only of the discounted sum of utilities but also the minimum utility across time.<fnref n="3" /> Formally, when a consumer lives for two periods, we assume that her

intertemporal utility is given by

$$ \text{Intertemporal utility: } U(c, d) = (1 - \lambda)(u(c) + \beta u(d)) + \lambda \min\{u(c), u(d)\}, \quad (1) $$

where c, d represent the consumer's consumption in the present and the future respectively, β is the rate of time preference.<fnref n="4" /> The parameter λ ∈ [0, 1] represents the wariness degree of the household: the higher the value of γ, the more the household cares about her minimum utility across time. When λ = 0, we recover the standard case. When λ = 1, the agent only cares about the minimum of her consumptions over time.

To understand whether wariness harms or promotes economic development, we incorporate the modeling (1) of wariness in a standard overlapping generations model (OLG model, henceforth). We then

<footnote><sup>a</sup> Department of Mathematics, International University, Hochiminh city, Viet Nam</footnote>

<footnote><sup>b</sup> Vietnam National University, Hochiminh city, Viet Nam</footnote>

<footnote><sup>c</sup> EM Normandie Business School, Métis Lab., France</footnote>

<footnote>☆ The authors would like to thank the co-Editor Alexis Akira Toda, an Associate Editor and two Referees for their helpful comments and suggestions. The authors also thank Cuong Le Van, Raouf Boucekkine and participants at the conferences VEAM 2023 and PET 2024 for comments.</footnote>

<footnote>* Correspondence to: EM Normandie (campus Caen), 9 Rue Claude Bloch, 14000 Caen, France.</footnote>

<footnote>E-mail addresses: phha@hcmiu.edu.vn (H.H. Pham), npham@em-normandie.fr, pns.pham@gmail.com (N.-S. Pham).</footnote>

<footnote n="1">¹ Address: Linh Trung, Thu Duc, Hochiminh city, Viet Nam.</footnote>

<footnote n="2">² See Chabot (2022) and Kim (2023) for the luxury industry during the period 2021–2022.</footnote>

<footnote n="3">³ This is related to the maxmin expected utility (Gilboa and Schmeidler, 1989).</footnote>

<footnote n="4">⁴ Our modeling of wariness in (1) can be considered as a finite-horizon version of the utility function $\sum_{i \ge 0} \beta^i u(c_i) + a \inf_{t \ge 0} u(c_t)$ described in Section 3.1.1 in Araujo et al. (2011) or Example 1 in Araujo (2015) and Araujo et al. (2019b), where wariness can be interpreted as a type of ambiguity aversion on a set of discount factors. Recall that when working with preferences on $\ell_+^\infty$, they define wariness as “the willingness to neglect gains but not losses in distant sets of dates or in events with very small likelihood”.</footnote>

<footnote>We can rewrite (modeling) as follows: $U(c, d) = u(c) + (1 - \lambda)\beta u(d)$ if $c \le d$, and $U(c, d) = (1 - \lambda)u(c) + ((1 - \lambda)\beta + \lambda)\beta u(d)$ if $c > d$. By the way, our modeling can be also interpreted as a model with endogenous discounting.</footnote>
study the role of wariness on the dynamics of capital stocks and its interplay with economic development.

First, we study how the saving function of the household (with the capital return given) is influenced by wariness. Interestingly, we find that a household's savings in the presence of wariness are not necessarily lower than in the absence of wariness. It depends on the interplay between the capital return and the wariness level. The intuition behind this is the following. When the capital return is low, the household expects a lower income in old age and a higher income when young. Therefore, in the presence of wariness, the household cares more about her consumption when old, leading to reduced consumption and increased savings when young compared to a scenario without wariness. By contrast, when the capital return is high, the household saves less.

Second, we investigate the dynamics of intertemporal equilibrium to understand the effects of wariness. Under mild conditions, we demonstrate that the equilibrium capital path converges to a unique steady state. Additionally, we provide comparative statics. We define the stationary capital return $R^*$ the capital return of the economy with stationary consumption (i.e., the consumption when young equals the consumption when old and it does not depend on time). We show that when the wariness is high enough, the steady-state capital stock of the economy with wariness equals the capital stock of the economy with stationary consumption. The intuition behind this is that when households really care about the worst consumption over two periods of life, they should choose the same level of consumption when young and old.

The more intriguing case is when the wariness level is quite low. In this case, the steady-state capital stock increases (respectively, decreases) with the wariness level if the stationary capital return is low (respectively, high). Interestingly, the steady-state capital stock in the economy with wariness may be higher than the steady-state capital stock in the economy without wariness. It is higher if either (1) the wariness level and the stationary capital return are low or (2) the wariness level is high and the capital return of the economy without wariness is low. We complement our theoretical findings with several simulations.

Third, we analyze the effects of wariness in two endogenous growth models (à la Romer, 1986 and à la Barro, 1990). We point out that when the wariness level is quite low, the growth rate of capital is increasing (respectively, decreasing) in the wariness level if the productivity is low (respectively, high). However, when the wariness level is high, the growth rate of capital does not depend on the wariness. We also prove that the capital growth rate in the economy with wariness is higher than in the economy without wariness if and only if the capital return is low.

Our results in both exogenous and endogenous growth models lead to an important implication: Taking into account the worst situation does not necessarily harm economic growth because it may enhance savings and economic growth, particularly in cases of low capital returns.

## Related literature

Our paper is related to the literature on endogenous discounting. Araujo et al. (2011), Araujo (2015) and Araujo et al. (2019a,b), consider the utility function of the form $\sum_{t \ge 0} \beta^t u(c_t) + a \inf_{t \ge 0} u(c_t)$ in infinite-horizon general equilibrium models. However, they study on the effects of parameter $a$ on asset bubbles while we examine the growth issue.

Ha-Huy and Nguyen (2022) investigate the optimal capital path of an infinite-horizon model with Ramsey-Rawls criterion where the objective function is $\sum_{t \ge 0} \beta^t u(c_t) + a \inf_{t \ge 0} u(c_t)$. They interpret parameter $a$ as the importance of equity in the inter-temporal generational evaluation in the choice of the economic agent. They provide conditions under which the optimal capital stock is constant over time or it coincides with the solution of the Ramsey problem (i.e., when the parameter $a = 0$). Our paper differs from Ha-Huy and Nguyen (2022)

in several points. First, we employ a OLG model instead of a model with an infinitely-lived agent as in Ha-Huy and Nguyen (2022); by consequence, the economic meaning of the wariness in our paper is quite different from that of parameter $a$ in Ha-Huy and Nguyen (2022). Second, and more importantly, the capital path of equilibrium in our paper is never constant over time (excepted the case where the initial capital is at the steady state). Third, we also investigate the issue of endogenous growth while Ha-Huy and Nguyen (2022) do not.

Our paper is also related to the literature on the impact of uncertainty because uncertainty may generate a wariness. Several papers document a negative relationship between high uncertainty and firms' investment. Kumar et al. (2023) use a survey of firms in New Zealand and show that firms' perceived macroeconomic uncertainty leads them to reduce their investment. Bloom et al. (2024) use a survey measure of subjective uncertainty collected by the U.S. Census Bureau for approximately 25,000 manufacturing plants and find that there is a strong and robust negative relationship between investment and high subjective uncertainty of plant managers.

Fukuda (2008) uses a OLG model but focuses on the profit maximization of the firm under Knightian uncertainty. He considers that the productivity of firm is random and the producer faces “uncertainty” in its productivity. By analyzing the effects of this uncertainty on the dynamic properties of equilibrium, he finds that there may be multiple steady states and a poverty trap may arise. Bianchi et al. (2023) find sizable negative effects of stochastic changes in the volatility of demand-side (rate of time preference) and supply-side (TFP) shocks on consumption, investment, and output.

Different from these papers focusing on the behavior of firms, we study how the wariness affects the savings of households and the economic growth. Our point is that if uncertainties generate a wariness in the households' preferences, then, in some cases, it may be beneficial to the economy because households may save more, and, by the way, improve the investment and economic growth.

The remainder of the paper is organized as follows. In Section 2, we introduce wariness in an exogenous growth model and provide basic properties of households' saving. Section 3 presents the effects of wariness in the exogenous growth model while Section 4 discusses the role of wariness in endogenous growth models (à la Romer (1986) or à la Barro (1990)). Section 5 concludes. Formal proofs are gathered in the Appendix section.

## 2. An OLG model with wariness

We introduce wariness in a standard two-period OLG model.<fnref n="5" />

**Household and wariness.** At period $t$, $N_t$ individuals are born. We assume that the population growth rate is constant over time and denote $n \equiv N_{t+1}/N_t$. Each consumer-worker lives two periods. When young, she supplies one unit of labor, earns a labor income, consumes $c_t$ and saves $s_t$. When old, she receives the income from her saving and consumes $d_{t+1}$.

Assume that the utility of this agent represents by the following function

$$ (1 - \lambda)(u(c_t) + \beta u(d_{t+1})) + \lambda \min\{u(c_t), u(d_{t+1})\} $$

where $\lambda \in [0, 1]$ represents the wariness of this individual. When $\lambda = 0$, we recover the standard case. When $\lambda = 1$, the agent only cares about the minimum of her(his) consumption $\min(u(c_t), u(d_{t+1}))$.

When $\lambda < 1$, we denote $\gamma = \frac{\lambda}{1-\lambda}$. Then, $\gamma$ varies between 0 and $+\infty$. It also represents the wariness of the household. The maximization problem of household born at date $t$ is given by

$$ (P_{c,t}) : \max_{(c_t, d_{t+1}, s_t)} [U(c_t, d_t) \equiv u(c_t) + \beta u(d_{t+1}) + \gamma \min(u(c_t), u(d_{t+1}))] $$

<footnote n="5">⁵ See de la Croix and Michel (2002) for an introduction of OLG models.</footnote>
$$c_t + s_t \le w_t \qquad (2)$$

$$d_{t+1} \le R_{t+1}s_t \qquad (3)$$

$$c_t, d_{t+1}, s_t \ge 0, \qquad (4)$$

where $R_t$ represents the capital return between time $t$ and $t+1$.

We assume that:

**Assumption 1.** The function $u$ is twice continuously differentiable, strictly increasing, strictly concave. $u'(0) = \infty$ and $u'(\infty) = 0$.

**Production.** Technology is represented by a constant returns to scale production function $F(K, L)$ where $K_t$ and $L_t$ are the aggregate capital and the labor forces.

The firm's profit maximization problem is

$$(P_{f,t}) : \max_{K_t, L_t \ge 0} (F(K_t, L_t) - R_t K_t - w_t L_t) \qquad (5)$$

where $R_t$ and $w_t$ represent the return on capital and the wage rate.

Denote $k_t \equiv K_t/L_t$ denotes the capital intensity, $f(k_t) \equiv F(k_t, 1)$. We require standard assumptions.

**Assumption 2.** The function $f$ is twice continuously differentiable, strictly increasing, strictly concave.

**Assumption 3.** $f(0) = 0, f(\infty) = \infty, f'(0) = \infty$ and $f'(\infty) = 0$.

The profit maximization implies that

$$R_t = f'(k_t) \text{ and } w_t = \omega(k_t), \text{ where we denote } \omega(k) \equiv f(k) - k f'(k), \forall k. \qquad (6)$$

Under a Cobb-Douglas technology represented by $F(K, L) \equiv AK^\alpha L^{1-\alpha}$ where $A > 0$ represents the total factor productivity. In this case, we have $R_t = \alpha Ak_t^{\alpha-1}$ and $w_t = (1-\alpha) Ak_t^\alpha$.

## 2.1. Intertemporal equilibrium

**Definition 1.** An intertemporal equilibrium is a positive sequence $(R_t, w_t, c_t, d_{t+1}, s_t, K_{t+1}, L_t)_{t \ge 0}$ which satisfies the following conditions:

(1) given the sequence $(R_t, w_t)$, the allocation $(K_t, L_t)$ is a solution to the problem $(P_{f,t})$ and the allocation $((c_t, s_t, d_{t+1})$ is a solution to the problem $(P_t)$,

(2) market clearing conditions:

physical capital : $K_{t+1} = N_t s_t$

labor : $L_t = N_t$

consumption good : $s_t + c_t + d_t/n = f(k_t)$,

## 2.2. Properties of the individual saving function

**Definition 2.** Given $\beta > 0, w > 0, R > 0$, we define $s_\beta(w, R)$ the unique solution of the following equation $u'(w-s) = \beta Ru'(Rs)$.

**Lemma 1.** Let **Assumption 1** be satisfied. Let $w > 0, R > 0$ be given. If $\beta > \beta' > 0$, then $s_\beta(w, R) > s_{\beta'}(w, R)$.

**Proof.** Denote $H(s, \beta) = u'(w-s) - \beta Ru'(Rs)$. Then, we have $\frac{\partial H}{\partial s}(s, \beta) = -u''(w-s) - \beta R^2 u''(Rs) > 0$. We also observe that $\frac{\partial H}{\partial \beta}(s, \beta) < 0$. So, $s_\beta(w, R)$ is strictly increasing in $\beta$. $\square$

Denote $\beta_1 \equiv \beta + \gamma, \beta_2 \equiv \frac{\beta}{1+\gamma}$. The following result shows the optimal solution of households.

**Proposition 1.** Let **Assumption 1** be satisfied.

(1) For $\lambda \in [0, 1)$ (or, equivalently, $\gamma \in [0, \infty)$), the optimal saving of the household problem $(P_{c,t})$, denoted by $s_t = s(w_t, R_{t+1})$, is given by

$$s_t = s(w_t, R_{t+1}) = \begin{cases} s_{\beta_1}(w_t, R_{t+1}) & \text{if } R_{t+1} < \frac{1}{\gamma+\beta} \\ \frac{w_t}{1+R_{t+1}} & \text{if } \frac{1}{\gamma+\beta} \le R_{t+1} \le \frac{1+\gamma}{\beta} \\ s_{\beta_2}(w_t, R_{t+1}) & \text{if } R_{t+1} > \frac{1+\gamma}{\beta} \end{cases} \qquad (7)$$

(2) When households only care about the lower level over the life-cycle (i.e. when $\gamma = +\infty$, or equivalently, $\lambda = 1$), we have $s_t = \frac{w_t}{1+R_{t+1}}$.

**Proof.** See Appendix. $\square$

When there is no wariness, the optimal saving is $s_t = s_\beta(w_t, R_{t+1})$. Since $\beta_1 \ge \beta \ge \beta_2$, **Lemma 1** implies that $s_{\beta_1}(w_t, R_{t+1}) \ge s_\beta(w_t, R_{t+1}) \ge s_{\beta_2}(w_t, R_{t+1})$. It means that the saving of the household under the presence of wariness can be higher or lower than that in the case without wariness. It depends on the relationship between the interest rate $R_t$ and the wariness level $\gamma$.

Let us explain the intuition in (7). The first regime is when $R_{t+1} < \frac{1}{\gamma+\beta}$. This condition can be rewritten as $R_{t+1} < \frac{1}{\beta}$ (the capital return is low) and $\gamma < \frac{1}{\gamma+\beta} - \beta$ (the wariness is low). In this case, the saving is $s_{\beta_1}(w_t, R_{t+1})$ which is higher than the saving in the case without wariness $s_\beta(w_t, R_{t+1})$. Indeed, when the capital return is low, the household's expected income when old $d_{t+1}$ would be low while her income when young $c_t$ would be high. So, in the presence of low wariness, the household cares more about her consumption when old. This implies that the household consumes less and saves more when young.

A similar interpretation applies for the third regime, i.e., when $R_{t+1} > \frac{1+\gamma}{\beta}$ (the capital return is high $\beta R_{t+1} - 1 > 0$ and the wariness is low $\gamma < \beta R_{t+1} - 1$).

The intermediate regime (i.e., when $\frac{1}{\gamma+\beta} \le R_{t+1} \le \frac{1+\gamma}{\beta}$ which is equivalent to $\gamma \ge \max(\beta R_{t+1} - 1, 1 - \beta R_{t+1})$) can be interpreted as the high wariness. In this case, the consumptions when young and old are the same, and the saving equals $\frac{w_t}{1+R_{t+1}}$.

Observe that if the degree of wariness increases, the intermediate regime enlarges and the difference between the two saving functions $s_{\beta_1}$ and $s_{\beta_2}$ raises.

By using **Proposition 1**, we get the following result exploring the effects of the wariness on the optimal saving.

**Corollary 1.** Let **Assumption 1** be satisfied. Given $\omega_t > 0$ and $R_{t+1} > 0$. We denote $s_t(\gamma)$ the optimal saving of the household with the wariness level $\gamma$. Let $\gamma_1 < \gamma_2$. We have different situations.

1. If $R_{t+1} < \frac{1}{\gamma_1+\beta}$, then $s_t(\gamma_1) < s_t(\gamma_2)$.

2. If $\frac{1}{\gamma_1+\beta} \le R_{t+1} \le \frac{1+\gamma_1}{\beta}$, then $s_t(\gamma_1) = s_t(\gamma_2) = \frac{w_t}{1+R_{t+1}}$.

3. If $\frac{1+\gamma_1}{\beta} < R_{t+1}$, then $s_t(\gamma_1) > s_t(\gamma_2)$.

**Proof.** See Appendix. $\square$

By comparing $\beta R_{t+1}$ with 1, we obtain the following result showing the monotonicity of the saving function with respect to the wariness level $\gamma$.

**Corollary 2.** Let **Assumption 1** be satisfied.

(1) If $\beta R_{t+1} < 1$ then the optimal saving is increasing in $\gamma$ for $\gamma > 0$.

(2) If $\beta R_{t+1} > 1$ then the optimal saving is decreasing in $\gamma$ for $\gamma > 0$.

**Proof.** See Appendix. $\square$

In some particular cases, we can explicitly compute the household's saving.
**Corollary 3.**

(1) If $u(c) = \ln(c)$, then $s_β(w_t, R_{t+1}) = \frac{\beta w_t}{1+\beta}$, $\forall \beta > 0$. In this case, we obtain the optimal saving

$$s_t = \begin{cases} \frac{\beta_1 w_t}{1+\beta_1} & \text{if } R_{t+1} < \frac{1}{\gamma+\beta} \\ \frac{w_t}{1+R_{t+1}} & \text{if } \frac{1}{\gamma+\beta} \le R_{t+1} \le \frac{1+\gamma}{\beta} \\ \frac{\beta_2 w_t}{1+\beta_2} & \text{if } R_{t+1} > \frac{1+\gamma}{\beta}. \end{cases}$$

(2) If $u(c) = \frac{c^{1-\sigma}}{1-\sigma}$, where $\sigma > 0$, then, by solving the equations $(w_t - s)^{-\sigma} = \beta R_{t+1} (R_{t+1}s)^{-\sigma}$, we obtain that $s_β(w_t, R_{t+1}) = \frac{w_t}{1+R_{t+1} - \frac{c}{\beta}(1+\gamma)}$. $\forall \beta$. By consequence, the optimal saving is determined

as follows:

$$s_t = \begin{cases} \frac{w_t}{1+R_{t+1} - \frac{c}{\beta}(1+\gamma)\beta_1^\sigma} & \text{if } R_{t+1} < \frac{1}{\gamma+\beta} \\ \frac{w_t}{1+R_{t+1}} & \text{if } \frac{1}{\gamma+\beta} \le R_{t+1} \le \frac{1+\gamma}{\beta} \\ \frac{w_t}{1+R_{t+1} - \frac{c}{\beta}(1+\gamma)\beta_2^\sigma} & \text{if } R_{t+1} > \frac{1+\gamma}{\beta}. \end{cases}$$

**3. Wariness and exogenous growth**

**3.1. Intertemporal equilibrium: convergence and steady state**

We now investigate the intertemporal equilibrium. The market clearing condition requires that $nk_{t+1} = s_t$ which becomes

$$nk_{t+1} = s(\omega(k_t), R_{t+1}) = \begin{cases} s_{\beta_1}(\omega(k_t), R_{t+1}) & \text{if } R_{t+1} < \frac{1}{\gamma+\beta} \\ \frac{\omega(k_t)}{1+R_{t+1}} & \text{if } \frac{1}{\gamma+\beta} \le R_{t+1} \le \frac{1+\gamma}{\beta} \\ s_{\beta_2}(\omega(k_t), R_{t+1}) & \text{if } R_{t+1} > \frac{1+\gamma}{\beta}. \end{cases} \quad (8)$$

where $k_0$ is exogenously given.

The following results shows the convergence of the equilibrium capital path ($k_t$) and the computing of the steady state.

**Proposition 2.** Let Assumptions 1, 2, 3 be satisfied.

(1) Assume that $\lambda \in [0, 1)$. Assume that $f'(k)u'(nkf'(k))$ is decreasing in $k$ for all $k > 0$. Then, in equilibrium, $k_{t+1}$ is an increasing function, denoted by $G(k_t)$, of $k_t$. By consequence, the capital path ($k_t$) monotonically converges.

Assume, in addition, that the function $\frac{\omega(k)}{k(1+f'(k))}$ is strictly decreasing in $k$. Then, the steady state $k^*$ is determined as follows.

(a) If $f'(k_3^*) < \frac{1}{\gamma+\beta}$, then $k^*$ is a solution to the equation $nk = s_{\beta_1}(\omega(k), f'(k))$.

(b) If $f'(k_3^*) > \frac{1+\gamma}{\beta}$, then $k^*$ is a solution to the equation $nk = s_{\beta_2}(\omega(k), f'(k))$.

(c) If $\frac{1+\gamma}{\beta} \ge f'(k_3^*) \ge \frac{1}{\gamma+\beta}$, then $k^* = k_3^*$, where $k_3^*$ is the unique solution to the equation $n = \frac{\omega(k)}{k(1+f'(k))}$.<fnref n="6" />

(2) Assume that $\lambda = 1$. In this case, we have $nk_{t+1}(1 + f'(k_{t+1})) = f(k_t) - k_t f'(k_t)$, $\forall t$. Therefore, $k_t$ converges. If the function $\frac{\omega(k)}{k(1+f'(k))}$ is strictly decreasing in $k$, then $k_t$ monotonically converges to the unique steady state $k_3^*$.

**Proof.** See Appendix. ☐

**Proposition 2** provides a general condition for the global convergence of the equilibrium capital stock. The intuition of our proof is to show that $k_{t+1}$ can be expressed as an increasing function $G$ of $k_t$ in each case in (8). The function $G$ is continuous but not differentiable at two points because there are three cases inf (8). Moreover, **Assumption 2** on the boundary allows us to prove that $G(k)$ runs from 0 to $\infty$ when $k$ runs from 0 to $\infty$. However, **Proposition 2** does not necessarily show the uniqueness of the steady state, which can be ensured under additional assumptions as we will show in Section 3.2.

It should be noticed that Assumptions in **Proposition 2** are satisfied in standard specifications. Indeed, if the function $cu'(c)$ is increasing in $c$ or $kf'(k)$ is increasing in $k$, then $f'(k)u'(nkf'(k))$ is decreasing in $k$.<fnref n="7" /> Moreover, if $f(k) = Ak^\alpha$, then $\frac{\omega(k)}{k(1+f'(k))} = \frac{(1-\alpha)Ak^{\alpha-1}}{1+\alpha Ak^{\alpha-1}}$, which is strictly decreasing in $k$ because $\alpha < 1$.

**3.2. Effects of wariness on the equilibrium capital path**

In the following, we study the effect of wariness on the equilibrium capital path and the steady state.

**3.2.1. General case**

**Definition 3.** Denote $k^s$ the steady state in the standard economy without wariness, i.e., $k^s$ is determined by

$$nk^s = s(\omega(k^s), f'(k^s)).$$

Following de la Croix and Michel (2002) (Proposition 1.11), we introduce an additional assumption.

**Assumption 4.** For each $\beta > 0$, the function $\frac{s_β(\omega(k), f'(k))}{k}$ is strictly decreasing in $k$ and

$$\lim_{k \to \infty} \frac{s_β(\omega(k), f'(k))}{k} < n < \lim_{k \to 0} \frac{s_β(\omega(k), f'(k))}{k} \quad (9)$$

Since $s \le \omega(k) \le f(k)$ and $\lim_{k \to \infty} \frac{f(k)}{k} < n$, the first inequality in (9) always holds. It should also be noticed that **Assumption 4** holds under standard specifications. Indeed, when $f(k) = Ak^\alpha$, $\alpha \in (0, 1)$ and $u(c) = \ln(c)$, we have $\frac{s_β(\omega(k), f'(k))}{k} = \frac{\beta(1-\alpha)Ak^{\alpha-1}}{1+\beta}$, which decreases from $\infty$ to 0 when $k$ increases from 0 to $\infty$. When $f(k) = Ak^\alpha$, $\alpha \in (0, 1)$ and $u(c) = \frac{c^{1-\sigma}}{1-\sigma}$ with $\sigma > 0$, we have $\frac{s_β(\omega(k), f'(k))}{k} = \frac{(1-\alpha)Ak^{\alpha-1}}{1+(\alpha Ak^{\alpha-1})^{-\frac{1}{\sigma}}+\frac{1}{\sigma}}$, which decreases from $\infty$ to 0 when $k$ increases from 0 to $\infty$.

Under **Assumption 4**, we have the following property.

**Lemma 2.** Let $\beta > 0$ be given. Under **Assumption 4**, then there exists a unique $k$ satisfying $n = \frac{s_β(\omega(k), f'(k))}{k}$. Moreover, this value $k$ is increasing in $\beta$.

We are now ready to state the main result of this section, which provides a full comparative statics showing the effect of the wariness on the capital steady state.

**Proposition 3.** Let assumptions in part 1 of **Proposition 2** be satisfied.

1. If $f'(k_3^*) < \frac{1}{\gamma+\beta}$ and **Assumption 4** holds, then the capital path converges to $k^* = k_1^*$ which is the unique solution to the equation $nk = s_{\beta_1}(\omega(k), f'(k))$. Moreover, $k_1^*$ is strictly increasing in the wariness level $\gamma$ and $k_1^* > k^s$.

<footnote n="6">⁶ It should be noticed that $k_3^*$ does not depend on $\gamma$.</footnote>

<footnote n="7">⁷ Observe that $\frac{\partial(f'(k)u'(nkf'(k))}{\partial k} = f''(k)u'(nkf'(k)) + f'(k)u''(nkf'(k))n(f'(k) + kf''(k))$. This is negative if $f'(k) + kf''(k) > 0$. Moreover, if $u'(c) + cu''(c) > 0, \forall c$, then we have</footnote>

<footnote>$$\frac{\partial(f'(k)u'(nkf'(k))}{\partial k} < u''(nkf'(k))nf'(k)(-kf''(k) + f'(k) + kf''(k)) < 0.$$</footnote>
2. If $f'(k_3^*) > \frac{1+\gamma}{\beta}$ and Assumption 4 holds, then the capital path converges to $k^* = k_2^*$ which is the unique solution to the equation $nk = s\beta_2(\omega(k), f'(k))$. Moreover, $k_2^*$ is strictly decreasing in the wariness level $\gamma$ and $k_2^* < k^s$.

3. If $\frac{1+\gamma}{\beta} \ge f'(k_3^*) \ge \frac{1}{\gamma+\beta}$, then the capital path converges to $k^* = k_3^*$, where $k_3^*$ is the unique solution to the equation $n = \frac{\omega(k)}{k(1+f'(k))}$.

Moreover, $k_3^* > k^s$ if and only if the interest rate of the economy without wariness is low (in the sense that $\beta f'(k^s) < 1$).

**Proof.** See Appendix. $\square$

First, condition $f'(k_3^*) < \frac{1}{\gamma+\beta}$ can be rewritten as $\beta f'(k_3^*) < 1$ and $\gamma < \frac{1}{f'(k_3^*)} - \beta$, which can be interpreted as low capital return and low wariness respectively. Second, condition $f'(k_3^*) > \frac{1+\gamma}{\beta}$ is equivalent to $0 < \gamma < \beta f'(k_3^*) - 1$, which can be interpreted as high capital return and low wariness. Note that $k_3^*$ is the capital return of the economy with stationary consumption (i.e., the consumption when young equals the consumption when old and it does not depend on time).

Therefore, Proposition 3 indicates that under low wariness and low capital return (respectively, high capital return), the steady state capital stock is increasing (respectively, decreasing) in the wariness level.

However, if the wariness level is high enough (in the sense that $\frac{1+\gamma}{\beta} \ge f'(k_3^*) \ge \frac{1}{\gamma+\beta}$, or equivalently, $\gamma \ge \max(\beta f'(k_3^*) - 1, \frac{1-\beta f'(k_3^*)}{\beta f'(k_3^*)})$), then the steady state capital stock equals $k_3^*$ which does not depend on the wariness level. The intuition behind this is that when households really care about the worst consumption over two periods of life, they should choose the same level of consumption when young and old.

Notice that $k_3^*$ is lower than the steady state $k^s$ of the economy without wariness if and only if the interest rate of the economy without wariness is high (in the sense that $\beta f'(k^s) > 1$).

**Proposition 3** leads to a direct consequence which compares the steady states of the economies with and without wariness.

### Corollary 4.

1. $k^* \ge k^s$ if and only if one of the two following conditions hold:

(1) $f'(k_3^*) < \frac{1}{\gamma+\beta}$; (2) $\frac{1+\gamma}{\beta} \ge f'(k_3^*) \ge \frac{1}{\gamma+\beta}$ and $\beta f'(k^s) \le 1$.

2. $k^* < k^s$ if and only if one of the two following conditions hold: (1)

$f'(k_3^*) > \frac{1+\gamma}{\beta}$;

(2) $\frac{1+\gamma}{\beta} \ge f'(k_3^*) \ge \frac{1}{\gamma+\beta}$ and $\beta f'(k^s) > 1$.

Condition $\beta f'(k^s) > 1$ (respectively, $\beta f'(k^s) < 1$) can be named as “high (respectively, low) real interest rate condition”. Indeed, we can write $\beta = \frac{1}{1+\rho}$ where $\rho$ is the subjective interest rate, and $f'(k^s) = 1+r_s$ where $r_s$ represents the real capital return at the steady state of the economy without wariness. By consequence, $\beta f'(k^s) > 1$ becomes $\frac{1+r_s}{1+\rho} > 1$, or equivalently, $r_s > \rho$.

### 3.2.2. Tractable cases and simulations

In this section, we consider some tractable cases where we can explicitly find the equilibrium capital path and compute the steady state.

Denote $\sigma_1 \equiv \frac{1}{1+\beta+\gamma}$, $\sigma_2 \equiv \frac{1+\gamma}{1+\beta+\gamma}$. Observe that $\sigma_1$ is decreasing but $\sigma_2$ is increasing in $\gamma$.

The following result is a consequence of Proposition 2.

**Corollary 5.** Assume that $f(k) = Ak^\alpha$ and $u(c) = \ln c$. At equilibrium, the dynamics of capital is characterized as follows:

$$nk_{t+1} = \begin{cases} (1-\sigma_2)(1-\alpha)Ak_t^\alpha & \text{if } k_t < a_2 \\ (1-\sigma_1)(1-\alpha)Ak_t^\alpha & \text{if } k_t > a_1 \end{cases} \quad (10)$$

$$nk_{t+1} + n\alpha Ak_{t+1}^\alpha = (1-\alpha)Ak_t^\alpha \quad \text{if } k_t \in [a_2, a_1] \quad (11)$$

<figure><img src="image_5.png" /><figcaption>**Fig. 1.** Graphs of $G(k)$ with and without wariness.</figcaption></figure>

$$\text{where } a_i \equiv \left( \frac{n \left( \frac{\alpha A (1-\sigma_i)}{\sigma_i} \right)^{\frac{1}{1-\alpha}}}{(1-\sigma_i)(1-\alpha)A} \right)^{\frac{1}{\alpha}} \quad \text{for } i = 1, 2. \quad 8$$

The unique steady state is determined by

$$k^* = \begin{cases} k_1^*(\gamma) \equiv \left( \frac{(1-\sigma_1)(1-\alpha)A}{n} \right)^{\frac{1}{1-\alpha}} & \text{if } f'(k_3^*) < \frac{1}{\gamma+\beta}, \quad \text{i.e., } \frac{1-\alpha}{\alpha n} \ge 1+\beta+\gamma \\ k_3^* \equiv \left( \frac{(1-\alpha-n\alpha)A}{n} \right)^{\frac{1}{1-\alpha}} & \text{if } \frac{1+\gamma}{\beta} \ge f'(k_3^*) \ge \frac{1}{\gamma+\beta}, \quad \text{i.e., } \\ & 1+\frac{\beta}{1+\gamma} < \frac{1-\alpha}{\alpha n} < 1+\beta+\gamma \\ k_2^*(\gamma) \equiv \left( \frac{(1-\sigma_2)(1-\alpha)A}{n} \right)^{\frac{1}{1-\alpha}} & \text{if } f'(k_3^*) > \frac{1+\gamma}{\beta}, \quad \text{i.e., } \frac{1-\alpha}{\alpha n} \le 1+\frac{\beta}{1+\gamma}. \end{cases} \quad (12)$$

**Proof.** See Appendix. $\square$

When there is no wariness (i.e., $\gamma = 0$), we have $\sigma_1 = \sigma_2 = \sigma = \frac{1}{1+\beta}$ and hence we recover the standard system $nk_{t+1} = \frac{\beta}{1+\beta}(1-\alpha)Ak_t^\alpha$ and the sequence $(k_t)$ converges to $k^s \equiv (\frac{\beta(1-\alpha)A}{n(1+\beta)})^{\frac{1}{1-\alpha}}$.

The relationship $k_{t+1} = G(k_t)$ in Corollary 5 and Proposition 2 is illustrated in Fig. 1. Here, we draw the graphics with $A = 4$, $\alpha = 0.3$, $n = 1.1 \approx 1.003^{35}$, $\beta = 0.7 \approx 0.99^{35.9}$. We consider two cases: (i) $\gamma = 0$ (without wariness) and (ii) $\gamma = 0.5$. We can see that in the case with $\gamma = 0.5$, the function $G$ is continuous but not differentiable at two points $a_1 = 1.9$ and $a_2 = 0.45$. This is consistent with (10) and (11).

Thanks to Corollary 5 and Proposition 3, we can provide a full comparative statics concerning the effects of $\gamma$.

**Proposition 4 (Comparative Statics).** Assume that $f(k) = Ak^\alpha$ and $u(c) = \ln c$.

1. When $f'(k_3^*) < \frac{1}{\gamma+\beta}$ (i.e., $\gamma \le \frac{1-\alpha}{\alpha n} - (1+\beta)$, then the equilibrium capital path $(k_t)$ converges to $k_1^*(\gamma) = (\frac{(1-\sigma_1)(1-\alpha)A}{n})^{\frac{1}{1-\alpha}}$ which is higher than $k^s$.

2. When $f'(k_3^*) > \frac{1+\gamma}{\beta}$, i.e., $\frac{1-\alpha}{\alpha n} \le 1+\frac{\beta}{1+\gamma}$, the equilibrium capital path $(k_t)$ converges to $k_2^*(\gamma) = (\frac{(1-\sigma_2)(1-\alpha)A}{n})^{\frac{1}{1-\alpha}}$ which is lower than $k^s$.

<footnote n="8">8 $a_i$ is decreasing in $\sigma_i$, for $i = 1, 2$. Therefore, $a_1$ is increasing but $a_2$ is decreasing in $\gamma$.</footnote>

<footnote n="9">9 Here, we consider that the length of each period in our OLG model is 35 years, the population growth rate is 0.3% and the annual discount factor is 0.99.</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 2. The equilibrium capital path with different wariness levels $\gamma$. Parameters are $\beta = 0.7$, $\alpha = 0.3$, $A = 4$, $n = 1.1$.</figcaption></figure>

3. $\frac{1+\gamma}{\beta} \ge f'(k_3^*) \ge \frac{1}{\gamma+\beta}$, (i.e., $1 + \frac{\beta}{1+\gamma} < \frac{1-\alpha}{an} < 1+\beta+\gamma$), the equilibrium capital path ($k_t$) converges to $k_3^* = \left(\frac{(1-\alpha-\alpha A)}{n}\right)^{\frac{1}{1-\alpha}}$. In this case, we observe that: $k_3^* \ge k^s \Leftrightarrow \frac{an}{1-\alpha} \le \frac{1}{1+\beta} \Leftrightarrow \beta f'(k^s) \le 1$.

Combining these three cases, we have that: $k^*(\gamma) \ge k^s$ if and only if $1+\beta \le \frac{1-\alpha}{an}$, or, equivalently, $\beta f'(k^s) \le 1$.

**Proof.** $k_2^*(\gamma)$ is decreasing in $\gamma$ because it is decreasing in $\sigma_2$ and $\sigma_2$ is increasing in $\gamma$. $k_1^*(\gamma)$ is increasing in $\gamma$ because it is decreasing in $\sigma_1$ and $\sigma_1$ is decreasing in $\gamma$.

At the steady state, $k^*(\gamma) \ge k^s$ if and only if one of the two following conditions hold: (1) $\gamma \le \frac{1-\alpha}{an} - (1+\beta)$; (2) $1 + \frac{\beta}{1+\gamma} < \frac{1-\alpha}{an} < 1+\beta+\gamma$ and $\frac{1-\alpha}{an} \ge 1+\beta$. Condition (2) is equivalent to $1+\beta \le \frac{1-\alpha}{an} < 1+\beta+\gamma$. By consequence, we obtain that: $k^*(\gamma) \ge k^s$ if and only if $1+\beta \le \frac{1-\alpha}{an}$, or equivalently $\beta f'(k^s) \le 1$. $\square$

Under the specifications in Proposition 4, we can explicitly compute the unique steady state. We complement our theoretical results by numerical simulations. In our simulation, we assume logarithmic utility and Cobb-Douglas production functions.

Fig. 2 displays the capital paths for various wariness levels. Here, we set that $A = 4$, $\alpha = 0.3$, $n = 1.1$, $\beta = 0.7$ as in Fig. 1. In Fig. 2(a), starting from the same initial capital level $k_0 = 0.1$, the capital path of the economy without wariness increasingly converges to a steady state $k^s = 1.07$. With the wariness degree $\gamma = 0.1$ ($\gamma = 0.3$, respectively), the capital path converges to the steady state $k_1^*(\gamma = 0.1) = 1.19(k_1^*(\gamma = 0.3) = 1.41$, respectively); this is consistent with point 1 of Proposition 4. Then, in Fig. 2(b), we set two high levels of wariness $\gamma = 0.5$ and $0.7$. We can see that $\lim_{t \to \infty} k_t = k_3^* = 1.53 > k^s = 1.07$; this is consistent with point 3 of Proposition 4. In both graphics, $\lim_{t \to \infty} k_t > k^s$ because $1+\beta = 1.7 < 2.12 = \frac{1-\alpha}{an}$.

In Fig. 2, it should be noticed that at the period 1, the capital stock of the economy with wariness is below the capital stock of the economy without wariness because we choose $k_0$ low enough so that $k_0 < a_2$ which ensures that $k_1 = \frac{(1-\sigma_2)(1-\alpha)}{n} Ak_0^\alpha = \frac{1-\alpha}{(1+\beta+\gamma)n} Ak_0^\alpha$ (see (10)). Observe that $\frac{1-\alpha}{(1+\beta+\gamma)n} Ak_0^\alpha$ is lower than that $\frac{1-\alpha}{(1+\beta)n} Ak_0^\alpha$ which is the capital at date 1 of the economy without wariness.

We now look at Fig. 3 where $\lim_{t \to \infty} k_t < k^s$ for different values of the wariness level $\gamma$. We again take that $\beta = 0.7$, $\alpha = 0.3$, $A = 4$, but we choose $n = 1.8$ to ensure that $1+\beta = 1.7 < 1.23 = \frac{1-\alpha}{an}$. We take the initial capital $k_0 = 2.5$. In all cases, we see that $k_t$ decreasingly converges. First, in the economy without wariness ($\gamma = 0$), the capital path converges to $k^s = 0.53$. Second, in Fig. 3(a), with the wariness degree $\gamma = 0.1$ ($\gamma = 0.3$, respectively), the capital path converges to

the steady state $k_2^*(\gamma = 0.1) = 0.49(k_2^*(\gamma = 0.3) = 0.42$, respectively); this is consistent with point 2 of Proposition 4. Then, in Fig. 3(b), with two high levels of wariness ($\gamma = 2$ and $\gamma = 3$), we see that $\lim_{t \to \infty} k_t = k_3^* = 0.23 < k^s = 0.53$; this is consistent with point 3 of Proposition 4.

When we work with a CRRA utility function, we have the following result which is a direct consequence of Propositions 2 and 3.

**Corollary 6.** If $f(k) = Ak^\alpha$ with $0 < \alpha < 1$, and $u(c) = \frac{c^{1-\sigma}}{1-\sigma}$ with $0 < \sigma \ne 1$, then at equilibrium, the dynamics of capital is characterized as follows

$$nk_{t+1} = s(\omega_t, R_{t+1}) = \begin{cases} \frac{(1-\alpha)Ak_t^\alpha}{1+(\alpha Ak_{t+1}^{\alpha-1})^{-\frac{1}{\alpha}}\beta_t^{-\frac{1}{\alpha}}} & \text{if } k_t > a_1 \\ \frac{(1-\alpha)Ak_t^\alpha}{1+\alpha Ak_{t+1}^{\alpha-1}} & \text{if } a_2 \le k_t \le a_1 \\ \frac{(1-\alpha)Ak_t^\alpha}{1+(\alpha Ak_{t+1}^{\alpha-1})^{-\frac{1}{\alpha}}\beta_2^{-\frac{1}{\alpha}}} & \text{if } k_t < a_2. \end{cases} \quad (13)$$

$$\text{where } a_i \equiv \left( \frac{n \left( \frac{\alpha A (1-\sigma_i)}{\sigma_i} \right)^{\frac{1}{1-\alpha}}}{(1-\sigma_i)(1-\alpha)A} \right)^{\frac{1}{\alpha}} \text{ for } i=1,2.$$

The unique steady state is determined by

$$k^* = \begin{cases} k_1^* & \text{if } \frac{1-\alpha}{an} > 1 + \beta + \gamma \\ k_3^* \equiv \left( \frac{(1-\alpha-\alpha A)}{n} \right)^{\frac{1}{1-\alpha}} & \text{if } 1 + \frac{\beta}{1+\gamma} \le \frac{1-\alpha}{an} \le 1 + \beta + \gamma \\ k_2^* & \text{if } \frac{1-\alpha}{an} \le 1 + \frac{\beta}{1+\gamma}. \end{cases} \quad (14)$$

$$\text{where } k_i^* \text{ is the positive solution of the equation } nk_i^* = \frac{(1-\alpha)A(k_i^*)^\alpha}{1+(\alpha A(k_i^*)^{\alpha-1})^{-\frac{1}{\alpha}} + \beta_i^{-\frac{1}{\alpha}}}, \quad \text{for } i=1,2.$$

1. When $\sigma_1 \equiv \frac{1}{1+\beta+\gamma} > \frac{an}{1-\alpha}$, i.e., $\gamma$ is low: $\gamma < \frac{1-\alpha}{an} - (1+\beta)$, then the capital path ($k_t$) converges to $k_1^*$ which is higher than $k^s$.

2. When $\sigma_2 \le \frac{an}{1-\alpha}$, i.e., $\frac{1-\alpha}{an} \le 1 + \frac{\beta}{1+\gamma}$, the capital path ($k_t$) converges to $k_2^*$ which is lower than $k^s$.

3. When $\sigma_1 < \frac{an}{1-\alpha} < \sigma_2$, i.e., $1 + \frac{\beta}{1+\gamma} < \frac{1-\alpha}{an} < 1 + \beta + \gamma$, the capital path ($k_t$) converges to $k_3^*$. Moreover,

$$k_3^* \ge k^s \Leftrightarrow \beta f'(k^s) \le 1 \Leftrightarrow \frac{an}{1-\alpha} \le \frac{1}{1+\beta}. \quad (15)$$

By combining 3 cases, we observe that: $k^* \ge k^s$ if and only if $1+\beta \le \frac{1-\alpha}{an}$, or equivalently $\beta f'(k^s) \le 1$.
<figure><img src="image_7.png" /><figcaption>Fig. 3. The equilibrium capital path with different wariness levels $\gamma$. Parameters are $\beta = 0.7$, $\alpha = 0.3$, $A = 4$, $n = 1.8$.</figcaption></figure>

### 3.3. Effect of wariness on dynamic efficiency

Following Malinvaud (1953) and Cass (1972), we define the notion of dynamic efficiency.

**Definition 4.** 1. A sequence of capital stock ($k_t$) is a feasible path if $f(k_t) - nk_{t+1} \ge 0, \forall t \ge 0$.

2. A feasible path ($k_t$) is efficient if there does not exist another feasible path ($k'_t$) satisfying $f(k'_t) - nk'_{t+1} \ge f(k_t) - nk_{t+1}, \forall t$ and there exists at least $t_0 \ge 0$ with $f(k'_{t_0}) - nk'_{t_0+1} > f(k_{t_0}) - nk_{t_0+1}$.

3. An intertemporal equilibrium is dynamically efficient if its capital path is efficient.

According to the consumption good market clearing condition $nk_{t+1} + c_t + \frac{\beta}{n} = f(k_t)$, any equilibrium capital path is feasible. An interesting question is whether it is efficient.

By using the same argument in Proposition 2.4 in de la Croix and Michel (2002), we obtain the following result.

**Lemma 3.** Under conditions in Proposition 2, an intertemporal equilibrium is dynamically efficient if $f'(k^*) > n$, and dynamically inefficient if $f'(k^*) < n$.

When $f'$ is strictly decreasing and $f'(0) > n > f'(\infty)$, there exists a unique so-called golden rule capital stock $k = k_{GR}$ determined by $f'(k_{GR}) = n$ (note that $k_{GR}$ does not depend on $\gamma$). In such a case, the conditions $f'(k^*) > n$ and $f'(k^*) < n$ are respectively equivalent to $k^* < k_{GR}$ and $k^* > k_{GR}$. Hence, we can investigate dynamic efficiency by comparing the steady state $k^*$ with the golden rule capital stock.

The following result is a direct consequence of Proposition 3 and Lemma 3.

**Proposition 5.** Let assumptions in Proposition 2 be satisfied.

1. If $f'(k_3^*) < \frac{1}{\gamma+\beta}$ and Assumption 4 holds, then the intertemporal equilibrium is dynamically efficient if $f'(k_1^*) > n$ and dynamically inefficient if $f'(k_3^*) < n$.
By consequence, if $f'(k_3^*) < \frac{1}{\gamma+\beta} < n$, then intertemporal equilibrium is dynamically inefficient.

2. If $f'(k_3^*) > \frac{1+\gamma}{\beta}$ and Assumption 4 holds, then the intertemporal equilibrium is dynamically efficient if $f'(k_2^*) > n$ and dynamically inefficient if $f'(k_3^*) < n$.
By consequence, if $f'(k_3^*) > \frac{1+\gamma}{\beta} > n$, then intertemporal equilibrium is dynamically efficient.

3. If $\frac{1+\gamma}{\beta} \ge f'(k_3^*) \ge \frac{1}{\gamma+\beta}$, the intertemporal equilibrium is dynamically efficient if $f'(k_3^*) > n$ and dynamically inefficient if $f'(k_3^*) > n$.

Proposition's point 3 indicates that when the capital return $f'(k_3^*)$ and the wariness have middle values, the equilibrium is dynamically efficient (inefficient) if there is under-accumulation (over-accumulation) at $k_3^*$.

Points 1 and 2 are more interesting because $k_1^*$ and $k_2^*$ depend on the wariness level $\gamma_2$. Proposition's point 1 suggests that when the capital return $f'(k_3^*)$ is lower than $\frac{1}{\gamma+\beta}$, the equilibrium is dynamically efficient if there is under-accumulation at $k_1^*$, i.e., $k_1^* < k_{GR}$. According to Proposition 3, $k_1^*$ is strictly increasing in the wariness level $\gamma$. So, in this case, the equilibrium is more likely to be dynamically efficient if the wariness is low enough. By contrast, when the capital return $f'(k_3^*)$ is higher than $(1+\gamma)/\beta$, then the equilibrium is more likely to be dynamically efficient if the wariness is quite high.

When $f(k) = Ak^a$, we find that $k_{GR} = (\frac{\alpha A}{n})^{\frac{1}{1-a}}$. Therefore, we can explicitly find conditions for the dynamic efficiency.

**Corollary 7.** Assume that $f(k) = Ak^a$, $u(c) = \frac{u^{1-\sigma}}{1-\sigma}$, $v_0 < \sigma \ne 1$, and $u(c) = \ln(c)$, for $\sigma = 1$. We have that:

1. If $\frac{1-\alpha}{n\alpha} > 1+\gamma+\beta$, then the inter-temporal equilibrium is dynamically efficient if $\frac{1-\alpha}{\alpha} < 1 + n^{1-\frac{1}{\sigma}}(\frac{\beta}{1+\gamma})^{-\frac{1}{\sigma}}$ and dynamically inefficient if $\frac{1-\alpha}{\alpha} > 1 + n^{1-\frac{1}{\sigma}}(\frac{\beta}{1+\gamma})^{-\frac{1}{\sigma}}$.

2. If $\frac{1-\alpha}{n} < 1 + \frac{\beta}{1+\gamma}$, then the inter-temporal equilibrium is dynamically efficient if $\frac{1-\alpha}{\alpha} < 1 + n^{1-\frac{1}{\sigma}}(\beta + \gamma)^{-\frac{1}{\sigma}}$ and dynamically inefficient if $\frac{1-\alpha}{\alpha} > 1 + n^{1-\frac{1}{\sigma}}(\beta + \gamma)^{-\frac{1}{\sigma}}$.

3. If $1 + \frac{\beta}{1+\gamma} \le \frac{1-\alpha}{n} \le 1+\gamma+\beta$, then the inter-temporal equilibrium is dynamically efficient if $\alpha > \frac{1}{n+2}$ and inefficient if $\alpha < \frac{1}{n+2}$.

**Proof.** See Appendix. ☐

## 4. Wariness and endogenous growth

### 4.1. Wariness and endogenous growth à la (Romer, 1986)

We now introduce endogenous growth à la (Romer, 1986). Assume that the production function at date $t$ is $f_t(k) = A_t f(k)$, where the productivity $A_t$ is taken as given by the firm. But, in equilibrium, due to a positive externality, $A_t$ is a function of the aggregate capital $k_t$:
$A_t = A(k_t)$. By consequence, we obtain the dynamical system of $(k_t)$

$$nk_{t+1} = s_t = \begin{cases} s_{\beta_1}(w_t, R_{t+1}) & \text{if } R_{t+1} < \frac{1}{\gamma+\beta} \\ \frac{w_t}{1+R_{t+1}} & \text{if } \frac{1}{\gamma+\beta} \le R_{t+1} \le \frac{1+\gamma}{\beta} \\ s_{\beta_2}(w_t, R_{t+1}) & \text{if } R_{t+1} > \frac{1+\gamma}{\beta} \end{cases}$$

$$= \begin{cases} s_{\beta_1} \left( A(k_t) \left( f(k_t) - k_t f'(k_t) \right), \right. \\ \qquad \left. A(k_{t+1}) f'(k_{t+1}) \right) & \text{if } A(k_{t+1}) f'(k_{t+1}) < \frac{1}{\gamma+\beta} \\ \\ \frac{A(k_t) \left( f(k_t) - k_t f'(k_t) \right)}{1+A(k_{t+1}) f'(k_{t+1})} & \text{if } \frac{1}{\gamma+\beta} \le A(k_{t+1}) f'(k_{t+1}) \le \frac{1+\gamma}{\beta} \\ \\ s_{\beta_2} \left( A(k_t) \left( f(k_t) - k_t f'(k_t) \right), \right. \\ \qquad \left. A(k_{t+1}) f'(k_{t+1}) \right) & \text{if } A(k_{t+1}) f'(k_{t+1}) > \frac{1+\gamma}{\beta} \end{cases}$$

For the sake of simplicity, we consider the logarithmic utility function $u(c) = \ln(c)$ and the Cobb-Douglas production function $f(k) = k^a$. In this case, the wage equals $w_t = (1-\alpha)A_t k^a$ and the capital return equals $R_t = \alpha A_t k_t^{a-1}$.

According to Corollary 3, the saving of household is given by

$$s_t = \begin{cases} \frac{\beta_1 w_t}{1+\beta_1} & \text{if } R_{t+1} < \frac{1}{\gamma+\beta} \\ \frac{w_t}{1+R_{t+1}} & \text{if } \frac{1}{\gamma+\beta} \le R_{t+1} \le \frac{1+\gamma}{\beta} \\ \frac{\beta_2 w_t}{1+\beta_2} & \text{if } R_{t+1} > \frac{1+\gamma}{\beta} \end{cases}$$

where recall that $\beta_1 \equiv \beta + \gamma$, $\beta_2 \equiv \frac{\beta}{1+\gamma}$.

Assume that the endogenous productivity $A_t$ is determined by $A_t = Ak_t^{1-\alpha}$, where $A > 0$ represents the exogenous productivity of our model. Hence, the wage is $w_t = (1-\alpha)Ak_t^{1-\alpha}k_t^\alpha = (1-\alpha)Ak_t$ while the capital return equals $R_t = \alpha Ak_t^{1-\alpha}k_t^{\alpha-1} = \alpha A$, $\forall t$. Therefore, we obtain the following result.

**Proposition 6.** Consider an endogenous growth model described above. The rate of growth of the capital stock per capita equals

$$\frac{k_{t+1}}{k_t} = g(\gamma) \equiv \begin{cases} \frac{\beta_1(1-\alpha)A}{n(1+\beta_1)} & \text{if } \alpha A < \frac{1}{\gamma+\beta} \\ \frac{(1-\alpha)A}{n(1+\alpha A)} & \text{if } \frac{1}{\gamma+\beta} \le \alpha A \le \frac{1+\gamma}{\beta} \\ \frac{\beta_2(1-\alpha)A}{n(1+\beta_2)} & \text{if } \alpha A > \frac{1+\gamma}{\beta} \end{cases} \quad (16)$$

Notice that when there is no wariness ($\gamma = 0$), we recover the standard saving $s_t = \frac{\beta_1 w_t}{1+\beta_1}$ and the rate of capital growth equals $g_0 \equiv \frac{\beta(1-\alpha)A}{n(1+\beta_1)}$. So, our result leads to interesting implications.

1. If $(\gamma + \beta)\alpha A < 1$ (the wariness and productivity are quite low), then the growth rate of physical per capital $g(\gamma)$ is increasing in the wariness level $\gamma$ and $g(\gamma) > g_0$.

2. If $\alpha A > \frac{1+\gamma}{\beta}$ (the wariness is quite low but the productivity is quite high), then the growth rate of physical per capital $g(\gamma)$ is decreasing in the wariness level $\gamma$ and $g(\gamma) < g_0$.

3. If $\frac{1}{\gamma+\beta} \le \alpha A \le \frac{1+\gamma}{\beta}$ (the productivity is moderate and the wariness is high), then $g(\gamma) \ge g_0$ if and only if $\alpha\beta A \le 1$ (i.e., the capital return is low).

Observe that $g(\gamma) \le g_0$ if and only if one of the following conditions holds: (1) $\alpha A > \frac{1+\gamma}{\beta}$; (2) $\frac{1}{\gamma+\beta} \le \alpha A \le \frac{1+\gamma}{\beta}$ and $\alpha\beta A \ge 1$. The second condition means that $\max(\frac{1}{\gamma+\beta}, \frac{1}{\beta}) = \frac{1}{\beta} \le \alpha A \le \frac{1+\gamma}{\beta}$.

Inversely, $g(\gamma) > g_0$ if and only if one of the following conditions holds: (1) $(\gamma + \beta)\alpha A < 1$; (2) $\frac{1}{\gamma+\beta} \le \alpha A \le \frac{1+\gamma}{\beta}$ and $\alpha\beta A < 1$. The second condition means that $\frac{1}{\gamma+\beta} \le \alpha A < \frac{1}{\beta}$.

To sum up, we get the following result:

**Corollary 8.** (1) The growth rate of capital stock per capita in the economy with wariness is less than that in the economy without wariness, i.e., $g(\gamma) \le g_0$ if and only if the capital return is high in the sense that $\alpha\beta A \ge 1$.

(2) $g(\gamma) > g_0$ if and only if $\alpha\beta A < 1$ (the capital return is low).

## 4.2. Wariness, public spending, and endogenous growth

We introduce a government which levies taxes on labor and capital incomes to finance public investment good (Bosi and Pham, 2016). The public investment $G_t$ at date $t$ is

$$G_t = \tau w_t + \tau_k \frac{R_t s_{t-1}}{n} \quad (17)$$

where $\tau$ and $\tau_k$ are the tax rates on labor and capital incomes respectively.

The representative firm maximizes its profit: $\max_{K_t, L_t \ge 0} F_{G_t}(K_t, L_t) - R_t K_t - w_t L_t$.

We focus on a simple model of public investment (R&D, for instance) in the spirit of Barro (1990):

$$F_{G_t}(K_t, L_t) = A(G_t)F(K_t, L_t), \text{ where } F(K_t, L_t) = K_t^\alpha L_t^{1-\alpha}, \\ A_t = \theta G_t^{1-\alpha}, \forall t.$$

Here, the public investment $G_t$ affects the TFP and this functional specification promotes endogenous growth dynamics.

Households faces two budget constraints (one per period):

$$c_t + s_t \le (1-\tau) w_t, \quad d_{t+1} \le (1-\tau_k) R_{t+1} s_t$$

to maximizes her utility by choosing $s_t$, $c_t$ and $d_{t+1}$.

**Definition 5.** Given positive rates $\tau$, $\tau_k$, an equilibrium is a positive sequence $(R_t, w_t, c_t, d_{t+1}, s_t, K_{t+1}, L_t, G_t)_{t \ge 0}$ satisfying two conditions:

(1) given the sequence $(R_t, w_t)$, the allocation $(K_t, L_t)$ is a solution to the problem $(P_{f,t})$ and the allocation $(c_t, s_t, d_{t+1})$ is a solution to the problem $(P_t)$;

(2) the market clearing conditions: $K_{t+1} = N_t s_t$, $L_t = N_t$, $s_t + c_t + d_t/n + G_t = f_t(k_t)$.

In equilibrium, we have

$$G_t = \tau w_t + \tau_k R_t k_t = \tau (1-\alpha) \theta G_t^{1-\alpha} k_t^\alpha + \tau_k \alpha \theta G_t^{1-\alpha} k_t^\alpha \\ = [\alpha \tau_k + (1-\alpha) \tau] \theta G_t^{1-\alpha} k_t^\alpha$$

which implies that

$$(\frac{G_t}{k_t})^\alpha = \theta [\alpha \tau_k + (1-\alpha) \tau]$$

$$R_t = \alpha A_t k_t^{\alpha-1} = \alpha \theta G_t^{1-\alpha} k_t^{\alpha-1} = R(\theta), \text{ where}$$

$$R(\theta) \equiv \alpha \theta^{\frac{1}{\alpha}} [\alpha \tau_k + (1-\alpha) \tau]^{\frac{1-\alpha}{\alpha}}.$$

We now look at the dynamical system of $(k_t)$

$$nk_{t+1} = s_t = \begin{cases} s_{\beta_1}((1-\tau)w_t, (1-\tau_k)R_{t+1}) & \text{if } (1-\tau_k)R_{t+1} < \frac{1}{\gamma+\beta} \\ \frac{(1-\tau)w_t}{1+(1-\tau_k)R_{t+1}} & \text{if } \frac{1}{\gamma+\beta} \le (1-\tau_k)R_{t+1} \le \frac{1+\gamma}{\beta} \\ s_{\beta_2}((1-\tau)w_t, (1-\tau_k)R_{t+1}) & \text{if } (1-\tau_k)R_{t+1} > \frac{1+\gamma}{\beta} \end{cases}$$

For the sake of tractability, we assume that $u(x) = \ln(x)$, $\forall x > 0$. By consequence, we obtain that

$$nk_{t+1} = s_t = \begin{cases} \frac{\beta_1}{1+\beta_1}(1-\tau)w_t & \text{if } (1-\tau_k)R_{t+1} < \frac{1}{\gamma+\beta} \\ \frac{(1-\tau)w_t}{1+(1-\tau_k)R_{t+1}} & \text{if } \frac{1}{\gamma+\beta} \le (1-\tau_k)R_{t+1} \le \frac{1+\gamma}{\beta} \\ \frac{\beta_2}{1+\beta_2}(1-\tau)w_t & \text{if } (1-\tau_k)R_{t+1} > \frac{1+\gamma}{\beta} \end{cases}$$

Since $w_t = (1-\alpha)A_t k_t^\alpha = \frac{1-\alpha}{\alpha} R_t k_t$, we can compute the growth rate of capital. We summarize our result in the following result.
**Proposition 7.** Consider an endogenous growth model described above. The rate of growth of the capital stock per capita equals

$$ \frac{k_{t+1}}{k_t} = g(\gamma) = \begin{cases} \frac{\beta_1}{1+\beta_1} \frac{(1-\tau)(1-\alpha)R(\theta)}{na} & \text{if } (1-\tau_k)R(\theta) < \frac{1}{\gamma+\beta} \\ \frac{(1-\tau)}{1+(1-\tau_k)R(\theta)} \frac{(1-\alpha)R(\theta)}{na} & \text{if } \frac{1}{\gamma+\beta} \le (1-\tau_k)R(\theta) \le \frac{1+\gamma}{\beta} \\ \frac{\beta_2}{1+\beta_2} \frac{(1-\tau)(1-\alpha)R(\theta)}{na} & \text{if } (1-\tau_k)R(\theta) > \frac{1+\gamma}{\beta} \end{cases} $$

where recall that $\beta_1 \equiv \beta + \gamma$, $\beta_2 \equiv \frac{\beta}{1+\gamma}$, $R(\theta) \equiv \alpha\theta^{\frac{1}{\alpha}}[\alpha\tau_k + (1-\alpha)\tau]^{\frac{1-\alpha}{\alpha}}$.

Therefore, we get a similar insight as obtained in Section 4.1. Precisely, we have that:

1. If $(1 - \tau_k)R(\theta) < \frac{1}{\gamma+\beta}$ (the wariness and capital return are quite low), then the growth rate of physical per capital $g(\gamma)$ is increasing in the wariness level $\gamma$.

2. If $(1 - \tau_k)R(\theta) > \frac{1+\gamma}{\beta}$ (the wariness is quite low but the capital return is quite high), then the growth rate of physical per capital $g(\gamma)$ is decreasing in the wariness level $\gamma$.

3. If $\frac{1}{\gamma+\beta} \le (1-\tau_k)R(\theta) \le \frac{1+\gamma}{\beta}$ (the capital return is moderate and the wariness is high), then $g(\gamma) \ge g_0$ if and only if $\alpha\beta A \le 1$ (i.e., the capital return is low).

## 5. Conclusion

We have investigated the effects of wariness on the economic dynamics of the economy in OLG models. In our exogenous growth economy with standard assumptions of utility and production functions, we have proved the capital stock of intertemporal equilibrium converges to a steady state. We have also studied how this steady state depends on the wariness. We have shown that whether the effects are positive or negative depend on the structure of the economy, specially the interest rate of the economy without wariness. This insight holds not only for the exogenous growth model but also for an endogenous growth framework à la Romer (1986) or à la Barro (1990).

We consider our paper as a starting point for understanding the role of wariness in economic dynamics within OLG models. Let us mention some potential avenues for future research.

First, we have worked with a separable utility function. A natural question arises: how can we introduce the notion of wariness for a non-separable lifecycle utility? To address this issue, we assume that the utility function of the household is $U(c_t, d_{t+1}) = H(c_t, d_{t+1}, \gamma \min(c_t, d_{t+1}))$ where $H$ is increasing in each component, $\gamma > 0$ represents the wariness degree. Under mild conditions on the function $H$, such as concavity and differentiability, we would be able to establish the existence and the uniqueness of the solution to the household's problem. However, the global dynamics of equilibrium in this case would be much richer because it depends on the form of the function $H$.<fnref n="10" />

Second, **Assumption 3** and conditions in part 1 of **Proposition 2** may not hold under more general production functions such as CES production functions. It is well known that in a canonical OLG model with a CES production function, if the factors of production are poorly substitutable, then there may exist two steady states, and a poverty trap may appear (see Galor and Ryder (1989), de la Croix and Michel (2002) among others). It would be valuable to investigate the effects of wariness on economic growth and poverty traps, and how these effects depend on the elasticity of substitution.<fnref n="11" />

Another promising avenue for future research involves extending our work to a stochastic economy with production and incomplete markets (Magill and Quinzii, 2002).<fnref n="12" />

Last, but not least, observe that the three regimes in (7) depend on the relationship between the capital return and the wariness degree. Therefore, it would be interesting to study how wariness influences the impacts of government debt (or social security) on the competitive equilibrium.

## CRediT authorship contribution statement

**Hai Ha Pham:** Formal analysis, Writing – original draft, Writing – review & editing, Methodology, Project administration, Software.

**Ngoc-Sang Pham:** Conceptualization, Formal analysis, Methodology, Writing – original draft, Writing – review & editing, Software.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

No data was used for the research described in the article.

## Acknowledgments

This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number C2024-28-11.

## Appendix

**Proof of Proposition 1.** For the sake of simplicity, we remove the subscript and consider the problem

$$ P(w, R) : \max_{(c,d,s)} \left\{ U(c, d) \equiv u(c) + \beta u(d) + \gamma \min(u(c), u(d)) \right\} \quad (18) $$

$$ c + s \le w, \quad d \le R_s, \quad c, d, s \ge 0, \quad (19) $$

Denote $U^*$ the maximum value of the problem $P(w, R)$. We also observe that

$$ U^* = \max \left( \begin{array}{l} \max_{\substack{c+s \le w, \\ d \le R_s \\ c \ge d}} \{u(c) + (\beta + \gamma)u(d)\}, \\ \max_{\substack{c+s \le w, \\ d \le R_s \\ c \le d}} \{(1 + \gamma)(u(c) + \frac{\beta}{1 + \gamma}u(d))\} \end{array} \right). \quad (20) $$

Denote $\beta_1 \equiv \beta + \gamma$, $\beta_2 \equiv \frac{\beta}{(1+\gamma)}$. We have $\beta_1 > \beta > \beta_2$.

Consider the first problem

$$ (P_1) : \max_{(c,d,s)} u(c) + \beta_1 u(d) \quad (21) $$

$$ c + s \le w, \quad d \le R_s, \quad c \ge d \ge 0 \quad (22) $$

In optimal, we must have $c + s = w$ and $d = R_s$. So, solving $(P_1)$ is equivalent to solving the following problem

$$ (P'_1) : \max_s u(w - s) + \beta_1 u(R_s) \quad (23) $$

$$ w - s \ge R_s \ge 0 \Leftrightarrow 0 \le s \le \frac{w}{1 + R} \quad (24) $$

<footnote n="10">¹⁰ Note that under a non-separable utility, present and future goods may be not normal; see Van and Pham (2020) among others. By consequence, the dynamical systems of capital may be non-monotonic and the convergence of capital path may not hold. See Guesnerie and Woodford (1993) for an excellent survey on endogenous fluctuations.</footnote>

<footnote n="11">¹¹ We may need mathematical tools concerning piecewise-smooth dynamical systems (see Sushko et al. (1989), Zhusubaliyev and Mosekilde (2003) and references therein).</footnote>

<footnote n="12">¹² Ohtaki and Ozaki (2015) consider a pure-endowment stationary stochastic overlapping generations economy with Knightian uncertainty, in which agents have maxmin expected utility preferences as in Gilboa and Schmeidler (1989). Then, they show that there may exist multiple stationary monetary equilibria.</footnote>
Denote $G(s) \equiv u(w-s) + \beta_1 u(Rs)$. We have $G'(s) = -u'(w-s) + \beta_1 Ru'(Rs)$ and $G''(s) = u''(w-s) + \beta_1 R^2 u''(Rs) < 0$.

Observe that $G'(0) = +\infty$ and $G'(\frac{Rw}{1+R}) = u'(\frac{Rw}{1+R})(\beta_1 R - 1)$. By consequence, we obtain that:

1. If $\beta_1 R \ge 1$, then $c = d = w \frac{R}{1+R}$ and $s = \frac{w}{1+R}$.

2. If $\beta_1 R < 1$, then $s = s_{\beta_1}$ which is a unique solution to the equation $G'(s) = 0$.

Consider the second problem

$$ (P_2): \max_{(c,d,s)} u(c) + \beta_2 u(d), \quad (25) $$

$$ c+s \le w, \quad d \le Rs, \quad 0 \le c \le d. \quad (26) $$

By using the similar argument, the solution of (P₂) is given by

1. If $\beta_2 R \le 1$, then $c = d = w \frac{R}{1+R}$ and $s = \frac{w}{1+R}$.

2. If $\beta_2 R > 1$, then $s = s_{\beta_2}$ which is a unique solution to the equation $-u'(w-s) + \beta_2 Ru'(Rs) = 0$.

We now come back to the original problem.

$$ U^* = \max \left( \begin{array}{l} \max_{\substack{c+d \le w, \\ d \le Rs \\ c \le d}} \{u(c) + (\beta + \gamma)u(d)\}, \\ \max_{\substack{c+d \le w, \\ d \le Rs \\ c \le d}} \{(1+\gamma)(u(c) + \frac{\beta}{1+\gamma}u(d))\} \end{array} \right). \quad (27) $$

We consider three cases.

1. If $\beta_1 R < 1$, then the solution of (P₁) satisfies $s = s_{\beta_1}$. Since $\beta_1 > \beta_2$, we have $\beta_2 R < 1$ which implies that the solution of (P₂) satisfies $c = d = w \frac{R}{1+R}$ and $s = \frac{w}{1+R}$. Note that the allocation $c = d = w \frac{R}{1+R}$ and $s = \frac{w}{1+R}$ is feasible for the problem (P₁). This follows that the maximum value of the problem (P₁) is higher than $(1+\gamma)(u(w \frac{R}{1+R}) + \frac{\beta}{1+\gamma}u(w \frac{R}{1+R})) = (1+\gamma + \beta)u(\frac{Rw}{1+R})$. So, the solution of the problem (P(w, R)) satisfies $s = s_{\beta_1}$.

2. If $\beta_2 R > 1$, then by using the same argument, we obtain that $s = s_{\beta_2}$.

3. If $\beta_1 R \ge 1 \ge \beta_2 R$, then $c = d = w \frac{R}{1+R}$ and $s = \frac{w}{1+R}$. $\square$

**Proof of Corollary 1.** Given $\gamma$, denote $\beta_1(\gamma) \equiv \beta + \gamma$, $\beta_2(\gamma) \equiv \frac{\beta}{1+\gamma}$.

Since $0 < \gamma_1 < \gamma_2$, we have $\frac{1}{\gamma_2+\beta} < \frac{1}{\gamma_1+\beta} < \frac{1+\gamma_1}{\beta} < \frac{1+\gamma_2}{\beta}$. So, we consider 5 cases.

1. If $R_{t+1} < \frac{1}{\gamma_2+\beta}$ then $R_{t+1} < \frac{1}{\gamma_1+\beta}$. According to Proposition 1, we have $s_t(\gamma_1) = s_{\beta_1}(\gamma_1)(\omega_t, R_{t+1})$ and $s_t(\gamma_2) = s_{\beta_1}(\gamma_2)(\omega_t, R_{t+1})$. Thanks to Lemma 1, we have

$$ s_{\beta_1}(\gamma_1)(\omega_t, R_{t+1}) < s_{\beta_1}(\gamma_2)(\omega_t, R_{t+1}). $$

Hence $s_t(\gamma_1) < s_t(\gamma_2)$.

2. If $\frac{1}{\gamma_2+\beta} \le R_{t+1} < \frac{1}{\gamma_1+\beta}$ then we have $s_t(\gamma_1) = s_{\beta_1}(\gamma_1)(\omega_t, R_{t+1}) < \frac{w_t}{1+R_{t+1}}$. Since $\frac{1}{\gamma_2+\beta} \le R_{t+1} < \frac{1+\gamma_2}{\gamma}$, Proposition 1 implies that $s_t(\gamma_2) = \frac{w_t}{1+R_{t+1}}$. So $s_t(\gamma_1) < s_t(\gamma_2)$.

3. It is a direct consequence of Proposition 1.

4. If $\frac{1+\gamma_1}{\beta} < R_{t+1} \le \frac{1+\gamma_2}{\beta}$, we have $s_t(\gamma_1) = s_{\beta_2}(\gamma_1)(\omega_t, R_{t+1})$ and $s_t(\gamma_2) = \frac{w_t}{1+R_{t+1}}$. So, $s_t(\gamma_1) > s_t(\gamma_2)$.

5. If $R_{t+1} > \frac{1+\gamma_2}{\beta}$, by using the similar argument as point 1, we get that $s_t(\gamma_1) > s_t(\gamma_2)$. $\square$

**Proof of Corollary 2.** (1) Let $\beta R_{t+1} < 1$. We consider two cases:

• For $\gamma \in (0, \frac{1}{R_{t+1}} - \beta)$, Proposition 1 implies that $s_t(\gamma) = s_{\beta_1}(\gamma)$ which is increasing in $\gamma$.

• For $\frac{1}{R_{t+1}} - \beta \le \gamma$, we have $\frac{1}{\beta+\gamma} \le R_{t+1} < \frac{1}{\beta} < \frac{1+\gamma}{\beta}$. Then Proposition 1 implies that $s_t = \frac{w_t}{1+R_{t+1}}$ which does not depend on $\gamma$.

(2) Let $\beta R_{t+1} > 1$. We consider two cases:

• For $\gamma < \beta R_{t+1} - 1$, i.e., $\frac{1+\gamma}{\beta} < R_{t+1}$, Proposition 1 implies that $s_t(\gamma) = s_{\beta_2}(\gamma)$ which is decreasing in $\gamma$.

• For $\gamma \ge \beta R_{t+1} - 1$, we have $\frac{1}{\beta} < R_{t+1} < \frac{1+\gamma}{\beta}$. Then Proposition 1 implies that $s_t = \frac{w_t}{1+R_{t+1}}$ which does not depend on $\gamma$. $\square$

**Proof of Proposition 2.** Part 2 is a direct consequence of Proposition 1. Let us prove part 1. Part 1. Let us start with intermediate steps.

**Lemma 4.** Assume that the function $k f'(k)$ is increasing on the interval $(0, \infty)$. Given $k_t > 0$, let $k_{t+1,1}$ be determined by the Euler equation $u'(\omega(k_t) - nk_{t+1,1}) = \beta_1 R_{t+1}u'(R_{t+1}nk_{t+1,1})$ and $R_{t+1} = f'(k_{t+1,1}) < \frac{1}{\gamma+\beta}$, where recall that $\omega(k) \equiv f(k_t) - k_t f'(k_t)$. Then $k_{t+1,1}$ is a strictly increasing function of $k_t$.

**Proof.** It suffices to prove that $w_t$ is increasing in $k_{t+1}$. Taking the derivative of both sides of the equation $u'(\omega(k_t) - nk_{t+1,1}) = \beta_1 R_{t+1}u'(R_{t+1}nk_{t+1,1})$ with respect to $k_{t+1}$ and noting that $R_{t+1} = f'(k_{t+1,1})$, we have

$$ \left( \frac{\partial \omega(k_t)}{\partial k_{t+1}} - n \right) u''(c_t) = \beta_1 \frac{\partial (R_{t+1} u' (R_{t+1} n k_{t+1}))}{\partial k_{t+1}} \\ \text{or, equivalently, } \frac{\partial \omega(k_t)}{\partial k_{t+1}} u''(c_t) = n u''(c_t) + \beta_1 \frac{\partial (R_{t+1} u' (R_{t+1} n k_{t+1}))}{\partial k_{t+1}} $$

Since $u'' < 0$ and $f'(k)u'(nkf'(k))$ is decreasing in $k$ for any $k > 0$, we have $\frac{\partial \omega(k_t)}{\partial k_{t+1}} > 0$. Note that $\omega(k_t) = f(k_t) - k_t f'(k_t)$ is increasing in $k_t$ (because the function $f$ is strictly concave) and $\omega(0) = 0$ (because $f(0) = 0$) and $\omega(\infty) = \infty$ (because of $f'(\infty) = 0$). By consequence, we get that $k_{t+1,1}$ is an increasing function of $k_t$. $\square$

**Lemma 5.** Assume that $f'(k)u'(nkf'(k))$ is decreasing in $k$ for any $k > 0$. Then $k_{t+1}$ determined by (8) is a strictly increasing, continuous function, denoted by $G(k_t)$, of $k_t$ (with $G(0) = 0, G(\infty) = \infty$). By consequence, the capital path $(k_t)$ converges.

**Proof.** By using the same argument in Lemma 4, we can prove that $k_{t+1}$ determined by (8) is an increasing function of $k_t$ in each case in the formula (8). We can check that $G(0) = 0, G(\infty) = \infty$ (because $u'(\infty) = 0$). $\square$

**Lemma 6.** Assume that $f'(k)u'(nkf'(k))$ is decreasing in $k$ for any $k > 0$. At equilibrium, the dynamics of capital is characterized by the following system

$$ nk_{t+1} = s(\omega(k_t), R_{t+1}) = \begin{cases} s_{\beta_1}(\omega(k_t), R_{t+1}) & \text{if } k_t > a_1 \\ \frac{\omega(k_t)}{1+R_{t+1}} & \text{if } a_2 \le k_t \le a_1 \\ s_{\beta_2}(\omega(k_t), R_{t+1}) & \text{if } k_t < a_2 \end{cases} \quad (28) $$

for any $t$, where $a_1 \equiv G^{-1}\left((f')^{-1}\left(\frac{1}{\gamma+\beta}\right)\right) > 0$ and $a_2 \equiv G^{-1}\left((f')^{-1}\left(\frac{1+\gamma}{\beta}\right)\right) > 0$.

**Proof.** Since $R_{t+1} = f'(k_{t+1})$, Assumptions 2, 3 imply that conditions $R_{t+1} < \frac{1}{\gamma+\beta}$ and $R_{t+1} > \frac{1+\gamma}{\beta}$ are respectively equivalent to $k_{t+1} > (f')^{-1}\left(\frac{1}{\gamma+\beta}\right)$ and $k_{t+1} > (f')^{-1}\left(\frac{1+\gamma}{\beta}\right)$.

By Lemma 5, the function $G$ is continuous, strictly increasing and $G(0) = 0, \lim_{x \to \infty} G(x) = \infty$. So there exist the inverse function $G^{-1}$ from $[0, \infty)$ to $[0, \infty)$. By consequence, conditions $k_{t+1} > (f')^{-1}\left(\frac{1}{\gamma+\beta}\right)$ and
$k_{t+1} > (f')^{-1} \left(\frac{1+\gamma}{\beta}\right)$ are respectively equivalent to $k_t > a_1$ and $k_t > a_2$, where $a_1$ and $a_2$ are well defined and positive. Therefore, the system (8) becomes (28). $\square$

**Lemma 7 (Finding Steady State).** Assume that the function $\frac{\omega(k)}{k(1+f'(k))}$ is strictly decreasing in $k$. Let $k_1^*, i = 1, 2, 3$ be satisfied the following equations

$$nk_1^* = s_{\beta_1}(\omega(k_1^*) - f'(k_1^*))$$

$$nk_2^* = s_{\beta_2}(\omega(k_2^*) - f'(k_2^*))$$

$$nk_3^* = \frac{\omega(k_3^*)}{1 + f'(k_3^*)}.$$

We have that: (1) $f'(k_3^*) < \frac{1}{\gamma+\beta}$ if and only if $f'(k_1^*) < \frac{1}{\gamma+\beta}$, and (2) $f'(k_3^*) > \frac{1+\gamma}{\beta}$ if and only if $f'(k_2^*) > \frac{1+\gamma}{\beta}$.

The level $k_1^*$ and $k_2^*$ represent the capital steady state of the economy with discount rate $\beta_1$ and $\beta_2$ respectively. $k_3^*$ is the capital steady state of the economy where the consumptions of young and old generations are the same.

**Proof of Lemma 7.** For the sake of simplicity, we write $s^{(i)}(k_i^*)$ instead of $s_{\beta_i}(\omega(k_i^*) - f'(k_i^*))$, for $i = 1, 2$.

1. If $f'(k_1^*) < \frac{1}{\gamma+\beta}$.

By definition of $s^{(1)}(k_1^*)$, we have that

$$u'(\omega(k_1^*) - s^{(1)}(k_1^*)) = \beta_1 f'(k_1^*) u'(f'(k_1^*) s^{(1)}(k_1^*))$$

Then we have $u'(\omega(k_1^*) - s^{(1)}(k_1^*)) < u'(f'(k_1^*) s^{(1)}(k_1^*))$ which is equivalent to $\omega(k_1^*) - s^{(1)}(k_1^*) > f'(k_1^*) s^{(1)}(k_1^*)$, i.e., $s^{(1)}(k_1^*) = nk_1^* < \frac{\omega(k_1^*)}{1+f'(k_1^*)}$.

If the function $\frac{\omega(k)}{k(1+f'(k))}$ is strictly decreasing for any $k > 0$, then we get that $k_1^* < k_3^*$ because $n = \frac{\omega(k_3^*)}{k_3^*(1+f'(k_3^*))} < \frac{\omega(k_1^*)}{k_1^*(1+f'(k_1^*))}$. So, $f'(k_3^*) < f'(k_1^*) < \frac{1}{\gamma+\beta}$.

2. By using the same argument, we have that: If $f'(k_1^*) > \frac{1}{\gamma+\beta}$, then $f'(k_3^*) > f'(k_1^*) > \frac{1}{\gamma+\beta}$.

3. If $f'(k_1^*) = \frac{1}{\gamma+\beta}$, then $n = \frac{\omega(k_3^*)}{k_3^*(1+f'(k_3^*))} = \frac{\omega(k_1^*)}{k_1^*(1+f'(k_1^*))}$, then $k_1^* = k_3^*$.

4. If $f'(k_2^*) > \frac{1+\gamma}{\beta}$, then $s^{(2)}(k_2^*) = nk_2^* > \frac{\omega(k_2^*)}{1+f'(k_2^*)}$. So, $n = \frac{\omega(k_2^*)}{k_2^*(1+f'(k_2^*))} > \frac{\omega(k_2^*)}{k_2^*(1+f'(k_2^*))}$. Thus, $k_2^* > k_3^*$, $f'(k_2^*) < f'(k_2^*)$.

5. If $f'(k_2^*) < \frac{1+\gamma}{\beta}$, then $s^{(2)}(k_2^*) = nk_2^* < \frac{\omega(k_2^*)}{1+f'(k_2^*)}$. So, $n = \frac{\omega(k_2^*)}{k_2^*(1+f'(k_2^*))} < \frac{\omega(k_2^*)}{k_2^*(1+f'(k_2^*))}$. Thus, $k_3^* > k_2^*$, $f'(k_3^*) < f'(k_2^*) < \frac{1+\gamma}{\beta}$.

6. If $f'(k_2^*) = \frac{1+\gamma}{\beta}$, then $s^{(2)}(k_2^*) = nk_2^* = \frac{\omega(k_2^*)}{1+f'(k_2^*)}$. So, $k_2^* = k_3^*$. $\square$

We now come back to the proof of part 1 of Proposition 2. We have that $k_t$ converges to $k^*$ satisfying $nk^* = s(\omega(k^*), f'(k^*))$. There are three cases.

(i) If $f'(k^*) < \frac{1}{\gamma+\beta}$, then, according to Proposition 1, we get that $nk^* = s(\omega(k^*), f'(k^*)) = s_{\beta_1}(\omega(k^*), f'(k^*))$. Point 1 of Lemma 7 implies that $f'(k_3^*) < \frac{1}{\gamma+\beta}$.

(ii) If $f'(k^*) > \frac{1+\gamma}{\beta}$, then, according to Proposition 1, we get that $nk^* = s(\omega(k^*), f'(k^*)) = s_{\beta_2}(\omega(k^*), f'(k^*))$. Point 2 of Lemma 7 implies that $f'(k_3^*) > \frac{1+\gamma}{\beta}$.

(iii) If $\frac{1+\gamma}{\beta} \ge f'(k^*) \ge \frac{1}{\gamma+\beta}$, then, according to Proposition 1, we get that $nk^* = s(\omega(k^*), f'(k^*)) = \frac{\omega(k)}{(1+f'(k))}$. So, we have $k^* = k_3^*$, where $k_3^*$ is the unique solution to the equation $n = \frac{\omega(k)}{k(1+f'(k))}$.

We now finish our proof by considering three cases.

1. If $f'(k_3^*) < \frac{1}{\gamma+\beta}$, then cases (ii) and (iii) above cannot happen. By consequence, $k^*$ is a solution to the equation $nk = s_{\beta_1}(\omega(k), f'(k))$.

2. If $f'(k_3^*) > \frac{1+\gamma}{\beta}$, then cases (i) and (iii) above cannot happen. By consequence, $k^*$ is a solution to the equation $nk = s_{\beta_2}(\omega(k), f'(k))$.

3. If $\frac{1+\gamma}{\beta} \ge f'(k_3^*) \ge \frac{1}{\gamma+\beta}$, then cases (i) and (ii) above cannot happen. By consequence, $k^* = k_3^*$. $\square$

**Proof of Proposition 3.**

1. If $f'(k_3^*) < \frac{1}{\gamma+\beta}$, then $k^*$ is a solution to the equation $nk = s_{\beta_1}(\omega(k), f'(k))$. By Lemma 2, we get that $k^* = k_1^*$ is increasing in $\beta_1$ and hence in $\gamma$.

2. If $f'(k_3^*) > \frac{1+\gamma}{\beta}$, we use the same argument as in point 1.

3. Denote $H(k) \equiv \frac{\omega(k)}{k(1+f'(k))}$. Since $H(k)$ is strictly decreasing and $H(k^*) = n$, condition $k_3^* > k^s$ is equivalent to $H(k_s) > H(k_3^*) = n$, i.e., $\omega(k^s) - nk^s > f'(k^s)nk^s$. This is equivalent to $u'(\omega(k^s) - s) < u'(f'(k^s)s)$, i.e., $\beta f'(k^s) < 1$. $\square$

**Proof of Corollary 5.** First we verify that the conditions Lemma 6 hold when $u(c) = \ln c$ and $f(k) = Ak^a$. In fact, we have $\omega(k) = (1-\alpha)Ak^a$. Thus the function $h(k) = \frac{\omega(k)}{k} = (1-\alpha)Ak^{a-1}$ is decreasing in $k$ and then $\frac{\omega(k)}{k(1+f'(k))}$ is also decreasing in $k$. The function $f'(k)u'(nkf'(k)) = \frac{1}{nk}$ is also decreasing in $k$. The rate of return

$$R_{t+1} = f'(k_{t+1}) = \alpha Ak_{t+1}^{a-1}.$$

From Lemma 6, the dynamic of capital at equilibrium satisfies the following equation

$$nk_{t+1} = \begin{cases} \frac{\beta_1 \omega(k_t)}{1+\beta_1} = \frac{\beta_1(1-\alpha)Ak_t^a}{1+\beta_1} & \text{if } \alpha Ak_{t+1}^{a-1} < \frac{1}{\gamma+\beta} \\ = (1-\sigma_1)(1-\alpha)Ak_t^a & \text{if } \frac{1}{\gamma+\beta} \le \alpha Ak_{t+1}^{a-1} \le \frac{1+\gamma}{\beta} \\ \frac{(1-\alpha)Ak_t^a}{1+\alpha Ak_{t+1}^{a-1}} & \text{if } \alpha Ak_{t+1}^{a-1} > \frac{1+\gamma}{\beta} \end{cases}$$

When $\alpha Ak_{t+1}^{a-1} < \frac{1}{\gamma+\beta}$, we have $k_{t+1} = G(k_t)$ with $G(k) = G_1(k) = \frac{1}{n}(1-\sigma_1)(1-\alpha)Ak^a$. By consequence, we can check that condition $\alpha Ak_{t+1}^{a-1} < \frac{1}{\gamma+\beta}$ becomes $k_1 > a_1$. Similarly, $\alpha Ak_{t+1}^{a-1} > \frac{1+\gamma}{\beta}$ becomes $k_t < a_2$ while $\frac{1}{\gamma+\beta} \le \alpha Ak_{t+1}^{a-1} \le \frac{1+\gamma}{\beta}$ becomes $a_2 \le k_t \le a_1$. As a consequence, we obtain (10) and (11). Since $k_3^*$ is the solution to the equation $\frac{\omega(k)}{k(1+f'(k))} = n$, we have that $n = \frac{(1-\alpha)Ak_3^{a-1}}{1+f'(k_3^*)}$, i.e., $f'(k_3^*) = \frac{an}{1-a-an}$. Therefore, we see that

$$\frac{1+\gamma}{\beta} \ge f'(k_3^*) \ge \frac{1}{\gamma+\beta} \iff \frac{\beta}{1+\gamma} + 1 \le \frac{1-\alpha}{an} \le 1+\gamma+\beta. \square$$

**Proof of Corollary 6.** Remind that $k_1^*, k^s$ and $k_2^*$ are respectively the solution of the following equations

$$g_1(k) \equiv \frac{(1-\alpha)Ak^{a-1}}{1 + (\alpha Ak^{a-1})^{-\frac{1}{\sigma}} + \beta_1^{-\frac{1}{\sigma}}} = n$$

$$g_3(k) \equiv \frac{(1-\alpha)Ak^{a-1}}{1 + (\alpha Ak^{a-1})^{-\frac{1}{\sigma}} + \beta_3^{-\frac{1}{\sigma}}} = n$$

$$g_2(k) \equiv \frac{(1-\alpha)Ak^{a-1}}{1 + (\alpha Ak^{a-1})^{-\frac{1}{\sigma}} + \beta_2^{-\frac{1}{\sigma}}} = n$$

Since the function $\alpha Ak^{a-1}$ is strictly decreasing in $k$ and the function $\frac{x}{1+\beta} = \frac{1}{x^{-1+\beta}x^{-\frac{1}{\sigma}}x^{-\frac{1}{\sigma}}}$ is strictly increasing in $x > 0$, then the function $g_1(k)$, $g_3(k)$, $g_2(k)$ are strictly decreasing in $k$. Since $\beta_1 = \beta + \gamma$ and $\beta_2 = \frac{\beta}{1+\gamma}$, we obtain that $k_1^*$ is strictly decreasing in $\gamma$ and $k_2^*$ is strictly increasing in $\gamma$. Moreover, $k_1^* > k^s > k_2^*$. $\square$

**Proof of Corollary 7.** We firstly consider the case $u(c) = \ln c$.
1. If $\frac{1-\alpha}{na} > 1 + \gamma + \beta$, then from Corollary 5, we have $k^* = k_1^* = \left(\frac{(1-\sigma_1)(1-\alpha)A}{n}\right)^{\frac{1}{1-\alpha}}$. The condition $k_{GR} > k^*$ is equivalent to
$$ \frac{(1-\sigma_1)(1-\alpha)A}{n} < \frac{\alpha A}{n}. $$

Using the fact that $\sigma_1 = \frac{1}{1+\gamma+\beta}$, we obtain the equivalent condition $\frac{1-\alpha}{\alpha} < 1 - \sigma_1 = 1 + \frac{1}{\beta+\gamma}$. Hence $k^* < k_{GR}$ if and only if $\frac{1-\alpha}{\alpha} < 1 + \frac{1}{\beta+\gamma}$. Inversely, we see that $k^* > k_{GR}$ if and only if $\frac{1-\alpha}{\alpha} > 1 + \frac{1}{\beta+\gamma}$.

2. If $\frac{1-\alpha}{an} \le 1 + \frac{\beta}{1+\gamma}$, then from Corollary 5, we have $k^* = k_2^* = \left(\frac{(1-\sigma_2)(1-\alpha)A}{n}\right)^{\frac{1}{1-\alpha}}$. So, we find

$$ f'(k^*) = \alpha A \left( \frac{(1-\sigma_2)(1-\alpha)A}{n} \right)^{\frac{1-\alpha}{1-\alpha}} = \frac{an}{(1-\sigma_2)(1-\alpha)}. $$

By consequence, condition $f'(k^*) > n$ is equivalent to $\frac{1-\alpha}{\alpha} < \frac{1}{1-\sigma_2}$. Recall that $\sigma_2 \equiv \frac{1+\gamma}{1+\beta+\gamma}$. So, $\frac{1}{1-\sigma_2} = \frac{1+\beta+\gamma}{\beta} = 1 + \frac{1+\gamma}{\beta}$. Hence, $f'(k^*) > n$ becomes $\frac{1-\alpha}{\alpha} < 1 + \frac{1+\gamma}{\beta}$ while $f'(k^*) < n$ becomes $\frac{1-\alpha}{\alpha} > 1 + \frac{1+\gamma}{\beta}$.

3. If $1 + \frac{\beta}{1+\gamma} \le \frac{1-\alpha}{n} \le 1 + \gamma + \beta$, then $k^* = k_3^* = \left(\frac{(1-\alpha-n\alpha)A}{n}\right)^{\frac{1}{1-\alpha}}$. Hence the condition $k^* < k_{GR}$ satisfies if and only if $\frac{(1-\alpha-n\alpha)A}{n} < \frac{\alpha A}{n}$ which is equivalent to $\alpha > \frac{1}{n+2}$. So, we obtain our result.

We now consider the case $u(c) = \frac{u-1-\alpha}{1-\sigma}$, $\forall 0 < \sigma \ne 1$.

1. If $\frac{1-\alpha}{n} > 1 + \gamma + \beta$, then $k^* = k_1^*$ is the solution of the equation
$$ g_1(k) \equiv \frac{(1-\alpha)Ak^{\alpha-1}}{1+(\alpha Ak^{\alpha-1})^{-\frac{1}{\alpha}}+\beta_1^{-\frac{1}{\alpha}}} = n. $$
Here, recall that $\beta_1 \equiv \beta + \gamma$, $\beta_2 \equiv \frac{\beta}{1+\gamma}$. We can verify that $g_1(k)$ is a strictly decreasing function and

$g_1(k_{GR}) = \frac{1-\alpha}{\alpha} \frac{n}{1+n^{-\frac{1}{\alpha}}\beta_1^{-\frac{1}{\alpha}}} < n = g_1(k_1^*)$ if and only if $\frac{1-\alpha}{\alpha} \left(1+n^{-\frac{1}{\alpha}}\beta_1^{-\frac{1}{\alpha}}\right) < 1$. Hence $k_1^* < k_{GR}$ and then the inter-temporal equilibrium is dynamically efficient if $\frac{1-\alpha}{\alpha} < 1 + n^{1-\frac{1}{\alpha}}\beta_1^{-\frac{1}{\alpha}}$. Inversely, the inter-temporal equilibrium is inefficient.

2. If $\frac{1-\alpha}{n} < 1 + \frac{\beta}{1+\gamma}$, then $k^* = k_2^*$ is the solution of the equation

$$ g_2(k) \equiv \frac{(1-\alpha)Ak^{\alpha-1}}{1+(\alpha Ak^{\alpha-1})^{-\frac{1}{\alpha}}+1\beta_2^{-\frac{1}{\alpha}}} = n. $$

We can verify that $g_2(k)$ is a strictly decreasing function and then we have $g_2(k_{GR}) = \frac{n}{1+n^{-\frac{1}{\alpha}}\beta_2^{-\frac{1}{\alpha}}} < n = g_2(k_2^*)$ if and only if $\frac{(1-\alpha)}{\alpha} \left(1+n^{-\frac{1}{\alpha}}\beta_2^{-\frac{1}{\alpha}}\right) < 1$. Hence $k_2^* < k_{GR}$ and then the inter-temporal

equilibrium is dynamically efficient if $\frac{1-\alpha}{\alpha} < 1 + n^{1-\frac{1}{\alpha}}\beta_2^{-\frac{1}{\alpha}}$ and inefficient if $\frac{1-\alpha}{\alpha} > 1 + n^{1-\frac{1}{\alpha}}\beta_2^{-\frac{1}{\alpha}}$.

3. If $1 + \frac{\beta}{1+\gamma} \le \frac{1-\alpha}{n} \le 1 + \gamma + \beta$, we can prove our result by using the same argument in the proof of part 3 of the case $u(c) = \ln(c)$. $\square$

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