2.1. Present-biased preferences and utility

2.1.1. Present bias

Suppose that the agent is risk averse and displays present bias in the form of quasi-hyperbolic discounting (Harris and Laibson, Reference Harris and Laibson2013). Time is divided into two periods: the current and the future. The agent exponentially discounts consumption flows in the current period with a discount rate $\rho \gt 0$ and further discounts future consumption flows with an additional discount factor $\beta \in (0,1)$ , which undermines the present value of consumption flows in future periods.

We model the agent as a sequence of temporal selves. Self $1$ is born at time $t_1$ . She regards $[t_{1},t_2)$ as the current period and treats $[t_{2},\infty )$ as the future periods, and in the current period, she controls the consumption-saving decisions. At time $t_2$ , self $2$ replaces self $1$ . She takes control of the consumption-saving choices in the new current period $[t_2,t_{3})$ and treats $[t_{3},\infty )$ as the new future periods. Thus, $T_n=t_{n+1}-t_n$ is the lifespan for self $n$ . Assume that the lifespan of each self is exponentially distributed with the constant parameter $\xi \gt 0$ . That is, the transition of selves is regulated by a Poisson process with a hazard rate $\xi$ . Repeating the process, we can obtain a sequence of selves $n\in \{1,2,\ldots \}$ . They live in a sequence of time intervals $\{[t_1,t_{2}),[t_2,t_{3}),\ldots \}$ , which are independent and identically distributed (i.i.d.).

Denote by $D_n(t,u)$ the quasi-hyperbolic discount function for self $n$ ; thus, we have:

(1) \begin{eqnarray} D_n(t,u)=\left \{ \begin{array}{l@{\quad}c} e^{-\rho (u-t)}, &{\textit{if }\ u\in [t_n,t_{n+1})}\\ \beta e^{-\rho (u-t)}, &{\textit{if }\ u\in [t_{n+1},\infty )} \end{array} \right . \end{eqnarray}

where $u\gt t$ and $t_n\lt t_{n+1}$ . For tractability, we focus on the limiting case when $\xi \rightarrow \infty$ , known as the instantaneous gratification (IG) model pioneered by Harris and Laibson (Reference Harris and Laibson2013). In this case, the above discount function (1) becomes:

(2) \begin{eqnarray} D(t,u)=\left \{ \begin{array}{l@{\quad}c} 1, &{\textit{if }\ u=t}\\ \beta e^{-\rho (u-t)}, &{\textit{if }\ u\gt t} \end{array} \right . \end{eqnarray}

A higher magnitude of present bias is measured by a smaller $\beta$ . If $\beta =1$ , equation (2) reduces to the traditional exponential discount function.

Models related to present bias require an assumption about the extent to which agents are aware of their future self-control problems (O’Donoghue and Rabin, Reference O’Donoghue and Rabin1999, Reference O’Donoghue and Rabin2001). In this paper, we focus on two types of present-biased agents: (i) completely sophisticated agents, where the current self correctly perceives her future selves’ present bias (Laibson, Reference Laibson1997), and (ii) fully naive agents, where the current self (falsely) believes that her future selves will act in a time-consistent manner (Akerlof, Reference Akerlof1991).

2.1.2. Utility

Suppose that an agent is born at age $0$ and dies at a random nonnegative age $\tau$ that is exponentially distributed with a constant mortality hazard rate $\lambda$ . Across agents, random deaths are statistically independent. To sustain a constant population, we assume that the newborn arrives at a constant rate $\lambda$ per unit of time. The agent accrues constant relative risk aversion utility over positive consumption flows $\{C_t\}_{t=0}^\tau$ :

(3) \begin{eqnarray} U(C)= \frac{C^{1-\gamma }}{1-\gamma },\quad{\gamma \gt 0\text{ and }\gamma \neq 1}, \end{eqnarray}

where $\gamma$ denotes the coefficient of relative risk aversion.

2.2. Labor income and household borrowing

In this section, we introduce two interrelated “regimes.” In a “normal” regime, the household decides her optimal consumption strategy and services her debt obligations. Once the household defaults on existing debt, she enters a “default” regime from which she cannot leave.

2.2.1. Normal regime

We model the labor income in the normal regime, $Y_t$ , as a diffusion process. All agents start life with identical labor earnings $Y_0\gt 0$ . The law of motion for labor income is given by:

(4) \begin{eqnarray} dY_t=\mu Y_tdt+\sigma Y_td\mathcal{B}_t,\quad Y_0\gt 0, \end{eqnarray}

where $\mathcal{B}_t$ is a standard Brownian motion, $\mu$ is an expected earnings growth rate, and $\sigma$ is a volatility of earnings.

Let $X_t$ be an agent’s wealth at time $t$ , and the initial wealth for all agents is equal to zero, that is, $X_0=0$ . With liquid wealth $X_t\gt 0$ , the agent can trade a risk-free financial asset that offers a constant risk-free rate of return $r$ and purchase an actuarially fair “reverse-life-insurance” contract that transfers end-of-life wealth $X_{\tau ^-}$ at time $\tau$ to an insurance company in exchange for flow payoffs $\lambda X_t\gt 0$ until death. Conversely, $X_t\lt 0$ implies that the agent borrows on a credit card, she can transfer end-of-life debt to an insurance company by paying flow expenditures $\lambda X_t\lt 0$ ; in addition, the agent should pay a borrowing wedge of $w^{r}\gt 0$ over the sum of the risk-free rate and the mortality hazard rate $r+\lambda$ . Accordingly, the wealth process is given by:

(5) \begin{eqnarray} dX_t=&\underbrace{\left [(r+\lambda +w^{r}\unicode{x1D7D9}_{\{X_{t-}\lt 0\}}) X_{t-}+Y_{t-}-C_{t-}\right ]}_{\text{saving rate, }G_{t-}}dt -X_{t-}d\mathcal{J}_t, \end{eqnarray}

where $\mathcal{J}$ is a pure jump process with a constant arrival rate of $\lambda$ and $\unicode{x1D7D9}_{\{X_{t-}\lt 0\}}$ is an indicator function that equals one when $X_{t-}\lt 0$ , zero otherwise. On the right-hand side of (5), the first term represents the saving rates $G_{t-}$ , and the second term captures a one-time transfer $X_{\tau ^-}$ from the agent to the insurance company at the stochastic death moment $\tau$ when $d\mathcal{J}_{\tau }=1$ .

2.2.2. Default regime

Suppose that an agent’s labor income deteriorates sufficiently due to an economic downturn. In that case, the agent reneges on her debt and is henceforth shunned from credit markets, forcing her savings (and debts) to equal zero. This feature reflects that a majority of households go through bankruptcy file Chapter 7 — in which case debtors forfeit nonexempt assets in exchange for a discharge of eligible debts (Dobbie and Song, Reference Dobbie and Song2015).

Let $\tau _d$ denote the default arrival time and $\widehat Y_t$ denote the labor income in the default regime. Following Bos et al. (Reference Bos, Breza and Liberman2018), we assume that at time $\tau _d$ , the labor income immediately drops from $Y_{\tau _d-}=\lim _{u\uparrow \tau _d-}Y_u$ , the pre-default income level, to $\widehat Y_{\tau _d}=\kappa Y_{\tau _d-}$ , where the constant $(1-\kappa )\in (0,1)$ measures the degree of income loss incurred by default, and then the agent permanently enters a default regime. In this regime, the agent can no longer borrow or save $(X_t=0)$ , and the earnings $\widehat Y_t$ follow the same process (4) as earnings in the normal regime:

(6) \begin{eqnarray} d\widehat Y_t=\mu \widehat Y_tdt+\sigma \widehat Y_td\mathcal{B}_t. \end{eqnarray}

This process starts at time $\tau _d$ with the value of $\widehat Y_{\tau _d}=\kappa Y_{\tau _d-}$ .

3.1. Normal regime

3.1.1. Valuation

Under sophisticated IG preferences, the agent correctly anticipates her future selves’ self-control problems. In this situation, the actual continuation-value function is given by:

(7) \begin{eqnarray} F^S(X_t,Y_t)=\mathbb{E}_t\left [\int ^{\tau \wedge \tau ^S_d}_0 e^{-\rho (u-t)}U\left (C^S_u\right )du\right ], \end{eqnarray}

where $C^S_t$ and $\tau ^S_d$ denote the consumption rate and default policy adopted by the sophisticated agent, respectively. Then, the actual current-value function is equal to:

(8) \begin{eqnarray} V^S(X_t,Y_t)=\beta F^S(X_t,Y_t). \end{eqnarray}

The intuition for equation (8) is as follows. The current self discounts the utility of all future selves by $\beta$ . Nevertheless, in continuous time, the current self lives only for a vanishingly short time interval ( $\xi \rightarrow \infty$ ); thus, the utility received by the current self has no measurable impact on the overall value function.

Under naive IG preferences, the household (incorrectly) believes that her future selves will behave according to standard exponential discounting. In this situation, the naive agent’s perceived continuation-value function satisfies:

(9) \begin{eqnarray} F^E(X_t,Y_t)=\mathbb{E}_t\left [\int ^{\tau \wedge \tau ^E_d}_0 e^{-\rho (u-t)}U\left (C^E_u\right )du\right ], \end{eqnarray}

where $C^E_t$ and $\tau ^E_d$ denote the consumption rate and default strategy adopted by the exponential agent, respectively. The perceived current-value function is

(10) \begin{eqnarray} V^E(X_t,Y_t)=\beta F^E(X_t,Y_t). \end{eqnarray}

Define $F^N(X,Y)$ and $C^N(X,Y)$ as the naive agent’s actual continuation-value function and consumption policy, respectively. Next, we show how to derive the optimal consumption policies in the presence of IG preferences.

3.1.2. Optimal consumption rules

Under a stationary MPE, the Bellman equation for the type- $j$ agent is given by:

(11) \begin{eqnarray} \begin{split} (\rho +\lambda )F^j(X,Y)=& \underbrace{U(C^j)}_{\text{utility}} +\underbrace{G^j(X,Y){F^j_X(X,Y)}}_{\text{evolution of }dX} \\[5pt]&+\underbrace{\mu Y{F^j_Y(X,Y)}+\frac{\sigma ^2}{2}Y^2{F^j_{YY}(X,Y)}}_{\text{evolution of }dY},\quad j\in \{S,N\}, \end{split} \end{eqnarray}

which shows that the instantaneous change in value due to discounting and mortality risk, $ (\rho +\lambda )F^j(X,Y)$ , must equal the current utility flow, $U(C^j)$ , plus the expected instantaneous change in the value function, $\left (\mathbb{E}\left (dF^j\right )/dt\right )$ .

The consumption optimality conditions under sophistication and naivete are given by, respectively:

(12) \begin{eqnarray} U^{\prime}(C^S)=\beta F^S_X(X,Y),\quad U^{\prime}(C^N)=\beta F^E_X(X,Y), \end{eqnarray}

where the marginal utility of consumption on the left-hand side, $U^{\prime}(C^S)$ or $U^{\prime}(C^N)$ , equals the marginal value of current liquid wealth on the right-hand side, $\beta F^S_X(X,Y)$ or $\beta F^E_X(X,Y)$ .

Two observations can be drawn from equation (12). First, present bias alters the consumption optimality condition by a multiplicative factor of $\beta$ . To see this, note that a standard exponential agent would determine consumption according to:Footnote ²

(13) \begin{eqnarray} U^{\prime}(C^E)=F^E_X(X,Y). \end{eqnarray}

Evidently, under IG preferences, the marginal value of current liquid wealth is discounted by $\beta$ , which arises because wealth is consumed by future selves whose utility is discounted by $\beta$ . Second, the optimality condition under sophistication uses the value function of the present-biased agent $F^S(X,Y)$ . In contrast, the condition under naivete uses the value function of the exponential-discounting agent $F^E(X,F)$ . This difference reflects the naive agent’s (incorrect) belief.

3.2. Default regime

In the default regime, wealth (or debt) is zero, and the household can neither borrow nor save. Hence, consumption equals labor income, and wealth is not an argument of the value function. Under this circumstance, behavioral bias cannot affect consumption rules, and thus, all agents behave as exponential discounters, whose value function is denoted by $\widehat F^E(\widehat Y)$ . This function satisfies the following differential equation:

(14) \begin{eqnarray} \begin{split} (\rho +\lambda ){\widehat F^E(\widehat Y)}=U(\widehat C^E)+\mu \widehat Y\widehat F^{E^{\prime}}(\widehat Y) +\frac{\sigma ^2}{2}\widehat Y^2\widehat F^{E^{\prime\prime}}(\widehat Y), \end{split} \end{eqnarray}

where $\widehat C^E=\widehat Y$ . The first term on the right-hand side of (14) gives the utility received by the household in the default regime. The second and third terms capture the impact of drift and volatility of income on the household’s value function $\widehat F^E(\widehat Y)$ , respectively.

3.2.1. Debt capacity $\underline{X}_t$

In the two regimes, the value functions $F^j(X,Y)$ and $\widehat F^E(\widehat Y)$ are connected by the transition from the normal to the default regime. To see this, note that upon default, the labor earnings decrease from $Y_{t-}$ to $Y_t=\kappa Y_{t-}$ , and the wealth jumps from $X_{t-}\lt 0$ to $X_t=0$ . As a result, the value of wealth at the default time, which we denote by $\underline{X}_t$ and makes the household indifferent between repaying her debt and defaulting, satisfies the following value-matching condition:

(15) \begin{eqnarray} F^j(\underline{X}_t,Y_t)=\widehat F^E(\kappa Y_t),\quad j\in \{S,N\}. \end{eqnarray}

We refer to $\underline{X}_t$ as the household’s debt capacity since it is the maximum amount of debt the household can borrow without triggering default in equilibrium. For tractability, we assume that when the household’s income is less than the interest payments to debt, that is, $Y_t\lt -(r+\lambda +w^r)X_t$ , the household defaults. Therefore, the debt capacity $\underline{X}_t$ is a function of $Y_t$ and equals:

(16) \begin{eqnarray} \underline{X}_t=-\frac{Y_t}{r+\lambda +w^r}. \end{eqnarray}

3.3. Exploiting the homogeneity property

To solve the Bellman equation (11), a key step in our derivation is to establish that the value function in the normal regime, $F^j(X,Y)$ , takes the following form:

(17) \begin{eqnarray} F^j(X,Y)=\frac{\left (b^jP^j(X,Y)\right )^{1-\gamma }}{1-\gamma },\quad j\in \{S,N\}, \end{eqnarray}

where the constant $b^j$ is defined in Appendix A. The function $P^j(X,Y)$ is interpreted as the certainty-equivalent wealth (CEW), which is the total wealth that makes the household indifferent between the status quo (with liquid wealth $X$ and labor income $Y$ ) and having a wealth level $P(X,Y)$ with no labor earnings:

(18) \begin{eqnarray} F^j(X,Y)=F^j(P^j(X,Y),0). \end{eqnarray}

Similarly, we show that the value function in the default regime, $\widehat F^E(\widehat Y)$ , is given by:

(19) \begin{eqnarray} \widehat F^E(\widehat Y)=\frac{\left (b^E\widehat P^E(\widehat Y)\right )^{1-\gamma }}{1-\gamma }, \end{eqnarray}

where the constant $b^E$ is given in Appendix A and $\widehat P^E(\widehat Y)$ is the CEW in the default regime. By linking the agent’s value function to her CEW, we can transform the agent’s payoff from the value function to the CEW.

Applying the homogeneity property, we can analyze the (scaled) CEW $p^j(x)=P^j(X,Y)/Y$ and the consumption-income ratio $c^j(x)=C^j(X,Y)/Y$ in terms of a single state variable, $x=X/Y$ , which denotes the wealth-income ratio. Equation (16) shows that the (scaled) debt capacity $\underline{x}=\underline{X}/Y=-1/(r+\lambda +w^r)$ is a constant and independent of the present bias parameter $\beta$ . By using Ito’s lemma, the law of motion for the wealth-income ratio process is given by:

(20) \begin{eqnarray} dx^j_t=\underbrace{\left [1+\left (r+\lambda +w^{r}\unicode{x1D7D9}_{\left \{x^j_{t-}\lt 0\right \}} -\mu +\sigma ^2\right )x^j_{t-}-c^j_{t-}\right ]}_{\text{saving rate, }g^j_{t-}}dt -\sigma x^j_{t-}d\mathcal{B}_t -x^j_{t-}d\mathcal{J}_t. \end{eqnarray}

The following proposition summarizes the main results in this section.

Proposition 1. (Present bias and overconsumption) The optimal consumption rules for sophisticated and naive agents are given by, respectively:

(21) \begin{eqnarray} c^S(x)=\frac{\gamma }{\gamma +\beta -1}c^E(x),\quad c^N(x)={\beta }^{-\frac{1}{\gamma }}c^E(x), \end{eqnarray}

where $c^{E}(x)=\left (b^E\right )^{1-1/\gamma }p^{E}(x)\left (p^{E^{\prime}}(x)\right )^{-1/\gamma }$ denotes the consumption-income ratio adopted by the exponential-discounting agent and the scaled CEW $p^E(x)$ in the normal regime ( $x\geq \underline{x}$ ) solves the following ordinary differential equation (ODE):

(22) \begin{eqnarray} \begin{split} 0=&\left [\frac{\gamma \left (b^Ep^{E^{\prime}}(x)\right )^{1-1/\gamma }-(\rho +\lambda )}{1-\gamma }+\mu -\frac{\gamma \sigma ^2}{2}\right ]p^E(x) +p^{E^{\prime}}(x) \\&+\left (r+\lambda +w^{r}\unicode{x1D7D9}_{\{x\lt 0\}}-\mu +\gamma \sigma ^2\right )xp^{E^{\prime}}(x) +\frac{\sigma ^2x^2}{2}\left [p^{E^{\prime\prime}}(x)-\gamma \frac{\left (p^{E^{\prime}}(x)\right )^2}{p^{E}(x)}\right ]. \end{split} \end{eqnarray}

The boundary conditions for (22) are

(23) \begin{eqnarray} p^E(\underline{x})=\kappa \hat p^E, \lim _{x\rightarrow \infty }p^{E}(x)=x+q, \end{eqnarray}

where the constant $q={1}/\left ({r+\lambda -\mu }\right )$ and the scaled value in the default regime, $\hat p^E=\widehat P^E(\widehat Y)/\widehat Y$ , is:

(24) \begin{eqnarray} \hat p^E=\left (b^E\right )^{-1}\left [\rho +\lambda -(1-\gamma ) \left (\mu -\frac{\gamma \sigma ^2}{2}\right )\right ]^{\frac{1}{\gamma -1}}. \end{eqnarray}

To ensure that $\hat p^E$ is nonnegative, we impose the following condition:

(25) \begin{eqnarray} \rho +\lambda -(1-\gamma ) \left (\mu -\frac{\gamma \sigma ^2}{2}\right )\gt 0. \end{eqnarray}

The economic interpretation of Proposition1 is as follows. First, equation (21) implies that relative to the exponential-discounting agent, sophistication (naivete) induces overconsumption by a multiplicative factor of $\frac{\gamma }{\gamma +\beta -1}\gt 1$ ( ${\beta }^{-\frac{1}{\gamma }}\gt 1$ ), since $\beta \in (0,1)$ . This result is intuitive: Under IG preferences, the desire for instantaneous gratification drives the agent to overconsume. Second, the first condition in equation (23) follows from the value-matching condition (15) at the default threshold. The second condition implies that when the wealth $X$ approaches infinity, the CEW $\lim _{X\rightarrow \infty }P^E(X,Y)=X+qY$ is simply the sum of wealth $X$ and the “human” wealth $qY$ , which is proportional to contemporaneous income. This is because the agent can use holdings of the single risk-free asset completely to buffer all idiosyncratic labor-earning shocks.

4.1. Stationary distribution

4.1.1. Stationary probability density function

Let $X^j_t$ and $Y^j_t$ denote the type- $j$ agent’s wealth and labor earnings at time $t$ , respectively. To calculate the cross-sectional stationary probability density of labor earnings $\phi ^j_{Y}(Y)$ and wealth $\phi ^j_{X}(X)$ , we start by solving the cross-sectional joint distribution of wealth and labor earnings $\phi ^j_{XY}(X,Y)$ . After obtaining $\phi ^j_{XY}(X,Y)$ , we can compute the cross-sectional stationary distributions of wealth and income by integrating over $Y$ and $X$ , respectively:

(26) \begin{eqnarray} \phi ^j_X(X)=\int _0^{+\infty }\phi ^j_{XY}(X,Y)dY,\quad \phi ^j_Y(Y)=\int _{-\infty }^{+\infty }\phi ^j_{XY}(X,Y)dX. \end{eqnarray}

Define $X_t$ and $Y_t$ as the aggregate wealth and labor income in the economy, respectively. After obtaining the probability density function for each group, we can compute the expectation of aggregate wealth $\mathbb{E}(X)$ and aggregate earnings $\mathbb{E}(Y)$ according to:

(27) \begin{eqnarray} \mathbb{E}(X)=\sum _{j=\{N,S\}}\eta ^j\mathbb{E}(X^j),\quad \mathbb{E}(Y)=\sum _{j=\{N,S\}}\eta ^j\mathbb{E}(Y^j), \end{eqnarray}

where $\mathbb{E}(X^j)=\int _{-\infty }^{+\infty } X\phi ^j_X(X)dX$ and $\mathbb{E}(Y^j)=\int _{0}^{+\infty } Y\phi ^j_Y(Y)dY$ are the average wealth and income for type- $j$ group of agents, respectively.

4.1.3. Wealth inequality

To numerically illustrate the impacts of preference heterogeneity on wealth inequality, we introduce two widely used measures of inequality. First, the Lorenz curve of wealth $\mathcal{L}_X(z)$ measures the proportion of aggregate wealth captured by the bottom $z$ percent of people, which is given by:

(28) \begin{eqnarray} \mathcal{L}_X(z)=\frac{\int _{0}^z\Phi _X^{-1}(z)dz}{\int _0^1\Phi _X^{-1}(z)dz}, \end{eqnarray}

where $\Phi _X^{-1}({\cdot})$ is the inverse of the cumulative distribution function (CDF), $\Phi _X({\cdot})$ . Second, previous literature also applies the Gini coefficient to measure the magnitude of wealth inequality, which equals:

(29) \begin{eqnarray} \Gamma _X=2\int _0^1(z-\mathcal{L}_X(z))dz. \end{eqnarray}

4.2. Equilibrium interest rate

Assume that a representative firm operates a production function $F(K,L)=AK^\alpha L^{1-\alpha }$ , where $\alpha \in (0,1)$ , $A\gt 0$ represents productivity, $K$ denotes the aggregate capital stock that depreciates at a constant rate $\delta \gt 0$ , and $L$ is the aggregate labor stock. The firm rents capital and labor in competitive markets. In the stationary equilibrium, capital demand equals capital supply:

(30) \begin{eqnarray} K=\mathbb{E}(X). \end{eqnarray}

Suppose each agent inelastically supplies $H\gt 0$ h of labor. In equilibrium, labor demand equals labor supply $L=H$ . Define $w=\mathbb{E}(Y)/H$ as an average wage rate across all agents. Since aggregate labor cost $wL$ equals aggregate labor earnings $wH$ , the law of large numbers implies

(31) \begin{eqnarray} wL=wH=\mathbb{E}(Y). \end{eqnarray}

Now, the steady-state equilibrium interest rate $r$ can be stated as follows.

Proposition 2. (Equilibrium interest rate with preference heterogeneity). In a competitive equilibrium, the firm’s optimization problem indicates that the equilibrium interest rate $r$ is given by:

(32) \begin{eqnarray} r &=& F_K(K,L)-\delta = A\alpha \left (\frac{K}{L}\right )^{\alpha -1}-\delta = \frac{\alpha }{1-\alpha }\frac{wH}{K}-\delta = \frac{\alpha }{1-\alpha }\frac{\mathbb{E}(Y)}{\mathbb{E}(X)}-\delta, \end{eqnarray}

where the third equality follows from the expression for the equilibrium average wage rate:

(33) \begin{eqnarray} w &=& F_L(K,L)=A(1-\alpha )\left (\frac{K}{L}\right )^\alpha = A(1-\alpha )\left (\frac{\mathbb{E}(X)}{H}\right )^\alpha . \end{eqnarray}

Given the interest rate $r$ , the equilibrium wage rate $w$ , initial labor earnings $Y_0$ , and aggregate capital $K$ are jointly determined by equations (31) through (33).Footnote ³

Proposition3 shows that a higher expected wealth $\mathbb{E}(X)$ incurs a lower equilibrium interest rate. To understand this result, note that the equilibrium interest rate is determined by the representative firm’s demand for capital and household savings. A higher $\mathbb{E}(X)$ implies that household saving increases, thus raising the capital supply. To strengthen the firm’s demand for capital and clear the asset market, the equilibrium interest rate $r$ must decrease. By contrast, a higher expected labor income $\mathbb{E}(Y)$ leads to a higher equilibrium interest rate. This is because a higher $\mathbb{E}(Y)$ strengthens the agent’s incentive to consume; thus, the representative firm will demand more capital for production. As a result, the equilibrium interest rate increases to encourage savings.

5.1 Calibration

First, we follow Prescott (Reference Prescott1986) and Cooley and Prescott (Reference Cooley and Prescott1995) and set the capital share of income $\alpha$ to 0.36. We set an annual depreciation rate of $\delta$ to $1.6\%$ to match an aggregate capital-output ratio $K/F(K,L)$ to $4.5$ as reported in Piketty (Reference Piketty2014). We set the productivity parameter $A$ to $0.774$ , so that the wage rate $w$ for an agent with the average labor efficiency equals unity. We set the coefficient of risk aversion $\gamma$ to $1.5$ , as in Attanasio et al. (Reference Attanasio, Banks, Meghir and Weber1999) and Gourinchas and Parker (Reference Gourinchas and Parker2002). Following Castañeda, Díaz-Giménez, and Ríos- Rull (Reference Castañeda, Díaz-Giménez and Ríos-Rull2003), we set the hazard parameter $\lambda =0.0167$ to target an agent’s expected lifetime at $1/\lambda =60$ years. We assume that the present bias parameter $\beta$ ranges from $0.5$ to $1$ . This is consistent with the literature, such as Laibson (Reference Laibson1997), Harris and Laibson (Reference Harris and Laibson2013), and Laibson et al.  (Reference Laibson, Lee, Maxted, Repetto and Tobacman2024).

Second, we calibrate the expected income growth $\mu$ and labor earnings growth volatility $\sigma$ by targeting a pair of quantities: a mean labor income of $\$106,250$ (see 2019 Survey of Consumer Finances, SCFFootnote ⁴) and a Gini coefficient for the cross-sectional labor earnings of $0.63$ (Castañeda, Díaz–Giménez, and Ríos– Rull, Reference Castañeda, Díaz-Giménez and Ríos-Rull2003). Similarly, De Nardi (Reference De Nardi2004) targets the Gini coefficient of labor earnings by calibrating income growth volatility. The calibrated values are $\mu =1.28\%$ and $\sigma =9.76\%$ . These values are also in line with estimates reported by Meghir and Pistaferri (Reference Meghir and Pistaferri2011).

Finally, we want an average equilibrium wealth of $\$746,820$ as in the 2019 SCF, which in light of equation (32) leads to an equilibrium interest rate $r$ equals $6.4\%$ per annum. With respect to household borrowing and default, we set the borrowing wedge $w^r$ to $12.46\%$ and the income loss parameter $\kappa$ to $0.3$ .

5.2 Optimal consumption policies and present bias

Figure 1.

Effects of present bias on consumption-saving rules. This picture depicts the optimal consumption and saving rules. Solid blue lines represent the exponential-discounting case, red dashed lines represent the sophisticated case ( $\beta ^S=0.783$ ), and yellow dotted lines represent the naive case ( $\beta ^N=0.783$ ). Under naive (sophisticated) beliefs, the steady-state wealth-income ratio is $\tilde x^N=-0.085$ ( $\tilde x^S=-0.071$ ). Other parameter values are reported in Table 1.

First, Panel A of Figure 1 plots the optimal consumption-income ratio against the wealth-income ratio $x$ .Footnote ⁵ When the credit limit is distant (i.e., $x\geq 0$ ), the consumption-income ratio $c(x)$ is increasing and concave in $x$ . This is due to the optimal response to buffer shocks and avoiding reliance on costly credit. Specifically, the instantaneous MPC $c^{\prime}(x)$ depicted in Panel B of Figure 1 is given by:Footnote ⁶

(34) \begin{eqnarray} c^{\prime}(x)=m\left (p^{\prime}(x)\right )^{-\frac{1}{\gamma }}\frac{\gamma _e(x)}{\gamma },\text{ where }\gamma _{e}(x)=-\frac{F_{XX}}{F_{X}}P(X,Y)=\gamma p^{\prime}(x)-\frac{p(x)p^{\prime\prime}(x)}{p^{\prime}(x)}. \end{eqnarray}

Equation (34) shows that the instantaneous MPC $c^{\prime}(x)$ is jointly determined by (i) the complete-market MPC $m$ , a constant given in Appendix A, (ii) the endogenous risk aversion $\gamma _e(x)\gt 0$ , and (iii) the marginal value of liquidity $p^{\prime}(x)\gt 0$ ; a higher level of endogenous risk aversion or a lower marginal value of liquidity induce a higher instantaneous MPC $c^{\prime}(x)$ . As the wealth-income ratio $x$ increases, self-insurance against income shocks becomes more effective so that $\gamma _e(x)$ and $p^{\prime}(x)$ jointly decline. Since the impact of $\gamma _e(x)$ plays a dominant role, consumption becomes less responsive to changes in wealth, explaining that $c^{\prime}(x)$ is decreasing in $x$ . In the limit $x\rightarrow \infty$ , self-insurance is sufficiently effective at achieving the complete-market risk sharing; hence, $\lim _{x\rightarrow +\infty }\gamma _{e}(x)=\gamma$ and $\lim _{x\rightarrow +\infty }p^{\prime}(x)=1$ .

Table 1.

Parameter values

Once the agent enters the debt region $x\lt 0$ , the instantaneous MPC spikes due to additional borrowing costs. In this situation, the precautionary motive is strong because reducing consumption can avoid costly borrowing in the near future. However, if the household accumulates more debt (i.e., a smaller $x$ ), the effect of the default option takes place. When the default option is sufficiently in-the-money, the household prioritizes consumption smoothing and responds to negative shocks by default. This reflects the diversification benefits of risky debt because default allows households to reduce leverage. Our result extends the standard concave consumption function to a more general incomplete-market environment with the consumer default option. This option effect also leads to a convex consumption function (see Panel B of Figure 1).

Second, consistent with equation (21), Panel A of Figure 1 confirms that the present-biased agent overconsumes more than her exponential-discounting counterpart, and the degree of overconsumption is proportional to $c^{E}(x)$ ; thus, it might influence the household with high values of $x$ more than her counterparts with low values of $x$ . As shown in Panel C of Figure 1, a critical consequence of overconsumption is the negative saving rate $g^j(x)\lt 0$ ( $j\in \{S,N\}$ ). Therefore, exponential-discounting households continue saving since they always have positive saving rates; by contrast, present-biased households dissave so that they are expected to move toward the steady-state wealth-income ratio $\tilde x^j$ , at which $g^j(\tilde x^j)=0$ . This negative steady-state wealth-income ratio $\tilde x^j\lt 0$ indicates that severely present-biased households will eventually concentrate in the debt region. This result can also be seen from the following Euler equation for the optimal consumption rule:Footnote ⁷

(35) \begin{eqnarray} -\frac{\mathbb{E}(dU(C))}{U(C)}=\left \{ \begin{array}{l@{\quad}c} r+w^r\unicode{x1D7D9}_{\{x\lt 0\}}-\rho \overbrace{-\gamma \left (1-\beta ^{1/\gamma }\right )\frac{\partial c^j(x)}{\partial x}}^{\text{ dis-saving motive, }-\Lambda ^j\lt 0}, &{\textit{if } j=N}\\ r+w^r\unicode{x1D7D9}_{\{x\lt 0\}}-\rho \underbrace{-(1-\beta )\frac{\partial c^j(x)}{\partial x}}_{\text{ dis-saving motive, }-\Lambda ^j\lt 0}, &{\textit{if } j=S} \end{array} \right . \end{eqnarray}

Compared to the case with exponential discounting, equation (35) shows that IG preferences induce two competing forces in saving incentives. First, present bias induces a novel term $-\Lambda ^j$ , which is negative since the instantaneous MPC $c^{j^{\prime}}(x)\gt 0$ (see Panel B of Figure 1). This term captures the additional consumption motive or, equivalently, the dissaving motive. Second, lower wealth accumulation incurred by overconsumption reduces the aggregate capital supply in the economy, driving up the equilibrium interest rate (see equation (32)), which in turn encourages saving. This section mainly focuses on the first channel, and Section 6 demonstrates the crucial role of equilibrium interest rate changes in welfare analysis.

Note that sophistication partially offsets the overconsumption and reduces the level of the agent’s indebtedness, since the steady-state wealth-income ratio increases from $\tilde x^N=-0.085$ to $\tilde x^S=-0.071$ . The economic intuition for this result is as follows.Footnote ⁸ The sophisticated agent correctly anticipates that her future selves’ overconsumption incentives will lead to poverty and costly borrowing, and thus, she acts conservatively by saving more to alleviate future financial distress. By contrast, the naive agent falsely believes that her future selves will behave as exponential discounters. As a consequence, underestimating the probability of future financial distress undercuts her incentives to save.

Finally, we draw a comparison between present bias and patience heterogeneity. First, equation (35) indicates that present bias generates within-individual impatience heterogeneity, and this state-dependent discount rate is:

(36) \begin{eqnarray} \rho ^j(x)=\rho +\Lambda ^j(x), \end{eqnarray}

where the instantaneous MPC $c^{j^{\prime}}(x)$ plays a vital role. When the distance to default is sufficiently large, lower liquid wealth (i.e., a smaller $x$ ) always exacerbates the financial constraint and raises $c^{j^{\prime}}(x)$ , thus incurring a larger $\rho ^{j}(x)$ . However, if the agent is near the default threshold, the default-option effect weakens the precautionary motive, and thus, the lower the wealth-income ratio $x$ , the more patient households are. This heterogeneity is endogenously created by the state-dependent effect of the default option and financial constraint. In contrast, patience heterogeneity is exogenously assumed. Second, Ramsey’s conjecture and the related literature (Ramsey, Reference Ramsey1928; Becker, Reference Becker1980; Bewley, Reference Bewley1982; Mitra and Sorger, Reference Mitra and Sorger2013) indicate a strict positive correlation between patience and capital ownership. Nevertheless, wealthy households in our model (manifested by high values of $X_t$ ) do not necessarily exhibit high levels of patience. To see this, consider the situation in which wealthy households also have high labor income $Y_{t}$ , which decreases the wealth-income ratio $x_{t}=\frac{X_{t}}{Y_{t}}$ , thus leading to a high instantaneous MPC $c^j(x_t)$ and a high subjective discount rate $\rho ^j(x_t)$ .

5.3. Quarterly MPC and wealth distribution

Column 1 of Table 2 reports the average quarterly MPC and wealth statistics from data. Following Achdou et al. (Reference Achdou, Han, Lasry, Lions and Moll2022), we define the quarterly MPC $QM^j(X)$ as the change in cumulative consumption over a quarter $\Delta t$ following a liquidity injection of size $\chi$ :

(37) \begin{eqnarray} QM^j(X)=\frac{C^j_{\Delta t}(X+\chi, Y)-C^j_{\Delta t}(X,Y)}{\chi },j\in \{E,S,N\}, \end{eqnarray}

where $C^j_{\Delta t}(X,Y)=\mathbb{E}\left [\int _u^{u+\Delta t}C^j(X_t,Y_t)\mid X_u=X,Y_u=Y\right ]$ is cumulative consumption over a quarter $\Delta t$ . In this paper, we focus on the quarterly MPC out of $\$500$ , that is, $\chi =\$500$ . The baseline model in Column 2 of Table 2 hits the mean wealth target with the chosen discount rate $\rho =4.99\%$ . Nevertheless, compared to the data, the distribution of aggregate wealth in the baseline model has a thinner right tail, manifested by a smaller Gini coefficient of wealth $\Gamma _X=0.78\lt 0.85$ , and the share of wealth-poor households is smaller. For example, exponential-discounting households with wealth less than $\$1,000$ only account for $0.86\%$ , which is much smaller than the $15.1\%$ in the data. Moreover, the average quarterly MPC is approximately $7.71\%$ in the baseline model, which is much smaller than the empirical estimate ( $16\%$ ). Intuitively, when default is distant (i.e., $x\gt 0$ ), a higher $x$ weakens households’ consumption-smoothing motives, implying the smaller MPC (see Panel B of Figure 1). In other words, having insufficient wealth-poor households in the model understates the aggregate quarterly MPC.

Table 2.

Wealth distribution and quarterly MPC. Column 1 contains some key moments of the wealth distribution computed from the 2019 SCF. Column 2: baseline model with exponential discounters, and the discount factor is chosen to match the mean wealth. Column 3: model with ex ante present bias homogeneity ( $\beta ^S=0.783$ ). Column 4: model with ex ante heterogeneity in present-biased preferences. The equilibrium interest rate is $r=6.4\%$ . Other parameter values are reported in Table 1

Next, we consider the economy in which all households hold sophisticated beliefs with the present bias parameter $\beta ^S=0.783$ . As shown in Column 3 of Table 2, the average quarterly MPC is approximately $7.67\%$ , which is still substantially lower than the $16\%$ in the data. The intuition is the following. As emphasized in Kaplan and Violante (Reference Kaplan and Violante2022), although present bias can generate more wealth-poor households due to overconsumption incentives, this effect is substantially attenuated by a lower subjective discount rate $\rho =3.39\%\lt 4.99\%$ when we match the mean wealth, since reducing $\rho$ in turn strengthens households’ saving motives. Panel B of Table 2 shows that after incorporating homogeneous present bias, the share of wealth-poor households ( $X\lt \$1,000$ ) only increases by $0.12\%$ (i.e., from $0.86\%$ to $0.98\%$ ).

Finally, we consider ex ante heterogeneity in present-biased preferences: one group is sophisticated and exhibits mild present bias ( $\beta ^S=0.98$ ), while the other group holds naive beliefs and exhibits severe present bias ( $\beta ^N=0.783$ ). The share of naive agents is $\eta ^N=13.6\%$ , and they are all HtM; the share of sophisticated agents is $\eta ^S=86.4\%$ , of which $0.6\%$ are HtM agents. Therefore, the proportion of HtM agents in this economy is $14.2\%$ , as reported in the 2019 SCF and Kaplan and Violante (Reference Kaplan and Violante2022). By comparing Columns 2 and 4, we can see that present bias heterogeneity generates a larger quarterly MPC of $15.68\%\gt 7.71\%$ than the baseline model. To understand this result, the following decomposition categorizes the changes in the quarterly MPC into three aspects:

(38) \begin{eqnarray} \begin{split} QM=&\underbrace{QM^E}_{\text{Baseline MPC}} +\underbrace{\int \left [QM(X)-QM^E(X)\right ]d\Phi ^E(X)}_{\text{Consumption Function}} +\underbrace{\int QM^E(X)\left [d\Phi _X(X)-d\Phi ^E(X)\right ]}_{\text{Distribution}} \\&+\underbrace{\int \left [QM(X)-QM^E(X)\right ]\left [d\Phi _X(X)-d\Phi ^E(X)\right ]}_{\text{Interaction}}, \end{split} \end{eqnarray}

where $QM$ and $\Phi _X(X)$ ( $QM^E$ and $\Phi ^E(X)$ ) denote the average quarterly MPC and wealth distribution in the heterogeneous-preferences model (baseline model), respectively. In the above equation (38), the component labeled “Consumption Function” captures the difference in average MPC that arises because the consumption functions are different in the two models. The component labeled “Distribution” characterizes the differences in average MPC that arise because the stationary distributions of the two models put mass in different parts of the wealth status. The last component labeled “Interaction” arises because the above two effects may reinforce or offset each other. Panel A of Table 2 shows that the larger MPC arises mainly because of the distributional channel. Panel B of Table 2 confirms this intuition since the model with heterogeneous present-biased households has a stationary distribution with a higher share of wealth-poor households: $14.32\%$ of households whose wealth is less than $\$1,000$ , compared with $0.86\%$ in the baseline model. As noted by Parker (Reference Parker2017), HtM status might be situational or reflective of persistent household traits (i.e., behavioral characteristics). Our model with present bias implies a relatively high persistence of HtM status, which is consistent with the empirical evidence from Aguiar et al. (Reference Aguiar, Bils and Boar2024).

Mounting evidence has shown that ex ante heterogeneity in discount rates is a crucial driving force for wealth inequality (Epper et al. Reference Epper, Fehr, Fehr-Duda, Kreiner, Lassen, Leth-Petersen and Rasmussen2020; Sargent et al. Reference Sargent, Wang and Yang2021).Footnote ⁹ We emphasize two main advantages of present bias heterogeneity. First, present-biased preferences help match the median debt-income ratio of $-0.085$ and the average quarterly MPC of $16\%$ simultaneously,Footnote ¹⁰ while high degrees of impatience can hardly do both. Specifically, if we match the median level of the debt-income ratio in the model with heterogeneous discount rates, the quarterly MPC for highly impatient households drops to $48.71\%$ , implying that the average MPC drops to $13.27\%$ . Second, since wealth-poor households account for higher quarterly MPCs, Kaplan and Violante (Reference Kaplan and Violante2022) emphasize that the one-asset model with preference heterogeneity in discount rates or elasticity of intertemporal substitution generally encounters a tradeoff between a reasonable MPC and an excessively polarized wealth distribution. They term the latter shortfall the “missing middle” problem: overstating the households who are not quite poor enough to be HtM but still have very little wealth (i.e., underestimating the median wealth level). We show that heterogeneity in present bias addresses the above problem because approximately $50\%$ of households hold wealth lower than $\$100,000$ as in the data (see Column 4 of Table 2).

5.4. Targeted fiscal stimulus

As noted in the previous section 5.2, under the premise of costly borrowing, households’ MPC exhibits a hump shape with respect to the wealth-income ratio $x$ . This feature has important implications for the design of fiscal stimulus policies. To this end, we use our model with ex ante present bias heterogeneity discussed above to study the efficiency of targeted fiscal stimulus.

Table 3 reports the aggregate one-year consumption response to budget-equivalent fiscal stimulus policies that target different fractions of the wealth distribution. At one extreme, all households are given a stimulus payment of $\$500$ (see the last line in Table 3), while at the other extreme, the bottom $2\%$ of households in the wealth distribution are given a stimulus payment of $\$25,000$ (see the first line in Table 3). At either extreme, the stimulus is less effective because only approximately $10\%-15\%$ of stimulus payments are consumed within the year of disbursal. Why do wealth-poor households consume conservatively when they are targeted by stimuli? A larger size of windfall decreases the MPC, and this size effect is in line with Fagereng et al. (Reference Fagereng, Holm and Natvik2021). We observe that the consumption response gradually rises as the stimulus target moves from all households to the bottom $50\%$ of the wealth distribution. At the optimum, when $\$1,250$ is distributed to the households in the bottom $50\%$ of the wealth distribution, we observe that approximately $41\%$ of the stimulus payments is consumed within one year. This result indicates that fiscal stimulus can produce a much larger consumption response when it is targeted toward households in the bottom half of wealth distribution. In contrast, during the Great Recession, under the Economic Stimulus Act of 2008, the US government gave tax rebates to approximately $80\%-85\%$ of households, with an average stimulus payment of $\$600-\$1,200$ . More recently, the Coronavirus Aid, Relief, and Economic Security Act of 2020 tilted payment toward unemployed workers (Carroll et al. Reference Carroll, Crawley, Slacalek, Tokuoka and White2020).

Table 3.

Wealth-targeted stimulus payments. Column 1 shows the bottom percentage of wealth distribution targeted. Column 2 represents the corresponding fiscal stimulus payments. Column 3 displays the aggregate one-year consumption response to fiscal stimulus

6.1. Welfare analysis of present bias

Figure 3.

Short-run welfare changes from removing present bias. The left panel displays the short-run impacts of removing present bias for naive households. The right panel depicts the welfare changes of removing present bias for sophisticated households ( $\beta ^S=0.98$ ). In the left panel, the steady-state wealth-income ratios are $\tilde x_{\beta ^N=0.783}=-0.11$ and $\tilde x_{\beta ^N=0.88}=2.36$ . The interest rate is fixed at $r=6.2\%$ . Other parameter values are provided in Table 1.

Figure 3 depicts the short-run welfare changes (i.e., with a fixed interest rate $r$ ) from removing present bias $\omega ^j_{PB}(x)$ against the wealth-income ratio $x$ , where $j\in \{S,N\}$ . Two observations can be drawn from the figure. First, when the interest rate is fixed, removing present bias is always beneficial for both naifs and sophisticates, since $\omega ^j_{PB}(x)\gt 0$ holds for any given $x$ . Second, the welfare gains $\omega ^j_{PB}(x)$ increase with the wealth-income ratio $x$ . To understand this result, note that equation (21) shows that the higher $x$ is, the higher the level of overconsumption induced by present bias. As a result, present-biased households with a larger wealth-income ratio can benefit more from restraining stronger overconsumption incentives. These results are robust with varying values of $\beta ^N$ .

We now proceed to explore the long-run impacts of present bias removal. Figure 4 depicts the CDF of long-run welfare changes in the naive and sophisticated cases $\Phi ^j(\Delta x^j_{PB})$ .

Figure 4.

Long-run welfare changes from removing present bias. The left panel displays the long-run welfare changes of removing present bias for naive households, and the right panel displays the long-run welfare changes of removing present bias for sophisticated households ( $\beta ^S=0.98$ ). In the left panel, the steady-state wealth-income ratios are $\tilde x_{\beta ^N=0.783}=-0.085$ and $\tilde x_{\beta ^N=0.88}=4.24$ . Other parameter values are provided in Table 1.

Consider first the naive case. Panel A of Figure 4 states that when the degree of present bias is relatively small, removing naive present bias even results in welfare costs, as manifested by $\Delta x^N_{PB}\lt 0$ . This result can be explained as the interaction of the following two competing effects. On the one hand, removing households’ present bias directly strengthens their incentives to save. On the other hand, the resulting stronger saving motives raise the capital supply in the economy; to stimulate the firm’s demand for capital and clear the asset market, the equilibrium interest rate $r$ must decrease. For instance, with $\beta ^N=0.783$ , removing naive present bias decreases the equilibrium interest rate to $6.2\%$ from $6.4\%$ . The lower interest rate in turn makes saving less appealing. With severe present bias, all households’ overconsumption incentives are sufficiently strong that they are indebted; in this situation, the additional saving motive dominates, and removing naive present bias is always beneficial (see the solid blue line in Panel A of Figure 4). If, in contrast, with mild present bias, most naive households have positive saving rates, the lower interest rate induced by the interest rate channel undermines their saving incentives, thus leading to welfare costs (see the red dashed and yellow dotted lines in Panel A of Figure 4). With $\beta ^N=0.88$ , approximately $18.75\%$ of naive households suffer from welfare losses after removing their present bias.

Consider next the sophisticated case. As shown in Panel B of Figure 4, removing sophisticated present bias also generates a negative externality on households’ welfare. The economic intuition is the same as in Panel A of Figure 4. Specifically, removing present bias alleviates the scarcity of capital; the resulting decrease in the interest rate $r$ limits the return from holding wealth. Since weaker wealth accumulation incentives partially offset the benefit of removing present bias, more than $55\%$ of the sophisticates lose welfare in the long run (see the solid blue line in Panel B of Figure 4). This finding shares similar insights with Guiso et al. (Reference Guiso, Pozzi, Tsoy, Gambacorta and Mistrulli2022), in which they incorporate present bias heterogeneity in a mortgage market and show that reducing the present bias of one group harms the welfare of others. Moreover, a higher value of $\beta ^N$ mitigates the negative effects on sophisticates’ welfare, as the drop in the interest rate $r$ is less drastic.

6.2. Value of sophistication

Figure 5.

Sophistication value. The left and right panels depict short- and long-run welfare changes of removing naivete, respectively, where the superscript $j\in \{N,S\}$ . Under naive beliefs, the steady-state wealth-income ratios are $\tilde x_{\beta ^N=0.783}=-0.085$ , $\tilde x_{\beta ^N=0.8}=-0.053$ , and $\tilde x_{\beta ^N=0.82}=-0.023$ ; under sophisticated beliefs, the steady-state wealth-income ratios are $\tilde x_{\beta ^S=0.783}=-0.071$ , $\tilde x_{\beta ^S=0.8}=-0.04$ , and $\tilde x_{\beta ^S=0.82}=-0.01$ .

Similar to the definition of welfare changes from present bias removal (42), we define the welfare changes of removing naivete, $\mathcal{W}_{NS}$ , as the maximum transfer a naive household is willing to pay to become sophisticated:

(45) \begin{eqnarray} \mathbf{b}^{N}\mathbf{P}^N\left (X_u,Y_u\right )= \mathbf{b}^{S}\mathbf{P}^S\left (X_u-\mathcal{W}_{NS},Y_u\right ), \end{eqnarray}

and the scaled welfare changes of removing naivete $\omega _{NS}$ is obtained by solving:

(46) \begin{eqnarray} \mathbf{b}^{N}\mathbf{p}^N\left (x_u\right )= \mathbf{b}^{S}\mathbf{p}^S\left (x_u-\omega _{NS}\right ). \end{eqnarray}

The short- and long-run welfare changes of removing naivete are captured by $\omega _{NS}$ and $\Delta x_{NS}$ , respectively.

Suppose that naive households become sophisticated. Figure 5 displays the welfare changes of removing naivete under three different levels of the present bias parameter. It is evident from this figure that removing naivete generates welfare gains: $\omega _{NS}(x)\gt 0$ and more than $95\%$ of naive households still benefit in the long run. Intuitively, compared to naive households, sophisticates correctly anticipate future selves’ splurging motives and thus save more to prevent potential poverty in the future. For example, with $\beta ^j=0.82$ , removing naivete raises the steady-state wealth-income ratio by $0.013$ . Moreover, a smaller degree of present bias reduces the sophistication value, as manifested by smaller welfare gains.

6.3. Potential remedies for present bias

Although welfare gains from removing present bias and naivete are substantial for indebted households, it is challenging to implement commitment devices in the first place. To guard against self-control problems and promote savings, prior studies suggest that illiquidity serves as a powerful commitment device, such as illiquid retirement accounts (Beshears et al. Reference Beshears, Choi, Laibson, Madrian and Milkman2015; Beshears et al. Reference Beshears, Choi, Laibson, Madrian and Skimmyhorn2022) and illiquid private equity investment (Acharya et al. Reference Acharya, Jimenez Gomez, Rachinskii and Rivera2022). However, the usage of such commitment devices is limited, and their effectiveness has been called into question. For instance, Beshears et al. (Reference Beshears, Choi, Laibson, Madrian and Skimmyhorn2022) find that present-biased agents spend some or all of their new savings shortly after they separate from each employer. In this section, we explain how to mitigate the negative impacts of present bias and naivete with illiquid investment in human capital and borrowing cost penalty.

6.3.1. Illiquidity

To study the value of illiquidity, we incorporate a linear investment decision. Specifically, given an investment policy $i_t\in \{0,\bar{i}\}$ , the household’s labor income in the normal regime, $Y_t$ , evolves according to:

(47) \begin{eqnarray} \frac{dY_t}{Y_t}=\left (\mu +i_t\right ) dt+\sigma d\mathcal{B}_t, \end{eqnarray}

and the process in the default regime is still given by equation (6). The above equation (47) shows that if the household chooses illiquid investment, that is, $i_t=0$ , income grows at the rate $\mu$ as before. If the household prefers liquid investment by choosing $i_t=\bar{i}\gt 0$ , the income growth rate increases. Simultaneously, the household must pay the investment costs; thus, the household’s saving rate, $G^{inv}(X_t,Y_t)$ , is given by:

(48) \begin{eqnarray} G^{inv}(X_t,Y_t)=(r+\lambda +w^{r}\unicode{x1D7D9}_{\{X_{t}\lt 0\}}) X_t+\left (1-\theta i_t\right )Y_t-C^{inv}_t, \end{eqnarray}

where $C^{inv}_t$ is the consumption rate in the presence of liquid investment. The parameter $\theta \gt 0$ measures the degree of investment costs, and a larger $\theta$ denotes the illiquidity of human capital investment.

Proposition 3 (Optimal investment policy for present-biased households). For the type- $j$ household, the optimal investment policy is given by:

(49) \begin{eqnarray} i^j(x)=\left \{ \begin{array}{r@{\quad}c@{\quad}l} 0, &{\text{if }\ F^{j}(X,Y)\gt F^{j,inv}(X,Y)}\\[3pt] \overline{i}, &{\text{if }\ F^{j,inv}(X,Y)\gt F^j(X,Y)} \end{array}\right . \end{eqnarray}

Proof. All proofs are provided in Appendix D.

The economic intuitions for Proposition3 are as follows. In equation (49), $F^j(X,Y)$ denotes the value function when the agent chooses illiquid investment and $F^{j,inv}(X,Y)$ represents the value function when the agent chooses liquid investment. When the investment cost parameter is low ( $\theta =3$ ) so that the value in the case of liquid investment is higher, that is, $F^{j,inv}(X,Y)\gt F^j(X,Y)$ , the agent prefers liquid investment. If, in contrast, $\theta$ is so large ( $\theta =15$ ) that $F^j(X,Y)\gt F^{j,inv}(X,Y)$ , the agent turns to illiquid investment.Footnote ¹¹

Figure 6.

Effects of illiquidity. Panels a and B depict the optimal consumption and probability density function of wealth, respectively. Panels C and D display the CDF of the long-run welfare changes from removing naive present bias and naivete, respectively. Solid blue lines represent the case with liquid investment, and red dashed lines represent the case with illiquid investment. The present bias parameter is set to $\beta ^j=0.783$ , and the liquid investment parameter is set to $\bar{i}=0.13\%$ , where $j\in \{N,S\}$ . Other parameters are reported in Table 1.

Since illiquidity impedes consumption smoothing, the consumption-income ratio drops when $\theta$ increases in the short run, as portrayed by Panel A of Figure 6. Panel B of Figure 6 depicts the probability density function of wealth. As can be seen, illiquid investment moves the wealth distribution rightward. The economic intuitions are that illiquid investment weakens households’ borrowing incentives to finance their investment in human capital. Consequently, they have stronger saving motives and reduce their indebtedness level.

Next, we show how illiquidity alters the welfare changes. Panel C of Figure 6 shows that illiquidity reduces the welfare gains from removing naive present bias. This result is straightforward: as mentioned above, illiquidity curbs the present-biased households’ overconsumption incentives (see Panel A of Figure 6) and, thus, weakens the gains from present bias removal. Moreover, Panel D of Figure 6 reflects that incorporating illiquidity decreases the sophistication value, which indicates that illiquidity partially substitutes for the role of sophistication in curbing excessive borrowing.Footnote ¹²

Figure 7.

Effects of borrowing wedge. Panel A depicts the optimal consumption-income ratio for three different levels of the borrowing wedge. Panels B and C display the welfare changes from removing naive present bias and naivete, respectively. The present bias parameters are set to $\beta ^N=\beta ^S=0.783$ . Other parameter values are reported in Table 1.