A solution method for heterogeneity involving present bias

Abstract

When solving for optimal strategies, a financial engineer needs to take into consideration of preferences heterogeneities, which involve not only present bias, but also future-focused preferences. We provide a reusable tool (i.e. algorithm) for explicitly solving optimal strategy in the presence of preferences variation over time, decision-makers and goods. In this framework, a new discount function identifies many hitherto unknown preference heterogeneities, a non-standard HJB yields sophisticated solution, a behavior equation produces naive and precommitted solutions. An application of the framework shows that a decision-maker can be neither impatient nor patient, and that sophisticated paradigm engenders immediate gratification in earlier phases of life cycle and self-control in later phases of life cycle.

Appendices

Appendix 1: Proof of Theorem 4.1

• 1. Time is partitioned into N intervals. The grid size is specified as equidistant $\Delta {t_j} = t_{j + 1} - t_j = \varepsilon = \dfrac{T}{N} \rightarrow 0^ +$ such that all subintervals shrink to zero in the limit as $N \rightarrow \infty$ or $\varepsilon \rightarrow 0^ +$. We can find $t_j = j \times \Delta t = j\varepsilon$ such that ${\hat{X}}_k = {\hat{X}} ( k\varepsilon ), ~{\hat{u}}_k = {\hat{u}} (k\varepsilon ),~~ {\hat{V}}_j \buildrel \Delta \over = {\hat{V}}({\hat{x}},j\varepsilon ), ~~ {\hat{V}}_N \buildrel \Delta \over = {\hat{V}}({\hat{X}}_N,N\varepsilon ) ={\hat{F}}( {\hat{X}}_N, N\varepsilon )$. For $j=0,\ldots ,N-1$ and $k=j,\ldots ,N-1$, the original problem (6−7) becomes

  $$\begin{aligned}{\hat{V}}_j &= \inf _{{\hat{u}} } {\mathbb {E}}^{ {\hat{x}}}_{j\varepsilon } \left( L({\hat{X}}_j,{\hat{u}}_j,j\varepsilon ) \varepsilon \nonumber \right. \\&\quad \left. + \sum \limits _{i = 1}^{N - j - 1} {\mathbb {D}} ( {0,i\varepsilon }) L({\hat{X}}_{i + j}, {\hat{u}}_{i + j }, (i + j)\varepsilon )\varepsilon +e^{ - (N - j)\rho \varepsilon } {\hat{V}}_N \right) \end{aligned}$$

  (15)

  subject to

  $$\begin{aligned} \left\{ \begin{aligned} {\hat{X}}_{k + 1}&= {\hat{X}}_k + {\hat{\eta }} ( {\hat{X}}_k, {\hat{u}}_k, k\varepsilon ) \varepsilon + \sum \limits _b^m {\hat{\sigma }} ^b ( {\hat{X}}_k, {\hat{u}}_k, k\varepsilon ) (z_{k + 1}^b - z_k^b)\\ {\hat{X}}_j&= {\hat{x}}\\ \end{aligned} \right. \end{aligned}$$

  (16)

  where ${\mathbb {E}}^{ {\hat{x}}}_{j\varepsilon }= {\mathbb {E}} \left[ \cdot \mid {\hat{X}}_j={\hat{x}} \right]$ is the conditional expectation. Our choice of functions guarantees (cf. Kloeden and Platen 1995; Ngo and Taguchi 2017) that Eq. (15) (resp. Eq. 16) converges to Eq. (6) (resp. Eq. 7).

• 2. On one hand, multiplication of Eq. (15) by $e^{ - (N - j-1)\rho \varepsilon }$ implies

  $$\begin{aligned} \begin{array}{l} e^{ - (N - j - 1)\rho \varepsilon } e^{ - (N - j )\rho \varepsilon } {\hat{V}}_N = \\ \inf_{\hat u } {\mathbb {E}}^{ {\hat{x}}}_{j\varepsilon } \left( \begin{array}{l} e^{ - (N - j-1)\rho \varepsilon } {\hat{V}}_j - e^{ - (N - j-1)\rho \varepsilon } L({\hat{X}}_j,{\hat{u}}_j,j\varepsilon ) \varepsilon +\\ \sum \limits _{i = 1}^{N - j - 1} e^{ - (N - j-1)\rho \varepsilon } {\mathbb {D}} ( {0,i\varepsilon })L \Big ({\hat{X}}_{i + j}, {\hat{u}}_{i + j }, (i + j) \varepsilon \Big )\varepsilon \\ \end{array} \right) \end{array} \end{aligned}$$

  (17)

  On the other hand, Eq. (15) implies

  $$\begin{aligned}e^{ - (N - j-1)\rho \varepsilon } {\hat{V}}_N &= \inf _{{\hat{u}} } {\mathbb {E}}^{ {\hat{x}}}_{(j+1)\varepsilon } \left( {\hat{V}}_{j+1} \nonumber \right. \\&\quad \left. - \sum \limits _{i = 1}^{N - j - 1} {\mathbb {D}} ( {0,(i-1)\varepsilon }) L \Big ({\hat{X}}_{i + j}, {\hat{u}}_{i + j }, (i + j)\varepsilon \Big )\varepsilon \right) \end{aligned}$$

  (18)

• 3. Inserting Eq. (18) into Eq. (17)

  $$\begin{aligned} {\hat{V}}_{j+1} -{\hat{V}}_j=\inf _{{\hat{u}} } {\mathbb {E}}^{ {\hat{x}}}_{j\varepsilon } \left[ {{\hat{K}}({\hat{x}},j\varepsilon )+ \rho }{\hat{V}}_j \varepsilon - L ( {\hat{X}}_j, {{\hat{u}}}_j, j\varepsilon )\varepsilon + o(\varepsilon ) \right] \end{aligned}$$

  (19)

  where

  $$\begin{aligned}{\hat{K}}({\hat{x}},j\varepsilon ) &= \left\{ \sum \limits _{i = 1}^{N - j - 1} L \Big ({\hat{X}}_{i + j}, {\hat{u}}_{i + j }, (i + j)\varepsilon \Big ) \nonumber \right. \\&\left. \quad \Big [ {\mathbb {D}} ( {0,(i-1)\varepsilon }) (1 - \rho \varepsilon ) - {\mathbb {D}} ( {0,i\varepsilon }) \Big ] \right\} \varepsilon \end{aligned}$$

  (20)

• 4. By the SQH algorithm (e.g. Harris and Laibson 2013) and the discretizing algorithm (e.g. Karp 2007), Eq. (20) can be rewritten as

  $$\begin{aligned}{\hat{K}}({\hat{x}},j\varepsilon ) &= \left\{ \sum \limits _{i = 1}^{N - j - 1} \Big [ ( 1 - \beta ) \lambda e^{ - \lambda i\varepsilon } \varepsilon + (r - \rho )\varepsilon \Big ] e^{ - ri\varepsilon } \nonumber \right. \\&\quad \left. L \Big ({\hat{X}}_{i + j}, {\hat{u}}_{i + j }, (i + j)\varepsilon \Big ) \right\} \varepsilon \end{aligned}$$

  (21)

  LHS of Eq. (19) can be rewritten as (e.g. Yang and Cheng (2013), equation 2.16),

  $$\begin{aligned} \begin{aligned}V ( x (t+\varepsilon ), t+\varepsilon ) - V(x,t) &= \frac{\partial V(x,t) }{\partial t}\varepsilon + \nabla _x V(x,t) \cdot \eta ( x,u,t) \varepsilon \\&\quad + \frac{1}{2} trace \Big ( \sigma ^{\mathrm{T}} (x,u,t) \cdot \sigma (x,u,t) \cdot \nabla _{xx} V(x,t) \Big ) \varepsilon \\&\quad + \nabla _x V(x,t) \cdot \sigma (x,u,t) \cdot \Big [ z(t+\varepsilon ) -z(t) \Big ] + o(\varepsilon ) \end{aligned} \end{aligned}$$

  (22)

• 5. Inserting (21−22) into (19)

  $$\begin{aligned} \begin{aligned}&\frac{\partial V(x,t) }{\partial t}\varepsilon + \nabla _x V(x,t) \cdot \eta ( x,u,t) \varepsilon \\&\qquad + \frac{1}{2} \Big ( \sigma ^{\mathrm{T}} (x,u,t) \cdot \sigma (x,u,t) \cdot \nabla _{xx} V(x,t) \Big ) \varepsilon \\&\qquad + \nabla _{x} V(x,t) \cdot \sigma (x,u,t) \cdot \Big [z(t+\varepsilon ) -z(t)\Big ] +o(\varepsilon )\\&\quad = \rho {\hat{V}}_j \varepsilon - L ( {\hat{X}}_j, {{\hat{u}}}_j, j\varepsilon )\varepsilon + o(\varepsilon )\\&\qquad +\sum \limits _{i = 1}^{N - j - 1} \Big [ ( 1 - \beta ) \lambda e^{ - \lambda i\varepsilon } \varepsilon \\&\qquad + (r - \rho )\varepsilon \Big ] e^{ - ri\varepsilon } L \Big ({\hat{X}}_{i + j}, {\hat{u}}_{i + j }, (i + j)\varepsilon \Big )\varepsilon \end{aligned} \end{aligned}$$

  (23)

  Dividing by $\varepsilon$ and taking the limit $\varepsilon \rightarrow 0$ complete the proof.

$\square$

Appendix 2: Proof of Remark (4.2)

This appendix will explain how the SHQH discount function includes other discount functions as its special cases.

1.1 Reduction to exponential discount function and its HJB

The exponential discount function at time t used to evaluate a payoff at time s is

$$\begin{aligned} e^{-r (s - t)}, ~~~~~~~~~s \in [t, \infty ) \end{aligned}$$

(24)

The stochastic optimal control problem under the exponential discounting (24) is

$$\begin{aligned} V(x,t)= \inf _{u\in {\mathcal {U}} } {\mathbb {E}} ^x_t \left( \int \limits _t^T { e^{-r (s - t)} L \Big ( {X(s),u(s),s} \Big ) ds} + e^{-r (T - t)} F \Big (X(T),T \Big ) \right) \end{aligned}$$

subject to the constraint (7).

The HJB associated with the exponential discounting (24) is

$$\begin{aligned} \begin{aligned}r V(x,t) - \frac{\partial V(x,t) }{\partial t} &= \inf _{u\in {\mathcal {U}} } \left[ L(x,u,t) + \Big \langle \eta (x,u,t), \nabla _x V(x,t) \Big \rangle \right. \\&\quad \left. + \frac{1}{2} trace \Big ( \sigma ^{\mathrm{T}} (x,u,t) \cdot \sigma (x,u,t) \cdot \nabla _{xx} V(x,t)\Big ) \right] \end{aligned} \end{aligned}$$

(25)

When

$$\begin{aligned} \left\{ \begin{aligned}&\rho =r \\&\beta =1 ~~ or~~ \lambda = 0 \end{aligned} \right. \end{aligned}$$

in Eq. (6), the SHQH discount function (6) and its HJB (8) reduce to the exponential discount function (24) and its HJB (25).

1.2 Reduction to the HG discount function and its HJB

The heterogeneous (HG) discount function (e.g. Marín-Solano and Patxot 2012; Marín-Solano et al. 2013; de-Paz et al. 2014) at time t used to evaluate a payoff at time s is

$$\begin{aligned} d^{HG}(t,s) = {\left\{ \begin{array}{ll} e^{-r (s - t)}, &{} s <T \\ e^{-\rho (s - t)}, &{} s=T \end{array}\right. } \end{aligned}$$

(26)

The stochastic optimal control problem under the HG discounting is

$$\begin{aligned} V(x,t)= \inf _{u\in {\mathcal {U}} } {\mathbb {E}} ^x_t \left( \int \limits _t^T { d^{HG}(t,s) L \Big ( {X(s),u(s),s} \Big ) ds} + d^{HG}(t,T) F \Big (X(T),T \Big ) \right) \end{aligned}$$

subject to the constraint (7).

The HJB associated with the HG discount function (26) is

$$\begin{aligned} \begin{aligned}&\rho V(x,t) + K(x,t) - \frac{\partial V(x,t) }{\partial t}=\\&\quad \inf _{u\in {\mathcal {U}} } \left[ L(x,u,t) + \eta (x,u,t)\cdot \nabla _x V(x,t) + \frac{1}{2} tr \Big ( \sigma ^{\mathrm{T}} (x,u,t) \cdot \sigma (x,u,t) \cdot \nabla _{xx} V(x,t)\Big ) \right] \end{aligned} \end{aligned}$$

(27)

where

$$\begin{aligned} K(x,t) = {\mathbb {E}}^x_t \left( \int \limits _t^T { e^{-r(s-t)}( r -\rho ) L ( u^*(s) )ds} \right) \end{aligned}$$

When

$$\begin{aligned} \beta =1 ~~ or~~ \lambda = 0 \end{aligned}$$

in Eq. (6), the SHQH discount function (6) and its HJB (8) reduce to the HG discount function (26) and its HJB (27). $\square$

1.3 Reduction to the SQH discount function and its HJB

The SQH discount function (e.g. Harris and Laibson 2013) at time t used to evaluate a payoff at time s is

$$\begin{aligned} {\mathbb {D}}(t,s) = {\left\{ \begin{array}{ll} e^{-r (s - t)} , &{} s \in [ t,t + \varsigma ) \\ \beta e^{-r (s - t)}, &{} s \in [ t + \varsigma ,\infty ) \end{array}\right. } \end{aligned}$$

(28)

The stochastic optimal control problem under the SQH discounting (28) is

$$\begin{aligned} V(x,t)= \inf _{u\in {\mathcal {U}} } {\mathbb {E}} ^x_t \left( \int \limits _t^T { {\mathbb {D}} (t,s) L \Big ( {X(s),u(s),s} \Big ) ds} + {\mathbb {D}} (t,T) F \Big (X(T),T \Big ) \right) \end{aligned}$$

subject to the constraint (7).

The HJB associated with the SQH discount function (SQH-HJB) is

$$\begin{aligned} \begin{aligned}&rV(x,t) + K(x,t) - \frac{\partial V(x,t) }{\partial t} \\&\quad =\inf _{u\in {\mathcal {U}} } \left[ L(x,u,t) + \eta (x,u,t)\cdot \nabla _x V(x,t)+ \frac{1}{2} tr \Big ( \sigma ^{\mathrm{T}} (x,u,t) \cdot \sigma (x,u,t) \cdot \nabla _{xx} V(x,t)\Big ) \right] \end{aligned} \end{aligned}$$

(29)

where

$$\begin{aligned} K(x,t) = {\mathbb {E}}^x_t \left( \int \limits _t^T e^{-r (s - t)} \lambda ( 1 - \beta ) e^{ - \lambda (s - t)} L ( u^*(s) )ds \right) \end{aligned}$$

$\square$

When

$$\begin{aligned} \rho = r \end{aligned}$$

in Eq. (6), the SHQH discount function (6) and its HJB (8) reduce to the SQH discount function (28) and its HJB (29).

Appendix 3: Proof of Lemma 5.1

Proof of Lemma 5.1

Time is partitioned into N intervals. The grid size is specified as equidistant $\Delta {t_j} = t_{j + 1} - t_j = \varepsilon = \dfrac{T}{N} \rightarrow 0^ +$ such that all subintervals shrink to zero in the limit as $N \rightarrow \infty$ or $\varepsilon \rightarrow 0^ +$. We can find $t_j = j \times \Delta t = j\varepsilon$ such that ${\hat{W}}_k = {\hat{W}} ( k\varepsilon ), ~{\hat{c}}_k = {\hat{c}} (k\varepsilon ),~~ {\hat{V}}_j \buildrel \Delta \over = {\hat{V}}({\hat{w}},j\varepsilon ), ~~ {\hat{V}}_N \buildrel \Delta \over = {\hat{V}}({\hat{W}}_N,N\varepsilon ) ={\hat{F}}( {\hat{W}}_N, N\varepsilon )$. For $j=0,\ldots ,N-1$ and $k=j,\ldots ,N-1$, the original problem (10−11) becomes

$$\begin{aligned} {\hat{V}}_j= \inf _{{\hat{c}} } {\mathbb {E}}^{ {\hat{w}}}_{j\varepsilon } \left( U({\hat{c}}_j ) \varepsilon + \sum \limits _{i = 1}^{N - j - 1} {\mathbb {D}} ( {0,i\varepsilon }) U({\hat{c}}_{i + j})\varepsilon +e^{ - (N - j)\rho \varepsilon } {\hat{V}}_N \right) \end{aligned}$$

(30)

subject to

$$\begin{aligned} \left\{ \begin{aligned} {\hat{W}}_{k + 1}&= {\hat{W}}_k + \left[ {\hat{W}}_k {{\hat{\mu }} _0} + {\hat{W}}_k {\hat{\pi }}_k({\hat{\mu }} - {{\hat{\mu }} _0})- {\hat{c}}_k \right] \varepsilon +\sum \limits _b^m {\hat{W}}_k {\hat{\pi }}^b_k {\hat{\sigma }} (z_{k + 1}^b - z_k^b)\\ {\hat{W}}_j&= {\hat{w}}\\ \end{aligned} \right. \end{aligned}$$

(31)

where ${\mathbb {E}}^{ {\hat{w}}}_{j\varepsilon }= {\mathbb {E}} \left[ \cdot \mid {\hat{W}}_j={\hat{w}} \right]$ is the conditional expectation. On one hand, multiplication of Eq. (30) by $e^{ - (N - j-1)\rho \varepsilon }$ implies

$$\begin{aligned}&e^{ - (N - j - 1)\rho \varepsilon } e^{ - (N - j )\rho \varepsilon } {\hat{V}}_N = \inf _{{\hat{c}} } \nonumber \\&\quad {\mathbb {E}}^{ {\hat{w}}}_{j\varepsilon } \left( \begin{array}{l} e^{ - (N - j-1)\rho \varepsilon } {\hat{V}}_j \\ - e^{ - (N - j-1)\rho \varepsilon } U({\hat{c}}_j ) \varepsilon +\\ \sum \limits _{i = 1}^{N - j - 1} e^{ - (N - j-1)\rho \varepsilon } {\mathbb {D}} ( {0,i\varepsilon })U ({\hat{c}}_{i + j} )\varepsilon \\ \end{array} \right) \end{aligned}$$

(32)

Following the same procedure as Appendix 1 leads to

$$\begin{aligned} \begin{aligned}&\frac{\partial V(w,t) }{\partial t} \varepsilon + \frac{\partial V(w,t) }{\partial w} \left[ w { \mu _0} + w \pi (t)( \mu - { \mu _0})- c(t) \right] \varepsilon \\&\quad + \frac{1}{2} \frac{\partial ^2 V(w,t) }{\partial w^2 } w^2 \pi ^2 (t) \sigma ^2 \varepsilon \\&\quad + \frac{\partial V(w,t) }{\partial w} w \pi (t) \sigma \Big [z(t+\varepsilon ) -z(t)\Big ] +o(\varepsilon ) = \rho {\hat{V}}_j \varepsilon - U ( {\hat{c}}_j )\varepsilon + o(\varepsilon )\\&\quad +\sum \limits _{i = 1}^{N - j - 1} \Big [ ( 1 - \beta ) \lambda e^{ - \lambda i\varepsilon } \varepsilon + (r - \rho )\varepsilon \Big ] e^{ - ri\varepsilon } U ({\hat{c}}_{i + j} )\varepsilon \end{aligned} \end{aligned}$$

(33)

Dividing by $\varepsilon$ and taking the limit $\varepsilon \rightarrow 0$ complete the proof. $\square$

Appendix 4: Proof of Table 1

This section solves the heterogeneity problem (10–11) under the logarithmic utility explicitly.

1.1 Precommitted and naive paradigms

Let a and b be constants. Under logarithmic utility, the functions in the SHQH behavior Eq. (14) are of the form

$$\begin{aligned} \left\{ \begin{array}{lr} U \Big (c^{P,N}(t_0) \Big )=\ln c^{P,N}(t_0) \\ V^{P,N} ( w,t_0 ) = \alpha ^{P,N} (t_0)\ln w + \psi ^{P,N}(t) \\ F(w) = b \ln w \end{array} \right. \end{aligned}$$

(34)

Using Eq. (14) and Eq. (34) (e.g. Shreve 2004), one arrives at the following plans made by the precommitted and naive decision-makers standing at any arbitrary point in the life cycle $t_0\in [0,T]$

$$\begin{aligned} \dfrac{c^{P,N} (t_0)}{W(t_0)} =\dfrac{1}{\alpha ^{P,N}(t_0)} \end{aligned}$$

(35)

where

$$\begin{aligned} \alpha ^{P,N}(t_0) = e^{- \rho (T-t)} b + \int \limits _{t_0}^T e^{ - r(\iota - t_0) } d\iota \end{aligned}$$

(36)

The precommitted decision-maker of cohort $t_0$ governs the behavior of other selves. Her optimal policy is obtained by evaluating her planned consumption program (35−36) at $t=0$

$$\begin{aligned} \dfrac{c ^P(t_0)}{W(t_0)} = \dfrac{1}{\alpha ^P (t_0)} \end{aligned}$$

where

$$\begin{aligned} \alpha ^P(t_0)= e^{ - \rho T} b + \int \limits _{t_0}^T e^{ - r(\iota - t_0) } d \iota \end{aligned}$$

For the naive decision-maker, the pain incurred by a old plan is continuously overwhelmed by the happiness from a new plan. Her optimal strategy is found by evaluating her planned consumption program (35−36) at $t=t_0$ and at $t_0=t$

$$\begin{aligned} \dfrac{c^N (t)}{W(t)} = \dfrac{1}{\alpha ^N (t)} \end{aligned}$$

where

$$\begin{aligned} \alpha ^N(t) = e^{ - \rho (T-t)} b + \int \limits _t^T e^{ - r(\iota - t) } d\iota \end{aligned}$$

1.2 Sophisticated paradigm

Under logarithmic utility, the functions in the SHQH non-standard HJB (12) are of the form

$$\begin{aligned} \left\{ \begin{array}{lr} U \Big (c^S(t) \Big )=\ln c^S(t) \\ V^S ( w,t ) = \alpha ^S (t) \ln w + \psi ^S(t) \\ F(w) = b\ln w \end{array} \right. \end{aligned}$$

(37)

Differentiating the SHQH non-standard HJB (12) with respect to c and Q and using Eq. (37), one arrives at the following first order condition (FOC):

$$\begin{aligned} \dfrac{c^S (t)}{W(t)} = \dfrac{1}{\alpha ^S (t)} \end{aligned}$$

(38)

Applying Ito’s lemma (e.g. Shreve 2004) on the wealth dynamics (11), and substituting the resulting equation into the K term (13) give

$$\begin{aligned} \begin{array}{l} K(w,t) = \int \limits _t^T e^{ - r(s - t)} \Big [\lambda ( 1 - \beta ) e^{ - \lambda (s - t) } + r - \rho \Big ] \ln \left( \dfrac{1 }{\alpha ^S(s)} \right) ds\\ ~~~~~~~ + \int \limits _t^T \Bigg \{ e^{ - r(s - t)} \Big [ \lambda ( 1-\beta ) e^{-\lambda (s-t)} + r - \rho \Big ] \underbrace{ ( \ln W(t)+ \int \limits _t^s \Gamma ^{\log } (\iota ) d\iota ) }_{{\mathbb {E}} (\ln W(s) ) } \Bigg \}ds \end{array} \end{aligned}$$

(39)

where

$$\begin{aligned} \Gamma ^{\log } (\iota )= \mu _0+ \frac{ (\mu - \mu _0 )^2}{2\sigma ^2} -\frac{1 }{\alpha ^S(\iota )} \end{aligned}$$

Inserting the K term (39) into the SHQH non-standard HJB (12) gives,

$$\begin{aligned} \alpha ^S(t) = b e^{ - \rho (T-t)} + \int \limits _t^T { \left\{ 1 - \int \limits _{\iota }^T { e^{ - r(s - \iota )} \Big [\lambda ( 1 - \beta ) e^{ - \lambda (s - \iota )} + r - \rho \Big ] } ds \right\} e^{ -\rho (\iota - t)} } d \iota \end{aligned}$$

Appendix 5: Proof of Table 2

This section solves the heterogeneity problem (10−11) under the CRRA utility explicitly.

1.1 Precommitted and naive paradigms

Under the CRRA utility, the functions in the SHQH behavior Eq. (14) are of the form

$$\begin{aligned} \left\{ \begin{aligned}&U\Big (c^{P,N}(t_0) \Big )= \dfrac{ \Big (c^{P,N} ( t_0 ) \Big )^\gamma }{\gamma } \\&V^{P,N}( w,t_0 ) = \dfrac{\alpha ^{P,N} ( t_0 ) w^\gamma }{\gamma }\\&F(w) = \dfrac{ b w^\gamma }{\gamma } \end{aligned} \right. \end{aligned}$$

(40)

Combining(e.g. Shreve 2004) the conjecture (40) and the SHQH behavior Eq. (14), one arrives at the plans made by the precommitted and naive decision-makers standing at any arbitrary point in the life cycle $t_0\in [0,T]$

$$\begin{aligned} \dfrac{c^{P,N} (t_0)}{W(t_0)}= \left( \dfrac{1}{\alpha ^{P,N} (t_0)} \right) ^{\frac{1}{1 - \gamma }} \end{aligned}$$

(41)

where

$$\begin{aligned} \left\{ \begin{aligned}&\alpha ^{P,N} (t_0) = \Big ( \upsilon (t_0)\Big ) ^{\gamma - 1} \times \left[ \left( e^{ - \rho (T-t)} e^{r(T - t_0)} b \right) ^{\frac{1}{1 - \gamma }} \upsilon (T) + \int \limits _{t_0}^T \upsilon (\iota ) d\iota \right] ^{1-\gamma }\\&\upsilon ( t_0 ) = \exp \left[ - \frac{1}{1 - \gamma }\int \limits _0^{t_0} \left( r - \frac{ (\mu - \mu _0) ^2 \gamma }{2 ( 1 - \gamma ) \sigma ^2} - \mu _0 \gamma \right) d\iota \right] \\ \end{aligned} \right. \end{aligned}$$

The precommitted decision-maker of cohort $t_0$ governs the behavior of other selves. Her optimal policy is obtained by evaluating Eq. (41) at $t=0$

$$\begin{aligned} \dfrac{c^P (t_0)}{W(t_0)} = \left( \dfrac{1}{\alpha ^P (t_0)} \right) ^{\frac{1}{1 - \gamma }} \end{aligned}$$

where

$$\begin{aligned} \alpha ^P (t_0) = \Big ( \upsilon (t_0) \Big ) ^{\gamma - 1} \times \left[ \left( e^{ - \rho T} e^{r(T - t_0)} b \right) ^{\dfrac{1}{1 - \gamma }} \upsilon (T) + \int \limits _{t_0}^T \upsilon (\iota ) d\iota \right] ^{1-\gamma } \end{aligned}$$

The naive decision-maker of cohort t re-optimizes at each instant. Her optimal policy is obtained by evaluating her planned consumption program (41) at $t=t_0$ and $t_0=t$,

$$\begin{aligned} \dfrac{c^N (t)}{W(t)} = \Big ( \dfrac{1}{\alpha ^N (t)} \Big )^{\frac{1}{1 - \gamma }} \end{aligned}$$

where

$$\begin{aligned} \alpha ^N (t) = \Big ( \upsilon (t) \Big )^{\gamma - 1} \times \left[ \left( e^{ - (\rho -r)(T - t)} b \right) ^{\frac{1}{1 - \gamma }} \upsilon (T) + \int \limits _{t}^T \upsilon (\iota ) d\iota \right] ^{1-\gamma } \end{aligned}$$

1.2 Sophisticated paradigm

Under CRRA utility, the functions in the SHQH non-standard HJB (12) are of the form

$$\begin{aligned} \left\{ \begin{array}{lr} U \Big (c^S(t) \Big )= \dfrac{ \Big (c^S (t) \Big )^\gamma }{\gamma } \\ V^S ( w,t ) = \dfrac{ \alpha ^S (t) w^\gamma }{\gamma } \\ F(w) = \dfrac{ b w^\gamma }{\gamma } \end{array} \right. \end{aligned}$$

(42)

Differentiating the SHQH non-standard HJB (12) with respect to c and using equation (42), one arrives at the following FOC,

$$\begin{aligned} \dfrac{c^S(t)}{W(t)} = \Big ( \dfrac{1}{\alpha ^S (t)}\Big )^{\frac{1}{1 - \gamma } } \end{aligned}$$

(43)

Applying Ito’s lemma on the wealth dynamics (11) and substituting the resulting equation into the K term (13) yield

$$\begin{aligned} K(w,t) = \frac{1}{\gamma } \int \limits _t^T { \left[ \begin{array}{lr} e^{ - r(s - t)} [\lambda ( 1 - \beta ) e^{ - \lambda (s - t)} + r - \rho ] \times \left( \dfrac{1}{\alpha ^S(s)} \right) ^{\frac{\gamma }{1 - \gamma }} \\ \times \underbrace{ (W(t))^\gamma \exp \big (\gamma \int \limits _t^s {\Gamma ^{CRRA } (\iota ) d\iota }\big ) }_{ {\mathbb {E}} \left( W(s) \right) ^\gamma } \end{array} \right] }ds \end{aligned}$$

(44)

where

$$\begin{aligned} \Gamma ^{CRRA } (\iota )= \mu _0+ \frac{ (\mu - \mu _0 )^2}{2(1-\gamma )\sigma ^2} - \left( \dfrac{1}{\alpha ^S(\iota )} \right) ^{\frac{\gamma }{1 - \gamma }} \end{aligned}$$

Or, equivalently,

$$\begin{aligned} K(w,t) = \frac{(W(t))^\gamma }{\gamma } H ^{CRRA}(t) \end{aligned}$$

(45)

where

$$\begin{aligned} H ^{CRRA}(t) = \int \limits _t^T { \left[ \begin{array}{lr} e^{ - r(s - t)} [\lambda ( 1 - \beta ) e^{ - \lambda (s - t)} + r - \rho ] \times \left( \dfrac{1}{\alpha ^S(s)} \right) ^{\frac{\gamma }{1 - \gamma }} \\ \times \exp \big (\gamma \int \limits _t^s {\Gamma ^{CRRA } (\iota ) d\iota }\big ) \end{array} \right] }ds \end{aligned}$$

Substituting the K term (45) into the SHQH non-standard HJB (12) gives,

$$\begin{aligned} \frac{\partial \alpha ^S(t)}{\partial t}- (\rho - M) \alpha ^S(t) = H ^{CRRA}(t) - (1 - \gamma ) \left( \alpha ^S(t) \right) ^{\frac{\gamma }{ \gamma -1 }} \end{aligned}$$

(46)

where

$$\begin{aligned} M= \gamma \mu _0+ \frac{\gamma (\mu - \mu _0 )^2}{2 (1-\gamma ) \sigma ^2} \end{aligned}$$

Appendix 6: Optimal strategies under CARA utility

This section solves the heterogeneity problem (10−11) under the CARA utility explicitly.

1.1 Precommitted and naive paradigms

Under the CARA utility, the functions in the SHQH behavior Eq. (14) are of the form

$$\begin{aligned} \left\{ \begin{aligned}&U \Big (c^{P,N}(t_0) \Big )= \dfrac{-1 }{\gamma } e^ { -\gamma c^{P,N}(t_0 ) } \\&V ^{P,N} ( w,t_0 ) = -e^ { - \gamma \big [ \alpha ^{P,N} (t_0 ) + \psi ^{P,N} (t_0 )w \big ]} \\&F(w) = -b e^ {- \gamma w } \end{aligned} \right. \end{aligned}$$

(47)

Combining (e.g. Shreve (2004), page 453) the conjecture (47) and the SHQH behavior equation (14), we arrive at the plans made by the precommitted and naive decision-makers standing at any arbitrary point in the life cycle $t_0\in [0,T]$

$$\begin{aligned} c^{P,N} (t_0) = \alpha ^{P,N} (t_0) + \psi ^{P,N} (t_0) W(t_0) - \dfrac{1}{\gamma } \ln [ \gamma \psi (t_0)] \end{aligned}$$

(48)

where

$$\begin{aligned} \left\{ \begin{array}{lr} \alpha ^{P,N} (t_0)= \dfrac{1}{ \gamma } \left\{ \begin{array}{l} \Big [ \rho (T - t)- r(T - t_0)- \ln b \Big ] e^{\int \limits _{t_0}^T - \psi ^{P,N}(\nu ) d\nu }\\ - \int \limits _{t_0}^T \vartheta ^{P,N} (\iota ) e^{\int \limits _{t_0}^{\iota } { - \psi ^{P,N} (\nu ) d\nu } } d\iota \end{array} \right\} \\ \vartheta ^{P,N}(\iota ) = \psi ^{P,N}(\iota )\Big \{ 1 - \ln [ \gamma \psi ^{P,N} (\iota ) ] \Big \} - r - \dfrac{(\mu -\mu _0 ) ^2}{2 \sigma ^2} \\ \psi ^{P,N}(t_0) = \dfrac{\mu _0}{ (\mu _0 - 1) e^{ - \mu _0(T - t_0) } + 1} \end{array} \right. \end{aligned}$$

The precommitted decision-maker of cohort $t_0$ governs the behavior of other selves. Her optimal policy is obtained by evaluating her planned consumption program (48) at $t=0$

$$\begin{aligned} c^P (t_0) = \alpha ^P (t_0) + \psi ^P (t_0) W(t_0) - \dfrac{1}{\gamma } \ln [ \gamma \psi ^P (t_0) ] \end{aligned}$$

where

$$\begin{aligned} \left\{ \begin{array}{lr} \alpha ^P(t_0)= \dfrac{1}{ \gamma } \left( \Big [ \rho T - r(T - t_0)- \ln b \Big ] e^{\int \limits _{t_0}^T - \psi ^P (\nu ) d\nu } -\int \limits _{t_0}^T \vartheta ^P(\iota ) e^{\int \limits _{t_0}^m { - \psi ^P (\nu ) d\nu } } d\iota \right) \\ \vartheta ^P(\iota ) = \psi ^P (\iota ) \Big \{ 1 - \ln [ \gamma \psi ^P (\iota ) ] \Big \} - r - \dfrac{(\mu -\mu _0 ) ^2}{2 \sigma ^2} \\ \psi ^P(t_0) = \dfrac{\mu _0}{ (\mu _0 - 1) e^{ - \mu _0(T - t_0) } + 1} \end{array} \right. \end{aligned}$$

For the naive decision-maker, her planned consumption program continues to become invalid and suboptimal with the progression of time . Her optimal policy is obtained evaluating her planned strategies (48) at $t=t_0$ and $t_0=t$

$$\begin{aligned} c^N (t) = \alpha ^N (t) + \psi ^N (t) W(t) - \dfrac{1}{\gamma } \ln [ \gamma \psi ^N(t) ] \end{aligned}$$

where

$$\begin{aligned} \left\{ \begin{array}{lr} \alpha ^N(t)= \dfrac{1}{ \gamma } \left( \Big [ \rho (T - t)- r(T - t)- \ln b \Big ] e^{\int \limits _t^T - \psi ^N (\nu ) d\nu } - \int \limits _{t}^T \vartheta ^N(\iota ) e^{\int \limits _t^m { - \psi ^N (\nu ) d\nu } } d\iota \right) \\ \vartheta ^N(\iota ) = \psi ^N (\iota ) \Big \{ 1 - \ln [\gamma \psi ^N (\iota ) ] \Big \} - r - \dfrac{(\mu -\mu _0 ) ^2}{2 \sigma ^2}\\ \psi ^N(t) = \dfrac{\mu _0}{ (\mu _0 - 1) e^{ - \mu _0(T - t) } + 1} \end{array} \right. \end{aligned}$$

1.2 Sophisticated paradigm

Under the CARA utility, the functions in the SHQH non-standard HJB (12) are of the form

$$\begin{aligned} \left\{ \begin{array}{lr} U(c^S(t))= \dfrac{-1 }{\gamma } e^ { - \gamma c^S(t) } \\ V ^S ( w,t ) = -e^ { - \gamma \Big [ \alpha ^S (t) + \psi ^S (t)w \Big ]} \\ F(w) = -b e^ {- \gamma w } \end{array} \right. \end{aligned}$$

(49)

Plugging equation (49) into the SHQH non-standard HJB (12) and collecting similar terms (e.g. Shreve 2004), one arrives at optimal consumption–wealth ratio,

$$\begin{aligned} c^S (t) = \alpha ^S(t) + \psi ^S(t) W(t) - \dfrac{1}{\gamma } \ln [ \gamma \psi ^S (t) ] \end{aligned}$$

(50)

where

$$\begin{aligned} \psi ^S(t) = \dfrac{\mu _0}{ (\mu _0 - 1) e^{ - \mu _0(T - t) } + 1} \end{aligned}$$